Gravitational Condensate Stars: An Alternative to Black Holes

Abstract

A new final endpoint of complete gravitational collapse is proposed. By extending the concept of Bose–Einstein condensation to gravitational systems, a static, spherically symmetric solution to Einstein’s equations is obtained, characterized by an interior de Sitter region of $p=-\rho$ gravitational vacuum condensate and an exterior Schwarzschild geometry of arbitrary total mass M. These are separated by a phase boundary with a small but finite thickness ℓ, replacing both the Schwarzschild and de Sitter classical horizons. The resulting collapsed cold, compact object has no singularities, no event horizons, and a globally defined Killing time. Its entropy is maximized under small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, which is of order $k_B\ell Mc/\hslash$, instead of the Bekenstein–Hawking entropy, $S_{BH}=4\pi k_{B}G{M}^2/\hslash c$. Unlike BHs, a collapsed star of this kind is consistent with quantum theory, thermodynamically stable, and suffers from no information paradox.

1. Introduction 1

The vacuum Einstein equations of classical general relativity (GR) possess a well-known solution for an isolated mass M, with the static, spherically symmetric line element

$$d{s}^2=-f\left(r\right)d{t}^2+\frac{d{r}^2}{h\left(r\right)}+r^2\left(d{\theta }^2+{sin}^2\theta d2\right)$$

(1)

where the functions $f\left(r\right)$ and $h\left(r\right)$ in this case are equal and given by

$$f\left(r\right)=h\left(r\right)=1-\frac{2GM}{r}=1-\frac{r_{{}_M}}{r}$$

(2)

in units where $c=1$. The dynamical singularity of the Schwarzschild metric (1) at $r=0$ with its infinite tidal forces clearly signals a breakdown of the vacuum Einstein equations. The kinematical singularity at the Schwarzschild radius $r_{{}_M}\equiv 2GM$ is of a different sort, corresponding to an infinite blue shift of the frequency of an infalling light wave with respect to its frequency far from the black hole (BH). Since the local curvature tensor is finite at $r=r_{{}_M}$, the singularity of the metric (1)–(2) there can be removed by a suitable (and singular) change of coordinates in the classical theory. A classical point test particle freely falling through the event horizon is said to experience nothing special at $r=r_{{}_M}$. Whether or not a true event horizon of this kind where light itself becomes trapped can be realized in a physical collapse process remains open to question.

This question becomes much more acute in quantum theory. For when $\hslash \ne 0$, a photon of finite asymptotic frequency $\omega$ (even if arbitrarily small) acquires a local energy $E=\hslash \omega {\left[f\left(r\right)\right]}^{-\frac{1}{2}}$, which diverges at $r=r_{{}_M}$. Since the effective coupling in gravity is ${(G/\hslash )}^{\frac{1}{2}}E$, proportional to energy, the kinematical singularity at $r_{{}_M}$ may be responsible for strong gravitational interactions between elementary quanta as their energy approaches the Planck energy $M_{Pl}={(\hslash /G)}^{\frac{1}{2}}$, by which point it can no longer be taken for granted that quantum effects on the classical geometry can be safely neglected.

In the semi-classical approximation, when a massless field, such as that of the photon, is quantized in the fixed Schwarzschild background, with certain boundary conditions corresponding to a regular future horizon, one finds that the BH radiates these quanta with a thermal spectrum at the asymptotic Hawking temperature, $T_H=\hslash /8\pi k_{B}GM$ [3]. It is usually assumed that the backreaction effect of this radiation on the classical geometry must be quite small. However, detailed calculations of the energy-momentum of the radiation show that its $\langle T_t^t\rangle$ and $\langle T_r^r\rangle$ components have an $f^{-2}$ infinite blue shift factor at the horizon which divergences are arranged to exactly cancel in free-falling coordinates [4,5]. Anything other than this exact cancellation of the separately diverging energy density and pressure in the semi-classical Einstein equations would significantly change the geometry near $r=r_{{}_M}$ from the classical Schwarzschild solution (2). The wavelengths contributing to these quantum stresses are of order $r_{{}_M}$ and, hence, quite non-local on the scale of the BH. Unlike the classical kinematic singularity in (2), such non-local semi-classical backreaction effects near $r_{{}_M}$ depending on the quantum state of the field theory, cannot be removed by a local coordinate transformation.

