Gravitational Condensate Stars: An Alternative to Black Holes
Pawel O. Mazur$^1$
and Emil Mottola$^2$
{A new solution for the endpoint of gravitational collapse is proposed. By extending the concept of Bose-Einstein condensation to gravitational systems, a cold, compact object with an interior de Sitter condensate phase and an exterior Schwarzschild geometry of arbitrary total mass $M$ is constructed. These are separated by a phase boundary with a small but finite thickness $\ell$ of fluid with eq. of state $p=+\rho c^2$, replacing both the Schwarzschild and de Sitter classical horizons. The new solution has no singularities, no event horizons, and a global time. Its entropy is maximized under small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, which is of order $k_{_B}\ell Mc/\hbar$, instead of the Bekenstein-Hawking entropy, $S_{_{BH}}= 4\pi k_{_B} G M^2/\hbar c$. Unlike black holes, a collapsed star of this kind is thermodynamically stable and has no information paradox.}
star.tex
\documentstyle[prl,aps,twocolumn]{revtex}
\input{title2.tex}
\begin{document}
\def\simlt{\stackrel{<}{{}_\sim}}
\def\simgt{\stackrel{>}{{}_\sim}}
\title{Gravitational Condensate Stars: An Alternative to Black Holes}   
\author{Pawel O. Mazur$^1$, and 
Emil Mottola$^2$} 
\preprint{LA-UR-01-5067}
\address{$^1$ Department of Physics and Astronomy, University of South
Carolina, Columbia, SC 29208, USA}
\address{$^2$ Theoretical Division, T-8, Los Alamos National 
Laboratory,
MS B285, Los Alamos, NM 87545, USA}
%\date{\today}
\abstract
{\small{A new solution for the endpoint of gravitational collapse is proposed.
By extending the concept of Bose-Einstein condensation to gravitational systems,
a cold, compact object with an interior de Sitter condensate phase 
and an exterior Schwarzschild geometry of arbitrary total mass $M$ is constructed. 
These are separated by a phase boundary with a small but finite thickness 
$\ell$ of fluid with eq. of state $p=+\rho c^2$, replacing both the Schwarzschild 
and de Sitter classical horizons. The new solution has no 
singularities, no event horizons, and a global time. Its entropy is maximized under 
small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, 
which is of order $k_{_B}\ell Mc/\hbar$, instead of the Bekenstein-Hawking entropy,
$S_{_{BH}}= 4\pi k_{_B} G M^2/\hbar c$. Unlike black holes, 
a collapsed star of this kind is thermodynamically stable and has no information 
paradox.}}
\pacs{04.40.-b, 04.60.-m, 04.70.-s, 73.43.Nq \qquad LA-UR-01-5067}
\maketitle2
\narrowtext

{\it Introduction.} The vacuum Einstein eqs. of classical general relativity possess a 
well-known solution for an isolated mass $M$, with the static, spherically symmetric line element,
\begin{equation}
ds^2 = -f(r)\, dt^2 + {dr^2 \over h(r)} + r^2\left( d\theta^2 + \sin^2\theta\,d\phi^2\right)
\,,
\label{sphsta}
\end{equation}
where the functions $f(r)$ and $h(r)$ are given by
\begin{equation}
f(r) = h(r) = 1 - {2GM\over r}\,,
\label{Sch}
\end{equation}
in units where $c=1$.  The dynamical singularity of this Schwarzschild metric at $r=0$ 
with its infinite tidal forces clearly signals a breakdown of the vacuum Einstein eqs. 
The kinematical singularity at the Schwarzschild radius $R_{_S} = 2GM$ 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. Since the curvature tensor is finite at $r=R_{_S}$, the 
singularity of the metric (\ref{sphsta})-(\ref{Sch}) there can be removed by a
suitable change of coordinates in the classical theory. A classical point test particle
freely falling through the event horizon experiences nothing special at $r=R_{_S}$.

