# Speed of sound and liquid-gas phase transition in nuclear matter

Wei-bo He  MOE Key Laboratory for Non-equilibrium Synthesis and Modulation of
Condensed Matter, School of Physics, Xi’an Jiaotong University, Xi’an 710049,
China  School of Physics, Peking University, Beijing, 100871, China  Guo-yun
Shao<EMAIL_ADDRESS>MOE Key Laboratory for Non-equilibrium Synthesis
and Modulation of Condensed Matter, School of Physics, Xi’an Jiaotong
University, Xi’an 710049, China  Chong-long Xie  MOE Key Laboratory for Non-
equilibrium Synthesis and Modulation of Condensed Matter, School of Physics,
Xi’an Jiaotong University, Xi’an 710049, China

###### Abstract

We investigate the speed of sound in nuclear matter at finite temperature and
density (chemical potential) in the nonlinear Walecka model. The numerical
results suggest that the behaviors of sound speed are closely related to the
the nuclear liquid-gas (LG) phase transition and the associated spinodal
structure. The adiabatic sound speed is nonzero at the critical endpoint (CEP)
in the mean field approximation. We further derive the boundary of vanishing
sound velocity in the temperature-density phase diagram, and point out the
region where the sound wave equation is broken. The distinction between the
speed of sound in nuclear matter and that in quark matter contains important
information about the equation of state of strongly interacting matter at
intermediate and high density. We also formulate the relations between
differently defined speed of sound using the fundamental thermodynamic
relations.

##  I introduction

Exploring the equation of state (EOS) of strongly interacting matter is an
important topic in both the theoretical and experimental nuclear physics
Gupta11 ; Bazavov17 ; Borsanyi20 ; Fukushima04 ; Ratti06 ; Costa10 ; Sasaki12
; Ferreira14 ; Schaefer10 ; Skokov11 ; Qin11 ; Gao16 ; Fischer14 ; Fu20 ;
Stephanov ; Shao2018 ; Shao2019 ; Luo2014 ; Luo2016 ; Luo2017 ; Adam21 ;
Song11 ; Song112 ; Deb16 ; Leonhardt2020 . During the space-time evolution of
the newly formed matter in heavy-ion collisions, the speed of sound is one of
the crucial physical quantities to describe the variation of EOS. The
dependence of speed of sound on environment (temperature, density, chemical
potential, etc.) carries important information in describing the evolution of
the fireball and final observables. Recently, the studies in Gardim20 ; Sahu21
; Biswas20  show that the speed of sound as a function of charged particle
multiplicity $\langle dN_{ch}/d\eta\rangle$ can be extracted from heavy-ion
collision data. In Sorensen21  the authors try to build a connection between
the sound velocity and the baryon number cumulants to study the quantum
chromodynamics (QCD) phase structure.

It is also an interesting topic to study the speed of sound during the QCD
phase transition in the early universe by observing the induced gravitational
wave. Although the propagation of gravitational wave is not sensitive to sound
velocity, the value of sound velocity affects the dynamics of primordial
density perturbations, and the induced gravitational waves by their horizon
reentry can then be an indirect probe on both the EOS and sound velocity,
which can provide useful knowledge of the evolution in the era of QCD phase
transition Abe21 .

Besides in heavy-ion collision experiments and early universe, the speed of
sound in neutron star matter also receives a lot of attention (e.g.,Reed20 ;
Kanakis20 ; Han20 ). The density dependent behavior of sound velocity
influences the mass-radius relation, the tidal deformability, and provides a
sensitive probe of the EOS of neutron star matter and the hadron-quark phase
transition in the dense core. To obtain a massive neutron star, some studies
show that it is essential for neutron star matter to have a density range
where the EOS is very stiff and the corresponding squared speed of sound is
significantly larger than $1/3$ Tews18 ; Greif19 ; Forbes19 ; Drischler20 ;
Essick20 ; Han19 ; Kojo ; Altiparmak2022 ; Moustakidis2017 ; Zhang2019 ;
Reed2020 ; Kanakis2020 ; Huang2022 . In addition, the study in Jaikumar21
indicates that the speed of sound is crucial for the gravitational wave
frequencies induced by the $g$-mode oscillation of a neutron star.

As an important quantity in describing the evolution of strongly interacting
matter, the speed of sound in QCD matter has been calculated, e.g., in LQCD
Aoki06 ; Borsanyi14 ; Bazavov14 ; Philipsen13 ; Borsanyi20 , (P)NJL model
Motta18 ; Ghosh06 ; Marty13 ; Deb16 ; Saha18 , quark-meson coupling model
Schaefer10 ; Abhishek18 , hadron resonance gas (HRG) model Venugopalan92 ;
Bluhm14 , field correlator method (FCM) Khaidukov18 ; Khaidukov19  and
quasiparticle model Mykhaylova21 . In most studies, the main focus is put on
the region of high temperature and vanishing or small chemical potential.

In Ref. he2022 , we give an intensive study on the speed of sound of QCD
matter in the full phase diagram. The numerical results suggest that the
dependence of sound speed on temperature, density and chemical potential are
closely related to the QCD phase structure. In particular, the value of
adiabatic sound speed is not zero at the CEP in the mean-field approximation.
In the region of chiral crossover region, the speed of sound increases quickly
with the rising temperature. The value of squared sound speed approaches to
1/3 after the chiral restoration at high temperature. Some new features of
speed of sound are also discovered, for example, the hierarchy phenomenon of
sound velocity for $u(d)$ and $s$ quark at low temperature with the increasing
chemical potential. In addition to the adiabatic sound velocity, the behaviors
of speed of sound under other conditions are also discussed.

The nuclear LG phase transition was discovered in experiments many years ago,
and the experimental phenomenon was explained with the spinodal decomposition
mechanism Chomaz2004 ; Pochodzalla95 ; Borderie01 ; Botvina95 ; Agostino99 ;
Srivastava02 ; Elliott02 ; Ma1999 ; Deng2022 . Recently, there are some
interesting investigations on baryon number fluctuations as induced in the
phase diagram due to the presence of the nuclear LG phase transition
Vovchenko17 ; Shao20202 . Since the nuclear LG transition and quark chiral
first-order phase transition belong to the same universality class, it is
interesting to study whether the speed of sounds in quark matter and in
nuclear matter behave in a similar way. Moreover, the study on the parameter
dependence of the speed of sound in nuclear matter is crucial for
investigating the transport phenomenon of hadronic matter Deb16 . In this
study, we will detailedly explore the behavior of sound speed in nuclear
matter and compare it with that in quark matter. It is expected that the
correlation between the behaviors of speed of sound and nuclear LG phase
transition can be revealed.

The paper is organized as follows. In Sec. II, we introduce briefly the
nonlinear Walecka model and the formulae of speed of sound under different
definitions. In Sec. III, we present the numerical results of squared sound
speed and discuss the relations with the nuclear LG transition. A summary is
finally given in Sec. IV.

##  II the nonlinear Walecka model and speed of sound

The Lagrangian density for the nucleons-meson system in the nonlinear Walecka
modelGlendenning1997  is

$\displaystyle\cal{L}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\sum_{N}\bar{\psi}_{N}\\!\big{[}i\gamma_{\mu}\partial^{\mu}\\!-\\!(\\!m_{N}\\!-\\!g_{\sigma}\sigma)\\!\\!-\\!g_{\omega}\gamma_{\mu}\omega^{\mu}\big{]}\\!\psi_{N}$
(1) $\displaystyle+\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2}\right)\\!-\\!\frac{1}{3}bm_{N}\,(g_{\sigma}\sigma)^{3}-\frac{1}{4}c\,(g_{\sigma}\sigma)^{4}$
$\displaystyle+\frac{1}{2}m^{2}_{\omega}\omega_{\mu}\omega^{\mu}-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}$

where $\omega_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$.
The interactions between nucleons are mediated by $\sigma,\,\omega$ mesons.
The $\mathrm{\rho}$ meson is not included since in this work we only consider
the behavior of speed of sound in symmetric nuclear matter. The model
parameters, $g_{\sigma},g_{\omega},b$ and $c$, are fixed in the mean-field
approximation with the compression modulus $K=240\,$MeV, the symmetric energy
$a_{sym}=31.3\,$MeV, the effective nucleon mass
$m^{*}_{N}=m_{N}-g_{\sigma}\sigma=0.75m_{N}$ ($m_{N}$ is the nucleon mass in
vacuum) and the binding energy $B/A=-16.0\,$MeV at nuclear saturation density
with $\rho_{0}=0.16\,fm^{-3}$.

