11institutetext: Emil Mottola (✉) 22institutetext: Dept. of Physics &
Astronomy, Univ. of New Mexico 22email<EMAIL_ADDRESS>
# Gravitational Vacuum Condensate Stars

Emil Mottola

###### Abstract

Gravitational vacuum condensate stars, proposed as the endpoint of
gravitational collapse consistent with quantum theory, are reviewed.
Gravastars are cold, low entropy, maximally compact objects characterized by a
surface boundary layer and physical surface tension, instead of an event
horizon. Within this thin boundary layer the effective vacuum energy
$\Lambda_{\rm eff}$ changes rapidly, such that the interior of a non-rotating
gravastar is a non-singular static patch of de Sitter space with eq. of state
$p=-\rho$. Remarkably, essentially this same result is obtained by
extrapolating Schwarzschild’s 1916 constant density interior solution to its
compact limit, showing how the black hole singularity theorems and the
Buchdahl compactness bound are evaded. The surface stress tensor on the
horizon is determined by a modification of the Lanczos-Israel junction
conditions for null hypersurfaces, which is applied to rotating gravastar
solutions as well. The fundamental basis for the quantum phase transition at
the horizon is the stress tensor of the conformal anomaly, depending upon a
new light scalar field in the low energy effective action for gravity. This
scalar conformalon field allows the effective value of the vacuum energy,
described as a condensate of an exact 4-form abelian gauge field strength
$F=dA$, to change at the horizon. The resulting effective theory thus replaces
the fixed constant $\Lambda$ of classical general relativity, and its
apparently unnaturally large sensitivity to UV physics, with a dynamical
condensate whose ground state value in empty flat space is $\Lambda_{\rm
eff}=0$ identically. This provides both a solution to the cosmological
constant problem and an effective Lagrangian dynamical framework for the
boundary layer and interior of gravitational condensate stars. The status of
present observational constraints and prospects for detection of gravastars
through their gravitational wave and echo signatures are discussed.

##  1 The Issue of the Final State of Gravitational Collapse

The possibility that a gravitationally bound system could be so compact that
not even light can escape its surface was first imagined in the eighteenth
century by Rev. J. Michell Michell:1784 , and P.-S. Laplace Laplace:1796 ;
HawkEllis:1973 . Each reasoned (apparently independently) that since the
escape velocity $\rm v$ from a spherical mass $M$ of radius $r$ is given by
${\rm v}^{2}\\!=\\!2GM/r$ in Newtonian gravitation, light traveling at speed
$c$ would not be able to escape the surface if emitted from a radius less than
$r_{\\!{}_{M}}\\!=\\!2GM/c^{2}$. Since this radius is much smaller than the
actual size of astronomical bodies known at the time, such as the sun or the
planets, and with no astronomical evidence of the existence of the highly
compact objects for which $r_{\\!{}_{M}}$ would be relevant, the possibility
of such ‘dark stars’ from which light itself could not escape was not taken
very seriously, nor developed further for well over a century.

In 1915 A. Einstein introduced the theory of general relativity (GR). Soon
thereafter, K. Schwarzschild found an exact solution of the sourcefree
Einstein’s eqs. in the case of spherical symmetry. The general static,
spherically symmetric metric line element in GR can be expressed in the form

$ds^{2}=-f(r)\,c^{2}dt^{2}+\frac{dr^{2}\\!\\!}{\,h(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\vspace{-2mm}$
(1)

in terms of two functions of radius, $f(r)$ and $h(r)$. For the sourcefree
asymptotically flat Schwarzschild solution, $f(r)$ and $h(r)$ are equal and
given by

$f(r)=h(r)=1-\frac{2GM}{c^{2}\,r}=1-\frac{r_{\\!{}_{M}}\\!}{\\!r}$ (2)

determining the static geometry of empty sourcefree space exterior to a
spherically symmetric body of mass $M$. In these $(t,r)$ coordinates the
metric has two apparent singularities, one at the origin $r=0$ of spherical
coordinates, and the other of quite a different character on a sphere of
finite radius $r=r_{\\!{}_{M}}$, the same radius discussed by Michell and
Laplace in Newtonian gravity.

The singularity at $r=0$ is similar to that of a point charge in Coulomb’s law
where the electric field diverges. Likewise for (1)-(2) the Riemannian
curvature and Kretschmann scalar
$R_{\alpha\beta\gamma\lambda}R^{\alpha\beta\gamma\lambda}=12r_{\\!{}_{M}}^{2}/r^{6}$
diverge as $r\to 0$. In contrast the curvature remains finite at
$r=r_{\\!{}_{M}}$, and of order $1/r_{\\!{}_{M}}^{2}$, which is relatively
small for large $M$. However a light wave emitted from any $r\geq
r_{\\!{}_{M}}$ with local frequency $\omega_{\,\rm loc}(r)$ is gravitationally
redshifted according to the relation

$\omega_{\infty}=\omega_{\,\rm loc}(r)\,f^{\frac{1}{2}}=\omega_{\,\rm
loc}(r)\sqrt{1-\frac{r_{\\!{}_{M}}\\!}{\\!r}}=const.$ (3)

Thus light emitted at $r=r_{\\!{}_{M}}$ with any finite local frequency
$\omega_{\,\rm loc}(r_{\\!{}_{M}})$ becomes redshifted to zero frequency and
cannot propagate outwardly at all. The radius $r=r_{\\!{}_{M}}$ therefore
defines the critical surface or event horizon from beyond which no event can
ever be observed by an outside observer. Thus the wave description of light in
GR recovers exactly the ‘dark star’ surface that Michell and Laplace had
imagined within a Newtonian framework. Conversely and equivalently, (3) also
implies that an inwardly directed light wave with the finite frequency
$\omega_{\infty}$ far from the horizon is blueshifted to an infinite local
frequency upon reaching $r=r_{\\!{}_{M}}$.

One would expect to be able to remove at least the singularity at $r=0$, and
the infinite curvatures it produces by allowing for a stress tensor source to
Einstein’s eqs. of finite extent, rather than one concentrated all at the
origin. Indeed Schwarzschild also found such a regular interior solution
Schwarzs:1916 , shortly after finding the well-known exterior solution (1)-(2)
most often associated with his name.

To simplify matters, Schwarzschild assumed a model of the interior matter
composed of an incompressible fluid with isotropic pressure and constant mass
density

$\rho=\bar{\rho}\equiv\frac{3M}{4\pi
r_{\\!{}_{S}}^{3}}\equiv\frac{\,3H^{2}\\!\\!}{8\pi G}\qquad{\rm so\ \
that}\qquad H^{2}=\frac{r_{\\!{}_{M}}}{r_{\\!{}_{S}}^{3}}$ (4)

where $r_{\\!{}_{S}}$ is the radius of the star’s surface.111Here and often in
the following, units are employed where the speed of light $c=1$, unless
otherwise needed for emphasis. At this surface the pressure
$p(r_{\\!{}_{S}})=0$, and the interior solution for $r\leq r_{\\!{}_{S}}$ is
matched to the exterior one (1)-(2) for $r\geq r_{\\!{}_{S}}$ by requiring
$f(r_{\\!{}_{S}})=h(r_{\\!{}_{S}})=1-r_{\\!{}_{M}}/r_{\\!{}_{S}}>0$ to be
continuous there (although their derivatives need not be). With these boundary
conditions the metric for Schwarzschild’s interior solution can be expressed
in the form (1), but with

$\displaystyle h(r)=1-H^{2}r^{2}$ (5a) $\displaystyle
f(r)=\frac{D^{2}\\!}{\\!\\!4}\,\geq 0$ (5b) where $D\equiv
3\sqrt{\smash[b]{1-H^{2}r_{\\!{}_{S}}^{2}}}-\sqrt{1-H^{2}r^{2}}$ (5c) is
assumed to be non-vanishing, so that the pressure
$p(r)=\frac{\raisebox{1.0pt}{$\displaystyle\bar{\rho}$}}{\raisebox{-1.0pt}{$\displaystyle
D$}}\,\left[\sqrt{1-H^{2}r^{2}}-\sqrt{\smash[b]{1-H^{2}r_{\\!{}_{S}}^{2}}}\,\right]$
(5d)

is also finite (and positive) on the interval $r\in[0,r_{\\!{}_{S}}]$. This
assumption holds for

$r_{\\!{}_{S}}>\frac{9}{8}\,r_{\\!{}_{M}}$ (6)

in which case the solution (5d) is everywhere non-singular and finite, and the
first derivative $f^{\prime}(r_{\\!{}_{S}})=r_{\\!{}_{M}}/r_{\\!{}_{S}}^{2}$
also turns out to be continuous at the surface $r=r_{\\!{}_{S}}$.

Conversely, if the inequality (6) is not satisfied, then $D$ vanishes at

$r_{0}=3r_{\\!{}_{S}}\,\sqrt{1-\frac{8}{9}\frac{r_{\\!{}_{S}}}{r_{\\!{}_{M}}}}\quad\in\
[0,r_{\\!{}_{S}}]\quad{\rm for}\quad r_{\\!{}_{M}}\leq
r_{\\!{}_{S}}\leq\frac{9}{8}\,r_{\\!{}_{M}}$ (7)

where $f(r_{0})=0$, and the pressure (5d) $p(r_{0})\to\infty$ diverges. In
this case the solution (5d) becomes singular and may no longer be deemed
acceptable. Since $M=4\pi\bar{\rho}r_{\\!{}_{S}}^{3}/3$, the lower bound (6)
on $r_{\\!{}_{S}}$ translates to an upper bound on the mass

$\overline{\\!M}_{\rm crit}=\frac{4c^{3}\\!\\!}{9G}\left(\\!\frac{1}{3\pi
G\bar{\rho}}\\!\right)^{\\!\\!\frac{1}{2}}\,,$ (8)

that is, $M<\overline{\\!M}_{\rm crit}$ is the condition for a non-singular
Schwarzschild star of mass $M$ to exist with a given fixed constant density
$\rho=\bar{\rho}$ interior.

An incompressible fluid would seem to be matter that is the most resistant to
further gravitational contraction, and it is already extreme at that for
implying an infinite speed of sound (v${}_{s}>c$), violating relativistic
causality. Thus $\overline{\\!M}_{\rm crit}$ would seem to be an upper bound
for the mass of any self-gravitating body to contain a non-singular interior.
If $\bar{\rho}$ is taken to be $5$ times the density $2.3\times 10^{14}{\rm
gm/cm}^{3}$ of normal nuclear matter, about the value expected in the cores of
neutron stars, then $\overline{\\!M}_{\rm crit}\simeq 3.4\,M_{\odot}$. Any
higher values of $\bar{\rho}$ lead only to smaller values of
$\overline{\\!M}_{\rm crit}$ from (8). Thus any self-gravitating system
composed of ultra dense nuclear matter with a mass larger than a few solar
masses would appear to be unstable and liable to complete gravitational
collapse to the singular state of the original solution (1)-(2), in which all
the matter is crushed to infinite density at the central singularity.

J. A. Wheeler and his collaborators carried out calculations utilizing models
of the eqs. of state of nuclear matter, and their stability, obtaining upper
bounds of order of a solar mass $M_{\odot}$ HarThorWakWheel:1965 . Current
theoretical values for this Tolman-Oppenheimer-Volkoff (TOV) limit have been
estimated to lie between $2.2\,M_{\odot}$ and $2.9\,M_{\odot}$ KaloBaym:1996 ,
the range due to the still considerable uncertainty in the high density
nuclear eq. of state. The LIGO/LSC observation of GW170817, the first
gravitational wave (GW) event due to merging neutron stars, placed the limit
in the range of $2.01\,M_{\odot}$ to $2.17\,M_{\odot}$ RezzMostWeih:2018 ;
ChoNSmass:2018  suggesting that higher central nuclear densities and the lower
end of the theoretically allowed mass range is preferred by the GW data.

The key point Wheeler made clear, and emphasized, is the reason that such an
upper TOV bound exists at all in GR is due to the fact that unlike Newtonian
gravity, positive internal pressure $p>0$ only increases gravitational
attraction in Einstein’s theory Wheeler:1964 . Formally taking the limit
$c\to\infty$ in (8) shows that the upper bound on $M$ is removed to infinity
in the non-relativistic Newtonian limit where this pressure effect disappears.
Thus in full GR even very ‘hard’ eqs. of state of the internal matter such as
Schwarzschild’s incompressible fluid only serve to promote rather than resist
gravitational collapse, making singularities of the kind exhibited in (1)-(2)
for the final state seem to be inevitable. Wheeler hoped that consistent
inclusion of quantum effects might provide an escape from this difficulty,
since as he said: “proper physical variables do not and cannot go to
infinity." Wheeler:1964

Wheeler thus brought to many physicists’ attention the seriousness of the
issue of the final state of complete gravitational collapse in classical GR.
Despite strong misgivings, but unable to see any apparent way out of the
difficulty, Wheeler came to believe in the inevitability of singularities in
gravitational collapse, and then popularized the term black hole to both
physicists and non-physicists alike.

Figure 1: The Carter-Penrose conformal diagram of the maximal Kruskal analytic
extension of the Schwarzschild geometry. Radial light rays are represented in
this diagram as $45^{\circ}$ lines. The angular coordinates $\theta,\phi$ are
suppressed.

The nature of the Schwarzschild singularity at $r=r_{\\!{}_{M}}$ as a causal
boundary only became clear from the work of D. Finkelstein Finkel:1958 , who
showed that the coordinate singularity in the line element (1)-(2) at
$r=r_{\\!{}_{M}}$ could be removed by a change of coordinates associated with
the trajectories of freely falling light rays (a set of coordinates actually
found earlier by A. S. Eddington Edding:1924 ). Since the worldlines of
timelike trajectories starting at finite $r>r_{\\!{}_{M}}$ also arrive at the
horizon at a finite proper time, it seems natural to extend the coordinates
through the horizon to the interior, $r<r_{\\!{}_{M}}$, so that the proper
time of such a timelike observer could continue uninterrupted. This leads to
the maximal analytic extension of the Schwarzschild geometry in Kruskal-
Szekeres coordinates MTW , illustrated in the Carter-Penrose conformal diagram
of Fig. 1. The causal boundary at which light hovers indefinitely, unable to
escape to infinity, which in the Schwarzschild solution (1)-(2) is the
spherical surface of radius $r=r_{\\!{}_{M}}$, at $T=X$ in Fig. 1 is the
(future) event horizon, following terminology introduced by W. Rindler
Rindler:1956 .

In order to remove the coordinate singularity at $r=r_{\\!{}_{M}}$, the
transformation from (1) to Eddington-Finkelstein or Kruskal-Szekeres $(T,X)$
coordinates must itself be singular at the horizon. Although not often stated
explicitly, this means that the geometries before and after the singular
coordinate transformation are not necessarily physically equivalent at
$r=r_{\\!{}_{M}}$, or if extended beyond it. This is because the mathematical
procedure of analytic continuation through a null hypersurface involves a
physical assumption, namely that the stress-energy tensor $T_{\mu}^{\ \nu}$ is
exactly vanishing there. Even in classical GR, the hyperbolic character of
Einstein’s eqs. allows generically for sources and discontinuities on the
horizon which would violate this assumption, invalidating analytic
continuation and potentially altering also the geometry of the interior from
what the analytic continuation of Fig. 1 would predict.

It is relevant here to recall that Einstein’s original conception of the
Equivalence Principle between gravitational and inertial mass, and its
subsequent mathematical formulation in terms of Riemannian geometry, requires
physics to be independent of real and regular local coordinate transformations
Einstein:1912 . Analytic continuation through BH horizons appends to this the
much stronger mathematical hypothesis of continuation in the space of complex
metrics, admitting singular coordinate transformations that lead to globally
extended spacetimes as in Fig. 1, which may or may not be realized in Nature.
Occasional statements that free fall through the event horizon implied by
continuation of worldlines to $r<r_{\\!{}_{M}}$ is required by the Equivalence
Principle are incorrect, just because of the possibility of sources and
discontinuities on the horizon, which would interrupt that free fall. The
Equivalence Principle does not prevent Einstein’s elevator from coming to an
abrupt end of its free fall at the surface of the earth, or a surface at
$r=r_{\\!{}_{M}}$, if a surface exists there.

When R. Kerr found an exact solution of the sourcefree Einstein eqs. for a
rotating BH Kerr:1963 , the interior was found to contain not only
singularities where the curvature diverges, located now on a ring of finite
radius, but also closed timelike curves HawkEllis:1973 . It is significant
that this paradoxical and acausal feature, widely believed to be unphysical,
occurs at macroscopic distance scales of order of the horizon scale
$r_{\\!{}_{M}}\\!\simeq\\!3\,(M/M_{\odot})$ km, far larger than the
microscopic Planck scale $L_{\rm Pl}\\!=\\!\\!\sqrt{\hslash
G/c^{3}}\\!\simeq\\!1.6\times\\!10^{-33}$ cm, at which quantum gravity effects
might be expected to become important or come to the rescue. The maximal
analytic extension of the Kerr solution also results in global features such
as an infinite number of asymptotically flat regions HawkEllis:1973 , beyond
the two in Fig. 1. Thus endowing collapsing matter with angular momentum not
only does not avoid the appearance of spacetime singularities, as Einstein
apparently had hoped Einstein:1939 , but rather leads to additional unphysical
features of analytically extended BH solutions.