The physics at the event horizon may be quite different when $\hbar \neq 0$.
Consider the same photon of finite asymptotic frequency $\omega$ far from the black hole. 
Even if $\hbar\omega$ is arbitrarily small, the infrared photon acquires 
a local energy $E=\hbar \omega f^{-{1\over 2}}$, which diverges at $r=R_{_S}$. 
Hence there is no {\it a priori} small parameter controlling deformation of the local
geometry near $r=R_{_S}$ due to quantum effects.

In the semi-classical approximation, when a massless field such as that of the photon 
is quantized in the fixed Schwarzschild background, one finds that the black hole radiates 
these quanta with a thermal spectrum at the asymptotic Hawking temperature,
$T_{_H} = {\hbar/8\pi k_B GM}$ \cite{Haw}. 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 $f^{-1}$ infinite blue shift factor at the horizon \cite{BirDav}. 
The wavelengths contributing to these quantum stresses are of order $R_{_S}$ and
hence are non-local on the scale of the hole. One has only to insert such a diverging 
energy density and pressure into the semi-classical Einstein eqs. (\ref{Eins}) below 
to see that the geometry near $r=R_{_S}$ is changed significantly from the classical 
Schwarzschild form. Unlike the classical kinematic 
singularity in (\ref{Sch}), such non-local semi-classical backreaction 
effects near $R_{_S}$ cannot be removed by a local coordinate 
transformation. 

Furthermore, the inverse dependence of $T_{_H}$ on $M$ implies that a black hole in thermal 
equilibrium with its own Hawking radiation has negative specific heat and 
therefore is unstable to thermodynamic fluctuations \cite{Hawb}. Energy conservation plus a 
thermal radiation spectrum also imply that the black hole has an enormous entropy,
$S_{_{BH}}\simeq 10^{77} k_{_B}(M/M_{\odot})^2$ \cite{Bek}, far in excess of a typical 
stellar progenitor. The associated information paradox has been conjectured to require 
an alteration in the principles of relativity, or quantum mechanics, or both. 

In lieu of such fundamental revision of the laws of physics, it is reasonable 
to examine alternatives to the strictly classical view of the event horizon as a 
harmless kinematic singularity, when $\hbar \neq 0$ 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 \cite{ZP,tH}. In fact,
$S = 4\,{\kappa + 1\over 7 \kappa + 1} S_{_{BH}}$, for a fluid with the eq. of state, 
$p =\kappa \rho$, becoming equal to the Bekenstein-Hawking entropy 
for $\kappa = 1$. Despite such intriguing features, these models have 
huge (Planckian) energy densities near $R_{_S}$ and a negative mass singularity at 
$r=0$, so that the Einstein eqs. are not reliable in the interior region. 

On the other hand, instead of abandoning quantum mechanics, demanding that its 
principles apply to self-gravitating systems implies that their specific heat 
$\propto \langle (\Delta E)^2 \rangle$ must be positive.
This leads to the proposal that quantum black hole entropy is carried 
{\it not} by Planckian but by {\it soft} quanta with $E \sim \hbar/GM$ \cite{Maz}.
More recently another proposal for incorporating quantum effects has been made, 
{\it viz.} that the horizon becomes a critical surface of a phase transition
in the quantum theory, supported by an interior region with eq. of state,  
$p = - \rho < 0$ \cite{CHLS}. Such a vacuum eq. of
state, first proposed by Gliner \cite{Glin} for the endpoint of gravitational 
collapse, is equivalent to a positive cosmological constant in Einstein's
eqs., and does not satisfy the energy condition $\rho + 3p \ge 0$
needed to prove the classical singularity theorems. 

In this Letter we show that an explicit static solution of Einstein's 
eqs. taking quantum considerations into account exists, with the critical 
surface of ref. \cite{CHLS} 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 black holes as the endpoint of
gravitational collapse, possibly with different observational signatures.

