#  Thermodynamics of frozen stars

Ram Brustein(1), A.J.M. Medved(2,3), Tamar Simhon(1)
${}^{\textrm{\normalsize(1)\ Department of Physics, Ben-Gurion University,
Beer-Sheva 84105, Israel}}$ ${}^{\textrm{\normalsize(2)\ Department of Physics
\& Electronics, Rhodes University, Grahamstown 6140, South Africa}}$
${}^{\textrm{\normalsize(3)\ National Institute for Theoretical Physics
(NITheP), Western Cape 7602, South Africa}}$<EMAIL_ADDRESS>j.medved<EMAIL_ADDRESS>
###### Abstract

The frozen star is a recent proposal for a non-singular solution of Einstein’s
equations that describes an ultracompact object which closely resembles a
black hole from an external perspective. The frozen star is also meant to be
an alternative, classical description of an earlier proposal, the highly
quantum polymer model. Here, we show that the thermodynamic properties of
frozen stars closely resemble those of black holes: Frozen stars radiate
thermally, with a temperature and an entropy that are perturbatively close to
those of black holes of the same mass. Their entropy is calculated using the
Euclidean-action method of Gibbons and Hawking. We then discuss their
dynamical formation by estimating the probability for a collapsing shell of
“normal” matter to transition, quantum mechanically, into a frozen star. This
calculation followed from a reinterpretation of a transitional region between
the Euclidean frozen star and its Schwarzschild exterior as a Euclidean
instanton that mediates a phase transition from the Minkowski interior of an
incipient Schwarzschild black hole to a microstate of the frozen star
interior. It is shown that, up to negligible corrections, the probability of
this transition is $e^{-A/4}$, with $A$ being the star’s surface area. Taking
into account that the dimension of the phase space is $e^{+A/4}$, we conclude
that the total probability for the formation of the frozen star is of order
unity. The duration of this transition is estimated, which we then use to
argue, relying on an analogy to previous results, about the scaling of the
magnitude of the off-diagonal corrections to the number operator for the
Hawking-like particles. Such scaling was shown to imply that the corresponding
Page curve indeed starts to go down at about the Page time, as required by
unitarity.

##  1 Introduction

The frozen star was originally meant to provide a classical, geometric
alternative to the collapsed polymer model [1], which describes a highly
excited configuration of long, closed, fundamental strings that is lacking a
semiclassical geometric description. The polymer model is premised on the
notion that the interior of any regular object which successfully mimics a
black hole (BH) should be in a strongly non-classical state [2], as follows
from the idea that the uncertainty principle prevents the collapse of matter
to a singularity, much in the same way that the quantum hydrogen atom is
stable against collapse. A strongly non-classical state is tantamount to
having maximal entropy density, which in turn means maximally positive radial
pressure for a state of fixed energy density. The frozen star model aims to
mimic the properties of this highly quantum state in terms of a classical
geometry, which amounts to “flipping” the radial pressure from maximally
positive to maximally negative [3, 4]. The name “frozen star” was first
adopted in a later article [5] as an homage to classic literature [6]. The
current version of the frozen star model, along with many technical details,
can be found in [7] (see also [8]).

The relevant aspects of the model will be reviewed in the next section, but
let us take immediate note of some of the important features of the frozen
star. It has been known for quite a while that large negative pressure
provides a way for stabilizing ultracompact objects against further collapse
[9, 10, 11, 12] and for evading singularity theorems [13, 14]. Lesser known is
that a star with maximally negative radial pressure will be ultrastable
against perturbations [3, 5, 8]. Another feature is that the modifications
from general relativity extend over horizon-sized scales. This happens to be
essential for the self-consistency of the model once quantum-induced
evaporation is included; otherwise, the radiated energy will far exceed the
original mass of the object [15, 16].

As much of the focus of the current paper is on the thermodynamics of a frozen
star, let us recall some of the lore pertaining to “conventional” BHs.
Bekenstein [17] first proposed that BHs possess an entropy $S_{BH}$ that is
proportional to the surface area of the horizon $A$, and Hawking [18] later
fixed the constant of proportionality according to $\;S_{BH}=\frac{A}{4}\;$.
The latter contribution came about when Hawking discovered that a BH of mass
$M$, emits thermal radiation with a temperature of $\;T_{H}=\frac{1}{8\pi
M}\;$. Hawking was able to subsequently show that the density matrix of the
emitted radiation is exactly thermal, leading to what is famously known as the
information-loss problem [19]. Closely related to this paradox is the no-hair
theorem [20], as well as the transPlanckian problem [21], and less closely
related is the BH species problem [22]. Although the mathematics is
consensually beyond dispute (however, see [23]), interpreting the
thermodynamic properties of BHs and resolving the associated problems remains
a challenge in spite of intense efforts.

As the BH entropy is a critical element of the thermodynamic paradigm, any
viable BH mimicker should be able to account for its exceptionally large
value. We will be able to show that the frozen star can do just that! This is
a highly non-trivial challenge, as even the most compact of stars, when
composed of conventional matter, would have an associated entropy $S$ that is
many orders of magnitude smaller than that determined by the
Bekenstein–Hawking area law. Indeed, out of the many proposals for
ultracompact, almost-black objects (See, for example, [24]), none of these, as
far as we know, can replicate the entropy–area law — including the factor of
1/4 — in an unambiguous way. To understand the difficulty, suppose that we
replace the infinite redshift of the BH horizon with a finite but large
redshift $1/\varepsilon^{2}$, then what is the line of demarcation between a(n
almost) BH and a very compact star? Meanwhile, the BH entropy can be viewed as
a geometric construct that is associated with (perfectly black) horizons and
cannot be assigned generically to regular stars, no matter how compact they
might be. This argument is presented at the end of Gibbons and Hawking’s paper
on BH thermodynamics via Euclidean actions and partition functions [25]. As we
will show, Gibbons and Hawking made a couple of strong assumptions regarding
the composition of the stellar material that are not valid for the frozen
star; thus our results are not in contradiction with theirs.

An outstanding question about the frozen star is how a collapsing shell of
matter might dynamically transition into a geometry that deviates so
dramatically from the Schwarzschild geometry over a horizon-sized length
scale. If this transition does transpire, such a scenario is rather suggestive
of a quantum-induced phase transition. Here, we take a first step in support
of this idea by applying Mathur’s argument [26] that such a transition is
indeed possible because the large entropy means there is an exponentially
large number of microstates associated with the final-state configuration. And
so we evaluate the probability of transition $\Gamma$, from the Minkowski
interior of collapsing shell of matter, or incipient BH, geometry to that of a
frozen star and then, following Mathur, view this as the quantum-transition
probability from the initial state to a single microstate of the frozen star.
Our calculation requires us to interpret a transitional layer straddling the
the outer surface of the star as a Euclidean instanton that mediates the
stated transition. We find that the action of this instanton is equal to the
BH entropy up to perturbatively small corrections and, by the above reasoning,
conclude that the probability to transition to a single microstate is
$\;\Gamma\sim e^{-S_{BH}}\;$. Finally, applying Mathur’s argument about
multiplying the single-state probability by the large degeneracy of states, we
are then led to a total probability of order unity, suggesting that such a
transition is not only possible but likely. To our knowledge, this is the
first time such an estimate of the transition probability has been
successfully carried out for an ultracompact, horizonless object.

The subsequent sections are organized as follows: The next section provides a
very brief review on the frozen star. Longer discussions on its conceptual
origins and important physical features can be found in [3, 4, 5, 7, 8]. In
Section 3, we explain why the temperature of emitted radiation from the star
would be perturbatively close to the Hawking value [18]. Included is a
heuristic calculation that is premised on the Schwinger effect [27], which was
originally presented in [28]. See [29] for a recent discussion on the
temperature of compact objects that is similarly based on the Schwinger
effect. Section 4 begins with a general discussion on Euclidean BH
thermodynamics, following the Euclidean-action method of Gibbons and Hawking
[25]. We go on to confirm that ,at leading order, $\;S=A/4\;$ for the frozen
star by adopting their method, albeit in an unorthodox way as the outer
surface of the star is no longer a boundary for the Euclidean spacetime like
it is in the Schwarzschild case. This calculation of the entropy is novel and
one of the main results in the paper. We continue on, in Section 5, to
reinterpret a narrow interpolating layer — smoothly connecting the
Schwarzschild exterior to the frozen star interior — as a Euclidean instanton
which facilitates the transition from the Minkowski interior of an incipient
BH to a single microstate of the frozen star geometry. We are able to confirm
that the probability of this transition is precisely $e^{-A/4}$, at leading
order in our perturbative parameters, just as advertised. The duration of this
transition is also estimated, and this result is used in Section 6 to argue
that the number operator and thus also the density matrix for the frozen star
radiation could have non-trivial off-diagonal elements. We further argue,
following an earlier treatment [30], that the magnitude of the off-diagonal
elements scales so as to reproduce Page’s famous curve describing a unitary
evaporation process [31]. Finally, a brief overview is presented in Section 7.
There, we point out that the Euclidean picture of the frozen star closely
resembles a BH sourced by a condensate of stringy winding modes [32] (see
also, [33]) and remark on the significance of this “coincidence”.

##  2 The frozen star geometry

Here, we will be reviewing the relevant aspects of the frozen star geometry,
especially as it pertains to the following analysis. See [5, 7] for formal
details.

Let us first discuss the frozen star metric. We assume that it is static and
spherically symmetric,
$\;ds^{2}=f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}\;$, where the
equality $\;g_{tt}=-g^{rr}\;$ follows from the star’s characteristic equation
of state $\;\rho+p_{r}=0$ .

As common to textbook discussions, one defines a mass function,

$m(r)\;=\;4\pi\int\limits_{0}^{r}dx\,x^{2}\rho(x)\;\;\;{\rm for}\;\;\;r\leq
R\;,$ (1)

leading to a Schwarzschild-like form for the $t$–$r$ sector,

$f(r)\;=\;1-\frac{2m(r)}{r}\;.$ (2)

If $R$ is the radial size of the star, as implied in Eq. (1), then the total
mass is defined as $\;M=m(R)\;$.

We now deviate from Schwarzschild by setting $\;f(r)=\varepsilon^{2}\;$,
leading to the following line element,

$ds^{2}=-\varepsilon^{2}dt^{2}+\frac{1}{\varepsilon^{2}}dr^{2}+r^{2}d\Omega^{2}.$
(3)

The dimensionless parameter $\varepsilon^{2}$ should be regarded as a very
small number 111What we now call $\varepsilon^{2}$ had, until recently, been
called $\varepsilon$. The change was made to emphasize its positivity.
(recently estimated to be smaller than $10^{-22}$ [34]) but always finite.
With this choice, the mass function is

$m(r)\;=\frac{r}{2}(1-\varepsilon^{2})\;,$ (4)

and so

$f(r)\;=\;\varepsilon^{2}\;.$ (5)

To understand the significance of the perturbative parameter
$\varepsilon^{2}$, it is useful to consider the limit $\;\varepsilon^{2}\to
0\;$. In this limiting case, $\;g_{rr}\to\infty\;$ at the surface of the star,
so that we can identify this limit with the classical one $\;l_{P}\to 0\;$
($l_{P}$ is the Planck length). In other words,
$\;\varepsilon^{2}\sim\frac{l_{P}}{R}\;$.

The matter densities, or diagonal stress-tensor components, then take the
forms

$\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle\frac{1-(rf)^{\prime}}{r^{2}}\;=\;\frac{1-\varepsilon^{2}}{r^{2}}\;,$
(6) $\displaystyle 8\pi p_{r}$ $\displaystyle=$
$\displaystyle-\frac{1-(rf)^{\prime}}{r^{2}}\;=\;-\frac{1-\varepsilon^{2}}{r^{2}}\;,$
(7) $\displaystyle 8\pi p_{\perp}$ $\displaystyle=$
$\displaystyle\frac{(rf)^{\prime\prime}}{2r}\;=\;0\;,$ (8)

where a prime represents a radial differentiation and note that all of the
off-diagonal components are vanishing.

To ensure that the metric of the frozen star interior joins smoothly to the
external Schwarzschild metric, we have added an outer layer $T_{out}$ of width
$2\lambda R$, with $\;\lambda\ll 1\;$, that allows for a continuous transition
from one geometry to the other [5]. As the density in the bulk of the frozen
star goes as $\;\sim\frac{1}{r^{2}}\;$ and so is formally divergent at the
star’s center, there is also a need to smooth out the density in the central
region [5]. We have accomplished this by adding a second transitional layer
$T_{in}$ of width $\eta R$ that connects the bulk geometry to a regularized
core of width $\eta R$, where $\;\eta\ll 1\;$. The geometry of the frozen star
is depicted in Fig. 1.