An open question at the time was whether singularities might be a special
property of highly symmetric exact solutions of Einstein’s eqs. To address
this question, in 1965 R. Penrose introduced the notion of a closed trapped
surface Penrose:1965 , within which outgoing light waves are actually bent
backwards by gravity and forced to become ingoing, such as occurs in (1)-(2)
for $r<r_{\\!{}_{M}}$. He then showed that if such a trapped surface exists,
and if matter within it satisfies the weak energy condition $\rho+p_{i}\geq
0$, then Einstein’s eqs. imply that a singularity would necessarily result,
independently of any symmetries. Here $p_{i},i=1,2,3$ are any of the three
principal pressure components of the matter stress tensor. The important
assumption made in Penrose:1965  is that some continuation of the geometry
beyond the horizon into the interior is necessary to have a trapped surface at
all. Related singularity theorems were subsequently proven by S. W. Hawking,
and Hawking and Penrose, with slightly different assumptions, but also relying
either on the existence of a closed trapped surface or a stronger condition
$\rho+\sum_{i=1}^{3}p_{i}\geq 0$ on the matter stress tensor (the strong
energy condition) HawkEllis:1973 . Since these singularity theorems are
independent of any symmetries and the existence of trapped surfaces in
gravitational collapse was widely assumed, they reinforced the conviction that
BH singularities will arise inevitably in the final state of gravitational
collapse in classical GR.

Also in 1965, a quite different possibility for the final state of
gravitational collapse was proposed by E. Gliner, based on his speculation
that superdense matter could arrive at an eq. of state with all three
principal pressures equal and negative, i.e. $p_{i}=-\rho<0$ Gliner:1966 .
This corresponds to a cosmological $\Lambda$ term in Einstein’s eqs., but
localized within the high density matter, rather than everywhere constant.
With this eq. of state and negative pressure, the strong energy condition is
violated, i.e. $\rho+3p<0$. This has the consequence that timelike geodesics
become defocused rather than focused down to a singularity, or in more
colloquial terms, that gravitational attraction in GR becomes effectively
repulsion instead. The key point about such an eq. of state is that it
reverses the sign of the pressure effect that led previously to the critical
mass bound (8). The $p=-\rho<0$ eq. of state is that which A. D. Sakharov had
speculated also could arise to avoid a singularity at the initial conditions
of the universe Sak:1966 , implying instead a non-singular expanding de Sitter
phase (well before the term ‘inflation’ was coined). Today this eq. of state
is believed to be that of the cosmological dark energy driving the Hubble
expansion to accelerate, rather than decelerate, again intuitively a kind of
repulsion.

With an interior $p=-\rho>0$ eq. of state, BH singularities and the
conclusions of the Penrose singularity theorem can be avoided, but if
Einstein’s eqs. hold, only if no true closed trapped surface is ever formed.
This shows that the crucial assumption of the Penrose theorem involves the
physics of the near-horizon region, since once any trapped surface at any
$r<r_{\\!{}_{M}}$ is assumed, it is ‘game over,’ and a singularity must occur
in GR. The logical loopholes in all the singularity theorems and resolution of
Wheeler’s difficulty for the final state of gravitational collapse can be
realized in Einstein’s theory, or its semi-classical extension, only if the
stress tensor of quantum matter can produce a large enough effective positive
$\Lambda_{\rm eff}$ term with negative pressure, and prevent the formation of
a true trapped surface in the collapse.

##  2 What’s the (Quantum) Matter with Black Holes?

Additional difficulties beyond the singularities and acausal closed timelike
curves, arising in strictly classical GR, appear when quantum effects in BH
spacetimes are considered. The genesis of these is Hawking’s argument that a
non-rotating, uncharged BH would emit thermal radiation, at a temperature
Hawking:1974 ; Hawk:1975

$T_{{}_{H}}=\frac{\hslash\kappa}{2\pi ck_{{}_{B}}}=\frac{\hslash c^{3}}{8\pi
Gk_{{}_{B}}M}$ (9)

where $\kappa=c^{4}/4GM$ is the surface gravity. With the temperature (9), it
was argued that thermodynamics could be applied to BHs, and a ‘First Law’
BarCarHawk:1973 ; HartHawk:1976

$dE=dMc^{2}=\frac{\kappa\,c^{2}}{8\pi
G}\,dA_{{}_{H}}=T\\!_{{}_{H}}\,dS\\!_{{}_{BH}}$ (10)

was deduced, with the BH assigned the Bekenstein-Hawking BH entropy,

$S\\!_{{}_{BH}}=k_{{}_{B}}\frac{A_{{}_{H}}}{4\,L_{\rm Pl}^{2}\\!}\,,$ (11)

equal to $1/4$ of its area $A_{{}_{H}}=4\pi r_{\\!{}_{M}}^{2}$ in Planck
units. This identification of entropy with area supported J. Bekenstein’s
speculation that BHs be assigned an entropy in order to avoid violations of
the second law of thermodynamics Beken:1973 , since ordinary matter with non-
zero entropy apparently can fall into BHs and so disappear without a trace.
The hypothesis that BH entropy is proportional to the area of its event
horizon was guided by D. Christodoulou’s classical result that horizon area
can never decrease in any process of interaction with matter, and will
positively increase if the process irreversibly sends matter with positive
mass-energy into the BH Christo:1970 .

The curious feature of (10) is that $\hslash$ cancels out between
$T\\!_{{}_{H}}$ and $dS\\!_{{}_{BH}}$. This is because the first form of (10)
is nothing but the classical Smarr formula Smarr:1973

$dM=\frac{\kappa}{8\pi G}\,dA_{{}_{H}}$ (12)

relating the change of a BH mass $M$ to the change in area $A_{{}_{H}}$ of its
horizon, a simple geometric relation in which $\hslash$ does not appear at
all. As in Christodoulou’s area law Christo:1970 , quantum mechanics plays no
role in the classical Smarr relation (12). Multiplying and dividing by
$\hslash$ should not be expected to turn a stictly classical relation into a
valid quantum one.

Related to this, if the identification of the classical area rescaled by
$k_{{}_{B}}/4L_{\rm Pl}^{2}$ with entropy and the thermodynamic interpretation
of (10) is to be valid in the quantum theory, then the classical limit
$\hslash\to 0$ (with $M$ fixed) which yields an arbitrarily low Hawking
temperature (9), assigns to the BH an arbitrarily large entropy, completely
unlike the zero temperature limit of any other cold quantum system.

Another paradoxical consequence of temperature in (9) inversely proportional
to $M=E/c^{2}$ is the implication that the heat capacity of a Schwarzschild BH

$\frac{\\!dE}{\,dT\\!_{{}_{H}}}=-\frac{8\pi Gk_{{}_{B}}M^{2}}{\hslash
c}=-\frac{Mc^{2}}{T\\!_{{}_{H}}}<0$ (13)

is negative, as was pointed out by Hawking himself HawkSpecHeat:1976 , and
incidentally also very large in magnitude. In statistical equilibrium the heat
capacity of a system is related to the energy fluctuations about its mean
value $\langle E\rangle$ by

$c_{{}_{V}}=\left(\frac{d\langle
E\rangle}{dT}\right)_{\\!\\!{}_{V}}=\frac{1}{\,k_{{}_{B}}T^{2}\\!}\
\Big{\langle}\big{(}E-\langle E\rangle\big{)}^{2}\Big{\rangle}>0$ (14)

at constant volume $V$. If pressure or some other thermodynamic variable is
held fixed there is an analogous relation. The positivity of the statistical
average in (14) requires only the existence of a well defined ground state
upon which the thermal equilibrium canonical ensemble is defined, but is
otherwise quite independent of the details of the system or its interactions.
Hence on general grounds of quantum statistical mechanics, the heat capacity
of any system in stable equilibrium at a fixed temperature must be positive.
Although a negative heat capacity can be observed in self-gravitating systems
in the microcanonical ensemble, that situation is temporary and not a true
equilibrium state, but one subject to large fluctuations and indicative of a
phase transition involving a large latent heat LyndBell:1977 ; Padman:1990 .

The physical basis of the instability of a BH in equilibrium with its own
Hawking radiation at $T=T_{{}_{H}}$ is easy to see. If by a small thermal
fluctuation the BH should absorb slightly more radiation in a short time
interval than it emits, its mass would increase, $\Delta M>0$, and hence from
(9) its temperature would decrease, $\Delta T<0$, so that it would now be
cooler than its surroundings and be favored to absorb more energy from the
heat bath than it emits in the next time step, decreasing its temperature
further and driving it further from equilibrium. In this way a runaway process
of the BH growing to absorb all of the surrounding radiation in the heat bath
would ensue. Likewise, if the initial fluctuation has $\Delta M<0$, the BH
temperature would increase, $\Delta T>0$, so that it would now be hotter than
its surroundings and favored to emit more energy than it absorbs from the heat
bath in the next time step, increasing its temperature further. Then a runaway
process toward hotter and hotter evaporation of all its mass to its
surroundings would ensue. In either case, the initial equilibrium is clearly
unstable, and hence cannot be a candidate for the quantum equilibrium ground
state for the system. The instability has also been verified from the negative
eigenvalue of the fluctuation spectrum of a BH in a box of (large enough)
finite volume GrossPerryYaffe:1982 ; Allen:1984 ; York:1986 .

The time scale for this unstable runaway process to grow exponentially is the
time scale for fluctuations away from the mean value of the Hawking flux, not
the much longer time scale associated with the lifetime of the BH under
continuous emission of that flux. The time scale for thermal fluctuations is
easily estimated to be the typical time between emissions of a single quantum
with typical energy (at infinity) of $k_{{}_{B}}T$, of a source whose energy
emission per unit area per unit time is of order
$(k_{{}_{B}}T_{{}_{H}})^{4}/\hslash^{3}c^{2}$. Multiplying by the area and
dividing by the typical energy $k_{{}_{B}}T$, the average number of quanta
emitted per unit time is found. The inverse of this, namely

$\Delta t\sim\frac{1}{A_{{}_{H}}}\frac{(\hslash
c)^{3}}{(k_{{}_{B}}T_{{}_{H}})^{3}}\sim\frac{r_{\\!{}_{M}}}{c}\sim
10^{-5}\left(\frac{M}{M_{\odot}}\right)\,{\rm sec}$ (15)

is the typical time interval (as measured by a distant observer) between
successive emissions of individual Hawking quanta. This time scale is quite
short. Any tendency for the system to become unstable or undergo a phase
transition to a stable phase would be expected to show up on this short a time
scale, governing the fluctuations in the mean flux, which is of order of the
collapse time itself and before a steady state flux could even be established.
The interpretation of the classical relation of energy to area (12) as an
equilibrium thermodynamic relation (10) of any kind is therefore placed in
serious doubt.

It is also instructive to evaluate $S\\!_{{}_{BH}}$ for typical astrophysical
BHs. For a solar mass $M_{\odot}\simeq 2\times 10^{33}$ gm

$S\\!_{{}_{BH}}\simeq 1.05\times
10^{77}\,k_{{}_{B}}\,\left(\frac{\\!\\!M}{M_{\odot}\\!}\right)^{\\!\\!2}$ (16)

which is truly an enormous entropy. For comparison, the entropy of the sun as
it is, a hydrogen burning main sequence star, is given to good accuracy by the
entropy of a non-relativistic perfect fluid. This is of the order
$Nk_{{}_{B}}$ where $N$ is the number of nucleons in the sun $N\sim
M_{\odot}/m_{{}_{N}}\sim 10^{57}$, times a logarithmic function of the density
and temperature profile which may be estimated to be of the order of $20$ for
the sun. Hence the entropy of the sun is approximately $S\\!_{\odot}\sim
2\times 10^{58}\,k_{{}_{B}}$ or nearly $19$ orders of magnitude smaller than
(16).

A simple scaling argument shows that the entropy of a gravitationally bound
relativistic fluid should scale like $M^{3/2}$ and be of order of ZelNovbook ;
EMZak:2010

$S\sim k_{{}_{B}}\left(\frac{\\!\\!M}{M_{\rm
Pl}\\!}\right)^{\\!\\!\frac{3}{2}}\sim
10^{57}\,k_{{}_{B}}\left(\frac{\\!\\!M}{M_{\odot}\\!}\right)^{\\!\\!\frac{3}{2}}$
(17)

which is less than $S_{\odot}$ because the relativistic radiation pressure in
the sun is small compared to the non-relativistic fluid pressure. However, the
entropy from the relativistic radiation pressure (17) grows with the $3/2$
power of the mass, whereas the non-relativistic fluid entropy $S_{\odot}$
grows only linearly with $M$. For stars with masses greater than about
$50\,M_{\odot}$ which are hot enough for their pressure to be dominated by the
photon $T^{4}$ radiation pressure, (17) gives the correct order of magnitude
estimate of such a star’s entropy at a few times $10^{59}\,k_{{}_{B}}$
ZelNovbook .

On the other hand, the BH entropy (16) is proportional to $M^{2}$, so the
discrepant factor with (17) is $(M/M_{\rm Pl})^{1/2}\sim 10^{19}$ for
$M=M_{\odot}$. Since (16) makes no reference to how the BH was formed, and a
BH may always be theoretically idealized as forming from an adiabatic process,
which keeps the entropy constant, (16) states that this entropy must suddenly
jump by a factor of order $10^{19}$ for a solar mass BH at the instant the
horizon forms at $r=r_{\\!{}_{M}}$. When Boltzmann’s formula

$S=k_{{}_{B}}\ln W(E)$ (18)

is recalled, relating the entropy to the total number of microstates in the
system $W(E)$ at the fixed energy $E$, the number of such microstates of a BH
satisfying (16) must jump by $\exp(10^{19})$ at that instant that the event
horizon is reached, a truly staggering proposition. How and from where this
enormous number of microstates can appear, when the horizon is supposed to be
nothing but a mathematical surface only, with no independent degrees of
freedom or stress tensor of its own, has remained perplexing and the subject
of numerous investigations spanning five decades Preskill:1992 ; Mathur:2009 ;
AMPS:2013 ; Giddings:2013 ; MottVauPT ; Marolf:2017 ; AHMST:2020 ; AHMald:2021
, often with the suggestion that new physics or modifying the laws of physics
themselves might be required.

Hawking radiation emerging from the BH at temperature (9) in a mixed thermal
state also seems to imply a breakdown of quantum unitary evolution, that is
difficult if not impossible to recover at the late or final stages of the BH
evaporation process, resulting in a severe ‘information paradox’ HawkUnit:1976
; Preskill:1992 ; Page:1993 ; UnruhWald:2017 .

As a consequence of the gravitational redshift (3), it also follows that
although the Hawking temperature $T\\!_{{}_{H}}$ of the radiation far from the
BH is very low, the local temperature of the radiation (19) is arbitrarily
high when extrapolated back

$T\\!_{\rm loc}(r)=\ \,\frac{T\\!_{{}_{H}}}{\sqrt{\raisebox{-1.0pt}{$1\ $--}\
\mbox{\small$\displaystyle\frac{\raisebox{-5.81247pt}{$r_{M}$\\!}}{r}$}}}\hskip
28.45274pt\vspace{-8mm}$ (19)

to the vicinity of the horizon $r\\!\to\\!r_{\\!{}_{M}}$, even becoming
transplanckian in this limit. Unlike classical test particles, when
$\hslash\neq 0$ such extremely blue shifted photons are necessarily present in
the vacuum as virtual quanta. Their effects upon the geometry depend upon the
quantum state of the vacuum, defined by boundary conditions on the wave
equation in a non-local way over all of space, and $\langle T_{\mu}^{\
\nu}\rangle$ may be large at $r=r_{\\!{}_{M}}$, notwithstanding the smallness
of the local classical curvature there Boul:1975 ; ChrFul:1977 . Since the
limits $r\to r_{\\!{}_{M}}$ and $\hslash\to 0$ do not commute, non-analytic
behavior near the event horizon, quite different from that in the strictly
classical ($\hslash\equiv 0$) situation is possible in the quantum theory.
Since gravity couples to all energies, thermal fluctuations at large
transplanckian temperatures and energies would be expected to have significant
backreaction effects on the classical spacetime geometry near the horizon,
giving rise to just the sources and discontinuities that would violate the
analytic continuation to the interior and global geometry in Fig. 1 it leads
to. Thus it is by no means clear why assuming a rigidly fixed classical BH
background, neglecting quantum fluctuations in the stress tensor down to
$r\\!=\\!r_{\\!{}_{M}}$, as in Hawking’s original semi-classical treatment,
should be valid Jacobson:1991 .

These myriad difficulties and BH paradoxes suggest that some important element
is missing, and needed to describe quantum effects or degrees of freedom on
the horizon scale in BH spacetimes, which are not present in classical GR.
Since the various problems and paradoxes arise at the macroscopic scale of the
BH horizon, these additional degrees of freedom require an addition or
modification of Einstein’s theory at that scale, well before the microscopic
Planck scale $L_{\rm Pl}$ or Planck scale curvatures are reached. The problem
of the magnitude of cosmological $\Lambda$ vacuum energy on the very largest
macroscopic Hubble scale of the universe carries with it a similar
implication. Together with the observation that the $p=-\rho$ eq. of state can
provide a resolution of both the BH and Big Bang singularities, this indicates
that the two problems at the interface of classical GR and quantum theory,
involving the nature of the quantum vacuum and vacuum energy, at macroscopic
scales are related EMEFT:2022 .

##  3 The Proposed Solution: Gravitational Condensate Stars

###  3.1 Background and Motivations for the First Gravastar Model

One model that attempted to take backreaction of Hawking radiation on the BH
into account, imagined it to be immersed in a Hawking radiation atmosphere,
with an effective equation of state, $p=\kappaup\rho$ tHooft:1998 . It was
found that due to the blue shift effect (19), the backreaction of such an
atmosphere on the metric near $r=r_{\\!{}_{M}}$ is enormous, with the interior
region quite different from the vacuum Schwarzschild solution. A large entropy
of order of $S_{{}_{BH}}$ is obtained from the hot fluid alone in such a
model, with $S=4\,\frac{\kappaup+1}{7\kappaup+1}S_{{}_{BH}}$, becoming equal
to the BH entropy (11) for $\kappaup=1$. This suggested that the maximally
stiff equation of state consistent with the causal limit $p=+\rho$ may play a
role in the quantum theory of fully collapsed objects.