The assumption required for this solution to exist is that gravity undergoes a vacuum 
rearrangement phase transition in the vicinity of $r=R_{_S}$. Specifically, in
this region quantum fluctuations on the scale $R_{_S}$ dominate the 
$T^t_{\ t} \sim T^r_{\ r}$ components of 
the stress tensor, which grow so large that the eq. of state approaches the most extreme one 
allowed by causality, $\rho =p$.  As this causal limit is reached, the 
interior spacetime becomes unstable to the formation of a new kind of gravitational 
Bose-Einstein condensate (GBEC) described by a non-zero macroscopic order parameter $\Psi$. 
If $\vert \Psi\vert^2$ is a constant scalar, it must couple to Einstein's eqs. in
the same way as a cosmological term, and the eq. of state of the interior region must be 
$\rho_{_V} = -p_{_V}= V(\vert\Psi\vert^2)$. A suggestion for the effective theory 
incorporating the effects of quantum anomalies that could give rise to both 
this interior GBEC phase and the $\rho = p$ shell has been presented elsewhere \cite{MM}. 
Here we forego any discussion of the details of the phase transition or collapse 
process and consider only the compact, stable endpoint of gravitational collapse, 
by solving the static Einstein's eqs. with the specified eqs. of state. 
For a recent review of investigations of other non-singular quasi-black-hole (QBH)
models see ref. \cite{Dym}.

{\it Solution of Eqs.} The eqs. to be solved are the Einstein eqs. for a
perfect fluid at rest in the coordinates (\ref{sphsta}), {\it viz.}
\begin{mathletters}
\begin{eqnarray}
-G_{\ t}^t &=&  {1\over r^2} {d\over dr}\left[r\left(1 - h\right)\right]
= -8\pi G\, T_{\ t}^t = 8\pi G \rho\,,\\
G_{\ r}^r &=& {h\over r f} {d f\over dr}  + {1\over r^2} \left(h -1\right) =
8\pi G \,T_{\ r}^r = 8\pi G p\,,
\end{eqnarray}
\label{Eins}
\end{mathletters}
\vspace{-.2cm}

\noindent 
together with the conservation eq., 
\begin{equation}
\nabla_a T^a_{\ r} = {d p\over dr} + {\rho + p\over 2f} \,{d f\over dr} = 0\,,
\label{cons}
\end{equation}
which ensures that the other components of the Einstein eqs. are satisfied.
These three first order eqs. for the four unknown functions, $f, h, \rho$, and $p$ 
become closed when an eq. of state for the fluid, relating $p$ and $\rho$ is specified.
Because of the considerations above we allow for three different regions
with the three different eqs. of state,
\begin{equation}
\begin{array}{clcl}
{\rm I.}\ & {\rm Interior:}\ & 0 \le r < r_1\,,\ &\rho = - p \,,\\
{\rm II.}\ & {\rm Shell:}\ & r_1 < r < r_2\,,\ &\rho = + p\,,\\
{\rm III.}\ & {\rm Exterior:}\ & r_2 < r\,,\ &\rho = p = 0\,.
\end{array}
\end{equation}

At the interfaces $r=r_1$ and $r=r_2$, we require the metric coefficients
$r,f$ and $h$ to be continuous, although the first derivatives of $f$, $h$ 
and $p$ must be discontinuous from the first order eqs. (\ref{Eins}) and 
(\ref{cons}).

In the interior region $\rho = - p$ is a constant from 
(\ref{cons}). Let us call this constant $\rho_{_V} = 3H_0^2/8\pi G$. If we
require that the origin is free of any mass singularity then the
interior is determined to be a region of de Sitter spacetime in static coordinates, {\it i.e.}
\begin{equation}
{\rm I.}\qquad f(r) = C\,h(r) = C\,(1 - H_0^2\,r^2)\,,\quad 0 \le r \le r_1\,.
\label{deS}
\end{equation}
The unique solution in the exterior vacuum region which
approaches flat spacetime as $r \rightarrow \infty$ is a region of 
Schwarzschild spacetime (\ref{Sch}),
{\it viz.}
\begin{equation}
{\rm III.} \qquad f(r) = h(r) = 1 - {2GM\over r}\,,\qquad  r_2 \le r\,.
\end{equation}
The integration constants $C, H_0$ and $M$ are arbitrary.