Figure 1: The frozen star geometry. The core $r\leq\eta R$ is denoted by $I$,
the transitional layer $\eta R<r<2\eta R$ by $T_{in}$, the frozen star bulk
$2\eta R\leq r\leq R(1-\lambda)$ by $B$ and the outer transitional layer
connecting the star to the Schwarzschild region $R(1-\lambda)\leq r\leq
R(1+\lambda)$ by $T_{out}$. Both transitional layers and the core are not
drawn to scale, so they can be distinguished in the figure. The actual
relative scale is smaller by many orders of magnitude.

To understand the significance of the perturbative parameter $\lambda$, we
call upon the polymer model and its string-theoretical perspective. In this
picture, the string energy density is decreasing in the transitional layer
from its finite value in the bulk to zero at the outer surface. This
transitional layer needs to still be introduced in the limit
$\;\varepsilon^{2}\to 0\;$, indicating that $\lambda$ and $\varepsilon^{2}$
are independent perturbative parameters. Since we have already identified
small values of $\varepsilon^{2}$ with the classical limit, it follows that
$\lambda$ controls the strength of classical stringy effects; that is, the
effects of $\alpha^{\prime}$ or the inverse of the string tension. It can then
be concluded that $\;\lambda^{2}\sim\frac{\alpha^{\prime}}{R^{2}}\;$.

The non-regularized density leads to small corrections of order $\eta$, when
integrated over the entirety of this central region; meaning that the
distinction between the bulk and regularized expressions is irrelevant to our
leading-order calculations in Section 4. Further note that, in both layers,
the condition $\;\rho+p_{r}=0\;$ is maintained and so it is true throughout
the entirety of the star’s interior. As an immediate consequence, the stress-
tensor conservation equation reduces from its general static form
$\;p^{\prime}_{r}+\frac{1}{2}(\ln{f})^{\prime}\left(\rho+p_{r}\right)+\frac{2}{r}\left(p_{r}-p_{\perp}\right)=0\;$
to

$p^{\prime}_{r}\;+\;\frac{2}{r}\left(p_{r}-p_{\perp}\right)\;=\;0\,$ (9)

or, equivalently,

$p_{\perp}\;=\;\frac{1}{2r}\partial_{r}(r^{2}p_{r})\;.$ (10)

We define the interior bulk of the frozen star as the region that ends at
$\;r=R\left(1-\lambda\right)\approx
2M\left(1-\lambda+\varepsilon^{2}\right)\;$, as can be verified by evaluating
$\;M=m(R)\;$ via Eqs. (1) and (6). The exterior geometry formally begins at
$\;r=R\left(1+\lambda\right)\approx
2M\left(1+\lambda+\varepsilon^{2}\right)\;$. The outer transitional layer is
interpolating between the exterior geometry and the interior bulk.

For future reference, one can view the leading-order scaling relation
$\;m=\frac{r}{2}\;$ as meaning that each spherical slice of the interior is,
up to small corrections, like a BH horizon.

##  3 Hawking radiation and temperature

The thermal distribution of outgoing particle emissions from a BH and their
associated temperature [18] is traditionally calculated under two assumptions:
(1) A future, eternal event horizon forms and (2) the exterior geometry can be
treated as static. However, it turns out that these assumptions can be
relaxed; in which case, compact objects emit Hawking-like thermal radiation
provided that (a) they are compact enough so that the redshift near their
surface is sufficiently large or, equivalently, the gravitational force
$\;F_{\rm G}=\frac{G_{N}ME}{r^{2}}\;$ near their surface is strong enough to
produce particles with energy $\;E\sim 1/M\;$ in Planck units and (b) a
suitable adiabatic condition is satisfied. These properties were formally
defined in [35], which stated that horizonless objects will indeed emit
thermal radiation but with two provisions: ($i$) There is an approximately
exponential relation between the affine parameters for the null generators on
past and future null infinity and ($ii$) the surface gravity of the object
changes at a slow-enough rate. The frozen star obeys both requirements and is,
therefore, expected to emit radiation with properties that are similar to
those of BH-emitted Hawking radiation. In the following, we first recall a
heuristic argument from [28] (see also [29]) that relies on a gravitational
Schwinger process and then apply the more formal ideas of [35] to the frozen
star.

###  3.1 Gravitational particle production analogue of the Schwinger effect

Let us first consider a pair-production event in the atmosphere of a massive,
compact, horizonless object, which is induced by the gravitational analogue of
the Schwinger effect [28]. To recall, the Schwinger equation [27] predicts the
rate per unit volume ${\cal R}_{PP}^{\rm E\ell}$ of electron–positron pair
production in an electric field ${\cal E}$, such that $\;{\cal R}^{\rm
E\ell}_{PP}=\frac{\alpha^{2}}{\pi^{2}}{\cal E}^{2}e^{-\frac{\pi
m_{e}^{2}}{e{\cal E}}}\;$, where $\;\alpha=e^{2}/(4\pi)\;$, $e$ is the
electron’s charge and $m_{e}$ is its mass. Now suppose that one exchanges the
electric force $\;F_{\rm E}={\cal E}e\;$ with the gravitational force $F_{\rm
G}$ in Schwinger’s equation. If $M$ is the mass of the compact object and $E$
is the relativistic energy of the positive-energy particle, the gravitational
force is approximately given by $\;F_{\rm G}=\frac{G_{N}ME}{r^{2}}\;$, where
we restored the dependence on Newton’s constant $G_{N}$. If the object is
compact enough so that its radius satisfies $\;R\approx 2MG_{N}\;$, then
$\;F_{G}\sim\frac{1}{2}\frac{E}{R}\;$. Substituting $\;m_{e}^{2}\to E^{2}\;$
and $\;e{\cal E}\to\frac{1}{2}\frac{E}{R}\;$ into ${\cal R}_{PP}^{\rm E\ell}$,
one finds that the resulting rate of the gravitational pair production per
unit volume ${\cal R}^{\rm G}_{PP}$ in the proximity of the object’s outer
surface is

${\cal R}_{PP}^{\rm
G}\;\sim\;\frac{\hbar}{R^{4}}\left(\frac{RE}{2\pi\hbar}\right)^{2}e^{-\frac{2\pi
RE}{\hbar}}\;.$ (11)

This rate is maximized when $\;E=\frac{\hbar}{\pi R}\;$, which is of the order
of the Hawking temperature $\;T_{H}=\frac{\hbar}{4\pi R}\;$ of a Schwarzschild
BH; in which case, $\;{\cal R}^{\rm G}_{PP}\sim\frac{\hbar}{R^{4}}\;$. This
estimate suggests that one Hawking-like pair of particles, each with energy
$\;E\sim T_{H}\;$, is produced per light-crossing time $R$ from a volume
$R^{3}$, which is the standard Hawking emission result. Meanwhile, away from
the outer surface, the rate of pair production becomes exponentially small;
indicating that Hawking-like radiation can originate near the outer surface of
a compact object, even if it is not formally a horizon.

###  3.2 Temperature and spectrum of emitted radiation

####  3.2.1 Formal conditions for the validity of the calculation

For a more formal derivation, we recall the results of [35], which reveals
that horizonless objects will indeed emit thermal radiation under certain
conditions.

Following [35], let us first consider null curves starting from past infinity
$\mathcal{J}^{-}$ and arriving at future infinity $\mathcal{J}^{+}$. There
will be some invertible relation between their respective affine parameters
$U$ and $u$; namely, $\;U=p(u)\;$ and $\;u=p^{-1}(U)\;$. The surface gravity
can then be defined formally as
$\;\kappa(u)=-\dfrac{\ddot{p}(u)}{\dot{p}(u)}\;,$ where dots denote
derivatives with respect to $u$. Next, picking one particular null curve and
labelling it by $U_{*}$ on $\mathcal{J}^{-}$ and $u_{*}$ on $\mathcal{J}^{+}$,
one can then write

$U\;=\;U_{*}+C_{*}\int_{u_{*}}^{u}e^{-\int_{u_{*}}^{\bar{u}}\kappa(\tilde{u})d\tilde{u}}d\bar{u}\;,$
(12)

for some dimensionless constant $C_{*}$.

Assuming that $\kappa(u)$ satisfies the following adiabatic condition :

$\frac{\left|\dot{\kappa}(u_{*})\right|}{\kappa^{2}(u_{*})}\;\ll\;1\;,$ (13)

one can further show that outgoing particles reaching $\mathcal{J}^{+}$ at
$u_{*}$ obey a Planckian distribution. In which case, Hawking-like radiation
exists with a temperature of

$T(u_{*})\;=\;\frac{\kappa(u_{*})}{2\pi}\;.$ (14)

These ideas can be applied explicitly to the frozen star because they rely
solely on the nature of spacetime outside the star. Importantly, the exterior
region for $\;r>R_{star}\equiv R\left(1+\lambda\right)\approx
2M\left(1+\lambda+\varepsilon^{2}\right)\;$ is described by the Schwarzschild
metric; thus the affine parameters are the Eddington–Finkelstein outgoing
coordinate $\;u=t-r_{*}\;$, where $r_{*}$ is the usual tortoise coordinate
that is defined by $\;dr_{*}=\frac{dr}{\sqrt{-g_{tt}g^{rr}}}\;$, and $U$ is a
Kruskal-like coordinate that is related to $u$ in the standard way. One can
write $r_{*}$ for both the interior and exterior regions and, since $r^{*}$ is
continuous across the inner and outer surfaces of the translational layer (and
throughout the layer itself), $u$ and $U$ are continuous as well.

The surface gravity for the Schwarzschild metric, when defined at the surface
of a star of radius $R_{star}$, is

$\kappa^{2}\;=\;-\frac{1}{2}\left(\nabla_{\mu}\xi_{\nu}\right)\left(\nabla^{\mu}\xi^{\nu}\right)|_{r=R_{star}}\;,$
(15)

where $\xi$ is the timelike Killing vector. As in the case of a Schwarzschild
BH, $\kappa$ can be expressed as $\;\kappa(r)=\frac{M}{r^{2}}\;$, so that, on
the surface $\;r=R_{star}\;$ of a frozen star, the leading-order result is
simply $\;\kappa=\frac{M}{R^{2}}\;$.

With the above relations in hand, we can now define the affine parameters $U$
on $\mathcal{J}^{-}$ and $u$ on $\mathcal{J}^{+}$ for the frozen star,

$U\;\approx\;U^{*}_{H}-A_{*}e^{-\kappa_{*}u}\;,$ (16)
$u\;=\;-\frac{1}{\kappa_{*}}\ln\left(\frac{U^{*}_{H}-U}{A_{*}}\right)\;,$ (17)

where $A_{*}$ is a dimensional constant, the best estimate for the would-be
horizon is at $\;U^{*}_{H}=R_{star}\approx R\;$ and notice that $\kappa_{*}$
satisfies $\;\kappa_{*}=-\ddot{U}/\dot{U}\;$.

The surface gravity naively vanishes in the interior of the frozen star
because of the constancy of $g_{tt}$ and $g^{rr}$. However, as we will argue
later, each spherical slice of the interior maintains the same properties that
one would attribute to a would-be horizon of that radial size. The physical
reason that these inner slices do not directly contribute to the radiation is
not that their Killing vector vanishes but because of the large interior
redshift greatly suppressing the propagation of particles in the bulk of the
frozen star. So that, even if such particles are produced, they are
effectively trapped within the star.

The adiabatic condition (13) for the frozen star model can be verified by by
evaluating the derivatives of the surface gravity. In general, for a star of
radius $R_{star}$,

$\dot{\kappa}\;=\;\frac{d\kappa}{du}\;=\;\frac{\partial\kappa}{\partial
t}-f(r)\frac{\partial\kappa}{\partial r}|_{r=R_{star}}\;.$ (18)

In the frozen star case, $\;\frac{\partial\kappa}{\partial t}=0\;$ and
$\;\kappa(r)=\frac{M}{r^{2}}\;$, and so

$\dot{\kappa}\;=\;\frac{2M}{r^{3}}f(r)|_{r=R_{star}}\;,$ (19)

which at leading order becomes

$\dot{\kappa}\;=\;\frac{\varepsilon^{2}+\lambda}{R^{2}}\;.$ (20)

On the other hand, at $\;r=R_{star}\;$, another leading-order result is

$\kappa^{2}\;=\;\frac{1}{4R^{2}}\;,$ (21)

so that the adiabatic condition (13) translates into

$\frac{|\dot{\kappa}|}{\kappa^{2}}\;=\;\frac{1}{4}\left(\varepsilon^{2}+\lambda\right)\;\ll\;1\;,$
(22)

meaning that the adiabatic condition for the frozen star is easily satisfied.