The thermodynamic potential derived under the mean-field approximation is

$\displaystyle\Omega\\!$ $\displaystyle\\!=$
$\displaystyle\\!-\\!\beta^{-1}\sum_{N}2\\!\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\\!\bigg{[}\ln\\!\left(1+e^{-\beta\left(E_{N}^{*}(k)-\mu_{N}^{*}\\!\right)}\right)\\!$
(2)
$\displaystyle+\\!\ln\\!\left(1\\!+\\!e^{-\beta\left(E_{N}^{*}(k)+\mu_{N}^{*}\right)}\right)\\!\bigg{]}\\!+\\!\frac{1}{2}m_{\sigma}^{2}\sigma^{2}\\!+\\!\frac{1}{3}bm_{N}\left(g_{\sigma}\sigma\right)^{3}$
$\displaystyle+\frac{1}{4}c\left(g_{\sigma}\sigma\right)^{4}-\frac{1}{2}m_{\omega}^{2}\omega^{2},$

where $\beta=1/T$, $E_{N}^{*}=\sqrt{k^{2}+m_{N}^{*2}}$. The effective chemical
potential $\mu_{N}^{*}$ is defined as $\mu_{N}^{*}=\mu_{N}-g_{\omega}\omega$
for nucleons.

By minimizing the thermodynamical potential

$\frac{\partial\Omega}{\partial\sigma}=\frac{\partial\Omega}{\partial\omega}=0,$
(3)

the meson field equations are derived as

$g_{\sigma}\sigma\\!=\\!\left(\frac{g_{\sigma}}{m_{\sigma}}\right)^{2}\\!\left[\rho_{p}^{s}+\rho_{n}^{s}-bm_{N}\left(g_{\sigma}\sigma\right)^{2}-c\left(g_{\sigma}\sigma\right)^{3}\right],$
(4)
$g_{\omega}\omega=\left(\frac{g_{\omega}}{m_{\omega}}\right)^{2}\left(\rho_{p}+\rho_{n}\right).$
(5)

In Eqs.(4)-(5), the nucleon number density for proton or neutron is

$\rho_{N}\\!=\\!2\\!\int\\!\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\bigg{[}f\left(E_{N}^{*}(k)-\mu_{N}^{*}\right)\\!-\\!\bar{f}\left(E_{N}^{*}(k)+\mu_{N}^{*}\right)\bigg{]},$
(6)

and the scalar density

$\displaystyle\rho_{N}^{s}$ $\displaystyle=$ $\displaystyle
2\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{m_{N}^{*}}{E_{N}^{*}(k)}\bigg{[}f\left(E_{N}^{*}(k)-\mu_{N}^{*}\right)$
(7) $\displaystyle\hskip
80.0pt+\bar{f}\left(E_{N}^{*}(k)+\mu_{N}^{*}\right)\bigg{]},$

where $f(E_{N}^{*}(k)-\mu_{N}^{*})$ and $\bar{f}(E_{N}^{*}(k)+\mu_{N}^{*})$
are the fermion and antifermion distribution functions.

The pressure $p$ and energy density $\epsilon$ can be derived using the
thermodynamic relations in the grand canonical ensemble as

$p=-\Omega,\,\,\,\,\,\,\,\,\epsilon=-p+Ts+\sum\mu_{N}\rho_{N},$ (8)

where $s$ is the entropy density. The general definition of speed of sound is

$c^{2}_{X}=\left(\frac{\partial p}{\partial\epsilon}\right)_{X}.$ (9)

A specific constant quantity $X$ is required to describe the propagation of
the compression wave through a medium. To indicate the different profiles of
the EOS of nuclear matter, $X$ can be chosen as
$s/\rho_{B},s,\rho_{B},T,\mu_{B}$.

For an ideal fluid, it evolves with a constant entropy density per baryon
$s/\rho_{B}$ under the adiabatic evolution. This conclusion can be drawn in
hydrodynamics with the conservation of energy and baryon number, therefore it
is most meaningful to calculate the speed of sound along the isentropic curve

$c_{s/{\rho_{B}}}^{2}=\bigg{(}\frac{\partial
p}{\partial\epsilon}\bigg{)}_{s/{\rho_{B}}}.$ (10)

The dependence of $c_{s/{\rho_{B}}}^{2}$ on parameters, e.g., temperature,
density and chemical potential, can indicate the variation of sound speed
during the evolution and provide important knowledge of interaction, phase
transition and the EOS.

The definitions of sound velocity under other conditions are also taken in
literature in dealing with different problems. The speed of sound with
constant baryon number density or entropy density are taken in describing the
intermediate process of hydrodynamic evolution Deb16 ; Albright16 ,

$c_{\rho_{B}}^{2}=\bigg{(}\frac{\partial
p}{\partial\epsilon}\bigg{)}_{\rho_{B}},$ (11)

and

$c_{s}^{2}=\bigg{(}\frac{\partial p}{\partial\epsilon}\bigg{)}_{s}.$ (12)

For example, the space-time derivatives of temperature and chemical potential
are functions of $c_{\rho_{B}}^{2}$ and $c_{s}^{2}$ with

$\partial_{0}\mu_{B}=-c_{s}^{2}\mu_{B}\,\mathbb{\nabla}\cdot\mathbb{u},$ (13)

and

$\partial_{0}T=-c_{\rho_{B}}^{2}T\,\mathbb{\nabla}\cdot\mathbb{u},$ (14)

where $\mathbb{u}$ denotes the space component of four-velocity.

It is also interesting to calculate the sound velocity with a fixed
temperature or chemical potential with

$c_{T}^{2}=\bigg{(}\frac{\partial p}{\partial\epsilon}\bigg{)}_{T},$ (15)

and

$c_{\mu_{B}}^{2}=\bigg{(}\frac{\partial
p}{\partial\epsilon}\bigg{)}_{\mu_{B}}.$ (16)

$c_{T}^{2}$ is widely used in calculating the sound speed in neutron star
matter. In Sorensen21  the authors make a connection between the logarithmic
derivative of $c_{T}^{2}$ with respect to baryon density and baryon number
fluctuations, which can be measured in experiment.

In this study, we will explore the speed of sound in nuclear matter under
different definitions in the full temperature-density and temperature-chemical
potential spaces. Since the general definitions given above can only be used
to calculate the speed of sound along special trajectories, it is necessary to
derive the corresponding formulae as functions of $T$ and $\mu_{B}$
($\rho_{B}$) to perform the calculation for any given temperature and density
(chemical potential).

Using the fundamental thermodynamic relations we derive the sound speed
formulae under different conditions in terms of $T$ and $\mu_{B}$ as

$c_{s/\rho_{B}}^{2}=\frac{s\rho_{B}\left(\frac{\partial
s}{\partial\mu_{B}}\right)_{T}-s^{2}\left(\frac{\partial\rho_{B}}{\partial\mu_{B}}\right)_{T}-\rho_{B}^{2}\left(\frac{\partial
s}{\partial
T}\right)_{\mu_{B}}+s\rho_{B}\left(\frac{\partial\rho_{B}}{\partial
T}\right)_{\mu_{B}}}{\left(sT+\mu_{B}\rho_{B}\right)\left[\left(\frac{\partial
s}{\partial\mu_{B}}\right)_{T}\left(\frac{\partial\rho_{B}}{\partial
T}\right)_{\mu_{B}}-\left(\frac{\partial s}{\partial
T}\right)_{\mu_{B}}\left(\frac{\partial\rho_{B}}{\partial\mu_{B}}\right)_{T}\right]},$
(17)
$c_{\rho_{B}}^{2}=\frac{s\left(\frac{\partial\rho_{B}}{\partial\mu_{B}}\right)_{T}-\rho_{B}\left(\frac{\partial\rho_{B}}{\partial
T}\right)_{\mu_{B}}}{T\left[\left(\frac{\partial s}{\partial
T}\right)_{\mu_{B}}\left(\frac{\partial\rho_{B}}{\partial\mu_{B}}\right)_{T}-\left(\frac{\partial
s}{\partial\mu_{B}}\right)_{T}\left(\frac{\partial\rho_{B}}{\partial
T}\right)_{\mu_{B}}\right]},$ (18) $c_{s}^{2}=\frac{s\left(\frac{\partial
s}{\partial\mu_{B}}\right)_{T}-\rho_{B}\left(\frac{\partial s}{\partial
T}\right)_{\mu_{B}}}{\mu_{B}\left[\left(\frac{\partial\rho_{B}}{\partial
T}\right)_{\mu_{B}}\left(\frac{\partial
s}{\partial\mu_{B}}\right)_{T}-\left(\frac{\partial s}{\partial
T}\right)_{\mu_{B}}\left(\frac{\partial\rho_{B}}{\partial\mu_{B}}\right)_{T}\right]},$
(19) $c_{T}^{2}=\frac{\rho_{B}}{T\left(\frac{\partial
s}{\partial\mu_{B}}\right)_{T}+\mu_{B}\left(\frac{\partial\rho_{B}}{\partial\mu_{B}}\right)_{T}},$
(20) $c_{\mu_{B}}^{2}=\frac{s}{T\left(\frac{\partial s}{\partial
T}\right)_{\mu_{B}}+\mu_{B}\left(\frac{\partial\rho_{B}}{\partial
T}\right)_{\mu_{B}}}.$ (21)

The corresponding sound speed formulae derived in terms of $T$ and $\rho_{B}$
are