The only non-vacuum region is region II. Let us define the dimensionless 
variable $w$ by $w\equiv 8\pi G r^2 p$, so that
eqs. (\ref{Eins})-(\ref{cons}) with $\rho = p$ may be recast in the form,
\begin{mathletters}
\begin{eqnarray}
{dr\over r} &=& {dh \over 1-w-h}\,,\\
{dh\over h} &=& -{1-w-h \over 1 + w - 3h}\, {dw\over w}\,.
\label{ueqb}
\end{eqnarray}
\label{ueq}
\end{mathletters}

\vspace{-.2cm}

\noindent 
together with $p f \propto wf/r^2$ a constant.
The first of eqs. (\ref{ueq}) is equivalent to the definition of the (rescaled)
Tolman mass function by $h = 1 - \mu/r$ and $d\mu(r) = 2\,G\,d m(r) = 8\pi
G\, \rho r^2\, dr = w\, dr$ within the shell. The second eq. (\ref{ueqb}) 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 (\ref{ueqb}) to leading order, and integrate
it immediately to obtain 
\begin{equation}
h \equiv 1- {\mu\over r} \simeq  \epsilon\ {(1 + w)^2\over w \ }\ll 1\,,
\label{hshell}
\end{equation}
in region II, where $\epsilon$ is an integration constant. 
Because of the condition $h \ll 1$ we require $\epsilon \ll 1$, with $w$
of order unity. Making use of eqs. (\ref{ueq}) and (\ref{hshell}) we have
\begin{equation}
{dr \over r} \simeq - \epsilon\, dw\, {(1 + w)\over w^2}\,.
\label{req}
\end{equation}
Because of the approximation $\epsilon \ll 1$, the radius $r$ hardly changes within 
region II, and $dr$ is of order $\epsilon \,dw$. The final
unknown function $f$ is given by $f = (r/r_1)^2 (w_1/w) f(r_1)
\simeq  (w_1/w) f(r_1)$ for small $\epsilon$.

Now requiring continuity of the metric coefficients $f$ and $h$ at both $r_1$ and $r_2$
gives the conditions,
\begin{mathletters}
\begin{eqnarray}
h(r_1) &=& 1-H_0^2 r_1^2 \simeq \epsilon \,{(1+ w_1)^2 \over
w_1}\,,\\ 
h(r_2) &=& 1 - {2GM\over r_2} \simeq  \epsilon \,{(1+ w_2)^2 \over w_2}\,,\\
{f(r_2)\over h(r_2)} &=& 1 \simeq {w_1\over w_2} {f(r_1)\over h(r_2)} = C
\left({1+w_1\over 1+w_2}\right)^2
\end{eqnarray}
\label{match}
\end{mathletters}

\vspace{-.2cm}

\noindent 
Together with $r_2/r_1$ from the solution of (\ref{req}) this gives four relations among
the eight integration constants $(r_1, r_2, w_1, w_2, H_0, M, C, \epsilon)$. Hence the
first four can be eliminated in favor of $H_0, M, C$ and $\epsilon \ll 1$, and
we have a four parameter family of static solutions. Assuming that 
$(r_1, r_2, w_1, w_2)$ remain finite as $\epsilon\rightarrow 0$, {\it i.e.}
are of order  $\epsilon^0$, we obtain from (\ref{req}) and (\ref{match})
that $r_1 \simeq H_0^{-1} \simeq r_2 \simeq 2GM$, but
$1 -H_0\,r_1,\ r_2-2GM, w_1-w_2$ and $C-1$ are all of order $\epsilon$, while 
$r_2 - r_1$ is ${\cal O}(\epsilon^2)$.