####  3.2.2 Hawking-like radiation from a frozen star

We are now well equipped to derive the thermal spectrum of the emitted
radiation for a frozen star by suitably adapting the derivation in [35].

Defining the modes on $\mathcal{J}^{-}$ as

$\xi(u)\;=\;\frac{1}{\sqrt{4\pi\omega}}e^{-i\omega u}$ (23)

and those on $\mathcal{J}^{+}$ as

$\varPhi(U)\;=\;\frac{1}{\sqrt{4\pi\Omega}}e^{-i\Omega U}\;,$ (24)

we can express the Bogoliubov coefficient

$\alpha_{\omega\Omega}\;=\;i\int
dU\left[\partial_{U}\varPhi^{*}\xi-\varPhi^{*}\partial_{U}\xi\right]\;$ (25)

as

$\alpha_{\omega\Omega}\;=\;-\frac{1}{4\pi\sqrt{\omega\Omega}}\int_{-\infty}^{U^{*}_{H}}dUe^{i\Omega
U}\left[\Omega\left(\frac{U^{*}_{H}-U}{A_{*}}\right)^{\frac{i\omega}{\kappa_{*}}}+\frac{\omega}{\kappa_{*}}\left(\frac{U^{*}_{H}-U}{A_{*}}\right)^{\frac{i\omega}{\kappa_{*}}-1}\right]\;.$
(26)

Recalling that $\;\beta_{\omega\Omega}\approx-i\alpha_{\omega\Omega}\;$ and
$\;i=e^{i\frac{\pi}{2}}\;$, we then find that

$\beta_{\omega\Omega}\;=\;-i\frac{1}{2\pi}\sqrt{\frac{\omega}{\Omega}}\frac{(\Omega)^{-\frac{i\omega}{\kappa_{*}}}}{\kappa_{*}}e^{\frac{\pi\omega}{2\kappa_{*}}}\Gamma\left(\frac{i\omega}{\kappa_{*}}\right)\;,$
(27)

which allows us to calculate the expectation value of the number operator for
the outgoing particles via the usual definition $\;\langle
0|b^{\dagger}b|0\rangle\;=\;\int_{0}^{\infty}d\Omega\beta_{\omega\Omega}\beta^{*}_{\omega^{\prime}\Omega}\;$.
The result is

$|\beta_{\omega\Omega}|^{2}=\frac{1}{e^{\frac{2\pi\omega}{\kappa_{*}}}-1}\delta(\omega-\Omega)\;.$
(28)

The frozen star temperature can now be readily identified as

$T_{FS}\;=\;\frac{\kappa_{*}}{2\pi}\;=\;\frac{1}{8\pi M}\;=\;T_{H}\;,$ (29)

to leading order in $\varepsilon^{2}$ and $\lambda$.

In summary, the temperature of the emitted radiation from a frozen star is the
same as Hawking temperature up to a perturbatively small correction.

##  4 Entropy

We start here with a calculation of the entropy of the frozen star by
integrating the first law, as was first done by Bekenstein and Hawking [17,
18]. Following this, the entropy of the frozen star will be calculated using
the Euclidean-action method as pioneered by Gibbons and Hawking [25].

###  4.1 Entropy from the first law

The first law states, in the current context, that

$dS\;=\;\beta(M)dM\;,$ (30)

where $\beta(M)$ is the inverse of the temperature.

To derive the entropy of a frozen star of mass $M$, it is helpful to first
consider a star of much-smaller mass $\;m_{0}\ll M\;$. From the calculation of
the temperature in the previous section, the inverse temperature at leading
order is

$\beta(m_{0})\;=\;8\pi m_{0}\;.$ (31)

One can now imagine increasing the mass of the frozen star by a small amount
$\Delta m$, so that $m_{0}+\Delta m$ is the new mass and and
$8\pi(m_{0}+\Delta m)$ is the new inverse temperature. It follows that

$S_{FS}(m_{0}+\Delta m)\;=\;S_{FS}(m_{0})+8\pi m_{0}\Delta m+{\cal O}(\Delta
m^{2})\;.$ (32)

Integrating the previous equation, with the key observation that $\beta(m)$
grows linearly with $m$, 222A similar scaling relation is used in [36] to
define an effective surface gravity.

$\beta(m)\;=\;8\pi m\;,$ (33)

one obtains

$\displaystyle S_{FS}(M)$ $\displaystyle=$ $\displaystyle
S_{FS}(m_{0})+\int_{m_{0}}^{M}dm\leavevmode\nobreak\ 8\pi
m\;=\;S_{FS}(m_{0})+4\pi\left(M^{2}-m_{0}^{2}\right)$ (34) $\displaystyle=$
$\displaystyle S_{FS}(m_{0})+S_{BH}(M)-S_{BH}(m_{0})\;.$

If we start out with the frozen star’s seed radius being close to its core
radius, then $S_{FS}(m_{0})$ and $S_{BH}(m_{0})$ are negligible in comparison
to $S_{BH}(M)$ . The conclusion is that, to leading perturbative order, the
frozen star entropy is equal to the Bekenstein–Hawking entropy of a same-mass
BH,

$S_{FS}(M)\;=\;S_{BH}(M)\;.$ (35)

###  4.2 Entropy from Euclidean action

Next is the calculation of the entropy of the frozen star using the method of
Gibbons and Hawking [25]. We start by calculating the action of the bulk of
the frozen star, which extends from the center up to the beginning of the
outer transitional layer. We then calculate the entropy of the bulk and show
that it is equal, at leading order in $\varepsilon^{2}$ and $\lambda$, to the
entropy of a same-mass BH. The action of the outer transitional layer will
then be calculated separately. In the next section, this outer layer will be
interpreted as a Euclidean instanton that mediates the transition between a
collapsing shell of ordinary matter with a Minkowski interior and the bulk of
a frozen star. The current section ends with a calculation of the entropy for
the total spacetime, which again agrees at leading order with that of a BH of
the same mass.

The Euclidean-action method then provides us with two ways of determining the
frozen star entropy; either by calculating the action of the bulk of the
interior or by evaluating the action for the total spacetime. This duality
then provides us, in turn, a means for realizing both of the competing
perspectives on the entropy, matter based and geometric, but with no ambiguity
in the final answer. Note that the geometric point of view follows
automatically if the limits $\;\varepsilon^{2}\to 0\;$ and $\;\lambda\to 0\;$
are imposed; in which case, the outer surface of the star tends to that of a
true horizon. However, it is more interesting and realistic to consider what
happens when $\varepsilon^{2}$ and $\lambda$ differ from zero.

Let us now recall how the calculation in [25] is performed. Gibbons and
Hawking started out by evaluating the action of the Euclidean BH solution
$I[g]$ and applying the identification

$I[g]\;=\;\ln{\cal Z\left[\beta\right]}\;=\;-\beta F\;=\;S-\beta M\;,$ (36)

where $\ln{\cal Z\left[\beta\right]}$ is the logarithm of the thermal
partition function and $F$ is the Helmholtz free energy.

The Euclidean Einstein action consists of both a volume and a surface
contribution,

$I[g]\;=\;-\frac{1}{16\pi}\int_{\cal M}\sqrt{g}{\cal
R}\;-\;\frac{1}{8\pi}\int_{\partial{\cal M}}\sqrt{h}[K]\;.$ (37)

The integrand of the volume portion trivially vanishes for the Schwarzschild
solution, and one is left only with the boundary contributions from the
horizon and timelike infinity, each of which is an integral of the trace of
the extrinsic curvature $K$ (or second fundamental form), where $[K]$
indicates that the contribution from flat space has been subtracted off. The
Euclidean geometry has zero area at the horizon and, once the conical
singularity at the horizon has been resolved by assigning the Euclidean time
$\tau$ with its appropriate periodicity, $\;\Delta\tau=\beta=8\pi M=4\pi R\;$,
the contribution from the horizon vanishes.

The only surviving contribution is then that of the asymptotic boundary
$I_{\infty}$. The Schwarzschild result goes as

$I_{\infty}\;=\;-\frac{1}{8\pi}\oint d\tau\sqrt{g^{rr}}\partial_{r}\left(4\pi
r^{2}\sqrt{g_{\tau\tau}}\right)_{|r\gg 2M}\;.$ (38)

After appropriately subtracting the divergent flat-space contribution and
taking the $\;r\to\infty\;$ limit at the end, one finds that

$I\;=\;-\frac{1}{2}\beta M\;.$ (39)

Now recalling Eq. (36), one arrives at an entropy of $\;S=\beta M/2\;$ or,
since $\;\beta=8\pi M\;$, $\;S=4\pi M^{2}\;$. This is simply
$\;S=S_{BH}=A/4\;$ and thus the Bekenstein–Hawking area law [17, 18]. For
future reference, one should notice that $\;I_{\infty}=-S_{BH}\;$.

The Gibbons–Hawking calculation relies on the fact that the Euclidean action
is a boundary term on the solution of the Einstein equations of motion.
Clearly, if the sole contribution to the Euclidean action comes from the
boundary term at infinity, one is guaranteed to get exactly the BH entropy. In
the original calculation, the interior of the BH is excised, the horizon
shrinks to a point and the conical singularity is removed. Therefore, the sole
contribution does indeed come from the asymptotic boundary term. However, in
Section 5 of [37], it was pointed out that the same result is still obtained
for a horizonless and non-singular spherical geometry for which the volume of
the sphere at the origin shrinks to zero. In this case, there is no
contribution from the inner boundary and the entropy is exactly equal to that
of a BH. We will encounter a similar situation in what follows.

####  4.2.1 Entropy of the frozen star bulk

To calculate the entropy of the bulk (again to leading order in
$\varepsilon^{2}$ and $\lambda$), it is important to realize that $\beta$
depends on $r$. From Eq. (33) and from the relation $\;m(r)=\frac{1}{2}r\;$ in
Eq. (4), it follows that

$\beta(r)\;=\;4\pi r\;,$ (40)

which is obviously different than the constant $\beta_{H}=8\pi M\;$ in [25]. A
radially dependent compactification scale may seem unusual, but then so is the
frozen star, which features large deviations from Schwarzschild over the
entirety of its interior. The Euclidean geometry of the frozen star is
depicted in Fig. 2. A similar geometry was found in [33].

It is possible to understand this peculiar dependence by considering the near-
horizon geometry of the Euclidean Schwarzschild BH, for which the relevant
part of the metric can be expressed as
$\;\frac{4\pi^{2}}{\beta_{H}^{2}}\tilde{r}^{2}d\tau^{2}+d\tilde{r}^{2}\;$.
Now, replacing the constant $\beta_{H}$ by $\;\beta\sim{r}\;$ and taking into
account that $\;\tilde{r}\sim r\;$ for the frozen star interior, one finds a
constant $g_{\tau\tau}$. In any case, such radial dependence inside of the
frozen star does not affect the validity of the method. As we have already
observed in Section 3, in the exterior of the frozen star, the temperature is
a constant and its inverse is equal at leading order to the standard value of
$\;\beta_{H}\;$.

Figure 2: The frozen star Euclidean geometry. The exterior geometry is
essentially the same as the Euclidean Schwarzschild geometry. The bulk has the
geometry of a cylinder of radius $\sqrt{\varepsilon^{2}}$ and the geometry of
the inner transitional layer, drawn here not to scale, expands back to the
asymptotic radius.

