Three-Dimensional Quantum Black Holes: A Primer

Abstract

We review constructions of three-dimensional ‘quantum’ black holes. Such spacetimes arise via holographic braneworlds and are exact solutions to an induced higher-derivative theory of gravity consistently coupled to a large-c quantum field theory with an ultraviolet cutoff, accounting for all orders of semi-classical backreaction. Notably, such quantum-corrected black holes are much larger than the Planck length. We describe the geometry and horizon thermodynamics of a host of asymptotically (anti-) de Sitter and flat quantum black holes. A summary of higher-dimensional extensions is given. We survey multiple applications of quantum black holes and braneworld holography.

1. Overview

Semi-classical gravity remains a useful proxy to study quantum effects in gravity from the perspective of a macroscopic observer. In this context, quantum fields live in a classical dynamical spacetime where the combined system is characterized by the semi-classical Einstein equations [1,2]

$$G_{ab}\left(g\right)+\Lambda g_{ab}=8\pi G_N\langle T_{ab}^{\mathrm{QFT}}\rangle .$$

(1)

On the left-hand side is the usual Einstein tensor for a classical spacetime $g_{ab}$ with cosmological constant $\Lambda$, while on the right-hand side $\langle T_{ab}^{\mathrm{QFT}}\rangle$ is the expectation value of the (renormalized) stress–energy tensor of the quantum field theory in some quantum state $|\Psi \rangle$. Semi-classical gravity should be viewed as an approximation and only valid in a certain regime. Indeed, the semi-classical approximation fails near the Planck scale, as at this level, quantum gravity effects become important, such that (1) can no longer be trusted. Further, the semi-classical fields in Equation (1) are not expected to be valid for generic quantum states $|\Psi \rangle$, e.g., macroscopic superpositions [3]. They are, however, known to be valid when $|\Psi \rangle$ is approximately classical, i.e., a coherent state.

Even in its regime of validity, solutions to semi-classical gravity, particularly black holes, are difficult to study consistently. Largely, this is because solving (1) amounts to solving the problem of a backreaction—how quantum matter influences the classical geometry and vice versa—which is a notoriously difficult and open problem, as it requires simultaneously solving a coupled system of geometry and quantum correlators. Often, in three spacetime dimensions and higher1, the problem is examined perturbatively, offering limited insight, especially when backreaction effects become large. These difficulties only compound when there are a large number of quantum fields, or when the field theory is strongly coupled, as is the case for quantum chromodynamics and the standard model of particle physics.

A context in which the physics of a large-N number of strongly interacting quantum fields may be probed is the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence [6]. Born out of studies in string theory, AdS/CFT is a non-perturbative candidate model of quantum gravity, where gravitational physics in a bulk $d+1$-dimensional asymptotically AdS spacetime has a dual description in terms of a CFT living on the d-dimensional conformal boundary of AdS. This duality is therefore a concrete realization of gravitational holography [7,8]. More specifically, in a large-N expansion, the planar diagram limit of the CFT, the bulk is well approximated by classical gravity (we will state this dictionary more precisely below). A powerful feature of the AdS/CFT correspondence is strong–weak coupling duality: coupling constants between the bulk and boundary theories are inversely related, $G_N\sim N^{-1}$. Thus, computations of strongly coupled field theories may instead be performed via a classical gravity calculation. While the boundary geometry on which the CFT lives may be curved (and even contain black holes [9]), it is fixed. Consequently, standard AdS/CFT holography alone is insufficient for addressing the semi-classical backreaction problem2.

Enter braneworld holography [11]. Historically introduced as a possible solution to the hierarchy problem [12,13], braneworld models treat the four-dimensional universe we experience as a membrane sitting in a spacetime with large extra-dimensions. When combined with holography, braneworlds function as a useful toolkit to address difficult problems in semi-classical gravity. In this framework, AdS/CFT duality is adapted to incorporate situations where a portion of the bulk, including its boundary, is removed by a d-dimensional Randall–Sundrum [13,14] or Karch–Randall [15,16] braneworld. Crucially, the geometry of the end-of-the-world (ETW) brane is dynamical, having an induced theory of gravity. More precisely, the brane serves as an infrared cutoff in the bulk, translating to a ultraviolet (UV) cutoff for the holographic CFT. As in holographic renormalization [17,18,19,20,21], a tower of higher-derivative corrections to the d-dimensional Einstein–Hilbert action are induced by the holographic cutoff ${\mathrm{CFT}}_d$. From the brane perspective, the induced theory may thus be interpreted as a semi-classical theory of gravity [22], where the higher-derivative corrections incorporate backreaction effects due to the CFT living on the brane, which is governed by

$$G_{ij}+{\Lambda }_d{g}_{ij}+\left(\mathrm{higher-curvature}\right)=8\pi G_d{\langle T_{ij}^{\mathrm{CFT}}\rangle }_{\mathrm{planar}}.$$

(2)

Here, ${\Lambda }_d$ and $G_d$ are induced cosmological and Newton constants on the brane, and the right-hand side indicates the holographic CFT is in its planar limit3.

At first glance, it would appear the braneworld has only complicated the situation with its higher-derivative corrections: solving the induced field Equation (2) requires solving the problem of backreaction in a complex higher-derivative theory of gravity. The computational advantage of braneworld holography, however, is that the semi-classical induced brane theory has an equivalent bulk description in terms of classical ${\mathrm{AdS}}_{d+1}$ gravity coupled to a brane obeying Israel junction conditions. Thus, exact spacetimes solving the classical bulk field equations with brane boundary conditions automatically correspond to exact solutions to the semi-classical brane equations of motion (2). Holographic braneworlds thus provide a means to exactly study the problem of backreaction without having to explicitly solve semi-classical field equations. In particular, classical ${\mathrm{AdS}}_{d+1}$ black holes which localize on the ETW brane are conjectured to precisely map to black holes in d-dimensions, including all orders of quantum backreaction [22], i.e., ‘quantum’ black holes.

The primary purpose of this article is to review the state of the art regarding such holographic quantum black holes. Emphasis is given to a particular class of analytic black holes which localize on an ${\mathrm{AdS}}_4$ braneworld, which were first uncovered by Emparan, Horowitz and Myers [23,24], corresponding to three-dimensional quantum black holes [22,25,26,27]. These braneworld black holes lead to an important observation: a backreaction can lead to the existence of black holes where there were none before. That is, famously, there are no black hole solutions in vacuum to classical Einstein gravity in three dimensions with positive or vanishing cosmological constant. Rather, the geometry of a point mass in ${\mathrm{Mink}}_3$ or ${\mathrm{dS}}_3$ is described as a conical defect without a black hole horizon [28,29]; Schwarzschild-${\mathrm{dS}}_3$, for example, is a conical defect with a single cosmological horizon but no black hole horizon. Quantum corrections due to backreaction alter the three-dimensional geometry in such a way that a black hole horizon is induced, leading to a type of (quantum) censorship of conical singularities. Meanwhile, classical black holes do exist in three-dimensional Einstein gravity with a negative cosmological constant [30,31], which is a consequence of the tendency for gravitational collapse afforded by the negatively curved geometry. Nonetheless, in such contexts, a backreaction yields behavior strikingly different from their classical counterparts.

A secondary goal of this review is to advertise a host of applications of holographic quantum black holes. These include the exact study of semi-classical horizon thermodynamics, holographic entanglement entropy and complexity, and the prospect of probing black hole singularities. Combined, quantum black holes serve as a theoretical laboratory to exactly test ideas in semi-classical/quantum gravity, deserving of further exploration.

Road Map and High-Level Summary

Backreaction without holography. Section 2 sets the stage by exploring quantum corrections to three-dimensional geometries at the perturbative level. This demonstrates that one need not appeal to AdS/CFT or holographic braneworlds to see how a quantum backreaction alters classical three-dimensional geometry. For example, a conformally coupled scalar field in Schwarzschild-(A)dS${ }_3$ produces a Casimir effect with negative energy density such that, upon solving the semi-classical Einstein Equation (1), the linear-order change to the $tt$-component of the metric in static patch coordinates goes like [32,33]

$$\delta g_{tt}={\displaystyle \frac{2L_{\mathrm{P}}F\left(M\right)}{r}},$$

(3)

where $L_{\mathrm{P}}=\hslash G_3$ is the Planck length in three dimensions4 and $F\left(M\right)$ is a positive function of the mass of the point particle generating the conical defect. The $1/r$-correction—which is solely a quantum effect—modifies the original blackening factor and its root structure, implying a black horizon may arise due to a backreaction. However, since the correction is on the order of the Planck length, this conclusion is tenuous, since quantum gravity effects are expected to play a role at this scale. If there is a large-c amount of such quantum fields, it is conceivable that their combined quantum effect is to produce a correction proportional to $c{L}_{\mathrm{P}}\gg L_{\mathrm{P}}$, for which quantum gravity effects can be neglected. Unfortunately, it is thus far unknown how to solve the backreaction problem with such a large number of fields via non-holographic methods. Thus, while the perturbative analysis is suggestive, it advocates for the holographic braneworld approach described above.

Holographic braneworlds and quantum black holes. In Section 3, we present a general portrait of braneworld holography and quantum black holes. Historically, this framework is a natural extension of holographic renormalization, and it is a prescription where bulk infrared (IR) divergences are eliminated by introducing an IR cutoff hypersurface near the AdS boundary, adding local counterterms, and employing a minimal subtraction scheme. In standard braneworld holography, the IR cutoff surface is replaced by an end-of-the-world brane $\mathcal{B}$ of tension $\tau$. The would-be divergent counterterms now combine with the brane action as seen from the bulk, leading to an induced theory of higher-derivative gravity on the brane (34)

$$\begin{array}{cc}\hfill I_{\mathrm{Bgrav}}& ={\displaystyle \frac{1}{16\pi G_d}}{\int }_{\mathcal{B}}{d}^{d}x\sqrt{-h}[R-2{\Lambda }_d+{\displaystyle \frac{L_{d+1}^2}{(d-4)(d-2)}}\left(R_{ij}^2-{\displaystyle \frac{d{R}^2}{4(d-1)}}\right)+\cdots ],\hfill \end{array}$$

(4)

coupled to a CFT with an ultraviolet cutoff. Here, $G_d$ and ${\Lambda }_d$ are induced scales dependent on the bulk Newton and cosmological constant and brane tension, and the ellipsis denotes an infinite tower of higher-derivative terms, which is proportional to increasing positive powers of the bulk AdS length scale $L_{d+1}$. A metric variation of this action results in the semi-classical equations of motion (2), the solutions of which are conjectured to constitute quantum-corrected geometries due to the backreaction of a large-c holographic CFT with a UV cutoff.

Benchmarking quantum black holes. Section 4 is devoted to constructing three-dimensional quantum black holes using holographic braneworlds, including static and rotating black holes in AdS${ }_3$ (Section 4.2), dS${ }_3$ (Section 4.3), and Minkowski${ }_3$ (Section 4.4). In all cases, the bulk geometry is taken to be the AdS${ }_4$ C-metric, with either a Karch–Randall (asymptotically AdS${ }_3$) or Randall–Sundrum (asymptotically flat or dS${ }_3$) end-of-the-world brane embedded inside. The neutral, static geometries induced on the brane include corrections of the type (3), supporting the intuition gained from the perturbative analysis. For example, the static quantum BTZ black hole is (64)

$$d{s}_{\mathrm{qBTZ}}^2=-\left({\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-8{\mathcal{G}}_{3}M-{\displaystyle \frac{\ell F\left(M\right)}{\overline{r}}}\right)d{\overline{t}}^2+{\left({\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-8{\mathcal{G}}_{3}M-{\displaystyle \frac{\ell F\left(M\right)}{\overline{r}}}\right)}^{-1}d{\overline{r}}^2+{\overline{r}}^{2}d{\overline{\varphi }}^2,$$

(5)

where M is the black hole mass, and ${\mathcal{G}}_3$ is a renormalized Newton’s constant due to an all-order resummation of the higher-derivative terms appearing in the induced action (4). One notable difference between the black hole (5) and the perturbative geometry (3) is that $\ell \sim c{L}_{\mathrm{P}}\gg L_{\mathrm{P}}$, where c is the central charge of the cutoff CFT${ }_3$ on the brane. Thus, the correction appears on a scale where quantum gravitational effects can be consistently ignored; the geometry (5) is a quantum-corrected black hole much larger than the Planck length. Further, the geometry (5) is an exact solution to the semi-classical brane theory.

Quantum black hole thermodynamics. Section 5 focuses on the horizon thermodynamics of quantum black holes. As with the geometry, the thermodynamics of the braneworld black hole is induced from the thermodynamics of the bulk AdS${ }_4$ black hole. For example, the bulk and brane horizon temperatures coincide. Meanwhile, the bulk entropy, given by the Bekenstein–Hawking area relation, is reinterpreted as generalized entropy

$$S_{\mathrm{BH}}^{\left(4\right)}⟺S_{\mathrm{gen}}^{\left(3\right)},$$

(6)

the sum of gravitational and matter entanglement entropies. Consequently, from the brane perspective, the first law of thermodynamics of quantum black holes is

$$dM=Td{S}_{\mathrm{gen}}^{\left(3\right)}+...,$$

(7)

where the ellipsis refers to possible additional variations such as rotation or charge. Thus, by accounting for a semi-classical backreaction, classical gravitational entropy is replaced by its semi-classical generalization. Since the bulk thermodynamic variables are exactly known, the first law holds to all orders of backreaction. This is highly non-trivial from the brane perspective. Indeed, the induced semi-classical theory includes an infinite tower of higher-derivative terms and the matter entropy of the cutoff CFT${ }_3$. To determine the entropy and mass non-perturbatively in a backreaction would require a resummation of the infinite tower of terms and knowledge of how to compute the von Neumann entropy of the CFT${ }_3$, which is a supremely challenging task. Via holography, this resummation is performed by the bulk.

Holographic braneworlds, moreover, provide a natural higher-dimensional origin of extended black hole thermodynamics (Section 5.6), where the cosmological constant is treated as a dynamical pressure. This is because the cosmological constant on the brane is partly induced by the brane tension. Tuning the brane tension alone amounts to varying the induced brane cosmological constant, such that mechanical work performed by the brane is interpreted as extended thermodynamics of the black hole on the brane. Intriguingly, all AdS${ }_3$ quantum black holes obey a semi-classical generalization of the reverse isoperimetric inequality, indicating quantum black holes have a maximal entropy state at fixed volume.

Quantum black holes can also have a wildly different thermal phase structure than their classical counterparts (Section 5.8). Specifically, in the case of the static quantum BTZ black hole, a large enough backreaction triggers reentrant phase transitions, e.g., a transition from thermal AdS to the quantum black hole back to thermal AdS. The first of these transitions can be understood as a quantum analog of the familiar first-order Hawking–Page transition of AdS black holes, while the second transition back to thermal AdS is entirely a consequence of non-perturbative backreaction effects.

Puzzles in higher-dimensions. Section 6 briefly reviews the history and status of higher-dimensional braneworld black holes and their semi-classical interpretation. Indeed, while we focus on three-dimensional quantum black holes, the conjecture [22,34] that classical bulk black holes correspond to quantum-corrected black holes on the brane, in principle, holds in arbitrary spacetime dimensions. After assessing arguments that imply four and higher-dimensional quantum black holes must be evaporating, we describe static four-dimensional braneworld black holes and their peculiar features when viewed as quantum black holes. We conclude with a short summary regarding the holographic duals of evaporating black holes.

Applications. In Section 7, we present a non-exhaustive list of applications and possible avenues for future research using the quantum black hole constructions described in this review. Historically, braneworlds, motivated by string theory, were conceived as possible resolutions to problems in high-energy phenomenology, e.g., the hierarchy problem of the standard model of particle physics [12,13]. Although there are yet to be any experimental signatures of extra dimensions or braneworlds, the view taken here is that braneworld holography provides an invaluable framework to test ideas in (non-perturbative) semi-classical gravity and beyond. For example, prescriptions in holographic information theory, i.e., entanglement entropy and complexity, can be tested in the context of quantum black holes. Moreover, classical ideas such as cosmic censorship can be revisited using quantum black holes as a guide.

Appendices. While unnecessary to follow the core narrative of this review, we include a number of appendices to be self-contained and pedagogical. The conventions used in the majority of this review are presented in Appendix A. In Appendix B, we provide further details about holographic regularization, including a detailed derivation of the local counterterms that give rise to the induced gravity theory on the brane. Appendix C summarizes the relevant history and physics of braneworlds. Geometric elements of the C-metric utilized in the main text are given in Appendix D. Appendix E presents a derivation of the thermodynamics of the static, neutral quantum BTZ black hole using a (bulk) canonical partition function.

2. Black Holes and Backreaction in 3D: A Perturbative Analysis

2.1. Three-Dimensional Black Holes and Conical Defects

In vacuum general relativity, black holes tend to disappear when lowering the spacetime dimension from four to three. This can be understood at the level of dimensional analysis. If the only dimensionful parameter is three-dimensional Newton’s constant $G_3$, then introducing a massive object of mass M does not introduce an additional length scale needed to characterize a black hole horizon solely in terms of its mass; indeed, $G_{3}M$ is dimensionless5. In fact, a massive point particle in flat $(2+1$)-dimensional general relativity is a conical defect with angular deficit $\delta =2\pi (1-\sqrt{1-8G_{3}M})$ and a conical singularity at the origin [28]. Moreover, while a cosmological constant $\Lambda$ will introduce another length scale, this alone is not sufficient to have a black hole horizon. Gravitational attraction is also required.

To elaborate, there are black holes in asymptotically ${\mathrm{AdS}}_3$. Namely, the Bañados–Teitelboim–Zanelli (BTZ) black hole [30,31]

$$d{s}^2=-N\left(r\right)d{t}^2+N^{-1}\left(r\right)d{r}^2+r^2{(d\varphi +N_{\varphi }dt)}^2,$$

(8)

with lapse and shift metric functions

$$N\left(r\right)\equiv -8G_{3}M+{\displaystyle \frac{r^2}{L_3^2}}+{\displaystyle \frac{{\left(4G_{3}J\right)}^2}{r^2}},N_{\varphi }\equiv -{\displaystyle \frac{4G_{3}J}{r^2}},$$

(9)

for mass M and spin J. The roots of the lapse,

$$r_{\pm }^2={\displaystyle \frac{L_3^2}{2}}\left[8G_{3}M\pm \sqrt{{\left(8G_{3}M\right)}^2-{\left({\displaystyle \frac{8G_{3}J}{L_3}}\right)}^2}\right],$$

(10)

characterize the outer ($r_{+}$) and inner/Cauchy ($r_{-}$) horizons, with $r_{+}\ge r_{-}\ge 0$, assuming $M{L}_3\ge J>0$ to avoid naked singularities. The reason we can interpret the BTZ metric as a ‘black hole’ is because the negatively curved geometry of ${\mathrm{AdS}}_3$ provides an innate geometric tendency for gravitational collapse (see, e.g., [35])6. Alternatively, there are no black holes in ${\mathrm{dS}}_3$; the positively curved ${\mathrm{dS}}_3$ background leads to an inability for collapse7. Consequently, a point mass in ${\mathrm{dS}}_3$, is described by a conical defect [29] with a single cosmological horizon.

To see this latter point, consider the Kerr–${\mathrm{dS}}_3$ metric. The line element formally takes the same form as (8) except now with lapse and shift functions

$$N\left(r\right)\equiv 1-8G_{3}M-{\displaystyle \frac{r^2}{R_3^2}}+{\displaystyle \frac{{\left(4G_{3}J\right)}^2}{r^2}},N_{\varphi }\equiv +{\displaystyle \frac{4G_{3}J}{r^2}},$$

(11)

where $R_3$ denotes the ${\mathrm{dS}}_3$ length scale and the ‘+’ sign in $N_{\varphi }$ denotes convention. Next, we introduce dimensionless parameters $\gamma \equiv r_{+}/R_3$ and $\alpha \equiv -4G_{3}J/\gamma R_3=i{r}_{-}/R_3$, where $r_{\pm }$ are

$$r_{\pm }^2={\displaystyle \frac{R_3^2}{2}}\left[(1-8G_{3}M)\pm \sqrt{{(1-8G_{3}M)}^2-{\left({\displaystyle \frac{8G_{3}J}{R_3}}\right)}^2}\right],$$

(12)

with only a single positive root, $r_{+}$, identified as the cosmological horizon. Then, the coordinate transformation [27,29,39]

$$\begin{array}{c}\hfill \tilde{t}=\gamma t+\alpha R_3\varphi ,\tilde{\varphi }=\gamma \varphi -\alpha t/R_3,\tilde{r}/R_3=\sqrt{{\displaystyle \frac{{(r/R_3)}^2+{\alpha }^2}{{\gamma }^2+{\alpha }^2}}}\end{array}$$

(13)

brings the Kerr–${\mathrm{dS}}_3$ geometry into an empty ${\mathrm{dS}}_3$ form, i.e.,

$$d{s}^2=-\left(1-{\displaystyle \frac{{\tilde{r}}^2}{R_3^2}}\right)d{\tilde{t}}^2+{\left(1-{\displaystyle \frac{{\tilde{r}}^2}{R_3^2}}\right)}^{-1}d{\tilde{r}}^2+{\tilde{r}}^{2}d{\tilde{\varphi }}^2.$$

(14)

Here, however, the coordinates $(\tilde{t},\tilde{\varphi })$ do not have the same periodicity as standard ${\mathrm{dS}}_3$, where $(t,r,\varphi )\sim (t,r,\varphi +2\pi )$. Rather,

$$(\tilde{t},\tilde{\varphi })\sim (\tilde{t}+2\pi R_3\alpha ,\tilde{\varphi }+2\pi \gamma ).$$

(15)

Thence, Kerr–${\mathrm{dS}}_3$ is a conical defect geometry with angular deficit $\delta =2\pi (1-\gamma )$.

It is worth pointing out that the ${\mathrm{AdS}}_3$ geometry (8) is not always a black hole. For $8G_{3}M<0$, the geometry is a conical defect, taking the form of empty ${\mathrm{AdS}}_3$ (the line element (14) with Wick rotation $R_3=-i{L}_3$), with the same periodicity (15), where now $\alpha \equiv r_{+}/L_3$ and $\gamma \equiv 4G_{3}iJ/L_3$8. In particular, when $J=0$, the states with $-1<8G_{3}M<0$ correspond to conical defects with angular deficit $\delta =2\pi (1-\sqrt{-8G_{3}M})$, while for $8G_{3}M<-1$, the geometry has a conical excess; at $8G_{3}M=-1$, the BTZ geometry is exactly empty ${\mathrm{AdS}}_3$. Furthermore, when $8G_{3}M<0$ (for arbitrary J), the metric components are well defined everywhere; i.e., there is no horizon and the conical singularity at $r=0$ is ‘naked’.

2.2. Backreaction and Quantum Dressing

Another way to introduce a dimensionful parameter is to allow for quantum effects. Namely, for $\hslash \ne 0$, there exists the three-dimensional Planck length $L_{\mathrm{P}}=\hslash G_3$ (though there is no notion of Planck mass in three dimensions). The question then is whether such quantum effects can modify the classical three-dimensional geometry so as to induce a (black hole) horizon when there was none before.

Evidence of this comes from perturbatively solving the semi-classical Einstein Equation (1) for a conformally coupled scalar field $\Phi$ [26,27,40,41,42,43,44,45], which is characterized by the action

$$I={\displaystyle \frac{1}{16\pi G_3}}\int d^{3}x\sqrt{-g}\left[R-2\Lambda \right]-{\displaystyle \frac{1}{2}}\int d^{3}x\sqrt{-g}\left[{(\nabla \Phi )}^2+{\displaystyle \frac{1}{8}}R{\Phi }^2\right].$$

(16)

In such a set-up, the first step is to determine the renormalized stress tensor $\langle T_{ab}\rangle$. This is accomplished by first constructing the Green function associated with the equation of motion for the scalar field $\Phi$ in ${\left(\mathrm{A}\right)\mathrm{dS}}_3$, $(\square -{\displaystyle \frac{1}{8}}R)\Phi =0$. Generically, the Green function is

$$G(x,x^{\prime })={\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{1}{|x-x^{\prime }|}}+{\displaystyle \frac{\lambda }{4\pi }}{\displaystyle \frac{1}{|x+x^{\prime }|}},$$

(17)

where $\lambda =0,1,-1$ corresponds to the scalar field obeying transparent, Neumann, or Dirichlet boundary conditions, respectively. For non-rotating backgrounds9 and assuming transparent boundary conditions, the renormalized stress tensor has the form

$$\langle T_b^a\rangle ={\displaystyle \frac{\hslash F\left(M\right)}{8\pi r^3}}\mathrm{diag}(1,1,-2),$$

(18)

where $F\left(M\right)$ is a positive function of the mass. The explicit expression for $F\left(M\right)$ is different depending on whether the background is conical (A)dS${ }_3$ or the BTZ geometry (see, e.g., [26,27,44,45] for details), while the radial dependence and diagonal tensorial structure will change when considering non-transparent boundary conditions [42]. In any case, the Green function of the quantum scalar field in the BTZ background is in the Hartle–Hawking state, satisfying the Kubo–Martin–Schwinger (KMS) condition at the black hole temperature [42].

Having the stress tensor (18) source the right-hand side of the semi-classical Einstein’s Equation (1), one finds for the static geometry the $\mathcal{O}\left(L_P\right)$ correction to the three-dimensional metric (8) is

$$N\left(r\right)={\displaystyle \frac{r^2}{L_3^2}}-8G_{3}M-\delta g_{tt},\delta g_{tt}={\displaystyle \frac{2L_{\mathrm{P}}F\left(M\right)}{r}},$$

(19)

for the ${\mathrm{AdS}}_3$ geometries, and

$$N\left(r\right)=1-8G_{3}M-{\displaystyle \frac{r^2}{R_3^2}}-\delta g_{tt},\delta g_{tt}={\displaystyle \frac{2L_{\mathrm{P}}F\left(M\right)}{r}},$$

(20)

for conical ${\mathrm{dS}}_3$. Notably, $\delta g_{tt}>0$, indicating gravitational attraction [32,33]10. This attractive effect suggests that a horizon induced due to a semi-classical backreaction might appear to dress the naked AdS${ }_3$ conical singularity or, in the case of conical dS${ }_3$, lead to a black hole horizon in addition to its classical cosmological horizon.

The ‘horizon’ radius, however, is proportional to the Planck length at the scale when quantum gravitational effects are expected to become important. Consequently, the above perturbative semi-classical analysis cannot be trusted, and we are not able to conclude the backreacted geometry results in a genuine horizon. This is not to say a semi-classical backreaction will always result in Planckian-sized horizons. Indeed, both gravitational and quantum effects are at play: a large $c\gg 1$ number of conformally coupled scalars results in a combined quantum effect $\propto c\hslash$ which may gravitate to yield large semi-classical black holes, i.e., those with horizon radius $\sim Gc\hslash =c{L}_{\mathrm{P}}\gg L_{\mathrm{P}}$, for which quantum gravity effects may be safely neglected. To verify this, the backreaction problem of a large number of fields must be non-linearly accounted for, and, thus far, perturbative methods have been unable to accomplish this consistently. Braneworld holography provides a framework for which the backreaction problem due to a large-c holographic conformal field theory can be exactly solved.

3. Braneworld Holography and Quantum Black Holes

The perturbative analysis above suggests that a semi-classical backreaction due to quantum fields can lead to the appearance of a black hole horizon when there was none before. Due to the limitations of the perturbative approach, however, the observation is cursory at best. Since the perturbative correction to the classical geometry is on the order of the Planck scale, we cannot definitively argue that a black horizon appears. Only if there are a large-c number of quantum fields present would this conclusion be plausible. The only known framework where a solution with these requirements can be consistently achieved is braneworld holography, where one innately works in a large-c limit. Below, we summarize the relevant aspects of holographic braneworlds.

3.1. AdS/CFT Dictionary and Holographic Renormalization

The AdS/CFT correspondence [6], in its strongest form, describes a duality between a theory of gravity and conformal field theory at the level of their partition functions, as summarized by Gubser, Klebanov, Polyakov and Witten (GKPW) [46,47]

$$\langle e^{-{\int }_{\partial \mathcal{M}}\mathcal{O}{\varphi }_{\left(0\right)}}{\rangle }_{\mathrm{CFT}}=Z_{\mathrm{grav}}\left[{\varphi }_{\left(0\right)}\right]{|}_{\mathcal{M}}.$$

(21)

On the right-hand side, we have the gravitational partition function of a bulk field $\Phi$ over an asymptotically $d+1$-dimensional AdS spacetime $\mathcal{M}$, with conformal boundary $\partial \mathcal{M}$, and ${\varphi }_{\left(0\right)}$ is the fixed boundary value of the bulk field $\Phi$. On the left-hand side is the generating functional for the dual d-dimensional CFT living on $\partial \mathcal{M}$, where $\mathcal{O}$ is the field theory operator dual to the bulk field. Taking variations with respect to ${\varphi }_{\left(0\right)}$ and then setting ${\varphi }_{\left(0\right)}=0$, one can obtain correlation functions of $\mathcal{O}$, which are sourced by ${\varphi }_{\left(0\right)}$. This equivalence of partition functions (21) is often dubbed the standard AdS/CFT dictionary, and, at least formally, it defines a model of non-perturbative quantum gravity.

One of the essential features of the AdS/CFT correspondence is that it can probe strongly coupled field theories on a non-dynamical background using weakly coupled, classical (super)gravity. This is because, typically, the holographic field theories are non-Abelian gauge theories with a gauge group of rank N and ’t Hooft coupling $\lambda$, where, at large-N and $\lambda \gg 1$ (the planar-diagram limit), the dynamics are effectively classical, with $\mathcal{O}(1/N)$ corrections in the dual field theory corresponding to bulk gravity quantum corrections $\mathcal{O}\left(G\right)$11. More generally, the dual field theory degrees of freedom are encoded in the central charge c, which, for known holographic theories scale with N, i.e., $c\sim N^{\alpha }$ for positive, real $\alpha$. Thus, the large-c limit coincides with the classical limit12 in the bulk, and the right-hand side of the dictionary (21) may be approximately given by a sum over classical saddles $\left\{{\Phi }_i\right\}$

$$\underset{c\to \infty }{lim}\langle e^{-{\int }_{\partial \mathcal{M}}\mathcal{O}{\varphi }_{\left(0\right)}}{\rangle }_{\mathrm{CFT}}=\sum _i{e}^{-I_{\mathrm{grav}}^{\mathrm{on-shell}}\left[{\Phi }_i\right]},$$

(22)

where each field configuration ${\Phi }_i$ is a solution to the bulk classical gravity equations of motion subject to the prescribed boundary conditions13. A particular case of interest is to turn off all sources ${\varphi }_{\left(0\right)}$ except those with the boundary value of the bulk metric. In such an event, at large c, it is consistent to turn off all bulk fields except the metric, such that the bulk is described by a pure theory of gravity, which is often taken to be the Einstein–Hilbert action.

Holographic Renormalization

The standard dictionary (21), however, requires special care in regard to divergences. Indeed, even at tree level (22), the gravity partition function exhibits long-distance infrared (IR) divergences, which correspond to ultraviolet (UV) divergences in the CFT correlation functions. These divergences may be removed via holographic renormalization, which is a prescription that adds appropriate local counterterms [17,18,19,20,21] in a minimal subtraction scheme. Since they will become relevant momentarily, let us outline the holographic renormalization procedure, leaving further computational details for Appendix B.

Consider a bulk asymptotically ${\mathrm{AdS}}_{d+1}$ spacetime $\mathcal{M}$ of curvature scale $L_{d+1}$ and cosmological constant ${\Lambda }_{d+1}=-d(d-1)/2L_{d+1}^2$, which is governed by classical Einstein gravity

$$I_{\mathrm{bulk}}={\displaystyle \frac{1}{16\pi G_{d+1}}}{\int }_{\mathcal{M}}{d}^{d+1}x\sqrt{-\widehat{g}}\left(\widehat{R}-2{\Lambda }_{d+1}\right)+{\displaystyle \frac{1}{8\pi G_{d+1}}}{\int }_{\partial \mathcal{M}}{d}^{d}x\sqrt{-h}K.$$

(23)

Here, $G_{d+1}$ is the $d+1$-dimensional Newton’s constant, ${\widehat{g}}_{ab}$ is the metric endowed on $\mathcal{M}$, and K in the Gibbons–Hawking York (GHY) boundary term is the trace of the extrinsic curvature of the boundary submanifold $\partial \mathcal{M}$ endowed with induced metric $h_{ij}$. Working in the large-c, planar-diagram limit, the bulk gravity theory has a dual holographic description in terms of a ${\mathrm{CFT}}_d$ living on the asymptotic conformal boundary $\partial \mathcal{M}$.

Asymptotically, the bulk AdS spacetime can be cast in a Fefferman–Graham gauge [48,49] such that near the boundary

$$d{s}^2={\widehat{g}}_{ab}d{x}^{a}d{x}^b=L_{d+1}^2\left({\displaystyle \frac{d{\rho }^2}{4{\rho }^2}}+{\displaystyle \frac{1}{\rho }}{g}_{ij}(x,\rho )d{x}^{i}d{x}^j\right),$$

(24)

where the d-dimensional metric has the expansion $g_{ij}(x,\rho )=g_{ij}^{\left(0\right)}\left(x\right)+\rho g_{ij}^{\left(2\right)}\left(x\right)+...+{\rho }^{d/2}{g}_{ij}^{\left(d\right)}\left(x\right)$. The conformal boundary is located at $\rho =0$. By perturbatively solving the bulk Einstein’s equations, the higher-order metric coefficients $g_{ij}^{(k>0)}\left(x\right)$ may be cast covariantly in terms of the metric $g_{ij}^{\left(0\right)}$ and derivatives thereof.

On shell, the bulk action (23) has IR divergences at $\rho =0$. To isolate and regulate these divergences, we introduce an IR cutoff $\rho =ϵ$, for $ϵ\ll 1$, near the asymptotic boundary, and integrate over bulk coordinate $\rho$ between $ϵ<\rho <{\rho }_c$, where ${\rho }_c>ϵ$ is some constant14. See Figure 1 for an illustration. This procedure produces a regulated bulk action,

$$I_{\mathrm{bulk}}^{\mathrm{reg}}={\displaystyle \frac{1}{16\pi G_{d+1}}}\left[{\int }_{\rho >ϵ}{d}^{d+1}x\sqrt{-\widehat{g}}\left(\widehat{R}-2{\Lambda }_{d+1}\right)+2{\int }_{\rho =ϵ}{d}^{d}x\sqrt{-h}K\right].$$

(25)

Using the perturbative expansion for $g_{ij}(x,\rho )$, the regulated action (25) may be divided into a contribution $I_{\mathrm{div}}$ which diverges in the limit $ϵ\to 0$ and a finite contribution $I_{\mathrm{fin}}$

$$I_{\mathrm{bulk}}^{\mathrm{reg}}=I_{\mathrm{div}}+I_{\mathrm{fin}}.$$

(26)

Schematically, the IR divergent contribution is (see Appendix B for details)

$$I_{\mathrm{div}}={\displaystyle \frac{L_{d+1}}{16\pi G_{d+1}}}\int d^{d}x\sqrt{g_{\left(0\right)}}\left[{ϵ}^{-d/2}{a}_{\left(0\right)}+{ϵ}^{-d/2+1}{a}_{\left(2\right)}+...+{ϵ}^{-1}{a}_{(d-2)}-logϵa_{\left(d\right)}\right],$$

(27)

with coefficients $a_{\left(0\right)},a_{\left(2\right)},...$ that are covariant combinations of $g_{ij}^{\left(0\right)}$ and its derivatives. In terms of the boundary metric $h_{ij}$, it may be cast as

$$I_{\mathrm{div}}={\displaystyle \frac{L_{d+1}}{16\pi G_{d+1}}}{\int }_{\partial \mathcal{M}}{d}^{d}x\sqrt{-h}\left[{\displaystyle \frac{2(d-1)}{L_{d+1}^2}}+{\displaystyle \frac{R}{(d-2)}}+{\displaystyle \frac{L_{d+1}^2}{{(d-2)}^2(d-4)}}\left(R_{ij}^2-{\displaystyle \frac{d{R}^2}{4(d-1)}}\right)+...\right]$$

(28)

where the ellipsis indicates higher-curvature and higher-derivative contributions (see, e.g., [17,21,51,52]). The finite contribution $I_{\mathrm{fin}}\sim \mathcal{O}\left({ϵ}^0\right)+\mathcal{O}\left(ϵ\right)...$ survives the $ϵ\to 0$ limit, though it will also typically include higher-curvature terms. Its interpretation will be given momentarily.

At this stage, the renormalized action is obtained by minimal subtraction,

$$I_{\mathrm{bulk}}^{\mathrm{ren}}=\underset{ϵ\to 0}{lim}(I_{\mathrm{bulk}}^{\mathrm{reg}}+I_{\mathrm{ct}}),$$

(29)

where a local counterterm action has been introduced, $I_{\mathrm{ct}}=-I_{\mathrm{div}}$, to precisely cancel the IR divergences. Then, via the standard AdS/CFT dictionary, variations with respect to the metric $h_{ij}$ of the renormalized action yields the quantum expectation value of the stress tensor of the holographic CFT,

$$\langle T_{ij}^{\mathrm{CFT}}\rangle =\underset{ϵ\to 0}{lim}\left(-{\displaystyle \frac{2}{\sqrt{\widehat{g}(x,\rho )}}}{\displaystyle \frac{\delta I_{\mathrm{bulk}}^{\mathrm{ren}}}{\delta {\widehat{g}}^{ij}(x,ϵ)}}\right)\equiv -{\displaystyle \frac{2}{\sqrt{h}}}{\displaystyle \frac{\delta W_{\mathrm{CFT}}\left[h\right]}{\delta h^{ij}}},$$

(30)

such that the renormalized bulk action is identified with the quantum effective action of the CFT, $W_{\mathrm{CFT}}\left[h\right]$. Thus, at leading order, the finite action $I_{\mathrm{fin}}$ characterizes the CFT.

