1. Introduction
Three decades ago, several (1+1)-dimensional black hole models were introduced to gain insight into the quantum nature of black hole radiation, with one of the most prominent and physically interesting models being the Callan–Giddings–Harvey–Strominger (CGHS) system [
1]. Simplified CGHS models, albeit with certain limitations, are exactly soluble and lead to many associated discoveries. New surprises related to complexity, temperatures, and entropy are still being found [
2,
3,
4,
5,
6].
Moving mirrors are accelerated boundaries that create energy, particles, and entropy. They are simplified (1+1)-dimensional versions of the dynamical Casimir effect [
7,
8]. Interesting in their own right, they also act as toy models for black hole evaporation [
9,
10,
11,
12,
13,
14,
15]. The general and physically relevant connections of moving mirrors to black hole physics can be found in canonical textbooks [
5,
16] and also in recent works, e.g., [
17,
18,
19,
20].
There have been a number of studies that relate different specific black hole models (e.g., the Schwarzschild [
21,
22,
23,
24] case) and their analog moving mirrors, including the extremal Reissner–Nordström (RN) [
25,
26], extremal Kerr [
27], RN [
28], Taub-NUT [
29] and Kerr [
30] black holes. In addition, de Sitter and anti-de Sitter cosmologies [
31] are also modeled by moving mirror trajectories. For appropriately chosen trajectories [
32], close comparisons can be made with the radiation emitted from dynamic spacetimes [
33,
34]. Such an equivalence between a mirror and a curved spacetime is called an accelerated boundary correspondence (ABC).
Our motivation in this paper is to synthesize and strongly link the well-known and important CGHS black hole model with its analog moving mirror counterpart. In the process, we want to derive the spectrum of particle production exactly and analytically, analyzing the close parallels between the two systems via the temperature, horizons and parameter analogs associated with the CGHS black hole mass and cosmological constant. Furthermore, we aim to initiate an investigation into the entanglement entropy of a generalized mirror system and its relationship to the self-force on the mirror and power of the emitted vacuum radiation. As we shall see, this link of inquiry reveals a close connection between the seemingly distinct concepts of self-force and information. Application of the results for the CGHS mirror reveals that the divergent self-force is directly a consequence of information loss.
Additionally, we push the correspondence further, by considering the close connection to classical electrodynamic analogies. The moving mirror is found to behave almost like a neutral particle coupled to the massless scalar field (similar to a charged particle coupled to the electromagnetic field). Hence some of the familiar radiation results in classical electrodynamics has direct correspondence in the mirror case.
The paper consists of three interrelated parts. In the first part, an overall equivalence between the CGHS black hole and the exponentially accelerated mirror in laboratory time is established: in
Section 2 we briefly review the CGHS action, the corresponding field equations of motion, and the formation of the CGHS black hole; in
Section 3, the details of the CGHS metric and the transformation of this coordinate system to an accelerated mirror trajectory are investigated; in
Section 4, we derive the particle flux radiated from the exponentially accelerated mirror in laboratory time and demonstrate its thermal characters for late times; and in
Section 5, the particle flux radiated from the CGHS black hole is reviewed.
In the second part, the relation between the mirror entanglement entropy and self-force is studied:
Section 6 is dedicated to derivation of the mirror entanglement entropy; in
Section 7, the moving mirror Larmor formula and Lorentz–Abraham–Dirac (LAD) force analogs are derived with an emphasis on entanglement entropy.
In the last part,
Section 8, the formulas derived in the previous section are used to find the radiative power and radiation reaction force for our particular moving mirror, the CGHS mirror.
In the conclusion, all three parts are summarized and some insight into future directions are provided in
Section 9. Additionally, we use
, except in the results of Equations (
90) and (
91) of
Section 8.