Furthermore, energy conservation plus a thermal radiation spectrum imply that a BH has an enormous entropy, $S_{BH}\simeq {10}^{77}{k}_{{}_B}{(M/M_{\odot })}^2$ [6], far in excess of a typical stellar progenitor. The application of thermodynamic arguments to BHs is itself put into doubt by the the inverse dependence of $T_{{}_H}$ on M implying that a BH in thermal equilibrium with its own Hawking radiation has negative specific heat and, therefore, is unstable to thermodynamic fluctuations [7]. On the other hand, requiring that basic thermodynamic principles apply to self-gravitating systems as well implies that their heat capacity must be proportional to their energy fluctuations, $\propto \langle {(\Delta E)}^2\rangle$, and hence must be positive. The assumption of a thermal mixed state of Hawking radiation emerging from a BH also leads to an ‘information paradox’ so severe that resolving it has been conjectured to require an alteration in the principles of relativity, or quantum mechanics, or both.

In light of the challenges BHs pose to quantum theory, and in lieu of revision to otherwise well-established fundamental laws of physics, it is reasonable to examine alternatives to the strictly classical view of the event horizon as a harmless kinematic singularity, when $\hslash \ne 0$ and the quantum wavelike properties of matter are taken into account. In earlier investigations which attempted to include the backreaction of the Hawking radiation in a self-consistent way, the entropy arises entirely from the radiation fluid [8,9]. In fact, $S=4[(\kappa +1)/(7\kappa +1)]S_{BH}$, for a fluid with the equation of state, $p=\kappa \rho$, becoming equal to the Bekenstein–Hawking entropy $S_{BH}$ for $\kappa =1$. Despite this suggestive feature, these fluid models have huge (Planckian) energy densities near $r_{{}_M}$ and a negative mass singularity at $r=0$, so that the Einstein equations are not reliable in either region.

A quite different proposal for incorporating quantum effects has been made in [10], viz. that the horizon becomes instead a critical surface of a phase transition in the quantum theory, supported by an interior region with equation of state, $p=-\rho <0$. Such a vacuum equation of state, first proposed by Gliner [11] for the endpoint of gravitational collapse, is equivalent to a positive cosmological term in Einstein’s equations, and does not satisfy the energy condition $\rho +3p\ge 0$ needed to prove the most general form of the classical singularity theorems [12].

In this paper [1], we show that an explicit static solution of Einstein’s equations taking quantum considerations into account exists, with the critical surface of ref. [10] replaced by a thin shell of ultra-relativistic fluid of soft quanta obeying $\rho =p$. Such a solution, lacking a singularity and an horizon is significant because it provides a stable alternative to BHs as the endpoint of gravitational collapse, with potentially different observational signatures.

The principal assumption required for this solution to exist is that gravity undergoes a vacuum rearrangement phase transition in the vicinity of $r=r_{{}_M}$, in which the vacuum energy density changes abruptly. Since a spatially homogeneous Bose-Einstein condensate (BEC) couples to Einstein’s equations in exactly the same way as an effective cosmological term ${\mathsf{\Lambda }}_{\mathrm{eff}}$, with equation of state $p=-\rho$, the existence of the interior region requires that general considerations of low temperature quantum BEC phase transitions can be extended to gravitation. It has been suggested that a phase transition could be induced by the equation of state of the compressed matter attaining the most extreme one allowed by causality, namely $p=+\rho$. The effective theory incorporating the low energy effects of quantum anomalies that could give rise to the interior $p=-\rho$ Gravitational Bose–Einstein condensate (GBEC) phase has been presented elsewhere [13,14,15]. In this paper, we forego any discussion of the details of the quantum phase transition and present only the solution of Einstein’s equations with the specified equations of state of the de Sitter (dS) interior and phase boundary layer, which is the model proposed in the original arXiv paper [1]. Developments since that original article are discussed in the Appendix A.