3.2. Braneworld Holography

In braneworld holography [11], the bulk IR cutoff surface $\partial \mathcal{M}$ is instead replaced by a d-dimensional end-of-the-world (ETW) Randall–Sundrum [13,14] or Karch–Randall [15,16] brane $\mathcal{B}$ at a small fixed distance away from the boundary (for a lightning review of braneworlds, see Appendix C). Hence, the physical space is cut off at the ETW brane and there are no longer IR divergences to be removed. For simplicity, assume the brane is purely tensional, having an action

$$I_{\tau }=-\tau {\int }_{\mathcal{B}}{d}^{d}x\sqrt{-h},$$

(31)

where $\tau$ is the brane tension. Since a portion of the bulk has been removed, to complete the space, a second copy of ${\mathrm{AdS}}_{d+1}$ with a brane is sewn to the first cutoff geometry along the cutoff surface (see Figure 2). This surgical procedure leads to a discontinuity in the extrinsic curvature $K_{ij}$ across the junction. The Israel junction conditons [53] relate this discontinuity to the brane stress tensor $S_{ij}$ via

$$∆K_{ij}-h_{ij}∆K=8\pi G_{d+1}{S}_{ij}=-8\pi G_{d+1}\tau h_{ij},$$

(32)

where $∆K_{ij}=K_{ij}^{+}-K_{ij}^{-}$ denotes the difference between the extrinsic curvature across either ‘+’ and ‘−’ sides of the brane (here, we take $K_{ij}^{+}=-K_{ij}^{-}$ such that $∆K_{ij}=2K_{ij}$), and the last equality follows from taking the metric $h_{ij}$ variation of the brane action (31), i.e., $S_{ij}\equiv -{\displaystyle \frac{2}{\sqrt{-h}}}{\displaystyle \frac{\delta I_{\tau }}{\delta h^{ij}}}$. Thus, the location of the brane in the completed space is determined by the junction conditions (32), which in the present case amounts to tuning the brane tension $\tau$.

Moreover, unlike the metric on the AdS boundary, the brane metric is dynamical, and it is governed by a holographically induced higher-curvature theory of gravity coupled to matter. Precisely, the induced brane theory is found by adding to the bulk theory (23) the brane action (31). Integrating out the bulk up to the ETW brane $\mathcal{B}$, as in holographic regularization, leads to an effective induced theory with action I

$$I\equiv I_{\mathrm{Bgrav}}\left[\mathcal{B}\right]+I_{\mathrm{CFT}}\left[\mathcal{B}\right],$$

(33)

where the brane gravity theory is (cf. [52,54])

$$\begin{array}{cc}\hfill I_{\mathrm{Bgrav}}& =2I_{\mathrm{div}}+I_{\tau }\hfill \\ & ={\displaystyle \frac{1}{16\pi G_d}}{\int }_{\mathcal{B}}{d}^{d}x\sqrt{-h}[R-2{\Lambda }_d+{\displaystyle \frac{L_{d+1}^2}{(d-4)(d-2)}}\left(R_{ij}^2-{\displaystyle \frac{d{R}^2}{4(d-1)}}\right)+\cdots ],\hfill \end{array}$$

(34)

where the factor of two accounts for integrating out the bulk on both sides of the brane, and the ellipsis corresponds to higher curvature densities, entering with higher powers of $L_{d+1}^2$. So far, the higher-derivative contributions have been computed up to quintic order in curvature for arbitrary d and sextic order for $d=3$ [52]. In principle, these results could be extended to arbitrary order, even though the calculations might be practically prohibitive. Here, $G_d$ represents the effective brane Newton’s constant induced from the bulk

$$G_d={\displaystyle \frac{d-2}{2L_{d+1}}}{G}_{d+1},$$

(35)

and ${\Lambda }_d=-(d-1)(d-2)/2L_d^2$ is an effective brane cosmological constant with an induced curvature scale $L_d$

$${\displaystyle \frac{1}{L_d^2}}={\displaystyle \frac{2}{L_{d+1}^2}}\left(1-{\displaystyle \frac{4\pi G_{d+1}{L}_{d+1}}{d-1}}\tau \right).$$

(36)

As written, it has been assumed the brane has a negative cosmological constant such that the bulk theory is coupled to a Karch–Randall brane [15]. When coupled to a Randall–Sundrum brane, the brane cosmological constant can be tuned to be positive or zero, as will be considered later.

Due to the presence of higher-derivative terms in the induced action (34), the brane theory of gravity is in general ‘massive’ since a massive graviton bound state will localize on the brane [15]. This brane graviton mass, however, will become negligible for a brane very near the boundary. Furthermore, general higher-derivative theories of gravity are often sick since they are typically accompanied by ghosts. In the present case, however, provided the series of higher-derivative terms is not truncated, the brane theory is not expected to inherit these usual pathologies, since the starting bulk theory and the procedure of integrating out bulk degrees of freedom are not pathological.

The action $I_{\mathrm{CFT}}\left[\mathcal{B}\right]$, meanwhile, describes the CFT theory, now living on the brane, and corresponds to the finite contribution to the regulated bulk action upon integrating out the bulk. To see this, note that upon integrating out the bulk degrees of freedom on both sides of the brane, we have

$$I\equiv 2I_{\mathrm{bulk}}^{\mathrm{reg}}+I_{\tau }.$$

(37)

Then, add and subtract the $2I_{\mathrm{div}}$, giving

$$I\equiv (2I_{\mathrm{div}}+I_{\tau })+(2I_{\mathrm{bulk}}^{\mathrm{reg}}-2I_{\mathrm{div}}),$$

(38)

where the first term in parentheses is recognized as $I_{\mathrm{Bgrav}}$ (34). The second term is simply $2I_{\mathrm{fin}}\equiv I_{\mathrm{CFT}}$, which, to leading order in the cutoff $ϵ$, is identified with the quantum effective action of the CFT, $I_{\mathrm{CFT}}=W_{\mathrm{CFT}}+\mathcal{O}\left(ϵ\right)$. In most cases of interest, we work in the limit that $ϵ$ is small, i.e., when the brane is close to the (now fictitious) ${\mathrm{AdS}}_{d+1}$ boundary, such that the matter on the brane has an approximate description as a large-c holographic CFT. Roughly speaking, a portion of the conformal ${\mathrm{AdS}}_{d+1}$ boundary has been pushed into the bulk, such that the dual ${\mathrm{CFT}}_d$ is now residing on the brane—however, this comes at a cost. Since the brane represents an IR cutoff surface, the CFT has a UV cutoff [55,56]. The cutoff, from the perspective of the boundary $g_{ij}^{\left(0\right)}$ metric, is denoted by $ϵ$, while from the induced brane metric $h_{ij}=(L_{d+1}^2/ϵ)g_{ij}^{\left(0\right)}$, the UV cutoff of the CFT is ${\delta }_{\mathrm{UV}}=L_{d+1}$.

3.3. Double Holography

A Karch–Randall braneworld15 has three equivalent descriptions: (i) bulk, (ii) intermediate, and (iii) boundary.

• Bulk: The bulk perspective is that of classical dynamical gravity in ${\mathrm{AdS}}_{d+1}$ coupled to an asymptotically ${\mathrm{AdS}}_d$ ETW brane of tension $\tau$. The simplest set-up assumes Einstein gravity plus a purely tensional brane; however, it is in principle possible to include higher-curvature corrections or fields to the bulk or brane actions. Israel junction conditions determine the location of the brane in the bulk, such that for a purely tensional brane, tuning the tension constitutes changing the position of the brane.
• Intermediate: The intermediate brane viewpoint describes induced dynamical gravity coupled to a UV cutoff ${\mathrm{CFT}}_d$ UV, which further communicates with a boundary CFT${ }_d$ (BCFT${ }_d$) via transparent boundary conditions. Bulk graviton fluctuations localize on the brane [13,14]. A subset of these graviton modes are light states with mass controlled by the tension; hence, the induced brane theory is an example of a massive theory of gravity. The remaining bulk graviton modes appear as a tower of Kaluza–Klein modes with masses set by the effective ${\mathrm{AdS}}_d$ brane length scale $1/L_d$.
• Boundary: Holographically, the bulk system has a dual description in terms of a ${\mathrm{CFT}}_d$ with a boundary (where the brane intersects the ${\mathrm{AdS}}_{d+1}$ boundary), i.e., a boundary ${\mathrm{CFT}}_d$. This set-up constitutes AdS/BCFT [57,58]. This perspective emerges when the brane gravity itself has a dual description in terms of a $(d-1)$-dimensional conformal defect. Upon replacing the brane gravity by a conformal defect (by integrating out the bulk and brane), the boundary perspective is characterized by the ${\mathrm{CFT}}_d$ on a fixed background, which is coupled to the defect. The boundary perspective is thus a UV/microscopic description of the bulk/brane gravity viewpoints.

Specific models exhibiting this type of ‘double holography’—holographic spacetimes dual to a BCFT that have three equivalent descriptions—have top–down string theoretic realizations [59]. We, however, will work with bottom–up constructions, for which their doubly holographic nature will play an important role in studying aspects of holographic entanglement and complexity, as we summarize in Section 716.

3.4. Holographic Quantum Black Holes: A Conjecture

Two equivalent ways to interpret the theory (33) are as follows. From the bulk perspective, I characterizes a theory of a $(d+1)$-dimensional system with dynamics ruled by Einstein gravity coupled to an end-of-the-world brane obeying appropriate boundary conditions. Meanwhile, from the (intermediate) brane perspective, I represents a specific higher-curvature gravity in d dimensions coupled to a large-c cutoff CFT that backreacts on the brane metric $h_{ij}$. The tower of higher-order derivative terms to the Einstein–Hilbert contribution represents quantum corrections induced by the backreaction of the ${\mathrm{CFT}}_d$. We refer to this higher-derivative tower as ‘corrections’ because in most cases of interest, one treats the brane action as an effective theory, assuming $L_d\gg L_{d+1}$ and guaranteeing the higher-derivative terms are suppressed by at least $\mathcal{O}(L_{d+1}^2/L_d^2)$17. Consistency between these two viewpoints implies solutions to the classical bulk equations satisfying proper brane boundary conditions exactly correspond to solutions to the semi-classical field equations on the brane. Therefore, the classical $(d+1)$-dimensional geometry encodes the entire series of quantum corrections to the d-dimensional brane geometry, accounting for all orders in the backreaction. Thus, holographic braneworlds provide a distinct computational advantage: rather than directly solving a complicated semi-classical theory of gravity, one may instead solve simpler classical gravitational field equations one dimension higher.

This philosophy, combined with the observation [62] that the $\sim 1/r^3$ corrections to the four-dimensional Newtonian potential due to massive Kaluza–Klein modes in the Randall–Sundrum model precisely coincide with corrections induced by one-loop quantum effects of the graviton propagator [63], suggests braneworld black holes from the brane perspective are quantum-corrected geometries. These insights in part motivated Emparan, Fabbri and Kaloper [22] to make the following conjecture:

Conjecture: Classical black holes which localize on a brane in ${\mathrm{AdS}}_{d+1}$ exactly map to d-dimensional quantum-corrected black holes including all orders of backreaction.

Such quantum-corrected black holes are dubbed ‘quantum’ black holes, though, they technically are solutions to the semi-classical theory induced on the brane. An illustration is given in Figure 3.

Explicit tests of this proposal include the exact localized ${\mathrm{AdS}}_4$ braneworld black holes discovered by Emparan, Horowitz, and Myers [23,24] with their projection onto the brane being reinterpreted as three-dimensional quantum black holes18. As we will see below, at least for the neutral, static geometries, the exact quantum black holes receive the same modifications to their geometry as suggested by the (non-holographic) perturbative analysis summarized in Section 2. The rotating and charged holographic quantum black holes, meanwhile, do not match the non-holographic perturbatively corrected counterparts (cf. [27,40,45]). In particular, a perturbative backreaction to the rotating BTZ black hole or Kerr–${\mathrm{dS}}_3$ solution due to a conformally coupled scalar field leads to a wildly more complicated radial dependence than that of a holographic CFT and a quantum stress tensor with a qualitatively different singularity structure, having an impact on the status of strong cosmic censorship. The remainder of this review will focus on these three-dimensional quantum black holes.

Before moving on to analyze the three-dimensional black holes, it is worth briefly commenting on the status of the proposal [22] in higher and lower dimensions. For brane dimensions $d\ge 4$, the most physically relevant case being $d=4$, there are still no known exact stationary solutions (see [64] for a review of analytic and numerical braneworld black holes)19. In fact, there is a no-go theorem [66] which alleges the exterior geometry on the brane in $d\ge 4$ cannot be static. The lack of exact solutions makes identifying the specific state of the ${\mathrm{CFT}}_d$ more difficult. In [22,34], for $d\ge 4$, it was qualitatively argued the obstruction to having static quantum black holes can be understood as a consequence of backreaction due to Hawking effects, such that any black hole that localizes on the brane must evaporate. However, static braneworld black holes in higher dimensions have been found numerically, e.g., [67,68,69,70,71,72], and the qualitative argument was shown to have flaws [69]. We review the status of the conjecture for higher-dimensional quantum black holes in Section 6.

In $d=2$ dimensions, the induced theory on the brane is characterized by a matter quantum effective action, which to leading order in expansion in the UV cutoff is characterized by the Polyakov action [4] coupled to a topological term and a cosmological constant. The precise form of the brane action likewise follows from a modification of holographic renormalization [19,50,73], where the counterterm action exactly truncates (see also [74,75]). Such a theory does not admit black hole solutions by itself. Thus, in order to find two-dimensional braneworld black holes, the theory must be modified by including a Dvali–Gabadadze–Porrati (DGP) term [76] to the tensional brane action (31). For example, one may replace the brane action (31) with a non-minimally coupled dilaton theory, e.g., Jackiw–Teitelboim (JT) gravity [77,78], as seen in [54]20. Such dilaton models of gravity admit exactly solvable black hole solutions including a semi-classical backreaction, e.g., [84,85,86,87].

Lastly, let us make some general remarks about static black holes localized on the brane. First, a brane with non-vanishing tension is an accelerated trajectory with respect to the bulk, i.e., the brane does not undergo geodesic motion. Thus, a black hole which localizes on the brane is in an accelerating frame, and it is the same for any observer glued to the brane. Next, a static black hole stuck to the brane will neither eat the brane nor slide off it. The reason is as follows [88]. To be static, the brane intersects the black hole orthogonally; otherwise, the black hole would grow by eating the brane21. Consequently, the brane bends to remain orthogonal to the black hole if the latter is being pulled off the brane (by, say, another black hole in the bulk). Thus, a static black hole localized on the brane experiences a restoring force due to the tension of the brane and does not slide off. Evaporating black holes, on the other hand, eventually slide off the brane.

4. Quantum Black Hole Taxonomy

Having laid the groundwork of braneworld holography in general dimensions, here we focus on the $d=3$ case, i.e., a holographic ${\mathrm{AdS}}_4$ bulk with a three-dimensional ETW brane. In this set-up, it is possible to write down a host of analytic braneworld black hole solutions to the bulk field equations plus Israel junction conditions. From the brane perspective, these black holes may be interpreted as ‘quantum’ black holes—those which exactly solve the induced semi-classical brane gravity including all orders of backreaction due to a large-c holographic ${\mathrm{CFT}}_3$ with a UV cutoff. Below, we classify known quantum black holes in three-dimensional (A)dS and Minkowski backgrounds. These include the quantum BTZ family of black holes (static, rotating, and charged) and their de Sitter/Minkowski analogs. Each of these braneworld black holes arise from specific parametrizations of the AdS${ }_4$ C-metric coupled to a Randall–Sundrum or Karch–Randall brane.

4.1. Bulk Geometry: AdS C-Metric

In this review, the bulk ${\mathrm{AdS}}_4$ gravity is taken to be Einstein–Maxwell theory. The most general metric solving the Einstein–Maxwell$-\Lambda$ gravity is the Plebiański–Demiański type-D metric [89]. We are interested in a specific sub-class of solutions, namely the AdS${ }_4$ C-metric, which can be interpreted as a single or pair of accelerating black holes in an ${\mathrm{AdS}}_4$ background, depending on the parameters of the solution. Here, we summarize the essentials for the braneworld construction (see Appendix D for a longer review). In particular, the neutral, non-rotating C-metric in Boyer–Lindquist-like coordinates has a line element (primarily following the conventions of [25])

$$d{s}^2={\displaystyle \frac{{\ell }^2}{{(\ell +xr)}^2}}\left[-H\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{H\left(r\right)}}+r^2\left({\displaystyle \frac{d{x}^2}{G\left(x\right)}}+G\left(x\right)d{\varphi }^2\right)\right],$$

(39)

with metric functions

$$H\left(r\right)={\displaystyle \frac{r^2}{{\ell }_3^2}}+\kappa -{\displaystyle \frac{\mu \ell }{r}},G\left(x\right)=1-\kappa x^2-\mu x^3.$$

(40)

We treat t and r as time and radial coordinates, respectively. However, the range for r is not the usual one for Boyer–Lindquist coordinates, with the ${\mathrm{AdS}}_4$ conformal boundary shifted from its familiar location, $r=\infty$. Rather, the position of the boundary depends on the coordinate x: for some values of x, the conformal boundary is closer than infinity, while for other values of x, the boundary is ‘further’ than infinity. That is, the radial coordinate has range $r\in (-\infty ,r_{\mathrm{bdry}})\cup (0,\infty )$, with $x{r}_{\mathrm{bdry}}=-\ell$ being the location of the asymptotic AdS${ }_4$ boundary, where the conformal factor diverges (this unfamiliar range for the radial coordinate can be seen more readily using the $(t,x,y,\varphi )$ coordinates in Appendix D). Further, $(x,\varphi )$ are angular variables, with $x\in [-1,1]$ analogous to $cos\theta$, and $\varphi$ is a general azimuthal coordinate whose periodicity will be discussed below.

Here, $\mu$, $\kappa$, ℓ and ${\ell }_3$ are real parameters characterizing the solution whose physical meaning will become more apparent momentarily. For now, $\kappa =\pm 1,0$ describes different possible slicings of the brane geometry, e.g., $\kappa =-1$ will recover the classical BTZ geometry on the brane; however, here, we leave $\kappa$ unspecified, thereby describing a family of braneworld black holes. The non-negative parameter $\mu$ is related to the mass of the bulk black hole. When interpreted as an accelerating black hole, the parameter $\ell \ge 0$ equals the inverse acceleration, $A={\ell }^{-1}$, and is related to the bulk ${\mathrm{AdS}}_4$ length scale $L_4$ via

$$L_4={\left({\displaystyle \frac{1}{{\ell }^2}}+{\displaystyle \frac{1}{{\ell }_3^2}}\right)}^{-1/2},$$

(41)

Lastly, the positive parameter ${\ell }_3$ will be related to the ${\mathrm{AdS}}_3$ brane curvature scale, however, with $0\le \ell <\infty$, such that ${\ell }_3>L_4$.

To gain further intuition for the global aspects of the metric (39), first set $\mu =0$ and perform the coordinate transformation [25]

$$cosh\left(\sigma \right)={\displaystyle \frac{{\ell }_3}{L_4}}{\displaystyle \frac{1}{|1+\frac{rx}{\ell }|}}\sqrt{1+{\displaystyle \frac{r^2{x}^2}{{\ell }_3^2}}},\widehat{r}=r\sqrt{{\displaystyle \frac{1-\kappa x^2}{1+\frac{r^2{x}^2}{{\ell }_3^2}}}}.$$

(42)

This brings the C-metric line element (39) to the form

$$d{s}^2=L_4^{2}d{\sigma }^2+{\displaystyle \frac{L_4^2}{{\ell }_3^2}}{cosh}^2\left(\sigma \right)\left[-\left(\kappa +{\displaystyle \frac{{\widehat{r}}^2}{{\ell }_3^2}}\right)d{t}^2+{\left(\kappa +{\displaystyle \frac{{\widehat{r}}^2}{{\ell }_3^2}}\right)}^{-1}d{\widehat{r}}^2+{\widehat{r}}^{2}d{\varphi }^2\right],$$

(43)

which is recognized to be empty ${\mathrm{AdS}}_4$, which is foliated by ${\mathrm{AdS}}_3$ slices with radius $L_{4}cosh\left(\sigma \right)$ at constant $\sigma$. Written this way, the role of $\kappa$ becomes apparent: for constant $\sigma$, $\kappa =+1,-1$ and 0, respectively, correspond to global ${\mathrm{AdS}}_3$, BTZ and Poincaré ${\mathrm{AdS}}_3$.

To obtain a better sense for the parameter ℓ and why the geometry (39) describes a black hole, perform the parameter and coordinate rescalings $\mu =2m/\ell$ and $r={\ell }_3\rho /\ell$. Keeping $m,L_4$, and $\rho$ finite, the limit $\ell \to \infty$ results in

$$d{s}^2=-f\left(\rho \right)d{t}^2+f^{-1}\left(\rho \right)d{\rho }^2+{\rho }^2\left({\displaystyle \frac{d{x}^2}{(1-\kappa x^2)}}+(1-\kappa x^2)d{\varphi }^2\right),f\left(\rho \right)=\kappa +{\displaystyle \frac{{\rho }^2}{L_4^2}}-{\displaystyle \frac{2m}{\rho }},$$

(44)

the metric for static, neutral ${\mathrm{AdS}}_4$ black holes (further clarified upon transforming $x=cos\theta$). Since ${\ell }^{-1}=A$, the limiting geometry (44) shows the acceleration distorts spherical surfaces parametrized by $(x,\varphi )$.

4.1.1. Horizons and Bulk Regularity

The limiting geometry (44) also shows black holes appear for $\mu \ne 0$. More generally, whether the C-metric (39) has a black hole depends on the root structure of the metric functions (40). In particular, the roots of $H\left(r\right)$ correspond to Killing horizons generated by the time translation Killing vector ${\partial }_t$, where we desire positive roots if we are to describe physical black hole horizons. Since the system is accelerating, it will also have non-compact Rindler-like acceleration horizons. To rid ourselves of the acceleration horizon, it is sufficient to work in the case where $\ell >L_4$. With this restriction, $H\left(r\right)$ has a single positive root $r_{+}$ representing a black hole horizon22, and the C-metric describes a single ‘slowly accelerating’ black hole suspended away from the center of ${\mathrm{AdS}}_4$ by a cosmic string attached at the horizon [90]. As with other spherical surfaces, the acceleration distorts the horizon into a conical shape.

Real roots of $G\left(x\right)$, meanwhile, correspond to symmetry axes of the Killing vector ${\xi }^a={\partial }_{\varphi }^a$, i.e., ${\xi }^2\sim G\left(x\right)$, vanishing at a zero of $G\left(x\right)$. These roots characterize the geometry of the horizon in the bulk. To ensure a finite black hole horizon in the bulk, one must be in the parameter space where there exists at least one positive root of $G\left(x\right)$, the smallest of which will be denoted $x_1$, and then work in the restricted range $0\le x\le x_1$. The general strategy is the following [23,24]. Treat the root $x_1$ as a primary parameter while $\mu$ is derived from $x_1$ via $G\left(x_1\right)=0$, i.e.,

$$\mu ={\displaystyle \frac{1-\kappa x_1^2}{x_1^3}}.$$

(45)

The desired parameter range follows from taking $x_1>0$ and

$$x_1\in (0,1]\mathrm{for}\kappa =+1,$$

(46)

$$x_1\in (0,\infty )\mathrm{for}\kappa =-1,0.$$

(47)

Observe $\mu$ monotonically decreases from $+\infty$ to zero, where $\mu =0$ coincides with the upper limit of $x_1$, e.g., when $\kappa =+1$, $\mu \to 0$ as $x_1\to 1$.

Further, for the range of $\mu$ and $\kappa$ we are interested in, $G\left(x\right)$ has three distinct zeros, each of which leads to a distinct conical singularity. The conical singularity at $x=x_1$, for example, is removed via the identification

$$\varphi \sim \varphi +∆\varphi ,∆\varphi ={\displaystyle \frac{4\pi }{|G^{\prime }\left(x_1\right)|}}={\displaystyle \frac{4\pi x_1}{3-\kappa x_1^2}}.$$

(48)

We see for the range of $x_1$, the function $G^{\prime }\left(x_1\right)=-{\displaystyle \frac{3-\kappa x_1^2}{x_1}}<0$, and that $∆\varphi$ is independent of ℓ and ${\ell }_3$. Moreover, $∆\varphi$ grows monotonically from 0 to $2\pi$. Fixing the period of $\varphi$ in this way, the spacetime will have conical singularities at the remaining roots of $G\left(x\right)$. The effect of these conical singularities is a distortion to the black hole horizon, which may be viewed as a cosmic string with a tension proportional to the angular deficit pulling the black hole away from the center of AdS${ }_4$ toward the boundary, generating the black hole acceleration (see Figure 4). Below, we will see how the spacetime surgery used to construct a ${\mathbb{Z}}_2$ braneworld will eliminate these conical singularities from the surgically complete bulk geometry.

4.1.2. Karch–Randall Braneworld Construction

The most advantageous geometric feature of the C-metric (39), for any $\mu$ and $\kappa$, is that the $x=0$ timelike hypersurface is umbilic. That is, the extrinsic curvature $K_{ij}$ of the hypersurface is proportional to the induced metric $h_{ij}$ on $x=0$. In particular, the outward unit normal to timelike surfaces of constant x is $n^i=-\left({\displaystyle \frac{x}{\ell }}+{\displaystyle \frac{1}{r}}\right)\sqrt{G\left(x\right)}{\partial }_x^i$, such that23 (see Appendix D.4)

$$K_{ij}={\displaystyle \frac{1}{\ell }}{h}_{ij},$$

(49)

at $x=0$. Umbilic surfaces automatically satisfy the Israel junction conditions (32). In this case, upon substituting (49), a brane at $x=0$ has tension

$$\tau ={\displaystyle \frac{1}{2\pi G_4\ell }}.$$

(50)

Thus, the tension is proportional to the acceleration of the bulk black hole. The tensionless limit corresponds to $\ell \to \infty$, where ${\ell }_3\to L_4$ by virtue of (41).

To see how tuning the tension amounts to changing the position of the brane, recall the empty ${\mathrm{AdS}}_4$ metric (43). The $x=0$ brane amounts to surfaces of fixed $\sigma ={\sigma }_b$ obeying

$$cosh\left({\sigma }_b\right)={\displaystyle \frac{{\ell }_3}{L_4}}=\sqrt{1+{\displaystyle \frac{{\ell }_3^2}{{\ell }^2}}},$$

(51)

and the brane geometry is ${\mathrm{AdS}}_3$ with curvature radius ${\ell }_3$, i.e., a Karch–Randall brane. In the tensionless limit, $\ell \to \infty$, then ${\sigma }_b=0$, cutting the bulk in half through the equator. Alternatively, as the tension becomes larger, $\ell \to 0$, it follows ${\sigma }_b\to \infty$, i.e., the brane is pushed to the asymptotic boundary of AdS${ }_4$. The solution is valid for all $0\le \ell <\infty$; however, we will be primarily interested in the case where the brane is near the ${\mathrm{AdS}}_4$ boundary (before it has been removed via the ETW brane). See Figure 5 for an illustration (and refer to Appendix D.5 for details projecting to the Poincaré disk).

More generally, the induced metric at $x=0$ simply follows from setting $x=0$ in the bulk C-metric (39), resulting in

$$d{s}^2{|}_{x=0}=-\left({\displaystyle \frac{r^2}{{\ell }_3^2}}+\kappa -{\displaystyle \frac{\mu \ell }{r}}\right)d{t}^2+{\left({\displaystyle \frac{r^2}{{\ell }_3^2}}+\kappa -{\displaystyle \frac{\mu \ell }{r}}\right)}^{-1}d{r}^2+r^{2}d{\varphi }^2,$$

(52)

For $\kappa =-1$, the boundary geometry ($\ell \to 0$) has a black hole with horizon radius $r_{+}={\ell }_3$. For $\ell \ne 0$, it is clear the geometry (52) is capable of describing a static black hole, which will ultimately be understood as the quantum BTZ black hole as we detail below.

Recall the bulk geometry will retain conical singularities at the remaining roots of $G\left(x\right)$ (not $x=x_1$). As it happens, there are no conical singularities in the restricted range $0\le x\le x_1$; for $\mu >0$, the remaining conical singularities live in the range $x<0$. Then, treating the $x=0$ hypersurface as a cutoff brane, excising the $x<0$ region leads to a spacetime free of conical singularities. To complete the space, a second copy of the $0\le x\le x_1$ region is glued along $x=0$, resulting in a ${\mathbb{Z}}_2$-symmetric double-sided Karch–Randall braneworld. Further, with the conical singularities removed, the final bulk solution no longer has a cosmic string. Nonetheless, the static black hole attached to the brane is in an accelerated frame.

4.1.3. Karch–Randall Braneworld Holography

Applying the formalism of Section 3, the induced brane action is (setting $d=3$ in (33))

$$I={\displaystyle \frac{1}{16\pi G_3}}{\int }_{\mathcal{B}}{d}^{3}x\sqrt{-h}\left[R-2{\Lambda }_3+L_4^2\left({\displaystyle \frac{3}{8}}{R}^2-R_{ij}^2\right)+...\right]+I_{\mathrm{CFT}},$$

(53)

with induced Newton’s constant

$$G_3={\displaystyle \frac{G_4}{2L_4}}$$

(54)

and effective (AdS${ }_3$) brane cosmological constant and curvature scale,

$${\Lambda }_3=-{\displaystyle \frac{1}{L_3^2}},{\displaystyle \frac{1}{L_3^2}}={\displaystyle \frac{2}{L_4^2}}\left(1-{\displaystyle \frac{L_4}{\ell }}\right),$$

(55)

where we used tension (50). Note that while $L_3$ appears in the action, the solutions on the brane are characterized by ${\ell }_3$.

In fact, we will primarily be interested in the case when $L_3\approx {\ell }_3$. This is because we are interested in the case when the three-dimensional graviton becomes effectively massless, i.e., when the brane is near the boundary. Hence, using the bulk length scale (41), it follows

$${\displaystyle \frac{1}{L_3^2}}={\displaystyle \frac{1}{{\ell }_3^2}}\left[1+{\displaystyle \frac{{\ell }^2}{4{\ell }_3^2}}+\mathcal{O}\left({\displaystyle \frac{{\ell }^4}{{\ell }_3^4}}+...\right)\right],$$

(56)

with $\ell \sim L_4\ll {\ell }_3$. In this limit, the induced theory is

$$I={\displaystyle \frac{1}{16\pi G_3}}{\int }_{\mathcal{B}}{d}^{3}x\sqrt{-h}\left[R+{\displaystyle \frac{2}{{\ell }_3^2}}+{\ell }^2\left({\displaystyle \frac{3}{8}}{R}^2-R_{ij}^2\right)+...\right]+I_{\mathrm{CFT}},$$

(57)

such that the higher-curvature expansion can be viewed as an expansion in an effective cutoff scale ℓ. Notice higher-curvature corrections enter at quadratic order.

The parameter ℓ also features into the coupling to the gravity theory of the cutoff ${\mathrm{CFT}}_3$ on the brane. To see this, first note the central charge $c_3$ of the ${\mathrm{CFT}}_3$ is normalized such that

$$c_3={\displaystyle \frac{L_4^2}{G_4}}={\displaystyle \frac{\ell }{2G_3\sqrt{1+{\nu }^2}}},$$

(58)

where we introduced the parameter $\nu \equiv \ell /{\ell }_3$. Expanding for small $(\ell /{\ell }_3)$ gives

$$2c_3{G}_3=L_4\approx \ell \left(1-{\displaystyle \frac{{\ell }^2}{2{\ell }_3^2}}+{\displaystyle \frac{3}{8}}{\displaystyle \frac{{\ell }^4}{{\ell }_3^4}}+...\right),$$

(59)

and $c_3\sim {\displaystyle \frac{\ell }{G_3}}$, which enters at a linear order in ℓ. Thus, the matter contributions enter at order $\mathcal{O}(\ell )$, while higher-curvature corrections appear at $\mathcal{O}\left({\ell }^2\right)$. Proceeding, the typical value of $I_{\mathrm{CFT}}$ is $|I_{\mathrm{CFT}}|\sim c_3$, while the typical value of the gravitational part of the action (53) is $|I_{\mathcal{B}\mathrm{grav}}|\sim {\displaystyle \frac{L_3}{G_3}}$. Combined, this leads to an effective dimensionless coupling $g_{\mathrm{eff}}$ quantifying the size of the effects the ${\mathrm{CFT}}_3$ has on the brane geometry [91]

$$g_{\mathrm{eff}}\sim {\displaystyle \frac{|I_{\mathrm{CFT}}|}{|I_{\mathcal{B}\mathrm{grav}}|}}\sim {\displaystyle \frac{G_3{c}_3}{L_3}}.$$

(60)

Thus, for small $\ell <{\ell }_3$, we see $g_{\mathrm{eff}}\approx \nu \ll 1$, backreaction effects are ‘small’, and the induced theory is effectively characterized by the action (57). Further, from (59), we see for a fixed $c_3$, gravity becomes weak ($G_3\to 0$) as $\ell \to 0$. Alternatively, when $\nu \ge 1$, backreaction effects are said to be large.

It is worth comparing the cutoff length scale to the three-dimensional Planck length $L_{\mathrm{P}}=\hslash G_3$ (where here, we have temporarily restored factors of ℏ). From (59), it follows

$$\ell \sim c_3\hslash G_3=c_3{L}_{\mathrm{P}}\gg L_{\mathrm{P}},$$

(61)

since the holographic cutoff ${\mathrm{CFT}}_3$ obeys $c_3\gg 1$24. Recalling the induced brane metric (52), notice the blackening factor includes a $\mathcal{O}\left(r^{-1}\right)$ term which goes like $\mu \ell$. We can anticipate this term as a semi-classical correction which sets the size of the braneworld black hole to be much larger than the Planck length.

4.2. Quantum BTZ Black Holes

Let us now delve into the geometry of quantum black holes. We begin with the simpler neutral, static quantum BTZ solution to establish its essential features, which we later enrich by adding rotation and charge.

4.2.1. Static Quantum BTZ

Recall the induced geometry at $x=0$ (52). Naively, we might refer to this as the quantum black hole; however, the angular coordinate $\varphi$ has period $∆\varphi$ (48). We thus rescale coordinates $(t,r,\varphi )$ to put the naive metric (52) in canonically normalized coordinates $(\overline{t},\overline{r},\overline{\varphi })$. Specifically,

$$t=\eta \overline{t},r={\displaystyle \frac{\overline{r}}{\eta }},\varphi =\eta \overline{\varphi },$$

(62)

with

$$\eta \equiv {\displaystyle \frac{∆\varphi }{2\pi }}={\displaystyle \frac{2x_1}{3-\kappa x_1^2}}.$$

(63)

The line element for the brane metric is now

$$d{s}_{\mathrm{qBTZ}}^2=-\left({\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-8{\mathcal{G}}_{3}M-{\displaystyle \frac{\ell F\left(M\right)}{\overline{r}}}\right)d{\overline{t}}^2+{\left({\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-8{\mathcal{G}}_{3}M-{\displaystyle \frac{\ell F\left(M\right)}{\overline{r}}}\right)}^{-1}d{\overline{r}}^2+{\overline{r}}^{2}d{\overline{\varphi }}^2.$$

(64)

Here, we have suggestively identified the three-dimensional mass

$$M\equiv -{\displaystyle \frac{\kappa }{8G_3}}{\displaystyle \frac{\ell }{L_4}}{\eta }^2=-{\displaystyle \frac{1}{2{\mathcal{G}}_3}}{\displaystyle \frac{\kappa x_1^2}{{(3-\kappa x_1^2)}^2}},{\mathcal{G}}_3\equiv G_3{\displaystyle \frac{L_4}{\ell }}={\displaystyle \frac{G_3}{\sqrt{1-{\nu }^2}}},$$

(65)

and

$$F\left(M\right)\equiv \mu {\eta }^3=8{\displaystyle \frac{(1-\kappa x_1^2)}{{(3-\kappa x_1^2)}^3}}.$$

(66)

The mass identification (65) is primarily motivated by the fact that for in Einstein–AdS gravity, the mass corresponds to the subleading constant term in the $g_{tt}$ metric in appropriate coordinates. However, the induced theory on the brane is a higher-derivative theory, and thus the definition of mass is expected to be modified due to higher-derivative corrections, starting at $\mathcal{O}\left({\ell }^2\right)$. When treated as corrections to Einstein gravity, the mass M may still be identified as the constant term in $g_{tt}$; however, now the classical Newton’s constant $G_3$ is ‘renormalized’ by the higher-derivative terms [92]

$$\begin{array}{cc}\hfill {\mathcal{G}}_3& =\left(1-{\displaystyle \frac{{\ell }^2}{2L_3^2}}+\mathcal{O}{\left({\displaystyle \frac{\ell }{L_3}}\right)}^4\right)G_3=\left(1-{\displaystyle \frac{{\nu }^2}{2}}\right){\displaystyle \frac{G_4}{2L_4}}+\mathcal{O}{\left({\displaystyle \frac{\ell }{L_3}}\right)}^4.\hfill \end{array}$$

(67)

Alternatively, here, it is assumed that the renormalized Newton’s constant be identified as ${\mathcal{G}}_3\equiv G_3{\displaystyle \frac{L_4}{\ell }}$ at all orders in ℓ, although this differs from (67) at order $\mathcal{O}{\left(\ell /L_3\right)}^4$25. As such, ${\mathcal{G}}_3$ in mass (67) is interpreted as an all-order resummation of the higher-derivative corrections to the mass [25], whilst $G_3$ is the ‘bare’ Newton constant, since it is the physical constant $G_{3}M$ which is being corrected. Further evidence for the identification of the mass will be given when we explore the horizon thermodynamics of the three-dimensional black hole. Lastly, the form function $F\left(M\right)$ (66) is purely a function of mass M because it depends on M only through $x_1$ and is otherwise independent of $\nu =\ell /{\ell }_3$.