2. Action and Field Equation
In this section, we will briefly summarize the action and field equations of the Callan–Giddings–Harvey–Strominger model in which a linear dilaton vacuum evolves into a black hole from matter injection. The CGHS action reads as
where
g is the metric tensor,
is the dilaton field,
is the cosmological constant, and
are the matter fields. To obtain the equations of motion, one may vary the action, Equation (
1), with respect to the metric
and the dilaton field
, respectively,
where
. Following [
5], one can readily solve Equation (
2) in the conformal gauge,
. The solution is
(gauge fixing) and
where the functions
and
are integrals depending on the stress–energy tensor of the matter field, connected to the mass of CGHS black hole and the event horizon, respectively. Assume that we start with a linear dilaton vacuum, then turn on the matter flux injected to the system at some time
and turn it off after the time
. Then when
, the geometry of the system will approach and finally settle down to the static CGHS black hole background. The value
becomes the mass of the black hole, and
gives the curve of the event horizon. Therefore, one can observe how the linear dilaton vacuum (before the time
) evolves into a CGHS black hole (after the time
) due to the matter injection.
3. CGHS Black Hole and Matching Condition
In this section, we concentrate on the CGHS black hole solution and some of the significant observable quantities within this model. The relevant metric for the CGHS black hole system can be cast in the following form [
35] (c.f. Schwarzschild gauge [
5]):
where
with
as the cosmological constant parametrization scale of the spacetime and
as the mass of the CGHS black hole. The curve of the event horizon function
is set to zero in Equation (
3) in order to obtain the metric for stationary CGHS black hole, Equation (
4). At the particular value of
, the static metric Equation (
4) has a divergence at the radial center location
.
For general
M and
, the horizon is at
. The surface gravity of the CGHS black hole can be obtained as [
36]
where for general
, consistency with the laws of black hole thermodynamics dictates that the temperature is
. For a double null coordinate system (
) with
and
, the associated tortoise coordinate
can be obtained in the usual way [
5], via
which yields
The absolute brackets are critical for real coordinate values.
Following Wilczek [
33], let us coincide the inner and outer regions of a collapsing null shell to form a black hole, where the exterior background is given by the CGHS metric,
and
is a light-like shell. In null coordinates, the system Equation (
9) can be rewritten as
So, the metric for the geometry describing the outside region of a collapsing shell takes the simplified form, .
The matching condition (see [
5,
30,
33]) with the flat interior geometry, described by the interior coordinates
is the trajectory corresponding to
, expressed in terms of the exterior function
. We can obtain this matching via the association
, and taking
along the light ray,
. This matching condition,
is the outside
u trajectory of the origin as a function of the inside coordinate
U. We can write this as
where
and
. The regularity condition of the modes requires that they vanish at
, which acts as a reflecting boundary in the black hole system. In the accelerated boundary correspondence (ABC) of the mirror system, the origin of the black hole functions as the mirror trajectory in flat spacetime. The position of the origin is a dynamic function
u with independent variable
U. Since the field vanishes (does not exist for
), the form of the field modes can be determined, allowing for the identification
(where
v is the flat spacetime advanced time in the moving mirror model) for the Doppler-shifted field modes. In the next section, we will define the analog mirror trajectory for the CGHS spacetime by making the identification
, which is a known function of the advanced time
v.
4. Exponentially Accelerated Mirror
In this section, we focus on the trajectory and particle flux radiation of the exponentially accelerated mirror in coordinate time to demonstrate their equivalence with the corresponding quantities in the CGHS black hole model.
In line with previous accelerated boundary correspondences, consider the exponentially accelerated mirror trajectory with proper acceleration [
37]:
where
is a parameter of the acceleration and
is the horizon in advanced time,
. The
x and
t are the usual lab coordinates of flat (1+1)-dimensional Minkowski spacetime. The trajectory in light cone coordinates as a function of advanced time is
where, identifying Equation (
14) with Equation (
12) as usual (see prior ABCs), the associated parameters in the CGHS system define the moving mirror’s null-ray horizon,
which is the location that the last incoming left-moving ray reflects off the mirror (and in the last equality we have set
for simplicity). The above expression is justified because when
,
. Past this position
, there is no more reflection and the left-mover modes never make it to an observer at right null-infinity,
. The mirror horizon couples the parameters
and
M, which are the cosmological constant and mass of the black hole, respectively, in the CGHS system. The fact that
, which is the finite
v for the mirror horizon, is also closely related to the CGHS black hole horizon, through
, corroborates the correspondence between the CGHS black hole and the exponentially accelerated mirror. A spacetime plot of this asymptotic light-like moving mirror is given in
Figure 1. A Penrose conformal diagram is given in
Figure 2.