$c^{2}_{s/\rho_{B}}=\frac{s^{2}+\rho_{B}^{2}\left[\left(\frac{\partial\mu_{B}}{\partial\rho_{B}}\right)_{T}\left(\frac{\partial
s}{\partial T}\right)_{\rho_{B}}-\left(\frac{\partial\mu_{B}}{\partial
T}\right)_{\rho_{B}}\left(\frac{\partial
s}{\partial\rho_{B}}\right)_{T}\right]+s\rho_{B}\left[\left(\frac{\partial\mu_{B}}{\partial
T}\right)_{\rho_{B}}-\left(\frac{\partial
s}{\partial\rho_{B}}\right)_{T}\right]}{\left(Ts+\mu_{B}\rho_{B}\right)\left(\frac{\partial
s}{\partial T}\right)_{\rho_{B}}},$ (22)
$c_{\rho_{B}}^{2}=\frac{s+\rho_{B}\left(\frac{\partial\mu_{B}}{\partial
T}\right)_{\rho_{B}}}{T\left(\frac{\partial s}{\partial
T}\right)_{\rho_{B}}},$ (23)
$c_{s}^{2}=\frac{\rho_{B}\left[\left(\frac{\partial s}{\partial
T}\right)_{\rho_{B}}\left(\frac{\partial\mu_{B}}{\partial\rho_{B}}\right)_{T}-\left(\frac{\partial
s}{\partial\rho_{B}}\right)_{T}\left(\frac{\partial\mu_{B}}{\partial
T}\right)_{\rho_{B}}\right]-s\left(\frac{\partial
s}{\partial\rho_{B}}\right)_{T}}{\mu_{B}\left(\frac{\partial s}{\partial
T}\right)_{\rho_{B}}},$ (24)
$c_{T}^{2}=\frac{\rho_{B}\left(\frac{\partial\mu_{B}}{\partial\rho_{B}}\right)_{T}}{T\left(\frac{\partial
s}{\partial\rho_{B}}\right)_{T}+\mu_{B}},$ (25)
$c_{\mu_{B}}^{2}=\frac{s\left(\frac{\partial\mu_{B}}{\partial\rho_{B}}\right)_{T}}{T\left[\left(\frac{\partial
s}{\partial
T}\right)_{\rho_{B}}\left(\frac{\partial\mu_{B}}{\partial\rho_{B}}\right)_{T}-\left(\frac{\partial\mu_{B}}{\partial
T}\right)_{\rho_{B}}\left(\frac{\partial
s}{\partial\rho_{B}}\right)_{T}\right]-\mu_{B}\left(\frac{\partial\mu_{B}}{\partial
T}\right)_{\rho_{B}}}.$ (26)

##  III Numerical results and discussions

In this section, we present the phase structure of symmetric nuclear matter
and the numerical results of the speed of sound under different definitions at
finite temperature and baryon density (chemical potential), and discuss the
relations between the speed of sound and nuclear LG phase transition.
Symmetric nuclear matter is considered in this study to describe the essential
behavior of speed of sound.

###  III.1 Nuclear liquid-gas phase transition and speed of sound at constant
$s/\rho_{B}$

Firstly, we demonstrate in Fig. 1 the phase structure of nuclear LG phase
transition and the isentropic curves for different values of $s/\rho_{B}$.
Fig. 1 (a) and (b) show the details of nuclear LG phase transition and the
corresponding spinodal structure. The black solid line is the first-order
phase transition line, and the blue dashed line is the boundary of spinodal
structure associated with the nuclear LG phase transition. Fig. 1 (b) shows
clearly that the two curves separate the phase diagram into the stable,
metastable and unstable phases. The spinodal phase decomposition plays a
dominant role in the experimental exploration of the first-order nuclear LG
transitionChomaz2004 ; Shao20202 . It has also inspired the anticipation to
identify the first-order chiral phase transition in high-energy heavy-ion
collisions through the spinodal phase separation Mishustin1999 ; Randrup2004 ;
Koch2005 ; Sasaki2007 ; Sasaki2008 ; Randrup2009 ; Steinheimer2012 ; Li2016 ;
Steinheimer2016 ; Steinheimer2017 .

Figure 1:  Phase diagram of nuclear liquid-gas transition and the isentropic
curves. (a) spinodal structure of LG phase transition the in $T-\mu_{B}$ plane
(b) phase structure of LG phase transition in $T-\rho_{B}$ plane (c)
isentropic trajectories in the full phase diagram, the black dashed line is
derived with $(\frac{\partial\mu_{B}}{\partial T})_{s/\rho_{B}}=0$.

The first-order transition line and the spinodal line meet at the CEP. The
isentropic curves in Fig. 1 indicate the evolutionary trajectories of an ideal
fluid under the adiabatic condition. Fig. 1 (b) shows that for smaller
$s/\rho_{B}$ the evolutionary trajectories pass through the stable, metastable
and unstable phases. It is expected that the important information on the
phase transition is carried by speed of sound. Fig. 1 (c) shows the nuclear LG
transition and isentropic trajectories in the full $T-\mu_{B}$ phase diagram.
In Fig. 1 (c), we also plot the boundary of $(\frac{\partial\mu_{B}}{\partial
T})_{s/\rho_{B}}=0$ (black dashed curve), which is closely related to the
behavior of sound speed at constant entropy density.

Figure 2:  Squared sound speed $c^{2}_{s/\rho_{B}}$ as functions of chemical
potential for several fixed temperatures.

We present the squared speed of sound $c^{2}_{s/\rho_{B}}$ as functions of
baryon chemical potential for different temperatures in Fig. 2 and the contour
map in Fig. 3. Fig. 2 shows that $c^{2}_{s/\rho_{B}}$ grows with the increase
of chemical potential for each temperature. It also indicates that, for
$\mu_{B}<1320\,$MeV, $c^{2}_{s/\rho_{B}}$ at a higher temperature is larger
than that at a lower temperature, because the pressure and energy density is
mainly driven by temperature for small chemical potential. For
$\mu_{B}>1320\,$MeV, the opposite happens, which is mainly attributed to the
density-driven effect with the decrease of dynamics mass of nucleon. For
$T=10\,$MeV, a small jump of $c^{2}_{s/\rho_{B}}$ appears on the boundary of
the first-order phase transition.

Figure 3:  Contour map of $c^{2}_{s/\rho_{B}}$ in the $T-\mu_{B}$ panel.

The contour map in Fig. 3 demonstrates the profiles of $c^{2}_{s/\rho_{B}}$ in
the full $T-\mu_{B}$ phase diagram with the nuclear LG phase transition. In
this figure we also plot the fitted chemical freeze-out line. The
parameterized formula of the freeze-out line is taken from Ref. Luo2017  with

$\mu_{B}(\sqrt{s_{NN}})=\frac{1.477}{1+0.343\sqrt{s_{NN}}}$ (27)

and

$T\left(\mu_{B}\right)=a-b\mu_{B}^{2}-c\mu_{B}^{4},$ (28)

where $a=0.158$ GeV, $b=0.14~{}\text{GeV}^{-1}$, and $c$ varies from 0.04 to
0.12. The chemical freeze-out curve in Fig. 3 is plotted for the intermediate
value of $c=0.08$. The value of $c^{2}_{s/\rho_{B}}$ along the freeze-out line
is given in Fig. 4. It shows $c^{2}_{s/\rho_{B}}$ on the freeze-out line
changes slightly for collision energy larger than 10 GeV. A small peak appears
at $\sqrt{s_{NN}}\simeq 6\,$GeV, and $c^{2}_{s/\rho_{B}}$ decreases with the
decline of collision energy.

Figure 4:  Values of $c^{2}_{s/\rho_{B}}$ along the chemical freeze-out line
as a function of collision energy.

We can also see that the chemical freeze-out line at low temperature is close
to the nuclear LG phase transition. We display the value of
$c^{2}_{s/\rho_{B}}$ on the boundary of the nuclear LG phase transition in
Fig. 5. The blue solid line is the value along the low-density side of the
first-order phase transition and the black solid line is the result on the
high-density side. The dash-dotted line is the “crossover” line, which is
derived with $\partial\sigma/\partial\mu_{B}$ (or $\partial
m^{*}_{N}/\partial\mu_{B}$) taking the extremum for a given temperature. This
is inspired by the description of quark chiral phase transition in quark
models. The aim of plotting the “crossover” line is to show that some
thermodynamic properties are sensitive to this line. Such a description is
also taken to discuss the net baryon number fluctuations induced by the
nuclear LG phase transition Shao20202 .

Figure 5:  Values of $c^{2}_{s/\rho_{B}}$ on the boundary of first-order
transition and the ”crossover” line.

Fig. 5 indicates that the square of sound speed is not zero but a small value
at the CEP. It is a feature of the mean field approximation. A similar
behavior exists for the quark chiral phase transition in the PNJL model with
the mean-field assumption he2022  . Fig. 5 also suggests that the value of
$c^{2}_{s/\rho_{B}}$ is small near the first-order transition region. Since
only the stable phase is considered in the above discussion, it cannot give a
complete demonstration of sound speed along the isentropic trajectories.
Therefore, we further present in Fig. 6 the contour map of
$c^{2}_{s/\rho_{B}}$ in the full $T-\rho_{B}$ panel, including all the stable,
metastable and unstable phases.