The metric (64) is highly reminiscent of the perturbative geometry (19) incorporating the semi-classical backreaction effects of a conformally coupled scalar26. We emphasize, however, that the $1/r$ correction in $g_{tt}$ (19) is perturbative, while it is exact in (64). In particular, the geometry (64) is an exact solution to the gravitational field equations of the semi-classical induced action (57)

$$\begin{array}{c}8\pi G_3\langle T_{ij}\rangle =R_{ij}-{\displaystyle \frac{1}{2}}{h}_{ij}\left(R+{\displaystyle \frac{2}{L_3^2}}\right)\\ +{\ell }^2[4R_i^k{R}_{jk}-{\displaystyle \frac{9}{4}}R{R}_{ij}-\square R_{ij}+{\displaystyle \frac{1}{4}}{\nabla }_i{\nabla }_{j}R+{\displaystyle \frac{1}{2}}{h}_{ij}\left({\displaystyle \frac{13}{8}}{R}^2-4R_{kl}^2+{\displaystyle \frac{1}{2}}\square R\right)],\end{array}$$

(68)

where $\square \equiv {\nabla }^2$. Decomposing the holographic stress tensor in an expansion in ${\ell }^2$, i.e., $\langle T_j^i\rangle ={\langle T_j^i\rangle }_0+{\ell }^2{\langle T_j^i\rangle }_2+...$, it follows

$$8\pi G_3{\langle T_j^i\rangle }_0=R_j^i-{\displaystyle \frac{1}{2}}{\delta }_j^i\left(R+{\displaystyle \frac{2}{{\ell }_3^2}}\right),$$

(69)

$$\begin{array}{cc}\hfill 8\pi G_3{\langle T_j^i\rangle }_2& =4R^{ik}{R}_{jk}-\square R_j^i-{\displaystyle \frac{9}{4}}R{R}_j^i+{\displaystyle \frac{1}{4}}{\nabla }^i{\nabla }_{j}R+{\displaystyle \frac{1}{2}}{\delta }_j^i\left({\displaystyle \frac{13}{8}}{R}^2-3R_{kl}^2+{\displaystyle \frac{1}{2}}\square R-{\displaystyle \frac{1}{2{\ell }_3^4}}\right).\hfill \end{array}$$

(70)

Substituting in the metric (64), we obtain for the renormalized stress-energy tensor

$${\langle T_j^i\rangle }_0={\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{\ell F\left(M\right)}{{\overline{r}}^3}}\mathrm{diag}\{1,1,-2\}$$

(71)

to leading order and

$${\langle T_j^i\rangle }_2={\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{\ell F\left(M\right)}{{\overline{r}}^3}}\left({\displaystyle \frac{1}{2{\ell }_3^2}}\mathrm{diag}\{1,-11,10\}-{\displaystyle \frac{24{\mathcal{G}}_{3}M}{{\overline{r}}^2}}\mathrm{diag}\{3,1,-4\}+{\displaystyle \frac{\ell F\left(M\right)}{2{\overline{r}}^3}}\right)$$

(72)

for the $\mathcal{O}\left({\ell }^2\right)$ corrections. Notice ${\langle T_{ij}\rangle }_2$ has a non-zero trace, which is a consequence of breaking the conformal symmetry due to the cutoff ℓ. In principle, one could compute $\langle T_{ij}\rangle$ at all orders of backreaction—unlike the non-holographic perturbative analysis—however, it proves cumbersome to do so.

Since the backreaction is turned off as $\ell \to 0$, the metric (64) in this limit may be interpreted as ‘classical’; indeed, the $1/r$ correction is eliminated, and the resulting geometry solves the three-dimensional vacuum Einstein equations. Hence, for $\ell \ne 0$, the geometry (64) is naturally understood as a quantum black hole in AdS${ }_3$, namely, the quantum BTZ (qBTZ) black hole. Notice further, in the limit of a small backreaction ($\nu <1$), the $1/r$ term in (64) is proportional to $\ell \sim c{G}_3\gg L_{\mathrm{P}}$ (61), such that the horizon size of the quantum BTZ black hole is large compared to the Planck length. Further, it is worth emphasizing that the solution (64) consistently solves the semi-classical brane equations of motion for any $\ell >0$, including large backreaction effects $(\nu >1)$.

Unlike the classical BTZ black hole, the quantum BTZ (64) has a curvature singularity at $\overline{r}=0$, as evidenced by the Kretschmann invariant,

$$R^{ijkl}{R}_{ijkl}={\displaystyle \frac{12}{{\ell }_3^4}}+{\displaystyle \frac{6F{\left(M\right)}^2{\ell }^2}{{\overline{r}}^6}}.$$

(73)

The curvature singularity descends from the curvature singularity of the bulk four-dimensional C-metric at $r=0$, and it is hidden behind the (bulk) horizon $r=r_{+}$. Here, the singularity at $\overline{r}=0$ sits behind the horizon at $\overline{r}={\overline{r}}_{+}$, the largest positive root of the metric function $H\left(\overline{r}\right)$, where the time-translation Killing vector ${\overline{\zeta }}^i=\eta {\partial }_{\overline{t}}^i$ goes null. Thus, at least for holographic quantum matter, the singularity structure of a classical black hole becomes dramatically altered due to a semi-classical backreaction. Note, moreover, relative to ${\overline{\zeta }}^i$, the surface gravity, defined via ${\overline{\zeta }}^i{\nabla }_i{\overline{\zeta }}_j={\kappa }_{+}{\overline{\zeta }}_j$, is

$${\kappa }_{+}={\displaystyle \frac{\eta }{2}}|H^{\prime }\left(r_{+}\right)|={\displaystyle \frac{1}{2}}\left|H^{\prime }\left({\overline{r}}_{+}\right)\right|={\displaystyle \frac{1}{2{\ell }_3^2}}|2{\overline{r}}_{+}+{\displaystyle \frac{\mu \ell {\eta }^3}{{\overline{r}}_{+}^2}}|,$$

(74)

In the limit $\ell \to 0$, this corresponds to the surface gravity of the classical BTZ black hole.

Aside from the match of the perturbed geometry, the leading order contribution to $\langle T_{ij}\rangle$ (71) agrees with the renormalized stress tensor of the free conformal scalar (except $L_{\mathrm{P}}\to \ell$). This match between the holographic and free field results is because transparent boundary conditions have been imposed on the CFT or the free scalar. In the former, these boundary conditions are naturally selcted because bulk fluctuations, which are dual to CFT excitations, move freely throughout the bulk. For the free field computation, however, transparent boundary conditions were an explicit choice. In either case, notice the leading order contribution to the stress tensor is not of the standard thermal type, i.e., $\mathrm{diag}\{-2,1,1\}$. Nonetheless, the CFT is in a thermal state. Indeed, the Green’s function from which $\langle T_{ij}\rangle$ is derived (in the perturbative treatment) is periodic in imaginary time with a period given by the (inverse) Tolman temperature [42]. Further, the Green function obeys analyticity properties shared by the Hartle–Hawking state, which is a state describing a black hole in thermal equilibrium with its own radiation.

A family of quantum black holes. From the mass identification (65), M takes values in the finite range [23,24],

$$-{\displaystyle \frac{1}{8{\mathcal{G}}_3}}\le M\le {\displaystyle \frac{1}{24{\mathcal{G}}_3}},$$

(75)

where ℓ is held fixed. The upper bound follows from studying for the end behavior of $M\left(x_1\right)$, with $x_1=\sqrt{3}$ (excluding $x_1=-\sqrt{3}$ on account of the parameter range (46)) at $\kappa =-1$ being a maximum. The lower bound, meanwhile, occurs for $\kappa =+1$ and $x_1=1$ (where $\mu =0$ according to (45)), and $M=0$ for either $x_1=0$ or $\kappa =0$. Thus, the qBTZ solution has negative and positive masses.

In fact, the solution may be categorized into physically distinct branches:

$$\begin{array}{cc}& \mathrm{Branch}1:-1<-\kappa x_1^2<3,\hfill \\ & \mathrm{Branch}2:3<-\kappa x_1^2<\infty .\hfill \end{array}$$

(76)

To understand how these branches arise, it is first natural to distinguish solutions with negative mass versus those with positive mass. All negative mass solutions occur for $\kappa =+1$ and for $x_1\in (0,1]$, while all positive mass solutions occur for $\kappa =-1$ and $x_1\in (0,\infty )$. Note further $M=0$ for either $x_1=0$ (for both $\kappa =\pm 1$) or $x_1\to \infty$ (and $\kappa =+1$). This suggests the positive mass solutions subdivide along the range $x_1\in (0,\sqrt{3})$ and $x_1\in (\sqrt{3},\infty )$, smoothly joining at $x_1=\sqrt{3}$. Further, since it is the combination $\kappa x_1^2$ that appears in all physical quantities of interest, the negative mass branch and the positive mass branch with $x_1\in (0,\sqrt{3})$ smoothly connect at $x_1=0$. Succinctly, the qBTZ solution is characterized by the two branches (76); however, a finer subdivision of the two branches is [25]

$$\begin{array}{cc}& \mathrm{Branch}1\mathrm{a}:\kappa =+1,0<x_1<1,\hfill \\ & \mathrm{Branch}1\mathrm{b}:\kappa =-1,0<x_1<\sqrt{3},\hfill \\ & \mathrm{Branch}2:\kappa =-1,\sqrt{3}<x_1<\infty .\hfill \end{array}$$

(77)

Although each of the mass branches (77) smoothly connects, each branch has a different physical interpretation. This can be better understood by considering the types of solutions belonging to each branch in the limit of vanishing backreaction. Firstly, branch 1a contains the ground state $M=-{\displaystyle \frac{1}{8{\mathcal{G}}_3}}$, where $F\left(M\right)=0$ (substitute $x_1=1$ into (66)). Geometrically, the qBTZ line element (64) takes the form of global ${\mathrm{AdS}}_3$. Notably, however, this ground state is valid for any $\ell \ge 0$, and thus for $\ell \ne 0$, it can be thought of as three-dimensional quantum anti-de Sitter spacetime, ${\mathrm{qAdS}}_3$, accounting for a large-c cutoff ${\mathrm{CFT}}_3$ living in AdS${ }_3$ with a vanishing renormalized stress tensor. Above this ground state, negative mass solutions (branch 1a) correspond to AdS${ }_3$ conical defects in the limit $\ell \to 0$, such that for $\ell \ne 0$, the negative mass family of solutions are understood to be quantum-corrected conical singularities, though are still referred to as quantum ‘black holes’. Classically, the conical defects are horizonless, while when $\ell >0$, the quantum Casimir stress tensor shrouds the conical singularity in a horizon.

Branches 1b and 2, having $M\ge 0$, both describe quantum corrections to the classical BTZ black hole geometry. The sources of the corrections of these two branches, however, differ. Since branch 1a and 1b connect at $x_1=0$ (where $M=0$ and $F\left(0\right)=8/27$) and 1a black holes form due to the backreaction of Casimir stress–energy, so too are the corrections resulting in the 1b black holes dominated by Casimir energy. Alternatively, the $M=0$ state of the branch 2 black holes (where $x_1\to \infty$) have zero $F\left(M\right)$ and hence stress–energy. This suggests the non-zero stress–energy of the $M>0$ states among the branch 2 black holes is due to Hawking radiation in thermal equilibrium with the finite temperature black hole (where the dominant Casimir energy has been subtracted from the quantum state appearing in $\langle T_{ij}\rangle$ [25]).

Classically, the negative mass conical singularities and positive mass BTZ black holes are disconnected sets of solutions to Einstein–AdS${ }_3$ gravity—there is a ‘mass gap’ [31]. Evidently, a semi-classical backreaction smoothly connects these solutions, such that the quantum BTZ geometry represents a family of quantum black holes. We illustrate this family in Figure 6.

It is natural to wonder if there exist any other bulk solutions which give rise to a BTZ black hole on the brane with masses $M\ge 0$, including those that exceed the upper bound in the range (75). Indeed, recall the AdS${ }_4$ geometry (43) for which $\mu =0$. When $\kappa =-1$, this four-dimensional geometry is dubbed the ‘BTZ black string’, since sections of constant $\sigma$ contain a BTZ black hole. Despite $\mu =0$, a BTZ black hole lives on the brane at $x=0$ with mass $M\ge 0$ in the canonically normalized coordinates $(\overline{t},\overline{r},\overline{\varphi })$; however, the geometry is uncorrected from its classical counterpart, since the quantum stress tensor vanishes (on account of $F\left(M\right)=0$ for all $M\ge 0$). Unlike the three branches (77), this braneworld black hole has no upper bound on M, and hence solutions with $M>1/24{\mathcal{G}}_3$ are induced by a BTZ black string localized on the AdS${ }_3$ brane.

4.2.2. Rotating Quantum BTZ

Bulk and Brane Geometry

It is reasonably straightforward to find rotating quantum BTZ black holes. The starting point is the rotating AdS C-metric describing accelerating Kerr–AdS${ }_4$ black holes,

$$\begin{array}{cc}\hfill d{s}^2={\displaystyle \frac{{\ell }^2}{{(\ell +xr)}^2}}[& -{\displaystyle \frac{H\left(r\right)}{\Sigma (x,r)}}{\left(dt+a{x}^{2}d\varphi \right)}^2+{\displaystyle \frac{\Sigma (x,r)}{H\left(r\right)}}d{r}^2\hfill \\ \hfill & +r^2\left({\displaystyle \frac{\Sigma (x,r)}{G\left(x\right)}}d{x}^2+{\displaystyle \frac{G\left(x\right)}{\Sigma (x,r)}}{\left(d\varphi -{\displaystyle \frac{a}{r^2}}dt\right)}^2\right)],\hfill \end{array}$$

(78)

where

$$\begin{array}{cc}& H\left(r\right)={\displaystyle \frac{r^2}{{\ell }_3^2}}+\kappa -{\displaystyle \frac{\mu \ell }{r}}+{\displaystyle \frac{a^2}{r^2}},G\left(x\right)=1-\kappa x^2-\mu x^3+{\displaystyle \frac{a^2}{{\ell }_3^2}}{x}^4,\hfill \\ & \Sigma (x,r)=1+{\displaystyle \frac{a^2{x}^2}{r^2}}.\hfill \end{array}$$

(79)

Here, a controls the rotation (the angular momentum per unit mass) and for $a=0$, we recover the static C-metric (39). By evaluating the Kretschmann scalar invariant $R^{abcd}{R}_{abcd}$, there is a curvature singularity when $r^2\Sigma =r^2+a^2{x}^2=0$, i.e., where $r=x=0$. This is the familiar ring singularity of Kerr black holes.

Despite complicating the geometry by including rotation, the $x=0$ hypersurface remains umbilic, such that $K_{ij}={\ell }^{-1}{h}_{ij}$ (49) and a Karch–Randall brane at $x=0$ has tension (50). The geometry at $x=0$ is

$$d{s}^2{|}_{x=0}=-H\left(r\right)d{r}^2+H^{-1}\left(r\right)d{r}^2+r^2{\left(d\varphi -{\displaystyle \frac{a}{r^2}}dt\right)}^2.$$

(80)

At first glance, this line element looks like a rotating black hole in Boyer–Lindquist-like coordinates. As in the static case, however, this geometry does not reflect the whole story. In particular, the ‘naive metric’ (80) unexpectedly no longer has a ring singularity but instead a curvature singularity at $r=0$. Of course, we should not yet expect the geometry (80) to describe a black hole, since bulk regularity conditions to deal with the conical nature of the bulk solution have not yet been imposed. Unlike the static system, however, bulk regularity conditions for the rotating solution are more subtle because they affect more than just the periodicity of angular coordinate $\varphi$. Our treatment below follows the analysis in [27].

Firstly, notice that the Killing vector ${\partial }_{\varphi }$ of the bulk C-metric (78) no longer has a vanishing norm at a zero $x_i$ of $G\left(x\right)$. Rather, it is the Killing vector

$${\xi }^b={\partial }_{\varphi }^b-a{x}_i^2{\partial }_t^b,$$

(81)

which obeys ${\xi }^2{|}_{x_i}=0$. Thus, avoiding conical defects at $x=x_i$ requires one to identify points along the integral curves of (81) with an appropriate period. To determine the correct periodicity, consider the rotating C-metric (78) near a zero $x=x_i$ such that $G\left(x\right)\sim G^{\prime }\left(x_i\right)(x-x_i)$. The removal of a conical singularity at, say $x=x_1$ (denoting the smallest positive root of $G\left(x\right)$), has us simultaneously perform a coordinate transformation $\tilde{t}=t+a{x}_i^2\varphi$ together with the periodicity condition for $\varphi$

$$\varphi \sim \varphi +∆\varphi ,∆\varphi ={\displaystyle \frac{4\pi }{|G^{\prime }\left(x_1\right)|}}={\displaystyle \frac{4\pi x_1}{3-\kappa x_1^2-{\tilde{a}}^2}},$$

(82)

where to arrive to the second equality, we recast the parameter $\mu$ in terms of $x_1$ and define

$$\mu ={\displaystyle \frac{1-\kappa x_1^2+{\tilde{a}}^2}{x_1^3}},\tilde{a}\equiv {\displaystyle \frac{a{x}_1^2}{{\ell }_3}}.$$

(83)

In other words, identifying points along the orbits of (81) are made on surfaces of constant

$$\tilde{t}\equiv t+a{x}_1^2\varphi .$$

(84)

As in the static braneworld construction, the remaining zeros $x_i\ne x_1$ are effectively removed by gluing a second copy of the spacetime with the end-of-the-world brane at $x=0$ such that the complete bulk spacetime has the restricted range $0\le x\le x_1$.

Now, return to the naive metric (80) and consider the limit $r\to \infty$. The metric is asymptotic to ‘rotating ${\mathrm{AdS}}_3$’, where the $dtd\varphi$ metric component is a constant. The coordinates $(t,r,\varphi )$, however, are not canonically normalized due to the periodicity in $\varphi$ (82). Further, since points along orbits of (81) are identified, the $\varphi$-periodicity returns one to a different point in time t: indeed, from (84), we see that assuming $\tilde{t}\sim \tilde{t}$, then $t\sim t-2\pi \eta a{x}_1^2$, for $\eta \equiv ∆\varphi /2\pi$. Unfortunately, this means we cannot merely rescale coordinates $(t,r,\varphi )\to (\overline{t},\overline{r},\overline{\varphi })$ as in the static case. Moreover, the periodicity alters the asymptotic form of the metric such that the $dtd\varphi$ would instead grow as $r^2$, implying a diverging angular momentum27.

The $r^2$-divergence can be ameliorated by changing coordinates $(t,\varphi )$ to $(\tilde{t},\tilde{\varphi })$ where,

$$t=\tilde{t}-a{x}_1^2\tilde{\varphi },\varphi =\tilde{\varphi }-{\displaystyle \frac{\tilde{a}}{{\ell }_3}}\tilde{t}.$$

(85)

The coefficient $-{\displaystyle \frac{\tilde{a}}{{\ell }_3}}$ in $\varphi$ is carefully chosen such that the $r^2$ divergence in the $\tilde{t}-\tilde{\varphi }$ component of the naive brane metric is eliminated. Even still, the angular coordinate $\tilde{\varphi }$ is not periodic in $2\pi$. Luckily, this is now easily resolved by the simple rescaling, $\tilde{t}=\eta \overline{t}$ and $\tilde{\varphi }=\eta \overline{\varphi }$, such that the transformation [25]

$$t=\eta (\overline{t}-\tilde{a}{\ell }_3\overline{\varphi }),\varphi =\eta \left(\overline{\varphi }-{\displaystyle \frac{\tilde{a}}{{\ell }_3}}\overline{t}\right),$$

(86)

places the brane geometry (80) in a more canonical form. It proves useful to also have the inverted coordinate transformation (86),

$$\overline{t}={\displaystyle \frac{1}{\eta \left(1-{\tilde{a}}^2\right)}}\left(t+\tilde{a}{\ell }_3\varphi \right),\overline{\varphi }={\displaystyle \frac{1}{\eta \left(1-{\tilde{a}}^2\right)}}\left(\varphi +{\displaystyle \frac{\tilde{a}}{{\ell }_3}}t\right).$$

(87)

From here, the Killing vectors transform as

$${\partial }_t={\displaystyle \frac{1}{\eta \left(1-{\tilde{a}}^2\right)}}\left({\partial }_{\overline{t}}+{\displaystyle \frac{\tilde{a}}{{\ell }_3}}{\partial }_{\overline{\varphi }}\right),{\partial }_{\varphi }={\displaystyle \frac{1}{\eta \left(1-{\tilde{a}}^2\right)}}\left({\partial }_{\overline{\varphi }}+\tilde{a}{\ell }_3{\partial }_{\overline{t}}\right).$$

(88)

Notice in the coordinates $(\overline{t},\overline{\varphi })$, the Killing vector (81) is now ${\xi }^b={\eta }^{-1}{\partial }_{\overline{\varphi }}$.

With the coordinate change (86), the brane metric does not quite have the canonical asymptotic form of a rotating AdS black hole. To do so, one introduces a normalized radial coordinate $\overline{r}$ such that [25].

$$r^2={\displaystyle \frac{{\overline{r}}^2-r_s^2}{(1-{\tilde{a}}^2){\eta }^2}},r_s\equiv {\ell }_3{\displaystyle \frac{\tilde{a}\eta }{x_1}}\sqrt{2-\kappa x_1^2}={\ell }_3{\displaystyle \frac{2\tilde{a}\sqrt{2-\kappa x_1^2}}{3-\kappa x_1^2-{\tilde{a}}^2}},$$

(89)

Altogether, having imposed bulk regularity conditions, the brane geometry (80) in the canonically normalized coordinates $(\overline{t},\overline{r},\overline{\varphi })$ is

$$\begin{array}{cc}\hfill d{s}^2{|}_{x=0}& =-\left(\kappa {\eta }^2\left(1+{\tilde{a}}^2-{\displaystyle \frac{4{\tilde{a}}^2}{\kappa x_1^2}}\right)+{\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-{\displaystyle \frac{\mu \ell {\eta }^2}{r}}\right)d{\overline{t}}^2\hfill \\ & +{\left(\kappa {\eta }^2\left(1+{\tilde{a}}^2-{\displaystyle \frac{4{\tilde{a}}^2}{x_1^2}}\right)+{\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-{\displaystyle \frac{\mu \ell {(1-{\tilde{a}}^2)}^2{\eta }^{4}r}{{\overline{r}}^2}}+{\displaystyle \frac{{\ell }_3^2{\tilde{a}}^2{\mu }^2{x}_1^2{\eta }^4}{{\overline{r}}^2}}\right)}^{-1}d{\overline{r}}^2\hfill \\ & +\left({\overline{r}}^2+{\displaystyle \frac{\mu \ell {\tilde{a}}^2{\ell }_3^2{\eta }^2}{r}}\right)d{\overline{\varphi }}^2-{\ell }_3\tilde{a}\mu x_1{\eta }^2\left(1+{\displaystyle \frac{\ell }{x_{1}r}}\right)(d\overline{\varphi }d\overline{t}+d\overline{t}d\overline{\varphi }),\hfill \end{array}$$

(90)

where we have kept both r and $\overline{r}$ when convenient (treating $r=r\left(\overline{r}\right)$).

Quantum Black Hole

As in the static case, we reexpress the metric (90) as

$$\begin{array}{cc}\hfill d{s}^2=& -\left({\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-8{\mathcal{G}}_{3}M-{\displaystyle \frac{\ell \mu {\eta }^2}{r}}\right)d{\overline{t}}^2+{\left({\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-8{\mathcal{G}}_{3}M+{\displaystyle \frac{{\left(4{\mathcal{G}}_{3}J\right)}^2}{{\overline{r}}^2}}-\ell \mu {(1-{\tilde{a}}^2)}^2{\eta }^4{\displaystyle \frac{r}{{\overline{r}}^2}}\right)}^{-1}d{\overline{r}}^2\hfill \\ & +\left({\overline{r}}^2+{\displaystyle \frac{\mu \ell {\tilde{a}}^2{\ell }_3^2{\eta }^2}{r}}\right)d{\overline{\varphi }}^2-8{\mathcal{G}}_{3}J\left(1+{\displaystyle \frac{\ell }{x_{1}r}}\right)d\overline{\varphi }d\overline{t}\hfill \end{array}$$

(91)

where we have suggestively identified the mass and angular momentum J of the black hole,

$$\begin{array}{cc}\hfill 8{\mathcal{G}}_{3}M& =-\kappa {\eta }^2\left(1+{\tilde{a}}^2-{\displaystyle \frac{4{\tilde{a}}^2}{\kappa x_1^2}}\right)=4{\displaystyle \frac{-\kappa x_1^2+{\tilde{a}}^2(4-\kappa x_1^2)}{{(3-\kappa x_1^2-{\tilde{a}}^2)}^2}}\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill 4{\mathcal{G}}_{3}J& ={\ell }_3{\eta }^2\tilde{a}\mu x_1={\displaystyle \frac{4{\ell }_3\tilde{a}(1-\kappa x_1^2+{\tilde{a}}^2)}{{(3-\kappa x_1^2-{\tilde{a}}^2)}^2}}.\hfill \end{array}$$

(93)

The above identifications are made on geometric grounds: in the asymptotic limit $\overline{r}\to \infty$, the terms proportional to ℓ decay faster than than the constant $-8{\mathcal{G}}_{3}M$ or the J in the $d\overline{t}d\overline{\varphi }$ component. Further, ${\mathcal{G}}_3\equiv L_4{G}_3/\ell$ is again the renormalized Newton’s constant, which now also plays the role of accounting for higher-derivative corrections to the angular momentum.

Horizon and singularity structure. In the static case, roots of the bulk metric function $H\left(r\right)$ correspond to the bulk Killing horizon of the time-translation Killing vector ${\partial }_t$. Including rotation in the bulk, the Killing vector

$${\zeta }^b={\partial }_t+{\displaystyle \frac{a}{r_i^2}}{\partial }_{\varphi },$$

(94)

for $r_i$ finite has modulus ${\zeta }^2=g_{ab}{\zeta }^a{\zeta }^b=-{\displaystyle \frac{{\ell }^2}{{(\ell +xr)}^2}}H\left(r_i\right)\Sigma (x,r_i)$ at $r=r_i$. Taking $r_i$ to be real, ${\zeta }^2=0$ when $H\left(r_i\right)=0$, where $r_i$ are positive real roots of $H\left(r\right)$ (79) (restricting to a coordinate range where there exists at least one real root). Let $r_{+}$ denote the largest positive, real root of $H\left(r\right)$, corresponding to the radius of the outer black hole. As with Kerr–AdS${ }_4$, there is a second positive real root $r_{-}$, satisfying $r_{-}<r_{+}$, which corresponds to the inner black hole horizon. Using $H\left(r_{\pm }\right)=0$ and assuming $r_{+}\ne r_{-}$, it is straightforward to express

$$\begin{array}{cc}& \mu \ell ={\displaystyle \frac{(r_{+}+r_{-})}{{\ell }_3^2}}\left[(r_{+}^2+r_{-}^2)+{\ell }_3^2\kappa \right],\hfill \\ & a^2={\displaystyle \frac{r_{+}{r}_{-}}{{\ell }_3^2}}\left[r_{+}^2+r_{-}^2+r_{+}{r}_{-}+{\ell }_3^2\kappa \right].\hfill \end{array}$$

(95)

The limit of vanishing angular momentum, $a\to 0$, coincides with $r_{-}\to 0$.

In canonically normalized coordinates $(\overline{t},\overline{r},\overline{\varphi })$, the inner and outer horizons of the quantum black hole are generated by orbits of the canonically normalized generator (94), i.e.,

$${\overline{\zeta }}_{\pm }^b\equiv {\displaystyle \frac{\eta (1-{\tilde{a}}^2)}{1+\frac{a^2{x}_1^2}{r_{\pm }}}}{\zeta }^b={\displaystyle \frac{\partial }{\partial \overline{t}}}+{\Omega }_{\pm }{\displaystyle \frac{\partial }{\partial \overline{\varphi }}},$$

(96)

where we used the Killing vector transformation (88). Here, ${\Omega }_{\pm }$ is the angular velocity of the horizons $r_{\pm }$ relative to a non-rotating frame at spatial infinity

$${\Omega }_{\pm }\equiv {\displaystyle \frac{a}{r_{\pm }^2+a^2{x}_1^2}}\left(1+{\displaystyle \frac{r_{\pm }^2{x}_1^2}{{\ell }_3^2}}\right).$$

(97)

For $x_1=1$, this coincides with the familiar angular velocity of Kerr–AdS${ }_4$ (e.g., [94,95]). Meanwhile, the angular velocity ${\Omega }^{\prime }$ relative to a rotating frame at spatial infinity is

$${\Omega }_{\pm }^{\prime }\equiv {\displaystyle \frac{a}{r_{\pm }^2+a^2{x}_1^2}}\left(1-{\displaystyle \frac{a^2{x}_1^4}{{\ell }_3^2}}\right),$$

(98)

obeying ${\Omega }_{\pm }-{\Omega }_{\pm }^{\prime }=a{x}_1^2/{\ell }_3^2$. Further, relative to ${\overline{\zeta }}^b$, the surface gravity of the inner and outer horizons is

$${\kappa }_{\pm }={\displaystyle \frac{\eta (1-{\tilde{a}}^2)}{(r_{\pm }^2+a^2{x}_1^2)}}{\displaystyle \frac{r_{\pm }^2}{2}}|H^{\prime }\left(r_{\pm }\right)|={\displaystyle \frac{\eta (1-{\tilde{a}}^2)}{(r_{\pm }^2+a^2{x}_1^2)}}{\displaystyle \frac{1}{2{\ell }_3^2{r}_{\pm }}}|{\ell }_3^2\mu \ell r_{\pm }+2r_{\pm }^4-2a^2{\ell }_3^2|,$$

(99)

where we used the definition of the surface gravity ${\kappa }_{\pm }$ on a Killing horizon generated by $\overline{\zeta }$, ${\overline{\zeta }}^b{\nabla }_b{\overline{\zeta }}^c\equiv {\kappa }_{\pm }{\overline{\zeta }}^c$. In particular, using parameters (95)

$$\begin{array}{cc}& {\kappa }_{+}={\displaystyle \frac{\eta (1-{\tilde{a}}^2)}{2{\ell }_3^2(r_{+}^2+a^2{x}_1^2)}}(r_{+}-r_{-})|(r_{+}^2+r_{-}^2+{\ell }_3^2\kappa )+2r_{+}(r_{+}+r_{-})|,\hfill \\ & {\kappa }_{-}={\displaystyle \frac{\eta (1-{\tilde{a}}^2)}{2{\ell }_3^2(r_{-}^2+a^2{x}_1^2)}}(r_{-}-r_{+})|(r_{+}^2+r_{-}^2+{\ell }_3^2\kappa )+2r_{-}(r_{+}+r_{-})|.\hfill \end{array}$$

(100)

In the limit of vanishing rotation, ${\kappa }_{+}$ becomes the surface gravity of the static qBTZ, (74). Further, the surface gravities vanish in the extremal limit, i.e., where the inner and outer black hole horizons coincide, $r_{+}=r_{-}$.

The horizons shroud a ring curvature singularity at $r=0$, i.e., $\overline{r}=r_s$. Further, near the ring singularity, there exists the possibility of closed timelike curves. Indeed, consider the norm of the axial Killing vector ${\partial }_{\overline{\varphi }}$ of (91) to be

$${\partial }_{\overline{\varphi }}^2=h_{\overline{\varphi }\overline{\varphi }}={\overline{r}}^2+{\displaystyle \frac{\mu \ell {\tilde{a}}^2{\ell }_3^2{\eta }^2}{r}}={\overline{r}}^2+{\displaystyle \frac{\mu \ell {\tilde{a}}^2{\ell }_3^2{\eta }^3}{\sqrt{{\overline{r}}^2-r_s^2}}}\sqrt{1-{\tilde{a}}^2},$$

(101)

meaning that if one has a too small and negative enough radii r, the axial vector becomes timelike, and its orbits are closed curves around the rotation axis. To ensure ${\partial }_{\varphi }^2\ge 0$ requires an additional restriction of the solution parameters: (i) $1-\widetilde{a^2}\ge 0$, and (ii) $\eta >0$, which implies $-\kappa x_1^2>{\tilde{a}}^2-3$. Notice the second condition is always implied by (i) if $\kappa =-1$.

Quantum stress tensor. As an exact solution to the full semi-classical theory (57), the metric (91) is known as the rotating quantum BTZ black hole. In the limit of vanishing backreaction, $\ell \to 0$, the classical rotating BTZ black hole is recovered for $\kappa =-1$, while for $\kappa =+1$, the geometry is that of a rotating AdS${ }_3$ conical defect. For $\ell \ne 0$, the non-vanishing components of the renormalized quantum-stress tensor (69) are found to be [25]

$$\begin{array}{cc}\hfill {\langle T_{\overline{t}}^{\overline{t}}\rangle }_0& ={\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{\ell \mu }{(1-{\tilde{a}}^2)r^3}}\left(1+2{\tilde{a}}^2+{\displaystyle \frac{3{\tilde{a}}^2{\ell }_3^2}{x_1^2{r}^2}}\right),\hfill \\ \hfill {\langle T_{\overline{r}}^{\overline{r}}\rangle }_0& ={\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{\ell \mu }{r^3}},\hfill \\ \hfill {\langle T_{\overline{\varphi }}^{\overline{\varphi }}\rangle }_0& =-{\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{\ell \mu }{(1-{\tilde{a}}^2)r^3}}\left(2+{\tilde{a}}^2+{\displaystyle \frac{3{\tilde{a}}^2{\ell }_3^2}{x_1^2{r}^2}}\right),\hfill \\ \hfill {\langle T_{\overline{\varphi }}^{\overline{t}}\rangle }_0& =-{\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{3{\ell }_3\ell \mu \tilde{a}}{(1-{\tilde{a}}^2)r^3}}\left(1+{\displaystyle \frac{{\tilde{a}}^2{\ell }_3^2}{x_1^2{r}^2}}\right),\hfill \\ \hfill {\langle T_{\overline{t}}^{\overline{\varphi }}\rangle }_0& ={\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{3\ell \mu \tilde{a}}{{\ell }_3(1-{\tilde{a}}^2)r^3}}\left(1+{\displaystyle \frac{{\ell }_3^2}{x_1^2{r}^2}}\right).\hfill \end{array}$$

(102)

These are most efficiently computed by first computing the components of the stress tensor in the naive coordinates (80) and then performing the appropriate coordinate transformations imposing bulk regularity. The $\mathcal{O}\left({\ell }^2\right)$ components ${\langle T_{ij}\rangle }_2$ (70) are cumbersome to express in canonically normalized coordinates; however, in naive coordinates, ${\langle T_{ij}\rangle }_2$ is not traceless, indicating the presence of a UV cutoff breaking the conformal symmetry of the CFT${ }_3$.

Notice the mass (92) and angular momentum (93) only depend on $\tilde{a}$ and $\kappa x_1^2$ and not on ℓ. Meanwhile, ℓ only enters as an overall prefactor in the stress tensor components (102). Hence, the leading-order stress tensor only depends on backreaction effects via ${\mathcal{G}}_3$ and not ${\mathcal{G}}_{3}M$ or ${\mathcal{G}}_{3}J$. Further, unlike the static solution (71), the stress tensor for the rotating qBTZ black hole is not characterized by a single function $F\left(M\right)$. Consider, however, the large-$\overline{r}$ asymptotics of the components (102). For example, substituting normalized coordinate (89) into ${\langle T_{\overline{t}}^{\overline{t}}\rangle }_0$, yields

$$\underset{\overline{r}\to \infty }{lim}{\langle T_{\overline{t}}^{\overline{t}}\rangle }_0\to {\displaystyle \frac{{\eta }^3\mu \ell \sqrt{1-{\tilde{a}}^2}}{16\pi G_3{\overline{r}}^3}}(1+2{\tilde{a}}^2).$$

(103)

Peeling off $\ell /16\pi G_3{\overline{r}}^3$, identify the form function

$$F(M,J)\equiv {\eta }^3\mu \sqrt{1-{\tilde{a}}^2}(1+2{\tilde{a}}^2)={\displaystyle \frac{8\sqrt{1-{\tilde{a}}^2}(1+2{\tilde{a}}^2)(1-\kappa x_1^2+{\tilde{a}}^2)}{{(3-\kappa x_1^2-{\tilde{a}}^2)}^3}},$$

(104)

such that for large $\overline{r}$

$${\langle T_{\overline{t}}^{\overline{t}}\rangle }_0\to {\displaystyle \frac{1}{16\pi G_3}}{\displaystyle \frac{\ell F(M,J)}{{\overline{r}}^3}}.$$

(105)

Due to the $\sim \mu /r^3$ behavior in each of the remaining components, up to unimportant factors, the large-$\overline{r}$ structure of (102) essentially depends on $F(M,J)$ as defined in (104). We will use $F(M,J)$ to better understand the family of rotating quantum black holes momentarily.

It is worth comparing the structure of the holographic stress tensor (102) to the stress tensor of a conformally coupled scalar field found by solving the semi-classical Einstein equations perturbatively (in particular, see [25,45]). The non-holographic stress tensor may be expressed as an infinite sum over images, e.g.,

$$8\pi G_3\langle T_{\overline{t}}^{\overline{t}}\rangle =\sum _{n=1}^{\infty }{\displaystyle \frac{1}{r_n^3}}\left(A_n+{\displaystyle \frac{{\overline{A}}_n}{r_n^2}}\right),$$

(106)

with $r_n=\sqrt{D_n{\overline{r}}^2+{\overline{D}}_n}$, where coefficients $A_n,{\overline{A}}_n$, $D_n$ and ${\overline{D}}_n$ are all complicated functions of M and J. An analogous structure holds for the remaining components. Coefficients aside, each term of the infinite sum has a similar radial dependence as the holographic stress tensor. The radial dependence of the entire infinite sum (106), however, is far more complicated than the radial dependence of the holographic stress tensor (102). Notably, while the holographic stress tensor is manifestly non-singular away from the ring singularity $\overline{r}=r_s$, it is not clear whether the same is true for the perturbative stress tensor.

Black hole branches revisited. In the non-rotating case, there are three branches of quantum black holes, branches 1a, 1b, and 2 (77). There is an analogous set of branches for non-vanishing J, where, in particular, branches 1b and 2 meet at a maximum value of M for fixed J. This occurs when

$$x_1^2+{\tilde{a}}^2=3,M={\displaystyle \frac{1}{8{\mathcal{G}}_3}}\left({\displaystyle \frac{12}{x_1^4}}-1\right),J={\displaystyle \frac{{\ell }_3}{{\mathcal{G}}_3}}{\displaystyle \frac{\sqrt{3-x_1^2}}{x_1^4}}.$$

(107)

At $x_1=\sqrt{2}$, one attains an extremal bound, where $M=J/{\ell }_3=1/4{\mathcal{G}}_3$. There is another extremal bound among the branch 2 black holes, founding by minimizing the mass M for fixed J,

$$\tilde{a}=1,M={\displaystyle \frac{J}{{\ell }_3}}={\displaystyle \frac{1}{{\mathcal{G}}_3(2+x_1^2)}},$$

(108)

which coincides with the bound (107) at $x_1=\sqrt{2}$. Classically, the rotating BTZ black hole obeys the extremality bound $M\ge J/{\ell }_3$. For the quantum black hole, however, for any value of J, this classical extremality bound is violated, $M\le J/{\ell }_3$, when $-\kappa x_1^2<2{\tilde{a}}^2$, giving rise to ‘super-extremal’ black holes among the branch 1 solutions. Pictorially, the branches of quantum black holes have a similar representation as Figure 6 (see Figure 6 of [25]).