{\it Principal Features.} If $\epsilon>0$ both $f$ and $h$ are of order 
$\epsilon$ and approximately constant in region II, but are nowhere vanishing. 
Hence there is no event horizon, and $t$ is a global time. A photon experiences 
a very large, ${\cal O}(\epsilon^{-{1\over 2}})$ but finite blue shift 
in falling into the shell from infinity.
The proper thickness of the shell,
\begin{equation}
\ell = \int_{r_1}^{r_2}\, dr\,h^{-{1\over 2}}  \simeq R_{_S} 
\epsilon^{1\over 2}\int_{w_2}^{w_1} dw\, w^{-{3\over 2}}
\sim \epsilon^{3\over 2} R_{_S}
\label{thickness}
\end{equation}
is small compared to $R_{_S}$. 
Although $\ell$ is arbitrary here, presumably it is fixed by microscopic physics and is
not greater than the Planck length by more than a few orders of magnitude. If
we assume that $\ell$ is a fixed multiple of the Planck length and independent
of $M$, then $\epsilon \sim (\ell/GM)^{2\over 3} \sim (M_{Pl}/M)^{2\over 3} \simeq 
10^{-25\pm 1}$ for a solar mass object, which certainly justifies the small 
$\epsilon$ approximation in this case. With $\ell$ fixed, $H_0^{-1} \simeq 2GM$
and $C = 1 + {\cal O}(\epsilon)$, the only remaining free parameter is $M$.

The entropy of the shell is obtained from the eq. of state, 
$p = \rho = (a^2/8\pi G) (k_{_B} T/\hbar)^2$, where we have introduced $G$ 
for dimensional reasons so that $a^2$ is a dimensionless constant. 
By the standard thermodynamic relation, $T s = p + \rho$ for a relativistic 
fluid with zero chemical potential, and the local specific entropy density 
$s(r) = a^2k_{_B}^2 T(r)/4\pi\hbar^2 G = a(k_{_B}/\hbar)(p/2\pi G)^{1\over 2}$ 
for local temperature $T(r)$. Thus $s = (ak_{_B}/4\pi\hbar Gr)\,w^{1\over 2}$ 
and the entropy of the fluid within the shell is 
\begin{equation}
S =4\pi \int_{r_1}^{r_2}s\,r^2\,dr\,h^{-{1\over 2}}\simeq 
{ak_{_B}R_{_S}^2 \over \hbar G}\ 
\epsilon^{1\over 2} \ \ln \left({w_1\over w_2}\right)\,.
\label{entsh}
\end{equation}
Using $w_1/w_2 = 1 + {\cal O}(\epsilon)$ and (\ref{thickness}) this is of order
\begin{equation}
S \sim a\,k_{_B}\, {M\over \hbar} \,R_{_S}\epsilon^{3\over 2}
\sim a\, k_{_B}{M\ell \over \hbar}
\ll S_{_{BH}}\,.
\label{entest}
\end{equation} 

Since the interior region I has $\rho_{_V} = -p_{_V}$, $T s$ vanishes there, 
as could be anticipated for a GBEC described by a macroscopic
single quantum state. The entropy of the entire quasi-black hole (QBH) is given then
by (\ref{entsh}) or (\ref{entest}), which is of order $10^{38}k_{_B}
(\ell/L_{Pl})$ for a solar mass object. This is some $38$ orders of magnitude
smaller than the Bekenstein-Hawking entropy for the same mass $M$, and  $20 \pm
1$ orders of magnitude lower than a typical stellar progenitor, which have
entropies  in the range of $10^{57} k_{_B}$ to $10^{59} k_{_B}$ for $M/m_{_N}
\sim 10^{57}$  nucleons. Since $w$ is of order unity in the shell while $r
\simeq R_{_S}$, the {\it local} temperature of the fluid within the shell is
of order $T_{_H}  \sim \hbar/k_{_B}GM$, {\it i.e.} the typical quanta are {\it soft}.  
Because of the absence of an event horizon, the QBH does not emit Hawking radiation. 
In fact, the thermal wavelength of the soft quanta is of order the linear dimension 
$R_{_S}$ of the object, so that quantum zero point fluctuations and finite
size effects are competitive with classical thermodynamics. If $w$ is somewhat smaller 
than unity, the shell need not emit any thermal radiation at all. A  discussion of the 
corrections to the classical hydrodynamic relations this entails must await a fuller 
quantum treatment. With no thermal emission the GBEC remnant then would be both 
ultracold and completely dark. 