In Fig. 6, besides the first-order phase transition line (black solid line)
and the spinodal line (red dashed line), we also derive the boundary of
vanishing sound speed, which is plotted with the yellow dashed line. At each
point of this boundary, it fulfills the condition of $(\frac{\partial
p}{\partial\epsilon})_{s/\rho_{B}}=0$. Inside this boundary (grey area), the
value of $c_{s/\rho_{B}}^{2}=(\frac{\partial
p}{\partial\epsilon})_{s/\rho_{B}}$ is negative. The sound wave equation is
broken in this situation, and becomes a decay function. It means that a
disturbance cannot be propagated in this region.

Figure 6:  Contour map of $c^{2}_{s/\rho_{B}}$ in the $T-\rho_{B}$ panel.

From figures 2, 3 and 6, we can see that the square of sound speed is quite
larger than $1/3$ at high density (large chemical potential), about 0.8 being
approached at very large chemical potential. However, the value of
$c^{2}_{s/\rho_{B}}$ in quark matter approaches to $1/3$ at high density
he2022 . To understand the variation of speed of sound in nuclear matter, we
plot the 3D map of $p/\epsilon$ as a function of the temperature and chemical
potential in Fig. 7. It indicates that the behavior of $p/\epsilon$ is
responsible for the growing speed of sound from low to high chemical
potential.

Figure 7:  3D map of $p/\epsilon$ as a function of temperature and chemical
potential.

Figure 8:  3D map of $(\epsilon-3p)/\epsilon$ as a function of temperature and
chemical potential. The black solid line is the nuclear LG phase transition
line. The black dashed line is the “crossover” line Shao20202

The interaction measurement or trace anomaly is defined as $\epsilon-3p$,
which can effectively describe the interaction in a thermal system. We
demonstrate the value of $(\epsilon-3p)/\epsilon$ of symmetric nuclear matter
in Fig. 8. Comparing it with the phase transition line, we find the behavior
of $(\epsilon-3p)/\epsilon$ is related to the nuclear phase transition. The
behavior of $(\epsilon-3p)/\epsilon$ reflects the variation of nucleon mass or
the $\sigma$ meson field, i.e. the interaction between nucleons. Moreover, we
can see that $(\epsilon-3p)/\epsilon$ in nuclear matter does not approach zero
at large chemical potential, quietly different from that of quark matter.

The calculation in the Walecka model and PNJL model indicate that the speed of
sound in nuclear matter is quite larger than that in quark matter at high
density. Theoretically, in the Walecka model, the $\omega$ meson interaction
is proportional to the baryon density, leading to a steady increase in the
speed of sound, with the limiting value of 1 at $\rho_{B}\to\infty$. In the
(P)NJL model, the value of the sigma mean-field, and therefore of the
corresponding interaction, decreases with density. In view of this the (P)NJL
model looks like a gas of non-interacting relativistic particles at
$\rho_{B}\to\infty$. To the extent that the Walecka model can be thought to
describe nuclear matter and the (P)NJL model can give one some insight of
high-density quark matter. If a phase transition from nuclear matter to quark
matter takes place with growing density, the value of speed of sound will have
a peak at a certain density. It is interesting that recent neutron star
research suggests that the value of the speed of sound first increases sharply
with $\rho_{B}$, exceeding the conformal value of $1/3$, then falls again
below $1/3$, and finally approaches $1/3$ at infinity from below. It is
anticipated to give a further analysis on the speed of sound of neutron star
matter with different phase transition mechanism and observation constraints.

###  III.2 Speed of sound at constant $\rho_{B}$ or $s$

We plot the contour map of the squared speed of sound $c^{2}_{\rho_{B}}$ in
the $T-\rho_{B}$ panel in Fig. 9 and $T-\mu_{B}$ panel in Fig. 10. The
numerical results indicate that the value of $c^{2}_{\rho_{B}}$ lies in the
range of $0\sim 1$. For most temperatures, $c^{2}_{\rho_{B}}$ shows a
nonmonotonous behavior with the increase of density or chemical potential. In
particular, a peak-like structure exists at intermediate density (chemical
potential).

Figure 9:  Contour map of $c^{2}_{\rho_{B}}$ in the $T-\rho_{B}$ panel.

Figure 10:  Contour map of $c^{2}_{\rho_{B}}$ in the $T-\mu_{B}$ panel.

We present the squared speed of sound $c^{2}_{s}$ at constant entropy density
in Fig. 11 and Fig. 12. The two figures indicate that the contour of
$c^{2}_{s}$ has a relatively complicated structure. The value of $c^{2}_{s}$
are divided into two parts by the yellow dashed curve. Outside the curve, the
value of $c^{2}_{s}$ is positive and smaller than 1. Inside the curve (gray
area), $c^{2}_{s}$ is negative. The value of $c^{2}_{s}$ vanishes on the
boundary given by the yellow dashed curve. Such a feature is possibly general
for a first-order phase transition in a interacting system with temperature
and density dependent fermion mass, and a similar structure for the speed of
sound in quark matter is found in Ref. he2022 .

Figure 11:  Contour map of $c^{2}_{s}$ in the $T-\rho_{B}$ panel. The yellow
dashed line is boundary where the value of $c^{2}_{s}$ vanishes.
$c_{s}^{2}=({\partial p}/{\partial\epsilon})_{s}$ is negative inside the
yellow dashed line.

Figure 12:  Contour map of $c^{2}_{s}$ in the $T-\mu_{B}$ panel. The yellow
dashed line is boundary where the value of $c^{2}_{s}$ vanishes.
$c_{s}^{2}=({\partial p}/{\partial\epsilon})_{s}$ is negative inside the
yellow dashed line.

As a matter of fact, the boundary (yellow dashed line) in Fig. 11 and Fig. 12
is just the curve shown in Fig. 1 (c). This can be proven with the
thermodynamics formula

$\bigg{(}\frac{\partial\mu_{B}}{\partial
T}\bigg{)}_{s/\rho_{B}}=\frac{\mu_{B}}{T}\frac{(\frac{\partial
p}{\partial\epsilon})_{s}}{(\frac{\partial
p}{\partial\epsilon})_{\rho_{B}}}=\frac{\mu_{B}}{T}\frac{c^{2}_{s}}{c^{2}_{\rho_{B}}}.$
(29)

Using Eqs. (29) we can obtain the boundary of vanishing $c^{2}_{s}$ by taking
$(\frac{\partial\mu_{B}}{\partial T})_{s/\rho_{B}}=0$. This is the physical
condition used to derive the dashed curve in Fig. 1 (c). Moreover, when the
condition $(\frac{\partial\mu_{B}}{\partial T})_{s/\rho_{B}}<0$ is fulfilled,
one of the two physical quantities $c_{s}^{2}$ and $c_{\rho_{B}}^{2}$ takes a
negative value. There indeed exists such a region in the phase diagram, as
shown in Fig. 1 (c) (inside the black dashed curve). Since $c_{\rho_{B}}^{2}$
is always positive, $c_{s}^{2}=({\partial p}/{\partial\epsilon})_{s}$ is
negative in this situation. The blue areas in Fig. 11 and Fig. 12 indicate the
negative region.

Comparing the result with that of quark matter in the PNJL model (Fig. 1 in
Ref. he2022  ), we find there are three regions in which
$(\frac{\partial\mu_{B}}{\partial T})_{s/\rho_{B}}<0$ for quark matter,
including the lower left of the $T-\mu_{B}$ phase diagram and a small
neighboring area of chiral-crossover transformation near the CEP, as well as a
part of the region where the chiral symmetry of $u,d$ quarks is restored but
the Polyakov loop is still confining. Correspondingly, $c_{s}^{2}$ or
$c_{\rho_{B}}^{2}$ is negative in the these regions. One can refer to Ref.
he2022  for details.

###  III.3 Speed of sound at constant $T$ or $\mu_{B}$

In the following, we study the speed of sound derived at constant $T$ or
$\mu_{B}$. Recently, the density dependent $c^{2}_{T}$ has been discussed for
neutron star matter under the equilibrium of weak interaction. The data from
observations support a large value of $c^{2}_{T}$ (larger than 1/3) at a few
times nuclear saturation density. In Sorensen21  the authors also try to
estimate $c^{2}_{T}$ at chemical freeze-out of quark-gluon plasma using the
baryon number fluctuations of the beam energy scan experiments at RHIC. We
explore here the behavior of $c^{2}_{T}$ and $c^{2}_{\mu_{B}}$ in nuclear
matter in the full $T-\rho_{B}$ and $T-\mu_{B}$ phase diagram.

Figure 13:  Contour map of $c^{2}_{T}$ in the $T-\rho_{B}$ panel. The yellow
dashed curve is the spinodal line and the boundary of $c^{2}_{T}=0$.
$c^{2}_{T}=(\partial p/\partial\epsilon)_{T}$ is negative inside the boundary.