4.2.3. Charged Quantum BTZ

Bulk and Brane Geometry

It is relatively straightforward to generalize the neutral quantum BTZ solutions to a charged system. Now, the bulk is characterized by the charged AdS${ }_4$ C-metric. For simplicity, setting rotation to zero, the line element is of the same form as (39), except the metric functions (40) receive an additional term:

$$d{s}^2={\displaystyle \frac{{\ell }^2}{{(\ell +xr)}^2}}\left[-H\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{H\left(r\right)}}+r^2\left({\displaystyle \frac{d{x}^2}{G\left(x\right)}}+G\left(x\right)d{\varphi }^2\right)\right],$$

(109)

$$H\left(r\right)={\displaystyle \frac{r^2}{{\ell }_3^2}}+\kappa -{\displaystyle \frac{\mu \ell }{r}}+{\displaystyle \frac{q^2{\ell }^2}{r^2}},G\left(x\right)=1-\kappa x^2-\mu x^3-q^2{x}^4.$$

(110)

This is a solution to four-dimensional Einstein–Maxwell gravity with negative cosmological constant $-2{\Lambda }_4=6/L_4^2$,

$$I={\displaystyle \frac{1}{16\pi G_4}}\int d^{4}x\sqrt{-\widehat{g}}\left[\widehat{R}+{\displaystyle \frac{6}{L_4^2}}-{\displaystyle \frac{{\ell }_{*}^2}{4}}{F}^2\right],{\ell }_{*}^2={\displaystyle \frac{16\pi G_4}{g_{*}^2}}$$

(111)

where ${\ell }_{*}$ is a coupling constant with dimensions of length, $g_{*}$ is the dimensionless gauge coupling constant, and $F_{ab}={\partial }_a{A}_b-{\partial }_b{A}_a$ is the Maxwell field tensor. The $U\left(1\right)$ gauge field $A_a$ for the charged C-metric (109)

$$A=A_{a}d{x}^a={\displaystyle \frac{2\ell }{{\ell }_{*}}}\left[e\left({\displaystyle \frac{1}{r_{+}}}-{\displaystyle \frac{1}{r}}\right)dt+g(x-x_1)d\varphi \right],$$

(112)

where e and g, respectively, denote the electric and magnetic charge parameters of the accelerating black hole such that $q^2=e^2+g^2$. A gauge has been chosen for the gauge potential (112) such that it remains regular at the largest root of $H\left(r\right)=0$, denoted $r_{+}$, and $x=x_1$.

As for the neutral black hole, real roots of $G\left(x\right)$ correspond to symmetry axes of the Killing vector ${\partial }_{\varphi }^a$, and we work in the restricted regime $0\le x\le x_1$. The conical singularity at $x=x_1$ is removed via the identification

$$\varphi \sim \varphi +∆\varphi ,∆\varphi ={\displaystyle \frac{4\pi }{|G^{\prime }\left(x_1\right)|}}={\displaystyle \frac{4\pi }{|-3+\kappa x_1^2-q^2{x}_1^4|}},$$

(113)

where we treat $\mu$ as a ‘derived’ parameter following $G\left(x_1\right)=0$,

$$\mu ={\displaystyle \frac{1-\kappa x_1^2-q^2{x}_1^4}{x_1^3}},$$

(114)

while $x_1$ and q are primary parameters.

Lastly, as usual, the $x=0$ hypersurface is totally umbilic, such that the Israel junction conditions fix the brane tension to be $\tau ={(2\pi G_4\ell )}^{-1}$. The Maxwell field strength also has junction conditions to obey. In particular, let $n^a$ denote the normal to the brane at $x=0$ (pointing toward increasing values of x) and let $e_i^a$ be a basis for tangent vectors to the brane. Then, projecting $F_{ab}$ onto the brane such that $F_{ij}\equiv F_{ab}{e}_i^a{e}_j^b$ and $f_i\equiv F_{ab}{e}_i^a{n}^b$, one has the following junction conditions [96] for a purely tensional brane

$$\begin{array}{cc}& ∆F_{ij}=F_{ij}^{+}-F_{ij}^{-}=0,\hfill \\ & ∆f_i=f_i^{+}-f_i^{-}=4\pi j_i,\hfill \end{array}$$

(115)

where $j_i$ is the electromagnetic surface current.

Induced Brane Theory

As in the neutral examples, the induced theory on the brane essentially follows from replacing the IR bulk cutoff in holographic renormalization with a brane. When the bulk theory of gravity is Einstein–Maxwell, the induced brane action (53) receives corrections due to the bulk Maxwell contribution, such that the induced brane action is now

$$I={\displaystyle \frac{1}{16\pi G_3}}{\int }_{\mathcal{B}}{d}^{3}x\sqrt{-h}\left[R-2{\Lambda }_3+L_4^2\left({\displaystyle \frac{3}{8}}{R}^2-R_{ij}^2\right)+...\right]+I_{\mathrm{EM}}+I_{\mathrm{CFT}},$$

(116)

where the electromagnetic term $I_{\mathrm{EM}}$ is

$$I_{\mathrm{EM}}=2\int d^{3}x\sqrt{-h}{A}_i{j}^i+I_{\mathrm{EM}}^{\mathrm{ct}}.$$

(117)

The first contribution is a boundary term with respect to the bulk Maxwell term necessary to keep the Dirichlet variational problem well posed. The second contribution describes local counterterms associated with the four-dimensional bulk Maxwell action that are included in holographic renormalization28 [97]

$$\begin{array}{cc}\hfill I_{\mathrm{EM}}^{\mathrm{ct}}={\displaystyle \frac{L_4{\ell }_{*}^2}{8\pi G_4}}& \int d^{3}x\sqrt{-h}[-{\displaystyle \frac{5}{16}}{F}^2+L_4^2({\displaystyle \frac{1}{288}}R{F}^2-{\displaystyle \frac{5}{8}}{R}_j^i{F}_{ik}{F}^{jk}\hfill \\ & +{\displaystyle \frac{3}{98}}{F}^{ij}({\nabla }_j{\nabla }^k{F}_{ki}-{\nabla }_i{\nabla }^k{F}_{kj})+{\displaystyle \frac{5}{24}}{\nabla }_i{F}^{ij}{\nabla }_k{F}_j^k)+\mathcal{O}\left(L_4^3\right)].\hfill \end{array}$$

(118)

Then, treating $\ell <{\ell }_3$ such that $L_4\sim \ell$, the effective theory on the brane (57) is now

$$\begin{array}{cc}\hfill I& ={\displaystyle \frac{1}{16\pi G_3}}\int d^{3}x\sqrt{-h}[R+{\displaystyle \frac{2}{{\ell }_3^2}}-{\displaystyle \frac{{\tilde{\ell }}_{*}^2}{4}}{F}^2+16\pi G_3{A}_i{j}^i\hfill \\ & +{\ell }^2\left({\displaystyle \frac{3}{8}}{R}^2-R_{ij}^2\right)+{\displaystyle \frac{4}{5}}{\ell }^2{\tilde{\ell }}_{*}^2({\displaystyle \frac{1}{288}}R{F}^2-{\displaystyle \frac{5}{8}}{R}_j^i{F}_{ik}{F}^{jk}\hfill \\ & +{\displaystyle \frac{3}{98}}{F}^{ij}({\nabla }_j{\nabla }^k{F}_{ki}-{\nabla }_i{\nabla }^k{F}_{kj})+{\displaystyle \frac{5}{24}}{\nabla }_i{F}^{ij}{\nabla }_k{F}_j^k)+\mathcal{O}\left({\ell }^3\right)]+I_{\mathrm{CFT}}.\hfill \end{array}$$

(119)

where in addition to the induced scales (54) $G_3=G_4/2L_4$, and $L_3$ (55), there is an effective three-dimensional gauge coupling

$${\tilde{\ell }}_{*}^2={\displaystyle \frac{16\pi G_3}{g_3^2}},g_3^2={\displaystyle \frac{2}{5}}{\displaystyle \frac{g_{*}^2}{L_4}}\approx {\displaystyle \frac{2}{5}}{\displaystyle \frac{g_{*}^2}{\ell }},$$

(120)

such that ${\ell }_{*}^2=4{\tilde{\ell }}_{*}^2/5$. It is clear that in the limit $\ell \to 0$, the coupling $g_3\to \infty$ becomes non-dynamical. Indeed, we will see how this ‘charge’ contribution to the metric disappears.

The metric equations of motion of the induced theory to order $\mathcal{O}\left({\ell }^2\right)$ are as in the neutral case, except now with an additional contribution coming from the $F^2$ contribution in the action. That is,

$$\begin{array}{cc}\hfill 8\pi G_3\langle T_{ij}\rangle & =G_{ij}-{\displaystyle \frac{1}{L_3^2}}{h}_{ij}-{\displaystyle \frac{{\tilde{\ell }}_{*}^2}{2}}\left(F_i^k{F}_{jk}-{\displaystyle \frac{1}{4}}{h}_{ij}{F}^2\right)+16\pi G_3{A}_k{j}^k{h}_{ij}+...,\hfill \end{array}$$

(121)

where the ellipsis constitutes terms at higher order in ℓ; the $\mathcal{O}\left({\ell }^2\right)$ contribution is precisely the same as in the neutral case (70), while $\mathcal{O}\left({\ell }^3\right)$ contributions arise from the ${\ell }^2{\tilde{\ell }}_{*}^2$ term in the action (119). Further, varying with respect to the gauge field $A_i$, we find the analog of the semi-classical Maxwell equations,

$$\begin{array}{cc}\hfill \langle J^j\rangle & =j^j+{\displaystyle \frac{{\tilde{\ell }}_{*}^2}{16\pi G_3}}\{{\nabla }_i{F}^{ji}+{\displaystyle \frac{16}{5}}{\ell }^2(-{\displaystyle \frac{1}{72}}R{\nabla }_i{F}^{ji}+{\displaystyle \frac{11}{18}}{F}_i^j{\nabla }^{i}R+{\displaystyle \frac{209}{294}}{R}^{ij}{\nabla }_k{F}_i^k\hfill \\ & +{\displaystyle \frac{5}{4}}{R}^{ik}{\nabla }_k{F}_i^j+{\displaystyle \frac{5}{4}}{F}^{ik}{\nabla }_k{R}_i^j+{\displaystyle \frac{317}{588}}{\nabla }_i{\nabla }^i{\nabla }_k{F}^{jk}+{\displaystyle \frac{317}{588}}{\nabla }^j{\nabla }^k{\nabla }_i{F}^{ik})+\mathcal{O}\left({\ell }^3\right)\}.\hfill \end{array}$$

(122)

Quantum Black Hole

Upon imposing bulk regularity conditions (equivalent to those for the neutral, static geometry), the induced geometry on the brane at $x=0$, in terms of canonically normalized coordinates $(t,r,\varphi )=(\eta \overline{t},{\eta }^{-1}\overline{r},\eta \overline{\varphi })$, is [98] (see also [99])

$$\begin{array}{cc}\hfill d{s}_{\mathrm{cqBTZ}}^2=& -H\left(\overline{r}\right)d{\overline{t}}^2+H^{-1}\left(\overline{r}\right)d{\overline{r}}^2+{\overline{r}}^{2}d{\overline{\varphi }}^2,\hfill \\ & H\left(\overline{r}\right)=-8M{\mathcal{G}}_3+{\displaystyle \frac{{\overline{r}}^2}{{\ell }_3^2}}-{\displaystyle \frac{\ell F(M,q)}{\overline{r}}}+{\displaystyle \frac{{\ell }^{2}Z(M,q)}{{\overline{r}}^2}}.\hfill \end{array}$$

(123)

Here, the mass is identified to be

$$M\equiv -{\displaystyle \frac{\kappa }{8{\mathcal{G}}_3}}{\eta }^2=-{\displaystyle \frac{\kappa }{8{\mathcal{G}}_3}}{\displaystyle \frac{4x_1^2}{{(3-\kappa x_1^2+q^2{x}_1^4)}^2}},$$

(124)

and form functions

$$F(M,q)\equiv \mu {\eta }^3=8{\displaystyle \frac{1-\kappa x_1^2-q^2{x}_1^4}{{(3-\kappa x_1^2+q^2{x}_1^4)}^3}},$$

(125)

$$Z(M,q)\equiv q^2{\eta }^4={\displaystyle \frac{16q^2{x}_1^4}{{(3-\kappa x_1^2-q^2{x}_1^4)}^4}}.$$

(126)

As an exact solution to the semi-classical theory (119), the geometry (123) is recognized as the ‘charged’ version of the quantum BTZ black hole. Notice, however, in the limit $\ell \to 0$ the q-dependent correction vanishes, indicating the charge of the braneworld black is a consequence of backreaction. Notably, the classical geometry ($\ell =0$) is not that of the charged BTZ metric [100]; we will return to this point momentarily.

Substituting the metric (123) into the stress tensor at the leading order in ℓ is equivalent to the neutral static qBTZ stress tensor (71), while the effects of charge arise at $\mathcal{O}\left({\ell }^2\right)$. Further, in coordinates $(\overline{t},\overline{r},\overline{\varphi })$, the projected components of the electromagnetic tensor are

$$F_{\overline{r}\overline{t}}={\displaystyle \frac{2e\ell }{{\ell }_{*}{\overline{r}}^2}}{\eta }^2,f_{\overline{\varphi }}=-{\displaystyle \frac{2g\ell }{{\ell }_{*}\overline{r}}}{\eta }^2.$$

(127)

Using the junction conditions (115), the induced current density is

$$j^{\overline{\varphi }}={\displaystyle \frac{g\ell {\eta }^2}{\pi {\ell }_{*}{\overline{r}}^3}}.$$

(128)

With these, the leading-order $\mathcal{O}(\ell )$ contribution to the semi-classical current density (122) has components

$$\begin{array}{cc}& \langle J^{\overline{t}}\rangle =-{\displaystyle \frac{\ell {\tilde{\ell }}_{*}^2}{8\pi G_3{\ell }_{*}}}{\displaystyle \frac{e{\eta }^2}{{\overline{r}}^3}}\propto {\displaystyle \frac{e\ell \sqrt{c}}{g_{*}{\overline{r}}^3}},\hfill \\ & \langle J^{\overline{\varphi }}\rangle ={\displaystyle \frac{\ell }{{\ell }_{*}}}{\displaystyle \frac{g{\eta }^2}{\pi {\overline{r}}^3}}\propto {\displaystyle \frac{g{g}_{*}\sqrt{c}}{{\overline{r}}^3}}.\hfill \end{array}$$

(129)

Interestingly, the temporal component vanishes in the limit $\ell \to 0$, while the azimuthal component is independent of backreaction. This is consistent with the three-dimensional dyonic defect in conical AdS${ }_3$ or a BTZ black hole [98].

Unlike the rotating case, the limit of vanishing backreaction does not return a classically AdS${ }_3$ geometry charged under three-dimensional Maxwell theory. Indeed, in three dimensions, the (electric) gauge field $A_t$ has a logarithmic dependence, $A_t\sim qlog\left(r\right)$, producing a logarithmic correction to the three-dimensional blackening factor of a classical charged BTZ [100]. This lack of logarithmic behavior arises from the fact the bulk four-dimensional gauge field does not localize on the brane in the same way as a gravity [98]. Still, the quantum black hole (123) is charged. As with mass or angular momentum, computing charge Q from the brane perspective would require a resummation of the infinite tower of higher-derivative terms appearing in the induced theory. Alternatively, the bulk theory performs this resummation, and the charge of the brane black hole is identified with the electric charge of the bulk black hole [98] (see also [101])

$$Q={\displaystyle \frac{2}{g_{*}^2}}\int ★F,$$

(130)

where the factor of two in the first equality is due to the ${\mathbb{Z}}_2$ symmetry, $★F$ refers to the Hodge dual of the bulk Maxwell tensor, and the integration is taken to be at the boundary. In particular, the electric charge is

$$Q_e={\displaystyle \frac{2e\ell }{g_{*}^2{\ell }_{*}}}{\int }_0^{2\pi \eta }d\varphi {\int }_0^{x_1}dx={\displaystyle \frac{8\pi \ell e\eta x_1}{g_{*}^2{\ell }_{*}}},$$

(131)

where $★F=r^2{F}_{rt}d\varphi dx$. Similarly, the magnetic charge $Q_g$ is

$$Q_g={\displaystyle \frac{2}{g_{*}^2}}\int F={\displaystyle \frac{8\pi \ell g\eta x_1}{g_{*}^2{\ell }_{*}}},$$

(132)

and $Q^2=Q_e^2+Q_g^2$. Notice $Q^2$ does not explicitly appear in the blackening factor (123).

Associated with the electric and magnetic charges are their respective potentials ${\mu }_e$ and ${\mu }_g$. Since, holographically, the bulk Einstein–Maxwell theory (111) has a dual interpretation in terms of a CFT${ }_3$ with a chemical potential, it is natural to refer to ${\mu }_e$ and ${\mu }_g$ as chemical potentials. In particular, the electric chemical potential equals the electric component of the bulk gauge field (112) at the boundary intersecting the brane at $x=0$ (where $r_{\mathrm{bdry}}\to -\infty$),

$$\underset{r_{\mathrm{bdry}}\to -\infty }{lim}{A}_{\overline{t}}\equiv {\mu }_e={\displaystyle \frac{2\ell e\eta }{r_{+}{\ell }_{*}}}.$$

(133)

Similarly, under electromagnetic duality of the bulk solution, the magnetic chemical potential ${\mu }_g$ has the same form as (133), except with $e\leftrightarrow g$. Notice ${\mu }_g{Q}_e-{\mu }_e{Q}_g=0$.

A Family of Charged Quantum Black Holes

Having included q, the parameter $x_1$ depends on q and belongs to a different range than the static (46) or rotating case. To determine this range, two conditions are imposed [98]: (i) the bulk has a horizon $r_0>0$, and (ii) $x_1$ is finite, having a maximum $x_1^{\max }$, such that constant $(t,r)$ surfaces are compact. The finite maximum value of $x_1$ is linked to the fact that the bulk (and brane) geometry has an extremal limit. Demanding there are no naked singularities requires $\mu$ to be bounded below by ${\mu }_{\mathrm{ext}}^{\left(\kappa \right)}$, i.e., the value of $\mu$ when the bulk black hole becomes extremal. The parameter $x_1=x_1^{\max }$ when $\mu ={\mu }_{\mathrm{ext}}^{\left(\kappa \right)}$.

Explicitly, for arbitrary $\kappa$, the extremal mass is [98]

$${\mu }_{\mathrm{ext}}^{\left(\kappa \right)}=\sqrt{{\displaystyle \frac{2}{3}}}{\displaystyle \frac{\sqrt{\sqrt{{\kappa }^2+12{\nu }^2{q}^2}-\kappa }\left(2\kappa +\sqrt{{\kappa }^2+12{\nu }^2{q}^2}\right)}{3\nu }}.$$

(134)

This can be derived as follows. First, the condition of extremality requires two positive real roots of $H\left(r\right)$ to be coincident; i.e., $r_{+}=r_{-}=r_0$ implies $H\left(r\right)={(r-r_0)}^{2}f\left(r\right)$ for some differentiable function $f\left(r\right)$ such that $H\left(r_0\right)=H^{\prime }\left(r_0\right)=0$. A little algebra yields

$${\mu }_{\mathrm{ext}}^{\left(\kappa \right)}\left(r_0\right)={\displaystyle \frac{2r_0}{\ell }}\left(\kappa +{\displaystyle \frac{2r_0^2}{{\ell }_3^2}}\right),q_{\mathrm{ext}}^{\left(\kappa \right)}\left(r_0\right)={\displaystyle \frac{r_0}{\ell }}\sqrt{\kappa +{\displaystyle \frac{3r_0^2}{{\ell }_3^2}}}.$$

(135)

Inverting $q_{\mathrm{ext}}^{\left(\kappa \right)}\left(r_0\right)$ for $r_0$ gives

$$r_0^{\left(\kappa \right)}={\displaystyle \frac{{\ell }_3\sqrt{\sqrt{{\kappa }^2+12{\nu }^2{q}^2}-\kappa }}{\sqrt{6}}}.$$

(136)

Substituting this into ${\mu }_{\mathrm{ext}}^{\left(\kappa \right)}$ (135) yields (134). Evidently, extremality occurs when a backreaction is non-vanishing ($\nu \ne 0$) or charge is non-zero for $\kappa =+1$. Alternatively, for $\kappa =-1$, there exists an extremal radius, $r_0^{(-1)}={\ell }_3/\sqrt{3}$, even when $q=0$ and where ${\mu }_{\mathrm{ext}}^{(-1)}<0$. In fact, generally, ${\mu }_{\mathrm{ext}}^{(-1)}$ can become negative (whenever $q<{\displaystyle \frac{1}{\sqrt{12}\nu }}$), as does $F(M,q)$, for $\kappa =-1$; meanwhile, $\kappa =+1,0$ gives ${\mu }_{\mathrm{ext}}^{(-1)}\ge 0$.

Furthermore, $x_1^{\max }$ is determined by substituting the extremal parameter (134) into (114) and solving for the resulting $x_1$ in terms of q and $\nu$. An analytic expression for $x_1^{\max }$ is possible but cumbersome and not particularly illuminating. Some intuition can be gained, however, by looking at the limit when both q and $\nu$ are small. Three cases are worth highlighting:

• $\kappa =1$: for $q=0$, $x_1$ is bounded above by 1, as $\mu \to 0$. For $q\ll 1$, this upper bound is lowered to

  $$x_1^{\max ,1}=1-q+2q^2.$$

  (137)
• $\kappa =-1$: for $q=0$, $x_1$ covers the whole real line, approaching $x_1=0$ from above as $\mu \to \infty$ and vice versa. Turning on $q\ll 1$, the maximum $x_1^{\max }$ is

  $$x_1^{\max ,0}={\displaystyle \frac{2}{3\sqrt{3}\nu q^2}}+{\displaystyle \frac{\sqrt{3}\nu }{2}}-{\displaystyle \frac{21\sqrt{3}{\nu }^3{q}^2}{8}}+....$$

  (138)
• $\kappa =0$: in the neutral case, the parameter range is equivalent to the $\kappa =-1$ case; however, for $q\ll 1$, then

  $$x_1^{\max ,0}={\displaystyle \frac{h\left(\nu \right)}{\sqrt{q}}},$$

  (139)

  with $h\left(\nu \right)=1-{\displaystyle \frac{\sqrt{\nu }}{3^{3/4}}}+{\displaystyle \frac{\nu }{2\sqrt{3}}}+\mathcal{O}\left({\nu }^{3/2}\right)+...$

A visual representation of the branches for the charged qBTZ black hole can be found in Figure 7. Notably, the branches have qualitatively similar features as the neutral qBTZ. This is because the finite mass range of the charged black holes is a subset of the mass range of the neutral qBTZ (75). The branch with $M<0$ corresponds to the classically horizonless charged defects, now with a horizon induced by quantum backreaction. Note, however, unlike the neutral or rotating quantum BTZ solutions, here, $F(M,q)$ can go negative for any $\nu$ when charge parameter q is large enough.

The limit of vanishing backreaction, $\nu \to 0$, is more subtle than the neutral quantum BTZ solutions. As is usual, in this limit, the brane is sent closer to the asymptotic AdS${ }_4$ boundary, where both gravity and the gauge theory become frozen. The geometry becomes that of a charged defect in conical AdS${ }_3$ or Mink${ }_3$ ($\kappa =+1$) or a black hole ($\kappa =-1$). Since the latter is a black hole geometry, the backreaction ($\nu \ne 0$) produces a quantum-corrected charged black hole; i.e., the horizon is not induced solely due to a backreaction. Alternatively, for the conical charged defects, horizons can only arise via quantum effects. Unlike the neutral set-up, however, whether a horizon appears depends on a balance between $\mu$ and q—a backreaction does not always dress the conical defects with a horizon. Specifically, when the non-backreacted solution has $q\ge \mu /2$, then the backreaction will result in a spacetime possessing a naked (timelike) singularity [98]. Meanwhile, for $q<\mu /2$, a backreaction will produce a quantum black hole provided the backreaction is not too large. As $\nu$ increases for a fixed q, an extremal black hole forms, and an even stronger backreaction results in naked singularities. For an illustration, see Figure 8.

4.3. Quantum dS Black Holes

In Section 2, we saw, perturbatively, that a semi-classical backreaction modifies conical dS${ }_3$ geometries to induce a black hole horizon (in addition to its classical cosmological horizon). This observation is bolstered using braneworld holography. The construction and analysis are similar to the quantum BTZ black hole; however, there are some essential differences.

4.3.1. Bulk Set-Up

The starting point in the bulk is again the AdS${ }_4$ C-metric (39) with metric functions (40) except with fixed $\kappa =+1$ and ${\ell }_3^2=-R_3^2$ (or Wick rotate ${\ell }_3\to i{R}_3$). The AdS${ }_4$ length scale is then

$${\displaystyle \frac{1}{L_4^2}}={\displaystyle \frac{1}{R_3^2}}-{\displaystyle \frac{1}{{\ell }^2}}.$$

(140)

Unlike the qBTZ set-up, keeping $L_4^2>0$ requires $R_3^2>{\ell }^2$, placing an upper bound on ℓ for fixed $R_3$. Further, rearranging (140) yields $\ell <L_4$—the opposite of the quantum BTZ construction. This condition is the first crucial difference between the quantum BTZ and dS set-ups, as maintaining $\ell <L_4$ implies the acceleration horizon is not effectively eliminated. To see this explicitly, consider the analog of the coordinate transformation (42) [26]

$$sinh\left(\sigma \right)={\displaystyle \frac{R_3}{L_4}}{\displaystyle \frac{1}{|1+\frac{rx}{\ell }|}}\sqrt{1-{\displaystyle \frac{x^2{r}^2}{R_3^2}}},\widehat{r}=r\sqrt{{\displaystyle \frac{1-x^2}{1-\frac{x^2{r}^2}{R_3^2}}}},$$

(141)

such that the line element of the C-metric (39) (with $\mu =0$) becomes

$$d{s}^2=L_4^{2}d{\sigma }^2+{\displaystyle \frac{L_4^2}{R_3^2}}{sinh}^2\left(\sigma \right)\left[-\left(1-{\displaystyle \frac{{\widehat{r}}^2}{R_3^2}}\right)d{t}^2+{\left(1-{\displaystyle \frac{{\widehat{r}}^2}{R_3^2}}\right)}^{-1}d{\widehat{r}}^2+{\widehat{r}}^{2}d{\varphi }^2\right].$$

(142)

Clearly, constant-$\sigma$ slices give dS${ }_3$ in static patch coordinates with radius $R_3=L_{4}sinh\left(\sigma \right)$ and a cosmological horizon at $\widehat{r}=R_3$.

The cosmological horizon at $\widehat{r}=R_3$ (for constant $\sigma$) is in fact identified with the bulk acceleration horizon. This can be seen via the coordinate transformation

$${\displaystyle \frac{t}{R_3}}={\displaystyle \frac{t_{\mathrm{R}}}{L_4}},cosh\left(\sigma \right)={\displaystyle \frac{\rho }{L_4}}cosh\left(\vartheta \right),{\displaystyle \frac{{\widehat{r}}^2}{R_3^2}}={\displaystyle \frac{{\rho }^2{sinh}^2\left(\vartheta \right)}{{\rho }^2{cosh}^2\left(\vartheta \right)-L_4^2}},$$

(143)

which brings the line element (142) to AdS${ }_4$–Rindler form,

$$d{s}^2=-\left({\displaystyle \frac{{\rho }^2}{L_4^2}}-1\right)d{t}_{\mathrm{R}}^2+{\left({\displaystyle \frac{{\rho }^2}{L_4^2}}-1\right)}^{-1}d{\rho }^2+{\rho }^2(d{\vartheta }^2+{sinh}^2\left(\vartheta \right)d{\varphi }^2).$$

(144)

Orbits of the Rindler-time translation Killing vector ${\partial }_{t_{\mathrm{R}}}$ correspond to uniformly accelerating trajectories. The cosmological horizon in (142) corresponds to the (non-compact) acceleration horizon $\rho =L_4$ with horizon temperature $T_{\mathrm{R}}={\left(2\pi L_4\right)}^{-1}$.

As in the qBTZ black hole, when $\mu \ne 0$, the root structure of the metric function $H\left(r\right)$ results in a black hole horizon. Meanwhile, real roots of $G\left(x\right)$ correspond to orbits of ${\partial }_{\varphi }$, and to ensure a finite black hole horizon, we again restrict to the range $0\le x\le x_1$, where $x_1$ is the smallest root of $G\left(x\right)$. Specifically, since $\kappa =+1$, it follows $x_1\in (0,1]$ (46), where $\mu$, via (45), is monotonically decreasing from $+\infty$ to zero, with $\mu =0$ for $x_1=1$. Further, as before, the conical singularity at $x=x_1$ is removed via the identification (48) with $\kappa =+1$.

The next essential difference between the quantum AdS and dS black hole constructions is that an asymptotically ${\mathrm{dS}}_3$ Randall–Sundrum brane is embedded AdS${ }_4$ at the umbilic $x=0$ hypersurface. Israel junction conditions again fix the brane tension to be $\tau ={(2\pi G_4\ell )}^{-1}$. In terms of the empty dS${ }_3$ geometry (142), the brane sits at

$$sinh\left({\sigma }_b\right)={\displaystyle \frac{R_3}{L_4}},$$

(145)

excluding the region $\sigma >{\sigma }_b$. Notice, moreover, the area of the bulk horizon at $\widehat{r}=R_3$ is finite,

$$2\pi L_4^2(cosh\left({\sigma }_b\right)-1)=2\pi R_3^2{\displaystyle \frac{\ell }{\ell +R_3}}.$$

(146)

That is, while the bulk acceleration horizon is generally non-compact (since it extends to the asymptotic boundary at $\sigma \to \infty$), its intersection with the brane is compact. Thus, the bulk acceleration horizon induces a (compact) cosmological horizon on the dS${ }_3$ brane. When $\mu \ne 0$, the bulk black hole horizon is projected onto the brane, resulting in an induced geometry

$$d{s}^2{|}_{x=0}=-\left(1-{\displaystyle \frac{r^2}{R_3^2}}-{\displaystyle \frac{\mu \ell }{r}}\right)d{t}^2+{\left(1-{\displaystyle \frac{r^2}{R_3^2}}-{\displaystyle \frac{\mu \ell }{r}}\right)}^{-1}d{r}^2+r^{2}d{\varphi }^2,$$

(147)

with a single black hole and cosmological horizon. See Figure 9 for an illustration.

Another caveat about the Randall–Sundrum brane construction is that the brane geometry will contain Big Bang and Big Crunch singularities in the asymptotic past and future. The reason follows because of the amount the brane radiates as it accelerates in the bulk. From the brane perspective, the time to reach the asymptotic past or future dS boundaries is infinite. Thus, the brane emits an infinite amount of radiation, thereby causing a piling of rays at future/past Cauchy horizons. The formation of the cosmological singularities is essentially the same phenomenon that enforces (strong) cosmic censorship at the inner Cauchy horizon in charged or rotating black holes. Alternatively, the Randall–Sundrum braneworld can be understood in terms of false vacuum decay [102,103,104] mediated by Coleman and de Luccia bubbles [105] of ‘nothing’. This vacuum decay process inevitably leads to Big Bang/Big Crunch singularities.

4.3.2. Quantum Schwarzschild–de Sitter

The naive metric (147) is not cast in canonically normalized coordinates, as $\varphi$ has periodicity $\varphi \sim \varphi +∆\varphi$. Rescaling coordinates $(t,r,\varphi )\to (\eta \overline{t},{\eta }^{-1}\overline{r},\eta \overline{\varphi })$ with $\eta$ given in (63) (for $\kappa =+1$), the geometry on the dS${ }_3$ brane is

$$d{s}_{\mathrm{qSdS}}^2=-H\left(\overline{r}\right)d{\overline{t}}^2+H{\left(\overline{r}\right)}^{-1}d{\overline{r}}^2+{\overline{r}}^{2}d{\overline{\varphi }}^2,H\left(\overline{r}\right)=\left(1-8{\mathcal{G}}_{3}M-{\displaystyle \frac{{\overline{r}}^2}{R_3^2}}-{\displaystyle \frac{\ell F\left(M\right)}{\overline{r}}}\right)$$

(148)

where we identify

$$8{\mathcal{G}}_{3}M\equiv 1-{\eta }^2=1-{\displaystyle \frac{4x_1^2}{{(3-x_1^2)}^2}},F\left(M\right)\equiv 8{\displaystyle \frac{(1-x_1^2)}{{(3-x_1^2)}^3}}$$

(149)

with renormalized Newton’s constant ${\mathcal{G}}_3\equiv G_3{\displaystyle \frac{L_4}{\ell }}$. The mass identification is motivated by how the mass of de Sitter black holes in Einstein–dS gravity is typically given by the subleading constant term in the $g_{tt}$ component of the metric.

Analogous to the quantum BTZ set-up, the dS metric (148) has the same form as the perturbative solution (20) to the semi-classical Einstein equations. Again, here, the $1/r$ correction is exact, and the metric is an exact solution to the whole tower of higher-derivative terms of the induced action. Substituting the metric into the gravitational equations of motion (see (69)–(70) with ${\ell }_3^2=-R_3^2$) yields the same structure of the stress–energy tensor ${\langle T_j^i\rangle }_0$ (71) and ${\langle T_j^i\rangle }_2$ (72). In the limit of vanishing backreaction, the geometry (148) takes the form of classical Schwarzschild–dS${ }_3$, which is a conical defect with a single cosmological horizon. For $\ell \ne 0$, the metric is interpreted as a static quantum black hole in dS${ }_3$, i.e., a three-dimensional quantum Schwarzschild–de Sitter black hole (qSdS) [26]. As in the qBTZ system, the qSdS black hole for $\ell >0$ has a curvature singularity at $\overline{r}=0$ that is hidden behind a black hole horizon induced by a semi-classical backreaction.

In contrast with the qBTZ system, the qSdS solution does not subdivide into several branches of quantum black holes. Indeed, since $\kappa =+1$, here, $0<x_1<1$. In this range, given the mass identification (149), the qSdS always has $M\ge 0$, with $M=1/8{\mathcal{G}}_3$ and $F=8/27$ at $x_1=0$, and $M=F=0$ at $x_1=1$. The $M=0$ solution with $\ell \ne 0$ is dubbed quantum dS${ }_3$, accounting for a large-c cutoff CFT living in dS${ }_3$. Meanwhile, the $M=1/8{\mathcal{G}}_3$ solution is the analog of the upper bound on the classical Schwarzschild–dS${ }_3$ conical defect.

The quantum SdS black hole, in fact, has a more stringent upper bound on mass than the conical defect mass. This is a consequence of the fact that the geometry (148) has both a black hole and cosmological horizon, and it is in complete analogy with classical higher-dimensional Schwarzschild–de Sitter black holes. It is well known that classical SdS black holes (in four or higher spacetime dimensions) have an upper bound on their mass in order to avoid the presence of a naked singularity. This largest mass black hole is known as the Nariai black hole [106] with mass $M_{\mathrm{N}}$. Geometrically, the Nariai black hole is one in which the (typically smaller) black hole horizon $r_h$ coincides with the (typically larger) cosmological horizon $r_c$, i.e., $r_h=r_c\equiv r_{\mathrm{N}}$, which is the Nariai horizon radius. In the context of the AdS${ }_4$ C-metric, the four-dimensional geometry has a Nariai limit that induces a Nariai geometry on the brane at $x=0$ [26]

$$d{s}_{\mathrm{N}}^2=-\left(1-{\displaystyle \frac{{\rho }^2}{{\overline{r}}_{\mathrm{N}}^2}}\right)d{\tau }^2+{\left(1-{\displaystyle \frac{{\rho }^2}{{\overline{r}}_{\mathrm{N}}^2}}\right)}^{-1}d{\rho }^2+{\overline{r}}_{\mathrm{N}}^{2}d{\overline{\varphi }}^2.$$

(150)

Here, $\tau$ and $\rho$ are time and radial coordinates, respectively, and ${\overline{r}}_{\mathrm{N}}=\eta r_{\mathrm{N}}$ with $r_{\mathrm{N}}=R_3/\sqrt{3}$. The (bulk) Nariai black hole places an upper bound on $(\mu \ell )$ in the C-metric; specifically, $(\mu \ell )\le {(\mu \ell )}_{\mathrm{N}}=2r_{\mathrm{N}}/3$. This bulk upper bound places an upper bound on the mass M of the qSdS black hole, which is denoted $M_{\mathrm{N}}$. The precise form of $M_{\mathrm{N}}$ is complicated; however, the Nariai mass was found to live in the finite range [26]

$${\displaystyle \frac{11}{27}}<8{\mathcal{G}}_{3}M<1,$$

(151)

where the upper limit corresponds to when $\ell /R_3\to 0$, while the lower limit occurs for $\nu \approx 1$. Therefore, the Nariai mass bound $M\le M_{\mathrm{N}}$ is generally more restrictive than the conical defect bound $M\le 1/8{\mathcal{G}}_3$. Notably, the classical SdS${ }_3$ conical defect does not have a Nariai limit. Thus, a semi-classical backreaction induces an upper limit on the amount of mass allowed in dS${ }_3$ which does not saturate the maximum conical deficit angle.

Finally, the Nariai solution puts a bound on the quantum backreaction due to the CFT for which a quantum dS${ }_3$ black hole exists. In particular, for non-vanishing backreaction $(\ell \ne 0)$, the form function $F\left(M\right)$ has a maximum value. Correspondingly, the angular deficit $∆\varphi$ has a maximum value. For too large angular deficits, a backreaction creates a black hole with a mass $M>M_{\mathrm{N}}$ such that it is too large to fit inside the dS${ }_3$ static patch. For such deficits, the quantum SdS solution no longer exists, and the resulting geometry is described by a naked conical defect spacetime, which is analogous to the braneworld geometry induced by the bulk BTZ black string.