The extremely 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, $\Delta p_1 \simeq H_0^2\, (3 + w_1)/8\pi G$ and
$\Delta p_2 \simeq -w_2/ 32\pi G^3M^2$, and are calculable by the Israel junction 
conditions \cite{Isr,Kuz}. We find the non-zero angular components of 
the surface tension to leading order in $\epsilon$,
\begin{mathletters}
\begin{eqnarray}
&&\quad \sigma_{\ \theta}^{\theta} = \sigma_{\ \phi}^{\phi} \simeq {1\over 32\pi G^2M}
{(3 + w_1)\over (1 + w_1)}\left({w_1\over \epsilon}\right)^{1\over 2}\,,\\
&&\quad \sigma_{\ \theta}^{\theta} = \sigma_{\ \phi}^{\phi} \simeq -{1\over
32\pi G^2M}  {w_2\over (1 + w_2)}\left({w_2\over \epsilon}\right)^{1\over 2}.
\end{eqnarray}
\label{surf}
\end{mathletters}

\vspace{-.2cm}

\noindent
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, 
{\it i.e.} both surface tensions exert a confining pressure on the shell region II. The
time component $\sigma_{\ t}^t =0$, corresponding to vanishing contribution
to the Tolman mass function $m(r)$ at the two interfaces. Since $\epsilon^{-{1\over 2}}
\sim (M/M_{pl})^{1\over 3}$, the surface tensions (\ref{surf}) are of order
$M^{-{2\over 3}}$ and far from Planckian. Hence 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. Resolving the interfaces will require going
beyond Einstein's eqs. to a more microscopic description of the quantum phase 
transition.

The energy within the shell (as measured at infinity),
\begin{equation}
E_{\rm II} = 4\pi \int_{r_1}^{r_2}\rho\,r^2 dr \simeq \epsilon M
\int_{w_2}^{w_1}{dw\over w}\,(1 + w)\sim  \epsilon^2 M 
\end{equation}
is also extremely small. Hence essentially all the 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.

{\it Stability.} In order to be a physically realizable endpoint of gravitational
collapse, any quasi-black hole candidate must be stable \cite{Maz}. 
Since only the 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,
\begin{equation}
S= {a k_{_B}\over \hbar G\,}\int_{r_1}^{r_2}r \,dr\, \left({d\mu\over dr}\right)^{1\over 2}
\left(1 - {\mu(r) \over r}\right)^{-{1\over 2}}\,,
\label{entropy}
\end{equation}
is maximized under all variations of $\mu (r)$ in region II with the endpoints $(r_1, r_2)$,
or equivalently $(w_1, w_2)$ fixed.

The first variation of this functional with the endpoints 
$r_1$ and $r_2$ fixed vanishes, {\it i.e.} $\delta S =0$ by the Einstein eqs.
(\ref{Eins}) for a static, spherically symmetric star. Thus any solution of 
eqs. (\ref{Eins})-({\ref{cons}) is guaranteed to be an extremum of $S$ \cite{Coc}.
This is also consistent with regarding Einstein's eqs. as a form of hydrodynamics, 
strictly valid only for the long wavelength, gapless excitations in gravity.

The second variation of (\ref{entropy}) is 
\begin{eqnarray}
\delta^2 S&=&{a k_{_B}\over 4\hbar G\,}\int_{r_1}^{r_2}r\,
dr\, \left({d\mu\over dr}\right)^{-{3\over 2}}
h^{-\frac{1}{2}}\,\times\nonumber\\
&&\left\{-\left[{d(\delta\mu)\over dr}\right]^2 +
{(\delta\mu)^2\over r^2h^2}{d\mu\over dr}\left(1+ {d\mu\over dr}\right)\right\}\,,
\label{varent}
\end{eqnarray}
when evaluated on the solution.
Associated with this quadratic form in $\delta\mu$ is a second order linear differential 
operator $\cal L$ of the Sturm-Liouville type, {\it viz.} 
\begin{eqnarray}
&& {d\over dr} \left\{r\left({d\mu\over dr}\right)^{-{3\over 2}}
h^{-{1\over 2}} {d\chi\over dr}\right\}+ {h^{-{5\over 2}}\over r} 
\left({d\mu\over dr}\right)^{1\over 2}\left(1+ {d\mu\over dr}\right)\chi \nonumber\\
&& \qquad \equiv {\cal L} \chi \,.
\label{stli}
\end{eqnarray}
This operator possesses two solutions satisfying ${\cal L}\chi_{_0} = 0$, obtained by
variation of the classical solution, $\mu (r; r_1, r_2)$ with respect
to the parameters $(r_1, r_2)$. Since these correspond to varying
the positions of the interfaces, $\chi_{_0}$ does not vanish at $(r_1, r_2)$
and neither function is a true zero mode. For example, it is easily verified that 
one solution is $\chi_{_0} = 1 -w$. However, we may set 
$\delta \mu \equiv \chi_{_0}\, \psi$, where $\psi$ does vanish at the endpoints and insert
this into the second variation (\ref{varent}). Integrating by parts, using
the vanishing of $\delta \mu$ at the endpoints and ${\cal L}\chi_{_0} = 0$ gives
\begin{equation}
\delta^2 S= -{a k_{_B}\over 4\hbar G\,}\int_{r_1}^{r_2}r\, dr\,
\left({d\mu\over dr}\right)^{-{3\over 2}}
h^{-\frac{1}{2}}\chi_{_0}^2\,\left({d\psi\over dr}\right)^2 < 0\,.
\label{secvar}
\end{equation}