Now, we will derive the thermal Planck distribution of the exponentially accelerated moving mirror particles detected by an observer on the right by use of the beta Bogolubov coefficient. The beta coefficient can be found via an integration [
17] by parts where we ignore non-contributing surface terms,
to obtain
where we utilize the Euler integral of the first kind as a Beta function,
, and
. Multiplying by its complex conjugate gives the particle count per
mode, per
mode:
or, equivalently expressed,
Thermal character results in the high frequency limit
approximation (a good explanation for how this corresponds to late times is given by Hawking [
9]). We can see by inspection that,
so that
at late times.
5. CGHS Particle Radiation
In this section, we calculate the particle flux radiated by the CGHS black hole and the corresponding beta Bogolubov coefficients, following the standard procedure (see, for example, [
5]). Remarkably, the beta Bogolubov coefficients match those corresponding ones obtained from the particle radiation of the exponentially accelerating mirror (
Section 4).
The standard Bogolubov procedure for calculating the Hawking radiation considers two relevant regions as mentioned in
Section 2—the linear dilaton vacuum and the CGHS black hole—described by the corresponding “in" and “out" coordinates, which can be connected via the Kruskal coordinates (see [
5] for details and elaboration on the standard notation). One then identifies plane wave modes for ingoing and outgoing sectors using null Minkowski coordinates
. They are related as
So, using
sector the plane wave modes have the following forms,
designated by
g (sometimes
u is used but here
u is already a retarded time null coordinate). The next step is to evaluate the beta Bogolubov coefficients by calculating scalar product between the plane wave modes. For our particular case, the corresponding integral is,
Substituting Equations (
23) and (
24) into Equation (
25), and also using Equation (
22), the above integral becomes
where
is the black hole event horizon location that is formed when injecting matter into linear dilaton vacuum. Note that Equation (
27) is similar to Equation (
15), i.e.,
. Calculation of Equation (
26) gives
The complex conjugate squared of Equation (
28) yields,
which is exactly Equation (
18) given that
. Interestingly, the mass of the CGHS black hole,
, does not appear in
, and
disappears when calculating
. This is because the spectrum of the CGHS black hole does not explicitly depend on its mass. The correspondence between the black hole mass and the curve of the event horizon,
, can be seen from the comparison of Equations (
15) and (
27). The curve of the event horizon is defined by the so-called “apparent horizon”,
, which is space-like or null and coincides with the event horizon after the matter injection has finished at time
, i.e., after time
when the geometry of the CGHS black hole is settled and the apparent horizon becomes the event horizon. The CGHS model is asymmetric with respect to left–right observers. As we have seen with the moving mirror model of the prior section, our observer is on the right side. The difference between the particle radiation for the CGHS model under left and right observers was explored in detail in Ref. [
38].
Hereinafter, due to the identical particle production between the exponentially accelerated mirror in coordinate laboratory time Equation (
18) and the CGHS system Equation (
29), for simplicity, we refer to this specific perfectly reflecting boundary trajectory, Equation (
14), as the CGHS mirror.
6. Mirror Entanglement Entropy
In this section, we review the derivation of (1+1)-dimensional entanglement (geometric) entropy in conformal field theory (CFT) and its connection to the rapidity of the moving mirror (see other derivations, e.g., [
26,
39]). For a recent related approach to calculating the time evolution of entanglement entropy in the presence of a moving mirror (particularly with horizons), see [
40].
Consider the entropy of a system in (1+1)-D CFT [
41],
where
L is the size of the system in general (and in our case, it is the mirror trajectory which measures the size of the system by the spacetime traversed accessible to the quantum field), and
is a UV cut-off.
For a general arbitrary moving mirror,
where
u and
are null coordinates that form the region in the system which we are considering, and
is asymmetrically smeared, i.e.,
. Here,
is the trajectory of the mirror in null coordinates (it is a function of retarded time
u). The smearing and dynamics of the mirror are related as
Substituting Equations (
32) into Equation (
30) yields the bare entropy of the system,
The vacuum entropy of the system can be found by considering a static mirror where
and
. Thus,
Even though the entropies above are defined in terms of smearing, this dependence can be removed by an intuitive renormalization via
Further simplification proceeds by a Taylor expansion of our arbitrary function
around
up to first order, that is,
Substituting Equation (
36) into Equation (
35) brings us to,
Moreover, for a static mirror,
and
, and therefore,
. As a result, Equation (
37) reduces to
where,
Equation (
38) is valid for any moving mirror that starts asymptotically static (zero velocity). Notable exceptions are the eternally thermal Carlitz–Willey mirror [
42] and the eternally uniformly accelerated mirror [
16]; however, most of the solved mirrors in the literature, by construction, do start statically, as they are often used to model gravitational collapse. The CGHS mirror is no exception.