4.3.3. Quantum Kerr–de Sitter

As in the static case, rotating quantum dS${ }_3$ black holes can be constructed in essentially the same way as the rotating qBTZ (91). Starting from the rotating C-metric (78) with $\kappa =+1$ and replacing $a\to -a$, and imposing bulk regularity, the metric on the brane in canonically normalized coordinates is [27]29

$$\begin{array}{cc}\hfill d{s}_{\mathrm{qKdS}}^2& =-\left(1-8{\mathcal{G}}_{3}M-{\displaystyle \frac{{\overline{r}}^2}{R_3^2}}-{\displaystyle \frac{\mu \ell {\eta }^2}{r}}\right)d{\overline{t}}^2\hfill \\ & +{\left(1-8{\mathcal{G}}_{3}M-{\displaystyle \frac{{\overline{r}}^2}{R_3^2}}+{\displaystyle \frac{{\left(4{\mathcal{G}}_{3}J\right)}^2}{{\overline{r}}^2}}-{\displaystyle \frac{\mu \ell {(1+{\tilde{a}}^2)}^2{\eta }^{4}r}{{\overline{r}}^2}}\right)}^{-1}d{\overline{r}}^2\hfill \\ & +\left({\overline{r}}^2+{\displaystyle \frac{\mu \ell {\tilde{a}}^2{R}_3^2{\eta }^2}{r}}\right)d{\overline{\varphi }}^2-4{\mathcal{G}}_{3}J\left(1+{\displaystyle \frac{\ell }{x_{1}r}}\right)(d\overline{\varphi }d\overline{t}+d\overline{t}d\overline{\varphi }).\hfill \end{array}$$

(152)

with mass and angular momentum identified as

$$8{\mathcal{G}}_{3}M\equiv 1-{\eta }^2\left(1-{\tilde{a}}^2+{\displaystyle \frac{4{\tilde{a}}^2}{x_1^2}}\right)=1-{\displaystyle \frac{4[x_1^2-{\tilde{a}}^2(x_1^2-4)]}{{(3-x_1^2+{\tilde{a}}^2)}^2}},$$

(153)

$$4{\mathcal{G}}_{3}J\equiv -R_3\tilde{a}\mu x_1{\eta }^2={\displaystyle \frac{4R_3\tilde{a}(x_1^2+{\tilde{a}}^2-1)}{{(3-x_1^2+{\tilde{a}}^2)}^2}}.$$

(154)

The metric (152) is dubbed the quantum Kerr–dS${ }_3$ black hole since when $\ell \to 0$, the geometry of the classical Kerr–ds${ }_3$ conical defect is recovered. There are a couple of limits worth noting: (i) when the parameter $\mu$ (83) vanishes, i.e., $x_1=\sqrt{1-{\tilde{a}}^2}$, both $M=J=0$, resulting in the empty ${\mathrm{dS}}_3$ geometry, and (ii) mass M also vanishes when $x_1=\sqrt{9-{\tilde{a}}^2}$; however, $J\ne 0$ and $\mu \ne 0$, resulting in a quantum rotating ${\mathrm{dS}}_3$.

Substituting the geometry (152) into the gravity field equations results in a holographic stress tensor with components of essentially the same form as (102)—see [27] for explicit details. Now, the function $F(M,J)$ from evaluating the large-$\overline{r}$ behavior of ${\langle T_{\overline{t}}^{\overline{t}}\rangle }_0$ is

$$F(M,J)\equiv \mu {\eta }^3\sqrt{1+{\tilde{a}}^2}(1-2{\tilde{a}}^2)={\displaystyle \frac{8\sqrt{1+{\tilde{a}}^2}(1-2{\tilde{a}}^2)(1-x_1^2-{\tilde{a}}^2)}{{(3-x_1^2+{\tilde{a}}^2)}^3}}.$$

(155)

Notice $F(M,J)$ will vanish either when $\mu =0$, i.e., $x_1^2=1-{\tilde{a}}^2$. The zero ${\tilde{a}}^2=1/2$, meanwhile, is unique only to the ${\langle T_{\overline{t}}^{\overline{t}}\rangle }_0$ component with the remaining components of the holographic stress tensor being non-zero for this value of $\tilde{a}$. This stands in contrast with the rotating qBTZ black hole, for which every component vanishes when ${\tilde{a}}^2=1$. Further, as in the qBTZ case, the radial dependence of the renormalized stress tensor due to a conformally coupled scalar field (found perturbatively in [27]) is significantly more complicated than the holographic stress tensor.

Horizon Structure

Horizons in the bulk correspond to positive real roots $r_i$ of $H\left(r\right)$, where the Killing vector

$${\zeta }^b={\partial }_t-{\displaystyle \frac{a}{r_i^2}}{\partial }_{\varphi }$$

(156)

becomes null. To classify the types of horizons, define the function $Q\left(r\right)\equiv r^{2}H\left(r\right)$. Since $Q\left(r\right)$ is a quartic polynomial in r, it will generally have either four, two, or zero real roots. Consider the case when there are four real roots: three positive roots correspond to the cosmological horizon $r_c$, the outer black hole horizon $r_{+}$, and inner black hole horizon $r_{-}$, obeying $r_{-}\le r_{+}\le r_c$, while the fourth root, $r_n=-(r_c+r_{+}+r_{-})<0$ and resides behind the singularity at $r=0$. Using $H\left(r_c\right)=0$, and $H\left(r_{\pm }\right)=0$, we can express

$$\begin{array}{cc}& R_3^2=r_c^2+r_{+}^2+r_c{r}_{+}+r_{-}(r_c+r_{+}+r_{-}),\hfill \\ & \mu \ell ={\displaystyle \frac{(r_c+r_{+})(r_c+r_{-})(r_{+}+r_{-})}{r_c^2+r_{+}^2+r_c{r}_{+}+r_{-}(r_c+r_{+}+r_{-})}},\hfill \\ & a^2={\displaystyle \frac{r_c{r}_{+}{r}_{-}(r_c+r_{+}+r_{-})}{r_c^2+r_{+}^2+r_c{r}_{+}+r_{-}(r_c+r_{+}+r_{-})}}.\hfill \end{array}$$

(157)

The limit $r_{-}\to 0$ coincides with $a=0$, while $r_{+}=r_{-}=0$ corresponds to $\mu \to 0$, resulting in the Kerr–${\mathrm{dS}}_3$ geometry with a single cosmological horizon.

The positive roots $r_i$ to $H\left(r\right)$ correspond to rotating horizons with rotation ${\Omega }_i$,

$${\Omega }_i\equiv {\displaystyle \frac{a}{R_3^2}}{\displaystyle \frac{(x_1^2{r}_i^2-R_3^2)}{(r_i^2+a^2{x}_1^2)}},$$

(158)

generated by the (canonically normalized) Killing vector

$${\overline{\zeta }}^b\equiv {\displaystyle \frac{\eta (1+{\tilde{a}}^2)}{\left(1+\frac{a^2{x}_1^2}{r_i^2}\right)}}{\zeta }^b={\partial }_{\overline{t}}^b+{\Omega }_i{\partial }_{\overline{\varphi }}^b.$$

(159)

Moreover, the surface gravity ${\kappa }_i$ associated with each horizon $r_i$ is given by

$${\kappa }_i={\displaystyle \frac{\eta (1+{\tilde{a}}^2)}{\left(r_i^2+a^2{x}_1^2\right)}}{\displaystyle \frac{r_i^2}{2}}|H^{\prime }\left(r_i\right)|={\displaystyle \frac{\eta (1+{\tilde{a}}^2)}{\left(r_i^2+a^2{x}_1^2\right)}}{\displaystyle \frac{1}{2R_3^2{r}_i}}|R_3^2\mu \ell r_i-2r_i^4-2a^2{R}_3^2|,$$

(160)

where we used the definition ${\overline{\zeta }}^b{\nabla }_b{\overline{\zeta }}^c=\kappa {\overline{\zeta }}^c$. Explicitly,

$$\begin{array}{cc}& {\kappa }_c=-{\displaystyle \frac{\eta (1+{\tilde{a}}^2)}{2R_3^2\left(r_c^2+a^2{x}_1^2\right)}}(r_c-r_{+})(r_c-r_{-})(r_{+}+r_{-}+2r_c),\hfill \\ & {\kappa }_{+}={\displaystyle \frac{\eta (1+{\tilde{a}}^2)}{2R_3^2\left(r_{+}^2+a^2{x}_1^2\right)}}(r_c-r_{+})(r_{+}-r_{-})(r_c+r_{-}+2r_{+}),\hfill \\ & {\kappa }_{-}=-{\displaystyle \frac{\eta (1+{\tilde{a}}^2)}{2R_3^2\left(r_{-}^2+a^2{x}_1^2\right)}}(r_c-r_{-})(r_{+}-r_{-})(r_c+r_{+}+2r_{-}).\hfill \end{array}$$

(161)

Notice the cosmological horizon surface gravity ${\kappa }_c$ vanishes when $r_c=r_{+}$ or $r_c=r_{-}$ and similarly for the other surface gravities. When $r_{-}\to 0$, i.e., ${\kappa }_c$ and ${\kappa }_{+}$ simplify to the surface gravities of the cosmological horizon and black hole horizon of the qSdS black hole [26].

In the naive coordinates, a computation of the Kretschmann scalar reveals a curvature singularity at $r=0$, corresponding to a ring singularity at $\overline{r}=r_s$ in canonically normalized coordinates. As in the rotating qBTZ black hole, closed timelike curves can appear near the ring singularity. These closed timelike curves can be eliminated via an appropriate periodic identification [107], such that constant $\overline{t}$ hypersurfaces are closed and span two black hole regions with opposite spin, cutting through intersections of $r=r_c$ and $r=r_{+}$.

Ergoregions. As with classical Kerr–dS spacetimes, the qKdS black hole has a stationary limit surface and two ergoregions associated with the outer black hole and cosmological horizons. Explicitly, the Killing vector ${\partial }_t$ in the naive metric has norm $\mathcal{N}$

$$\mathcal{N}=-H\left(r\right)+{\displaystyle \frac{a^2}{r^2}}.$$

(162)

Thus, at the outer and cosmological horizons, the time-translation Killing vector ${\partial }_t$ becomes spacelike. The locus of points where $\mathcal{N}=0$ yields a stationary limit surface, satisfying $r(R_3^2-r^2)=R_3^2\mu \ell$. Since there exist regions in between the outer and cosmological horizons where ${\partial }_t$ is timelike, there are two ergoregions, where an observer is forced to move in the direction of rotation of the outer black hole horizon or cosmological horizon. In principle, the Penrose process of energy extraction in the qKdS solution operates in morally the same way as a classical four-dimensional Kerr–de Sitter black hole (see, e.g., [108]).

Extremal limits. As with the four-dimensional Kerr–de Sitter black hole, the quantum Kerr–${\mathrm{dS}}_3$ has a number of limiting geometries. Specifically, (i) extremal or ‘cold’ limit, where $r_{+}=r_{-}$; (ii) rotating Nariai limit, where $r_c=r_{+}$, and (iii) the ‘ultracold’ limit where $r_c=r_{+}=r_{-}$. In the near-horizon regime, the geometries appear as fibered products of a circle and two-dimensional anti-de Sitter, de Sitter, and Minkowski space, respectively. For details, see [27]. These limiting geometries, moreover, have the same qualitative features as (warped) ${\mathrm{dS}}_3$ black hole solutions to topologically massive gravity, cf. [37,38,109]. There is also a ‘lukewarm’ limit, where the surface gravities ${\kappa }_c={\kappa }_{+}$ at a value different from the surface gravity of the Nariai black hole.

Adding charge. Similar to the charged quantum BTZ black hole, quantum de Sitter black holes can be charged (see [110]). In this context, as with the classical charged dS black holes in $3+1$-dimensions and higher, the charged black hole has three horizons: outer and inner black hole horizons, and the cosmological horizon. As such, the solution has three limiting geometries: (i) the extremal limit, where $r_{+}=r_{-}$; (ii) the charged Nariai limit, where $r_c=r_{+}$, and (iii) the ultracold limit, where $r_c=r_{+}=r_{-}$. Moreover, the physical parameter space of the solution is characterized by a ‘shark fin’ diagram for non-zero backreaction parameter $\nu$.

4.4. Quantum Black Holes in Flat Space

A point mass in three-dimensional Minkowski space, Mink${ }_3$, is described by a conical singularity with no horizon; the Schwarzschild solution in three dimensions is not a black hole. Again, quantum backreaction effects alter the geometry such that the black hole horizon appears to hide the conical singularity [32]. The only known way to consistently construct an exact quantum black hole in asymptotic Mink${ }_3$ space is via braneworld holography.

The bulk set-up is the same as for quantum dS${ }_3$ black holes: the AdS${ }_4$ C-metric with a two-sided Randall–Sundrum brane. In some respects, the asymptotically flat quantum black holes may be viewed as a special limit, $R_3\to \infty$, of the dS${ }_3$ black holes. Although they were the first exact three-dimensional braneworld black hole solution to be discovered [23], quantum black holes in Mink${ }_3$ have received less attention than their (A)dS cousins [22]. For this reason, we present a fresh take on quantum Mink${ }_3$ black holes.

4.4.1. Bulk Set-Up

Consider the static C-metric (39) with metric functions (40), except $\kappa =+1$ and without the (A)dS factor, such that $H\left(r\right)$ has the form

$$H\left(r\right)=1-{\displaystyle \frac{\mu \ell }{r}}.$$

(163)

It is straightforward to verify the bulk geometry is an Einstein metric with a negative cosmological constant where the bulk length scale is

$$L_4=\ell .$$

(164)

This is the first notable difference compared to the (A)dS examples and will have ramifications as we proceed. This means the acceleration horizon is present in the spacetime. However, it has a different effect than in the dS case where $\ell <L_4$.

For $\mu \ne 0$, the bulk geometry has a black hole horizon at $H\left(r_{+}\right)=0$,

$$r_{+}=\mu \ell ,$$

(165)

hiding the curvature singularity at $r=0$. There is a second horizon at $r\to \infty$ corresponding to the bulk AdS${ }_4$ horizon30. To see this, consider the bulk geometry with $\mu =0$ (such that $H\left(r\right)=1$ and $G\left(x\right)=1-x^2$). The coordinate transformation

$$w\equiv \ell +xr,\widehat{r}\equiv r\sqrt{1-x^2},$$

(166)

brings the bulk metric to empty AdS${ }_4$ in Poincaré form

$$d{s}^2={\displaystyle \frac{{\ell }^2}{w^2}}\left(-d{t}^2+d{\widehat{r}}^2+{\widehat{r}}^{2}d{\varphi }^2+d{w}^2\right).$$

(167)

Evidently, $r=0$ no longer represents a curvature singularity but instead a non-singular worldline $w=\ell$, $\widehat{r}=0$. Meanwhile, $xr=-\ell$ maps to the asymptotic boundary $w=0$ and $r\to \infty$ to the AdS${ }_4$ horizon $w\to \infty$.

As is now standard, real roots of $G\left(x\right)$ correspond to orbits of ${\partial }_{\varphi }$. Let $x_1$ denote the smallest root of $G\left(x\right)$ and again restrict to the range $0\le x\le x_1$. With $\kappa =+1$, $x_1\in (0,1]$ (46), where $\mu$ is defined in (45). The conical singularity at $x=x_1$ is removed via the identification (48) with $\kappa =+1$.

The $x=0$ hypersurface is umbilic. In Poincaré coordinates with $\mu =0$ (166), this surface is located at $w=\ell$. We place a two-sided ETW Randall–Sundrum brane at $x=0$, retaining only the region $0\le x\le x_1$ (Figure 10). To complete space, we glue another copy of the region $0\le x\le x_1$ along the brane at $x=0$. Israel junction conditions fix the brane tension to be $\tau ={(2\pi G_4\ell )}^{-1}$. Notably, since $\ell =L_4$, the tension is said to be at its critical value. As we will see momentarily, this condition leads to a vanishing induced cosmological constant. The induced geometry at $x=0$ is

$$d{s}_{x=0}^2=-\left(1-{\displaystyle \frac{\mu \ell }{r}}\right)d{t}^2+{\left(1-{\displaystyle \frac{\mu \ell }{r}}\right)}^{-1}d{r}^2+r^{2}d{\varphi }^2.$$

(168)

For $\ell \ne 0$, this geometry describes a black hole with horizon radius $r_{+}=\mu \ell$. In terms of coordinate w, the black hole horizon is located at $w_{+}=\ell +x{r}_{+}$. Therefore, the brane at $w=\ell$ cuts through the bulk black hole horizon such that the black hole extends only slightly off of the brane to a distance $w_{\max }=\ell +r_{+}{x}_1\le \ell +r_{+}$ (since $x_1\le 1$).

The induced brane theory is (53); however, since $L_4=\ell$, the induced brane cosmological constant (55) is zero, while the induced Newton’s constant (54) is $G_3=G_4/2\ell$. The central charge of the cutoff CFT${ }_3$ is $c_3={\displaystyle \frac{L_4^2}{G_4}}={\displaystyle \frac{\ell }{2G_3}}$. Higher-derivative terms enter at order $\mathcal{O}\left({\ell }^2\right)$, and treating the cutoff as small, $\ell \le 1$, the higher-derivative contributions serve as corrections to three-dimensional Einstein gravity.

4.4.2. Quantum Schwarzschild Black Hole

The angular coordinate $\varphi$ in the naive metric (168) has a periodicity of $\varphi \sim \varphi +∆\varphi$. Working with rescaled coordinates $(t,r,\varphi )\to (\eta \overline{t},{\eta }^{-1}\overline{r},\eta \overline{\varphi })$ for $\eta$ (63) (with $\kappa =+1$), the geometry on the Mink${ }_3$ brane is31

$$d{s}_{\mathrm{qSchw}}^2=-H\left(\overline{r}\right)d{\overline{t}}^2+H{\left(\overline{r}\right)}^{-1}d{\overline{r}}^2+{\overline{r}}^{2}d{\overline{\varphi }}^2,H\left(\overline{r}\right)=1-8G_{3}M-{\displaystyle \frac{\ell F\left(M\right)}{\overline{r}}}$$

(169)

where we identified the mass as in the de Sitter context (149)

$$M\equiv {\displaystyle \frac{1}{8G_3}}{\displaystyle \frac{(x_1^2-1)(x_1^2-9)}{{(3-x_1^2)}^2}},F\left(M\right)\equiv 8{\displaystyle \frac{(1-x_1^2)}{{(3-x_1^2)}^3}}.$$

(170)

Since $L_4=\ell$, the renormalized Newton’s constant ${\mathcal{G}}_3=G_3$.

The geometry (169) is an exact solution to the induced semi-classical equations of motion (68) (with ${\ell }_3\to \infty$) with a quantum stress tensor as in (71) and (72) (with ${\ell }_3\to \infty$). Thus, we interpret the solution as the three-dimensional quantum Schwarzschild black hole. A curvature singularity at $\overline{r}=0$ is hidden behind a black hole event horizon at

$${\overline{r}}_{+}={\displaystyle \frac{\ell F\left(M\right)}{(1-8G_{3}M)}}.$$

(171)

At the horizon, the time-translation Killing vector ${\overline{\zeta }}^i=\eta {\partial }_{\overline{t}}^i$ goes null, having surface gravity

$${\kappa }_{+}={\displaystyle \frac{1}{2}}\left|H^{\prime }\left({\overline{r}}_{+}\right)\right|={\displaystyle \frac{\ell F\left(M\right)}{{\overline{r}}_{+}^2}}={\displaystyle \frac{{(1-8G_{3}M)}^2}{\ell F\left(M\right)}}.$$

(172)

4.4.3. Quantum Kerr Black Hole

Rotating black holes in Mink${ }_3$ are constructed by embedding a Randall–Sundrum brane with critical tension inside the rotating C-metric (78), such that the metric functions are

$$\begin{array}{cc}& H\left(r\right)=1-{\displaystyle \frac{\mu \ell }{r}}+{\displaystyle \frac{a^2}{r^2}},G\left(x\right)=1-x^2-\mu x^3,\hfill \\ & \Sigma (x,r)=1+{\displaystyle \frac{a^2{x}^2}{r^3}}.\hfill \end{array}$$

(173)

Formally, the bulk regularity analysis is the same; however, some of the expressions are simpler because in this limit $\tilde{a}=0$, which is a consequence of the $G\left(x\right)$ function losing its quartic term compared to its (A)dS siblings. Thus, the metric on the brane in canonically normalized coordinates is

$$\begin{array}{cc}\hfill d{s}_{\mathrm{qKerr}}^2& =-\left(1-8G_{3}M-{\displaystyle \frac{\mu \ell {\eta }^2}{r}}\right)d{\overline{t}}^2+{\left(1-8G_{3}M+{\displaystyle \frac{{\left(4G_{3}J\right)}^2}{{\overline{r}}^2}}-{\displaystyle \frac{\mu \ell {\eta }^{4}r}{{\overline{r}}^2}}\right)}^{-1}d{\overline{r}}^2\hfill \\ & +\left({\overline{r}}^2+{\displaystyle \frac{\mu \ell a^2{x}_1^4{\eta }^2}{r}}\right)d{\overline{\varphi }}^2-4G_{3}J\left(1+{\displaystyle \frac{\ell }{x_{1}r}}\right)(d\overline{\varphi }d\overline{t}+d\overline{t}d\overline{\varphi }),\hfill \end{array}$$

(174)

with canonical coordinates

$$\begin{array}{cc}& \overline{t}={\eta }^{-1}(t-a{x}_1^2\varphi ),\overline{\varphi }={\eta }^{-1}\varphi ,\hfill \\ & {\overline{r}}^2={\eta }^2{r}^2+r_s^2,r_s\equiv -{\displaystyle \frac{2a{x}_1^2\sqrt{2-x_1^2}}{(3-x_1^2)}},\hfill \end{array}$$

(175)

where $\eta =2x_1/(3-x_1^2)$. Further, the angular momentum J is identified as

$$4G_{3}J\equiv {\displaystyle \frac{4a{x}_1^2(x_1^2-1)}{{(3-x_1^2)}^2}}.$$

(176)

while the mass is identified as in the static case (170) and is thus independent of the rotation parameter a. A ring singularity appears at $\overline{r}=r_s$.

The metric (174) is dubbed the quantum Kerr${ }_3$ black hole. In the limit of vanishing backreaction, one recovers the classical conical Kerr geometry. At leading order in a small-ℓ expansion, the stress tensor of the holographic CFT${ }_3$ is

$$\begin{array}{cc}& {\langle T_{\overline{t}}^{\overline{t}}\rangle }_0={\displaystyle \frac{\mu \ell }{16\pi G_3{r}^3}}\left(1+{\displaystyle \frac{3a^2{x}_1^2}{r^2}}\right),\hfill \\ & {\langle T_{\overline{r}}^{\overline{r}}\rangle }_0={\displaystyle \frac{\mu \ell }{16\pi G_3{r}^3}},\hfill \\ & {\langle T_{\overline{\varphi }}^{\overline{\varphi }}\rangle }_0=-{\displaystyle \frac{\mu \ell }{16\pi G_3{r}^3}}\left(2+{\displaystyle \frac{3a^2{x}_1^2}{r^2}}\right),\hfill \\ & {\langle T_{\overline{\varphi }}^{\overline{t}}\rangle }_0={\displaystyle \frac{3\mu \ell a{x}_1^2}{16\pi G_3{r}^3}}\left(1+{\displaystyle \frac{a^2{x}_1^2}{r^2}}\right),\hfill \\ & {\langle T_{\overline{t}}^{\overline{\varphi }}\rangle }_0=-{\displaystyle \frac{3\mu \ell a}{16\pi G_3{r}^5}}.\hfill \end{array}$$

(177)

Notice now that function $F(M,J)$ from evaluating the large-$\overline{r}$ behavior of ${\langle T_{\overline{t}}^{\overline{t}}\rangle }_0$ is exactly equal to $F\left(M\right)=\mu {\eta }^3$, as in the static case.

4.5. Horizon Structure

Bulk black hole horizons correspond to real roots $r_i$ of $H\left(r\right)$, i.e.,

$$r_{\pm }={\displaystyle \frac{1}{2}}(\mu \ell \pm \sqrt{{(\mu \ell )}^2-4a^2}).$$

(178)

These correspond to rotating horizons with rotation

$${\Omega }_{\pm }=-{\displaystyle \frac{a}{(r_{\pm }^2+a^2{x}_1^2)}},$$

(179)

generated by Killing vector ${\overline{\zeta }}^j={\partial }_{\overline{t}}^j+{\Omega }_{\pm }{\partial }_{\overline{\varphi }}^j$. The surface gravities are

$${\kappa }_{\pm }={\displaystyle \frac{\eta }{(r_{\pm }^2+a^2{x}_1^2)}}{\displaystyle \frac{r_{\pm }^2}{2}}\left|H^{\prime }\left(r_{\pm }\right)\right|={\displaystyle \frac{\eta }{2(r_{\pm }^2+a^2{x}_1^2)}}|\mu \ell -{\displaystyle \frac{2a^2}{r_{\pm }}}|.$$

(180)

There is an extremal black hole when the inner and outer horizons coincide, $r_{+}=r_{-}\equiv r_{\mathrm{ex}}$, i.e., when $\mu \ell =2a$. In this limit, the surface gravities (180) vanish.

5. Quantum Black Hole Thermodynamics

Black hole thermodynamics [111,112,113,114] reveals an interplay between geometry, quantum mechanics, and thermodynamics. This is encapsulated by the Bekenstein–Hawking entropy–area relation,

$$S={\displaystyle \frac{k_{\mathrm{B}}{c}^3}{\hslash G}}{\displaystyle \frac{A\left[H\right]}{4}},$$

(181)

for Boltzmann constant $k_{\mathrm{B}}$, and we have temporarily restored factors of ℏ and speed of light c. The area law states black holes carry a thermodynamic entropy proportional to the area of a codimension-2 cross-section of their event horizon H. Hence, black holes may be treated as genuine thermal systems with energy, entropy, and temperature, and other thermodynamic variables depending on the type of black hole. In the case of stationary black holes, the laws of black hole mechanics [115] may be reinterpreted as laws of thermodynamics. For example, the first law relates a variation in mass M to variations of the other thermodynamic variables

$$dM=TdS+...,$$

(182)

with Hawking temperature T, and where the ellipsis implies variations of other possible thermodynamic variables, e.g., electric/magnetic charge, rotation, and so forth.

As we review below, just as the bulk geometry imprints itself on the brane, so too does its thermal description. Thus, classical thermodynamics of the bulk black hole geometry is interpreted as semi-classical thermodynamics of the quantum black hole system. This allows for an exact study of quantum black hole thermodynamics at any order in a backreaction.

5.1. Bulk Thermodynamics

When we think of mapping out the properties of a thermodynamic system, we typically think of one near equilibrium. Likewise, in black hole thermodynamics, often the first task is to check whether the system has a well-defined notion of thermal equilibrium. Such is the case, for example, of Kerr–Newman black holes in AdS, where thermal equilibrium is unambiguously defined [116]. Alternatively, the thermodynamics of black holes in de Sitter space is conceptually more subtle because a static patch observer encounters a system with two horizons at different temperatures (except in special limits, e.g., Nariai), such that the system is not generally in equilibrium. Similarly, due to the presence of an acceleration and black hole horizon, the thermodynamics of the C-metric is generally more subtle than non-accelerating black holes—even when the accelerating black holes are embedded in AdS. Furthermore, although the C-metric line element does not display time dependence, uniformly accelerating black holes will deliver non-vanishing radiation at asymptotic infinity [117]. Thus, it is not obvious how an accelerating black hole could possibly be in equilibrium.

The AdS${ }_4$ C-metric, does, however, have a distinct advantage over the flat or dS C-metrics. Working in a particular regime of parameters, the negative cosmological constant has the effect of essentially removing the acceleration horizon such that it can be consistently neglected. This regime is precisely the ‘slowly accelerating’ black hole [90] where the (inverse) acceleration and AdS${ }_4$ length scale satisfy

$$\mathbf{Slowly}\mathbf{accelerating}:\ell >L_4.$$

(183)

In such situations, the AdS${ }_4$ C-metric is described by a single black hole suspended away from the origin by a cosmic string attached to the AdS boundary (recall Figure 4). Consequently, being able to ignore the acceleration allows for the temperature of the black hole system to be defined in a straightforward way. Even still, due to the presence of cosmic strings, the black hole is not isolated. This makes consistently carrying out the thermodynamic analysis of the slowly accelerating black hole non-trivial. Indeed, one must account for both the black hole and cosmic string, leading to a modified first law including variations of the tension of the cosmic string [118,119,120,121]32.

While the starting point is the same, the thermodynamics of the bulk black hole system used in the braneworld black hole constructions in Section 4 is ultimately different from the usual AdS${ }_4$ C-metric. Firstly, in the braneworld construction, there is no cosmic string; instead, there is a brane. For a purely tensional brane of constant tension, the brane does not play an obvious role in the bulk black hole thermodynamics (we will revisit this when we allow for variable tension). Second, the bulk geometry is regular in that the zeros to the metric function $G\left(x\right)$ are removed along with the conical defects they induce. These two facts make the thermodynamic analysis less subtle than the standard C-metric.

In general, however, the question of thermal equilibrium remains due to the presence of the bulk acceleration horizon. In fact, this question distinguishes the treatment of the Karch–Randall and Randall–Sundrum braneworld constructions. For the former, the bulk system started with a slowly accelerating black hole (183), such that the acceleration horizon does not imprint itself on the brane. Alternatively, for a dS${ }_3$ Randall–Sundrum brane, the bulk acceleration horizon cannot be ignored because one is not in the slow acceleration regime, thus complicating the thermal analysis. Instead, $\ell \le L_4$. The edge case $\ell =L_4$ corresponds to the flat Randall–Sundrum brane. Recall in this scenario the acceleration horizon does not localize on the brane (as in the dS${ }_3$ Randall–Sundrum brane) such that the acceleration horizon may be ignored. We will return to these differences momentarily.

Moving forward, let us first consider the thermodynamics of the bulk system with an AdS${ }_3$ Karch–Randall braneworld. In the literature on the exact three-dimensional braneworld black holes, typically, one assumes the Bekenstein–Hawking entropy Formula (181) and Hawking temperature. The energy is then identified by demanding the first law hold, which is subsequently found to coincide with the mass M identified geometric construction. For example, consider the static, neutral black hole construction (39). It proves useful to introduce the real and non-negative parameter [24]

$$z\equiv {\displaystyle \frac{{\ell }_3}{r_{+}{x}_1}},$$

(184)

for black hole horizon radius $r_{+}$. Given the range of $x_1$, generally, $z\in [0,\infty )$. It is possible to express the parameters $x_1,\mu$ and $r_{+}$ solely in terms of z and $\nu \equiv \ell /{\ell }_3$. In particular, solving $H\left(r_{+}\right)=0$ for $x_1^2$ yields

$$x_1^2=-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{(1-\nu z^3)}{z^2(1+\nu z)}}.$$

(185)

Rearranging z (184) and substituting for $x_1^2$ above gives

$$r_{+}^2=-{\ell }_3^2\kappa {\displaystyle \frac{(1+\nu z)}{(1-\nu z^3)}}.$$

(186)

Further, with (185), the parameter $\mu$ (45) obeys

$$\mu x_1=-\kappa {\displaystyle \frac{(1+z^2)}{(1-\nu z^3)}}.$$

(187)

The Hawking temperature T of the bulk black hole horizon is proportional to the surface gravity relative to the canonical timelike Killing vector ${\partial }_{\overline{t}}$ (74). Using (185)–(187), the temperature can be recast in terms of z and $\nu$,

$$T={\displaystyle \frac{{\kappa }_{+}}{2\pi }}={\displaystyle \frac{1}{2\pi {\ell }_3}}{\displaystyle \frac{z(2+3\nu z+\nu z^3)}{1+3z^2+2\nu z^3}}.$$

(188)

The Bekenstein–Hawking entropy, meanwhile, is (setting $c=\hslash =k_{\mathrm{B}}=1$)

$$\begin{array}{cc}\hfill S={\displaystyle \frac{\mathrm{Area}\left(r_{+}\right)}{4G_4}}& ={\displaystyle \frac{2}{4G_4}}{\int }_0^{2\pi \eta }d\varphi {\int }_0^{x_1}dx{r}_{+}^2{\displaystyle \frac{{\ell }^2}{{(\ell +x{r}_{+})}^2}}\hfill \\ & ={\displaystyle \frac{8\pi {\ell }_3^2}{4G_4}}{\displaystyle \frac{\nu z}{1+3z^2+2\nu z^3}},\hfill \end{array}$$

(189)

where the factor of two appearing in the second equality is because the brane is two-sided.

Keeping parameters ${\ell }_3$ and $\nu$ fixed, we can identify the energy E via

$${\partial }_{z}E=T{\partial }_{z}S,$$

(190)

such that the first law

$$dE=TdS$$

(191)

is obeyed. In particular, the energy E is explicitly found to be

$$E={\displaystyle \frac{\sqrt{1+{\nu }^2}}{2G_3}}{\displaystyle \frac{z^2(1-\nu z^3)(1+\nu z)}{{(1+3z^2+2\nu z^3)}^2}}=M,$$

(192)

where in the second equality, we substituted (185)–(187) into the identified mass (65). This confirms M should indeed be identified as the mass of the bulk black hole.

Gravitational Path Integral Approach

The above method is sufficient for studying the thermodynamics, assuming the identification of the thermodynamic variables is correct. A more fundamental approach would be to evaluate the quantum gravitational canonical partition function in the semi-classical limit via the on-shell Euclidean action [123]. Such an approach was taken in [124] (see Appendix E for details).

Formally, the partition function $Z\left(\beta \right)$ is given by a Euclidean path integral whose fixed boundary data on field configurations corresponds to thermodynamic data defining the thermal ensemble. At leading order in a stationary phase approximation, this becomes

$$Z\left(\beta \right)\approx e^{-I_{\mathrm{on-shell}}},$$

(193)

where $\beta$ is the (inverse) temperature of the system and $I_{\mathrm{on-shell}}$ is the on-shell Euclidean action. In say, the Schwarzschild black hole, $\beta$ is introduced as the periodicity of the Euclidean time circle to make the Euclidean solution regular at the horizon. In the case of the AdS${ }_4$ warped geometry, additional care is needed. Firstly, one Wick rotates the Lorentzian geometry (39) $t_E=it$ for Euclidean time $t_E$. To avoid a conical singularity at $r=r_{+}$, the Euclidean time direction is compactified into a circle, $t_E\sim t_E+∆t_E$, with period

$$∆t_E={\displaystyle \frac{4\pi }{|H^{\prime }\left(r_{+}\right)|}}.$$

(194)

Further, the bulk regularity conditions eliminating the conical singularity at $x_1$ must be respected. Combined, one works with Euclideanized canonically normalized time coordinate ${\overline{t}}_E$ with periodicity

$${\overline{t}}_E\sim {\overline{t}}_E+\beta ,\beta ={\displaystyle \frac{∆t_E}{\eta }}.$$

(195)

Working at fixed $\beta$ thus defines working in a canonical ensemble of fixed temperature $T={\beta }^{-1}$. Indeed, the periodicity (195) is the inverse of the temperature (188).

With respect to the partition function (193), the thermodynamic energy E and entropy S are defined as

$$\begin{array}{cc}& E\equiv -{\partial }_{\beta }logZ\approx {\partial }_{\beta }{I}_{\mathrm{on-shell}},\hfill \\ & S\equiv \beta E+logZ\approx (\beta {\partial }_{\beta }-1)I_{\mathrm{on-shell}}.\hfill \end{array}$$

(196)

With some work, the on-shell action for the static, neutral black hole is

$$I_{\mathrm{on-shell}}=-{\displaystyle \frac{2\pi {\ell }^{2}z}{G_4\nu }}{\displaystyle \frac{[1+2\nu z+\nu z^3(2+\nu z)]}{(2+3\nu z+\nu z^3)(1+3z^2+2\nu z^3)}},$$

(197)

from which one recovers the mass (192) and entropy (189). In summary, the thermodynamics of the bulk black hole directly follow from a Euclidean gravitational path integral.

Critical to this approach is being able to identify a system in thermal equilibrium. The black hole system in question has multiple non-degenerate horizons, as in the case with the dS${ }_3$ Randall–Sundrum brane. Thus, upon Wick rotating to Euclidean signature, the geometry will have multiple conical singularities: one associated with each horizon. In such cases, one removes a single conical singularity by fixing the periodicity of the Euclidean time coordinate for the associated horizon. Consequently, one is only able to treat a part of the entire system (neglecting the other horizons). Strictly speaking, however, in such situations, the complete system is not in thermal equilibrium, and it is not clear how to define a thermal partition function (without further modification).

5.2. Identifying Bulk and Brane Thermodynamics

The thermodynamics of the classical four-dimensional black hole is reinterpreted as the semi-classical thermodynamics of the three-dimensional quantum black hole. This is a by-product of the fact that the bulk black hole horizon localizes on the brane such that the temperature of the bulk black hole coincides with the temperature of the horizon induced on the brane. To wit, consider a $(d+1)$-dimensional static bulk geometry with line element

$$d{s}^2=-A\left(r\right)d{t}^2+A^{-1}\left(r\right)d{r}^2+r^{2}d{\Omega }_{d-1}^2,$$

(198)

where $r=r_h$ denotes the event horizon of the black hole, equal to the largest root of $A\left(r\right)=0$, and is generated by the time-translation Killing vector ${\partial }_t^a$. Let $\Phi =\Phi (r,{\varphi }^i)$ for $(d-1)$ Gaussian normal coordinates $\left\{{\varphi }^i\right\}$ denote the hypersurface equation of the brane $\mathcal{B}$. The induced metric on $\mathcal{B}$ is [125]

$$\begin{array}{cc}\hfill d{s}_{\mathcal{B}}^2& =-A\left(r\right)d{t}^2+\left(A^{-1}\left(r\right)+r^2{({\partial }_r\Phi )}^2\right)d{r}^2+r^2{\gamma }_{ij}d{\varphi }^{i}d{\varphi }^j\hfill \\ & \equiv -f\left(r\right)d{t}^2+g^{-1}\left(r\right)d{r}^2+r^2{\gamma }_{ij}d{\varphi }^{i}d{\varphi }^j,\hfill \end{array}$$

(199)

having identified

$$f\left(r\right)\equiv A\left(r\right),g\left(r\right)\equiv {\displaystyle \frac{A\left(r\right)}{1+A\left(r\right)r^2{({\partial }_r\Phi )}^2}}.$$

(200)

The bulk event horizon at $r=r_h$ corresponds to a horizon on $\mathcal{B}$, i.e., the bulk and brane blackening factors have the same root structure. Further, assuming ${\partial }_r\Phi$ is regular such that $A\left(r\right){({\partial }_r\Phi )}^2$ vanishes at $r=r_h$, the temperature of the bulk black hole coincides with the horizon induced on the brane:

$$T_{\mathrm{bulk}}^h={\displaystyle \frac{|A^{\prime }\left(r\right)|}{4\pi }}{|}_{r=r_h}={\displaystyle \frac{\sqrt{f^{\prime }\left(r\right)g^{\prime }\left(r\right)}}{4\pi }}{|}_{r=r_h}=T_{\mathcal{B}}^h.$$

(201)

Geometrically, the identification of the bulk and brane horizon temperatures is because the bulk time-translation Killing vector remains a time-translation Killing vector on the brane33.