2. Solution of Einstein Equations for Static, Spherical Symmetry

The general form of the stress-energy tensor in the static, spherically symmetric geometry of (1) is

$$T_{\mu }^{\nu }=\left(\begin{array}{cccc}-\rho & 0& 0& 0\hfill \\ 0& p& 0& 0\hfill \\ 0& 0& p_{\perp }& 0\hfill \\ 0& 0& 0& p_{\perp }\hfill \end{array}\right)$$

(3)

so that the Einstein equations in the static spherical coordinates of (1) are

$$\begin{array}{cc}\hfill -G_t^t& ={\displaystyle \frac{1}{r^2}}{\displaystyle \frac{d}{dr}}\left[r\left(1-h\right)\right]=-8\pi G{T}_t^t=8\pi G\rho ,\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill G_r^r& ={\displaystyle \frac{h}{rf}}{\displaystyle \frac{df}{dr}}+{\displaystyle \frac{1}{r^2}}\left(h-1\right)=8\pi G{T}_r^r=8\pi Gp\hfill \end{array}$$

(4b)

together with the conservation equation

$$\nabla {}_{\lambda }{T}_r^{\lambda }=\frac{dp}{dr}+\frac{\rho +p}{2f}\frac{df}{dr}+\frac{2}{r}(p-p_{\perp })=0$$

(5)

which ensures that the other components of Einstein’s equations are satisfied. In (5) the transverse pressure $p_{\perp }\equiv T_{\theta }^{\theta }=T_{\varphi }^{\varphi }$ is allowed to be different from the radial pressure $p\equiv T_r^r$. For a perfect fluid $p_{\perp }=p$ and the last term of (5) vanishes. In that case, (4)–(5) are three first order equations for the four functions, $f,h,\rho$, and p, which become closed when an equation of state for the fluid relating p and $\rho$ is specified.

Because of the considerations in the Introduction, as a first phenomenological model we allow for three different regions with the three different equations of state

$$\begin{array}{cccc}\mathrm{I}.& \mathrm{de}\mathrm{Sitter}\mathrm{Interior}:\hfill & 0\le r<r_1,& \rho =-p,\hfill \\ \mathrm{II}.& \mathrm{Thin}\mathrm{Shell}:\hfill & r_1<r<r_2,& \rho =+p,\hfill \\ \mathrm{III}.& \mathrm{Schwarzschild}\mathrm{Exterior}:\hfill & r_2<r,& \rho =p=0.\hfill \end{array}$$

(6)

At the interfaces $r=r_1$ and $r=r_2$, the metric functions f and h are required to be continuous, although the first derivatives of f, h and p must be discontinuous from the first order Equations (4) and (5).

In the interior region $\rho =-p$ is a constant from (5). Let us call this constant ${\rho }_{{}_V}=3H^2/8\pi G$. If we require that the origin is free of any mass singularity then the interior is determined to be a region of dS spacetime in static coordinates, i.e.,

$$\mathrm{I}.f\left(r\right)=Ch\left(r\right)=C\left(1-H^2{r}^2\right),0\le r\le r_1$$

(7)

where C and H are constants, which at this point are arbitrary.

The unique solution in the exterior vacuum region which approaches flat space as $r\to \infty$ is a region of Schwarzschild spacetime (2), viz.

$$\mathrm{III}.f\left(r\right)=h\left(r\right)=1-{\displaystyle \frac{2GM}{r}}=1-{\displaystyle \frac{r_{{}_M}}{r}},r_2\le r$$

(8)

where the mass M can take on any (positive) value.

The only non-vacuum region is region II. Defining the dimensionless variable w by

$$w\equiv 8\pi G{r}^{2}p$$

(9)

Equations (4) and (5) with $\rho =p$ may be recast in the form

$$\begin{array}{cc}\hfill \frac{dr}{r}& =\frac{dh}{1-w-h},\hfill \end{array}$$

(10a)

$$\begin{array}{cc}\hfill \frac{dh}{h}& =-\frac{1-w-h}{1+w-3h}{\displaystyle \frac{dw}{w}}.\hfill \end{array}$$