The Euclidean action of the bulk does not receive any contribution from the
extrinsic-curvature boundary terms because, as argued above, it vanishes at
$\;r=0\;$ due to the shrinking of the volume of the sphere to zero at the
origin and the outermost surface of the bulk is not a true boundary of
spacetime. The remaining volume integral is

$\displaystyle I_{Bulk}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{Bulk}d^{4}x\;\sqrt{g}\;{\cal R}\;,$ (41)

and using Eq. (40) for the compactification scale of the Euclidean time
direction, one can rewrite this as

$I_{Bulk}\;=\;-\frac{1}{4}\int_{0}^{R}dr\;4\pi r^{2}\;r{\cal R}\;.$ (42)

Let us next use the trace of Einstein’s equations, while recalling the form of
the stress tensor ${\cal T}^{a}_{\;\;b}$ from Section 2, to recast the scalar
curvature in terms of the energy density and pressure in the frozen star’s
bulk, $\;{\cal R}=-8\pi{\cal
T}=-8\pi\left(-\rho+p_{r}+2p_{\perp}\right)=16\pi\rho=\frac{2}{r^{2}}\;$. So
that

$I_{Bulk}\;=\;-2\pi\int_{0}^{R}dr\;r=-\pi R^{2}\;$ (43)

and, therefore,

$I_{Bulk}\;=\;-\frac{1}{2}\beta(R)M\;.$ (44)

Using the relationship (36) between the action and the entropy, we conclude
that $\;S_{Bulk}=S_{BH}\;$.

It is worthwhile to understand how we managed to evade Gibbons and Hawking’s
conclusion that a normal star or, for that matter, any horizonless object
could not have a geometric entropy, regardless of how compact it may be [25].
Looking at the discussion leading up to their Equation (3.16), one can
identify two critical assumptions: (1) The star is composed of “normal” matter
in thermal equilibrium and (2) its radius $R$ and temperature $T$ satisfy
$\;R^{3}\gg T^{-3}\;$. Note that $T$ here is not the Hawking temperature but
rather the actual thermodynamic temperature of the matter. The former
condition does not apply to our model, as the source of the frozen star is
exotic matter at zero temperature. 333Fundamentally, the frozen star would
consist of a state of strings close to Hagedorn temperature [1], but this
matter is no less exotic. As for the latter condition, this is clearly
violated as it fails whether we use the inverse of the Hawking temperature,
which is order $R$, or the actual inverse temperature, which formally
diverges.

####  4.2.2 Action of the outer transitional layer

It is straightforward to evaluate the action of the outer transitional layer
$I_{TL}$ using the metric and stress-tensor components as presented in Section
2. Recall that this transitional layer is a thin shell of width $2\lambda R$,
with $\;\lambda\ll 1\;$. The metric of this layer smoothly connects the bulk
metric to the exterior Schwarzschild metric. For simplicity, we evaluate the
action to leading order in $\varepsilon^{2}$ and $\lambda$ and similarly for
the rest of the calculations. Here, we only need to evaluate the volume term
in the action because the boundary terms are supported only on the true
boundaries at $\;r=0\;$ and $\;r\to\infty\;$, which are outside the domain of
integration. In this layer, it remains true that $\;p=-\rho\;$, but
$p_{\perp}$ is non-vanishing and large, so that now $\;{\cal R}=-8\pi{\cal
T}=16\pi(\rho-p_{\perp})\;$. It follows that

$I_{TL}\;=\;-4\pi\beta\int_{TL}\leavevmode\nobreak\ dr\;r^{2}(\rho-
p_{\perp})\;,$ (45)

where, in principle, $\beta$ could be a function of $r$; however, since the
width of the boundary layer scales as $\;\sim\lambda\;$ and the value of
$\beta$ at the outer edge of the layer is $\;\beta=4\pi R\;$, we can ignore
any possible $r$ dependence at leading order.

We now recall our first form of the frozen star conservation equation (9) and
also consider that
$\;p^{\prime}_{r}\sim\dfrac{|p_{r}|}{\lambda}\gg|p_{r}|,\rho\;$, from which it
can be deduced that
$\;p_{\perp}\sim\dfrac{p_{r}}{\lambda}\gg|p_{r}|\;,\rho\;$. Hence, we can
ignore the contribution from $\rho$ in Eq. (45) at leading order. Also
recalling our second form of the conservation equation (10) and observing that
the factors of $r$ appearing in Eq. (45) can be approximated as $R$’s, we can
reexpress the action $I_{TL}$ as

$I_{TL}\;=\;16\pi^{2}R\int\limits_{R(1+\varepsilon^{2}-\lambda)}^{R(1+\varepsilon^{2}+\lambda)}dr\leavevmode\nobreak\
\frac{R}{2}\leavevmode\nobreak\
\partial_{r}(r^{2}p)\;=\;8\pi^{2}R^{2}\leavevmode\nobreak\
(r^{2}p)\biggl{|}^{R(1+\varepsilon^{2}+\lambda)}_{R(1+\varepsilon^{2}-\lambda)}\;.$
(46)

The value of $p$ at the upper end is its Schwarzschild value of $\;p=0\;$,
while the value of $p$ at the lower end is its frozen star bulk value of
$\;r^{2}p=-\frac{1}{8\pi}\;$, meaning that

$I_{TL}\;=\;+\pi R^{2}\;=\;S_{BH}\;.$ (47)

####  4.2.3 Total entropy of the frozen star

The action of the Schwarzschild exterior is given by

$I_{EXT}\;=\;-\frac{1}{16\pi}\int_{r>R(1+\varepsilon^{2}+\lambda)}dx^{4}\sqrt{g}{\cal
R}\;\;-\;\frac{1}{8\pi}\int_{\partial{\cal M}}\sqrt{h}[K]\;.$ (48)

The volume term vanishes for the exterior Schwarzschild metric and the surface
contribution comes only from the boundary at infinity, as the boundary at
$\;r=0\;$ is outside the domain of integration.

The contribution of the boundary at infinity is given in Eq. (39) and is equal
to $-S_{BH}$, so that

$I_{EXT}\;=\;-\frac{1}{2}\beta M\;=\;-S_{BH}\;,$ (49)

which can be combined with our previous calculations to give

$I[\text{frozen
star}]\;=\;I_{Bulk}+I_{TL}+I_{EXT}\;=\;-S_{BH}+S_{BH}-S_{BH}\;=\;-S_{BH}\;.$
(50)

It follows that the frozen star has a leading-order entropy that is equal to
that of a BH with the same mass,

$S_{FS}\;=\;S_{BH}\;.$ (51)

Let us comment that, because the outer surface becomes a true horizon in the
$\;\varepsilon^{2},\lambda\to 0\;$ limits, it must be true that the zeroth-
order contributions to $I_{Bulk}$ and $I_{TL}$ exactly cancel, just as was
found above. This tells us, unequivocally, that the scaling relation
$\;\beta=4\pi r\;$ is the correct choice as anything else would have offset
the required cancellation.

##  5 Dynamical collapse of matter to a frozen star

As suggested in the Introduction, it is unreasonable to expect a “normal”
system of collapsing matter to slowly evolve into an ultracompact object whose
geometry deviates from the Schwarzschild geometry on horizon-order length
scales. What is rather needed is something akin to a quantum-induced phase
transition, where we have implicitly recalled that the final system of matter
is, in its fundamental description, a highly quantum state of strings at a
finite temperature. It is then natural, in the Euclidean picture, to regard
the transitional layer as a gravitational instanton that mediates the
transition from an infalling shell of matter with a Schwarzschild exterior and
an empty interior to a frozen star.

There is, indeed, a long tradition of using Euclidean instantons in just this
way, whether it has been to facilitate bounces between two phases of de Sitter
(e.g., [38]), to induce a quantum-tunneling process from nothingness to a
Lorentzian inflationary cosmology (e.g., [39]) or, in the guise of a domain
wall, to trigger the creation of a pair of BHs (e.g., [40]). For a general
discussion, see [41, 42]. The theme in all these works, in analogy to similar
ideas in quantum mechanics, is that the probability of transition is given (up
to a numerical prefactor) by $\;\Gamma=e^{-I_{Inst}}\;$, where the instanton
action $I_{Inst}$ is typically evaluated using either the on-shell solution or
at a saddle point of an appropriate partition function. The instanton
interpolates between an initial spacelike section of some specific solution,
which is typically the false vacuum solution, and a final spacelike section of
another solution, so that the action $I_{Inst}$ can be defined by a
subtraction procedure of the respective Euclidean actions of these two
solutions.

In our case, the initial-state geometry is defined on the outermost slice of
the transitional layer and corresponds to the solution when the collapsing
shell of matter, which can also be viewed as a domain wall, is just on the
brink of passing through its would-be horizon. The interior of the shell at
this initial time contains the false vacuum, which is simply flat Minkowski
space. All this is in agreement with our calculation at the end of the last
section, which makes it clear that the Euclidean action vanishes on the
initial-state spacelike slice, since a Schwarzschild exterior enclosing normal
matter will have a negligible free energy until a horizon forms.

Meanwhile, the final-state geometry is defined on the innermost slice of the
transitional layer, at a radius just inside the would-be horizon. 444One of
our working assumptions is that $\;\varepsilon^{2}<\lambda\;$ (see Section 2),
and so $\;R+\varepsilon^{2}-\lambda<R\;$. Once this time has been reached, the
shell and its once-empty interior has transitioned into a fully formed frozen
star. From this perspective, the inner-most slice represents a bubble
enclosing the true vacuum, as the frozen star represents a state of maximal
entropy or minimal free energy.

This picture depends on having, from an external perspective, a finite initial
thickness for the matter shell and a slow velocity for its descent. Given that
the shell’s width far exceeds that of the transitional layer, $2\lambda R$,
and that its collapse is slow, $\;\dfrac{dr}{dt}\sim\lambda\;$, the phase
transition can be triggered long before the entirety of the shell passes
through its Schwarzschild radius.

The standard prescription in the case of a bubble or bounce (B) mediated
transition at finite temperature from a false vacuum (FV) to the true one is
an exponent of the form [41, 42]

$\ln\Gamma\;=\;I_{B}-I_{FV}\;,$ (52)

which is, in our case,

$\displaystyle\ln\Gamma$ $\displaystyle=$ $\displaystyle
I\left[R(1+\varepsilon^{2}-\lambda)\right]-I\left[R(1+\varepsilon^{2}+\lambda)\right]$
(53) $\displaystyle=$ $\displaystyle-I_{TL}\;,$

and so

$\Gamma\;\sim\;e^{-S_{BH}}\;,$ (54)

as previously advertised. Formally, in the classical limit of $\;l_{P}\to
0\;$, the transition probability vanishes and the frozen star never forms; the
shell rather continues on to form a singular BH. As will soon be shown, there
is evidence that, if a frozen star does form, its evaporation is unitary,
painting a nicely consistent picture.

We now follow Mathur (e.g., [26]) and identify this as the probability for
transition to any one of the $\;e^{S_{FS}}=e^{S_{BH}}\;$ frozen star
microstates, so that the total probability is

$\Gamma_{Tot}\;\sim\;e^{+S_{BH}}\Gamma\;\sim 1\;\;.$ (55)

That is to say, the transition from an incipient BH to any frozen star
configuration of equal mass is a likely, if not certain, outcome.

###  5.1 How fast?

As for the duration of the transition from an external observer’s perspective,
this is simply the width of the transitional layer $2\lambda R$, divided by
the shell’s approximate radial velocity in the outer layer
$\;\lambda\pm\varepsilon^{2}\sim\lambda\;$. In other words, a time scale of
order $R$, the light-crossing time of the star. Consider though that the
analogue calculation for a BH is exponentially large. However, from the
perspective of a particle that exits the matter shell just before the frozen
star has fully formed, the relevant time scale is the proper-time equivalent
or, quite simply, the width of the transitional layer divided by a speed of
order unity, $\;T_{trans}\sim\lambda R\;\sim\sqrt{\alpha^{\prime}}\sim
l_{s}\;$, where we have recalled a discussion from Section 2 relating
$\lambda$ to the inverse of the string tension and $l_{s}$ is the fundamental
string length, which in weakly coupled string theory satisfies
$\;l_{P}=l_{s}g_{s}\;$ with $g_{s}$ being the string-coupling strength.

The scale $T_{trans}$ can be compared to the analogue time scale in general
relativity that is used in the calculation of the Hawking effect, where the
transition (or horizon formation) is assumed to be instantaneous. This string-
scale fuzziness in the transition time — or, equivalently, in the thickness of
the transitional layer — will be important in what follows.

##  6 Radiation revisited

In this section, we argue on the basis of previous discussions in [44, 30]
that, when the entropy $S_{BH}$ is finite, the evaporation of the radiation is
consistent with unitarity. The key observation is that this finiteness implies
that the would-be horizon is not a surface of infinite redshift but rather a
surface of large, finite redshift and, consequently, has some quantum width.
As a result, the number operator of the Hawking particles is no longer
diagonal and likewise for the density matrix of the emitted radiation.