Below, we review the thermodynamics for each type of quantum black hole explored in Section 4, starting with the static neutral quantum BTZ family of black holes.

5.3. Quantum BTZ Black Holes

5.3.1. Static Quantum BTZ

The thermodynamic variables of the static bulk black hole worked out above are

$$\begin{array}{cc}& M={\displaystyle \frac{\sqrt{1+{\nu }^2}}{2G_3}}{\displaystyle \frac{z^2(1-\nu z^3)(1+\nu z)}{{(1+3z^2+2\nu z^3)}^2}},\hfill \\ & T={\displaystyle \frac{1}{2\pi {\ell }_3}}{\displaystyle \frac{z(2+3\nu z+\nu z^3)}{1+3z^2+2\nu z^3}},\hfill \\ & S={\displaystyle \frac{8\pi {\ell }_3^2}{4G_4}}{\displaystyle \frac{\nu z}{1+3z^2+2\nu z^3}}.\hfill \end{array}$$

(202)

The claim is that these thermodynamic variables are to be interpreted as the thermodynamic quantities of the quantum black hole. Above, we already saw how the temperatures coincide. From the brane perspective, determining the mass is a highly non-trivial task due to the higher-derivative nature of the gravity action. Let us therefore first focus on the entropy S.

Entropy

From the bulk perspective, S is the classical four-dimensional Bekenstein–Hawking entropy, $S=S_{\mathrm{BH}}^{\left(4\right)}$ (189). Alternatively, from the brane point of view, S must be a sum of gravitational entropy and the von Neumann entropy of the holographic CFT${ }_3$. That is, S is identified as the generalized entropy [126]:

$$S_{\mathrm{BH}}^{\left(4\right)}\equiv S_{\mathrm{gen}}^{\left(3\right)}={\displaystyle \frac{4\pi {\ell }_3}{4G_3}}{\displaystyle \frac{z\sqrt{1+{\nu }^2}}{1+3z^2+2\nu z^3}},$$

(203)

where we replaced $G_4=2G_3{L}_4=2G_3\ell /\sqrt{1+{\nu }^2}$. Because we have the full bulk solution, $S_{\mathrm{gen}}^{\left(3\right)}$ is exact and valid for all $\nu$. To parse the gravitational and matter contributions to the entropy, however, it is useful to expand (203) in a small $\nu$ expansion,

$$S_{\mathrm{gen}}^{\left(3\right)}={\displaystyle \frac{4\pi {\ell }_{3}z}{4G_3(1+3z^2)}}-{\displaystyle \frac{8\pi {\ell }_3{z}^4}{4G_3{(1+3z^2)}^2}}\nu +{\displaystyle \frac{2\pi {\ell }_{3}z(1+6z^2+9z^4+8z^6)}{4G_3{(1+3z^2)}^3}}{\nu }^2+\mathcal{O}\left({\nu }^3\right).$$

(204)

The first term can be understood as the three-dimensional Bekenstein–Hawking entropy of the classical BTZ black hole34

$$S_{\mathrm{BTZ}}={\displaystyle \frac{4\pi {\ell }_{3}z}{4G_3(1+3z^2)}}={\displaystyle \frac{{\pi }^2{\ell }_3^2}{G_3}}{T}_{\mathrm{BTZ}}=\pi {\ell }_3\sqrt{{\displaystyle \frac{2M_{\mathrm{BTZ}}}{G_3}}}.$$

(205)

where $T_{\mathrm{BTZ}}\equiv {lim}_{\nu \to 0}T$ and similarly for $M_{\mathrm{BTZ}}$. The second term in (204) is proportional to $\ell \sim c_3$. Since higher-derivative contributions to the entropy enter at order ${\nu }^2$, the $\mathcal{O}\left(\nu \right)$ term can only correspond to the von Neumann entropy of the CFT${ }_3$ in the limit of weak backreaction. The $\mathcal{O}\left({\nu }^2\right)$ and higher-order terms are in principle a combination of the matter and higher-derivative effects that are difficult to distinguish.

Since the brane theory is in general a higher-derivative theory, in principle, the entire gravitational entropy can be computed using the Iyer–Wald entropy functional [127,128],

$$S_{\mathrm{IW}}\equiv -2\pi {\int }_H{d}^{D-2}x\sqrt{q}{\displaystyle \frac{\partial \mathcal{L}}{\partial R^{ijkl}}}{ϵ}_{ij}{ϵ}_{kl}.$$

(206)

Here, $q_{ij}$ is the induced metric of the codimension-2 cross-section H of the horizon with binormal ${ϵ}_{ij}$, and $\mathcal{L}$ is the scalar Lagrangian scalar characterizing the gravity theory, e.g., for the Einstein–Hilbert Lagrangian, $S_{\mathrm{IW}}={\displaystyle \frac{A\left[H\right]}{4G_D}}=S_{\mathrm{BH}}$. In the case of the semi-classical induced theory on the brane (57), the Iyer–Wald entropy is [25]35

$$S_{\mathrm{IW}}^{\left(3\right)}={\displaystyle \frac{1}{4G_3}}\int dx\sqrt{q}\left[1+{\ell }^2\left({\displaystyle \frac{3}{4}}R-g_{\perp }^{ij}{R}_{ij}\right)+\mathcal{O}({\ell }^4/{\ell }_3^6)\right],$$

(207)

with $g_{ij}^{\perp }=g_{ij}-q_{ij}$ being the metric in the directions orthogonal to the horizon36. The leading contribution is the three-dimensional Bekenstein–Hawking entropy,

$$S_{\mathrm{BH}}^{\left(3\right)}={\displaystyle \frac{1}{4G_3}}{\int }_{H}dx\sqrt{q}={\displaystyle \frac{1}{4G_3}}{\int }_0^{2\pi }d\overline{\varphi }{\overline{r}}_{+}={\displaystyle \frac{2\pi {\overline{r}}_{+}}{4G_3}}.$$

(208)

Substituting in parameters (185) and (186), notice

$$S_{\mathrm{BH}}^{\left(3\right)}={\displaystyle \frac{(1+\nu z)}{\sqrt{1+{\nu }^2}}}{S}_{\mathrm{gen}}^{\left(3\right)}=S_{\mathrm{BTZ}}+{\displaystyle \frac{\pi {\ell }_3{z}^2(1+z^2)}{G_3{(1+3z^2)}^2}}\nu +\mathcal{O}\left({\nu }^2\right).$$

(209)

Due to its dependence on $\nu$, $S_{\mathrm{BH}}^{\left(3\right)}$ contains semi-classical backreaction effects; only when $\nu =0$ does (208) coincide with the classical entropy in (204). Notice the difference

$$S_{\mathrm{BH}}^{\left(3\right)}-S_{\mathrm{BTZ}}={\displaystyle \frac{\nu z(1+z^2)}{1+3z^2}}{S}_{\mathrm{BTZ}},$$

(210)

It is natural to interpret this difference as the leading contribution to the CFT entropy.

Evaluating the Iyer–Wald entropy (207) on the quantum BTZ background (64) yields

$$S_{\mathrm{IW}}^{\left(3\right)}=\left[1-{\displaystyle \frac{{\nu }^2}{2}}-{\displaystyle \frac{z(1+z^2)}{1+\nu z}}{\nu }^3+\mathcal{O}\left({\nu }^4\right)\right]S_{\mathrm{BH}}^{\left(3\right)}.$$

(211)

Thus, the higher-derivative contributions to the gravitational entropy enter at order $\mathcal{O}\left({\nu }^2\right)$.

With the Iyer–Wald and generalized entropies, the CFT${ }_3$ entropy can be determined. This is because the generalized entropy associated with a black hole horizon is, generally,

$$S_{\mathrm{gen}}=S_{\mathrm{IW}}+S_{\mathrm{vN}}^{\mathrm{mat}},$$

(212)

where $S_{\mathrm{vN}}^{\mathrm{mat}}\equiv -\mathrm{tr}\rho log\rho$ is the von Neumann entropy of state $\rho$ of quantum fields living on the classical background confined to one side of the horizon. Typically, the matter entropy is UV divergent due to vacuum entanglement just across the horizon. The leading order divergence is of the form $A\left[H\right]/{ϵ}^{D-2}$ for UV regulator $ϵ$, while there will also be subleading divergences in $ϵ$. The Bekenstein–Hawking contribution to $S_{\mathrm{IW}}$ (with renormalized Newton’s constant) regularizes the area divergence of the matter entropy, while the subleading divergences are regulated via the higher-derivative contributions to $S_{\mathrm{IW}}$. Thence, the generalized entropy is UV finite and independent of the UV cutoff [130,131,132]. Therefore, the matter entanglement entropy can be formally computed by taking the difference of the generalized and gravitational entropies. Explicitly to leading order in $\nu$37

$$S_{\mathrm{CFT}}^{\left(3\right)}\equiv S_{\mathrm{gen}}^{\left(3\right)}-S_{\mathrm{IW}}^{\left(3\right)}\approx -{\displaystyle \frac{4\pi {\ell }_3{z}^2\nu }{4G_3(1+3z^2)}}=-\nu z{S}_{\mathrm{BTZ}}.$$

(213)

Note that the minus here does not imply the entanglement entropy is negative. Rather, here, $S_{\mathrm{CFT}}^{\left(3\right)}$ corresponds to the finite contribution to the entanglement entropy after the leading piece has been absorbed in a renormalization of $G_3$ (which differs from the renormalization of $G_3$ due to higher-derivatives effects on the mass).

Observe when $z\ll 1$, the matter entropy (213) takes the form

$$S_{\mathrm{CFT}}^{\left(3\right)}{|}_{z\ll 1}\approx -{\displaystyle \frac{\pi {\ell }_3\nu z^2}{G_3}}=-2\pi c_3{\left(\pi {\ell }_{3}T\right)}^2,$$

(214)

where we used $T_{\mathrm{BTZ}}{|}_{z\ll 1}\approx z/\pi {\ell }_3$ and $\ell \approx 2c_3{G}_3$. The proportionality to $T^2$ is consistent with the behavior of a $2+1$-dimensional conformal gas, implying the entropy is thermal. Meanwhile, for large-z but $\nu z\ll 1$,

$$S_{\mathrm{CFT}}^{\left(3\right)}{|}_{z\gg 1}\approx -{\displaystyle \frac{\pi {\ell }_3\nu }{3G_3}}=-{\displaystyle \frac{2\pi c_3}{3}},$$

(215)

a non-thermal entropy. Comparing the two limits of the matter entropy, we can infer $z\ll 1$ characterizes states where thermal effects dominate, while $z\gg 1$ describes states dominated by non-thermal effects. In fact, as we will see momentarily, the large and small-z limits, respectively, coincide with quantum states having large and small Casimir effects.

The First Law of Thermodynamics

Having identified the bulk horizon entropy with the generalized entropy on the brane (203), the bulk first law (191) from the brane perspective reads

$$dM=Td{S}_{\mathrm{gen}}^{\left(3\right)}.$$

(216)

This observation is consistent with two-dimensional quantum black holes (another context where the backreaction problem can be exactly solved) [133,134]. Thus, accounting for a semi-classical backreaction, the standard first law of horizon thermodynamics is modified by replacing ‘classical’ entropy with $S_{\mathrm{gen}}$. It is worth emphasizing this first law is exact and valid for all $\nu$, i.e., for small or large backreaction.

From the brane perspective, since the theory includes an infinite tower of higher-derivative contributions, an unambiguous definition of the mass is lacking. Therefore, the first law (216) provides another route to determining the mass. That is, we demand that the quantum BTZ black hole satisfies the semi-classical first via which the right-hand side defines the mass M (202). Notice that in the large-z limit, the mass hits its minimum value

$$\underset{z\to \infty }{lim}M=-{\displaystyle \frac{1}{8{\mathcal{G}}_3}}.$$

(217)

This mass can be thought of as the negative Casimir energy of the cutoff CFT${ }_3$, thus explaining the non-thermality of the matter entropy (215). Alternatively, for small $\nu$ and z, the Casimir effects are suppressed, leading to a thermal matter entropy (214).

Recall that the qBTZ solution parametrizes a family of quantum black holes with three branches (76)38. As depicted in Figure 11, all quantum black holes have higher temperatures than classical BTZ with the same mass. Incidentally, the branch 1a quantum dressed conical singularities ($-1/\left(8{\mathcal{G}}_3\right)<M<0$) have negative heat capacity, $\partial M/\partial T<0$, except at $M=0$, where the heat capacity diverges. In this range, where Casimir effects are dominant, the entropy S is most naturally understood to be entanglement entropy due to vacuum fluctuations across the horizon. Meanwhile, the largest black holes in branch 1b also have negative heat capacity, $M_{\mathrm{c}}<M<1/24{\mathcal{G}}_3$, where $M_{\mathrm{c}}\ne 0$ is some critical value of the mass where the heat capacity diverges. Further, the branch 1b black holes have a larger entropy than branch 2, which is likely due to the fact that the branch 1b black holes are formed due to backreaction of the Casimir energy.

5.3.2. Rotating Quantum BTZ

Using the analysis of the static quantum BTZ as a guide, it is in principle straightforward to analyze the thermal properties of the rotating quantum BTZ black hole (91). The essential new feature here is that there is an outer and inner black hole horizon, $r_{+}$ and $r_{-}$, with $r_{-}<r_{+}$. Thus, there is a parameter z (184) for each horizon. Below, we report only the thermodynamics for the outer horizon as the analysis of the inner horizon follows mutatis mutandis. In addition to $z={\ell }_3/r_{+}{x}_1$, we also introduce the rotation parameter

$$\alpha \equiv \tilde{a}/\sqrt{-\kappa x_1},$$

(218)

where, recall, $\tilde{a}\equiv a{x}_1^2/{\ell }_3$.

Following the logic leading to the parameters (185)–(187), now

$$\begin{array}{cc}& x_1^2=-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{1-\nu z^3}{z^2[1+\nu z-{\alpha }^2(z-\nu )]}},\hfill \\ & r_{+}^2=-{\ell }_3^2\kappa {\displaystyle \frac{1+\nu z-{\alpha }^{2}z(z-\nu )}{1-\nu z^3}},\hfill \\ & \mu x_1=-\kappa {\displaystyle \frac{(1+z^2)(1+{\alpha }^2(1-z^2))}{1-\nu z^3}}.\hfill \end{array}$$

(219)

Substituting these into the expressions for mass (92), rotation (93), surface gravity (100), and angular velocity (97), the thermodynamic variables are [24,25]

$$\begin{array}{cc}& M={\displaystyle \frac{\sqrt{1+{\nu }^2}}{2G_3}}{\displaystyle \frac{(1-\nu z^3)[z^2(1+\nu z)+{\alpha }^2(1+4\nu z^3(1+{\alpha }^2)-(1+4{\alpha }^2)z^4)]}{{[1+3z^2+2\nu z^3-{\alpha }^2(1+4\nu z^3+3z^4)]}^2}},\hfill \\ & T={\displaystyle \frac{1}{2\pi {\ell }_3}}{\displaystyle \frac{[z^2(1+\nu z)-{\alpha }^2(1-2\nu z^3+z^4)][2+3\nu z(1+{\alpha }^2)-4{\alpha }^2{z}^2+\nu z^3+{\alpha }^2\nu z^5]}{z(1+\nu z)[1+{\alpha }^2(1-z^2)][1+3z^2+2\nu z^3-{\alpha }^2(1-4\nu z^3+3z^4)]}},\hfill \\ & S={\displaystyle \frac{\pi {\ell }_3\sqrt{1+{\nu }^2}}{G_3}}{\displaystyle \frac{z(1+{\alpha }^2(1-z^2))}{[1+3z^2+2\nu z^3-{\alpha }^2(1+4\nu z^3+3z^4)]}},\hfill \\ & J={\displaystyle \frac{{\ell }_3\sqrt{1+{\nu }^2}}{G_3}}{\displaystyle \frac{\alpha z(1+z^2)[1+{\alpha }^2)(1-z^2)]\sqrt{(1-\nu z^3)[1+\nu z-{\alpha }^{2}z(z-\nu )]}}{{[1+3z^2+2\nu z^3-{\alpha }^2(1+4\nu z^3+3z^4)]}^2}},\hfill \\ & \Omega ={\displaystyle \frac{\alpha (1+z^2)}{{\ell }_3}}{\displaystyle \frac{\sqrt{(1-\nu z^3)[1+\nu z-{\alpha }^{2}z(z-\nu )]}}{z(1+\nu z)[1+{\alpha }^2(1-z^2)]}}.\hfill \end{array}$$

(220)

As with the static case, these variables serve as the thermal quantities of the bulk AdS${ }_4$ black hole and are identified to be the thermodynamic variables of the quantum black hole. In particular, the Bekenstein–Hawking entropy of the bulk black hole is

$$\begin{array}{cc}\hfill S_{\mathrm{BH}}^{\left(4\right)}& ={\displaystyle \frac{2}{4G_4}}{\int }_0^{2\pi \eta }d\varphi {\int }_0^{x_1}dx{\displaystyle \frac{r_{+}^2{\ell }^2}{{(\ell +r_{+}x)}^2}}\eta \left(1+{\displaystyle \frac{a^2{x}_1^2}{r_{+}^2}}\right)\hfill \\ & ={\displaystyle \frac{\pi }{G_4}}\eta {\displaystyle \frac{\ell x_1(r_{+}^2+a^2{x}_1^2)}{\ell +r_{+}{x}_1}},\hfill \end{array}$$

(221)

and is identified to be the generalized entropy of the black hole on the brane $S_{\mathrm{gen}}^{\left(3\right)}$. Notice that in the limit of vanishing backreaction, the entropy reduces to the classical BTZ entropy

$$\underset{\nu \to 0}{lim}{S}_{\mathrm{gen}}^{\left(3\right)}={\displaystyle \frac{\pi {\ell }_3}{\sqrt{2G_3}}}\left(\sqrt{M+{\displaystyle \frac{J}{{\ell }_3}}}+\sqrt{M-{\displaystyle \frac{J}{{\ell }_3}}}\right).$$

(222)

The Wald entropy is formally no different than before, and the matter entropy (213) obeys the same relation at the leading order in $\nu$.

It can be explicitly verified that the thermodynamic variables (220) satisfy (for fixed $\nu$)

$$\begin{array}{cc}& {\partial }_{z}M-T{\partial }_{z}S-\Omega {\partial }_{z}J=0,\hfill \\ & {\partial }_{\alpha }M-T{\partial }_{\alpha }S-\Omega {\partial }_{\alpha }J=0.\hfill \end{array}$$

(223)

Consequently, from the brane perspective, the semi-classical first law is

$$dM=Td{S}_{\mathrm{gen}}^{\left(3\right)}+\Omega dJ.$$

(224)

Again, n.b., mass M and angular momentum J, from the point of view of the brane, include the infinite tower of higher-derivative contributions in the gravity action (encoded in ${\mathcal{G}}_3$), which was exactly resummed due to our knowledge of the bulk theory.

While the first law holds for any range of parameters, not all ranges are physically sensible. Thus, restrictions are made when exploring the thermodynamics of the rotating solution [25]. In particular, for non-extremal solutions, one takes

$$0\le {\alpha }^2\le {\displaystyle \frac{1+\nu z}{z(z-\nu )}}.$$

(225)

The lower bound is chosen to avoid naked closed timelike curves ($\kappa =+1$ negative mass quantum-dressed cones belonging to branch 1a have ${\alpha }^2<0$ (218)). The upper bound follows from demanding the outer black hole event horizon $r_{+}$ be real and positive. For $\kappa =-1$ and $\nu <z<{\nu }^{-1/3}$, the upper bound implies $1+{\alpha }^2(1-z^2)>0$. Even with this bound, however, the temperature T of the black hole can go negative without further restricting the range of parameters. The classical BTZ extremal limit (108) occurs when

$${\alpha }^2={\alpha }_{\mathrm{ext}}^2={\displaystyle \frac{z^2(1+\nu z)}{1-2\nu z^3+z^4}},$$

(226)

where temperature $T=0$.

As noted in Section 4.2.2, the rotating quantum black holes can exist beyond the classical extremality bound (226). This is because the temperature T and angular momentum J are in general non-monotonic with respect to $\alpha$. There are thus two distinct types of rotating black holes: (i) those which respect the classical extremality bound, $0\le \alpha \le {\alpha }_{\mathrm{ext}}$, where T and J are monotonic in $\alpha$, and (ii) the ‘superextremal’, i.e., $\alpha >{\alpha }_{\mathrm{ext}}$, where T and J are non-monotonic. In Figure 12, we illustrate these distinct families, where the blue region corresponds to black holes respecting the extremality bound, while the green and red regions correspond to ‘superextremal’ solutions. The superextremal black hole solutions are possible due to the combined non-linear effects of the rotation and quantum backreaction, and they are consequently dubbed non-perturbative rotating black holes [135]. Among these solutions (those belonging to the red region) are black holes which are both thermodynamically unstable (having a negative heat capacity) and violate the so-called quantum reverse isoperimetric inequality (see Equation (277)).

5.3.3. Charged Quantum BTZ

As with the rotating case, the charged quantum BTZ black hole (123) has an outer and inner horizon. In addition to the parameter $z={\ell }_3/r_{+}{x}_1$, it is useful to introduce

$$\gamma \equiv q{x}_1^2,$$

(227)

along with ${\gamma }_e\equiv e{x}_1^2$ and ${\gamma }_g\equiv g{x}_1^2$, obeying ${\gamma }^2={\gamma }_e^2+{\gamma }_g^2$. Consequently, following the logic yielding parameters (185)–(187) gives [99]39

$$\begin{array}{cc}& x_1^2=-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{(1-\nu z^3+{\gamma }^2\nu z^3+{\gamma }^2{\nu }^2{z}^4)}{z^2(1+\nu z)}},\hfill \\ & r_{+}^2=-{\displaystyle \frac{\kappa {\ell }_3^2(1+\nu z)}{(1-\nu z^3+{\gamma }^2\nu z^3+{\gamma }^2{\nu }^2{z}^4)}},\hfill \\ & \mu x_1=-{\displaystyle \frac{\kappa (1+z^2-{\gamma }^2{z}^2+{\gamma }^2{\nu }^2{z}^4)}{(1-\nu z^3+{\gamma }^2\nu z^3+{\gamma }^2{\nu }^2{z}^4)}}.\hfill \end{array}$$

(228)

In terms of these parameters, the thermodynamic quantities are [98]

$$\begin{array}{cc}& M={\displaystyle \frac{\sqrt{1+{\nu }^2}}{2G_3}}{\displaystyle \frac{z^2(1+\nu z)[1-\nu z^3+{\gamma }^2\nu z^3(1+\nu z)]}{{[1+3z^2+2\nu z^3+{\gamma }^2{z}^2{(1+\nu z)}^2]}^2}},\hfill \\ & T={\displaystyle \frac{z}{2\pi {\ell }_3}}{\displaystyle \frac{2+3\nu z-\nu z^3({\gamma }^2{(1+\nu z)}^2-1)}{[1+3z^2+2\nu z^3+{\gamma }^2{z}^2{(1+\nu z)}^2]}},\hfill \\ & S={\displaystyle \frac{\pi {\ell }_3\sqrt{1+{\nu }^2}}{G_3}}{\displaystyle \frac{z}{[1+3z^2+2\nu z^3+{\gamma }^2{z}^2{(1+\nu z)}^2]}},\hfill \\ & Q_e=\sqrt{{\displaystyle \frac{16\pi }{5g_3^2{G}_3}}}{\displaystyle \frac{{\gamma }_e{z}^2(1+\nu z)\sqrt{1+{\nu }^2}}{[1+3z^2+2\nu z^3+{\gamma }^2{z}^2{(1+\nu z)}^2]}},\hfill \\ & {\mu }_e=\sqrt{{\displaystyle \frac{5g_3^2}{4\pi G_3}}}{\displaystyle \frac{{\gamma }_e\nu z^3(1+\nu z)}{[1+3z^2+2\nu z^3+{\gamma }^2{z}^2{(1+\nu z)}^2]}}.\hfill \end{array}$$

(229)

More specifically, parameters (228) are substituted into mass (124), electric charge (131) and potential (133) (the form of the magnetic charge $Q_g$ and potential ${\mu }_g$ are of the same form but with ${\gamma }_e\to {\gamma }_g$). Further, the temperature follows from the surface gravity, $T={\displaystyle \frac{{\kappa }_{+}}{2\pi }}={\displaystyle \frac{|H^{\prime }\left({\overline{r}}_{+}\right)|}{4\pi }}$, while the entropy is

$$\begin{array}{cc}\hfill S& =S_{\mathrm{BH}}^{\left(4\right)}={\displaystyle \frac{2}{4G_4}}{\int }_0^{2\pi \eta }d\varphi {\int }_0^{x_1}dx{\displaystyle \frac{{\ell }^2{r}_{+}^2}{{(\ell +x{r}_{+})}^2}}={\displaystyle \frac{4\pi \ell r_{+}^2{x}_1\eta }{G_4(\ell +r_{+}{x}_1)}},\hfill \end{array}$$

(230)

and is identified as the three-dimensional generalized entropy $S_{\mathrm{gen}}^{\left(3\right)}$. It can be easily verified that the thermodynamic quantities (229) satisfy

$$\begin{array}{cc}& {\partial }_{z}M-T{\partial }_{z}S-{\mu }_e{\partial }_z{Q}_e-{\mu }_g{\partial }_z{Q}_g=0,\hfill \\ & {\partial }_{\nu }M-T{\partial }_{\nu }S-{\mu }_e{\partial }_{\nu }{Q}_e-{\mu }_g{\partial }_{\nu }{Q}_g=0,\hfill \\ & {\partial }_{{\gamma }_i}M-T{\partial }_{{\gamma }_i}S-{\mu }_e{\partial }_{{\gamma }_i}{Q}_e-{\mu }_g{\partial }_{{\gamma }_i}{Q}_g=0,\hfill \end{array}$$

(231)

where ${\gamma }_i={\gamma }_e,{\gamma }_g$. Thus, from the brane perspective, the first law of thermodynamics is

$$dM=Td{S}_{\mathrm{gen}}^{\left(3\right)}+{\mu }_{e}d{Q}_e+{\mu }_{g}d{Q}_g.$$

(232)

Notice that in the limit $\nu \to 0$, the chemical potentials ${\mu }_e$ and ${\mu }_g$ vanish (as does the charge, which is readily apparent from (131)). This is indicative of the fact that the charge of the brane black hole is a quantum effect, and the charged quantum BTZ black hole does not reduce to the classical charged quantum BTZ black hole. Further, in the limit of vanishing backreaction $\nu \to 0$, the classical relation (205) between $S_{\mathrm{BTZ}},T_{\mathrm{BTZ}}$ and $M_{\mathrm{BTZ}}$ continues to hold. In fact, the generalized entropy has a qualitatively similar behavior as the entropy of the neutral, static qBTZ (Figure 11). The only essential difference between entropies of the charged and neutral black holes is that, rather than reaching zero, the charged black hole entropy ends at a finite, non-zero value—the entropy of the extremal black hole [98]. Meanwhile, in stark contrast with the neutral qBTZ, with $q\ne 0$, as the mass decreases monotonically below zero, the temperature reaches a finite maximum value and then quickly tends to zero at the extremal limit (shown in Figure 13). For a more complete treatment of the thermodynamics of the extremal black hole, see [98].

5.4. Quantum dS Black Holes

Geometrically, the chief difference between the quantum BTZ and dS${ }_3$ black holes is that the latter are equipped with a cosmological horizon. As with classical de Sitter black holes, the black hole horizon is generally hotter than the cosmological horizon. According to a static patch observer, then, the system is characterized by two generally unequal temperatures and thus not in thermal equilibrium. This complicates and enriches the thermodynamic analysis of quantum de Sitter black holes.

5.4.1. Quantum Schwarzschild–de Sitter

Along with $\nu \equiv \ell /R_3$, it is useful to introduce the dimensionless parameter [26]

$$z\equiv {\displaystyle \frac{R_3}{r_i{x}_1}},$$

(233)

where $r_i$ refers to either the black hole or cosmological horizon, $r_h$ and $r_c$, respectively. We can now express $x_1$, $\mu$, and $r_{+}$ solely in terms of $\nu$ and z

$$\begin{array}{cc}& x_1^2={\displaystyle \frac{1}{z^2}}{\displaystyle \frac{1+\nu z^3}{1+\nu z}},\hfill \\ & r_{+}^2=R_3^2{\displaystyle \frac{1+\nu z}{1+\nu z^3}},\hfill \\ & \mu x_1={\displaystyle \frac{z^2-1}{1+\nu z^3}}.\hfill \end{array}$$

(234)

These are the Wick rotated counterparts of the qBTZ solution parameters (185)–(187). Further, $G_4=2{\ell }_4{G}_3=2G_3\ell /\sqrt{1-{\nu }^2}$. In terms of parameters (234), the mass (149) is

$$M={\displaystyle \frac{1}{8G_3}}\sqrt{1-{\nu }^2}{\displaystyle \frac{(z^2-1)(9z^2-1+8\nu z^3)}{{(3z^2-1+2\nu z^3)}^2}}.$$

(235)

In the quantum de Sitter limit, where $z=1$, it follows $M=0$. Also note the mass M vanishes at large z, ${lim}_{z\to \infty }M\approx {\displaystyle \frac{1}{4{\mathcal{G}}_3\nu z}}+\mathcal{O}(1/z^2)$. For fixed $x_1\ne 1$ and for $z=z_h$, it is natural to think of the large $z_h$ limit as a small quantum Schwarzschild black hole, where $R_3\gg r_h{x}_1$.

Temperature

On the brane, the black hole and cosmological horizons will appear to emit radiation at the Hawking and Gibbons–Hawking temperatures $T_h,T_c$, respectively. Explicitly,

$$T_h=-{\displaystyle \frac{z_h}{2\pi R_3}}{\displaystyle \frac{2+3\nu z_h-\nu z_h^3}{3z_h^2-1+2\nu z_h^3}},T_c={\displaystyle \frac{z_c}{2\pi R_3}}{\displaystyle \frac{2+3\nu z_c-\nu z_c^3}{3z_c^2-1+2\nu z_c^3}}.$$

(236)

In the limit $\nu \to 0$, the black hole temperature vanishes, since $z_h$ diverges, while the cosmological horizon temperature reduces to

$$\underset{\nu \to 0}{lim}{T}_c={\displaystyle \frac{1}{\pi R_3}}{\displaystyle \frac{z_c}{3z_c^2-1}}\equiv T_{{\mathrm{SdS}}_3}.$$

(237)

Since in general $r_h<r_c$, and hence $z_c<z_h$, the black hole horizon has a higher temperature than the cosmological horizon, $T_h>T_c$. Thus, as usual for Schwarzschild–de Sitter spacetimes, the black hole and cosmological horizons are not in thermal equilibrium. Only in the Nariai limit do the temperatures of the two horizons coincide. There is more on this below.

Entropy and the First Law

The four-dimensional Bekenstein–Hawking entropy of the bulk black hole is

$$\begin{array}{cc}\hfill S_{\mathrm{BH}}^{\left(4\right)}& ={\displaystyle \frac{\mathrm{Area}\left(r_{+}\right)}{4G_4}}={\displaystyle \frac{2}{4G_4}}{\int }_0^{2\pi \eta }d\varphi {\int }_0^{x_1}dx{r}_{+}^2{\displaystyle \frac{{\ell }^2}{{(\ell +x{r}_{+})}^2}}\hfill \\ & ={\displaystyle \frac{\pi R_3}{G_3}}{\displaystyle \frac{z\sqrt{1-{\nu }^2}}{3z^2-1+2\nu z^3}}.\hfill \end{array}$$

(238)

As with the AdS${ }_3$ quantum black holes, from the brane perspective, the bulk entropy is interpreted as the three-dimensional generalized entropy $S_{\mathrm{gen}}^{\left(3\right)}$. In the limit $z=1$, the generalized entropy of the quantum de Sitter solution is proportional to the Gibbons–Hawking entropy of the ${\mathrm{dS}}_3$ cosmological horizon, i.e., the sum of gravitational entropy and entanglement entropy due to the CFT living outside of the cosmological horizon. Further, the generalized entropy is related to the three-dimensional Bekenstein–Hawking entropy $S_{\mathrm{BH}}^{\left(3\right)}$ of the horizon(s) on the brane via

$$S_{\mathrm{gen}}^{\left(3\right)}={\displaystyle \frac{\sqrt{1-{\nu }^2}}{1+\nu z}}{S}_{\mathrm{BH}}^{\left(3\right)},$$

(239)

where $S_{\mathrm{BH}}^{\left(3\right)}={\displaystyle \frac{2\pi r_{+}\eta }{4G_3}}$. In the limit of vanishing backreaction, the three-dimensional entropies coincide and are equal to the cosmological horizon entropy of classical Schwarzschild–dS${ }_3$ [136]

$$\underset{\nu \to 0}{lim}{S}_{\mathrm{gen}}^{\left(3\right)}={\displaystyle \frac{\pi R_3}{G_3}}{\displaystyle \frac{z_c}{3z_c^2-1}}={\displaystyle \frac{{\pi }^2{R}_3^2}{G_3}}{T}_{{\mathrm{SdS}}_3}={\displaystyle \frac{\pi R_3}{2G_3}}\sqrt{1-8G_{3}M}=S_{{\mathrm{SdS}}_3}.$$

(240)

In a perturbative series expansion of small $\nu$, the linear-order $\mathcal{O}\left(\nu \right)$ contribution to the generalized entropy (238) captures the CFT matter entropy, while quadratic and higher-order contributions include the effects of the higher-derivative corrections. Formally, the matter entropy is given by the difference of the generalized entropy and Iyer–Wald entropy, which, to leading order and for large-z, has the same non-thermal result as static quantum BTZ (215).

Putting together the mass (235), temperatures (236) and entropy (238), there is a separate first law for the black hole and cosmological horizons,

$$dM=T_{h}d{S}_{\mathrm{gen},h}^{\left(3\right)},$$

(241)

and

$$dM=-T_{c}d{S}_{\mathrm{gen},c}^{\left(3\right)}.$$

(242)

In combination, notice

$$0=T_{h}d{S}_{\mathrm{gen},h}^{\left(3\right)}+T_{c}d{S}_{\mathrm{gen},c}^{\left(3\right)}.$$

(243)

Thus, as the generalized entropy attributed to the black hole increases, the generalized entropy of the cosmological horizon decreases. This is a consequence of the minus sign appearing in the first law for the cosmological horizon (242), i.e., the entropy of the cosmological horizon decreases as the mass increases. Akin to classical de Sitter space, this suggests quantum ${\mathrm{dS}}_3$ represents a maximum entropy state with a finite number of degrees of freedom. Hence, quantum de Sitter black holes behave as instantons constraining the states of the original de Sitter degrees of freedom (see, e.g., [137,138,139]).

Nariai Limit

As noted above, in general, a static patch observer sees the quantum SdS${ }_3$ black hole as one with two unequal temperatures. A special limit where the temperatures coincide is the (quantum) Nariai solution (150), i.e., the largest mass black hole able to fit inside the cosmological horizon, where $r_h=r_c=r_{\mathrm{N}}$. Recall that in this limit, $(\mu \ell )$ attains a maximum

$${\mu }_{\mathrm{N}}={\displaystyle \frac{2}{3\sqrt{3}}}{\displaystyle \frac{1}{\nu }}.$$

(244)

such that

$$z_{\mathrm{N}}={\displaystyle \frac{R_3}{r_{\mathrm{N}}{x}_1^{\mathrm{N}}}}={\displaystyle \frac{\sqrt{3}}{x_1^{\mathrm{N}}}}.$$

(245)

In this case, $x_1^{\mathrm{N}}$ is the particular value of $x_1$ in the Nariai limit found by solving ${\mu }_{\mathrm{N}}=(1-x_1^2)/x_1^3$ for $x_1$. For arbitrary $\nu$, there will generally be one real solution for $x_1$ and one real solution, which for small $\nu$ takes the form

$$x_1^{\mathrm{N}}=\sqrt{3}{\left({\displaystyle \frac{\nu }{2}}\right)}^{1/3}-{\displaystyle \frac{\sqrt{3}}{2}}\nu +\mathcal{O}\left({\nu }^{5/3}\right),$$

(246)

vanishing in the limit $\nu \to 0$ (as one would expect). In terms of $x_1^{\mathrm{N}}$, the mass of the Nariai solution $M_{\mathrm{N}}$ is therefore defined by $M_{\mathrm{N}}{\equiv M|}_{z=z_{\mathrm{N}}}$. Similarly, the entropy of the Nariai solution is defined as $S_{\mathrm{N}}^{\left(3\right)}\equiv S_{\mathrm{gen}}^{\left(3\right)}{|}_{z=z_{\mathrm{N}}}$. See Figure 14 for a plot of the horizon entropies as a function of mass M normalized with respect to the Nariai mass.