(10b)

together with $pf\propto wf/r^2$ a constant. The first Equation (10a) is equivalent to the definition of the (rescaled) Tolman–Misner–Sharp mass function $\mu \left(r\right)=2Gm\left(r\right)$, with $h=1-\mu /r$ and $d\mu \left(r\right)=8\pi G\rho r^{2}dr=wdr$ within the shell. The second Equation (10b) can be solved only numerically in general. However, it is possible to obtain an analytic solution in the thin shell limit, $0<h\ll 1$, for in this limit we can set h to zero on the right side of (10b) to leading order, and integrate it immediately to obtain

$$h\equiv 1-\frac{\mu }{r}\simeq \epsilon \frac{{(1+w)}^2}{w}\ll 1$$

(11)

in region II, where $\epsilon$ is an integration constant. Because of the condition $h\ll 1$ we require $\epsilon \ll 1$, if w is of order unity. Making use of Equations (10) and (11) we have

$$\frac{dr}{r}\simeq -\epsilon dw\frac{(1+w)}{w^2}$$

(12)

so that because $\epsilon \ll 1$ the radius r hardly changes within region II, and $dr$ is of order $\epsilon dw$. The final unknown function f is given by (5) to be $f={(r/r_1)}^2(w_1/w)f\left(r_1\right)\simeq (w_1/w)f\left(r_1\right)$ to leading order in $\epsilon$ for $\epsilon \ll 1$.

Now requiring continuity of the metric coefficients f and h at both $r_1$ and $r_2$ gives the conditions

$$\begin{array}{cc}\hfill f\left(r_1\right)& =Ch\left(r_1\right)=C(1-H^2{r}_1^2)\simeq C\epsilon \frac{{(1+w_1)}^2}{w_1}\hfill \end{array}$$

(13a)

$$\begin{array}{cc}\hfill f\left(r_2\right)& =h\left(r_2\right)=1-\frac{2GM}{r_2}\simeq \epsilon \frac{{(1+w_2)}^2}{w_2}\hfill \end{array}$$

(13b)

which together with the solution for f evaluated at $r=r_2,w=w_2$ gives

$$C{(1+w_1)}^2={(1+w_2)}^2$$

(14)

and three independent relations among the eight integration constants $(r_1,r_2,w_1,w_2,H,M,C,\epsilon )$. Assuming that $(r_1,r_2,w_1,w_2,H,M,C)$ all remain finite as $\epsilon \to 0$, i.e., they are all of order ${\epsilon }^0$, we obtain from (12) and (13) that

$$\begin{array}{cc}\hfill & r_1=\frac{1}{H}\left[1-\epsilon \frac{{(1+w_1)}^2}{2w_1}\right]\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & r_2=r_{{}_M}\left[1+\epsilon \frac{{(1+w_2)}^2}{w_2}\right]\hfill \end{array}$$

(15b)

for two of the three relations, so that $r_1\to r_{{}_H}=H^{-1}$ and $r_2\to r_{{}_M}$ with $r_2-r_1=\Delta r$ of order $\epsilon$, and $r_{{}_H}\simeq r_{{}_M}$ to leading order in $\epsilon$. Thus, the boundary layer II straddles the location of the classical Schwarzschild and dS horizons, and $r_1\to r_2$ coincide at $r_{{}_H}=r_{{}_M}$, becoming no longer independent in the limit $\epsilon \to 0$. Since the mass M is a free parameter there remain three undetermined integration constants $C,w_1,w_2$ which satisfy the one relation (14) if $\epsilon \ne 0$, and lastly $0<\epsilon \ll 1$ itself.

The important feature of this solution is that for any $\epsilon >0$ both f and h are of order $\epsilon$ but nowhere vanishing. Hence there is no event horizon, and t is a global Killing time. A photon experiences a very large, $\mathcal{O}\left({\epsilon }^{-\frac{1}{2}}\right)$ but finite blue shift in falling into the shell from infinity.

The proper thickness of the shell in the metric (1) is

$$\ell ={\int }_{r_1}^{r_2}\frac{dr}{\sqrt{h}}=r_{{}_M}\sqrt{\epsilon }{\int }_{w_2}^{w_1}dw{w}^{-\frac{3}{2}}=2r_{{}_M}\sqrt{\epsilon }\left(w_2^{-\frac{1}{2}}-w_1^{-\frac{1}{2}}\right)$$

(16)

to leading order in $\epsilon$, and, hence ℓ is $\mathcal{O}\left({\epsilon }^{\frac{1}{2}}\right)$ and small compared to $r_{{}_M}$.