Fig. 13 and Fig. 14 demonstrate the contour maps of $c^{2}_{T}$ in the
$T-\rho_{B}$ and $T-\mu_{B}$ panel, respectively. The two figures show that
the value of $c^{2}_{T}$ increases with the rising density or chemical
potential in the stable phase, and the causality is always satisfied with
$c^{2}_{T}<1$. The value of $c^{2}_{T}$ is close to $c^{2}_{s/\rho_{B}}$ at
low temperature, since the isentropic trajectories are approximately parallel
to the density or chemical potential axis as indicated in Fig. 1. The behavior
of $c^{2}_{T}$ in nuclear matter at low temperature is to some extent similar
that of neutron star matter under $\beta$ equilibrium. Combined with the
behavior of $c^{2}_{T}$ in quark matter, if a hadron-quark phase transition
happens at high density, the value of $c^{2}_{T}$ will decrease in the mixed
phase or pure quark phase. The detailed study about the speed of sound in
neutron star matter is in progress.

Fig. 13 also demonstrates that there exists a region (gray area) where
$c^{2}_{T}=(\frac{\partial p}{\partial\epsilon})_{T}<0$. This region is just
the unstable phase derived with the thermodynamic stability conditions. On the
boundary of spinodal line including the CEP of the first-order transition, the
value of $c^{2}_{T}$ is zero. $c^{2}_{T}$ is positive in both the stable and
metastable phases.

Figure 14:  Contour map of $c^{2}_{T}$ in the $T-\mu_{B}$ panel.

We display the contour map of $c^{2}_{\mu_{B}}$ in Fig. 15 and Fig. 16.
Similar to Fig. 13, the contour map in Fig. 15 indicates that
$c^{2}_{\mu_{B}}=(\partial p/\partial\epsilon)_{\mu_{B}}<0$ inside the
spinodal line. $c^{2}_{\mu_{B}}$ is positive in the metastable and stable
phases. The contour maps in Fig. 15 and Fig. 16 also indicate that the value
of $c^{2}_{\mu_{B}}$ has a peak-like structure in the phase diagram.

Figure 15:  Contour map of $c^{2}_{\mu_{B}}$ in the $T-\rho_{B}$ panel. The
yellow dashed curve is the spinodal line and the boundary of
$c^{2}_{\mu_{B}}=0$. $c^{2}_{\mu_{B}}=(\partial p/\partial\epsilon)_{\mu_{B}}$
is negative inside the boundary.

Finally, we note that the numerical results at high temperature should be
treated with caution, since more hadronic degrees of freedom will appear at
high temperature. In this study, we show the numerical results in the full
phase diagram in order to demonstrate the whole changing trend of speed of
sound.

Figure 16:  Contour map of $c^{2}_{\mu_{B}}$ in the $T-\mu_{B}$ panel.

##  IV Summary

In this work, we studied the speed of sound in symmetric nuclear matter at
finite temperature and density (chemical potential) in the nonlinear Walecka
model. We derived the speed of sound under different definitions in the
complete phase diagram including the stable, metastable and unstable phases
associated with the first-order phase transition. We systematically discussed
the relations between the speed of sound and nuclear LG phase transition.

The numerical results indicate that the behavior of the speed of sound in the
phase diagram is closely related the phase structure of nuclear matter. From
the perspective of ideal fluid evolution, we focus on exploring the behavior
of adiabatic sound velocity at constant $s/\rho_{B}$. The calculation
indicates that the sound speed $c^{2}_{s/\rho_{B}}$ is nonzero at the CEP
under the mean field approximation, and the boundary of vanishing sound
velocity is derived. We also found that the value of $c^{2}_{s/\rho_{B}}$ is
quite larger than 1/3 at high density, different from that in quark matter
where $c^{2}_{s/\rho_{B}}$ approaches to 1/3 at high density and/or high
temperature.

We also explored the behaviors of sound speed under different physical
conditions, and analyzed the correlations with the nuclear LG phase
transition, as well as the relations between different definitions. The
calculation shows that it is natural in nuclear matter to have a sound speed
larger than $\sqrt{1/3}$ at a few times nuclear saturation density. A further
study on the speed of sound in neutron star matter with a hadron-quark phase
transition and observation constraints will be performed in the future.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
under Grant No. 11875213.

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# Re-examining the stability of rotating horizonless black shells mimicking
Kerr black holes

Ulf Danielsson  Suvendu Giri

###### Abstract

In arXiv:1705.10172 a string theory inspired alternative to gravitational
collapse was proposed, consisting of a bubble of AdS space made up of
ingredients from string theory. These ultra compact objects are $9/8$ times
the size of the corresponding Schwarzschild black hole, but being within the
photosphere are almost indistinguishable from them. Slowly rotating
counterparts of these black shells were constructed in arXiv:1712.00511, which
closely mimic a Kerr black hole, but have a quadrupole moment that differs
from Kerr. Recently, arXiv:2109.09814 studied the dynamical stability of the
stationary black shells against radial perturbations and accretion of matter,
and examined a two parameter family of fluxes required for stability. In this
paper, we re-examine the rotating black shells with particular attention to
the constraints imposed by this dynamical analysis for non-rotating shells.
Extrapolating these results to rotating shells, we find that they can indeed
support themselves at a critical point in the gravitational potential.
Additionally, requiring that they settle back to their new Buchdahl radius
after accreting matter, uniquely fixes the fluxes required for dynamical
stability. The flux parameters turn out to have an extremely simple form, and
fulfil one of the constraints for perturbative radial stability while exactly
saturating the other. The preferred quadrupole moment that we find, given some
physical assumptions, is $7\%$ less than Kerr.

##  1 Introduction

Black holes continue to challenge our uv understanding of physics. On one
hand, the existence of an event horizon gives rise to the information paradox
[1] (see [2] for a review), while on the other hand, the spacetime singularity
at the center of the black hole has evaded a satisfactory understanding in our
best developed theories of uv physics. The enormous entropy of black holes
seems to have an origin at the horizon [3], but being featureless à la the no-
hair theorem [4], we do not have a complete understanding of its thermodynamic
origin. While string theory has had success in accounting for the microscopic
entropy of supersymmetric black holes [5], it is less clear what happens in
the case of the non-supersymmetric black holes of the real world.

Attempts to understand how information can be preserved even in the presence
of a genuine horizon, such as the idea of black hole complementarity proposed
in [6], has run into problems. In fact, in [7], it was argued that full
compatibility with quantum mechanics requires the horizon to be replaced by a
firewall. For other discussions about possible new physics see e.g.[8, 9, 10,
11] and the review in [12]. This is the motivation behind proposals replacing
black holes with other objects that lack a horizon, and in this way evade the
information paradox.

Usually, the surface of such an object hides very close to where the horizon
would have been, making it very difficult to distinguish them from ordinary
black holes. In [13], however, we proposed a model where the space time is
unstable against the formation of bubbles of AdS, which then replace
Schwarzschild black holes. These bubbles have a radius equal to $9R_{s}/8$,
commonly known as the Buchdahl radius, which is macroscopically larger than
the Schwarzschild radius. In our paper we showed how the necessary matter
components to build such bubbles arise in a natural way from string theory.

In [14] we generalized our model to include slow rotation, and proposed that
Kerr black holes are replaced by rotating black shells. Since the black shells
lack horizons the exterior metric need not be the one of Kerr. In fact, by
considering metrics more general than Kerr we found a quadrupole moment that
differed from the one of Kerr.111The improved analysis of the present paper
picks a physically better motivated case, and gives a different prediction
than the one of [14].

An important part of our model is a source term that transfers energy between
the different components, preserving the total energy. Without such a term the
black shell cannot be stable. In [13] we argued for the existence of a
suitable source term in the quasi-static case. In [15] this analysis was
reconsidered in the case of physical accretion of matter on to the black
shell. Analytical constraints on the source term were obtained, and checked
against numerical calculations. It was shown that the source term can indeed
be tuned so that the black shell grows in radius and stabilizes at the new
Buchdahl radius when matter is falling on to it.

In this paper we revisit our analysis of the rotating black shell confronting
the constraints on the dynamics obtained in [15]. It turns out that the
combined analysis uniquely fixes the parameters determining the source.
Furthermore, despite the complexity of the expressions involved in describing
the rotating case, the final result is remarkably simple. The preferred value
for the quadrupole moment that we find in our analysis is, given some physical
assumptions, given by $-0.926a^{2}M$, and thus about $7\%$ less in magnitude
than the one of Kerr.

The outline of the paper is as follows. In section 2 we review the non-
rotating black shell as well as revisit the rotating one. In section 3 we
discuss the role of the source term making use of the results in [15].
Finally, we conclude in section 4 with an outlook towards future work.

##  2 Black holes as black shells

###  2.1 General principles and the non-rotating case

The starting point is a metastable Minkowski spacetime that can decay non-
perturbatively through tunneling, into a stable AdS vacuum. This takes place
through the nucleation of a bubble containing the true vacuum, via a Coleman-
de Luccia gravitational instanton [16]. In the vacuum, the probability of such
a nucleation event is exponentially suppressed. However, when matter
collapses, threatening to form a black hole, such a phase transition can be
triggered so that a bubble forms and captures the in-falling matter, which can
attach itself to the bubble wall and turn into a gas of open strings. This
increase in entropy can make the process inevitable rather than unlikely.