Equation (
38) is more intuitively written in spacetime coordinates using the relation between null and spacetime trajectories of the mirror as
Applying this relation into Equation (
38) yields,
where
is the time-dependent rapidity. It is simple to see that the magnitude of the entropy increases as the mirror moves faster. Unitarity in this context [
43] strictly requires that the entropy must achieve a constant value in the far past and far future.
This von Neumann entropy measure of the degree of quantum entanglement between the two subsystems (past and future) constitutes a two-part composite quantum system. It explicitly reveals the connection of information of entanglement to the dynamics (rapidity) of the moving mirror system. Allow us to speculate that a thermodynamic treatment of the system, and the corresponding macroscopic state of the entanglement, is characterized by a distribution of its microstates; then, it may be appropriate to coincide the Boltzmann entropy with the von Neumann entropy. In this conjectural case, a characteristic speed based on the smallest unit of an operable binary digit of information results.
In the next section, we will ultimately apply the above entanglement–rapidity relationship, Equation (
40), which can be expressed as
, independently of the coordinates of its argument, to gain insight into the self-force and Larmor power by reformulating them in terms of entropy.
8. CGHS Larmor Power and Self-Force
Having derived the power and self-force for any moving mirror in general, we now specialize to the exponentially accelerated mirror that has particle production which corresponds to the CGHS system. We apply the Larmor power and LAD force derived in previous sections to our particular CGHS mirror and find that a simple entanglement-over-distance relationship is revealed, connecting the entanglement–rapidity relationship to the space traversed. In addition, the loss of unitarity is explicitly manifest in the divergence of the power and self-force at the time the horizon forms in the proper frame.
Let us start from the trajectory of the CGHS mirror in spacetime coordinates [
37],
The Larmor power and self-force for the CGHS mirror are found using Equations (
45) and (
69), where
is the acceleration in proper time. The procedure of defining and deriving
is given in [
37]. Let us start from the connection between proper and coordinate times. For the CGHS mirror, it is obtained to be
The inverse of Equation (
86) yields
Applying it into Equation (
85) leads to the trajectory in proper time,
The next step to obtain
is to find celerity and then rapidity. Using the rapidity, the proper acceleration is found to be,
This result, Equation (
89), is found using a different method by Juárez-Aubry [
54] and is in agreement. Substituting Equation (
89) into Equations (
45) and (
69), we obtain corresponding Larmor power and radiation reaction force for the CGHS mirror as,
and
The terms on the right of Equations (
90) and (
91) have reinstated
ℏ and
c, noting that
has units of an acceleration in order to emphasize they are a quantum Larmor power and quantum self-force, respectively. The dependences of the CGHS mirror Larmor power and self-force on proper time and
, Equations (
90) and (
91), are demonstrated graphically in
Figure 3 and
Figure 4.
Figure 4 has lines corresponding to different values of
which intersect. This is explained by the fact that the dependence of the CGHS self-force, Equation (
91), on the single parameter of the system
is non-trivially different from the dependence of the power, Equation (
90).
Let us now consider the timespace trajectory of the mirror in natural units,
Using this form of the trajectory, we find rapidity in terms of space coordinate
x,
So, the rapidity, or the information defining dynamical quantity, in terms of
x has a surprisingly simple form: it linearly depends on the space coordinate. Equation (
93) is the simplest way to express the trajectory of the CGHS mirror.
The last interesting quantity we compute is the energy flux in terms of
x [
17]. The CGHS mirror flux is found to be
This form immediately clarifies that at late times (far-left positions,
), the energy flux is a constant associated with thermal emission that is in agreement with the thermal behavior of the CGHS black hole radiation,
. A graphical illustration of this flux is shown in
Figure 5.
9. Conclusions and Future Work
Overall, the equivalence between the CGHS black hole and the exponentially accelerating moving mirror in lab time can be seen from several explicit matching quantities: the matching condition for the CGHS black hole and the trajectory of the mirror in null coordinates, the spectra, and consequently, the temperatures. A one-to-one correspondence is ensured as long as the
requirement is met. The correspondence is summarized in
Table 1.