The magnitude of $\epsilon$ and hence of ℓ can be fixed only by consideration of the quantum effects that give rise to the phase transition boundary layer. A subsequent analysis of the stress tensor of the conformal anomaly [14,15], shows that these quantum vacuum polarization effects become significant when $r_2-r_{{}_M}$ is of order of the Planck length $L_{\mathrm{Pl}}={(\hslash G)}^{\frac{1}{2}}\simeq 1.6\times {10}^{-33}$ cm, so that $\epsilon \sim L_{\mathrm{Pl}}/r_{{}_M}$, and $\ell \sim \sqrt{L_{\mathrm{Pl}}{r}_{{}_M}}\gg L_{\mathrm{Pl}}$, making a a semi-classical mean field treatment of the boundary layer feasible.

The entropy of the thin shell is obtained from the equation of state, $p=\rho =(a^2/8\pi G){(k_{B}T/\hslash )}^2$, where we have introduced G for dimensional reasons so that $a^2$ is a dimensionless constant. By the standard thermodynamic Gibbs relation, $Ts=p+\rho$ for a relativistic fluid with zero chemical potential, and, hence, the local specific entropy density is

$$s\left(r\right)=\frac{a^2{k}_B^{2}T\left(r\right)}{4\pi {\hslash }^{2}G}=\frac{a{k}_B}{\hslash }{\left(\frac{p}{2\pi G}\right)}^{\frac{1}{2}}=\frac{a{k}_B}{4\pi \hslash Gr}{w}^{\frac{1}{2}}$$

(17)

for local temperature $T\left(r\right)$. The entropy of the fluid within the shell is thus

$$S=4\pi {\int }_{r_1}^{r_2}\frac{s{r}^{2}dr}{\sqrt{h}}=\frac{a{k}_{{}_B}{r}_{{}_M}^2}{\hslash G}\sqrt{\epsilon }ln\left(\frac{w_1}{w_2}\right)\sim a{k}_B\frac{M\ell }{\hslash }$$

(18)

and of order $k_{B}M\ell /\hslash$ to leading order in $\epsilon$, assuming $a,w_1,w_2$ are $\mathcal{O}\left(1\right)$. Since the interior region I has ${\rho }_{{}_V}=-p_{{}_V}$, ${\left(Ts\right)}_{{}_V}=p_{{}_V}+{\rho }_{{}_V}$ vanishes there. This is in accord with a GBEC having equation of state ${\rho }_{{}_V}=-p_{{}_V}$ being a single coherent macroscopic quantum state with zero entropy. Thus, the entropy of the entire compact quasi-black hole (QBH) is given by the entropy of the shell alone. By (18) this is of order $k_B{(r_{{}_M}/L_{\mathrm{Pl}})}^{\frac{3}{2}}$ for $\ell \sim \sqrt{L_{\mathrm{Pl}}{r}_{{}_M}}$, or $S\sim \sqrt{\epsilon }{S}_{BH}\ll S_{BH}$, far smaller than the Bekenstein–Hawking entropy. Its $M^{3/2}$ scaling, furthermore, makes it comparable to the entropy of typical stellar progenitors of mass M, in the range of ${10}^{57}{k}_B$ to ${10}^{59}{k}_B$ for a solar mass and $M_{\odot }/m_N\sim {10}^{57}$ nucleons. Thus there is no information paradox arising from an enormous entropy unaccountably associated with a BH horizon, if the horizon is replaced by a thin boundary layer of this kind.

Since w is of order unity in the shell, the local temperature of the fluid within the shell is of order $T_H\sim \hslash /k_B{r}_{{}_M}$, so that the typical quanta are soft with wavelengths of order $r_{{}_M}$, and there is no transplanckian problem. Because of the global timelike Killing field t and absence of either an event horizon or an interior singularity, there is no loss of unitarity or conflict with quantum theory. As a static solution, neither the interior nor the shell emit Hawking radiation. A gravitational condensate star is both cold and dark, and, hence, in its external geometry and its appearance to distant observers indistinguishable from a BH.