In [13] we showed how all required matter components can be obtained from
string theory. The bubble wall itself is in the form of a brane, for instance
a 5-brane wrapping an internal 3-cycle, which carries magnetic flux.
Alternatively, this flux can be thought of in terms of dissolved branes of a
lower dimensionality, for instance 3-branes wrapping internal 3-cycles.
Massless vibrations in the flux fields within the brane give rise to radiation
on top of the shell. It is into this radiation that the in-falling matter
converts when the bubble nucleates. The number of massless degrees of freedom
is furthermore governed by the rank of the magnetic fluxes, i.e., number of
dissolved branes. An important part of our model is that the radiation gas
will be heated to the local Unruh temperature when the system is in
equilibrium.

With these components, we were able to solve the Israel-Lanczos-Sen junction
conditions [17, 18, 19] and show that the black shell has a critical point at
a radius equal to $9R_{s}/8$, commonly known as the Buchdahl radius (where
$R_{s}\coloneqq 2M$ is the Schwarzschild radius). The junction conditions
relate the energy density on the shell with the geometry in the bulk. The
first of these ensures that the induced metric $(h_{ij})$ is well defined,
while the second gives the energy momentum tensor on the shell $(S_{ij})$ in
terms of its extrinsic curvature of the shell $(K_{ij})$ embedded in the
bulk222Throughout this paper Greek indices $\mu,\nu$ refer to bulk
coordinates, while Latin indices $i,j$ refer to those on the shell.

$\Delta h_{ij}=0,\quad S_{ij}=-\frac{1}{8\pi}\left(\Delta K_{ij}-\Delta
Kh_{ij}\right),$ (2.1)

where $\Delta$ denotes the difference in the corresponding quantities across
the shell, and $K\coloneqq K_{ij}h^{ij}$ is the trace of the extrinsic
curvature. These ensure that the composite spacetime which includes the
inside, the outside and the surface of the bubble together solves Einstein’s
equations. For a bubble of radius $r=R$ with AdS space (with radius $1/k$) in
its interior, and Schwarschild metric (with mass $M$) outside, the junction
conditions give

$\begin{split}S^{t}{}_{t}&=-\rho=\frac{1}{4\pi
R}\left(\sqrt{1-\frac{2M}{R}}-\sqrt{1+k^{2}R^{2}}\right),\\\
S^{\theta}{}_{\theta}=S^{\phi}{}_{\phi}&=p=\frac{1}{8\pi
R}\left(\frac{1-M/R}{\sqrt{1-2M/R}}-\frac{1+2k^{2}R^{2}}{\sqrt{1+k^{2}R^{2}}}\right),\end{split}$
(2.2)

where $\rho$ and $p$ denote the density and pressure respectively, of the
perfect fluid constituting the shell. The metric induced on the shell is

$ds^{2}_{\textrm{ind}}=-\left(1-\frac{2M}{R}\right)\,dt^{2}+R^{2}\,d\theta^{2}+R^{2}\,\sin^{2}\theta\,d\phi^{2}.$
(2.3)

Based on physical arguments presented in [13], we demand that the stress
tensor can be split into the constituents above, namely a brane, gas and stiff
matter with equations of state $p_{\tau}=-\rho_{\tau},p_{g}=\rho_{g}/2$, and
$p_{s}=\rho_{s}$ respectively. For the shell at the critical radius
$R=9R_{s}/8$, this gives the individual components as:

$\begin{split}\left(S_{\textrm{brane}}\right)^{i}\,_{j}&=\left(-\frac{k}{4\pi}+\frac{2}{27M\pi}-\frac{1}{81kM^{2}\pi}\right)\delta^{i}\,_{j},\\\
\left(S_{\textrm{gas}}\right)^{i}\,_{j}&=\frac{1}{18M\pi}u^{i}\,u_{j}+\frac{1}{54M\pi}\delta^{i}\,_{j},\\\
\left(S_{\textrm{stiff}}\right)^{i}\,_{j}&=\frac{2}{81kM^{2}\pi}u^{i}\,u_{j}+\frac{1}{81kM^{2}\pi}\delta^{i}\,_{j},\end{split}$
(2.4)

where $u^{i}$ is the normalized velocity vector of the stationary shell
$u^{i}=3\left(1,0,0\right)$. Note that the metric used is the induced metric
of eq (2.3) evaluated at $R=9R_{s}/8$. However, without any other ingredients
in the model, this critical point turns out to be unstable and with the
slightest perturbation, the shell will either start expanding or collapse into
a black hole. We proposed in our paper a coupling between the brane tension
and the gas such that energy can be transferred between the two, changing the
tension and the density of the gas in response to a perturbation.

This is accomplished through a source term in the continuity equation for the
components that preserves the total energy. It was argued that a natural
quantity that could govern such a source is the local Unruh temperature. Any
deviation in the temperature of the system away from the Unruh temperature
should initiate an energy transfer aimed at equilibrating the temperature. It
was furthermore shown how such a process achieved stability in the case of
quasi-static perturbations. In [13] we studied the quasi-static case and
argued that one can find such a source term.

In [15], the stability of a non-rotating black shell was carefully studied,
and it was shown that the quasi-static approach in [13] was not sufficient.
The reason is corrections to the Unruh temperature due to the motion and
acceleration of the shell. Any physical perturbation, such as the impact of
matter, will induce motion that is not quasi-static in any limit. In [15, eq
(86)], the source term was therefore generalized to

$j\equiv 3\rho_{g}\left(\alpha\ \dot{a}_{{}_{R}}/a_{{}_{R}}+\beta\
F/2\right),$ (2.5)

where $a_{{}_{R}}$ is the Unruh temperature, and the dot denotes the time
derivative. We have defined $F=\dot{A}/A$, where $A$ is the area of the shell.
$\alpha$ and $\beta$ are constants that parametrize respectively, the response
of the shell to a change in its Unruh temperature, and the response to a
change in its area under compression or relaxation. The quasi-static case
corresponds to $\alpha=\beta=1$. An analysis of linear perturbations
proportional to $e^{i\omega\tau}$ with this source, led to a frequency given
by [15, eq (B.18)]

$\omega\approx\frac{128\pi\zeta_{0}}{27(4-9\alpha)m}\big{(}i\pm\sqrt{-1+9(4-9\alpha)(6\alpha-\beta-2)/(16\pi\zeta_{0})^{2}}\big{)},$
(2.6)

where we have introduced a non-zero viscosity given by $\zeta_{0}$, and $m$ is
the mass of the bubble. We have taken the limit $m\gg 1/k$. From this one
finds that stability with damping requires $\alpha<4/9$ and $6\alpha-\beta>2$.
This is not all that is needed. Studying an impulsive accretion event, it was
found that one must impose (in the large $m$ limit)

$\alpha=\beta+2/3.$ (2.7)

This is to make sure that the shell, after accreting the in-falling matter,
settles at the new Buchdahl radius. These constraints will play an important
role for the rotating black shell. We will come back to this after reviewing
the rotating case.

###  2.2 Revisiting the rotating black shell

In [14] we studied a slowly rotating black shell. To do this we considered
solutions to the Einstein equations generalizing the Kerr-metric outside of
the rotating shell, and generalizing the AdS-metric inside of the rotating
shell. On the outside we add to the Kerr metric a superposition of a metric
similar to the one of Novikov and Manko [20] (parametrized by $q$) and the one
of Hartle and Thorne [21] (parametrized by $p$), given by:

$\begin{split}\mathrm{d}s^{2}=&\,-g_{tt}\,\left(1+a^{2}\left(q\,\lambda+\,p\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\right)\right)\,\mathrm{d}t^{2}+g_{rr}\,\left(1+a^{2}\,\left(q\,\nu-\,p\,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\right)\right)\,\mathrm{d}r^{2}\\\
&+g_{\theta\theta}\,\left(1+a^{2}\,\left(q\,\nu+p\,\psi\right)\right)\,\mathrm{d}\theta^{2}+g_{\phi\phi}\,\left(1-a^{2}\,\left(q\,\lambda-p\,\psi\right)\right)\,\mathrm{d}\phi^{2}+2g_{t\phi}\,\mathrm{d}t\,\mathrm{d}\phi\\\
&+\Delta g_{\mu\nu}dx^{\mu}dx^{\nu},\end{split}$ (2.8)

where $g_{\mu\nu}$ is the Kerr metric with mass $M$ and spin $a$, while the
perturbing functions
$\lambda,\nu,{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$
and $\psi$ are given by:

$\begin{split}\lambda&=\frac{1}{M^{2}}\left[(4M^{2}-4Mr+2r^{2})\mathscr{L}-4M(r-M)(3\cos^{2}\theta-1)-2\mathscr{L}(2M^{2}-6Mr+3r^{2})\cos^{2}\theta\right],\\\
\nu&=\frac{1}{M^{2}}\left[2\mathscr{L}\cos^{2}\theta\left(6M^{2}-10Mr+3r^{2}\right)-2\mathscr{L}\left(6M^{2}-6Mr+r^{2}\right)+4M\cos^{2}\theta(3r-7M)+4M(5M-r)\right],\\\
{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}&=-\frac{5}{8\,M^{2}}\frac{\left(3\cos^{2}\theta-1\right)}{r\left(r-2M\right)}\left[2M\left(r-M\right)\left(2M^{2}+6Mr-3r^{2}\right)-3r^{2}\left(r-2M\right)^{2}\mathscr{L}\right],\\\
\psi&=\frac{5}{8\,M^{2}}\frac{\left(3\cos^{2}\theta-1\right)}{r}\left[2M\left(2M^{2}-3Mr-3r^{2}\right)-3r\left(r^{2}-2M^{2}\right)\mathscr{L}\right].\end{split}$
(2.9)

where $\mathscr{L}\coloneqq\log\left(1-2M/r\right)$. Here we have also added
an extra contribution, which solves Einstein’s equations, to make sure that
the mass of the system remains $M$, given by

$\Delta
g_{\mu\nu}dx^{\mu}dx^{\nu}=-\frac{16}{3}a^{2}Mq\left(\frac{r-M}{r^{2}}\,dt^{2}+\frac{r-M}{\left(r-2M\right)^{2}}\,dr^{2}+r\,d\theta^{2}+r\sin^{2}\theta\,d\phi^{2}\right).$
(2.10)

To have full control over the metric that we use, we have calculated its
Geroch-Hansen multiple moments [22, 23]. The above metric has333Without the
added piece, $\mathscr{M}_{0}$ would have had an extra contribution
$(8/3)a^{2}Mq$.