Thus the entropy of the solution is maximized with respect to radial
variations that vanish at the endpoints, {\it 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 {\it dynamical} stability of a static, spherically symmetric solution of 
Einstein's eqs. 
\cite{Coc}.

{\it Conclusions.} A compact, non-singular solution of Einstein's eqs.
has been presented here as a possible stable alternative to black holes for the 
endpoint of gravitational collapse. Realizing this alternative requires that a 
quantum gravitational vacuum phase transition intervene before the classical event 
horizon 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 some $20$ orders of 
magnitude smaller than that of a typical stellar progenitor, there is no entropy 
paradox and instead a violent process of entropy shedding, as in a supernova, is 
needed to produce a cold GBEC or `grava(c)star' remnant. The shell with its maximally 
stiff eq. of state $p=\rho c^2$, where the speed of light is equal to the speed of 
sound could be expected to produce outgoing shock fronts when struck. 
These may serve to distinguish gravastars from black holes observationally, and
possibly provide a more efficient central engine for energetic
astrophysical sources. The spectra of gravitational radiation from a struck 
gravastar should bear the imprint of its fundamental frequencies of vibration. Finally,
we note that the interior de Sitter region with $p = -\rho c^2$ may be interpreted also
as a cosmological spacetime, with the horizon of the expanding universe replaced 
by a quantum phase interface.
\vspace{-.5cm}\break
\vspace{.25cm}

\noindent
Research of P. O. M. supported in part by an NSF grant.

\vspace{-.7cm}
\begin{references}
\vspace{-1.7cm}

\bibitem{Haw} S. W. Hawking, Nature {\bf 248}, 30 (1974).
\bibitem{BirDav} {\it Quantum fields in curved space}, N. D. Birrell and P. C. W.
Davies, Cambridge (1982), and references therein.
\bibitem{Hawb} S. W. Hawking, Phys. Rev. {\bf D13}, 191 (1976).
\bibitem{Bek} J. D. Bekenstein, Phys. Rev. {\bf D7}, 2333 (1973)
\bibitem{ZP} W. H. Zurek and D. N. Page, Phys. Rev. {\bf D29}, 628 (1984).
\bibitem{tH} G. 't Hooft, Nucl. Phys. Proc. Suppl. {\bf 68}, 174 (1998).
\bibitem{Maz} P. O. Mazur, Acta Phys. Polon. {\bf B27} 1849 (1996).
\bibitem{CHLS} G. Chapline, E. Hohlfield, R. B. Laughlin, and D. I. \break
Santiago, Phil. Mag. {\bf B81}, 235 (2001).
\bibitem{Glin} E. B. Gliner, Sov. Phys. JETP {\bf 22} , 378 (1966).
\bibitem{MM} P. O. Mazur and E. Mottola, Phys. Rev. {\bf D64}, 104022 (2001),
and references therein.
\bibitem{Dym} I. Dymnikova, e-print gr-qc/0010016 (unpublished).
\bibitem{Isr} W. Israel, N. Cimento {\bf B44}, 1 (1966); {\bf B48}, 463 (1967).
\bibitem{Kuz} V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys. Rev.
{\bf D36}, 2919 (1987).
\bibitem{Coc} W. J. Cocke, Ann. Inst. H. Poincar{\'e}, {\bf 2}, 283 (1965).