Despite the interesting result of tHooft:1998 , indicating the importance of
backreaction on the geometry, the model of tHooft:1998  cannot be viewed as a
satisfactory solution to the final state of the collapse problem, since it
involves huge Planckian energy densities near $r=r_{\\!{}_{M}}$, and a
negative mass singularity at $r=0$, indicating the breakdown of the semi-
classical approximation in both regions. The negative mass singularity arises
because a repulsive core is necessary to counteract the self-attractive
gravity of the dense relativistic fluid with positive energy. This suggests
again that an effectively repulsive eq. of state with a negative pressure
$p<0$ (rather than a negative mass-energy singularity) should play a role in
the BH interior.

A different proposal of a quantum phase transition at or near
$r=r_{\\!{}_{M}}$ was made in ChaplineHLS:2001 , based on a suggestive
condensed matter analogy with the liquid-vapor critical point of a non-
relativistic Bose fluid, where the wave eq. for acoustic excitations mimics
the wave eq. for light in the vicinity of a BH horizon. In this condensed
matter analog system the position dependent speed of sound ${\rm v}_{s}(r)$
takes the place of a position dependent speed of light $c_{\rm eff}(r)$, both
of which formally vanish at the critical surface, where $c^{2}_{\rm
eff}(r)=c^{2}f(r)$ is the $g_{tt}$ metric coefficient of $dt^{2}$ in (1). The
authors of ChaplineHLS:2001 ; Chap:2003  also argued for a negative pressure
on the interior side of the critical surface, and an effective $p=-\rho$ eq.
of state, as Gliner had 35 years earlier, but did not tackle the delicate
issue of joining a de Sitter interior to a Schwarschild exterior, in a fully
consistent treatment of the problem in GR.

Matching the Schwarzschild exterior solution to a non-singular de Sitter (dS)
interior had a long previous history. Continuous transitions between the two
were studied e.g. in Dymn:1992 , while it was recognized that joining the
exact Schwarzschild and dS geometries directly at their mutual horizons
$H^{-1}$ and $2GM$, requires some discontinuity or interposition of ‘non-
inflationary material’ PoisIsr:1988 . In addition to uncertainties of the
physics involved, the earlier GR formalism Lanczos:1924 ; OBrSyn:1952 ;
Israel:1966a ; Israel:1967  for dealing with singular hypersurfaces when the
normal to hypersurface becomes null, as it does at a BH horizon, was
recognized to be inadequate BarIsr:1991 . The necessity of some anisotropic
matter at the joining of the interior to exterior geometries was made explicit
in CatVis:2005 .

Motivated by these considerations, the first fully relativistic gravitational
vacuum condensate star model was proposed in gravastar:2001 ; Grav_Univ:2023 ,
with the critical surface of refs. ChaplineHLS:2001  replaced by a thin shell
of ultra-relativistic fluid with the maximally stiff eq. of state $p=\rho$ and
vacuum condensate interior with negative pressure, $p=-\rho$. Partly for the
reason of avoiding the technical difficulties associated with singular null
hypersurfaces, the proposal in the original paper gravastar:2001 ;
Grav_Univ:2023  made use of two timelike hypersurfaces at $r_{1}$ and $r_{2}$
with an interposed fluid boundary layer of ‘non-inflationary material’ obeying
the eq. of state $p=\rho$. In addition to the model of tHooft:1998 , the
choice of this eq. of state at the causal limit where the speed of sound
coincides with the speed of light, was motivated by physical considerations of
a quantum phase transition produced by the infrared effects of dimensional
reduction from $D=4$ to $D=2$ dimensions. Nevertheless the choice of $p=\rho$
in Grav_Univ:2023  is certainly an ansatz, illustrating a proof of principle,
but without a rigorous basis in fundamental physics. It therefore would be
subject to modification as that fundamental physics came more clearly into
view by subsequent developments EMVau:2006 ; EMZak:2010 ; EMEFT:2022 , and it
also became possible to treat joining at null horizon hypersurfaces directly
MazEM:2015 ; BelGonEM1:2022 .

###  3.2 The First Gravastar Model

The general form of the stress-energy tensor in the static, spherically
symmetric geometry of (1) for a non-rotating body is

$T^{\mu}_{\ \,\nu}=\left(\begin{array}[]{cccl}\\!\\!-\rho&0&0&0\\\ 0&p&0&0\\\
0&0&p_{\perp}&0\\\ 0&0&0&p_{\perp}\end{array}\right)$ (20)

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

$\displaystyle-G_{\ t}^{t}$
$\displaystyle=\mbox{\small$\displaystyle\frac{1}{r^{2}}$}\mbox{\small$\displaystyle\frac{d}{dr}$}\,\big{[}r\left(1-h\right)\big{]}=-8\pi
GT_{\ t}^{t}=8\pi G\,\rho\,,$ (21a) $\displaystyle G_{\ r}^{r}$
$\displaystyle=\mbox{\small$\displaystyle\frac{h}{rf}$}\mbox{\small$\displaystyle\frac{df}{dr}$}+\mbox{\small$\displaystyle\frac{1}{r^{2}}$}\,\big{(}h-1\big{)}=8\pi
GT_{\ r}^{r}=8\pi G\,p\ $ (21b)

together with the conservation equation

$\nabla\\!_{\lambda}\,T^{\lambda}_{\
r}=\frac{dp}{dr}+\frac{\rho+p}{2f}\,\frac{df}{dr}+\frac{2}{r}\,(p-p_{\perp})=0$
(22)

which ensures that the other components of Einstein’s eqs. are satisfied. In
(22) the transverse pressure $p_{\perp}\\!\equiv\\!T^{\theta}_{\
\theta}\\!=\\!T^{\phi}_{\ \phi}$ 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 (22) vanishes. In that case (21)-(22) are three first order
equations for the four functions, $f,h,\rho$, and $p$, which become closed
when an eq. of state for the fluid relating $p$ and $\rho$ is specified.

Because of the considerations of the previous sections, the first gravastar
model was assumed to contain three different regions with the three different
eqs. of state

$\begin{array}[]{clcl}{\rm I.}&{\rm de\ Sitter\ Interior:}&0\leq
r<r_{1}\,,&\rho=-p\,,\\\ {\rm II.}&{\rm Thin\
Shell:}&r_{1}<r<r_{2}\,,&\rho=+p\,,\\\ {\rm III.}&{\rm Schwarzschild\
Exterior:}&r_{2}<r\,,&\rho=p=0\,.\end{array}$ (23)

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 $p$ are generally discontinuous from the first order eqs. (21)
and (22).

In the interior region $\rho=-p$ is a constant from (22). Labelling this
constant $\rho_{{}_{V}}=3H^{2}/8\pi G$, and requiring that at the origin the
solution is free of any mass singularity determines the interior to be a
region of dS spacetime in static coordinates, i.e.

${\rm I.}\qquad f(r)=C\,h(r)=C\,\big{(}1-H^{2}\,r^{2}\big{)}\,,\quad 0\leq
r\leq r_{1}$ (24)

where $C$ and $H$ are constants, which at this point are arbitrary. Note that
the static dS horizon where both $f$ and $h$ vanish is at
$r=r_{\\!{}_{H}}=H^{-1}$.

The unique solution in the exterior vacuum region where $T_{\mu}^{\ \,\nu}=0$
that approaches flat Minkowski space as $r\to\infty$ is the Schwarzschild
solution (2)

${\rm III.}\qquad f(r)=h(r)=1-\frac{2GM}{r}=1-\frac{r_{\\!{}_{M}}}{r}\,,\qquad
r_{2}\leq r$ (25)

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

The only non-vacuum region is the thin shell interface region II. In this
region it proves useful to define the dimensionless variable $w$ by

$w\equiv 8\pi Gr^{2}p$ (26)

so that eqs. (21)-(22) with $\rho=p$ may be recast in the form

$\displaystyle\frac{dr}{r}$ $\displaystyle=\frac{dh}{1-w-h}\,,$ (27a)
$\displaystyle\frac{dh}{h}$
$\displaystyle=-\frac{1-w-h}{1+w-3h}\,\mbox{\small$\displaystyle\frac{dw}{w}$}\,.$
(27b)

together with $pf\propto wf/r^{2}$ a constant. The first eq. (27a) is
equivalent to the definition of the (rescaled) Misner-Sharp mass function
$\mu(r)=2Gm(r)$, with $h=1-\mu/r$ and $d\mu(r)=8\pi G\,\rho r^{2}\,dr=w\,dr$
within the shell. The second eq. (27b) can be solved only numerically in
general. However, it is possible to obtain an analytic solution in the thin
shell limit $0<h\ll 1$, since in this limit $h$ may be set to zero on the
right side of (27b) to leading order. Assuming $w$ remains finite, (27b) can
then be integrated immediately to obtain

$h\equiv 1-\frac{\mu}{r}\simeq\varepsilon\ \frac{(1+w)^{2}}{w\ }\ll 1$ (28)

in region II, where $\varepsilon$ is an integration constant. Because of the
condition $h\ll 1$, we require $\varepsilon\ll 1$, if $w$ is of order unity.
Making use of eqs. (27) and (28) then gives

$\frac{dr}{r}\simeq-\varepsilon\,dw\,\frac{(1+w)}{w^{2}}$ (29)

so that because $\varepsilon\ll 1$ the radius $r$ hardly changes within region
II, and $dr$ is of order $\varepsilon\,dw$. The final unknown function $f$ is
given by (22) to be $f=(r/r_{1})^{2}(w_{1}/w)f(r_{1})$ so that

$f\simeq\frac{w_{1}}{\\!w}\,f(r_{1})$ (30)

to leading order in $\varepsilon$ for $\varepsilon\ll 1$.

Continuity (${\pazocal C}^{0}$) of the metric functions $f$ and $h$ at $r_{1}$
and $r_{2}$ gives the conditions

$\displaystyle f(r_{1})$
$\displaystyle=C\,h(r_{1})=C\,(1-H^{2}r_{1}^{2})\simeq
C\,\varepsilon\,\frac{(1+w_{1})^{2}\\!\\!}{w_{1}}$ (31a) $\displaystyle
f(r_{2})$
$\displaystyle=h(r_{2})=1-\frac{2GM}{r_{2}}\simeq\varepsilon\,\frac{(1+w_{2})^{2}\\!\\!}{w_{2}}$
(31b)

which together with (30) evaluated at $r=r_{2},w=w_{2}$ provides

$C\,(1+w_{1})^{2}=(1+w_{2})^{2}$ (32)

and hence three independent relations among the eight integration constants
$(r_{1},r_{2},\\\ w_{1},w_{2},H_{0},M,C,\varepsilon)$. Assuming that
$(r_{1},r_{2},w_{1},w_{2},H_{0},M,C)$ all remain finite as $\varepsilon\to 0$,
i.e. they are all of order $\varepsilon^{0}$, then
$r_{1}\\!\to\\!r_{\\!{}_{H}}=H^{-1}$ and $r_{2}\\!\to\\!r_{\\!{}_{M}}$ with
$r_{2}-r_{1}=\Delta r$ of order $\varepsilon$, so that $r_{\\!{}_{H}}\simeq
r_{\\!{}_{M}}$ to leading order in $\varepsilon$. 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 $\varepsilon\to 0$. Since the mass $M$ is a free
parameter there remain three undetermined integration constants
$C,w_{1},w_{2}$ which satisfy the one relation (32) in addition to
$\varepsilon\ll 1$ itself.

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

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

$\ell=\int_{r_{1}}^{r_{2}}\frac{dr}{\\!\\!\\!\sqrt{h}}=r_{\\!{}_{M}}\\!\sqrt{\varepsilon}\int_{w_{2}}^{w_{1}}\\!dw\,w^{-\frac{3}{2}}=2r_{\\!{}_{M}}\\!\sqrt{\varepsilon}\
\Big{(}w_{2}^{-\frac{1}{2}}-w_{1}^{-\frac{1}{2}}\Big{)}$ (33)

to leading order in $\varepsilon$, and hence $\ell$ is ${\cal
O}(\varepsilon^{\frac{1}{2}})$ and small compared to $r_{\\!{}_{M}}$. The
magnitude of $\varepsilon$ and hence of $\ell$ can be fixed only by
consideration of the quantum effects that give rise to the phase transition
boundary layer, which will be discussed in Sec.6.

The entropy of the thin shell is obtained from its eq. of state, which is that
of a relativistic fluid in $1+1$ dimensions, and can be written in the form
$p=\rho=(a^{2}/8\pi G)(k_{B}T/\hslash)^{2}$, where $G$ is introduced for
dimensional reasons, so that $a^{2}$ is a dimensionless constant. By the
standard Gibbs relation, $Ts=p+\rho$ for a relativistic fluid with zero
chemical potential, the local specific entropy density is

$s(r)=\frac{a^{2}k_{B}^{2}T(r)}{4\pi\hslash^{2}G}=\frac{ak_{B}}{\hslash}\left(\frac{p}{2\pi
G}\right)^{\\!\\!\frac{1}{2}}=\frac{ak_{B}}{4\pi\hslash Gr}\,w^{\frac{1}{2}}$
(34)

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

$S=4\pi\\!\int_{r_{1}}^{r_{2}}\frac{s\,r^{2}\,dr}{\\!\\!\\!\sqrt{h}\
}=\frac{ak_{{}_{B}}r_{\\!{}_{M}}^{2}}{\hslash
G}\sqrt{\varepsilon}\,\ln\Big{(}\frac{w_{1}}{w_{2}}\Big{)}\sim
a\,k_{B}\frac{M\ell}{\hslash}$ (35)

and of order $k_{B}M\ell/\hslash$ to leading order in $\varepsilon$, assuming
$a,w_{1},w_{2}$ are ${\pazocal O}(1)$. Since the interior region I has
$\rho_{{}_{V}}=-p_{{}_{V}}$, $(Ts)_{{}_{V}}=p_{{}_{V}}+\rho_{{}_{V}}=0$ there.
This is in accord with a gravitational Bose-Einstein condensate (GBEC) being a
single 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 (35) this is of order $k_{B}(r_{\\!{}_{M}}/L_{\rm
Pl})^{\frac{3}{2}}$ for $\ell\\!\sim\\!\sqrt{L_{\rm Pl}r_{\\!{}_{M}}}$, or
$S\\!\sim\\!\sqrt{\varepsilon}S_{BH}\ll S_{BH}$, far smaller than the
Bekenstein-Hawking entropy (11). The $M^{3/2}$ scaling of (35) 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 $T_{{}_{H}}\sim\hslash/k_{B}r_{\\!{}_{M}}$ and quite cold,
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 $K_{(t)}=\partial_{t}$ and
absence of either an event horizon or an interior singularity, a gravitational
condensate star shows no loss of unitarity, information paradox, or any
conflict with either quantum theory or general principles of statistical
mechanics. As a static solution, neither the interior nor the thin shell
boundary layer emit Hawking radiation. A gravitational condensate star is both
cold and dark, and hence in its appearance to distant observers and in its
external geometry, in most respects indistinguishable from a BH.

###  3.3 The Lanczos-Israel conditions at the $r_{1}$ and $r_{2}$ boundaries

In the original gravastar model of gravastar:2001 ; Grav_Univ:2023 ;
MazEMPNAS:2004  the Lanczos-Israel junction conditions Lanczos:1922 ;
OBrSyn:1952 ; Israel:1966a ; Israel:1967

$[K_{a}^{\ b}]\equiv K_{a\,+}^{\ b}-K_{a\,-}^{\ b}=-4\pi G\,\Big{(}2S_{a}^{\
b}-\delta_{a}^{\ b}S_{c}^{\ c}\Big{)}$ (36)

were used to relate the discontinuity in the extrinsic curvature

$K_{ab}=-n_{\mu}\upsilon_{(b)}^{\ \nu}\nabla_{\\!\nu}\upsilon_{(a)}^{\
\mu}=-n_{\mu}\upsilon_{(b)}^{\ \nu}\big{(}\partial_{\nu}\upsilon_{(a)}^{\
\mu}+\Gamma^{\mu}_{\ \ \,\nu\lambda}\upsilon_{(a)}^{\ \lambda}\,\big{)}$ (37)

to the surface stress tensor $S_{a}^{\ b}$, where $n^{\mu}$ is the spacelike
normal to the surface at fixed $r$, normalized to

$n^{\mu}n_{\mu}=1$ (38)

in the full four-dimensional spacetime. The indices $a,b$ in (36) are
intrinsic to the surface and thus range over $t,\theta,\phi$ only, while
$\upsilon_{(a)}$ are a set of three mutually orthogonal basis vectors,
orthogonal also to $n_{\mu}$, i.e. satisfying $\upsilon_{(a)}^{\
\mu}n_{\mu}=0$, and normalized so as to project the four-dimensional metric
onto the hypersurface of fixed $r$. Thus

$\gamma_{ab}=\upsilon_{(a)}^{\ \mu}\upsilon_{(b)}^{\ \nu}g_{\mu\nu}$ (39)

is the induced metric on the three-dimensional hypersurface, the inverse of
which must be used to raise the indices $a,b,c$. Since the normal to the
surface at fixed $r$ and basis vectors so defined have components

$n^{\mu}=\sqrt{h(r)}\,\delta^{\mu}_{\
\,r}\,,\qquad\upsilon^{(a)}_{\mu}=\left\\{\begin{array}[]{cl}\delta^{\
a}_{\mu}\,,&\mu=t,\theta,\phi\\\ 0\,,&\mu=r\end{array}\right.$ (40)

in the coordinates (1), the non-vanishing components of $K_{a}^{\
b}=K_{ac}\gamma^{cb}$ in these coordinates are

$\displaystyle K_{t}^{\ t}$
$\displaystyle=\displaystyle{\frac{\\!\\!\sqrt{h}}{f}}\,\Gamma_{\,rtt}=\frac{\\!\\!\sqrt{h}}{2f}\,\frac{d\\!f\\!}{dr}$
(41a) $\displaystyle K_{\theta}^{\ \theta}=K_{\phi}^{\ \phi}$
$\displaystyle=-g^{\theta\theta}\sqrt{h}\,\Gamma_{r\theta\theta}=\displaystyle{\frac{\\!\\!\sqrt{h}}{r}}$
(41b)

with the result that the non-vanishing components of the surface stress tensor
from (36) are

$\displaystyle S\\!_{t}^{\ t}=\displaystyle{\frac{1}{4\pi
G}}\left[\displaystyle{\frac{\\!\\!\sqrt{h}}{r}}\right]$ (42a) $\displaystyle
S\\!_{\theta}^{\ \theta}=S\\!_{\phi}^{\ \phi}=\displaystyle{\frac{1}{8\pi
G}}\left(\left[\displaystyle{\frac{\\!\\!\sqrt{h}}{2f}\,\frac{d\\!f\\!}{dr}}\right]+\left[\displaystyle{\frac{\\!\\!\sqrt{h}}{r}}\right]\right)$
(42b)

on the timelike surface interfaces at $r_{1}$ and $r_{2}$. Since the metric
function $h$ is continuously matched at the interfaces, $[K_{\theta}^{\
\theta}]$ and $S_{t}^{\ t}$ vanish, while $S_{\theta}^{\ \theta}$ is of order
$\varepsilon^{-\frac{1}{2}}$. Making use of the eqs. (28)-(31)

$\frac{\sqrt{h}}{2f}\frac{d\\!f\\!}{dr}\simeq\frac{1}{2r_{\\!{}_{M}}}\sqrt{\frac{w}{\varepsilon}}\qquad\quad\rm
in\ region\ II$ (43)

to leading order in $\varepsilon$, which enables evaluation of the
discontinuities in (42). The non-zero angular components are then
Grav_Univ:2023  222The sign conventions in gravastar:2001 ; MazEMPNAS:2004
are such that $\sigma_{1,2}$ there are the negative of the surface stress
tensors $S^{\ \theta}_{\theta}=S^{\ \phi}_{\phi}$ properly defined here. Eqs.
(C5) and (C7) of MazEM:2015  also have an overall sign change from the
Lanczos-Israel eq. (36) for $S^{\ b}_{a}$, such that $\eta,\sigma$ of (C7)
have the same values as $\eta,\sigma$ in gravastar:2001 ; MazEMPNAS:2004 .