$\displaystyle\mathscr{M}_{0}$ $\displaystyle=M,$
$\displaystyle\mathscr{J}_{0}$ $\displaystyle=0,$ (2.11)
$\displaystyle\mathscr{M}_{1}$ $\displaystyle=0,$
$\displaystyle\mathscr{J}_{1}$ $\displaystyle=aM,$ (2.12)
$\displaystyle\mathscr{M}_{2}$
$\displaystyle=-a^{2}M-\frac{2}{15}a^{2}M^{3}\left(16q-15p\right),$
$\displaystyle\mathscr{J}_{2}$ $\displaystyle=0,$ (2.13)

with higher moments being of the form
$\mathscr{M}_{2k}=\left(-1\right)^{k}Ma^{2k}$, and
$\mathscr{J}_{2k+1}=\left(-1\right)^{k}Ma^{2k+1}$ (which come from the Kerr
metric) plus terms containing $p$ and $q$ that appear with $a^{n}$ where
$n>2$. Thus we see how $q$ and $p$ give rise to a deviation in the quadrupole
moment from the one of Kerr. Our goal is to determine these parameters.

Inside the bubble, we construct a generalization of the AdS metric of the
following form

$\begin{split}\mathrm{d}s^{2}=&\,-\left(1+k^{2}\,r^{2}\right)\,\left(1+a^{2}\left(c_{1}\mu_{1}+c_{2}\sigma_{1}\right)\right)\,\mathrm{d}t^{2}+\left(1+k^{2}\,r^{2}\right)^{-1}\,\left(1+a^{2}\left(c_{1}\mu_{2}+c_{2}\sigma_{2}\right)\right)\,\mathrm{d}r^{2}\\\
&+r^{2}\,\left(1+a^{2}\left(c_{1}\mu_{3}+c_{2}\sigma_{3}\right)\right)\,\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\,\left(1+a^{2}\left(c_{1}\mu_{4}+c_{2}\sigma_{4}\right)\right)\,\mathrm{d}\phi^{2},\end{split}$
(2.14)

where the functions $\mu_{i},\sigma_{i}$ are given by

$\begin{split}\mu_{1}&=f_{1}\,\mathrm{P}_{2}\left(\cos\theta\right),\,\mu_{2}=-f_{1}\,\mathrm{P}_{2}\left(\cos\theta\right),\,\mu_{3}=f_{3}\,\mathrm{P}_{2}\left(\cos\theta\right),\,\mu_{4}=f_{3}\,\mathrm{P}_{2}\left(\cos\theta\right),\\\
\sigma_{1}&=g_{1}\left(3\cos^{2}\theta-1\right),\,\sigma_{2}=\frac{r}{6}\,g_{3}^{\prime}\left(3\cos^{2}\theta-1\right),\,\sigma_{3}=g_{3}\cos^{2}\theta,\,\sigma_{4}=g_{3}\left(2\cos^{2}\theta-1\right),\end{split}$
(2.15)

with $f_{1},f_{3},g_{1}$ and $g_{3}$ further given by

$\begin{split}f_{1}&=\frac{5k^{2}r^{2}+3}{4k^{6}r^{4}+4k^{4}r^{2}}-\frac{3\left(k^{2}r^{2}+1\right)\textrm{arctan}(kr)}{4k^{5}r^{3}},\quad
f_{3}=\frac{4k^{2}r^{2}-3}{4k^{4}r^{2}}-\frac{3\left(k^{2}r^{2}-1\right)\textrm{arctan}(kr)}{4k^{5}r^{3}},\\\
g_{1}&=-\frac{2k^{4}r^{4}-2k^{2}r^{2}-3}{6k^{6}r^{4}+6k^{4}r^{2}}-\frac{\textrm{arctan}(kr)}{2k^{5}r^{3}},\quad
g_{3}=\frac{k^{2}r^{2}-3}{3k^{4}r^{2}}+\frac{\textrm{arctan}(kr)}{k^{5}r^{3}}.\end{split}$
(2.16)

We then move on to solve the first junction condition to order $a^{2}$ in the
spin for a shell that is located at coordinate radius $r_{\textrm{out}}$ from
the outside, and $r_{\textrm{in}}$ from the inside

$r_{\textrm{out}}=R+Ra^{2}\left(m_{1}+m_{2}\cos 2\theta\right),\quad
r_{\textrm{in}}=R+Ra^{2}\left(n_{1}+n_{2}\cos 2\theta\right).$ (2.17)

This relates parameters of the metric inside the bubble to those on the
outside outside, in terms of those of the shell. In particular, it gives
$c_{1}$ and $c_{2}$ in terms of $p,q$ and the difference
$\left(m_{1}-n_{1}\right)$:

$\begin{split}c_{1}&=\frac{k^{5/2}M}{96}\frac{26624-405M^{2}(1276-1323\ln
3)p+432M^{2}(28+621\ln
3)q+186624M^{2}\left(m_{1}-n_{1}\right)}{9kM(27kM^{2}-4)-(81kM^{2}-16)\arctan(9kM/4)},\\\
c_{2}&=\frac{4k^{5/2}M\left(-68+729M^{2}(4-5\ln
3)q\right)}{9kM(-16+27kM^{2})+64\arctan(9kM/4)}.\end{split}$ (2.18)

At this point, there is a large amount of mathematical freedom in the system,
and to proceed we need to impose physical constraints. Our goal is to to
uniquely fix the exterior spacetime, parametrized by $p$ and $q$, and thus
predict the value of, e.g., the quadrupole moment of the rotating shell. To do
this we need to find a way to fix $c_{1}$ and $c_{2}$. There is a simple
possibility that singles itself out: to simply set $c_{1}=c_{2}=0$ so that the
interior metric is pure undeformed AdS. The argument for such a choice would
be that the piece of fresh space time that is nucleated inside of the bubble,
should not depend on what sits outside of it. The junction conditions are then
fully taken care of by the matter components on top of the shell together with
the metric on the outside. One should note that from the point of view of the
inside AdS metric, the outside asymptotic universe will be viewed as rotating,
with the AdS glued to the frame dragged spacetime just outside of the shell.
We find that $c_{2}=0$ fixes $q$, while $c_{1}=0$ then gives a relation
between $p$ and $m_{1}-n_{1}$.444In [14] only $c_{1}$ was, implicitly, put to
zero while the freedom in $c_{2}$ was not observed. The case studied there had
a $c_{2}$ different from zero.

To get further constraints, we then use the second junction condition to
compute the energy momentum tensor $S_{ij}$ on the shell. Following the
physical motivation behind the construction of the black shells, this stress
tensor should be made of various components. As long as each component is a
perfect fluid (with corresponding equations of state), and they share the same
velocity vector $u^{i}$, the total stress tensor will also be of the form of a
perfect fluid:

$S^{ij}=\left(\rho_{\textrm{total}}+p_{\textrm{total}}\right)u^{i}u^{j}+p_{\textrm{total}}\,g^{ij}.$
(2.19)

This places a further restriction on the constants and determines
$p,q,(m_{1}-n_{1})$ uniquely. The results obtained are exact to all orders in
$k$, but too large to write down. So we present only the leading order result
in $1/\left(kM\right)$ for some of the quantities below.