The cold radiation fluid in the shell is confined to region II by the surface tensions at the timelike interfaces $r_1$ and $r_2$. These arise from the pressure discontinuities, $\Delta p_1\simeq H^2(3+w_1)/8\pi G$ and $\Delta p_2\simeq -w_2/32\pi G^3{M}^2$, and are calculable by the Lanczos–Israel junction conditions [16,17,18]. The non-zero angular components of the surface tension are 2

$$\begin{array}{cc}\hfill & {\mathcal{S}}_{\theta }^{\theta }{|}_{r=r_1}={\mathcal{S}}_{\varphi }^{\varphi }{|}_{r=r_1}\equiv -{\sigma }_1=\frac{1}{32\pi G^{2}M}\frac{(3+w_1)}{(1+w_1)}{\left(\frac{w_1}{\epsilon }\right)}^{\frac{1}{2}}\hfill \end{array}$$

(19a)

$$\begin{array}{cc}\hfill & {\mathcal{S}}_{\theta }^{\theta }{|}_{r=r_2}={\mathcal{S}}_{\varphi }^{\varphi }{|}_{r=r_2}\equiv -{\sigma }_2=-\frac{1}{32\pi G^{2}M}\frac{w_2}{(1+w_2)}{\left(\frac{w_2}{\epsilon }\right)}^{\frac{1}{2}}\hfill \end{array}$$

(19b)

to leading order in $\epsilon$, at $r_1$ and $r_2$, respectively. The signs correspond to the inner surface at $r_1$ exerting an outward force and the outer surface at $r_2$ exerting an inward force, i.e., both surface tensions exert a confining pressure on the shell region II. Clearly these large transverse surface tensions violate the perfect fluid ansatz at the interfacial boundaries. Nevertheless, since ${\epsilon }^{-\frac{1}{2}}\sim {(M/M_{\mathrm{pl}})}^{\frac{1}{2}}$, the surface tensions (19) are of order $M^{-\frac{1}{2}}$ and far from Planckian, so that the matching of the metric at the phase interfaces $r_1$ and $r_2$, analogous to that across stationary shocks in hydrodynamics, should be reliable. The time component of the surface stress tensor at $r_1$ and $r_2$ vanishes and makes no contribution to the Tolman–Misner–Sharp mass function $\mu \left(r\right)=2Gm\left(r\right)$ at either of the two interfaces.

The Misner–Sharp energy within the shell

$$E_{\mathrm{II}}=4\pi {\int }_{r_1}^{r_2}\rho r^{2}dr=\epsilon M{\int }_{w_2}^{w_1}\frac{dw}{w}(1+w)=\epsilon M\left[ln\left({\displaystyle \frac{w_1}{w_2}}\right)+w_1-w_2\right]$$

(20)

to leading order in $\epsilon$, is of order $M_{\mathrm{Pl}}$ and also extremely small. Hence essentially all the Misner–Sharp mass of the object comes from the energy density of the vacuum condensate in the interior, even though the shell is responsible for all of its entropy.

3. Stability

In order to be a physically realizable endpoint of gravitational collapse, any QBH candidate must be stable [20]. Since only region II is non-vacuum, with a ‘normal’ fluid and a positive heat capacity, it is clear that the solution is thermodynamically stable. The most direct way to demonstrate this stability is to work in the microcanonical ensemble with fixed total M, and show that the entropy functional

$$S=\frac{a{k}_{{}_B}}{\hslash G}{\int }_{r_1}^{r_2}rdr{\left(\frac{d\mu }{dr}\right)}^{\frac{1}{2}}{\left(1-\frac{\mu \left(r\right)}{r}\right)}^{-\frac{1}{2}}$$

(21)

for the $p=\rho$ fluid in region II is maximized under all variations of $\mu \left(r\right)$ with the endpoints $(r_1,r_2)$, equivalently $(w_1,w_2)$ fixed.

The first variation of this functional with the endpoints $r_1$ and $r_2$ fixed vanishes, i.e., $\delta S=0$ by the Einstein Equation (4) for a static, spherically symmetric star. Thus, any solution of Equations (4) and (5) is guaranteed to be an extremum of S [21]. This is also consistent with regarding Einstein’s equations as a form of hydrodynamics, strictly valid only for the long wavelength, gapless excitations in gravity. The second variation of (21) is

$${\delta }^{2}S=\frac{a{k}_B}{4\hslash G}{\int }_{r_1}^{r_2}rdr{\left(\frac{d\mu }{dr}\right)}^{-\frac{3}{2}}{h}^{-\frac{1}{2}}\left\{-{\left[\frac{d\left(\delta \mu \right)}{dr}\right]}^2+\frac{{\left(\delta \mu \right)}^2}{r^2{h}^2}\frac{d\mu }{dr}\left(1+\frac{d\mu }{dr}\right)\right\}$$