$\displaystyle S^{\ \theta}_{\\!\theta}\big{|}_{r=r_{1}}=S^{\
\phi}_{\\!\phi}\big{|}_{r=r_{1}}\equiv-\sigma_{1}=\frac{1}{32\pi
G^{2}M}\frac{(3+w_{1})}{(1+w_{1})}\left(\frac{w_{1}}{\varepsilon}\right)^{\\!\\!\frac{1}{2}}$
(44a) $\displaystyle S^{\ \theta}_{\\!\theta}\big{|}_{r=r_{2}}=S^{\
\phi}_{\\!\phi}\big{|}_{r=r_{2}}\equiv-\sigma_{2}=-\frac{1}{32\pi
G^{2}M}\frac{w_{2}}{(1+w_{2})}\left(\frac{w_{2}}{\varepsilon}\right)^{\\!\\!\frac{1}{2}}$
(44b)

respectively, to leading order in $\varepsilon$, at $r_{1}$ and $r_{2}$.

The signs of these surface stresses 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 surfaces exert a confining force on the thin shell layer in
region II. Clearly these large transverse surface stresses violate the perfect
fluid ansatz at the interfacial boundaries. Nevertheless, as it will turn out
from semi-classical estimates in Sec. 6,
$\varepsilon^{-\frac{1}{2}}\sim(M/M_{\rm Pl})^{\frac{1}{2}}$, so that the
surface tensions (44) are of order $M^{-\frac{1}{2}}$ and far from Planckian.
Thus 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 in the mean field semi-classical approximation. The time component of
the surface stress tensor at $r_{1}$ and $r_{2}$ vanishes and makes no
contribution to the Misner-Sharp mass function $\mu(r)=2Gm(r)$ at either of
the two interfaces.

The Misner-Sharp mass-energy within the shell

$E_{\rm II}=4\pi\\!\int_{r_{1}}^{r_{2}}\\!\rho\,r^{2}dr=\varepsilon
M\\!\int_{w_{2}}^{w_{1}}\frac{dw}{w}\,(1+w)=\varepsilon
M\,\Big{[}\\!\ln\Big{(}\mbox{\small$\displaystyle\frac{w_{1}}{w_{2}}$}\Big{)}+w_{1}-w_{2}\Big{]}$
(45)

to leading order in $\varepsilon$, is of order $M_{\rm Pl}$ and also extremely
small. In this accounting essentially all of 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.

###  3.4 Thermodynamic Stability

In Grav_Univ:2023 ; MazEMPNAS:2004  a thermodynamic argument for the stability
of the first gravastar model was provided. This was based on analyis of the
entropy functional which can be expressed in the form

$S=\frac{ak_{{}_{B}}}{\hslash
G}\int_{r_{1}}^{r_{2}}r\,dr\,\left(\frac{d\mu}{dr}\right)^{\\!\\!\frac{1}{2}}\left(1-\frac{\mu(r)}{r}\right)^{\\!\\!-\frac{1}{2}}$
(46)

in the thin shell region II where the eq. of state $p=\rho$, (34), and the
relation $d\mu(r)=8\pi G\rho r^{2}dr=wdr$ has been used. The first variation
of (46) with respect to $\mu(r)$ with the endpoints $r_{1}$ and $r_{2}$ fixed
vanishes, i.e. $\delta S=0$ by the Einstein eqs. (21) for a static,
spherically symmetric star. Thus any solution of eqs. (21)-(22) is guaranteed
to be an extremum of $S$ Cocke:1965 . This is consistent with regarding
Einstein’s eqs. as an effective field theory (EFT) of low energy
hydrodynamics, strictly valid only at long wavelengths.

The second variation of (46) is

$\delta^{2}S=\frac{ak_{B}}{4\hslash
G}\int_{r_{1}}^{r_{2}}\\!r\,dr\,\left(\frac{d\mu}{dr}\right)^{\\!\\!-\frac{3}{2}}h^{\\!-\frac{1}{2}}\left\\{-\left[\frac{d(\delta\mu)}{dr}\right]^{2}+\frac{(\delta\mu)^{2}}{r^{2}h^{2}}\frac{d\mu}{dr}\left(1+\frac{d\mu}{dr}\right)\right\\}$
(47)

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

${\pazocal
L}\,\chi=\frac{d}{dr}\left\\{r\left(\frac{d\mu}{dr}\right)^{\\!\\!-\frac{3}{2}}\\!h^{\\!-\frac{1}{2}}\frac{d\chi}{dr}\right\\}+\frac{h^{-\frac{5}{2}}}{r}\left(\frac{d\mu}{dr}\right)^{\\!\\!\frac{1}{2}}\left(1+\frac{d\mu}{dr}\right)\,\chi\,.$
(48)

This operator possesses two solutions satisfying ${{\pazocal
L}}\chi_{{}_{0}}=0$, 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 (28)-(29)
it is readily verified that one solution to ${\pazocal L}\chi_{{}_{0}}=0$ is
$\chi_{{}_{0}}=1-w$, from which the second linearly independent solution
$(1-w)\ln\,w+4$ 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, by setting
$\delta\mu=\chi_{{}_{0}}\,\psi$, where $\psi$ does vanish at the endpoints and
inserting this into the second variation (47) one obtians

$\delta^{2}S=-\frac{ak_{B}}{4\hslash
G}\int_{r_{1}}^{r_{2}}\\!r\,dr\,\left(\frac{d\mu}{dr}\right)^{\\!\\!-\frac{3}{2}}h^{-\frac{1}{2}}\chi_{{}_{0}}^{2}\,\left(\frac{d\psi}{dr}\right)^{\\!2}<0$
(49)

after integration by parts and using $\delta\mu=0$ at the endpoints and
${\pazocal L}\chi_{{}_{0}}=0$.

Since (49) is negative definite, the entropy of the solution is maximized with
respect to radial variations that vanish at the endpoints, 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 with the given boundary
conditions. On the other hand, the boundary conditions that the radial
variations $\delta\mu$ vanish at the interface boundaries at $r_{1}$ and
$r_{2}$ may themselves be questioned. This apparent shortcoming of the
thermodynamic stability proof provided in Grav_Univ:2023 ; MazEMPNAS:2004  can
only be addressed within a a fully dynamical theory where the boundary
conditions follow from the dynamical variational principle itself, rather than
being imposed, as they were in the original gravastar model of Grav_Univ:2023
; MazEMPNAS:2004 . The EFT introduced in EMEFT:2022  and discussed in Sec. 7
is expected to settle the question of stability with a full dynamical
treatment based on an effective Lagrangian whose Euler-Lagrange eqs. possess a
gravitational condensate star solution, removing any additional assumptions of
boundary conditions beyond those of a non-singular interior and an
asymptotically flat exterior.

Even from the preliminary analysis of Grav_Univ:2023 ; MazEMPNAS:2004 , it
should be clear that the main qualititative results do not depend crucially on
the specific eq. of state used in the thin shell transition region II. The
important features are the interior dS and exterior Schwarzschild regions and
the location of the thin shell at or straddling the mutual Schwarzschild and
dS horizons, where they are joined. The assumption of the specific eq. of
state $p=\rho$ of the thin shell in the first gravastar model of
gravastar:2001 ; MazEMPNAS:2004  should therefore be regarded as an ansatz,
illustrating a proof of principle of joining the interior dS geometry of a
GBEC condensate with $\Lambda_{\rm eff}>0$ to the exterior Schwarzschild
solution, with the possibility left open to modification of the thin shell
transition region when a more complete or fundamental theoretical framework
became available Grav_Univ:2023 . A proposal for that more fundamental EFT
framework will be described in Secs. 6 and 7.

##  4 The Schwarzschild Constant $\bar{\rho}$ Interior Solution Revisited:
Evading the Buchdahl Bound and Determination of $C$

A major step and refinement of the original gravastar proposal took place in
2015, with the re-analysis of the Schwarzschild constant density interior
solution (5d) in the limit of $r_{\\!{}_{S}}\to r_{\\!{}_{M}}$, i.e. for
$\varepsilon=0$, leading also to an improved understanding and formulation of
the Lanczos-Israel conditions for null hypersurfaces.

In 1959 H. A. Buchdahl proved that in order for a static, spherically
symmetric solution of Einstein’s eqs. of mass $M$ and radius $r_{\\!{}_{S}}$
to remain everywhere finite, its compactness $GM/c^{2}r_{\\!{}_{S}}$ must not
be greater than $4/9$, or equivalently

$r_{\\!{}_{S}}\geq\frac{9}{8}r_{\\!{}_{M}}$ (50)

the same as from the limiting case (6) of the constant density interior
Schwarzschild solution (5d). The Buchdahl bound (50) depends upon three
necessary conditions:

  1. (i)

The pressure is everywhere isotropic: $p_{\perp}(r)=p(r)$,

  2. (ii)

The density profile is monotonically non-increasing outward:
$\rho^{\prime}=\frac{d\rho}{dr}\leq 0$,

  3. (iii)

The metric functions $f(r),h(r)$ and the derivative $\frac{df}{dr}$ are
continuous at $r=r_{\\!{}_{S}}$,

in addition to Einstein’s eqs. holding everywhere. The first condition (i) is
satisfied by all ordinary fluid eqs. of state. The second condition (ii) is
physically reasonable since an outer layer of higher density material would
generally result in an instability to the denser material falling towards the
center. The third condition (iii) precludes a discontinuity at the surface of
the star.

When the inequality (ii) is saturated, i.e. $\rho=\bar{\rho}$ is a constant,
one recovers the Schwarzschild interior solution (5d), which actually
disallows the equality in (50), since in that marginal case the pressure (5d)
diverges at $r_{0}=0$. This is consistent with the bound on the central
pressure MartinVisser:2003

$p(0)\geq\bar{\rho}\,\left[\frac{1-\sqrt{1-r_{\\!{}_{M}}/r_{\\!{}_{S}}}}{3\sqrt{1-r_{\\!{}_{M}}/r_{\\!{}_{S}}}-1}\right]$
(51)

when condition (ii) holds, since this lower bound diverges when
$r_{\\!{}_{S}}=\frac{9}{8}r_{\\!{}_{M}}$.

Because the bound for the general spherically symmetric static solution
obeying conditions (i)-(iii) is saturated by the limiting case of
$\rho^{\prime}=0$, the behavior of the constant density solution (5d) itself
is of fundamental interest as the limit (50) is reached and $r_{\\!{}_{S}}$ is
then reduced further.

When one does consider the constant density solution (5d) for
$r_{\\!{}_{S}}<\frac{9}{8}r_{\\!{}_{M}}$, some of its rather remarkable
features quickly become apparent. First, the pressure divergence which first
appears at the origin moves out to a spherical surface of finite radius
$r_{0}$ given by (7), and a new regular solution for $0\leq r<r_{0}$ opens up
behind it, with $D<0$, negative pressure, and $\bar{\rho}+3p<0$, violating the
strong energy condition. The metric functions $f,h$ remain non-negative in the
interior, with $f(r_{0})=0$ at $r=r_{0}$ only, cf. Figs. 2-4.

Figure 2: Pressure (in units of $\bar{\rho}$) as a function of $r$ (in units
of $r_{\\!{}_{S}}$) of the interior Schwarzschild solution (5d) for various
values of the parameter $r_{\\!{}_{S}}/r_{\\!{}_{M}}<9/8=1.125$. The upper
plot shows $p(r)$ for the values
$r_{\\!{}_{S}}/r_{\\!{}_{M}}=1.124,1.087,1.053$ (brown, orange, red curves),
where the divergence in the pressure occurs at
$r_{0}/r_{\\!{}_{S}}=0.106,0.552,0.761$ respectively. The lower plot shows
$p(r)$ for the values $r_{\\!{}_{S}}/r_{\\!{}_{M}}=1.053,1.010,1.001$ (red,
green, blue curves), where the divergence in the pressure occurs at
$r_{0}/r_{\\!{}_{S}}=0.761,0.959,0.996$ respectively. For $r<r_{0}$ the
pressure and $\bar{\rho}+3p$ are negative. Note the change of vertical scale
in the plots (the red curves are the same in each) and the pointwise approach
of the negative interior pressure $p\rightarrow-\bar{\rho}$ as $r_{\\!{}_{S}}$
approaches the Schwarzschild radius $r_{\\!{}_{M}}$ from above and $r_{0}$
approaches $r_{\\!{}_{M}}$ from below. Figure 3: The redshift factor
$\sqrt{f}$ as a function of $r$ (in units of $r_{\\!{}_{S}}$) of the interior
Schwarzschild solution (5d) for the same values of the parameter
$r_{\\!{}_{S}}/r_{\\!{}_{M}}<9/8=1.125$ as in Figs. 2. The brown, orange, red,
green and blue curves are for the values
$r_{\\!{}_{S}}/r_{\\!{}_{M}}=1.124,1.087,1.053,1.010,1.001$ respectively. Note
the approach of the zero of $\sqrt{f}$ at $r_{0}$ towards $r_{\\!{}_{S}}$ from
below as $r_{\\!{}_{S}}$ approaches the Schwarzschild radius $r_{\\!{}_{M}}$
from above. In this limit $\sqrt{f(0)}\to 1/2$. Figure 4: The metric function
$h$ as a function of $r$ (in units of $r_{\\!{}_{S}}$) of the interior and
exterior Schwarzschild solution for the values of the parameter
$r_{\\!{}_{S}}/r_{\\!{}_{M}}=2.500,1.667,1.250,1.111,1.000$ (brown, orange,
red, green and blue curves) respectively. The minimum of $h$ approaches zero
at $r=r_{\\!{}_{S}}$, as $r_{\\!{}_{S}}$ approaches the Schwarzschild radius
$r_{\\!{}_{M}}$ from above. Note the cusp-like discontinuity of the
derivatives $dh/dr$ at $r_{\\!{}_{S}}$ and $df/dr$ at $r_{0}$ in this and the
previous figure, which coincide at $r_{0}=r_{\\!{}_{S}}$, when
$r_{\\!{}_{S}}=r_{\\!{}_{M}}$.

As the star is compressed beyond the Buchdahl bound and its radius approaches
the Schwarzschild radius $r_{\\!{}_{S}}\rightarrow r_{\\!{}_{M}}^{+}$ from
outside, (7) shows that the radius of the sphere where the pressure diverges
and $f(r_{0})=0$ moves from the origin to the outer edge of the star, i.e.
$r_{0}\to r_{\\!{}_{M}}^{-}$, and in that limit the interior solution with
negative pressure comes to encompass the entire interior region $0\leq
r<r_{\\!{}_{S}}$, excluding only the outer boundary at
$r_{\\!{}_{S}}=r_{\\!{}_{M}}$. Finally, and most remarkably of all, since in
this limit $H^{2}r_{\\!{}_{S}}^{2}=r_{\\!{}_{M}}/r_{\\!{}_{S}}\rightarrow 1$,
inspection of (5d) shows that the entire interior solution then has constant
negative pressure

$p=-\bar{\rho}\,,\qquad{\rm for}\qquad
r<r_{\\!{}_{S}}=r_{0}=r_{\\!{}_{M}}=2GM$ (52)

with the metric functions

$f(r)=\mbox{\small$\displaystyle\frac{1}{4}$}\,(1-H^{2}r^{2})=\mbox{\small$\displaystyle\frac{1}{4}$}\,h(r)=\mbox{\small$\displaystyle\frac{1}{4}$}\,\Big{(}1-\mbox{\small$\displaystyle\frac{r^{2}}{r_{\\!{}_{M}}^{2}}$}\Big{)},\qquad
H=\frac{1}{r_{\\!{}_{M}}}$ (53)

corresponding to a regular static patch of pure dS space, although one in
which $g_{tt}$ is $\frac{1}{4}$ its usual value, so that the passage of time
in the interior is modified from what would be expected in the usual static
coordinates of dS space. In other words the constant density interior solution
which Schwarzschild found in 1916 becomes essentially the gravitational
condensate star solution of 2001 and Sec. 3, in which the thin shell boundary
layer is of infinitesimal thickness ($\varepsilon=0$), residing exactly at the
null hypersurface $r=r_{\\!{}_{M}}$, fixing the value of $C=\frac{1}{4}$ of
(24) in the interior dS patch unambiguously.