It proves to be a worthwhile exercise to think about the radiation process of
the frozen star in the context of Hawking’s seminal calculation of the thermal
spectrum [18]. In Hawking’s original model, one considers a collapsing shell
of matter, viewed as an incipient BH, and focuses on a certain class of null
rays. Namely, those rays which enter the interior of the shell when its radius
$R_{shell}$ is substantially larger than its Schwarzschild radius $R$ but exit
the interior at a time close to that of horizon formation, so that
$\;(R_{shell}-R)_{in}\gg(R_{shell}-R)_{out}$. Because the ray exits when $\
(R_{shell}-R)_{out}/R\ll 1\;$, it undergoes an exponentially large redshift
after passing through the interior region. From the perspective of an
asymptotic observer, it is this strong time-dependent effect that leads to the
creation of Hawking particles. Hawking was able, of course, to make this
description of particle emission from an incipient BH quantitatively precise.

The analogous situation for the frozen star is that the null ray must enter
the collapsing matter shell before the transition (as described in Section 5)
takes place but then exit the shell only after the transition has transpired.
This is the only way in which the ray can experience the effect of a suitably
large enough redshift. It can then be deduced that, if the transition occurs
at an advanced time of $\;v=v_{trans}\;$, the prospective Hawking mode should
have entered the matter distribution no later than this time. What should now
be clear is that $v_{trans}$ is the analogue of Hawking’s $v_{0}$, which is
defined by the last incoming ray that can exit before the horizon finally
forms. For now on, we will denote both of these cutoff times by $v_{\star}$.

If the transition from ordinary matter to a frozen star was an instantaneous
process, this would be the end of the story, as the mathematics would
basically replicate those of the Hawking calculation. However, as discussed in
the previous section, the time scale of the transition is, locally, given by
$\;T_{trans}\sim\lambda R\sim l_{s}\;$; 555We use the local time scale
$R\lambda$ and not the asymptotic scale $R$ as what is required is the
analogue of Hawking’s instantaneous-collapse model. meaning that $v_{\star}$
in the frozen star model should not be regarded as precise but should rather
be smeared over this scale. Alternatively, one could view the smearing as an
uncertainty in the location of the would-be horizon. Either way, this smearing
modifies the Hawking calculation in a subtle but important manner.
Fortunately, we are well positioned to account for just such an effect, as
will now be explained.

Let us recall an earlier discussion [44], where the effects of a quantum-
uncertain horizon position were incorporated into the otherwise standard
Hawking calculation. Here, we will just sketch the argument, leaving a precise
calculation to a future publication.

The key revision of Hawking’s calculation in [44] is based on an integral that
appears in Hawking’s original calculation of the number operator for the
outgoing particles,

$I_{C}\;=\;\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}dv\;e^{iv(\omega^{\prime}-\omega^{\prime\prime})}\;=\;\delta(\omega^{\prime}-\omega^{\prime\prime})\;,$
(56)

where $\omega^{\prime}$ and $\omega^{\prime\prime}$ are incoming-particle
frequencies and the advanced time is defined as $\;v=t+r_{*}\;$. For a formal
definition of the number operator, see Section 3. This integral is to be
adapted so that it includes the effects of semiclassical physics. In [44],
these effects resulted from the quantum blurring of the horizon position;
while here, they result from the string-scale blurring in the transition time.
The resulting modified integral depends on an additional parameter
$v_{shell}$,

$\displaystyle{I}_{SC}(\omega^{\prime}-\omega^{\prime\prime};v_{shell})$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}dv\;e^{i(v-v_{shell})(\omega^{\prime}-\omega^{\prime\prime})}\;,$
(57)

where $v_{shell}$ is to be regarded as a variable that is to be integrated
against a normalized Gaussian probability distribution $P(v_{shell})$ with
width $\sigma$. In the case of the frozen star, $\;\sigma\propto T_{trans}\;$.

The diagonal and off-diagonal contributions are then separated,

$\displaystyle\langle{I}_{SC}(\omega^{\prime}-\omega^{\prime\prime};v_{shell})\rangle$
$\displaystyle=$
$\displaystyle\delta(\omega^{\prime}-\omega^{\prime\prime})+\left\langle\frac{1}{2\pi}\int\limits_{0}^{v_{shell}}dv^{\prime}\;e^{iv^{\prime}(\omega^{\prime}-\omega^{\prime\prime})}\right\rangle,$
(58)

where $\;\langle f(v)\rangle=\int dvP(v)f(v)\;$.

Having mapped out a formally similar framework to that of [44], we can now
simply recall the end result. After an extremely long calculation, it was
revealed that the number operator for outgoing particles picks up an off-
diagonal part 666It is explained in [44] as to why the other two types of two-
point functions, that of ingoing particles and mixed, are unaffected by the
inclusion of quantum blurring. that is a rather complicated combination of
Gamma functions and exponential factors whose arguments contain the two out-
frequencies $\omega$, ${\widetilde{\omega}}$ and their difference
$\;\Delta\omega=\omega-{\widetilde{\omega}}\;$. Most important for current
considerations is that the off-diagonal contribution scales with $\sigma/R$,
which, on general grounds, should scale as $1/\sqrt{S_{BH}}$ [30].

Although the formal steps has been glossed over, it is useful to consider the
actual mechanism that is responsible for the off-diagonal terms in the number
operator. First, in the Hawking calculation, the number operator is measuring
the overlap of two incoming waves at the same advanced time $v$. Such waves
form an orthogonal basis in frequency-space, leading to adiagonal expression
in frequency space $\;I_{C}\sim\delta(\omega^{\prime}-\omega)\;$. This
diagonal character is then preserved by the outgoing waves in the number
operator after performing a subsequent Fourier transform. On the other hand,
if these modes become wavepackets that are smeared over a finite time scale,
then the orthogonality condition is modified in such a way that the would-be
delta function becomes a Gaussian and the strength of the off-diagonal
elements in the number operator scales linearly with the duration of the
smearing scale. This happens in part because there are two subsequent Fourier
transorms that are each weighted by a Gaussian. The first is the “expectation
value” appearing in Eq. (58) and the second transforms a function of
$\omega^{\prime}-\omega^{\prime\prime}$ into a function of
$\;\Delta\omega=\omega-{\widetilde{\omega}}\;$, namely, the number oeprator.

Another long calculation, this time in [30], reveals that an off-diagonal
correction of this strength in the number operator translates into off-
diagonal elements in the multi-particle density matrix that are suppressed by
a factor of $(\sigma/R)^{2}$ with respect to the diagonal elements.
Consequently, the off-diagonal contributions are exponentially suppressed at
early times but will become significant and eventually dominate, once the
number of emitted particles $\;N\lesssim S_{BH}\;$ is enough to compensate for
the suppression factor $\;(\sigma/R)^{2}\gtrsim 1/S_{BH}\;$. This tipping
point occurs near the midway point of evaporation, the so-called Page time
[31]. Moreover, this type of relation between diagonal and off-diagonal
elements is exactly what is needed to reproduce the famous Page curve [31],
which can be viewed as evidence of a unitary evaporation process. That this is
so has already been verified in [30]. Here, we recall our previous form of the
entropy-versus-time plot, reproduced in Fig. 3 and refer back to the original
source for mathematical details.

Figure 3: Dependence of the Rényi entropy $H_{2}$ of the BH radiation on the
number of emitted particles $N$ for the case of off-diagonal elements
discussed in the text (dashed). For comparison, the solid line depicts the
prediction of the Page model. Figure reproduced from [30].

Tracking this flow of information over the lifetime of the star is
operationally unfeasible. Hence, this form of density matrix is in agreement
with a previous argument that a standard BH and a BH mimicker, such as the
frozen star, should be observationally indistinguishable when in equilibrium
[45]. Observational differences would rather require an out-of-equilibrium
event such as a BH merger in which the BH is emitting a macroscopic amount of
energy in the form of gravitational waves.

##  7 Conclusion

Our main lesson is that the frozen star may be unique among ultracompact but
horizonless objects in that it is able to reproduce, up to perturbative
corrections, the same entropy–area law as that of a conventional BH. We have
also shown that there is a natural choice for a Euclidean instanton which
mediates the phase transition between the interior geometries of an incipient
BH and a frozen star. Standard techniques were used to reveal a probability of
transition that goes as $e^{-S}$, just as expected via statistical reasoning.
Moreover, we have argued, following Mathur, that the large degeneracy of the
frozen star predicts a total transition probability of order unity. Meaning
that the transition from a collapsing matter system to a frozen star would be
commonplace if it is at all possible.

An unexpected consequence of the probability-of-transition calculation is that
it yields an extremely large action for the intervening layer, being of the
same order as the bulk action of the frozen star itself. The actual matter
content, however, is suppressed by a factor of $\lambda$, the shell width in
Schwarzschild units, as a rather simple calculation reveals. A possible
interpretation is that this matter might be in a highly excited state, which
is suggestive of a BH “firewall” [43]. If this interpretation of the large
action is indeed correct, it would be interesting to understand why a
firewall-like structure is associated with a horizonless object.

It should be reemphasized that our calculation of the entropy is fundamentally
geometric in that it is based directly on the Euclidean-action method. The
energy density appears but has no obvious connection to matter fields in the
context of the frozen star model. The microscopic origin of the matter
densities must rather be gleaned from the polymer model which is a string-
theoretic description of the same compact object. The frozen star and
collapsed polymer models are supposed to be two different perspectives of the
same physical object. That the two models are describing the same object is
supported by the similarities of this Euclidean geometry with that of the
punctured cigar which is discussed in [32]. In the punctured cigar, the
Euclidean time circles are decreasing with $r$ from the outer surface to the
origin and are cut off, or “punctured”, close to the $\;r=0\;$ tip. The frozen
star geometry puncture comes about because of its small, central, regularized
zone. This similarity is relevant because the polymer model, once
Euclideanized, can also be viewed as a condensate for the winding modes of the
thermal scalar [46, 47, 48]. We hope to solidify this connection in the
future, as BH physics — which is becoming more and more accessible as
gravitational-wave data collection and analysis improves — could then provide
a window into fundamental string theory.

## Acknowledgments

We thank Yoav Zigdon for discussions. The research is supported by the German
Research Foundation through a German-Israeli Project Cooperation (DIP) grant
“Holography and the Swampland” and by VATAT (Israel planning and budgeting
committee) grant for supporting theoretical high energy physics. The research
of AJMM received support from an NRF Evaluation and Rating Grant 119411 and a
Rhodes Discretionary Grant SD07/2022. AJMM thanks Ben Gurion University for
their hospitality during his visit.

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# Vector fields on non-compact manifolds

Tsuyoshi Kato  Department of Mathematics, Kyoto University, Kyoto 606-8502,
Japan<EMAIL_ADDRESS>,  Daisuke Kishimoto  Faculty of Mathematics,
Kyushu University, Fukuoka 819-0395, Japan<EMAIL_ADDRESS>and
Mitsunobu Tsutaya  Faculty of Mathematics, Kyushu University, Fukuoka
819-0395, Japan<EMAIL_ADDRESS>
###### Abstract.

Let $M$ be a non-compact connected manifold with a cocompact and properly
discontinuous action of a discrete group $G$. We establish a Poincaré-Hopf
theorem for a bounded vector field on $M$ satisfying a mild condition on
zeros. As an application, we show that such a vector field must have
infinitely many zeros whenever $G$ is amenable and the Euler characteristic of
$M/G$ is non-zero.

###### Key words and phrases:

vector field, non-compact manifold, bounded cohomology, Poincaré-Hopf theorem

###### 2010 Mathematics Subject Classification:

57R25, 58K45

##  1\. Introduction

Let $M$ be a non-compact connected manifold. Then $M$ admits a non-vanishing
vector field as $M$ admits a vector field with isolated zeros which can be
swept out to infinity. However, the resulting non-vanishing vector field is
not satisfactory because it may not be bounded in the following sense. A
vector field $v$ on a Riemannian manifold is _bounded_ if both $|v|$ and
$|dv|$ are bounded, where $dv$ denotes the derivative of $v$. Note that our
boundedness condition is different from the one in [6] and its related works.
Bounded vector fields appear in the study of manifolds of bounded geometry.
Now we ask whether or not a non-vanishing bounded vector field exists on $M$.
Weinberger [11, Theorem 1] proved that a manifold $M$ of bounded geometry has
a non-vanishing vector field $v$ with $|v|$ constant and $|dv|$ bounded if and
only if the Euler class of $M$ in the bounded de Rham cohomology
$\widehat{H}^{*}(M)$ is trivial. As one may think of the Poincaré-Hopf theorem
as a refinement of the Euler class criterion for the existence of a non-
vanishing vector field on a compact orientable manifold, it is natural to ask
whether or not one can establish the Poincaré-Hopf theorem for bounded vector
fields on non-compact manifolds of bounded geometry.