Regarding the temperature, the Nariai limit is subtle. Naively, setting $r_c=r_h$ leads to the horizon temperatures (236) vanishing. This is a consequence of working with surface gravities ${\kappa }_h$ and ${\kappa }_c$ defined with respect to the time-translation Killing vector $\xi ={\partial }_{\overline{t}}$:

$$\begin{array}{cc}& {\kappa }_h={\displaystyle \frac{1}{2}}{H}^{\prime }\left({\overline{r}}_h\right)={\displaystyle \frac{1}{2{\overline{r}}_h{r}_{\mathrm{N}}^2}}({\overline{r}}_{\mathrm{N}}^2-{\overline{r}}_h^2),{\kappa }_c=-{\displaystyle \frac{1}{2}}{H}^{\prime }\left({\overline{r}}_c\right)=-{\displaystyle \frac{1}{2{\overline{r}}_c{r}_{\mathrm{N}}^2}}({\overline{r}}_{\mathrm{N}}^2-{\overline{r}}_c^2),\hfill \end{array}$$

(247)

which clearly vanish when ${\overline{r}}_{h,c}\to {\overline{r}}_{\mathrm{N}}={\displaystyle \frac{R_3}{\sqrt{3}}}\eta$. However, for Schwarzschild–de Sitter, there is a more natural choice of normalization of the time-translation Killing vector, owed to Bousso and Hawking [140], resulting in a non-vanishing Nariai temperature. Technically, the Bousso–Hawking normalization is chosen such that ${\xi }^2=-1$ at the radius ${\overline{r}}_0$ where the blackening factor $H\left(\overline{r}\right)$ obtains a maximum,

$$H^{\prime }\left({\overline{r}}_0\right)=0⟹{\overline{r}}_0^3={\displaystyle \frac{{\overline{r}}_i}{2}}(3{\overline{r}}_{\mathrm{N}}^2-{\overline{r}}_i^2).$$

(248)

for a positive real root ${\overline{r}}_i$ of $H\left(\overline{r}\right)$. Physically, the radius ${\overline{r}}_0$ corresponds to the location where an observer can stay in place without accelerating, i.e., the position where the acceleration due to the gravitational attraction from the black hole balances out the cosmological acceleration. This is consistent with the asymptotically flat Schwarzschild black hole where ${\overline{r}}_0\to \infty$ (since $R_3\to \infty$), and it is empty de Sitter where ${\overline{r}}_0\to 0$ (via ${\overline{r}}_i\to \eta R_3$).

In terms of radius ${\overline{r}}_0$, where

$$\begin{array}{cc}\hfill H\left({\overline{r}}_0\right)& ={\displaystyle \frac{1}{r_{\mathrm{N}}^2}}({\overline{r}}_{\mathrm{N}}^2-{\overline{r}}_0^2).\hfill \end{array}$$

(249)

the analogous Bousso–Hawking temperature is

$$\overline{T}={\displaystyle \frac{T}{\sqrt{H\left({\overline{r}}_0\right)}}}.$$

(250)

In terms of the horizon radii ${\overline{r}}_{h,c}$,

$$\begin{array}{cc}& {\overline{T}}_{h,c}=\mp {\displaystyle \frac{1}{4\pi r_{\mathrm{N}}{\overline{r}}_{h,c}}}{\displaystyle \frac{{\overline{r}}_{\mathrm{N}}^2-{\overline{r}}_h^2}{\sqrt{{\overline{r}}_{\mathrm{N}}^2-{\left(\frac{{\overline{r}}_{h,c}}{2}(3{\overline{r}}_{\mathrm{N}}^2-{\overline{r}}_{h,c}^2)\right)}^{2/3}}}},\hfill \end{array}$$

(251)

where the minus sign corresponds to the black hole temperature, and the plus sign corresponds to the cosmological horizon temperature. Carefully taking the limit ${\overline{r}}_{\mathrm{N}}\approx {\overline{r}}_{h,c}$, the temperature ${\overline{T}}_{h,c}$ of both horizons approaches the Nariai temperature $T_{\mathrm{N}}=1/\left(2\pi r_{\mathrm{N}}\right)$ (see, e.g., Appendix B of [134]). The expression for $\overline{T}$ in terms of z and $\nu$ is cumbersome to leading order, for small $\nu$ is equal to the dS${ }_3$ Gibbons–Hawking temperature [26]. Figure 14 displays the temperatures $T_c,T_h$, and $\overline{T}$ as a function of mass M normalized by the Nariai mass. The behavior is essentially identical to classical four-dimensional Schwarzschild–de Sitter black holes (see, e.g., Figure 2 of [137]), reflecting the holographic character of the induced geometry.

5.4.2. Quantum Kerr–de Sitter

The new feature of the quantum Kerr–de Sitter black hole (152) compared to the quantum SdS is that now there is an outer and inner black hole horizon obeying $r_{-}<r_{+}<r_c$. In addition to z (233), we introduce the rotation parameter

$$\alpha \equiv {\displaystyle \frac{a{x}_1}{R_3}}={\displaystyle \frac{\tilde{a}}{x_1}},$$

(252)

where $\left\{r_i\right\}=\{r_{\pm },r_c\}$. We can express $x_1$, $\mu$ and $r_i$ solely in terms $z,\nu ,$ and $\alpha$:

$$\begin{array}{cc}& x_1^2={\displaystyle \frac{1+\nu z^3}{z^2[1+\nu z+{\alpha }^{2}z(z+\nu )]}},\hfill \\ & r_i^2=R_3^2{\displaystyle \frac{1+\nu z+{\alpha }^{2}z(z+\nu )}{1+\nu z^3}},\hfill \\ & \mu x_1={\displaystyle \frac{(z^2-1)(1+{\alpha }^2(1+z^2))}{1+\nu z^3}}.\hfill \end{array}$$

(253)

Using the parameters (253), the thermodynamic variables are [27]

$$\begin{array}{cc}& M={\displaystyle \frac{1}{8G_3}}\sqrt{1-{\nu }^2}{\displaystyle \frac{(z^2-1)[1+{\alpha }^2(1+z^2)][9z^2-1+8\nu z^3+{\alpha }^2(9z^4-1+8\nu z^3)]}{{(3z^2-1+2\nu z^3+{\alpha }^2(1+4\nu z^3+3z^4))}^2}},\hfill \\ & J={\displaystyle \frac{\alpha R_3}{G_3}}\sqrt{1-{\nu }^2}{\displaystyle \frac{z(z^2-1)[1+{\alpha }^2(1+z^2)]\sqrt{(1+\nu z^3)(1+\nu z+{\alpha }^{2}z(z+\nu ))}}{{(3z^2-1+2\nu z^3+{\alpha }^2(1+4\nu z^3+3z^4))}^2}},\hfill \\ & {\Omega }_i={\displaystyle \frac{\alpha }{R_3}}{\displaystyle \frac{(z^2-1)\sqrt{(1+\nu z^3)(1+\nu z+{\alpha }^{2}z(z+\nu ))}}{z(1+\nu z)(1+{\alpha }^2(1+z^2))}},\hfill \\ & T_i={\displaystyle \frac{1}{2\pi R_3}}{\displaystyle \frac{(z^2(1+\nu z)+{\alpha }^2(1+2\nu z^3+z^4))\left|(2+3\nu z-\nu z^3+{\alpha }^2(4z^2+\nu z(z^4+3)))\right|}{z(1+\nu z)(1+{\alpha }^2(1+z^2))(3z^2-1+2\nu z^3+{\alpha }^2(1+3z^4+4\nu z^3))}},\hfill \\ & S={\displaystyle \frac{\pi R_3}{G_3}}{\displaystyle \frac{\sqrt{1-{\nu }^2}z(1+{\alpha }^2(1+z^2))}{(3z^2-1+2\nu z^3+{\alpha }^2(1+3z^4+4\nu z^3))}},\hfill \end{array}$$

(254)

which follow from substituting the parameters (253) into mass M (153), angular momentum J (154), rotation ${\Omega }_i$ (158), surface gravity (160), and the Bekenstein–Hawking area formula40. Collectively, the variables obey the first law

$$dM=T_{i}d{S}_i+{\Omega }_{i}dJ,$$

(255)

for all values of the parameters. On the brane, S has the usual interpretation as the generalized entropy $S_{\mathrm{gen}}^{\left(3\right)}$.

The quantum Kerr–dS${ }_3$ black hole has limits where two or more horizons become degenerate. These include the extremal ($r_{+}=r_{-}$) black hole, where $T_{\mathrm{ext}}=0$, the lukewarm black hole ($T_c=T_{+}$), and the Nariai geometry ($r_c=r_{+}$). As with quantum SdS${ }_3$, the Nariai black hole will have a vanishing temperature using the standard normalization of the Killing vector. However, in its near horizon geometry, the temperature of the Nariai black hole and cosmological horizons are equal to the same non-zero temperature $T_{\mathrm{N}}$.

Lastly, two limiting cases include the quantum de Sitter, at $z=1$ or $\mu =0$, leading to

$$M=J={\Omega }_i=0,$$

(256)

$$S={\displaystyle \frac{2\pi R_3}{4G_3}}{\displaystyle \frac{\sqrt{1-{\nu }^2}}{1+\nu }},T_c={\displaystyle \frac{1}{2\pi R_3}},$$

(257)

and the limit of vanishing backreaction, where

$$\begin{array}{cc}& M={\displaystyle \frac{1}{8G_3}}{\displaystyle \frac{(z^2-1)(1+{\alpha }^2(1+z^2))(9z^2-1+{\alpha }^2(9z^4-1))}{{(3z^2-1+{\alpha }^2(1+3z^4))}^2}},\hfill \\ & J={\displaystyle \frac{\alpha R_3}{G_3}}{\displaystyle \frac{z(z^2-1)(1+{\alpha }^2(1+z^2))\sqrt{1+{\alpha }^2{z}^2}}{{(3z^2-1+{\alpha }^2(1+3z^4))}^2}},\hfill \\ & {\Omega }_c={\displaystyle \frac{\alpha (z^2-1)\sqrt{1+{\alpha }^2{z}^2}}{R_{3}z(1+{\alpha }^2(1+z^2))}},\hfill \\ & T_c={\displaystyle \frac{1}{2\pi R_3}}{\displaystyle \frac{2(1+2{\alpha }^2{z}^2)(z^2+{\alpha }^2(1+z^4))}{z(3z^2-1+{\alpha }^2(1+3z^4))(1+{\alpha }^2(1+z^2))}},\hfill \\ & S_c={\displaystyle \frac{\pi R_3}{G_3}}{\displaystyle \frac{z(1+{\alpha }^2(1+z^2))}{(3z^2-1+{\alpha }^2(1+3z^4))}},\hfill \end{array}$$

(258)

with $z=z_c$, since there are no black holes. It is straightforward to verify that the resulting thermodynamic variables (258) agree with classical Kerr–${\mathrm{dS}}_3$ [27].

5.5. Quantum Black Holes in Flat Space

As with the quantum de Sitter black holes, the thermodynamics of the asymptotically flat quantum black holes follow from the bulk AdS${ }_4$ black hole geometry with a Randall–Sundrum brane. Notably, however, in this case, the tension is tuned to its critical value (since $L_4=\ell$) such that no cosmological horizon appears. Therefore, only the thermodynamics of the black hole horizons in the bulk are imprinted on the brane.

5.5.1. Quantum Schwarzschild Black Hole

In a sense, the thermodynamics of the quantum Schwarzschild black hole (169) is given by the $R_3\to \infty$ limit of the thermodynamics of the quantum Schwarzschild–de Sitter. Some care must be taken, however, since in this limit, $z=R_3/r_{+}{x}_1$ diverges, while $\nu =\ell /R_3$ goes to zero. To be more illustrative, recall the mass (170). Rearranging $\mu x_1=(1-x_1^2)/x_1^2$ such that $x_1^2=1/(1+\mu x_1)$ yields

$$M={\displaystyle \frac{\ell \widehat{x}(8+9\widehat{x})}{16G_4{\left(1+\frac{3}{2}\widehat{x}\right)}^2}},$$

(259)

where

$$\widehat{x}\equiv \mu x_1.$$

(260)

It is easy to verify ${\mu }^2={\widehat{x}}^2(1+\widehat{x})$. Meanwhile, from the surface gravity (172), the temperature of the bulk horizon is

$$T={\displaystyle \frac{1}{4\pi \ell \widehat{x}}}{\displaystyle \frac{1}{\left(1+\frac{3}{2}\widehat{x}\right)}}.$$

(261)

Further, the four-dimensional Bekenstein–Hawking entropy is

$$\begin{array}{cc}\hfill S_{\mathrm{BH}}^{\left(4\right)}& ={\displaystyle \frac{4\pi \eta }{4G_4}}{\int }_0^{x_1}dx{r}_{+}^2{\displaystyle \frac{{\ell }^2}{{(\ell +x{r}_{+})}^2}}={\displaystyle \frac{4\pi }{4G_4}}{\displaystyle \frac{r_{+}^2\ell x_1}{\ell +r_{+}{x}_1}}\hfill \\ & ={\displaystyle \frac{4\pi {\ell }^2}{4G_4}}{\displaystyle \frac{{\widehat{x}}^2}{\left(1+\frac{3}{2}\widehat{x}\right)}},\hfill \end{array}$$

(262)

where to reach the final line, we used $r_{+}=\mu \ell$.

In summary, the thermodynamic variables of the bulk AdS${ }_4$ black hole are mass (259), temperature (261), and entropy (262), and they obey the first law41

$$dM=Td{S}_{\mathrm{BH}}^{\left(4\right)}.$$

(263)

From the brane perspective, these thermodynamic quantities are identified as the thermodynamic variables of the quantum Schwarzschild black hole, upon substituting $G_4=2\ell G_3$, and where $S_{\mathrm{BH}}^{\left(4\right)}=S_{\mathrm{gen}}^{\left(3\right)}$.

It is easy to verify the mass is a monotonically increasing function of $\widehat{x}$ with a minimum $M=0$ (at $\widehat{x}=r_{+}=0$) and a maximum at $M={\displaystyle \frac{1}{8G_3}}$, at $\widehat{x}\to \infty$. Further, for small $\widehat{x}$, the temperature goes like $T\sim 1/\widehat{x}$ and $T\sim 1/{\widehat{x}}^2$ for large $\widehat{x}$, while the entropy behaves as

$$\begin{array}{cc}& S_{\mathrm{gen}}^{\left(3\right)}\approx {\displaystyle \frac{\pi \ell {\widehat{x}}^2}{2G_3}}\left(\mathrm{small}\widehat{x}\right),\hfill \\ & S_{\mathrm{gen}}^{\left(3\right)}\approx {\displaystyle \frac{\pi \ell \widehat{x}}{3G_3}}\left(\mathrm{large}\widehat{x}\right).\hfill \end{array}$$

(264)

It is useful to think about these limits in terms of the bulk parameter $\mu$, where $\widehat{x}\approx \mu$ for small $\widehat{x}$ and $\widehat{x}\approx {\mu }^{2/3}$ for large $\widehat{x}$42. So, for small $\mu$, the entropy $S_{\mathrm{BH}}^{\left(4\right)}\approx {\displaystyle \frac{\pi {(\mu \ell )}^2}{G_4}}$, having the behavior of a classical four-dimensional Schwarzschild black hole of mass $2M{G}_4=\mu \ell$ and temperature $T={(4\pi \mu \ell )}^{-1}$. Alternatively, for $\mu \gg 1$, the black hole extends a distance ℓ off of the brane, looking like a ‘flattened pancake’ [23].

Lastly, notice that in the limit of large ℓ, the temperature (261) vanishes while the entropy (262) diverges, whereas for small ℓ, the temperature diverges and the entropy decreases. The latter is consistent with the fact that for a vanishing backreaction, there is no black hole.

5.5.2. Quantum Kerr Black Hole

The (outer) horizon thermodynamics of the quantum Kerr black hole (174) are readily worked out to be

$$\begin{array}{cc}& M={\displaystyle \frac{(r_{+}^2+a^2)x_1[8r_{+}\ell +9x_1(r_{+}^2+a^2)]}{8G_3{[3x_1(r_{+}^2+a^2)+2r_{+}\ell ]}^2}},\hfill \\ & T={\displaystyle \frac{\ell }{2\pi r_{+}}}{\displaystyle \frac{(r_{+}^2-a^2)(x_1(r_{+}^2+a^2)+r_{+}\ell )}{(r_{+}^2+a^2)x_1(r_{+}{x}_1+\ell )[3(r_{+}^2+a^2)x_1+2r_{+}\ell ]}},\hfill \\ & S_{\mathrm{gen}}^{\left(3\right)}={\displaystyle \frac{\pi r_{+}(r_{+}^2+a^2)x_1^2}{G_3[3x_1(r_{+}^2+a^2)+2r_{+}\ell ]}}\hfill \\ & J={\displaystyle \frac{a{r}_{+}(r_{+}^2+a^2)x_1^2}{G_3{[3x_1(r_{+}^2+a^2)+2r_{+}\ell ]}^2}}\sqrt{{\displaystyle \frac{\ell [x_1(r_{+}^2+a^2)+r_{+}\ell ]}{r_{+}}}},\hfill \\ & \Omega ={\displaystyle \frac{a}{x_1(r_{+}^2+a^2)(r_{+}{x}_1+\ell )}}\sqrt{{\displaystyle \frac{\ell [x_1(r_{+}^2+a^2)+r_{+}\ell ]}{r_{+}}}}.\hfill \end{array}$$

(265)

Alternatively, using $(r_{+}^2+a^2)=r_{+}\mu \ell$, $\widehat{x}\equiv \mu x_1$, and introducing $\widehat{a}\equiv a/(\mu \ell )$, the thermodynamic variables become

$$\begin{array}{cc}& M={\displaystyle \frac{\widehat{x}(8+9\widehat{x})}{32G_3{\left(1+\frac{3}{2}\widehat{x}\right)}^2}},\hfill \\ & T={\displaystyle \frac{(1+\widehat{x})}{4\pi \ell \widehat{x}\left(1+\frac{3}{2}\widehat{x}\right)}}{\displaystyle \frac{(1-4{\widehat{a}}^2+\sqrt{1-4{\widehat{a}}^2})}{\left[1+\frac{\widehat{x}}{2}(1+\sqrt{1-4{\widehat{a}}^2})\right]\left[1-2{\widehat{a}}^2+\sqrt{1-4{\widehat{a}}^2}\right]}},\hfill \\ & S_{\mathrm{gen}}^{\left(3\right)}={\displaystyle \frac{\pi \ell }{4G_3}}{\displaystyle \frac{{\widehat{x}}^2}{\left(1+\frac{3}{2}\widehat{x}\right)}}(1+\sqrt{1-4{\widehat{a}}^2}),\hfill \\ & J={\displaystyle \frac{\widehat{a}\ell {\widehat{x}}^2}{4G_3{\left(1+\frac{3}{2}\widehat{x}\right)}^2}}\sqrt{1+\widehat{x}},\hfill \\ & \Omega ={\displaystyle \frac{4\widehat{a}\sqrt{1+\widehat{x}}}{\widehat{x}\ell [\widehat{x}(1+\sqrt{1-4{\widehat{a}}^2})+2][1+\sqrt{1-4{\widehat{a}}^2}]}},\hfill \end{array}$$

(266)

and obey the first law

$$dM=Td{S}_{\mathrm{gen}}^{\left(3\right)}+\Omega dJ.$$

(267)

The quantum Schwarzschild black hole thermodynamics is recovered in the $\widehat{a}\to 0$ limit. As before, $T\sim 1/\widehat{x}$ for small $\widehat{x}$ and $T\sim 1/{\widehat{x}}^2$ for large $\widehat{x}$, while $S_{\mathrm{gen}}^{\left(3\right)}$ has the same $\widehat{x}$-dependence as in the non-rotating case (264).

5.6. Extended Black Hole Thermodynamics

While black holes have a thermodynamic description, they are peculiar in that the first law (182) lacks a pressure–volume work term. This is because for general black holes, there is no clear notion of pressure or volume. For black holes in spacetimes with a cosmological constant, the situation changes dramatically. A first glimpse of this comes from Euler’s theorem of homogeneous functions, which implies black holes with a non-zero cosmological constant ${\Lambda }_{d+1}$ obey [141]

$$(d-2)G_{d+1}M=(d-1)TS-2P_{d+1}V+...,$$

(268)

where T refers to temperature, S refers to Bekenstein–Hawking entropy, and the ellipsis refers to other possible thermodynamic variables, e.g., $\Omega dJ$. The essential new feature is used to identify the cosmological constant as thermodynamic pressure

$$P_{d+1}\equiv -{\displaystyle \frac{{\Lambda }_{d+1}}{8\pi G_{d+1}}}.$$

(269)

Then, V is the conjugate variable to the pressure, which is dubbed the ‘thermodynamic volume’ [141,142,143] formally equal to43

$$V\equiv {\left({\displaystyle \frac{\partial M}{\partial P}}\right)}_{S,...}.$$

(270)

For spacetimes with a vanishing cosmological constant, the relation (268) reduces to the Smarr formula; however, for ${\Lambda }_{d+1}\ne 0$, the $P-V$ term is required for consistency.

Going one step further, treating the cosmological constant as a dynamical variable leads to an extended framework of black hole thermodynamics, resulting in the first law of extended black hole thermodynamics

$$dM=TdS+Vd{P}_{d+1}+....$$

(271)

Glossing over the details, allowing for the cosmological constant to be a dynamical pressure means that the thermodynamics of anti-de Sitter black holes, in particular, acquire a richer structure than their asymptotically flat counterparts, behaving as Van der Waals fluids [145,146], polymers [147] and allowing for the construction of black hole heat engines [148]. Thus, the extended thermodynamics of AdS black holes offer a rich gravitational perspective on everyday phenomena (for a review, see [149]).

Extended black hole thermodynamics is not without its criticisms. A common critique is treating the cosmological constant as a variable pressure. Indeed, while the $P-V$ term in the Smarr Formula (268) is required for consistency, its appearance does not imply the pressure should be made variable, at least from a gravitational perspective. Assuming AdS/CFT duality, it is more natural to allow for varying the cosmological constant since variations in ${\Lambda }_{d+1}$ are dual to variations in the number of degrees of freedom of the dual theory [141,148,150,151,152,153,154,155].

Holographic braneworlds suggest a higher-dimensional origin for extended black hole thermodynamics [156]. In particular, a dynamical cosmological constant on the brane naturally follows from tuning the brane tension. In fact, keeping other bulk parameters $L_{d+1}$ and $G_{d+1}$ fixed, varying the tension alone corresponds to varying the induced brane cosmological constant on the brane ${\Lambda }_d$:

$$\delta \tau ={\displaystyle \frac{\delta {\Lambda }_d}{8\pi G_d}}.$$

(272)

Hence, classical black hole thermodynamics in the bulk including work completed by the brane induces extended thermodynamics of quantum black holes on the brane.

This observation can be made explicit in the context of the braneworld constructions considered here. Specifically, when treating the brane tension $\tau$ as a thermodynamic variable akin to the surface tension of liquids, the first law of the bulk black hole is

$$dM=TdS+A_{\tau }d\tau ,$$

(273)

where $A_{\tau }\equiv {\left({\displaystyle \frac{\partial M}{\partial \tau }}\right)}_S$ is the variable conjugate to $\tau$. Consequently, tension variation $d\tau$ induces extended thermodynamics on the brane. Particularly, the bulk first law (273) maps to the extended first law on the brane

$$dM=Td{S}_{\mathrm{gen}}+Vd{P}_d+...,$$

(274)

thus extending the quantum first law, where the pressure $P_d$ is the pressure of the quantum black hole and V is its conjugate thermodynamic volume.

In summary, since the scales of the brane theory are induced, holographic braneworlds provide a gravitational motivation for treating the cosmological constant as a variable. For example, the pressure and volume of the static quantum BTZ black hole are [156]

$$\begin{array}{cc}& P_3\equiv -{\displaystyle \frac{{\Lambda }_3}{8\pi G_3}}={\displaystyle \frac{\sqrt{1+{\nu }^2}}{4\pi {\nu }^2{\ell }_3^2{G}_3}}(\sqrt{1+{\nu }^2}-1),\hfill \\ & V\equiv {\left({\displaystyle \frac{\partial M}{\partial P_3}}\right)}_{S_{\mathrm{gen}},c_3}=-{\displaystyle \frac{2\pi {\ell }_3^2{z}^2[-2+{\nu }^2+3{\nu }^3{z}^3+{\nu }^4{z}^4+\nu z({\nu }^2-4)]}{{(1+3z^2+2\nu z^3)}^2}}.\hfill \end{array}$$

(275)

Together with the standard thermodynamic variables (202), it is straightforward to verify the extended first law (274) is obeyed. Extended thermodynamics for charged and rotating qBTZ black holes were computed in [135] (see also [99] for charged BTZ).

5.7. Quantum Reverse Isoperimetric Inequality

One of the more puzzling aspects of extended black hole thermodynamics is the thermodynamic volume V. In simple cases, e.g., $D\ge 4$ AdS-RN, volume V coincides with the geometric volume of the black hole, i.e., the amount of spacetime volume excluded by the black hole horizon. In general, however, the thermodynamic volume is not the geometric volume, cf. [142,143,144]44. Nonetheless, the thermodynamic volume plays a crucial role in understanding black hole thermodynamics. A sharp example of this is that there is strong evidence that AdS black holes obey the reverse isoperimetric inequality [143]

$$\mathcal{R}\equiv {\left({\displaystyle \frac{(D-1)V}{{\Omega }_{D-2}}}\right)}^{{\displaystyle \frac{1}{D-1}}}{\left({\displaystyle \frac{{\Omega }_{D-2}}{A_{\mathrm{BH}}}}\right)}^{{\displaystyle \frac{1}{D-2}}}\ge 1.$$

(276)

Here, ${\Omega }_{D-2}$ is the volume of a unit $(D-2)$ sphere, of D-dimensional AdS, and $A_{\mathrm{BH}}$ is the area of the black hole horizon. Notably, it is the thermodynamic volume for which this inequality holds, not the geometric volume. Refined generalizations of the inequality (276), inspired by the classical Penrose inequality [157,158], have been conjectured and tested for a plethora of examples [159].

Physically, the reverse isoperimetric inequality states an asymptotically AdS black hole with fixed thermodynamic volume has an entropy no larger than Schwarzschild–AdS of the same volume, i.e., Schwarzschild–AdS is a maximal entropy state at fixed (thermodynamic) volume. While there is no general proof of the inequality (276), there are very few known counterexamples45 and all such ‘superentropic’ black holes have a negative heat capacity at constant volume, $C_{\mathrm{V}}<0$, and are thus thermodynamically unstable [165]. Via AdS${ }_3$/CFT${ }_2$ duality, black hole superentropicity can be microscopically understood as an overcounting of the (naive) Cardy entropy of the CFT${ }_2$ [166].

When semi-classical quantum effects are accounted for, the classical inequality (276) is known to be violated [156]. A natural quantum generalization of (276) is proposed to be [135]

$${\mathcal{R}}_{\mathrm{Q}}\equiv {\left({\displaystyle \frac{(D-1)V_{\mathrm{th}}}{{\Omega }_{D-2}}}\right)}^{{\displaystyle \frac{1}{D-1}}}{\left({\displaystyle \frac{{\Omega }_{D-2}}{4{\mathcal{G}}_D{S}_{\mathrm{gen}}}}\right)}^{{\displaystyle \frac{1}{D-2}}}\ge 1.$$

(277)

Here, the classical area has been replaced by the generalized entropy and $V_{\mathrm{th}}$ is the Casimir-subtracted thermodynamic volume,

$$V_{\mathrm{th}}=V-V_{\mathrm{cas}},$$

(278)

where, in analogy with Casimir mass, $V_{\mathrm{cas}}$ is the thermodynamic volume assigned to empty AdS space. In the case of $D=3$, the quantum reverse isoperimetric inequality (277) has been shown to hold for all AdS${ }_3$ quantum black holes at all orders of backreaction—except for a subspace of rotating black holes (those belonging to the red region in Figure 12), which are all found to be thermodynamically unstable, in accordance with Johnson’s conjecture [165] regarding the classical inequality. This implies there exists a maximum entropy state among thermodynamically stable quantum black holes at fixed volume.

5.8. Phase Transitions of Quantum Black Holes

As with ordinary thermodynamic systems, black holes in AdS can undergo phase transitions. A paradigmatic example is the Hawking–Page (HP) phase transition [116]: below a certain temperature $T_{\mathrm{HP}}$, large AdS black holes in equilibrium with their radiation transition to thermal AdS. At the level of the quantum gravitational partition function, this transition signals an exchange between dominant contributions in the Euclidean path integral. The classical BTZ black hole also undergoes a HP phase transition [167,168]. To wit, the canonical free energy $F_{\mathrm{BTZ}}=-TlogZ$ of a static BTZ black hole is

$$F_{\mathrm{BTZ}}=M-TS=-{\displaystyle \frac{{\pi }^2{\ell }_3^2}{2G_3}}{T}^2,$$

(279)

for temperature $T=r_{+}/2\pi {\ell }_3^2$. Comparing with the free energy of thermal AdS, $F_{\mathrm{AdS}}=M_{\mathrm{AdS}}=-1/8G_3$, a first-order phase transition occurs at a temperature

$$T_{\mathrm{HP}}={\displaystyle \frac{1}{2\pi {\ell }_3}}.$$

(280)

When $T<T_{\mathrm{HP}}$, thermal AdS has a lower free energy than the black hole and is, therefore, the dominant contribution to the partition function; for $T>T_{\mathrm{HP}}$, the black hole has lower free energy and becomes the dominant contribution to the partition function.

Quantum AdS black holes also undergo thermal phase transitions [124,169]. In fact, large backreaction effects can trigger new transitions unseen by their classical counterparts. Consider, for example, static, neutral quantum BTZ. Working in the standard canonical ensemble, where c and P are held fixed46, the free energy is

$$\begin{array}{cc}\hfill F_{\mathrm{qBTZ}}& \equiv M-T{S}_{\mathrm{gen}}=-{\displaystyle \frac{z^2\sqrt{1+{\nu }^2}}{2G_3}}{\displaystyle \frac{[1+2\nu z+\nu z^3(2+\nu z)]}{{(1+3z^2+2\nu z^3)}^2}}.\hfill \end{array}$$

(281)

As with the classical scenario, compare to ‘quantum’ thermal AdS${ }_3$ (qTAdS), i.e., pure ${\mathrm{AdS}}_3$ including a backreaction due to the cutoff ${\mathrm{CFT}}_3$, with free energy

$$F_{\mathrm{TqAdS}}=M_{\mathrm{qTAdS}}=-{\displaystyle \frac{1}{8{\mathcal{G}}_3}}.$$

(282)

In Figure 15, we present a side-by-side comparison of the canonical free energy of the classical BTZ black hole and quantum BTZ for large backreaction, $\nu >1$. Focusing on the quantum BTZ black hole, as the temperature monotonically increases, there are reentrant phase transitions from thermal AdS to qBTZ and back to thermal AdS: (i) for temperatures up to a critical temperature (where $∆F\equiv F_{\mathrm{qBTZ}}-F_{\mathrm{qTAdS}}=0$), thermal AdS has lower free energy, until at the critical temperature, there is a discontinuity in the slope of the free energy, i.e., a first-order phase transition and a quantum analog of the Hawking–Page phase transition. (ii) After this temperature, the qBTZ black hole has a lower free energy until there is a jump discontinuity in the free energy, a zeroth-order phase transition, beyond which TAdS always has a lower free energy. The reentrant phase transition is the combination of the first- and zeroth-order phase transitions as the temperature monotonically varies, and it is exhibited by an (inverse) swallow tail.

As displayed in Figure 15, reentrant phase transitions do not occur for the classical BTZ black hole. Reentrant phase transitions, however, are known to appear in (classical) higher-derivative theories of gravity in higher dimensions and Born–Infeld gravity [172,173,174,175]. In all such cases, the transitions are between different phases of the black hole, e.g., large to small and back to large black holes. For the quantum black hole, the reentrant phase is a reentrant Hawking–Page phase transition, moving between thermal AdS and the same black hole. Again, the zeroth-order phase transitions only occur for a large enough backreaction. In this regime, the brane has decreasing tension, and the gravitational theory on the brane becomes more massive and effectively four-dimensional. Meanwhile, for a small backreaction, thermal AdS dominates at all temperatures until the $\nu =0$ limit where one recovers the phase behavior of classical BTZ.

Even though the phase structure admits a region where the black hole is thermodynamically favored, it may not contribute to the Euclidean path integral if it is thermally unstable. The heat capacity serves as a diagnostic to determine whether the black hole is stable against thermal fluctuations. For example, classically, the Hawking–Page transition is between thermal AdS and stable black holes (small AdS black holes are unstable). Likewise, in the case of quantum BTZ, a computation of the heat capacity [176] reveals reentrant phase transitions occur in the canonical ensemble between thermal AdS and a branch of thermally stable quantum black holes. More precisely, a study of the temperature of the qBTZ solution reveals three branches47 of continuously connected black holes: (A) ‘cold’ black hole, (B) ‘intermediate’ black hole, and (C) ‘hot’ black hole. Black holes belonging to branches A and C have a negative heat capacity and are thus unstable, while branch B black holes have a positive heat capacity and are thermally stable. The reentrant phase transitions occur only between thermal AdS and these intermediate black holes.

6. Braneworld Black Holes in Higher Dimensions

The majority of this review focused on exact constructions of three-dimensional quantum black holes. From the perspective of the non-holographic perturbative treatment, working in three spacetime dimensions, $d+1\ge 4$ was solely to simplify the problem. Indeed, the expectation value of the renormalized quantum stress tensor $\langle T_{ij}\rangle$ is not known in general. Exceptions do exist48. For example, for even-dimensional homogeneous and isotropic FRW cosmologies, the trace anomaly for conformal quantum fields is known [178,179]. In the case of static, spherically symmetric backgrounds, however, the renormalized stress tensor is only known up to an arbitrary function of an appropriate radial coordinate [5].

One possible way to circumvent this issue is to instead work in a truncated s-wave sector of the matter theory and use the two-dimensional conformal anomaly to fix the form of the quantum-stress tensor and find quantum corrections to, say, the Schwarzschild black hole [180,181,182,183]. Another approach is to solve the (semi-classical) Tolman–Oppenheimer–Volkoff equations for a massless conformally coupled scalar field [184]. Working perturbatively in ℏ, one can construct analytic asymptotically flat, static, spherically symmetric solutions, while numerics yields non-perturbative corrections in ℏ49. Thus, the situation for solving the backreaction problem in $d+1\ge 4$ is more complicated and subject to quantum gravitational corrections. This again motivates the use of holographic techniques as employed in the three-dimensional case. However, as we now review, the status of finding higher-dimensional quantum black holes via braneworlds is more complicated.

6.1. Higher-Dimensional Quantum Black Holes?

Given the success of studying three-dimensional quantum black holes starting from the AdS${ }_4$ C-metric, it is natural to wonder if a similar analysis follows from a higher-dimensional C-metric. Notably, however, no exact solutions to Einstein’s equations describing accelerating black holes in dimension $D=d+1>4$ are known50. As reasoned in [192], this is because in $D\ge 5$, the string generating the acceleration has a singularity along its symmetry axis passing through the black hole, which is distinctly less mild than in $D=4$. Put another way, in four dimensions, there exist more general accelerating black holes accelerated by Levi–Civita strings or rods. The C-metric is a special limit of such a geometry, which is singled out as a metric where the string has a milder singularity along its axis and better-behaved asymptotics than its brethren. In $D\ge 5$, it seems there is no such special limit for accelerating black holes. Alternately, one can perturbatively construct a C-metric in $D>4$ by perturbing a higher-dimensional Schwarzschild black hole to give it uniform acceleration [193]. However, it was found that such a solution with constant string tension does not allow for localized braneworld black holes (they do not satisfy the Israel junction conditions). Moreover, allowing for non-uniform string tension results in infinitely many localized braneworld black hole solutions.

Historically, the first attempt at finding a four-dimensional braneworld black hole embedded in a five-dimensional bulk was carried out by Chamblin, Hawking and Reall [194]. Their starting point takes the original Randall–Sundrum model and replaces the Minkowski line element on the brane with a four-dimensional Ricci flat metric, e.g., the Schwarzschild black hole,

$$d{s}_5^2=d{y}^2+a^2\left(y\right)g_{ij}^{\mathrm{Schw}}d{x}^{i}d{x}^j.$$

(283)

Here, y denotes the bulk extra direction, and $a^2\left(y\right)$ denotes the warp factor (for the RS-II scenario, $a^2\left(y\right)=e^{-2\left|y\right|/L_5}$ for bulk AdS${ }_5$ length scale $L_5$). The brane is located at $y=0$, such that from the brane perspective, the geometry is exactly the static, four-dimensional Schwarzschild black hole. Note, however, the five-dimensional Chamblin, Hawking, Reall ‘black string’ (283) suffers from a classical dynamical instability [195] analogous to the Gregory–Laflamme instability of Kaluza–Klein black strings [196,197]. We will return to this instability momentarily. Further, the black string extends to the AdS${ }_5$ horizon at $y=\infty$ and where the black hole horizon becomes singular with diverging scalar curvature invariants.

The Chamblin, Hawking, Reall braneworld black hole thus cannot describe the end state of gravitational collapse. In fact, whilst searching for examples of Oppenheimer–Synder gravitational collapse of braneworld black holes, a no-go theorem was posed [66]: the exterior geometry of the dust cloud cannot be static. This theorem, and the dearth of evidence of exact static black holes localizing in the RS-II construction (circa 2002), in part motivated Tanaka [34] and Emparan, Fabbri, and Kaloper [22] to conjecture that higher-dimensional braneworld black holes must be time-dependent (see also [198]). Their reasoning utilized AdS/CFT holography and was argued to be consistent with the proposal that black holes that localize on the brane may be interpreted as quantum black holes.

6.2. Predictions from Holography

Let us review the arguments [22,34] predicting higher-dimensional braneworld black holes must be time-dependent. Take the bulk to be classical AdS${ }_5$ general relativity with a four-dimensional RS brane. According to braneworld holography, the induced theory on the brane describes a higher-derivative theory of gravity coupled to a large-N gauge theory in the ’t Hooft planar limit with large ’t Hooft coupling $\lambda =N{g}_{\mathrm{YM}}^2$ (we refer to this matter theory as a large-c CFT${ }_4$ with an ultraviolet cutoff owed to the presence of the brane). In particular, the full AdS/CFT duality in this case is between type IIB string theory on ${\mathrm{AdS}}_5\times S^5$ and $\mathcal{N}=4$ super Yang–Mills $SU\left(N\right)$ gauge theory with AdS length $L_5=L_{\mathrm{P}}^{\left(10\right)}{\left(g_{s}N\right)}^{1/4}$ (for string coupling $g_s$ and ten-dimensional Planck length $L_{\mathrm{P}}^{\left(10\right)}$) and ’t Hooft coupling $\lambda =g_{s}N$. The effective number of CFT degrees of freedom is51

$$c\sim N^2={\left({\displaystyle \frac{L_5}{L_{\mathrm{P}}^{\left(5\right)}}}\right)}^3={\left({\displaystyle \frac{L_5}{L_{\mathrm{P}}^{\left(4\right)}}}\right)}^2,$$

(284)

such that large $N=L_5/L_{\mathrm{P}}^{\left(4\right)}$ coincides with the four-dimensional Planck length going to zero. Further, notice the combination

$$N^2\hslash ={\left({\displaystyle \frac{L_5}{L_{\mathrm{P}}^{\left(4\right)}}}\right)}^2{\displaystyle \frac{{\left(L_{\mathrm{P}}^{\left(4\right)}\right)}^2}{G_4}}={\displaystyle \frac{L_5^2}{G_4}},$$

(285)

remains fixed as $\hslash \to 0$ and keeping $L_5,G_4$ fixed.