Typical profiles of the pressure $p(r)$ and $f^{\frac{1}{2}}$ for values of
the radius $r_{\\!{}_{S}}$ in the range
$r_{\\!{}_{M}}<r_{\\!{}_{S}}<\frac{9}{8}r_{\\!{}_{S}}$ and the approach of
$r_{\\!{}_{S}}\to r_{\\!{}_{M}}$ are shown in Figs. 2 and 3. Insight into the
pressure divergence at $r=r_{0}$ and vanishing of $f(r_{0})$ is obtained by a
careful examination of the covariant Komar mass-energy for a time independent
geometry possessing the static Killing field $K_{(t)}=\partial_{t}$, with
components $K^{\mu}_{(t)}=\delta_{t}^{\ \mu}$ in coordinates (1). This Komar
mass is defined by

$M=\int_{V}\\!\big{(}\\!-2\,T_{\mu}^{\ \nu}+T_{\lambda}^{\
\lambda}\,\delta_{\mu}^{\ \nu}\big{)}\,K^{\mu}_{(t)}\,d^{3}\Sigma_{\nu}\ +\
\frac{1}{4\pi G}\int_{\partial V_{-}}\\!\\!\\!\kappa\,dA$ (54)

expressing the total mass-energy of the system $M$ in terms of a three-volume
integral of the matter stress-energy, plus a possible surface flux
contribution from the inner two-surface, where

$\kappa=-\nabla^{\mu}\\!K^{\nu}_{(t)}\,e^{0}_{\ [\mu}e^{1}_{\
\nu]}=\displaystyle\frac{1}{2}\sqrt{\frac{h}{f}}\ \frac{d\\!f\\!}{dr}$ (55)

is the surface gravity, defined in terms of the vierbeins $e^{0}_{\
t}=\sqrt{-g_{tt}}=f^{\frac{1}{2}},\,e^{1}_{\
r}=\sqrt{g_{rr}}=h^{-\frac{1}{2}}$ with the areal integration measure

$dA=r^{2}\sin\theta\,d\theta\,d\phi$ (56)

on the spherical two-surface of constant $t$ and $r$.

Since the volume integration measure in (54) is

$d^{3}\Sigma_{\nu}=\delta^{t}_{\ \nu}\,\displaystyle{\sqrt{\frac{f}{h}}}\
r^{2}\sin\theta\,dr\,d\theta\,d\phi$ (57)

it contains a factor of $\sqrt{f}$ which vanishes at exactly the same
$r=r_{0}$ as where $p(r_{0})$ diverges, so that the pressure singularity at
$r=r_{0}$ and in fact the total volume integrand in (54) is MazEM:2015

$\frac{1}{G}\frac{d}{dr}\big{[}r^{2}\kappa(r)\big{]}=4\pi\sqrt{\frac{f}{h}}\
r^{2}\,\big{(}\rho+p+2p_{\perp}\big{)}=4\pi r^{2}\bar{\rho}\ {\rm
sgn}(D)+\frac{8\pi}{3}\,r_{0}^{\ 3}\,\bar{\rho}\ \delta(r-r_{0})$ (58)

with a well-defined $\delta$-function contribution having support at
$r=r_{0}$, which when integrated over $dr$ gives a finite result. This
$\delta$-function contribution to the volume integral in (54) is a result of
the discontinuity in the surface gravities

$\kappa_{\pm}\equiv\lim_{r\to r_{0}^{\pm}}\kappa(r)=\pm\frac{4\pi
G}{3}\,\bar{\rho}\,r_{0}=\pm\frac{r_{M}\,r_{0}}{2r_{\\!{}_{S}}^{3}}$ (59)

on either side of the $r=r_{0}$ surface, which results in a transverse
pressure and contribution to the integrand (58) of

$8\pi\sqrt{\frac{f}{h}}\
r^{2}\,(p_{\perp}-p)=\frac{8\pi}{3}\,\bar{\rho}r_{0}^{\
3}\,\delta(r-r_{0})=2M\,\bigg{(}\\!\frac{r_{0}}{r_{\\!{}_{S}}}\\!\bigg{)}^{\\!\\!3}\,\delta(r-r_{0})$
(60)

localized on the spherical surface at $r=r_{0}$, and hence a breakdown of the
isotropic pressure assumption on that surface. This gives the finite
contribution

$E_{{}_{S}}=2M\,\bigg{(}\\!\frac{r_{0}}{r_{\\!{}_{S}}}\\!\bigg{)}^{\\!\\!3}\to
2M$ (61)

to the Komar energy (54) of the surface, which becomes $2M$ in the limit
$r_{0}\to r_{\\!{}_{S}}$.

The Schwarzschild constant density interior solution (5d) in the limit
$r_{\\!{}_{S}}\to r_{\\!{}_{M}}$ therefore has the physical interpretation of
a gravastar with a well defined surface tension energy (61) localized at
$r=r_{\\!{}_{M}}=r_{\\!{}_{H}}$, precisely where the BH and dS horizons would
be, and the two geometries are joined instead.

The dynamical stability of this sequence of constant density Schwarzschild
stars beyond the Buchdahl limit has been studied in PosChir:2019 .

###  4.1 Redshift Modified Boundary Conditions on a Null Hypersurface

It is clear that the presence of the $\sqrt{f}\to 0$ factor in the volume
measure (57) is critical in rendering the Komar energy on the null surface at
$r=r_{0}$ finite. This points to the modification of the Lanczos-Israel
junction conditions necessary to give a finite, well-defined and physical
surface stress tensor on a null surface where the normal $n^{\mu}$ itself
becomes null and cannot be normalized as in (38). To handle this case one
should define the redshifted extrinsic curvature tensor

${\bf K}_{a}^{\ b}=-\gamma^{bc}{\bf n}_{\mu}\upsilon_{(c)}^{\
\nu}\nabla_{\\!\nu}\upsilon_{(a)}^{\ \mu}=-\gamma^{bc}{\bf
n}_{\mu}\upsilon_{(c)}^{\ \nu}\Gamma^{\mu}_{\ \ \,\nu\lambda}\upsilon_{(a)}^{\
\lambda}$ (62)

where the normal ${\bf n}$ to the surface at constant $r$ approaching $r_{0}$
satisfies

${\bf n}\cdot{\bf n}=({\bf n}^{r})^{2}g_{rr}=f(r)\,,\qquad{\rm so\
that}\qquad{\bf n}^{r}=\sqrt{fh}$ (63)

which goes to zero on the null surface, instead of (38) and the first member
of (40). This gives a redshift modified extrinsic curvature tensor (62) with
the non-vanishing components

$\displaystyle{\bf K}_{t}^{\ t}$
$\displaystyle=\displaystyle{\sqrt{\frac{h}{f}}\,\Gamma_{\,rtt}=\frac{1}{2}\sqrt{\frac{h}{f}}\,\frac{d\\!f\\!}{dr}=\kappa}$
(64a) $\displaystyle{\bf K}_{\theta}^{\ \theta}={\bf K}_{\phi}^{\ \phi}$
$\displaystyle=-g^{\theta\theta}\sqrt{fh}\,\Gamma_{r\theta\theta}=\displaystyle{\frac{\\!\\!\sqrt{fh}}{r}}\to
0$ (64b)

instead of (41). This redshifted extrinsic curvature has the finite
discontinuities

$\displaystyle\big{[}{\bf K}_{t}^{\ t}\big{]}$ $\displaystyle=[\kappa]$ (65a)
$\displaystyle\big{[}{\bf K}_{\theta}^{\ \theta}\big{]}$
$\displaystyle=\big{[}{\bf K}_{\phi}^{\ \phi}\big{]}=0$ (65b)

on the null surface, so that applying the standard Lanczos-Isreal conditions,
but to this redshift modified extrinsic curvature ${\bf K}_{a}^{\ b}$, gives
the surface stress tensor density

${}^{(\Sigma)}\\!T_{a}^{\ \,b}\,\sqrt{\frac{f}{h}}={\pazocal S}_{a}^{\
\,b}\,\delta(r-r_{0})\,,\qquad 8\pi G\,{\pazocal S}_{a}^{\ b}=-\big{[}{\bf
K}_{a}^{\ b}\big{]}+\delta_{a}^{\ b}\,\big{[}{\bf K}_{c}^{\ c}\big{]}$ (66)

instead of (42), and a finite surface tension

${\pazocal S}_{\theta}^{\ \theta}={\pazocal S}_{\phi}^{\
\phi}=\tau_{{}_{S}}=\frac{\Delta\kappa}{8\pi
G}=\frac{M\,r_{0}}{8\pi\,r_{\\!{}_{S}}^{3}}$ (67)

of the null surface at $r=r_{0}$, instead of (44) which diverges as
$\varepsilon\to 0$.

The factor of $\sqrt{f/h}$ in (67) and modified extrinsic curvature (62) is
clearly necessitated by the covariant integration measure (57) in the Komar
energy integral (54). Once ${\bf K}_{a}^{\ b}$ is defined as in (62), with the
normal ${\bf n}$ normalized as in (63) so as to become a null vector in the
horizon limit, the surface stress tensor (67) on a null hypersurface can be
determined unambiguously by application of the junction conditions (66), which
are of the same form as the standard Lanczos-Israel conditions, but for ${\bf
K}_{a}^{\ b}$ rather than $K_{a}^{\ b}$ of (37), and crucially for ${\bf
K}_{a}^{\ b}$ with one covariant and one contravariant index. Raising or
lowering these indices would introduce factors of $f$ or $1/f$ that would
again render the formalism empty or ambiguous on a null hypersurface where
$f(r_{0})$ vanishes. There is no need for introducing a ‘transverse’ normal
and ‘transverse’ curvature tensor as proposed in BarIsr:1991 .

This procedure can also be justified and verified explicitly from the form of
the Einstein tensor in the spacetime metric (1). Since the metric functions
$f,h$ must be equal on the null hypersurface approached from either side, but
their derivatives may not be, the first derivatives $f^{\prime},h^{\prime}$
can contain Heaviside step functions. Thus Dirac $\delta$-functions signaling
integrable surface contributions can only appear from second derivative terms
$f^{\prime\prime},h^{\prime\prime}$. The angular components $G_{\theta}^{\
\theta}=G_{\phi}^{\ \phi}$ in the metric (1) indeed contain a second
derivative of $f$ with respect to $r$, but they contain singular
$f^{\prime}/f,h^{\prime}/h$ and $h/f$ terms as well, which become infinite or
ill-defined on the null hypersurface where $f(r_{0})=0$. However, the tensor
densities

$\displaystyle\displaystyle{\sqrt{\frac{f}{h}}\,G^{\
\theta}_{\theta}=\sqrt{\frac{f}{h}}\,G^{\
\phi}_{\phi}=\frac{1}{2}\frac{d}{dr}\left(\sqrt{\frac{h}{f}}\frac{df}{dr}\right)+\frac{1}{2r}\left(\sqrt{\frac{f}{h}}\frac{dh}{dr}+\sqrt{\frac{h}{f}}\frac{df}{dr}\right)}$
(68)
$\displaystyle=\displaystyle{\left[\frac{1}{2}\sqrt{\frac{h}{f}}\,\frac{df}{dr}\right]\delta(r-r_{0})+\dots}$
(69)

combine these terms into a total $r$ derivative. It is clear that the
remaining terms of (68), although discontinuous, do not contain any
$\delta$-function contributions to the surface stress tensor at $r=r_{0}$,
while the $\delta$-function term coming from the total derivative first term
is precisely the well-defined discontinuity in the surface gravity $\kappa$ of
(55). The $\delta$-function comes automatically with the correct measure
factor of the Komar mass integral, so that the surface stress tensor in the
angular components on the null hypersurface at $r=r_{0}$ is defined by (66),
directly from Einstein’s eqs.

Thus once the technical issue of the correct modification of the Lanczos-
Israel junction conditions on a null hypersurface is determined, with the
finitely integrable distributional surface tensor at $r=r_{0}$ given by (66),
the Schwarzschild constant density interior solution becomes not a
pathological case arguing for the necessary collapse to a BH singularity for
masses violating the Buchdahl bound (50), but on the contrary a well-defined
and physically sensible solution to Einstein’s eqs. for
$r_{\\!{}_{S}}=r_{\\!{}_{M}}$, which shows that evading the Buchdahl bound
requires discarding the first condition (i) of isotropy assumed in its proof,
and the appearance of a physical surface with surface tension (67), as well as
allowing a negative pressure interior. Furthermore and remarkably, in the
limit $r_{\\!{}_{S}}\to r_{\\!{}_{M}}^{+}$ from above, the spherical null
surface becomes coincident with the Schwarzschild radius itself $r_{0}\to
r_{\\!{}_{M}}^{-}$ from below, and the negative pressure of the interior also
becomes a constant $p=-\bar{\rho}$, which is just the $\varepsilon\to 0$
classical limit of the gravitational condensate star of Grav_Univ:2023 ;
MazEMPNAS:2004 . Thus this viable non-singular alternative for the final state
of gravitational collapse might have been found just a few months after
Einstein introduced GR.

In the gravastar limit $r_{\\!{}_{S}}\to r_{\\!{}_{M}}$ the surface stress
tensor and surface tension becomes

${\pazocal S}_{\theta}^{\ \theta}={\pazocal S}_{\phi}^{\
\phi}=\tau_{{}_{S}}=c^{2}\frac{\Delta\kappa}{8\pi
G}=\frac{Mc^{2}}{8\pi\,r_{\\!{}_{M}}^{2}}=\frac{c^{6}}{32\pi\,G^{2}M}$ (70)

after restoring the factors of $c$. The discontinuity of the derivatives
$f^{\prime},h^{\prime}$ at $r=r_{\\!{}_{M}}$ and non-zero stress tensor there
in this limit clearly does not satisfy the analytic continuation assumption,
or ‘uneventful free fall’ hypothesis used to obtain Fig. 1. The conformal
diagram of the idealized classical gravastar in which $\varepsilon=0$
identically is shown in Fig. 5 instead.

It is important that in this limit $f(r_{\\!{}_{M}})=h(r_{\\!{}_{M}})=0$,
vanishing at one radius only, but never becoming negative. Hence light rays
can hover there indefinitely, but there is no trapped surface on which
outgoing light rays are bent inward, so this critical assumption of the 1965
Penrose theorem is not satisfied. There is no singularity in the interior,
only a regular static patch of dS space. Since $p=-\bar{\rho}$ also violates
the strong energy condition, the singularity theorems relying on this
condition instead of a trapped surface are also evaded.

Figure 5: Conformal diagram for the gravastar with angular variables
$(\theta,\phi)$ suppressed, in the idealized case $\varepsilon\\!\\!=\\!\\!0$.
Schwarzschild exterior (diamonds) and regular de Sitter static patch interior
(triangles), are joined at their mutual null horizons at
$r\\!\\!=R(=\\!r_{\\!{}_{M}}\\!\\!=\\!r_{\\!{}_{H}})$. A few typical constant
$r$ timelike worldlines (red curves) and constant $t$ spacelike hypersurfaces
(green curves) are also shown.

More realistically one may expect that $f,h$ become small, ${\pazocal
O}(\varepsilon)$, but never exactly vanishing when quantum effects are
included, as in the first gravastar model. Then light rays can also escape the
interior in principle, and the multiple copies of the conformal diagram of
Fig. 5 would no longer appear. This shows again that global properties are
very sensitive to small changes in the near-horizon region. The conformal
diagram of a gravastar with any finite $\varepsilon>0$ (however small) is one
triangular wedge only of Fig. 5, resembling that of any regular body such as a
star, albeit with a highly redshifted timelike tube, which is the thin
transitional boundary layer between the Schwarzschild exterior and dS
interior.

To relate (70) to (44) of the earlier first gravastar model of Sec. 3, one
observes that multiplying (43) by the redshift factor of $\sqrt{f(r)}$ needed
in the definitions of ${\bf K}$ and $\kappa$ for null hypersurfaces gives

$\kappa_{\,\rm
II}\\!\simeq\frac{1}{2r_{\\!{}_{M}}}\sqrt{\frac{wf}{\varepsilon}}\simeq\frac{1}{2r_{\\!{}_{M}}}\sqrt{\frac{w_{1}f(r_{1})}{\varepsilon}}\simeq\frac{1+w_{1}}{4r_{\\!{}_{M}}}$

in the entire region II, which is both finite and constant as $\varepsilon\to
0$. Thus adding the contributions to the surface stress tensor from the
discontinuities of ${\bf K}_{a}^{\ b}$ at the two surfaces at $r_{1}$ and
$r_{2}$, give equal and opposite contributions from the intermediate region
II, which cancel. This leaves only the total discontinuity from the interior
dS region I to exterior Schwarzschild region III, i.e.