A typical manifold of bounded geometry is a covering space of a compact
manifold, which we equip with a lift of a metric on the base compact manifold.
We say that an action of a group $G$ on a space $X$ is properly discontinuous
if every point $x\in X$ has a neighborhood $U$ such that $gU\cap
U\neq\emptyset$ implies $g=1$. Equivalently, the quotient map $X\to X/G$ is a
covering. So we consider a connected non-compact manifold $M$ on which a
cocompact and properly discontinuous action of a group $G$ is given, and will
establish the Poincaré-Hopf theorem for a bounded vector field on $M$. To
state it, we set notation. Let $\ell^{\infty}(G)$ denote the vector space of
bounded functions $G\to\mathbb{R}$, and let $G$ act on $\ell^{\infty}(G)$ from
the left by

$(g\phi)(h)=\phi(hg)$

for $g,h\in G$ and $\phi\in\ell^{\infty}(G)$. We define the module of
coinvariants of $\ell^{\infty}(G)$ by

$\ell^{\infty}(G)_{G}=\ell^{\infty}(G)/\langle\phi-g\phi\mid\phi\in\ell^{\infty}(G),\,g\in
G\rangle$

where $\langle S\rangle$ denotes the vector subspace of $\ell^{\infty}(G)$
generated by a subset $S\subset\ell^{\infty}(G)$. Let
$\mathbbm{1}\in\ell^{\infty}(G)$ denote the constant function with value $1$.
Let $D\subset M$ be a fundamental domain (its definition is given in Section
3). We will define the index $\mathrm{ind}(v)$ of a bounded vector field $v$
on $M$ as an element of $\ell^{\infty}(G)_{G}$, and will prove

$\operatorname{ind}(v)(g)=\sum_{x\in\mathrm{Zero}(v)\cap
gD}\mathrm{ind}_{x}(v)$

whenever $v$ is strongly tame, which is a mild condition on the zeros of $v$
defined in Definition 5.5, where $\operatorname{ind}_{x}(v)$ denotes the local
index of a vector field $v$ at the zero $x\in\operatorname{Zero}(v)$. Here,
the above equality means that there is a representative
$\phi\in\ell^{\infty}(G)$ of $\operatorname{ind}(v)\in\ell^{\infty}(G)_{G}$
such that $\phi(g)$ is the right hand side of the equality. Now we are ready
to state the Poincaré-Hopf theorem for a bounded vector field on $M$.

######  Theorem 1.1.

Let $M$ be a non-compact connected manifold equipped with a cocompact and
properly discontinuous action of a group $G$ such that $M/G$ is orientable. If
a vector field $v$ on $M$ is strongly tame and bounded, then we have

$\mathrm{ind}(v)=\chi(M/G)\mathbbm{1}\quad\text{in}\quad\ell^{\infty}(G)_{G}.$

As an application of Theorem 1.1, we will prove:

######  Theorem 1.2.

Let $M$ and $G$ be as in Theorem 1.1. If $G$ is amenable and $\chi(M/G)\neq
0$, then every tame bounded vector field on $M$ must have infinitely many
zeros.

Let $M$ and $G$ be as in Theorem 1.1. Then by the above mentioned result of
Weinberger [11, Theorem 1] together with [2], one can deduce that a vector
bundle $v$ on $M$ with $|v|$ constant and $|dv|$ bounded must have a zero. But
one cannot deduce further information on zeros, such as their numbers, from
these results. As an application of Theorem 1.2, we will get the following
result, where tameness of a diffeomorphism is defined in Definition 5.6 in an
analogous way to tameness of a vector field.

######  Corollary 1.3 (cf. [11, Corollary to Theorem 1]).

Let $M$ and $G$ be as in Theorem 1.1. If $G$ is amenable and $\chi(M/G)\neq
0$, then every tame diffeomorphism of $M$ which is $C^{1}$ close to the
identity map must have infinitely many fixed points.

######  Example 1.4.

Let $L$ be the non-compact surface called Jacob’s ladder, a surface with
infinite genus and two ends, which admits an infinite cyclic covering map onto
the closed oriented surface of genus $2$. Then we can apply Corollary 1.3 to
$L$, and conclude that any tame diffeomorphism of $L$ which is $C^{1}$ close
to the identity map must have infinitely many zeros. This can be easily
generalized to the infinite connected sum $M=\sharp^{\infty}N$ of a closed
connected oriented even dimensional manifold $N$ with $\chi(N)\neq 2$.

We briefly describe the strategy of our proof, as well as some of the tools we
exploit. Let $M$ and $G$ be as in Theorem 1.1. Recall that the Poincaré-Hopf
theorem for a compact manifold can be proved by using a suitable integral in
top dimensional de Rham cohomology. Motivated by the compact case, we will
define the integral

(1.1) $\int_{M}\colon\widehat{H}^{n}(M)\to\ell^{\infty}(G)_{G}$

where $\dim M=n$, and will prove Theorem 1.1 by using it similarly to the
compact case. So our approach is an extension of the classical case by means
of $\ell^{\infty}(G)_{G}$. However, unlikely to the compact case, the target
module $\ell^{\infty}(G)_{G}$ of the integral has some interesting algebraic
properties we will use to deduce Theorem 1.2.

Let us observe possible connections of our results to other contexts. Our
results could be connected to the index theory on open manifolds by Roe [10].
More specifically, our index could be related to the index of the Dirac
operator

$d+d^{\ast}\colon\widehat{\Omega}^{\mathrm{even}}(M)\to\widehat{\Omega}^{\mathrm{odd}}(M)$

on the bounded de Rham complex $\widehat{\Omega}^{\ast}(M)$, which lives in
the operator $K$-theory $K_{\ast}(C^{\ast}_{u}(|G|))$ of the uniform Roe
algebra $C^{\ast}_{u}(|G|)$. There is another possible connection. In [9], the
pushforward of a vector bundle on $M$ to $M/G$ is defined, and its structure
group is the group of unitary operators with finite propagation on the Hilbert
space of square integrable functions $G\to\mathbb{C}$. On the other hand, as
in [7, 8], the module of coinvariants of bounded functions
$\mathbb{Z}\to\mathbb{Z}$ appears in the homotopy groups of such a group of
unitary operators of finite propagation for $G=\mathbb{Z}$. Then our results
could be connected to the obstruction theory for the pushforward of $TM$ onto
$M/G$.

As mentioned in [3], there is an isomorphism $\widehat{H}^{n}(M)\cong
H^{\mathrm{uf}}_{0}(M)$ (see [1] for the proof), where $\dim M=n$ and
$H^{\mathrm{uf}}_{*}(M)$ denotes the uniformly finite homology of $M$ as in
[3]. Since uniformly finite homology is a quasi-isometry invariant, there is
an isomorphism $H^{\mathrm{uf}}_{0}(M)\cong H^{\mathrm{uf}}_{0}(G)$. On the
other hand, as in [5], there is an isomorphism
$H^{\mathrm{uf}}_{0}(G)\cong\ell^{\infty}(G)_{G}$. Then we get an isomorphism

$\widehat{H}^{n}(M)\cong\ell^{\infty}(G)_{G}.$

However, this isomorphism is not explicit as it is given by a zig-zag of
several isomorphisms. We believe that the integral (1.1) gives a direct and
explicit description of this isomorphism. Our intuition relies on the case
$M=\mathbb{R}$ and $G=\mathbb{Z}$ which we treat in Proposition 4.5, and we
propose the following conjecture:

######  Conjecture 1.5.

The integral (1.1) is an isomorphism.

Throughout this paper, manifolds will be smooth and without boundary, unless
otherwise specified, and group actions on manifolds will be smooth too.

### Acknowledgement

The authors were partially supported by JSPS KAKENHI Grant Numbers JP22H01123
(Kato), JP22K03284 and JP19K03473 (Kishimoto), JP22K03317 (Tsutaya). The
authors deeply appreciate the referees’ useful advice and comments.
Especially, Section 2 was significantly improved by the referees’ advice.

##  2\. Module of coinvariants

In this section, we collect properties of the module of coinvariants
$\ell^{\infty}(G)_{G}$ that we are going to use. Block and Weinberger [3]
introduced the uniformly finite homology $H^{\mathrm{uf}}_{*}(X)$ of a metric
space $X$, and showed basic properties of it. Later, Brodzki, Niblo, and
Wright [5] studied amenability of discrete groups by using the uniformly
finite homology, where every discrete group will be equipped with a word
metric. They observed that if $G$ is finitely generated, then the uniformly
finite chain complex $C_{*}^{\mathrm{uf}}(G)$ is naturally isomorphic to the
chain complex $C_{*}(G;\ell^{\infty}(G))$. Then since
$H_{0}(G,\ell^{\infty}(G))=\ell^{\infty}(G)_{G}$, there is a natural
isomorphism

(2.1) $H_{0}^{\mathrm{uf}}(G)\cong\ell^{\infty}(G)_{G}$

whenever $G$ is finitely generated.

######  Proposition 2.1.

Let $G$ and $H$ be finitely generated groups. Then a quasi-isometric
homomorphism $G\to H$ induces an isomorphism

$\ell^{\infty}(H)_{H}\xrightarrow{\cong}\ell^{\infty}(G)_{G}.$

###### Proof.

By [3, Corollary 2.2], a quasi-isometric homomorphism $G\to H$ induces an
isomorphism $H^{\mathrm{uf}}_{*}(G)\xrightarrow{\cong}H^{\mathrm{uf}}_{*}(H)$.
Then the statement follows from (2.1). ∎

######  Corollary 2.2.

If $G$ is a finite group, then

$\ell^{\infty}(G)_{G}\cong\mathbb{R}.$

###### Proof.

Let $1$ denote the trivial group. Since $G$ is finite, the inclusion $1\to G$
is a quasi-isometry. Then since $\ell^{\infty}(1)_{1}\cong\mathbb{R}$, the
statement is proved by Proposition 2.1. ∎

######  Proposition 2.3.

Let $G$ be a finitely generated infinite group, and let
$\phi\in\ell^{\infty}(G)$. If $\phi(g)=0$ for all but finitely many $g\in G$,
then $\phi$ is zero in $\ell^{\infty}(G)_{G}$.

###### Proof.

In [12, Theorem 7.6], Whyte gave a necessary and sufficient condition for an
element of $C_{0}^{\mathrm{uf}}(G)$ to be trivial in $H^{\mathrm{uf}}_{0}(G)$.
Through the natural isomorphism $C_{0}^{\mathrm{uf}}(G)\cong
C_{0}(G;\ell^{\infty}(G))=\ell^{\infty}(G)$, this condition is stated as
follows. An element $\phi\in\ell^{\infty}(G)$ is zero in
$\ell^{\infty}(G)_{G}$ if and only if there are $C>0$ and $r>0$ such that for
any finite subset $S\subset G$,

$\left|\sum_{g\in S}\phi(g)\right|\leq C\cdot\sharp\\{g\in G\mid 0<d(g,S)\leq
r\\}$

where $d$ denotes a word metric of $G$. If $G$ is infinite, then for any non-
empty finite subset $S\subset G$, we have $\sharp\\{g\in G\mid 0<d(g,S)\leq
1\\}\geq 1$. Suppose $\phi\in\ell^{\infty}(G)$ satisfies $\phi(g)=0$ for all
but finitely many $g\in G$. Then if we set $C=\sum_{g\in G}|\phi(g)|$ and
$r=1$, the above inequality holds for any finite subset $S\subset G$, and so
$\phi$ is zero in $\ell^{\infty}(G)_{G}$. ∎

Recall that a mean on a group $G$ is a linear map

$\mu\colon\ell^{\infty}(G)\to\mathbb{R}.$

such that $\mu(\mathbbm{1})=1$ and $\mu(\phi)\geq 0$ whenever $\phi(g)\geq 0$
for all $g\in G$, where $\mathbbm{1}\in\ell^{\infty}(G)$ denotes the constant
function with value $1$ as in Section 1. A group $G$ is _amenable_ if it
admits a $G$-invariant mean. The proof of [3, Theorem 3.1] together with (2.1)
implies the following.