Now, according to the conjecture [22], the braneworld black hole must be a solution to the semi-classical theory sourced by the CFT stress tensor $\langle T_{ij}\rangle$ in some quantum state. For a black hole background, in principle, the quantum state could be in any of the common choices of the vacuum state, i.e., the Hartle–Hawking, Boulware, or Unruh state. Recall that the Hartle–Hawking state describes a black hole in a thermal bath in equilibrium with its own radiation, producing a non-dynamical configuration. On the other hand, the Unruh vacuum describes a time-dependent evaporating black hole (there is a thermal flux of radiation at future null infinity). To infer whether a black hole in the Hartle–Hawking state is possible—without finding an explicit solution—one can instead estimate the evaporation time of a radiating black hole; if the time for evaporation is finite, then a static black hole solution is not possible. To this end, consider a weakly coupled CFT in four dimensions. In particular, the trace anomaly for $\mathcal{N}=4$$SU\left(N\right)$ super Yang–Mills theory to leading order in ℏ is

$$\langle T_i^i\rangle ={\displaystyle \frac{\hslash (N^2-1)}{32{\pi }^2}}\left(R_{ij}^2-{\displaystyle \frac{1}{3}}{R}^2\right)\approx {\displaystyle \frac{\hslash N^2}{32{\pi }^2}}\left(R_{ij}^2-{\displaystyle \frac{1}{3}}{R}^2\right),$$

(286)

taking $N\gg 1$. Thus, the anomaly for such a CFT is simply the free field result enhanced by a $\mathcal{O}\left(N^2\right)$ factor. With this in mind, the power emitted by Hawking quanta modeled by the large-N CFT will be

$${\displaystyle \frac{dM}{dt}}\sim {\displaystyle \frac{N^2\hslash }{R_0^2}},$$

(287)

for initial horizon radius $R_0=2\sim G_{4}M$. Then, the time for evaporation $t_{\mathrm{evap}}$ heuristically is

$$t_{\mathrm{evap}}^{-1}\equiv {\displaystyle \frac{1}{R_0}}{\displaystyle \frac{d{R}_0}{dt}}={\displaystyle \frac{2G_4}{R_0}}{\displaystyle \frac{dM}{dt}}\sim {\displaystyle \frac{2}{R_0^3}}\hslash G_4{N}^2={\displaystyle \frac{2L_5^2}{R_0^3}},$$

(288)

where we substituted in (285). Thus, the evaporation time is finite (even as $\hslash \to 0$). Moreover, such a black hole would evaporate rapidly due to the $\mathcal{O}\left(N^2\right)$ enhancement of the free theory result. Altogether, this suggests black holes cannot remain static on the brane: they shrink and evaporate in a finite time.

Assuming the bulk/brane correspondence holds, the semi-classical evaporation of the black hole localized on the brane should have a classical bulk signature. One possibility, originally put forth in [34] (see also [199]), is that the semi-classical evaporation is linked to a classical instability of the bulk five-dimensional solution. The intuition is as follows. Consider the five-dimensional black Chamblin, Hawking, Reall black string (283), and have a Randall–Sundrum brane intersect it. The radius of the black hole will exponentially shrink with the AdS${ }_5$ length $L_5$ as one moves away from the brane. Further, the black string suffers from a Gregory–Laflamme instability when the horizon radius becomes smaller than $L_5$. As a result of the instability, a portion of the horizon localized on the brane will pinch off and fall into a region of the bulk black hole away from the brane. Thus, while the horizon area of the bulk black hole does not shrink, the area of the horizon localized on the brane shrinks. This five-dimensional deformation due to dynamical instability implies a type of classical evaporation of braneworld black holes.

Another view is that semi-classical radiation corresponds to gravitational radiation of the bulk black hole. Indeed, from the brane perspective, the Hawking quanta are modeled by the large-c CFT, corresponding to bulk Kaluza–Klein gravitons. Further, a black hole stuck to a brane is accelerating away from the center of AdS, thus producing gravitational waves. Thus, Hawking radiation from the brane perspective corresponds to bulk gravitational bremsstrahlung [22]. Qualitatively, moreover, the classical gravitational waves emitted into the bulk have a characteristic frequency $\omega$ which from the brane perspective is estimated to be $\omega \sim {\left(G_{4}M\right)}^{-1}$, which is the four-dimensional Hawking temperature. This reasoning suggests why the classical bulk gravitational radiation appears as thermal radiation on the brane.

6.3. Counterexamples

The essential problem of the holographic argument leading to the finite evaporation time (288) is that the result assumes the large-N CFT is weakly coupled. Of course, for the holographic construction of the bulk/brane set-up to be valid, the cutoff CFT on the brane is strongly coupled, i.e., large ’t Hooft coupling. Thus, as first recognized by Fitzpatrick, Randall, and Wiseman [69], it is not clear the radiation has access to all of its $\mathcal{O}\left(N^2\right)$ degrees of freedom. Importantly, being at strong coupling can lead to a reductionin the accessible degrees of freedom, from $\mathcal{O}\left(N^2\right)$ to $\mathcal{O}\left(1\right)$ via confinement (see also [200])52. Such a reduction thus makes it plausible that static braneworld black holes in higher dimensions do exist.

In fact, there are several examples of higher-dimensional static braneworld black holes, both analytically and numerically constructed [67,68,69,70,71,72,201,202,203,204,205,206,207,208]. The question is whether these static solutions are to be viewed as quantum black holes on the brane. To explore this point, reconsider the 5D/4D Randall–Sundrum construction with the five-dimensional Schwarzschild black string (283). Despite its dynamical instability, this is an example of a static braneworld black hole, where the black hole on the brane is simply the four-dimensional Schwarzschild geometry. Seemingly then, the cutoff CFT on the brane does not modify the geometry: from the looks of it, the brane geometry consists of a non-trivial zero mode and no excited Kaluza–Klein modes [69]. Nonetheless, Fitzpatrick, Randall, and Wiseman argued the dual quantum black hole description might be consistent with the existence of such static localized black hole solutions. It is simply that the quantum corrections to the geometry are suppressed53.

To appreciate this last point, consider the set-up of [202], consisting of an asymptotically AdS${ }_5$ bulk spacetime, with two positive tension Karch–Randall branes with an AdS${ }_5$ Schwarzschild black string stretching between the two branes (see Figure 16). In particular, the bulk geometry has line element

$$d{s}_5^2={\displaystyle \frac{L_5^2}{{cos}^2\left(u\right)}}\left[d{u}^2+{\displaystyle \frac{1}{{\ell }_4^2}}{g}_{ij}d{x}^{i}d{x}^j\right],$$

(289)

with bulk AdS${ }_5$ radius $L_5$, and induced four-dimensional radius ${\ell }_4=L_5\sec \left(u_0\right)$. The AdS${ }_5$ boundary is located at $u=\pm \pi /2$, while the two AdS${ }_4$ branes are at $u=\pm u_0$, each with the same positive tension $\tau ={\displaystyle \frac{6sin\left(u_0\right)}{8\pi G_5{L}_5}}$, and separated a finite distance apart. Further, take the four-dimensional metric $g_{ij}$ to be the AdS${ }_4$–Schwarzschild black hole,

$$g_{ij}d{x}^{i}d{x}^j=-f\left(r\right)d{t}^2+f^{-1}\left(r\right)d{r}^2+r^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2),f=1+{\displaystyle \frac{r^2}{{\ell }_4^2}}-{\displaystyle \frac{2G_{4}M}{r}}.$$

(290)

Without branes, this system has an instability when $r_{+}/{\ell }_4<1$ (small black holes of horizon radius $r_{+}$), while it is stable for large black holes, $r_{+}/{\ell }_4>1$ [210]. The stability structure remains the same with branes included [202].

The classical bulk is stable for large mass black holes that localize on the branes (focus on only a single brane as a being living on one brane would not directly experience the other brane). From the brane perspective, the induced geometry is simply the AdS${ }_4$–Schwarzschild black hole. Seemingly, the geometry has no quantum corrections due to the backreacting holographic field theory living on the brane. This runs counter to what would be expected based on a weakly coupled analysis of the quantum-stress tensor54. Indeed, recall the trace anomaly of weakly coupled $\mathcal{N}=4$ SU(N) super Yang–Mills theory (286). Substituting in the AdS${ }_4$–Schwarzschild geometry (290), the anomaly reads

$$\langle T_i^i\rangle =-{\displaystyle \frac{3\hslash (N^2-1)}{8{\pi }^2{\ell }_4^4}}.$$

(291)

Since $\langle T_i^i\rangle$ is non-vanishing, one expects the AdS${ }_4$–Schwarzschild black hole to become quantum-corrected due to backreaction. In view of the conjecture of holographic braneworld black holes, the question thus becomes in what sense can the AdS${ }_4$–Schwarzschild black hole on the brane be viewed as a ‘quantum’ black hole.

The resolution of this apparent tension with the conjecture is the following. The quantum stress tensor of the strongly coupled CFT (as required for a consistent holographic description of the bulk gravity) simply does not correct the geometry, at least at leading order. To see this, consider the boundary CFT stress tensor, which can be computed in full using holographic renormalization methods, leading to [202]

$$\langle T_{ij}\rangle =-{\displaystyle \frac{3\hslash N^2}{32{\pi }^2{\ell }_4^4}}{g}_{ij}.$$

(292)

Taking the trace returns the anomaly (291). The main takeaway is that the stress tensor is proportional to the metric. In such an instance, the CFT only renormalizes the effective brane cosmological constant and thus does not lead to any quantum-corrected geometry. Notice that in the limit, the brane cosmological constant vanishes, ${\ell }_4\to \infty$, and so too does the quantum stress tensor, $\langle T_{ij}\rangle \to 0$. This is consistent with the Schwarzschild black string with a Randall–Sundrum brane construction in [69].

A conclusion, then, is that formally, the AdS${ }_4$–Schwarzschild braneworld black hole can be interpreted as a quantum black hole. The cutoff CFT, however, is special in that it does not explicitly backreact on the brane geometry, and it only serves to renormalize the induced cosmological constant. Geometrically, this is a manifestation of the bulk spacetime (289) being foliated by the AdS${ }_4$–Schwarzschild black hole. Consequently, the classical Kaluza–Klein graviton modes are not excited in the background solution. Observe, moreover, the correction to the cosmological constant is independent of the black hole mass and hence its temperature. The brane black hole thus does not radiate, which is consistent with no excited Kaluza–Klein modes. This is at first puzzling from the perspective of AdS/CFT, as one would have expected a component of the holographic stress tensor to correspond to a thermal plasma of CFT degrees of freedom outside the black hole. As noted in [200], this is a consequence of the large-N super Yang–Mills theory confining to $\mathcal{O}\left(1\right)$ degrees of freedom.

It is worth emphasizing the implications of the static solution found from the AdS${ }_5$ black string [202]. There is no Hawking emission, and up to the term generating the Weyl anomaly, the quantum stress tensor is everywhere vanishing. Despite the lack of thermal radiation near infinity, the static state of the CFT is in thermal equilibrium with the black hole since they nonetheless obey the Kubo–Martin–Schwinger (KMS) condition. This suggests, from the bulk perspective, such braneworld black holes remain stuck to the brane, while from the brane perspective, the black hole evaporation is disallowed due to the large-N and large ’t Hooft coupling of the holographic CFT.

6.4. Evaporating Braneworld Black Holes

Putting the status of the higher-dimensional static quantum black holes aside, braneworld holography has proven useful in studying black hole evaporation. Indeed, while some bulk black holes can get stuck on the brane and never shrink (‘black droplets’) [71] and hence are dynamically stable, there are other bulk black solutions that evaporate due to a Gregory–Laflamme instability [72], as originally suggested in [22,34]. The essential physical insight is that evaporation takes place when the brane black hole is coupled to an appropriate thermal bath modeled by thermalized $\mathcal{O}\left(N^2\right)$ degrees of freedom of the CFT. From the bulk perspective, this means the horizon of the black hole on the brane is to be connected to another horizon in the bulk. The bulk system will have a Gregory–Laflamme-like instability such that the black hole attached to the brane slides off the brane into the bulk, forming a (typically) larger black hole horizon. Thus, while the horizon on the brane reduces its size (evaporation), the horizon in the bulk does not shrink, which is consistent with the classical evolution of black hole horizons.

There are in fact several examples of evaporating braneworld black holes (see [72] for details). In all cases, the bulk system considers D-dimensional black holes in the large-D limit [212,213,214] that localize on either one or a pair of Karch–Randall branes55. In particular, these holes include the following: (i) A small AdS black string is stretched between two branes with black holes localizing on both branes. The string instability triggers either the evaporation of both black holes into a single bulk black hole or the evaporation of one brane black hole into the other brane black hole. (ii) A small AdS black hole on the brane is connected to a larger bulk black hole (bath) such that the brane black hole evaporates entirely into the bulk. (iii) A large unstable AdS black droplet is connected to a non-gravitating boundary (bath) via a thin black funnel, such that the droplet evaporates into the boundary.

In short, the exhaustive large-D analysis reveals holographic CFTs coupled to black holes have distinctive traits over their weakly interacting counterparts. According to [72,215], their semi-classical dynamics are always dual to classical Gregory–Laflamme-like dynamics of bulk black holes.

7. Outlook and Applications

We have seen how holographic braneworlds provide a means to exactly study the problem of semi-classical backreaction. In particular, there is a precise construction of quantum-corrected black holes and an exploration of their horizon thermodynamics. But this is only a small cross-section of the utility of braneworld holography. Phenomenological considerations aside—a historical motivation to study braneworld physics—holographic braneworlds have a number of applications, particularly in the interdisciplinary field of holographic information theory. Below, we very briefly give a non-exhaustive and biased list of applications of quantum black holes and, more broadly, holographic braneworlds.

7.1. Applications

7.1.1. Holographic Entanglement and Gravitational Entropy

For a quantum mechanical system in state $\rho$, the von Neumann entropy is defined as

$$S_{\mathrm{vN}}\equiv -\mathrm{tr}[\rho log\rho ].$$

(293)

If the system is in a pure quantum state, the entropy (293) is identically zero. When the system comprises smaller subsystems, $S_{\mathrm{vN}}$ quantifies how entangled the subsystems are. In quantum field theory, subsystems are often chosen to be spatial regions $\Sigma$ of the entire quantum system. Intriguingly, for ground states of local Hamiltonians, the entanglement entropy, while divergent in the ultraviolet, generally adheres to an area law (in $d+1$-spacetime dimensions)

$$S_{\mathrm{vN}}=c_0{\displaystyle \frac{\mathrm{Area}[\partial \Sigma ]}{{ϵ}^{d-1}}}+...,$$

(294)

where $c_0$ refers to the constant, $ϵ$ refers to the UV regulator, and the ellipsis refers to subleading UV divergences whose precise terms are state-dependent. Thus, $S_{\mathrm{vN}}$ scales with the boundary area of the subsystem rather than its volume, which is in contrast to the volume law typically observed in thermal states. The entropy–area relation (294) is clearly reminiscent of the Bekenstein–Hawking entropy formula for black holes, suggesting black hole entropy arises from vacuum entanglement due to quantum fields across the horizon, tracing out the interior degrees of freedom [131,216,217,218,219,220]. In particular, in scenarios of gravity induced due to matter loops [221,222], the Bekenstein–Hawking entropy is solely due to entanglement due to vacuum fluctuations [223,224,225], where the UV-cutoff dependence can be absorbed in a renormalization of Newton’s constant [130].

Gravitational entropy, therefore, is imbued with an information–theoretic character. This is crystallized in AdS/CFT, where the entanglement entropy of a holographic CFT can be computed using the Ryu–Takayanagi (RT) formula [226,227]

$$S_A^{\mathrm{vN}}={\displaystyle \frac{\mathrm{Area}\left(\gamma \right)}{4G}}.$$

(295)

Here, $\gamma$ denotes a codimension-2 minimal surface in the bulk asymptotically AdS space, anchored to region A on the AdS boundary, such that $\partial \gamma =\partial A$ and $\gamma$ is homologous to A. The RT prescription generalizes the Bekenstein–Hawking entropy formula for black holes: a black hole horizon is an example of a minimal surface, and the bulk AdS spacetime need not contain a black hole56. As such, it has led to numerous insights into gravitational physics, notably in understanding the emergence of gravity from quantum entanglement [233,234,235,236,237,238]57 following analogous insights from coarse-grained thermodynamics [240,241,242,243,244,245].

When the bulk theory includes higher curvature and semi-classical quantum corrections, the RT Formula (295) generalizes to the following extremization prescription [246]

$$S_A^{\mathrm{vN}}=\underset{\gamma }{\min }\left\{\underset{\gamma }{\mathrm{ext}}\left[{\displaystyle \frac{\mathrm{Area}\left(\gamma \right)}{4G}}+{\mathcal{S}}_{\mathrm{DC}}\left(\gamma \right)+S_A^{\mathrm{bulk}}\left({\Sigma }_{\gamma }\right)\right]\right\}.$$

(296)

Here, ${\mathcal{S}}_{\mathrm{DC}}$ denotes the Dong–Camps anomaly terms [247,248] analogous to the Wald corrections to the Bekenstein–Hawking entropy, and $S_A^{\mathrm{bulk}}$ is the entanglement entropy of bulk fields across the entanglement wedge of A, ${\Sigma }_{\gamma }$, such that the quantity in brackets is recognized to be the generalized entropy $S_{\mathrm{gen}}\left(\gamma \right)$. Further, in the event there are multiple extremal surfaces, the prescription says the von Neumann entropy $S_A^{\mathrm{vN}}$ is given by choosing the extremal surface which minimizes the generalized entropy $S_{\mathrm{gen}}\left(\gamma \right)$. Notably, the von Neumann entropy can undergo phase transitions as extremal surface configurations exchange dominance.

Holographic braneworlds provide additional evidence that gravitational entropy can be interpreted as entanglement entropy. Indeed, the gravity on the brane is induced from the cutoff CFT58. In combination with the RT prescription, it readily follows that the area entropy of black holes localized on an ETW brane is exactly equal to entanglement entropy [55]—the minimal surface coincides with the bulk black hole horizon intersecting the brane. Additionally, the braneworld set-up naturally resolves various subtleties in identifying Bekenstein–Hawking and entanglement entropies [251]. In particular, the UV cutoff $ϵ$ for the CFT—equal to the bulk AdS length scale $L_{d+1}$— fixes the number of species of the dual CFT such that Newton’s constant is correctly reproduced (cf. Equation (203)), thus resolving the ‘species problem’. Heuristically, consider bulk AdS${ }_4$ with a black hole. The contribution of a single field to the CFT${ }_3$ entanglement entropy is of the order $S_A^{\mathrm{vN}}\sim {\displaystyle \frac{c_0}{L_4}}$. Then, for a large number of fields $c_3=L_4^2/G_4\sim L_4/G_3$, the total entanglement entropy is $S_A^{\mathrm{vN}}=c_3{c}_0/L_4\sim c_0/G_3$.

7.1.2. The Entropy of Hawking Radiation

Hawking’s discovery that black holes emit thermal radiation leads to the information puzzle [252]: do black holes evolve unitarily or not? According to Hawking’s original calculation, the fine-grained von Neumann entropy of radiation $S_{\mathrm{vN}}^{\mathrm{rad}}$ grows in time indefinitely, paradoxically surpassing the coarse-grained entropy of the black hole. Alternatively, if the black hole plus radiation obeys standard quantum principles, the radiation entropy instead follows a unitary Page curve [253], never exceeding the black hole entropy. Previously thought to be a problem only quantum gravity would solve, the paradox can be addressed in semi-classical gravity for which the extremization prescription (296) plays a prominent role [254,255,256]. In particular, a variant of (296) known as the ‘island formula’,

$$S_{\mathrm{vN}}\left({\Sigma }_X\right)=\underset{X}{\min }\left\{\underset{X}{\mathrm{ext}}\left[{\displaystyle \frac{\mathrm{Area}\left(X\right)}{4G}}+{\mathcal{S}}_{\mathrm{DC}}\left(X\right)+S_{\mathrm{vN}}^{\mathrm{sc}}\left({\Sigma }_X\right)\right]\right\},$$

(297)

can be used to explicitly compute unitary Page curves in evaporating or eternal black hole backgrounds. Here, $S_{\mathrm{vN}}\left({\Sigma }_X\right)$ is the fine-grained entropy in the full quantum theory, $S_{\mathrm{vN}}^{\mathrm{sc}}$ is the von Neumann entropy of bulk quantum fields in the semi-classical approximation, and ${\Sigma }_X$ is a codimension-1 slice bounded by a codimension-2 quantum extremal surface (QES) X and a cutoff surface. Generally, ${\Sigma }_X$ is disconnected, ${\Sigma }_X={\Sigma }_R\cup I$, where ${\Sigma }_R$ is the region outside the black hole collecting radiation and I is an ‘island’ (with $X=\partial I$) lying primarily inside the black hole.

Originally achieved for models of two-dimensional dilaton gravity59, holographic braneworlds confirm unitary Page curves arise in higher-dimensional gravity [65,260]. This is accomplished using ‘double holography’ (recall the description before Section 3.4). For simplicity, consider the boundary perspective, i.e., the vacuum state of the d-dimensional boundary CFT on $\mathbb{R}\times S^{d-1}$ coupled to a $(d-1)$-dimensional conformal defect along the equator of the sphere $S^{d-1}$. The entanglement entropy of the holographic boundary CFT can be computed using the RT prescription. Taking the bulk to be AdS${ }_{d+1}$ with an AdS${ }_d$ Dvali–Gabadadze–Porrati (DGP) brane [261] (where the brane action includes, e.g., its own Einstein–Hilbert term), the entropy of the boundary CFT vacuum reduced to a boundary region A is given by [54]

$$S_{\mathrm{vN}}\left(A\right)=\underset{\gamma }{\min }\left\{\underset{\gamma }{\mathrm{ext}}\left[{\displaystyle \frac{\mathrm{Area}\left(\gamma \right)}{4G_{d+1}}}+{\displaystyle \frac{\mathrm{Area}(\gamma \cap \mathrm{brane})}{4G_d}}\right]\right\},$$

(298)

for bulk extremal surface $\gamma$ homologous to A and $\partial A=\partial \gamma$. The first term on the right-hand side is the familiar RT formula, while the second term is the gravitational contribution that arises when the bulk RT surface intersects the DGP brane. There are topologically distinct configurations for the bulk extremal surface $\gamma$: surfaces $\gamma$ which do not intersect the brane and (ii) surfaces that intersect the brane; only with (ii) does the second term in the brackets contribute. As both are candidate extremal surfaces, the entropy $S_{\mathrm{vN}}\left(A\right)$ is computed using the surface which gives the smallest value on the right-hand side (298). Transitions between ‘disconnected’ and ‘connected’ phases will occur depending on the size of $\gamma$ and the brane tension and gravitational coupling.

The holographic entropy Formula (298) can be understood as a relation between the boundary and bulk perspectives. It is through the brane perspective that the right-hand side can be reinterpreted as the QES or island formula60. Qualitatively, the disconnected phase corresponds to the situation where, according to beings confined to the brane, no quantum extremal islands are formed, while in the connected phase, extremal islands appear61. By applying this doubly holographic reasoning to higher-dimensional topological black holes [65], black strings [264,265], and de Sitter space [82,266], the prescription (298) yields unitary Page curves except in cases where the horizon is extremal with a vanishing temperature.

Notably, thus far, all $3+1$-dimensional braneworld models that have been explored and exhibit island formation are described by massive gravity theories; islands disappear in the limit at which the graviton on the brane becomes massless [267] (see, e.g., [268] for a review). Further, long-range gravity theories have that the energy of an excitation localized to the island can be detected outside the island, which is inconsistent with the principle that operators in an entanglement wedge commute with operators in its complement [269]. Combined, this suggests the phenomenon of island formation is a feature of massive gravity, and it is even inconsistent in massless theories.

7.1.3. Holographic Complexity

Quantum complexity is another useful diagnostic in information theory. Loosely speaking, complexity measures the difficulty of constructing an arbitrary quantum state from a reference state using a specific set of resources. This concept has proven highly valuable in computer science and quantum computation, where the resources are given by some elementary operations (‘gates’) and the mapping between reference and target states defines a quantum circuit. Complexity has also been fruitful in gravitational physics, particularly as a tool to quantify the information processing of black holes. This application touches upon the foundational aspects of gravity and its interplay with quantum mechanics with some evidence suggesting that gravity itself may emerge from the minimization of complexity —that is, from efficient quantum computation [270,271,272,273,274,275,276].

Historically, computational complexity was proposed as a new entry in the holographic dictionary to address the puzzle of why black hole interiors continue to grow after scrambling [277]. Various conjectures for its precise duality have been proposed with the initial contenders being ‘complexity=volume’ (CV) [278,279] and ‘complexity=action’ (CA) [280,281]. CV posits that the complexity of a CFT state $|\Psi \rangle$ defined on a Cauchy slice $\sigma$ is given by the maximal volume of a codimension-one bulk surface $\Sigma$ anchored on the boundary slice $\sigma$,

$${\mathcal{C}}_V\left(|\Psi \rangle \right)={\displaystyle \frac{\mathrm{Vol}(\Sigma )}{GL}}.$$

(299)

On the other hand, CA asserts that complexity is defined by the on-shell gravitational action I evaluated within a codimension-zero bulk region known as the Wheeler–deWitt (WdW) patch $\mathcal{W}$, anchored at the given boundary slice $\sigma$,

$${\mathcal{C}}_A\left(|\Psi \rangle \right)={\displaystyle \frac{I\left(\mathcal{W}\right)}{\pi }}.$$

(300)

Conceptually, CV is arguably more intuitive than CA, resembling the computational complexity of a tensor network circuit that captures the entanglement structure of the CFT state [282,283,284,285]. For high-temperature thermofield double states, however, CV and CA complexities have the same late time behavior

$${\left.{\displaystyle \frac{d{\mathcal{C}}_{V,A}}{dt}}\right|}_{t\gg \beta }\sim TS,$$

(301)

which is consistent with the expectation of complexity in such states.

More generally, an infinite number of gravitational duals to holographic complexity are technically possible [286,287]. Dubbed ‘complexity=anything’ (CAny), these generalizations are designed to encapsulate key aspects of complexity, including the anticipated linear growth following scrambling and the observed ‘switchback effect’ —a delay in the onset of linear growth caused by perturbations. While rigorous proof connecting any of these proposals to a concrete field theory definition of complexity is currently lacking, the notion is that they may be associated with ambiguities inherent in its definition. These ambiguities include the arbitrary choice of elementary gates, cost factors, and the reference state, among others.

Aspects of holographic complexity proposals have been explored using braneworlds [82,91,288,289,290], including the influence of backreaction effects using the static quantum BTZ black hole as a guide [91]. In summary,

• In the braneworld effective theory, CV admits a semi-classical expansion of the form

  $${\mathcal{C}}_V\left(|\Psi \rangle \right)={\displaystyle \frac{\mathrm{Vol}(\Sigma )}{GL}}+{\displaystyle \frac{\delta \mathrm{Vol}(\Sigma )}{GL}}+\mathcal{V}(\Sigma )+{\mathcal{C}}_V^{\mathrm{bulk}}\left(|\varphi \rangle \right)+\cdots ,$$

  (302)

  with the leading term being the complexity of the classical black hole, $\delta \mathrm{Vol}(\Sigma )$ is the change in the volume due to the quantum backreaction, $\mathcal{V}(\Sigma )$ represents higher curvature terms akin to the Wald corrections to the Bekentein–Hawking entropy, and ${\mathcal{C}}_V^{\mathrm{bulk}}\left(|\varphi \rangle \right)$ is the complexity of the bulk state $|\varphi \rangle$ defined on $\Sigma$. Note that the structure of (302) resembles an expansion of the QES prescription (296). Furthermore, (302) does satisfy the late time growth (301), upon replacing S with the generalized entropy $S_{\mathrm{gen}}$.
• Conversely, CA does not simplify to the classical three-dimensional CA proposal plus corrections. The action involves cancellations among the bulk, boundary, and joint terms, making the late-time growth highly sensitive to quantum effects. As a result, the late time behavior (301) is not reproduced. This discrepancy arises because the WdW patch extends to the singularity, whose structure is significantly altered by quantum backreaction, leading to substantial quantum contributions to CA.

Analogous features have been observed for the rotating quantum BTZ [289]. Based on these preliminary studies, quantum black holes advocate the need for a semi-classical generalization of holographic complexity62.

7.1.4. Singularity Probes and Quantum Cosmic Censorship

Black hole singularities reflect the fact that classical gravity is incomplete. Further, the Hawking–Penrose singularity theorems imply their inevitability. Due to their unphysical consequences, Penrose conjectured there must exist a type of cosmic censorship [158], known as weak cosmic censorship, where a horizon must shroud the singularity from null infinity. Singularities, moreover, mark a lack of predictive power in an otherwise classically deterministic theory. The strong cosmic censorship conjecture [292,293], which is independent of weak cosmic censorship, asserts classical general relativity should remain deterministic. In technical terms, for generic smooth initial data, the Cauchy evolution of a Cauchy hypersurface is inextendible beyond the Cauchy horizon. For example, for charged and rotating black holes, whose inner horizons are Cauchy horizons, the past timelike singularity heralds a lack of predictivity beyond the inner horizon. Conceivably, however, the infinitely blue-shifted energy of a particle entering from the exterior of the black hole should result in a large backreaction so as to turn the smooth (regular) inner Cauchy horizon into a non-smooth barrier [294,295].

It is natural to wonder whether either form of cosmic censorship holds when semi-classical backreaction effects are accounted for. Thus far, there is agreement that the expectation value of the quantum stress tensor of a free scalar field diverges at the inner horizon of a RN or Kerr black hole (in $D=d+1=4$) on approach from the exterior [296,297,298,299], suggesting a type of strong cosmic censorship. A notable exception, however, is the classical rotating BTZ black hole, where the backreaction is mild enough such that the inner horizon does not become singular, indicating a violation in strong cosmic censorship [298,299]63. A limitation of these works is that the analysis was only accomplished at leading order in backreaction. The rotating quantum BTZ black hole, on the other hand, which exactly accounts for all orders of backreaction, has essentially the structure of an AdS${ }_4$–Kerr black hole. Consequently, as argued in [300], and verified in [301] using the techniques of [299], rotating quantum BTZ has a singular inner horizon. Thus, for the BTZ black hole, strong cosmic censorship still holds once backreaction effects beyond leading order are accounted for.

There is an intuitive sense that quantum effects induce a weak form of cosmic censorship. Indeed, backreaction of the Casimir stress tensor modifies the conical (A)dS${ }_3$ and Mink${ }_3$ defect geometries such that the naked singularities become hidden behind a horizon. A standard test of (classical) weak cosmic censorship is to try to overspin or overcharge near-extremal black holes such that they shed their horizons. So far, Kerr–Newman black holes do not [302,303]. Likewise, the rotating quantum BTZ black hole cannot be overspun [176].

Another test of weak cosmic censorship is the conjectured classical Penrose inequality [304]. Loosely speaking, assuming there are no naked singularities and collapsing matter settles to a Kerr black hole, then the total mass M for a $D\ge 4$-dimensional asymptotically flat [305,306,307,308] or AdS spacetime [309,310] with a marginally trapped surface $\sigma$ is bounded below by the area $A\left[\sigma \right]$

$${\displaystyle \frac{16\pi G_{D}M}{(D-2){\Omega }_{D-2}}}\ge {\left({\displaystyle \frac{A\left[\sigma \right]}{{\Omega }_{D-2}}}\right)}^{{\displaystyle \frac{D-3}{D-2}}}+{\ell }_D^{-2}{\left({\displaystyle \frac{A\left[\sigma \right]}{{\Omega }_{D-2}}}\right)}^{{\displaystyle \frac{D-1}{D-2}}}.$$

(303)

Here, $G_D$ and ${\ell }_D$ denote the D-dimensional Newton’s constant and curvature scale, respectively, and ${\Omega }_n\equiv 2{\pi }^{(n+1)/2}/\mathsf{\Gamma }[(n+1)/2]$ is the volume of a unit n-sphere. In the limit ${\ell }_D\to \infty$, the Penrose inequality for asymptotically flat space is recovered. While the Penrose inequality has not been proven in general, any spacetime violating (303) is believed to be a counterexample to weak cosmic censorship. Notably, the classical Penrose inequality can be violated at leading order in perturbative backreaction [311,312]. This motivates the need for a semi-classical generalization of (303), i.e., a quantum Penrose inequality [311,312]

$$\begin{array}{cc}\hfill {\displaystyle \frac{16\pi {\mathcal{G}}_{D}M}{(D-2){\Omega }_{D-2}}}& \ge {\left({\displaystyle \frac{4{\mathcal{G}}_D{S}_{\mathrm{gen}}}{{\Omega }_{D-2}}}\right)}^{{\displaystyle \frac{D-3}{D-2}}}+{\ell }_D^{-2}{\left({\displaystyle \frac{4{\mathcal{G}}_D{S}_{\mathrm{gen}}}{{\Omega }_{D-2}}}\right)}^{{\displaystyle \frac{D-1}{D-2}}}\hfill \end{array}$$

(304)

where the area $A\left[\sigma \right]$ has been replaced for the generalized entropy $S_{\mathrm{gen}}$ associated with a (quantum) marginally trapped surface. To test whether the quantum inequality (304) holds even for large backreaction effects, a three-dimensional inequality was proposed in [135]

$$8\pi {\mathcal{G}}_3(M-M_{\mathrm{cas}})\ge {\ell }_3^{-2}{\left({\displaystyle \frac{4{\mathcal{G}}_3{S}_{\mathrm{gen}}}{2\pi }}\right)}^2,$$

(305)

where $M_{\mathrm{cas}}=-1/8{\mathcal{G}}_3$ is the Casimir energy of backreacting quantum fields. All static and rotating quantum BTZ black holes were found to satisfy (305) at all orders of backreaction. This implies the existence of weak quantum cosmic censorship64 in non-perturbative semi-classical gravity for which the quantum Penrose inequality would be a logical input.

7.1.5. Imprints of Quantum Backreaction beyond Thermal Equilibrium

Realistic, astrophysical black holes are generally thought to be dynamical. Such non-stationary black holes lack an equilibrium thermodynamic description, and their horizons change shape and oscillate. One way to explore black hole dynamics is to consider (small) time-dependent perturbations to static or stationary black holes. Such perturbations display characteristic patterns of damped oscillations, dubbed quasi-normal modes (QNMs), that capture deviations away from equilibrium [314,315]. Furthermore, certain quasi-normal resonances serve as black hole signatures in gravitational wave astronomy.

Technically, QNMs are derived by examining the fluctuation equations of gauge-invariant perturbations. The simplest example involves the study of probe fields. For example, for a probe scalar field $\varphi$, the fluctuation equation is given by the Klein–Gordon equation,

$$(\square +m^2)\varphi =0,$$

(306)

where $\varphi$ is subsequently decomposed intro Fourier and harmonic modes, $\varphi =e^{i\omega t}{Y}_l^m(\Omega )\Phi \left(r\right)$. The characteristic quasi-normal mode frequencies $\omega$ that solve the fluctuation equation are generally complex, $\omega ={\omega }_R-i{\omega }_I$, with a negative imaginary part indicating the decay of perturbations. For holographic AdS black holes, quasi-normal modes appear as poles in relevant retarded two-point correlators of the dual CFT, providing a field-theoretic characterization of black holes out of thermal equilibrium [316].

It is natural to wonder how a backreaction modifies characteristic traits of black hole dynamics. A detailed study of QNMs of the quantum BTZ black hole provides some preliminary insights [317]. Considering probe fields of brane localized matter, dual to CFT operators with conformal dimension $∆\in [1,2]$ (accessible via the standard quantization) and spin $s=0,\pm 1/2$, it was found that the QNMs and their overtones exhibit qualitatively different behavior depending on the branch of the qBTZ solution selected. This distinction can be used to differentiate between the types of singularities cloaked by a horizon: dressed conical singularities versus genuine quantum-corrected black holes. Furthermore, the magnitude ${\omega }_I$ of the imaginary part of the leading mode generally decreases with the strength of backreaction for the quantum-corrected black hole branch. This implies the thermalization time, $t_{\mathrm{th}}\sim 1/{\omega }_I$, defined by the late-time decay of two-point correlators $\langle \mathcal{O}\left(0\right)\mathcal{O}\left(t\right)\rangle \sim e^{t/t_{\mathrm{th}}}$, accelerates due to semi-classical effects. This phenomenon can be explained from the perspective of the dual CFT: quantum matter in the bulk with a large central charge corresponds to coupling a large number of light operators in the boundary CFT, increasing the number of degrees of freedom a small perturbation needs to excite before thermalization is reached.

The pole structure of retarded two-point CFT correlators dual to quantum BTZ QNMs, moreover, exhibits a markedly different behavior than its classical counterpart [317]. This includes, for example, ‘pole-skipping’, i.e., points in momentum space where the retarded Green’s function is not unique. Specifically, for the quantum BTZ black hole, the momentum of the pole-skipping points exhibits a non-trivial dependence on the parameter controlling the strength of backreaction. This implies, given the connection between hydrodynamics and chaos in holographic CFTs [318,319], the chaotic dynamics of black holes are dramatically altered when backreaction is accounted for65. Further, by studying pole collisions in complex momentum space, quantum corrections have a significant impact on the analytic structure of the poles of retarded Green’s functions. In particular, the quantum corrections intertwine the tower of modes in a series of level-crossing events noticeably distinct from the level-touching events observed in the classical BTZ geometry [322].