$\displaystyle\sqrt{f(r_{1})}\,S^{\
\theta}_{\\!\theta}\big{|}_{r=r_{1}}+\sqrt{f(r_{2})}\,S^{\
\theta}_{\\!\theta}\big{|}_{r=r_{2}}=\displaystyle{\frac{3+w_{1}}{64\pi
G^{2}M}-\frac{w_{2}}{64\pi G^{2}M}\frac{1+w_{1}}{1+w_{2}}}$
$\displaystyle=\displaystyle{\frac{1}{16\pi G^{2}M}=\frac{\Delta\kappa}{8\pi
G}}$ (71)

where (32) with $C=1/4$, implying $w_{1}=1+2w_{2}$ has been used. Hence the
total surface tension is the very thin boundary layer on the null horizon is
obtained, to leading order in $\varepsilon\ll 1$ (even if non-zero), in which
no reference to the intermediate region II, its eq. of state or integration
constants $w_{1},w_{2}$ at all appear in the final result. Thus the
multiplication of the extrinsic curvature and surface stress tensor defined by
(36)-(42) by an additional factor of $\sqrt{f}$, as required by the volume
measure in the the Komar energy integral gives a finite result when the normal
in the definition (37) declines into tangency with the hypersurface and ${\bf
n}$ itself becomes a null vector BarIsr:1991 ; BelGonEM1:2022 . This result is
finite and universal, in the sense of being entirely independent of any
assumptions of the eq. of state or any other characteristics of the thin shell
boundary layer interposed between the exterior Schwarzschild and interior dS
geometries, joined at their mutual null horizons, in the classical limit
$\varepsilon\to 0$.

###  4.2 The ‘First Law’ for Spherically Symmetric Gravastars

For the sourcefree exterior BH solution of (1), the volume term in (54)
vanishes if integrated from $r=r_{\\!{}_{M}}$ to $r=\infty$, but since

$\displaystyle\kappa_{ext}(r)=\frac{GM}{r^{2}}\to\kappa_{+}=\frac{1}{4GM}=\frac{1}{2r_{\\!{}_{M}}}\qquad
as\qquad r\to r_{\\!{}_{M}}$ (72)

the surface integral at the horizon boundary gives the entire mass $M$ of the
BH. Already at the level of the Smarr relation (12), the coefficient of
$dA_{{}_{H}}$ appears to be some sort of surface tension, but this
interpretation is unclear since the BH horizon is assumed to be a causal
boundary only, with vanishing surface stress tensor.

On the other hand, for the full interior + exterior Schwarzschild solution in
the gravastar limit with $r_{\\!{}_{S}}=r_{\\!{}_{M}}$, there is a surface
stress tensor and physical surface tension (70) at $r=r_{\\!{}_{M}}$. Since
the solution is regular at the origin, one may integrate (54) from $r=0$ to
$r=\infty$. There is no contribution from the last explicit surface area
integral in (54), but the integrand (58) of the volume integral in (54) has a
Dirac $\delta$-function contribution at $r=r_{\\!{}_{M}}$ from the
discontinuity (59) in the surface gravities there, which becomes

$\Delta\kappa\equiv\kappa_{+}-\kappa_{-}=\frac{1}{2GM}=\frac{1}{r_{\\!{}_{M}}}$
(73)

for $r_{\\!{}_{S}}=r_{\\!{}_{M}}$. This surface tension itself gives rise to a
surface energy (61), which becomes just $2M$ in the gravastar limit. The first
term in (58) gives the volume contribution $-M$ to the Komar mass integral
from the negative pressure in the interior. The sum of these and result from
(54) is

$E=E_{v}+E_{{}_{S}}=-M+2M=M$ (74)

giving again just the total mass $+M$. The differential relation

$dE=dE_{v}+\tau_{\,S}\,dA_{{}_{H}}=-dMc^{2}+\tau_{\,S}\,dA_{{}_{H}}$ (75)

for the volume and surface terms in the Komar mass may be derived from this
MazEM:2015 , showing that identification of $\tau_{s}$, the coefficient of
$dA_{{}_{H}}$ with the surface tension of the physical surface located at
$r=r_{\\!{}_{M}}=r_{\\!{}_{H}}$ for the full (interior + exterior)
Schwarzshild solution is fully justified. Since $\tau_{{}_{S}}$ depends on the
discontinuity $\Delta\kappa=\kappa_{+}-\kappa_{-}=2\kappa_{+}$ of equal and
opposite surface gravities at the surface it contributes twice the value from
the Smarr BH formula (12), with the volume term of the interior, absent in
(12) making up the difference to the total $M$. Despite the different local
attributions of energy to the volume and surface in the Komar vis-a-vis the
Misner-Sharp definition of energy, the total mass of the gravastar is $M$ in
either definition, as it must be.

The constant density interior solution is the critical case that saturates and
exceeds the Buchdahl bound. Its behavior including negative pressure and the
dark energy vacuum eq. of state $p_{{}_{V}}=-\rho_{{}_{V}}$ are inherent in
and can be described quite satisfactorily in strictly classical GR, providing
an explicit counterexample to the singularity theorems. This maximally compact
classical solution has no trapped surface, assumed in the Penrose singularity
theorem. The discontinuity (73) of the surface gravities at the would-be
horizon is instead the location of the thin shell of a gravitational
condensate star, with a non-zero surface tension of a physical surface. This
possibility for a non-singular endpoint of gravitational collapse is and was
always allowed in classical GR, awaiting a more complete analysis of possible
discontinuities at null surfaces, and physical realization of a quantum phase
transition at the horizon capable of generating a change in the local vacuum
to $p_{{}_{V}}=-\rho_{{}_{V}}<0$.

A summary of the main results of the reanalysis of the Schwarzschild constant
density interior solution presented in this section is as follows.

  * •

The Buchdahl bound (50) on the compactness of a spherically symmetric self-
gravitating mass $M$ of any kind subject to Einstein’s eqs. is evaded and
$r_{\\!{}_{S}}\to r_{\\!{}_{M}}<9r_{\\!{}_{M}}/8$, if the assumption of
isotropy of the pressure everywhere is removed, and instead $p_{\perp}-p\neq
0$ exhibiting a $\delta$-function corresponding to a physical surface tension
at $r_{0}=r_{\\!{}_{M}}$ when $r_{\\!{}_{S}}=r_{\\!{}_{M}}$ coincide. The
constant negative pressure interior with $p=-\bar{\rho}$ is a non-singular
patch of static dS space, which is responsible for effective repulsion in
general relativity rather than the fatal attraction that leads to the critical
mass (8), thereby resolving Wheeler’s ‘difficulty’ of the final state of
complete gravitational collapse, which need not be a BH.

  * •

The $\varepsilon\to 0$ classical GR limit may be considered the universal
gravastar limit, in the sense that it is independent of any ansatz or
additional assumptions of an eq. of state of an interposed boundary layer of
‘non-inflationary material’ with a well-defined surface tension
$\tau\\!_{{}_{S}}$ of (70). This is a clear specific example of the
generalization of the Lanczos-Israel matching conditions to null
hypersurfaces, also dispensing with the need to restrict to matching on
timelike boundaries, as in the original gravastar proposal, and subsequent
papers.

  * •

The determination of $C=\frac{1}{4}$ is just that required to make the surface
gravities on the Schwarzschild exterior and dS interior sides of the surface
at $r=r_{\\!{}_{M}}$ equal and opposite, i.e. $\kappa_{-}=-\kappa_{+}$, which
is the physical condition of equality of the forces pressing inward and
outward on the surface in static equilibrium. This is seen also in the
Euclidean formulation as the equality of the Euclidean time periodicities
$\beta_{+}=2\pi\kappa_{+}=8\pi GM$ and $\beta_{-}=2\pi\kappa_{-}=4\pi/H$, so
that the inferred Rindler-Hawking ‘temperatures’ $1/\beta_{+}=1/8\pi GM$ and
$1/\beta_{-}=H/4\pi$ (rather than $H/2\pi$) are equal at
$r_{\\!{}_{H}}=H^{-1}=r_{\\!{}_{M}}=2GM$, just as would be expected in
equilibrium.

  * •

Because $\tau\\!_{{}_{S}}>0$, the ‘First Law’ of non-rotating gravastars (75)
shows that increasing the area by $A_{{}_{H}}$ by allowing for surface mode
perturbations with non-zero angular momenta costs energy, so that the static
equilibrium of the surface is stable to such perturbations, independently of
any eq. of state assumptions. Similar to the Bohr atom, these angular
perturbations are expected to have a discrete spectrum which will distinguish
gravastars with a surface from BH horizons in gravitational wave (GW)
signatures.

  * •

The differential relation (75) between gravastar solutions parametrized by $M$
is purely a classical mechanical relation, entirely within the domain of
classical GR, rather than a quantum or thermodynamic one. The area
$A_{{}_{H}}$ is simply the geometrical area of the condensate star surface
with no implication of entropy. As (76) shows there is no entropy at all
associated with a macroscopic condensate at zero temperature. The Planck
length $L_{\rm Pl}=\sqrt{\hslash G/c^{3}}$, Planck mass $M_{\rm
Pl}=\sqrt{\hslash c/G}$, or the Boltzmann constant $k_{B}$ do not enter these
classical relations at all at $\varepsilon=0$.

  * •

Further, by the Gibbs relation

$p+\rho=s\,T+\mu\,n$ (76)

and $\mu=0$ here, since no chemical potential corresponding to a conserved
quantum number has entered our classical considerations, the interior
Schwarzschild-de Sitter solution with $p+\rho=0$ is a zero entropy density
$s=0$ and zero temperature single macroscopic state, justifying its
designation as a gravitational Bose-Einstein condensate (GBEC).

  * •

The matching of the metric interior to the exterior solution for
$r=r_{\\!{}_{M}}$ has the cusp-like behavior shown in Figs. 3-4, with
discontinuous first derivative with respect to the original Schwarzschild
radial coordinate $r$. This is symptomatic of non-analytic behavior,
invalidating the assumption of metric analyticity needed for deriving
periodicity in complexified imaginary time $t\to-i\tau$ HartHawk:1976 . Unlike
in the analytically extended vacuum Schwarzschild solution, where $f(r)$
becomes negative and the Killing vector $K=\partial_{t}$ becomes spacelike in
the interior $r<r_{\\!{}_{M}}$ of a BH, in the negative pressure gravastar
solution $f(r)$ is everywhere non-negative, and there is no requirement of any
fixed periodicity in imaginary time of either the geometry or Green’s
functions of quantum fields in this geometry. The surface gravity $\kappa$
therefore carries no implication of temperature or thermal radiation. The zero
temperature vacuum state in the exterior Schwarzschild geometry is expected to
be the Boulware vacuum, which is the usual Minkowski vacuum in the
asymptotically flat region $r\to\infty$, whose Green’s functions also have
non-analytic cusp-like behavior at $r=r_{\\!{}_{M}}$ Boul:1975 , and which has
no Hawking radiation ChrFul:1977 .

  * •

Non-analyticity at $r=r_{\\!{}_{M}}$ is exactly the property suggested by the
analogy of BH horizons to phase boundaries and quantum critical surfaces in
condensed matter physics ChaplineHLS:2001 ; Chap:2003 , although the equality
of $|\kappa_{\pm}|$ was not obtained in the condensed matter analogies,
requiring as it does a proper general relativistic treatment of matching on
null hypersurfaces, which is given also for horizons with non-zero angular
momentum in BelGonEM1:2022

  * •

Since $f(r)=0$ corresponds to the ‘freezing’ of local proper time at
$r=r_{\\!{}_{S}}$, it also suggests critical slowing down characteristic of a
phase transition. The vanishing of the effective speed of light
$c^{2}_{eff}=c^{2}f(r)$ is analogous to the behavior of the sound speed
determined by the low energy excitations at a critical surface or phase
boundary. This and experience with Bose-Einstein condensates in other contexts
suggest that gravitation and spacetime itself are ‘emergent’ phenomena of a
more fundamental microscopic many-body theory Laugh:2003 ; Mazur:2007 ;
ChapMazur:2009 .

  * •

The interpretation of $c^{2}_{eff}\geq 0$ as the effective speed of light
squared, which must remain always non-negative, calls to mind Einstein’s
original papers on the local relativity principle for static gravitational
fields, which led him to the general theory from the Minkowski metric
$ds^{2}=-c^{2}\,dt^{2}+dx^{2}+dy^{2}+dz^{2}$, by allowing first the time
component $-g_{tt}=c^{2}$ and eventually all other components of the metric to
be functions of space (and then also time) Einstein:1912 . Thus it could be
argued that the non-negativity of $c^{2}_{eff}=c^{2}f(r)$ in a static geometry
is more faithful to Einstein’s original conception of the Equivalence
Principle, realized by real continuous coordinate transformations, rather than
complex analytic extension around a square root branch point that would allow
$c_{eff}^{2}<0$.

  * •

At the minimum, the matching of the $p=-\bar{\rho}$ dS interior to
Schwarzschild exterior provides a consistent alternative to analytic
extension, entirely within the framework of classical GR, provided only that
surface boundary layers on null boundary surfaces are admitted. A phase
boundary and non-singular ‘BH’ interior require no violation of the
Equivalence Principle, at least in its weak form.

  * •

Finally, since $K_{(t)}$ remains timelike for a gravastar, $t$ is a global
time and unlike in the analytic continuation hypothesis, the spacetime is
truly globally static. The $t=$ const. hypersurface is a Cauchy surface and is
everywhere spacelike. This is exactly the property of a static spacetime
necessary to apply standard quantum theory, for the quantum vacuum to be
defined as the lowest energy state of a Hamiltonian bounded from below, and
for the Schrödinger equation to describe unitary time evolution, thus avoiding
any conflict with unitarity or an ‘information paradox.’

##  5 Slowly Rotating Gravastars: Junction Conditions & Moment of Inertia

A serendipitous consequence of the reanalysis of the 1916 Schwarzschild
interior solution via the Komar mass integral (54) is that it provides an
explicit example of ‘gluing’ of two different geometries at their mutual null
horizons, in which the surface stresses can be unambiguously determined,
providing a clear interpretation of the surface tension of the null surface at
$r_{\\!{}_{M}}=r_{\\!{}_{H}}$. This example serves as a general template to
provide the general matching conditions for null surfaces with non-zero
angular momentum, necessary to describing rotating gravastars as well
BelGonEM1:2022 .

Consider the general axisymmetric and stationary metric line element
Lewis:1932 ; HartleSharp:1967 ; Bardeen:1970 ; FriedmanChandra:1972 ;
ChandraBH

$ds^{2}=-e^{2\nu}dt^{2}+e^{2\psi}\big{(}d\phi-\omega\,dt\big{)}^{2}+e^{2\alpha}dr^{2}+e^{2\beta}d\theta^{2}$
(77)

in coordinates adapted to the two Killing symmetries of time translation
$K_{(t)}=\partial_{t}$ and axisymmetry $K_{(\phi)}=\partial_{\phi}$ around the
axis of rotation. The five functions $\nu,\psi,\omega,\alpha,\beta$ are
functions of the remaining $(r,\theta)$ coordinates only. The choice of those
remaining two coordinates is still subject to some coordinate freedom, which
if fully exploited reduces the five functions appearing in (77) to just four
independent functions. However it is convenient to leave the remaining
coordinate freedom unfixed, in order to encompass various choices that may be
made.

Corresponding to the two Killing symmetries and Killing vectors $K_{(t)}$ and
$K_{(\phi)}$ are two conserved quantities which may be constructed by Komar’s
general method, viz. the total mass-energy

$\displaystyle M=\displaystyle\frac{1}{4\pi G}\int_{\partial
V_{+}}\\!(\kappa+\omega{\pazocal J})\,dA$
$\displaystyle=\displaystyle\int_{V}\sqrt{-g}\ \big{(}\\!\\!-T_{t}^{\
t}+T_{r}^{\ r}+T_{\theta}^{\ \theta}+T_{\phi}^{\
\phi}\big{)}\,dr\,d\theta\,d\phi\,+\,\frac{1}{4\pi G}\int_{\partial
V_{-}}\\!(\kappa+\omega{\pazocal J})\,dA$ (78)

and angular momentum

$\displaystyle\displaystyle J=\frac{1}{8\pi}\int_{\partial
V_{+}}\\!\\!{\pazocal J}\,dA=G\int_{V}\sqrt{-g}\
T_{t}^{\,\phi}\,dr\,d\theta\,d\phi+\frac{1}{8\pi}\int_{\partial
V_{-}}\\!\\!{\pazocal J}\,dA$ (79)

where BelGonEM1:2022

$\displaystyle\kappa=\displaystyle\mbox{\small$\displaystyle\frac{1}{2}$}\,e^{-\alpha-\nu}\
\frac{\partial e^{2\nu}\\!}{\\!\partial r}\,$ (80a) $\displaystyle{\pazocal
J}=\displaystyle-\mbox{\small$\displaystyle\frac{1}{2}$}\,e^{2\psi}\,e^{-\alpha-\nu}\
\frac{\partial\omega}{\partial r}$ (80b)

are the surface gravity and ($8\pi$ times) the angular momentum density per
unit area respectively. The volume measure factor and area element are given
by

$\displaystyle\sqrt{-g}=\exp\,(\nu+\psi+\alpha+\beta)$ (81) $\displaystyle
dA=\exp\,(\beta+\psi)\,d\theta\,d\phi\,.$ (82)

in the coordinates of (77). In the case of spherical symmetry $\omega,T_{t}^{\
\phi}$ and ${\pazocal J}$ vanish, (78)-(80a) reduce to (54)-(55), and (82)
becomes $\sqrt{f/h}\ r^{2}\sin\theta$.