######  Proposition 2.4.

For a finitely generated group $G$, the following statements are equivalent:

  1. (1)

$G$ is amenable;

  2. (2)

$\ell^{\infty}(G)_{G}\neq 0$;

  3. (3)

$\mathbbm{1}\in\ell^{\infty}(G)$ is non-zero in $\ell^{\infty}(G)_{G}$.

##  3\. Basic properties of fundamental domains

In this section, we define a fundamental domain of a manifold with a free
group action, and show its basic properties. Throughout this section, let $M$
be a connected manifold of dimension $n$, possibly with boundary, on which a
cocompact and properly discontinuous action of a group $G$ is given. Since $G$
is a quotient of the fundamental group of a compact manifold $M/G$, which is
finitely generated, $G$ is finitely generated.

We define a _fundamental domain_ $D$ of $M$ as the closure of an open set of
$M$ such that

$M=\bigcup_{g\in
G}gD\quad\text{and}\quad\mathrm{Int}(D)\cap\mathrm{Int}(gD)=\emptyset$

for all $1\neq g\in G$. Remark that $D$ needs not be connected. A manifold $M$
admits a fundamental domain. Indeed, given a triangulation of $M/G$, we can
lift it to get a triangulation of $M$ such that the $G$-action is free and
simplicial. We choose one lift of the interior of each maximal simplex of
$M/G$ to $M$, so the closure of the union of these open simplices of $M$ is a
fundamental domain of $M$. We choose such a fundamental domain, so that $D$ is
a simplicial complex such that each $D\cap gD$ is a subcomplex of $D$ and

(3.1) $\partial D=\left(\bigcup_{1\neq g\in G}D\cap
gD\right)\cup(D\cap\partial M).$

If $gD\cap hD$ is $(n-1)$-dimensional, then we call it a _facet_ of $gD$ (and
$hD$). We also call $gD\cap\partial M$ a facet of $gD$ when $\partial
M\neq\emptyset$. Then the boundary of $D$ is the union of its facets. Clearly,
the $G$-action on $M$ restricts to $\partial M$, and $D\cap\partial M$ is a
fundamental domain of $\partial M$.

We construct a generating set of $G$ by using a fundamental domain $D$. Let
$S$ be a subset of $G$ consisting of elements $g\in G$ such that $D\cap gD$ is
a facet of $D$.

######  Proposition 3.1.

The set $S$ is a symmetric generating set of $G$.

###### Proof.

Let $g\in G$ and $x\in\mathrm{Int}(D)$. Then $gx$ belongs to
$\mathrm{Int}(gD)$, and so since $M$ is connected, there is a path $\ell$ from
$x$ to $gx$ which passes $g_{0}D,g_{1}D,\ldots,g_{k}D$ in order for
$1=g_{0},g_{1},\ldots,g_{k-1},g_{k}=g\in G$ such that $g_{i}D\cap g_{i+1}D$ is
a facet and $\ell\cap g_{i}D\cap g_{i+1}D$ is a single point sitting in the
interior of a facet $g_{i}D\cap g_{i+1}D$ of $g_{i}D$ for $i=0,1,\ldots,k-1$.
Since $g_{i}D\cap g_{i+1}D=g_{i}(D\cap g_{i}^{-1}g_{i+1}D)$ is a facet of
$g_{i}D$, $D\cap g_{i}^{-1}g_{i+1}D$ is a facet of $D$, implying
$g_{i}^{-1}g_{i+1}\in S$. Thus since

$g=g_{k}=(g_{0}^{-1}g_{1})(g_{1}^{-1}g_{2})\cdots(g_{k-1}^{-1}g_{k}),$

we obtain that $S$ is a generating set of $G$. If $g\in S$, then $g(D\cap
g^{-1}D)=gD\cap D$ is a facet of $D$, and so $D\cap g^{-1}D$ is a facet of $D$
too. Hence $g^{-1}\in S$, that is, $S$ is symmetric, completing the proof. ∎

######  Corollary 3.2.

There is a partition $S=S^{+}\sqcup S^{-}\sqcup S^{0}$ such that
$(S^{+})^{-1}=S^{-}$ and $(S^{0})^{2}=\\{1\\}$.

###### Proof.

Let $S^{0}$ be the subset of $S$ consisting of elements of order two. Then the
statement follows because $S$ is symmetric. ∎

Let $S^{+}=\\{s_{1},\ldots,s_{k}\\}$ and $S^{0}=\\{t_{1},\ldots,t_{l}\\}$,
where $S^{+}$ and $S^{0}$ are finite because $G$ is finitely generated as
mentioned above. We put

$E=D\cap\partial M,\quad F_{i}^{+}=D\cap s_{i}D,\quad F_{i}^{-}=D\cap
s_{i}^{-1}D,\quad F_{j}^{0}=D\cap t_{j}D$

for $i=1,2,\ldots,k$ and $j=1,2,\ldots,l$.

######  Lemma 3.3.

The facets of $D$ are
$E,F_{1}^{+},\ldots,F_{k}^{+},F_{1}^{-},\ldots,F_{k}^{-},F_{1}^{0},\ldots,F_{l}^{0}$.

###### Proof.

The statement follows from Corollary 3.2. ∎

We consider an orientation of a facet of $gD$.

######  Lemma 3.4.

Suppose that $M$ is oriented. If $F=gD\cap hD$ is a facet for $g,h\in G$, then
the orientations of $F$ induced from $gD$ and $hD$ are opposite.

###### Proof.

An outward vector of $gD$ rooted at $F$ is an inward vector of $hD$. Then the
statement follows. ∎

##  4\. Integral in bounded cohomology

In this section, we define the integral in bounded cohomology. Let $M$ be a
connected Riemannian manifold of dimension $n$, possibly with boundary. As in
[10], we say that a differential form $\omega$ on $M$ is _bounded_ if both
$|\omega|$ and $|d\omega|$ are bounded. Let $\widehat{\Omega}^{p}(M)$ denote
the set of bounded $p$-forms on $M$. Then by definition,
$\widehat{\Omega}^{*}(M)$ is closed under differential, and so it is a
differential graded algebra. We define the _bounded de Rham cohomology_ of $M$
as the cohomology of $\widehat{\Omega}^{*}(M)$, which we denote by
$\widehat{H}^{*}(M)$. We record the following obvious fact.

######  Lemma 4.1.

If a map $f\colon M\to N$ between manifolds has bounded derivative, then it
induces a map $f^{*}\colon\widehat{\Omega}^{*}(N)\to\widehat{\Omega}^{*}(M)$.

Now we consider a cocompact and properly discontinuous action of a discrete
group $G$ on a manifold $M$, and choose a fundamental domain $D\subset M$. A
Riemannian metric of $M$ will be chosen to be the lift of a Riemannian metric
of $M/G$. We assume that $M/G$ is oriented. Then in particular, the
fundamental domain $D$ is oriented. We define the _integral_ of a bounded
differential form on $M$ by

(4.1)
$\int_{M}\colon\widehat{\Omega}^{n}(M)\to\ell^{\infty}(G),\quad\left(\int_{M}\omega\right)(g)=\int_{gD}\omega.$

We may think of the above integral as the external transfer of the covering
$M\to M/G$. Note that we can similarly define the integral for $\partial M$ by
using a fundamental domain $D\cap\partial M$ of $\partial M$. We prove Stokes’
theorem.

######  Proposition 4.2.

For $\omega\in\widehat{\Omega}^{n-1}(M)$, we have

$\int_{M}d\omega=\int_{\partial
M}\omega\quad\text{in}\quad\ell^{\infty}(G)_{G}.$

###### Proof.

We consider the facets of $D$ described in Lemma 3.3. Define
$\phi_{i}^{\pm},\,\phi^{0}_{j}\in\ell^{\infty}(G)$ by

$\phi_{i}^{\pm}(g)=\int_{gF^{\pm}_{i}}\omega\quad\text{and}\quad\phi^{0}_{j}(g)=\int_{gF^{0}_{i}}\omega$

for $i=1,2,\ldots,k$ and $j=1,2,\ldots,l$, where the orientations of
$gF^{\pm}_{i}$ and $gF^{0}_{i}$ are induced from $gD$. Then by Lemma 3.3 and
the usual Stokes’ theorem, we have

$\int_{gD}d\omega=\int_{gE}\omega+\sum_{i=1}^{k}(\phi_{i}^{+}(g)+\phi_{i}^{-}(g))+\sum_{j=1}^{l}\phi^{0}_{j}(g),$

where the orientation of $gE$ is induced from $gD$. Since
$gF_{i}^{-}=gs_{i}^{-1}F_{i}^{+}$, it follows from Lemma 3.4 that
$\phi_{i}^{-}(g)=-\phi_{i}^{+}(gs_{i}^{-1})$. Then we get

$\phi_{i}^{+}+\phi_{i}^{-}=\phi_{i}^{+}-s_{i}^{-1}\phi_{i}^{+}.$

Quite similarly, we can also get

$\phi_{j}^{0}=\frac{1}{2}(\phi_{j}^{0}-t_{j}^{-1}\phi_{j}^{0}).$

Thus since $E=D\cap\partial M$ is a fundamental domain of $\partial M$, we
obtain

$\int_{M}d\omega=\int_{\partial
M}\omega+\sum_{i=1}^{k}(\phi_{i}^{+}-s_{i}^{-1}\phi_{i}^{+})+\sum_{j=1}^{l}\frac{1}{2}(\phi_{j}^{0}-t_{j}^{-1}\phi_{j}^{0}).$

Therefore the proof is finished. ∎

The following is immediate from Proposition 4.2.

######  Corollary 4.3.

If $M$ is without boundary, then the integral (4.1) induces a map

$\int_{M}\colon\widehat{H}^{n}(M)\to\ell^{\infty}(G)_{G}.$

By considering $n$-forms with support in $gD$, we can easily see that the
integral in bounded cohomology is always surjective. We give two supporting
examples for Conjecture 1.5.

######  Proposition 4.4.

Conjecture 1.5 is true for $G$ finite.

###### Proof.

If $G$ is finite, then $M$ is compact, and so $\widehat{H}^{n}(M)$ coincides
with the usual $n$-th de Rham cohomology of $M$, which is isomorphic with
$\mathbb{R}$. On the other hand, by Corollary 2.2, we have
$\ell^{\infty}(G)_{G}\cong\mathbb{R}$. Then since the integral in bounded
cohomology is surjective as mentioned above, it is actually an isomorphism, as
stated. ∎

######  Proposition 4.5.

Conjecture 1.5 is true for $M=\mathbb{R}$ and $G=\mathbb{Z}$, where
$\mathbb{Z}$ acts on $\mathbb{R}$ by translation.

###### Proof.

We choose the interval $[0,1]\subset\mathbb{R}$ as a fundamental domain. Let
$g=1\in\mathbb{Z}$. Suppose that

(4.2) $\int_{\mathbb{R}}f(x)dx=\phi-g\phi$

for a bounded function $f(x)$ on $\mathbb{R}$ and
$\phi\in\ell^{\infty}(\mathbb{Z})$, where the integral is taken in the sense
of (4.1). The equation (4.2) is equivalent to the fact that the $1$-form
$f(x)dx$ belongs to the kernel of the integral in bounded cohomology because

$\phi-g^{n}\phi=(\phi+g\phi+\cdots+g^{n-1}\phi)-g(\phi+g\phi+\cdots+g^{n-1}\phi).$

Note that $(\phi-g\phi)(i)=\phi(i)-\phi(i+1)$. Now we define

$h(x)=\int_{0}^{x}f(t)dt.$

To see that the integral in bounded cohomology is injective, it is sufficient
to show that $h(x)\in\widehat{\Omega}^{0}(\mathbb{R})$. Since $dh(x)=f(x)dx$,
$dh(x)$ is bounded. For $0\leq n\leq x<n+1$, we have

$h(x)=\sum_{i=0}^{n-1}\int_{i}^{i+1}f(t)dt+\int_{n}^{x}f(t)dt=\phi(0)-\phi(n)+\int_{n}^{x}f(t)dt.$

Since $f(x)$ is bounded, $\int_{n}^{x}f(t)dt$ is bounded too as $x$ and $n$
vary. Then $h(x)$ is bounded for $x\geq 0$. Quite similarly, we can show that
$h(x)$ is bounded for $x<0$ too, and so we get
$h(x)\in\widehat{\Omega}^{0}(\mathbb{R})$. Thus we obtain the injectivity.
Since the integral in bounded cohomology is surjective as mentioned above, it
is an isomorphism. ∎

##  5\. Poincaré-Hopf theorem

In this section, we prove Theorems 1.1 and 1.2. Throughout this section, let
$M$ be a connected manifold of dimension $n$ equipped with a cocompact and
properly discontinuous action of a discrete group $G$ such that $M/G$ is
oriented. The metric of $M$ will be the lift of a metric of $M/G$.