(22)

when evaluated on the solution. Associated with this quadratic form in $\delta \mu$ is a second order linear differential operator $\mathcal{L}$ of the Sturm–Liouville type, viz.

$$\mathcal{L}\chi =\frac{d}{dr}\left\{r{\left(\frac{d\mu }{dr}\right)}^{-\frac{3}{2}}{h}^{-\frac{1}{2}}\frac{d\chi }{dr}\right\}+\frac{h^{-\frac{5}{2}}}{r}{\left(\frac{d\mu }{dr}\right)}^{\frac{1}{2}}\left(1+\frac{d\mu }{dr}\right)\chi .$$

(23)

This operator possesses two solutions satisfying $\mathcal{L}{\chi }_{{}_0}=0$, obtained by variation of the classical solution, $\mu (r;r_1,r_2)$ with respect to the parameters $(r_1,r_2)$. Indeed by changing variables from r to w and using the explicit solution (11) and (12) it is readily verified that one solution to $\mathcal{L}{\chi }_{{}_0}=0$ is ${\chi }_{{}_0}=1-w$, from which the second linearly independent solution $(1-w)lnw+4$ may be obtained. Since these correspond to varying the positions of the $r_1,r_2$ interfaces, neither ${\chi }_{{}_0}$ vanishes at $(r_1,r_2)$ and neither is a true zero mode. However, we may set $\delta \mu ={\chi }_{{}_0}\psi$, where $\psi$ does vanish at the endpoints and insert this into the second variation (22). Integrating by parts, using the vanishing of $\delta \mu$ at the endpoints and $\mathcal{L}{\chi }_{{}_0}=0$ one obtains

$${\delta }^{2}S=-\frac{a{k}_B}{4\hslash G}{\int }_{r_1}^{r_2}rdr{\left(\frac{d\mu }{dr}\right)}^{-\frac{3}{2}}{h}^{-\frac{1}{2}}{\chi }_{{}_0}^2{\left(\frac{d\psi }{dr}\right)}^2<0$$

(24)

which is negative definite.

Thus, the entropy of the solution is maximized with respect to radial variations that vanish at the endpoints, i.e., those with fixed total energy. Since deformations with non-zero angular momentum decrease the entropy even further, stability under radial variations is sufficient to demonstrate that the solution is stable to all small perturbations. In the context of a hydrodynamic treatment, thermodynamic stability is also a necessary and sufficient condition for the dynamical stability of a static, spherically symmetric solution of Einstein’s equations [21].

4. Conclusions

A compact, non-singular solution of Einstein’s equations has been presented here as a possible stable alternative to BHs for the endpoint of gravitational collapse [1]. Realizing this alternative requires that a quantum gravitational vacuum phase transition intervene and allow the vacuum energy ${\rho }_{{}_V}=-p_{{}_V}$ to change before the classical event horizon or a trapped surface can form. Although only the static spherically symmetric case has been considered, it is clear on physical grounds that axisymmetric rotating solutions should exist as well. Since the entropy of these objects is of the order of magnitude of a typical stellar progenitor, or less, there is no huge BH entropy to be explained and instead a process of entropy shedding, as in a supernova, is needed to produce a cold GBEC or ‘grava(c)star’ remnant.

In this paper we have assumed that the thin boundary layer where the quantum phase transition occurs can be described as a relativistic fluid with maximally stiff equation of state $p=+\rho$, where the speed of light is equal to the speed of sound. Although this is a phenomenological model, the possibility that such a boundary layer could be expected to produce excitations bearing the imprint of its fundamental normal mode vibration frequencies when struck should be robust and serve to distinguish gravastars from black holes observationally. These surface excitations may also provide a more efficient central engine for astrophysical sources to impart energy to accreting matter, producing ultra-high energy particles, gamma rays and gravitational radiation. Finally, the interior dS region with $p_{{}_V}=-{\rho }_{{}_V}$ may be interpreted also as a cosmological spacetime, with the horizon of the expanding universe replaced by a quantum phase interface.