A null horizon in the geometry (77) occurs when $e^{2\nu}\to 0$, which without
loss of generality may be taken to be at a fixed $r=r_{\\!{}_{H}}$. The
induced metric on the horizon

$ds^{2}\Big{|}_{r=r_{\\!{}_{H}}}=e^{2\psi_{\\!{}_{H}}}\big{(}d\phi-\omega_{{}_{\textit{H}}}dt\big{)}^{2}+e^{2\beta_{\\!{}_{H}}}d\theta^{2}$
(83)

involves the functions $\psi,\omega,\beta$ which must be continuous there,
whereas $\nu$ and $\alpha$ need not be. It is the inclusion of the$\sqrt{-g}$
factor of the volume integration measure in the Komar integrals that leads to
total second derivatives of $\nu,\alpha$ with respect to $r$. These total
derivative terms give rise both to the areal surface terms in (78)-(79), and
the distributional Dirac $\delta$-functions on null hypersurfaces from the
discontinuities of $\kappa$ and ${\pazocal J}$ there. Thus, as in (68)-(69) it
is necessary to consider the tensor density$\sqrt{-g}G_{\mu}^{\ \nu}$ and in
particular its potentially singular part $e^{\nu+\alpha}G_{\mu}^{\ \nu}$, to
find these $\delta$-function and surface contributions BelGonEM1:2022 .

This is the fundamental physical basis for defining the redshifted extrinsic
curvature for stationary, axisymmetric geometries by

${\bf K}^{a}_{\ b}=-\gamma^{ac}n^{\mu}\Gamma_{\mu
bc}=\mbox{\small$\displaystyle\frac{1}{2}$}e^{-\alpha+\nu}\gamma^{ac}\left(\frac{\partial
g_{bc}}{\partial r}-\frac{\partial g_{rc}}{\partial x^{b}}-\frac{\partial
g_{rb}}{\partial x^{c}}\right)\,,\quad\\{a,b,c\\}=t,\theta,\phi$ (84)

with the normal vector having the components

${\bf n}_{\mu}=\delta^{r}_{\ \mu}\,e^{\alpha+\nu}\,,\hskip 34.14322pt{\bf
n}^{\mu}=\delta^{\mu}_{\ r}\,e^{-\alpha+\nu}$ (85)

normalized to

${\bf n}\cdot{\bf n}=e^{2\nu}\to 0$ (86)

on the null hypersurface, instead of (38) for a timelike hypersurface. These
relations and the inverse $\gamma^{ac}$ of the induced metric are to be
evaluated for finite $e^{2\nu}$ and the horizon limit
$e^{2\nu}\sim\varepsilon\to 0$ taken at the end.

With ${\bf n}$ defined by (85), ${\bf K}^{a}_{\ b}$ has the components

$\displaystyle\displaystyle{{\bf K}^{t}_{\
t}=\mbox{\small$\displaystyle\frac{1}{2}$}\,e^{-\alpha-\nu}\,\frac{\partial}{\partial
r}e^{2\nu}-\mbox{\small$\displaystyle\frac{1}{2}$}\,\omega\,e^{2\psi}\,e^{-\alpha-\nu}\,\frac{\partial\omega}{\partial
r}=\kappa+\omega\,{\pazocal J}}$ (87a) $\displaystyle\displaystyle{{\bf
K}^{t}_{\
\phi}=\mbox{\small$\displaystyle\frac{1}{2}$}\,e^{2\psi}\,e^{-\alpha-\nu}\,\frac{\partial\omega}{\partial
r}=-{\pazocal J}}$ (87b) $\displaystyle\displaystyle{{\bf K}^{\phi}_{\
t}=\mbox{\small$\displaystyle\frac{1}{2}$}\,\omega\,e^{-\alpha-\nu}\,\frac{\partial}{\partial
r}e^{2\nu}-\mbox{\small$\displaystyle\frac{1}{2}$}\,\omega^{2}e^{2\psi}\,e^{-\alpha-\nu}\,\frac{\partial\omega}{\partial
r}-\mbox{\small$\displaystyle\frac{1}{2}$}\,e^{-2\psi}\,e^{-\alpha+\nu}}\,\frac{\partial}{\partial
r}\,\big{(}\omega e^{2\psi}\big{)}$
$\displaystyle=\displaystyle{\omega\,\kappa+\omega^{2}\,{\pazocal J}+{\pazocal
O}(\varepsilon)}$ (87c) $\displaystyle\displaystyle{{\bf K}^{\theta}_{\
\theta}=\mbox{\small$\displaystyle\frac{1}{2}$}\,e^{-\alpha+\nu-2\beta}\,\frac{\partial}{\partial
r}e^{2\beta}={\pazocal O}(\varepsilon)}$ (87d)
$\displaystyle\displaystyle{{\bf K}^{\phi}_{\
\phi}=\mbox{\small$\displaystyle\frac{1}{2}$}\,\omega\,e^{2\psi}\,e^{-\alpha-\nu}\,\frac{\partial\omega}{\partial
r}+\mbox{\small$\displaystyle\frac{1}{2}$}\,e^{-2\psi}\,e^{-\alpha+\nu}\,\frac{\partial}{\partial
r}e^{2\psi}=-\omega\,{\pazocal J}+{\pazocal O}(\varepsilon)}$ (87e)

with the non-listed components vanishing. Thus the discontinuities on the null
horizon surface are

$\displaystyle\left[{\bf K}^{t}_{\
t}\right]=[\kappa]+\omega_{{}_{\textit{H}}}[{\pazocal J}]$ (88a)
$\displaystyle\left[{\bf K}^{t}_{\ \phi}\right]=-[{\pazocal J}]$ (88b)
$\displaystyle\left[{\bf K}^{\phi}_{\
t}\right]=\omega_{{}_{\textit{H}}}[\kappa]+\omega_{{}_{\textit{H}}}^{2}\,[{\pazocal
J}]$ (88c) $\displaystyle\left[{\bf K}^{\theta}_{\ \theta}\right]=0$ (88d)
$\displaystyle\left[{\bf K}^{\phi}_{\
\phi}\right]=-\,\omega_{{}_{\textit{H}}}[{\pazocal J}]$ (88e)

where $\omega\\!=\\!\omega_{{}_{\textit{H}}}$ on the horizon. The null horizon
junction conditions are then of the same form as the Lanczos-Israel conditions
(with one contravariant and one covariant index), namely

$\displaystyle[{\bf K}_{a}^{\ b}]=-4\pi G\,\Big{(}2{\pazocal S}_{a}^{\
b}-\delta_{a}^{\ b}{\pazocal S}_{c}^{\ c}\Big{)}$ (89a) $\displaystyle 8\pi
G\,{\pazocal S}_{a}^{\ b}=-[{\bf K}_{a}^{\ b}]+\delta_{a}^{\ b}[{\bf K}_{c}^{\
c}]$ (89b)

but with this physical stress tensor on the null horizon hypersurface related
to the original Lanczos-Israel (LI) one by

$\big{(}{\pazocal S}_{a}^{\ b}\big{)}_{\rm here}=\lim_{e^{\nu}\to
0}\left\\{e^{\nu}\,\big{(}S_{a}^{\ b}\big{)}_{\rm LI}\right\\}$ (90)

which is finite in the horizon limit $e^{\nu}\to 0$ since the discontinuities
of ${\bf K}^{a}_{\ b}$ in (88) are finite in that limit. The surface stress
tensor density is then found to be

${}^{(\Sigma)}T_{a}^{\ \,b}\,e^{\alpha+\nu}={\pazocal S}_{a}^{\
\,b}\,\delta(r-r_{\\!{}_{H}})$ (91)

localized on the surface $r=r_{\\!{}_{H}}$ with the correct volume measure
factor (to be multiplied by the continuous $e^{\psi+\alpha}$ factors) for the
Komar mass and angular momentum integrals.

This improved understanding and generalization of the junction condition
formalism to null hypersurfaces and in particular rotating null horizons
appropriate for the Kerr geometry opens the way to finding rotating gravastar
solutions. The formalism has been applied in the case of slow rotation
following methods of Hartle:1967 ; HartleThorne:1968 ; ChandraMiller:1974  who
express the slowly rotating line element in the (77) form with

$\displaystyle
e^{2\nu}=f(r)\Big{[}1+2h_{0}(r)+2h_{2}(r)\,P_{2}(\cos\theta)\Big{]}$ (92a)
$\displaystyle
e^{2\psi}=r\sin\theta\,\Big{[}1+2k_{2}(r)\,P_{2}(\cos\theta)\Big{]}$ (92b)
$\displaystyle e^{2\alpha}=\displaystyle{\frac{1}{h(r)}\left[1+2\
\frac{m_{0}(r)+m_{2}(r)\,P_{2}(\cos\theta)}{r-2m(r)}\right]}$ (92c)
$\displaystyle e^{2\beta}=r\Big{[}1+2k_{2}(r)\,P_{2}(\cos\theta)\Big{]}\,.$
(92d)

The frame dragging angular velocity $\omega=\omega(r)$ is independent of
$\theta$ to lowest (first) order in the slow rotation expansion about the
spherically symmetric solution, while $h_{0},m_{0}$ are the monopole
($\ell=0$) and $h_{2},m_{2},k_{2}$ the quadupole ($\ell=2$) perturbations,
which are second order in that expansion, and multiplied by the appropriate
Legendre polynomials $P_{0},P_{2}$. The Einstein eqs. linearized around the
previously determined spherically symmetric solution are then solved assuming
that the eq. of state for the fluid composing a slowly rotating star is
unchanged from its form in the non-rotating solution. In the case of a slowly
rotating gravastar the solutions are separated again into an exterior
sourcefree region, and interior dS region with the same $p=-\rho$ eq. of
state, and the solutions are matched at their mutual horizons using the
discontinuities and surface stress tensor (89b).

Requiring that the solution be asymptotically flat as $r\to\infty$, the
induced metric on the horizon (83) to be continuous (${\pazocal C}^{0}$), and
the interior solution not to contain $\delta$-function source terms at the
origin $r=0$, severely restricts the solutions for the functions
$h_{0},m_{0},h_{2},m_{2},k_{2}$. With these conditions one finds
BelGonEM2:2022  333Here $J$ is the angular momentum in geometric units
$GJ/c^{3}$, with dimensions of (length)2.

$\displaystyle r>r_{\\!{}_{H}}:\qquad\qquad\omega$
$\displaystyle=\frac{2J}{r^{3}}$ (93a) $\displaystyle m_{0}$
$\displaystyle=J^{2}\left(\frac{1}{r_{\\!{}_{H}}^{3}\\!}-\frac{1}{r^{3}}\right)$
(93b) $\displaystyle h_{0}$
$\displaystyle=-\frac{m_{0}}{r-r_{\\!{}_{H}}}=-\frac{J^{2}\\!}{r_{\\!{}_{H}}^{3}r^{3}}\left(r^{2}+rr_{\\!{}_{H}}+r_{\\!{}_{H}}^{2}\right)$
(93c) $\displaystyle m_{2}$
$\displaystyle=\frac{J^{2}}{r^{3}}\left(r-r_{\\!{}_{H}}\right)\left(\frac{5}{r}-\frac{2}{r_{\\!{}_{H}}\\!}\right)$
(93d) $\displaystyle h_{2}$
$\displaystyle=\frac{J^{2}}{r^{3}}\bigg{(}\frac{1}{r}+\frac{2}{r_{\\!{}_{H}}\\!\\!}\,\bigg{)}$
(93e) $\displaystyle k_{2}$
$\displaystyle=-\frac{2J^{2}}{r^{3}}\bigg{(}\frac{1}{r}+\frac{1}{r_{\\!{}_{H}}\\!\\!}\,\bigg{)}$
(93f)

for the exterior solution $r>r_{\\!{}_{H}}$, and

$\displaystyle r<r_{\\!{}_{H}}:\qquad\qquad\omega$
$\displaystyle=\frac{2J}{r_{\\!{}_{H}}^{3}}=\omega_{H}$ (94a) $\displaystyle
m_{0}$ $\displaystyle=0$ (94b) $\displaystyle h_{0}$
$\displaystyle=C_{{}_{I}}$ (94c) $\displaystyle m_{2}$
$\displaystyle=-\frac{2J^{2}(r^{2}-r_{\\!{}_{H}}^{2})^{2}}{r^{2}r_{\\!{}_{H}}^{5}}$
(94d) $\displaystyle h_{2}$
$\displaystyle=\frac{2J^{2}}{rr_{\\!{}_{H}}}\left(\frac{1}{r^{2}}-\frac{1}{r_{\\!{}_{H}}^{2}}\right)$
(94e) $\displaystyle k_{2}$
$\displaystyle=-\frac{2J^{2}}{rr_{\\!{}_{H}}}\left(\frac{1}{r^{2}}+\frac{1}{r_{\\!{}_{H}}^{2}}\right)$
(94f)

for the interior solution, with $r_{\\!{}_{H}}=r_{\\!{}_{M}}$ unchanged from
the non-rotating solution to this order.

With these requirements the exterior solution for a slowly rotating gravastar
(93) is in fact the same as that of a slowly rotating Kerr BH, up to a
coordinate transformation from the $(r,\theta)$ Hartle-Thorne coordinates used
in slow rotation expansion of the metric in (92) to the Boyer-Lindquist
$(r_{{}_{\rm BL}},\theta_{{}_{\rm BL}})$ coordinates in which the Kerr
solution is more commonly expressed, and then expanded to second order in $J$.
This coordinate transformation is given explicitly by eqs. (4.12) of
BelGonEM2:2022 . In particular the ergosphere boundary is given by $r_{\rm
ergo}(\theta)=r_{\\!{}_{H}}[1+4J^{2}\sin^{2}\theta/r_{\\!{}_{M}}^{4}]\geq
r_{\\!{}_{H}}$ in the Hartle-Thorne coordinates of (92). The reason for the
two exterior solutions of a Kerr BH and slowly rotating gravastar to be
identical is that although the metric functions $\alpha,\nu,\omega$ have
discontinuous derivatives on the gravastar null horizon surface, the induced
surface metric (83) is continuous, and all the metric functions have been
required to be finite there, which are the same conditions required for the
horizon of a Kerr BH, that lead to the ‘no hair’ theorems Carter:1971 ;
Carter:1973 ; Carter:2009b ; Robinson:1975 .

The solution (93)-(94) is such that the discontinuities of two quantities
which appear in (88) and the surface stress-energy tensor (91) are

$\displaystyle\displaystyle{[\kappa]=\frac{1}{r_{\\!{}_{M}}}\left(1-\frac{3J^{2}}{r_{\\!{}_{M}}^{4}}+\frac{C_{{}_{I}}}{2}\right)}$
(95a) $\displaystyle[{\pazocal
J}]=\displaystyle{\frac{3J}{r_{\\!{}_{M}}^{2}}\,\sin^{2}\theta}$ (95b)

with the result that surface energy of the horizon boundary at
$r=r_{\\!{}_{H}}$ in the Komar mass-energy is

$\displaystyle\displaystyle{E_{{}_{S}}=\frac{1}{4\pi
G}\int_{H}\Big{(}[\kappa]+\omega[{\pazocal J}]\Big{)}dA=\frac{[\kappa]}{4\pi
G}\,A_{{}_{H}}+\frac{3\omega_{{}_{H}}\,J}{2\,G}\int_{0}^{\pi}d\theta\,\sin^{3}\theta}$
$\displaystyle\displaystyle{=\frac{[\kappa]r_{\\!{}_{M}}^{2}}{G}+\frac{2\omega_{{}_{H}}J}{G}=2M\left(1+\frac{C_{{}_{I}}}{2}+\frac{J^{2}}{r_{\\!{}_{M}}^{4}}\right)}$
(96)

where (94a) has been used. The Komar volume energy of the interior solution is

$E_{{}_{V}}=-M\,\left(1+C_{{}_{I}}\right)$ (97)

so that the total energy of the slowly rotating gravastar specified by
(93)-(94) is

$E\equiv{\pazocal
M}=E_{{}_{V}}+E_{{}_{S}}=M\left(1+\frac{2J^{2}}{r_{\\!{}_{M}}^{4}}\right)$
(98)

where $M$ is the mass of the non-rotating solution. Since this is the
irreducible mass for a non-rotating BH, (98) will be recognized as equivalent
to the Christodoulu formula

${\pazocal
M}^{2}=M^{2}+\frac{J^{2}}{4G^{4}M^{2}}=M^{2}\left(1+\frac{4J^{2}}{r_{\\!{}_{M}}^{4}}\right)$
(99)

when $E={\pazocal M}$ is expanded to first order in $J^{2}/r_{\\!{}_{M}}^{4}$.
Thus $C_{{}_{I}}$ drops out of the total mass-energy ${\pazocal M}$ of the
slowly rotating solution, when the volume and surface contributions are added,
which then obeys exactly the same relation as a Kerr BH of the same mass and
angular momentum to this order in $J^{2}$.

For the Komar angular momentum of the gravastar, the integrand of the volume
term is proportional to $T^{\ t}_{\phi}$, which vanishes in the interior under
the assumption of the $p=-\rho$ eq. of state being unchanged from the non-
rotating case in order to apply the formalism of Hartle:1967 ;
HartleThorne:1968 ; ChandraMiller:1974 . Using (95b), the entire angular
momentum is carried by the surface contribution

$J_{\rm tot}=J_{{}_{S}}=\frac{1}{8\pi}\int_{H}[{\pazocal
J}]\,dA=\frac{3J}{4}\,\int_{0}^{\pi}d\theta\,\sin^{3}\theta=J$ (100)

which is due to the surface stress tensor ${\pazocal S}^{\ t}_{\phi}$. From
(36) and (95) the components of the surface stress tensor are

$\displaystyle 8\pi{\pazocal S}^{t}_{\ t}$
$\displaystyle=-\frac{6J^{2}}{r_{\\!{}_{H}}^{5}}\,\sin^{2}\\!\theta$ (101a)
$\displaystyle 8\pi{\pazocal S}^{t}_{\ \phi}$
$\displaystyle=\frac{3J}{r_{\\!{}_{H}}^{2}}\,\sin^{2}\\!\theta\,$ (101b)
$\displaystyle 8\pi{\pazocal S}^{\phi}_{\ t}$
$\displaystyle=-\frac{2J}{\,r_{\\!{}_{H}}^{4}}$ (101c) $\displaystyle
8\pi{\pazocal S}^{\phi}_{\ \phi}$
$\displaystyle=\frac{1}{r_{\\!{}_{H}}}+\frac{C_{{}_{I}}}{2r_{\\!{}_{H}}}-\frac{3J^{2}}{r_{\\!{}_{H}}^{5}}+\frac{6J^{2}}{r_{\\!{}_{H}}^{5}}\,\sin^{2}\\!\theta$
(101d) $\displaystyle 8\pi{\pazocal S}^{\theta}_{\ \theta}$
$\displaystyle=\frac{1}{r_{\\!{}_{H}}}+\frac{C_{{}_{I}}}{2r_{\\!{}_{H}}}-\frac{3J^{2}}{r_{\\!{}_{H}}^{5}}\,.$
(101e)

by a straightforward application of the modified Lanczos-Israel junction
conditions on the null horizon hypersurface at $r_{\\!{}_{H}}=r_{\\!{}_{M}}$
BelGonEM1:2022 ; BelGonEM2:2022 .