Let $\Phi$ denote a representative of the Thom class of $M/G$. Then as in [4],
the support of $\Phi$ is compactly supported, and so $\Phi$ is a bounded
$n$-form on $T(M/G)$. Let $\pi\colon M\to M/G$ denote the projection. Then the
derivative of $\pi$ is bounded, and so by Lemma 4.1, we get the induced map
$\pi^{*}\colon\widehat{\Omega}^{*}(T(M/G))\to\widehat{\Omega}^{*}(TM)$. In
particular, $\pi^{*}(\Phi)$ is a bounded $n$-form on $TM$. Note that
$\pi^{*}(\Phi)$ represents the Thom class of $M$ in bounded cohomology. Let
$v$ be a vector field on $M$ with $|dv|$ bounded. Then by Lemma 4.1,
$v^{*}(\pi^{*}(\Phi))$ is a bounded $n$-form on $M$, and so we can define the
index of $v$ by

$\operatorname{ind}(v)=\int_{M}v^{*}(\pi^{*}(\Phi))\in\ell^{\infty}(G)_{G}.$

We remark that the index $\operatorname{ind}(v)$ is independent of the choice
of a representative $\Phi$ of the Thom class of $M/G$. Indeed, if $\Psi$ is
another representative of the Thom class of $M/G$, then $\Phi-\Psi=d\alpha$
for some compactly supported $(n-1)$-form $\alpha$ on $T(M/G)$, where $\Psi$
is compactly supported. Hence we get
$\pi^{*}(\Phi)-\pi^{*}(\Psi)=d\pi^{*}(\alpha)$, where all differential forms
are bounded, and so by Corollary 4.3, the indices of $v$ defined by $\Phi$ and
$\Psi$ are equal. We also remark that by Proposition 2.4, the index of a
bounded vector field on $M$ is always zero whenever $G$ is not amenable. (cf.
[11, Theorem 2])

We now show some properties of the index. Let $v_{0}$ denote the zero vector
field, that is, the zero section $M\to TM$. Then $v_{0}^{*}(\pi^{*}(\Phi))$ is
a representative of the Euler class $e(M)$ in bounded cohomology, which was
considered by Weinberger [11].

######  Proposition 5.1.

There is an equality

$\int_{M}e(M)=\chi(M/G)\mathbbm{1}.$

###### Proof.

Let $\bar{v}_{0}$ denote the zero vector field on $M/G$, so $v_{0}$ is the
lift of $\bar{v}_{0}$. Since the projection $\pi\colon\mathrm{Int}(gD)\to
M/G-\pi(\partial(gD))$ is a diffeomorphism and both $\partial(gD)$ and
$\pi(\partial(gD))$ have measure zero, we have

$\int_{gD}v_{0}^{*}(\pi^{*}(\Phi))=\int_{M/G}\bar{v}_{0}^{*}(\Phi)=\int_{M/G}e(M/G)=\chi(M/G).$

Thus the proof is finished. ∎

######  Lemma 5.2.

If vector fields $v$ and $w$ on $M$ with $|dv|$ and $|dw|$ bounded are
homotopic by a homotopy with bounded derivative, then

$\operatorname{ind}(v)=\operatorname{ind}(w).$

###### Proof.

Let $v_{t}\colon M\times[0,1]\to TM$ be a homotopy with bounded derivative
such that $v_{0}=v$ and $v_{1}=w$. Since the induced maps
$v_{t}^{*}\colon\widehat{\Omega}^{*}(TM)\to\widehat{\Omega}^{*}(M\times[0,1])$
and $\pi^{*}\colon\widehat{\Omega}^{*}(T(M/G))\to\widehat{\Omega}^{*}(TM)$
commute with the differential, we have

$\int_{M\times[0,1]}dv_{t}^{*}(\pi^{*}(\Phi))=\int_{M\times[0,1]}v_{t}^{*}(\pi^{*}(d\Phi))=0$

where $\Phi$ is a closed $n$-form representing the Thom class of $T(M/G)$. On
the other hand, by Proposition 4.2, we have

$\int_{M\times[0,1]}dv_{t}^{*}(\pi^{*}(\Phi))=\int_{M\times
1}w^{*}(\pi^{*}(\Phi))-\int_{M\times 0}v^{*}(\pi^{*}(\Phi)).$

Thus the statement is proved. ∎

######  Proposition 5.3.

Let $v$ be a bounded vector field on $M$. Then we have

$\operatorname{ind}(v)=\operatorname{ind}(v_{0}).$

###### Proof.

Clearly, $tv$ is a homotopy from $v_{0}$ to $v$ with bounded derivative. Then
by Lemma 5.2, the proof is done. ∎

We consider a mild condition on zeros of a vector field. Let $B_{\delta}(x)$
denote the open $\delta$-neighborhood of $x\in M$, and let $B_{\epsilon}(M)$
denote the open $\epsilon$-neighborhood of $M$ in $TM$.

######  Definition 5.4.

A vector field $v$ on a manifold $M$ is _tame_ if there are $\delta>0$ and
$\epsilon>0$ such that

  1. (1)

$B_{\delta}(x)\cap B_{\delta}(y)=\emptyset$ for $x\neq y\in\mathrm{Zero}(v)$,
and

  2. (2)

$v^{-1}(B_{\epsilon}(M))\subset\bigcup_{x\in\mathrm{Zero}(v)}B_{\delta}(x)$.

######  Definition 5.5.

A vector field $v$ on $M$ is _strongly tame_ if it is tame and there is
$\delta>0$ such that for each $x\in\mathrm{Zero}(v)$, $B_{\delta}(x)\subset
gD$ for some $g\in G$.

We also define a tame diffeomorphism, in analogy with tame vector fields.

######  Definition 5.6.

A diffeomorphism $f\colon M\to M$ is _tame_ if there are $\delta>0$ and
$\epsilon>0$ such that

  1. (1)

$B_{\delta}(x)\cap B_{\delta}(y)=\emptyset$ for $x\neq y\in\mathrm{Fix}(f)$,
and

  2. (2)

$d(x,f(x))>\epsilon$ for $x\in M-\bigcup_{y\in\mathrm{Fix}(y)}B_{\delta}(y)$

where $d$ stands for the metric of $M$.

We prove a technical lemma.

######  Lemma 5.7.

Let $f\colon\mathbb{R}^{n}\to T\mathbb{R}^{n}$ be a section of the tangent
bundle $T\mathbb{R}^{n}=\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$
such that for some $\delta,\epsilon>0$, $f^{-1}(B_{\epsilon}(0))\subset
B_{\delta}(0)$. Let $\omega$ be a representative of the Thom class of
$T\mathbb{R}^{n}$ such that $\mathrm{supp}(\omega)\subset\mathbb{R}^{n}\times
B_{\epsilon/2}(0)$. Then we have

$\mathrm{ind}_{0}(f)=\int_{B_{\delta}(0)}f^{*}(\omega).$

###### Proof.

Let $B$ denote the closure of $B_{\delta}(0)$. Then by definition,
$\mathrm{ind}_{0}(f)$ is the mapping degree of the composite

$\partial
B\xrightarrow{f}\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})\xrightarrow{p}\mathbb{R}^{n}\setminus\\{0\\}\xrightarrow{q}S^{n-1}$

where $p\colon
T\mathbb{R}^{n}=\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ denotes
the second projection and $q\colon\mathbb{R}^{n}\setminus\\{0\\}\to S^{n-1}$
denotes the natural projection onto the unit sphere. There is a function
$\rho\colon[0,\infty)\to\mathbb{R}$ such that $\rho(x)=0$ for $x$ sufficiently
close to $0$ and $\rho(x)=1$ for $x\geq\epsilon/2$. Let $\alpha$ be an
$(n-1)$-form on the unit sphere $S^{n-1}$ of $\mathbb{R}^{n}$ such that
$\int_{S^{n-1}}\alpha=1$. Now we define

$\eta=p^{*}(d\rho\wedge q^{*}(\alpha)).$

By definition, we have $\mathrm{supp}(\eta)\subset\mathbb{R}^{n}\times
B_{\epsilon/2}(0)$, and since $\int_{\mathbb{R}^{n}}d\rho\wedge
q^{*}(\alpha)=1$, $\eta$ represents the Thom class of $T\mathbb{R}^{n}$. Then
by the uniqueness of the Thom class, there is an $(n-1)$-form $\tau$ on
$T\mathbb{R}^{n}$ with vertically compact support such that

$\omega-\eta=d\tau.$

We have $\mathrm{supp}(\tau)\subset\mathbb{R}^{n}\times B_{\epsilon/2}(0)$,
implying $f^{*}(\tau)|_{\partial B}=0$. So by Stokes’ theorem, we get

$\int_{B}f^{*}(\omega)-\int_{B}f^{*}(\eta)=\int_{B}f^{*}(d\tau)=\int_{B}df^{*}(\tau)=\int_{\partial
B}f^{*}(\tau)=0.$

Note that $\eta=dp^{*}(\rho\cdot q^{*}(\alpha))$ and $\rho(f(\partial B))=1$.
Then by Stokes’ theorem, we have

$\int_{B}f^{*}(\eta)=\int_{B}df^{*}\circ p^{*}(\rho\cdot
q^{*}(\alpha))=\int_{\partial B}f^{*}\circ p^{*}(\rho\cdot
q^{*}(\alpha))=\int_{\partial B}f^{*}\circ p^{*}\circ
q^{*}(\alpha)=\mathrm{ind}_{0}(f).$

Thus since $\int_{B_{\epsilon}(0)}f^{*}(\omega)=\int_{B}f^{*}(\omega)$, the
proof is finished. ∎

We compute the index of a strongly tame bounded vector field.

######  Proposition 5.8.

Let $v$ be a strongly tame bounded vector field on $M$. Then

$\operatorname{ind}(v)(g)=\sum_{x\in\operatorname{Zero}(v)\cap
gD}\operatorname{ind}_{x}(v).$

###### Proof.

Let $\delta$ and $\epsilon$ be as in the definition of a tame vector field. As
in [4], we may assume that the support of $\pi^{*}(\Phi)$ is contained in
$B_{\epsilon/2}(M)$. Then we have

$\operatorname{ind}(v)(g)=\sum_{x\in\operatorname{Zero}(v)\cap
gD}\int_{B_{\delta}(x)}v^{*}(\pi^{*}(\Phi))$

for each $g\in G$. On the other hand, by tameness of $v$,
$\pi^{*}(\Phi)|_{B_{\delta}(x)}$ is compactly supported for
$x\in\mathrm{Zero}(v)$, and so by Lemma 5.7, we get

$\int_{B_{\delta}(x)}v^{*}(\pi^{*}(\Phi))=\operatorname{ind}_{x}(v)$

for each $x\in\operatorname{Zero}(v)$. Thus the statement is proved. ∎

Now we are ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Combine Propositions 5.1, 5.3 and 5.8. ∎

###### Proof of Theorem 1.2.

As mentioned at the beginning of Section 3, $G$ is finitely generated. Then we
can apply the results in Section 2. Let $v$ be a tame bounded vector field on
$M$, and suppose that $v$ has finitely many zeros. We can easily see that $v$
is homotopic to a strongly tame vector field by a homotopy with bounded
derivative. Then by Lemma 5.2, we may assume that $v$ itself is strongly tame,
and so by Theorem 1.1, we have

$\operatorname{ind}(v)=\chi(M/G)\mathbbm{1}.$

Since $\chi(M/G)\neq 0$, it follows from Proposition 2.4 that
$\chi(M/G)\mathbbm{1}$ is non-zero in $\ell^{\infty}(G)_{G}$. Then by
Proposition 2.3, we obtain that $v$ must have infinitely many zeros, a
contradiction. Thus $v$ must have infinitely many zeros. ∎

###### Proof of Corollary 1.3.

Observe that a tame diffeomorphism is the composite of a tame vector field and
the exponential map if it is $C^{1}$ close to the identity map. Then the
statement follows from Theorem 1.2. ∎

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