The result of studying this correspondence has shed light on the origin of particle creation. The moving mirror corresponds to the center of the black hole system (the origin itself). Interestingly, the CGHS mirror mimics the regularity condition (at ) that influences the field modes of the CGHS black hole. The advanced time horizon of the moving mirror corresponds to a quantity determined by the black hole parameters and M. The particular simplicity of the mathematics in the mirror case and the ease and utility in describing this exact spectrum via a accelerating trajectory in a flat-spacetime background appears to be a result of the particular matching function associated with the CGHS geometry.
More general considerations give us the Larmor power radiated by an arbitrary moving mirror and the LAD formula for the radiation reaction. The derivations utilize general dynamics of the mirror expressed in terms of proper acceleration and rapidity and lead naturally to an information interpretation by expressing the rapidity in terms of entanglement entropy. The power and force are found to have the same dynamic form as that in classical electrodynamics for a moving point charge. In terms of information, the entanglement power and entanglement self-force are interpreted in terms of first and second derivatives of the von Neumann entanglement entropy, respectively.
Specializing to our particular CGHS moving mirror, the Larmor power is found to diverge as . As proper time ticks to , the mirror is infinitely accelerating, reaching the speed of light. Consistently, the direction of the self-force is opposite the direction of the radiated Larmor power. It is worth emphasizing that as a guide, in SI units, both the Larmor power and LAD force for the CGHS mirror are proportional to ℏ, underscoring the fact that the power and self-force are quantum (not classical) measures.
Lastly, the CGHS mirror has two simplifying results when expressed in terms of space rather than time or light-cone coordinates: the trajectory rapidity is simply proportional to the distance traveled, and the radiative flux emitted by the CGHS mirror is seen by eye as thermal (constant emission) at far left positions (late times). In summary list form, the salient features of this work are as follows:
CGHS mirror ⇔ CGHS black hole center:
Section 4.
Future extensions of this work are foreseen. Hawking [
9] pointed out that at very early times of gravitational collapse, a star cannot be described by the no-hair theorem. So, in this context, a variety of different collapse situations correspond to different mirror trajectories, i.e., there are different spacetimes for different mirror dynamics. It is likely to be fruitful to consider modifications to the mirror trajectory (one of which was already done to CGHS, e.g., [
44]) made to provide different early time approaches to a thermal distribution, particularly those modifications that can afford unitarity and finite evaporation energy, modeling more realistic situations congruent with finite mass black holes and quantum purity. It is worth mentioning that mirrors with different accelerations can lead to the same physics. An example can be Möbius transformed CGHS mirror trajectory that gives the exact physics as CGHS spacetime (see [
55] for more details on Möbius transformations of moving mirrors).
The modifications that can take into account energy conservation, like those of the dilaton gravity models and moving mirror models, have had significant success in the laboratory for studying black hole evaporation. The physical problem in (1+1) dilaton gravity of the evaporating black hole and its modified emission extends to complete evaporation for the Russo, Susskind, and Thorlacius (RST) model [
56] and to partial evaporation, leaving a remnant for the Bose, Parker, and Peleg (BPP) model [
57]. The two-dimensional RST model for evaporating black holes is locally equivalent—at the full quantum level—to Jackiw–Teitelboim (JT) gravity that was recently shown to be unitary [
39].
The similarity of drifting moving mirrors (see, for example, Ref. [
17]) to the BPP model is striking in several qualitative aspects: NEF emission as a thunderpop (NEF emission from evaporating black holes, at least in the case of a (1+1)-dimensional dilaton gravity, has already been known in the literature for over 20 years [
58]), a leftover remnant, and finite total energy emission. It is also interesting that the mass of the remnant in the BPP model is independent of the mass
M of the infalling matter since, with respect to the issue of energy conservation, there is no known physical analog for
, the initial mass of the shockwave, in the mirror model.
We hope this work offers insight for future direction, using the formulas for mirror radiative power and radiation reaction force. Investigations of behavior of the Larmor power and LAD force of other existing moving mirror models will be used to compare results to better understand the specific physics of moving mirrors and the general physics of acceleration radiation from the quantum vacuum.