The moment of inertia $I$ is defined as the ratio of its angular momentum
$J/G$ to angular velocity $\omega_{{}_{\textit{H}}}$, and given by

$I=\frac{J/G}{\omega_{{}_{\textit{H}}}\\!\\!}=Mr_{\\!{}_{H}}^{2}=4G^{2}M^{3}$
(102)

identical to its value for a Kerr BH Posada:2017 ; BeltrPosad:2023 . This
might have been expected from the fact that the external geometry is identical
to the Kerr BH geometry, and the interior dS condensate carries no angular
momentum, all of $J$ being concentrated on the rotating null horizon surface
from (100). The result (102) may be compared to the moment of inertia of a
thin shell of uniform surface density with total mass $M$ and radius
$r_{\\!{}_{H}}$ in Newtonian mechanics, which is $2/3$ of (102). The factor of
$3/2$ larger for the moment of inertia of a rotating gravastar is evidently a
relativistic effect due to the surface density contributing to the mass-energy
localized on the surface being $2M$ rather than $M$ and the non-uniform
distribution of the surface stress tensor components $-{\pazocal S}^{t}_{\
\,t}+{\pazocal S}^{\theta}_{\ \,\theta}+{\pazocal S}^{\phi}_{\ \,\phi}$ (101)
contributing to the Komar surface energy (96). in the non-rotating inertial
frame. Nonetheless, with the moment of inertia (102) the contribution of the
rotational kinetic energy to the total mass ${\pazocal M}$ in (98) is

$E_{\rm
rot}=2M\,\frac{J^{2}}{r_{\\!{}_{M}}^{4}}=\frac{1}{2}\,I\omega_{{}_{\textit{H}}}^{2}$
(103)

exactly what would be expected from the Newtonian mechanics of a rotating
body. Unlike a BH where it is supposed that there is no mass-energy at all at
the horizon to rotate or to give rise to the large moment of inertia (102),
the rotating gravastar has a well-defined Komar stress-energy (101) and
angular momentum on the rotating surface at $r=r_{\\!{}_{H}}$ (100), where all
the angular momentum resides, and this gives rise to (102) and (103) by
straightforward evaluation of the relevant surface integrals (96), (100).

The results for the external solution (93), (102) and (103) being identical to
that of a slowly rotating Kerr BH is consistent with the BH ‘no hair’ theorems
being extended to rotating gravastars as well Carter:1971 ; Carter:2009b ;
Maztalk:2020 ; BarCarRubLib:2019 , at least in the strictly classical limit of
an infinitesimally thin shell ($\varepsilon\to 0$). This again is the result
of the boundary conditions on the metric perturbations in (92) at the null
horizon being finite and identical in the two cases. Thus in this limit slowly
rotating gravastars are expected to be indistinguishable from Kerr BH’s in
their external geometry, and have the same tidal deformability and Love number
as a Kerr BH, namely none at all ChirPosGued:2020 . Any gravastar ‘hair’ would
be limited to the very thin quantum phase transition boundary layer of
thickness $\ell$ (33), and be very short.

As welcome as these first results for rotating gravastars are, the interior
solution (94) contains a singularity at $r=0$, which leads to divergences in
the Weyl tensor at the origin proportional to $J^{2}/r^{5}$ in both the
monopole and quadrupole terms. For that reason (94) cannot be considered fully
satisfactory. Since all the perturbations to order $J^{2}$ are fully
determined by the boundary conditions of asymptotic flatness, matching of the
induced metric (83) on the null horizon surface, and absence of
$\delta$-function singularities in the Komar $M$ or $J$ at the origin, the
singular result at the origin cannot be avoided in the Hartle-Thorne method,
which requires that the eq. of state of the rotating matter, in this case
$p=-\rho$, be unchanged from the non-rotating solution. This assumption may be
questioned, and indeed considerations of rotating superfluid condensates
suggests that a non-vanishing vortex density should arise in a rotating
gravitational condensate as well. This would require a different interior
solution for a rotating gravastar, in which the $p=-\rho$ condition of
Hartle:1967 ; HartleThorne:1968 ; ChandraMiller:1974  is relaxed.

To summarize the results of this section, the conditions of:

  1. (i)

Asymptotic flatness,

  2. (ii)

Matching conditions on the slowly rotating null horizon,

  3. (iii)

Unchanged $p=-\rho$ eq. of state from the non-rotating gravastar, and

  4. (iv)

Absence of $\delta$-function singularities at the origin in the Komar mass and
angular momentum,

determine the slowly rotating gravastar solution to order $J^{2}$, up to one
undetermined constant $C_{{}_{I}}$. The exterior solution is identical to that
of a Kerr BH to this order, including the moment of inertia (102) which is
consistent with all of the angular momentum being carried by the rotating null
hypersurface at this lowest order in $J^{2}$. However the interior solution
(94) under these four requirements is singular at the origin, and therefore
not fully satisfactory. It appears that relaxing at least condition (iii) is
necessary to remove the $1/r^{5}$ power law singularity at $r=0$. It is
possible that then allowing a distribution of angular momentum within the
interior would also modify the exterior solution for a rapidly rotating
gravastar from that of a Kerr BH.

##  6 Macroscopic Effects of the Quantum Conformal Anomaly

The considerations of the previous sections are entirely classical, and show
that an endpoint of complete gravitational collapse different from a BH is
possible even in Einstein’s classical GR, with no violation of the Equivalence
Principle. However, it does not provide a mechanism by which a phase
transition from one value of $\Lambda_{\rm eff}$ outside a gravastar to
another value inside can occur, elucidate the special role of the event
horizon in triggering this transition, nor specify the physics determining the
value of $\varepsilon>0$. Addressing these questions requires consideration of
the effects of the quantum theory of the matter/radiation stress tensor on the
near-horizon geometry, and in particular, the macroscopic effects of the
conformal anomaly of the quantum stress tensor source for Einstein’s eqs. in
the near-horizon region.

At the smallest microscopic scales probed, the principles of quantum field
theory (QFT) hold, and matter under extreme pressures and densities in the
standard model (SM) of particle physics is clearly quantum in nature. Yet
Einstein’s classical GR remains unreconciled to quantum theory, and the
tension between quantum matter and classical gravity comes to the fore both in
the puzzles and paradoxes of BHs reviewed in Sec. 2, and in the nature and
value of cosmological dark energy driving the accelerated expansion of the
universe RiessSN:1998 ; PerlmutterSN:1999 ; DES:2019 .

The problems of reconciling classical GR with QFT first appear with the
stress-energy tensor $T^{\mu\nu}$, which is treated as a completely classical
source in Einstein’s eqs., whereas $\hat{T}^{\mu\nu}$ is a UV divergent
operator in QFT. Thus the minimal accomodation necessary to couple quantum
matter to Einstein’s theory is to replace the divergent $\hat{T}^{\mu\nu}$
operator by its renormalized expectation value
$\langle\hat{T}^{\mu\nu}\rangle$, in a semi-classical approximation. Defining
this expectation value by regularizing and renormalizing the contributions of
matter loops leads inevitably to consideration of the effects of quantum
matter on gravity itself, or in other words, to the effective field theory
(EFT) treatment of gravity, keeping track of quantum corrections to GR.

The quantum corrections are of two quite distinct kinds. The first are the
strictly short distance/high energy corrections contained in higher order
local curvature invariants which are needed in any case to renormalize
$\langle\hat{T}^{\mu\nu}\rangle$ DeWitt:1975 ; BirDav . The second are the
effects of higher point stress tensor correlators, which can contain light
cone singularities, that extend over macroscopic distance scales and provide
quantum corrections to GR even at low energy scales, and in particular on null
horizons.

Usually only quantum corrections of the first kind are considered Don:1994 ;
Bur:2004 ; Don:2012 , and the additions to the classical action of GR are
assumed to involve only an expansion in the local curvature invariants, such
as
$R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu},R_{\alpha\beta}R^{\alpha\beta},R^{2}$.
Since these are fourth order in derivatives of the metric, they become
significant only at the extreme UV Planck scale, but are negligibly small at
macroscopic distance scales or weak curvatures. The expansion in local
invariants is based on the assumption of decoupling of UV degrees of freedom
from the low energy EFT and strict separation of scales, familiar in other EFT
approaches AppCar:1975 ; Leut:1994 ; Alv:2012 .

On the other hand it has also been known for some time, even in flat space
QFT, that anomalies are not captured by such an expansion in higher order
local invariants, nor are they suppressed by any UV scale. Anomalies are
associated instead with the fluctuations of massless fields which do not
decouple, and which lead to $1/k^{2}$ poles in momentum space correlation
functions, that grow large on the light cone $k^{2}\to 0$ rather than the
extreme UV regime $k^{2}\sim M_{\mathrm{Pl}}^{2}$. Such massless light cone
poles are found in explicit calculations of the triangle anomaly diagrams of
$\langle\hat{J}_{5}^{\lambda}\hat{J}^{\alpha}\hat{J}^{\beta}\rangle,\langle\hat{T}^{\mu\nu}\hat{J}^{\alpha}\hat{J}^{\beta}\rangle$
in massless QED4 GiaEM:2009 ; ArmCorRose:2009 , and in the stress tensor
three-point correlator
$\langle\hat{T}^{\alpha\beta}\hat{T}^{\gamma\lambda}\hat{T}^{\mu\nu}\rangle$
of a general conformal field theory (CFT), by solution of the conformal Ward
Identities in momentum space BzoMcFSken:2018 ; TTTCFT:2019 .

Quite contrary to the decoupling hypothesis, quantum anomalies lead instead to
the principle of anomaly matching from UV to low energy EFT tHooft:1979 . In
the strong interactions, the chiral anomaly of the UV theory, QCD, survives to
low energies, requiring a specific Wess-Zumino (WZ) addition to the low energy
meson EFT WZ:1971 ; Leut:1994 , which is not suppressed by any high energy
scale, and without which the low energy $\pi^{0}\to 2\gamma$ decay rate, which
helped establish QCD as the UV theory of the strong interactions, does not
come out correctly FritGellLeut:1973 ; TreiJackGross:2015 ; Bertlbook .

In gravitation theory the relevant anomaly requiring attention is the
conformal anomaly of $T_{\mu}^{\ \nu}$. The classical stress-energy tensor of
conformal matter or radiation is traceless $T_{\mu}^{cl\,\mu}=0$. The
conformal (or trace) anomaly arises because it is impossible to maintain this
traceless condition at the quantum level, in an arbitrary metric background,
if the covariant conservation eq. $\nabla_{\nu}T_{\mu}^{\ \nu}=0$ is also to
be maintained. Since covariant conservation of $T_{\mu}^{\ \nu}$ is a
necessary requirement for the consistent coupling of matter to Einstein’s
eqs., it is the trace condition on $T_{\mu}^{\ \mu}$ that is given up and
which is ‘anomalous.’ This clash of symmetries, forcing one to choose between
them at the quantum level, is what is meant by an ‘anomaly.’ Although perhaps
a surprise when they were first discovered, it eventually became clear that
‘anomalies’ are a natural and necessary feature of QFT, and lead to
interesting and essential connections between the low energy (infrared:IR) and
high energy (ultraviolet:UV) limits of the theory, both in the SM and for GR
and the EFT of low energy gravity.

Intuitively, a conformal symmetry of the classical theory cannot be maintained
at the quantum level because QFT in all interesting cases involves
regularization and renormalization of UV divergent Feynman diagrams, which
introduce a scale, if only through logarithms, that necessarily breaks the
conformal and scale invariance of the classical theory. This is in fact the
reason that coupling ‘constants’ run with energy scale ChanEllis:1973 ;
GiaEM:2009 , a well-verfied experimental fact L3Running:2005 .

Since event horizons are null hypersurfaces with geometric significance as the
locus of points at which the timelike Killing field $K_{(t)}$ (or
$K=K_{(t)}+\omega K_{(\phi)}$ in the rotating case BelGonEM1:2022 ) become
null, signaling the critical condition for which a trapped surface can form if
$K^{2}$ changes sign, the light cone singularities of anomalous quantum
correlators such
$\langle\hat{T}^{\alpha\beta}\hat{T}^{\gamma\lambda}\hat{T}^{\mu\nu}\rangle$
can have large macroscopic effects there. Such light cone singularities imply
the existence of at least one additional light scalar (a priori massless)
degree of freedom in the low-energy EFT of macroscopic gravity, that is
clearly not accounted for in the classical action of GR, nor by the addition
of higher order local curvature invariants to the effective action.

It is $f(r)=-K_{(t)}\cdot K_{(t)}\to 0$ (or $e^{2\nu}=-K^{2}\to 0$ in the
rotating case of Sec. (5) which leads to the infinite blueshifting of
frequencies (3) or energies (19), as soon as $\hslash\neq 0$ is admitted. The
result is that the local quantum ‘vacuum’ near a BH horizon is infinitely
shifted in energy scales with respect to a state asymptotically far from the
BH, violating a naive expectation of decoupling of UV from IR scales. The
specification of the vacuum state involves non-local boundary conditions of
the entire Cauchy constant time slice of (1), which are not determined by any
simple expansion in the local curvature. On the contrary these boundary
conditions span a large range of scales between IR and UV, and are sensitive
to the conformal transformation properties of the effective action of quantum
matter coupled to gravity. The enormous entropy (16), the Hawking effect upon
which it is predicated, and the various BH paradoxes and conundra rely
crucially upon the specification of the quantum vacuum state for arbitrarily
high local frequencies and energies in the near-horizon region, precisely
where masses become irrelevant compared to local energies typified by (19),
making all finite mass scales negligible and the conformal anomaly relevant
there EMZak:2010 ; AntMazEM:2012 .

These observations lead to the important step in an EFT of gravity by the
taking into account the macroscopic effects of the conformal anomaly on
horizons in EMVau:2006 , where it was shown that the energy-momentum tensor
derived from the effective action of the conformal trace anomaly of massless
fields in curved space becomes large (indeed formally infinite) for generic
quantum states at both the Schwarzschild BH and dS static horizons.

###  6.1 The Two Dimensional Conformal Anomaly and Stress Tensor

In order to see the macroscopic effects of the conformal anomaly on horizons
in a simple case, it is useful to consider first $D=2$ spacetime dimensions.
In 2D it is straightforward to show by any covariant method preserving
coordinate invariance BirDav , that the renormalized expectation value

$\langle T_{\mu}^{\ \mu}\rangle\Big{|}_{\rm 2D}=\frac{N\hslash}{24\pi}\ R$
(104)

is non-vanishing in a general curved 2D spacetime, for $N=N_{s}+N_{f}$ the
number of free conformal scalar and Dirac fermion fields, i.e. in a conformal
field theory (CFT). By taking a variation of (104) with respect to the
arbitrary background metric, and then evaluating the result in flat space, one
can relate the anomaly to the existence of a massless $1/k^{2}$ pole in the
one-loop vacuum polarization diagram of $\Pi_{\mu\nu\alpha\beta}=i\langle
T_{\mu\nu}T_{\alpha\beta}\rangle$, cf. Fig. 6.

Figure 6: The one-loop stress tensor vacuum polarization of a 2D CFT, which
exhibits the massless $1/k^{2}$ pole of (105). In the equivalent tree graph at
right the conformalon scalar vertex $T^{(1)}_{\mu\nu}$ is given by the term
linear in $\varphi$ in (110), and the $\varphi$ propagator is obtained from
(108), cf. QuanGrav2D:2023 .

This tensor has the form in momentum space

$\displaystyle\displaystyle{\Pi_{\mu\nu\alpha\beta}(k)\Big{|}_{\rm
2D}=(\eta_{\mu\nu}k^{2}-k_{\mu}k_{\nu})(\eta_{\alpha\beta}k^{2}-k_{\alpha}k_{\beta})\,\frac{N\hslash}{12\pi
k^{2}}}$ (105) $\displaystyle\displaystyle{\Pi^{\ \mu}_{\mu\
\alpha\beta}(k)\Big{|}_{\rm
2D}=\frac{N\hslash}{12\pi}\,(\eta_{\alpha\beta}k^{2}-k_{\alpha}k_{\beta})}$
(106)

showing that the non-zero trace and coefficient on the right side of (104) is
directly related to the existence and residue of the $1/k^{2}$ pole in
$\Pi_{\mu\nu\alpha\beta}$. In fact, once the tensor index structure indicated
in (105) is fixed, as required by symmetries and the covariant conservation
law Ward identities $k^{\mu}\Pi_{\mu\nu\alpha\beta}(k)=0$ on any index, the
one-loop diagram of Fig. 6 is UV finite and completely determined, with (106)
the result BertlKohl:2001 . This shows that the conformal anomaly and pole is
independent of the regularization scheme and detailed UV behavior of the
quantum theory, provided that the identities following from the covariant
conservation law are maintained.

The essential point now is that the massless pole in (105) is a lightlike
singularity, indicating significant effects on the light cone, which extends
to macroscopic distance scales, and is particularly relevant on null horizons.