\end{references}


\end{document}

which gives to first order in $\epsilon$,
\begin{equation}
r \simeq r_1 \left[ 1 - \epsilon\ln \left({w\over w_1}\right) +
\epsilon \left( {1\over w} - {1\over w_1} \right)\right]\,.
\label{rshell}
\end{equation} 
By changing variables from $r$ to $w$ and using the
explicit solution (\ref{hshell})-(\ref{req}) it is readily verified that
$\chi_{_0}=1-w$ satisfies ${\cal L}\chi_{_0}=0$, from which the second
linearly independent solution $(1-w)\ln\,w + 4$ may be obtained. Substituting
then {\it e.g.} $\delta\mu = (1-w) \psi (w)$ in (\ref{varent}) brings 
$\delta^2 S$ into the form (\ref{secvar}) which is explicitly
negative definite.
Hence 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. 
\begin{mathletters}
\begin{eqnarray}
h(r_1) &=& 1-H_0^2 r_1^2 \simeq \epsilon \,{(1+ w_1)^2 \over
w_1}\,,\\ 
h(r_2) &=& 1 - {2GM\over r_2} \simeq  \epsilon \,{(1+ w_2)^2 \over w_2}\,,\\
{f(r_2)\over h(r_2)} &=& 1 \simeq {w_1\over w_2} {f(r_1)\over h(r_2)} = C
\left({1+w_1\over 1+w_2}\right)^2
\end{eqnarray}
\label{match}
\end{mathletters}

\noindent 
Earlier investigations which attempted to include the 
backreaction of the Hawking radiation in a self-consistent way demonstrate the potential 
importance of quantum effects on the geometry near $r=R_{_S}$Its entropy is a local maximum of the hydrodynamic 
entropy, whose modest value given by (\ref{entsh})-(\ref{entest}) below 
is easily attainable in a physical collapse process from a progenitor with a 
comparable mass. 


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gr-qc0109035.extracted_bib
\bibitem{Haw} S. W. Hawking, Nature {\bf 248}, 30 (1974).

\bibitem{BirDav} {\it Quantum fields in curved space}, N. D. Birrell and P. C. W.
Davies, Cambridge (1982), and references therein.

\bibitem{Hawb} S. W. Hawking, Phys. Rev. {\bf D13}, 191 (1976).

\bibitem{Bek} J. D. Bekenstein, Phys. Rev. {\bf D7}, 2333 (1973)

\bibitem{ZP} W. H. Zurek and D. N. Page, Phys. Rev. {\bf D29}, 628 (1984).

\bibitem{tH} G. 't Hooft, Nucl. Phys. Proc. Suppl. {\bf 68}, 174 (1998).

\bibitem{Maz} P. O. Mazur, Acta Phys. Polon. {\bf B27} 1849 (1996).

\bibitem{CHLS} G. Chapline, E. Hohlfield, R. B. Laughlin, and D. I. \break
Santiago, Phil. Mag. {\bf B81}, 235 (2001).

\bibitem{Glin} E. B. Gliner, Sov. Phys. JETP {\bf 22} , 378 (1966).

\bibitem{MM} P. O. Mazur and E. Mottola, Phys. Rev. {\bf D64}, 104022 (2001),
and references therein.

\bibitem{Dym} I. Dymnikova, e-print gr-qc/0010016 (unpublished).

\bibitem{Isr} W. Israel, N. Cimento {\bf B44}, 1 (1966); {\bf B48}, 463 (1967).

\bibitem{Kuz} V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys. Rev.
{\bf D36}, 2919 (1987).

\bibitem{Coc} W. J. Cocke, Ann. Inst. H. Poincar{\'e}, {\bf 2}, 283 (1965).