$\begin{split}p&=\frac{64\left(1557\ln 3-1028\right)}{10935(5\ln
3-4)D}\frac{1}{M^{2}}+\frac{32768\left(135\ln
3-172\right)}{32805D^{2}}\frac{1}{kM^{3}}+\frac{1}{M^{2}}\mathscr{O}\left(k^{-2}M^{-2}\right),\\\
q&=-\frac{68}{729(5\ln 3-4)}\frac{1}{M^{2}},\\\
\left(m_{1}-n_{1}\right)&=\frac{2\left(316144-726744\ln 3+410103(\ln
3)^{2}\right)}{6561(5\ln 3-4)D}\frac{1}{M^{2}}\\\ &-\frac{128(135\ln
3-172)(1323\ln
3-1276)}{59049D^{2}}\frac{1}{kM^{3}}+\frac{1}{M^{2}}\mathscr{O}\left(k^{-2}M^{-2}\right),\end{split}$
(2.20)

where we have defined $D=196-261\ln 3$, which will reappear throughout. The
normalized velocity vector $u^{i}=\left(3+\gamma a^{2},0,a\beta\right)$ is:

$\begin{split}\gamma&=-\frac{4D\left(729M^{2}n_{1}+64\right)+128(135\ln
3-172)\sin^{2}\theta}{243M^{2}D}\frac{1}{M^{2}}\\\
&\quad+\frac{256\left(19344-76136\ln 3+63153(\ln
3)^{2}\right)}{729D^{2}}\frac{1}{kM^{3}}+\frac{1}{M^{2}}\mathscr{O}\left(k^{-2}M^{-2}\right),\\\
\beta&=-\frac{64\left(8+27Mk\right)}{243M^{2}\left(4+9Mk\right)}.\end{split}$
(2.21)

The values of $p$ and $q$ numerically evaluate to

$p=-\frac{0.295}{M^{2}}-\frac{0.00287}{kM^{3}}+\frac{1}{M^{2}}\mathscr{O}\left(k^{-2}M^{-2}\right),\quad
q=-\frac{0.625}{M^{2}},$ (2.22)

which determines the quadrupole moment, via eq (2.13), to be

$\mathscr{M}_{2}=-a^{2}M-\frac{2}{15}a^{2}M^{3}\left(16q-15p\right)=-0.926a^{2}M-\frac{0.00575a^{2}}{k}+a^{2}M\mathscr{O}\left(k^{-2}M^{-2}\right)$
(2.23)

Thus, for this physically natural scenario, the expected quadrupole moment of
the black shell is about $7\%$ less than the one of a Kerr black hole.555This
is different from the result in [14], which corresponded to another possible
solution.

So far, all results are exact, and we have not made any expansion in $k$. To
verify that the shell is made of three components with equations of state of
tension, gas, and stiff matter we need to expand in large $k$. This allows us
to split the total $S_{ij}$ into its constituents:

$\begin{split}\left(S_{\textrm{brane}}\right)^{i}\,_{j}&=\left(-\frac{k}{4\pi}+\frac{2}{27M\pi}-\frac{1}{81kM^{2}\pi}-a^{2}\mathscr{Z}_{1}\right)\delta^{i}\,_{j},\\\
\left(S_{\textrm{gas}}\right)^{i}\,_{j}&=\left(\frac{1}{18M\pi}+a^{2}\,\mathscr{Z}_{2}\right)u^{i}u_{j}+\left(\frac{1}{54M\pi}+\frac{a^{2}}{3}\,\mathscr{Z}_{2}\right)\delta^{i}\,_{j},\\\
\left(S_{\textrm{stiff}}\right)^{i}\,_{j}&=\left(\frac{2}{81kM^{2}\pi}+a^{2}\mathscr{Z}_{3}\right)u^{i}\,u_{j}+\left(\frac{1}{81kM^{2}\pi}+\frac{a^{2}}{2}\mathscr{Z}_{3}\right)\delta^{i}\,_{j},\end{split}$
(2.24)

where the factors $\mathscr{Z}_{i}$ are:

$\begin{split}\mathscr{Z}_{1}&=-\frac{A_{1}+16A_{2}\cos 2\theta}{6561\pi
D}\frac{1}{M^{3}}+\frac{A_{3}+96A_{4}\cos 2\theta}{177147\pi
D}\frac{1}{kM^{4}}+\frac{1}{M^{3}}\mathscr{O}\left(k^{-2}M^{-2}\right),\\\
\mathscr{Z}_{2}&=\frac{B_{1}D+32B_{2}\cos 2\theta}{4374\pi
D}\frac{1}{M^{3}},\\\ \mathscr{Z}_{3}&=-\frac{4C_{1}D+192C_{2}\cos
2\theta}{59049\pi
D^{2}}\frac{1}{kM^{4}}+\frac{1}{M^{3}}\mathscr{O}\left(k^{-2}M^{-2}\right),\end{split}$
(2.25)

with

$\begin{split}A_{1}&=36032-54000\ln 3+1458n_{1}DM^{2},\\\ A_{2}&=292-765\ln
3,\\\ A_{3}&=4374n_{1}D^{2}M^{2}+256(133648+9\ln 3(17361\ln 3-31336)),\\\
A_{4}&=879536-2046648\ln 3+1248291(\ln 3)^{2},\\\
B_{1}&=352+2187n_{1}M^{2},\\\ B_{2}&=52+495\ln 3,\\\ C_{1}&=256(99\ln
3-92)+729n_{1}DM^{2},\\\ C_{2}&=299440-625080\ln 3+336771(\ln
3)^{2}.\end{split}$ (2.26)

In the next section we will examine the energy momentum tensor and its
properties more closely.

##  3 How the rotating shell supports itself

One can verify that the induced energy momentum tensor is covariantly
conserved with respect to the induced metric, i.e., $\nabla_{i}S^{i}{}_{j}=0$.
This is not automatically guaranteed. In general, the covariant derivative of
the extrinsic curvature is given by a projection of the bulk Einstein tensor
on the shell, using the Gauss-Codazzi equations [24, eq (3.42)]

$\nabla_{i}K^{i}{}_{j}-\partial_{j}K=G_{\mu\nu}e^{\mu}{}_{j}n^{\nu},$ (3.1)

where $G_{\mu\nu}$ is the four dimensional Einstein tensor in the bulk,
$e^{\mu}{}_{j}$ is the tangent vector, and $n^{\nu}$ is the vector normal to
the shell. In our case this vanishes trivially on the outside since
$G_{\mu\nu}=0$. It also vanishes on the inside due to the projection.
Vanishing of the covariant derivative implies that the combined system can
sustain itself such that the total pressure balances the gravitational (and
centrifugal) forces without the need for any external force acting from the
bulk. However, the covariant derivative of each component of the stress tensor
is non-vanishing and they cancel against each other non-trivially. Up to
$\mathscr{O}\left(k^{-2}M^{-2}\right)$, they read:

$\begin{split}\nabla_{i}\left(S_{\textrm{brane}}\right)^{i}{}_{\theta}&=\frac{32(292-765\ln
3)a^{2}\sin 2\theta}{6561\pi DM^{3}}-\frac{64(879536+3\ln 3(416097\ln
3-682216))a^{2}\sin 2\theta}{59049\pi D^{2}kM^{4}},\\\
\nabla_{i}\left(S_{\textrm{gas}}\right)^{i}{}_{\theta}&=-\frac{32(292-765\ln
3)a^{2}\sin 2\theta}{6561\pi DM^{3}}+\frac{128(19344+\ln 3(63153\ln
3-76136))a^{2}\sin 2\theta}{19683\pi D^{2}kM^{4}},\\\
\nabla_{i}\left(S_{\textrm{stiff}}\right)^{i}{}_{\theta}&=\frac{64(763472+9\ln
3(96597\ln 3-176648))a^{2}\sin 2\theta}{59049\pi D^{2}kM^{4}},\end{split}$
(3.2)

where $D=196-261\ln 3$, as defined below eq (2.20), and all other components
vanish. As promised, one can note how the nonzero terms add up to zero when
summed. The physical interpretation of this is that the individual terms exert
mutual forces on each other such that the system is stable. From where do
these source terms come? It is natural to assume that they have the same
origin as the source terms needed to achieve stability in the case of the non-
rotating black shell. More precisely, we argue that eq (2.5) should be
generalized to

$j_{i}\coloneqq 3\rho_{g}\left(\alpha\
\frac{\nabla_{i}a_{{}_{R}}}{a_{{}_{R}}}+\beta\ \frac{F_{i}}{2}\right).$ (3.3)

In [15], $F$ was determined in terms of the area. A convenient alternative
would be to express it in terms of the three dimensional Ricci scalar. In the
non-rotating case this is simply given by

$R^{(3)}=\frac{2+2\dot{R}^{2}+4R\ddot{R}}{R^{2}}\approx\frac{2}{R^{2}}.$ (3.4)

Hence, we can, to lowest order in any perturbation, use

$F_{i}=-\frac{\nabla_{i}R^{(3)}}{R^{(3)}}.$ (3.5)

We can now evaluate the source term, as given above in eq (3.3) and demand
that it equals the flux from the non-zero covariant derivative of the
brane/gas in eq (3.2). At leading order in $\left(1/kM\right)$ this gives,

$\begin{split}j_{\theta}&=-\frac{1}{9M\pi}\left(\alpha\frac{40(100+243\ln
3)}{243D}+\beta\frac{2560(21\ln 3-4)}{243D}\right)\frac{a^{2}\sin
2\theta}{M^{2}}\\\ &=\frac{40\alpha(100+243\ln 3)+1280\beta(21\ln
3-4)}{2187\pi D}\frac{a^{2}\sin 2\theta}{M^{3}}\\\
&\overset{!}{=}\frac{32A_{2}}{6561\pi D}\frac{a^{2}\sin
2\theta}{M^{3}},\end{split}$ (3.6)

where $A_{2}=292-765\ln 3$, and $D=196-261\ln 3$, as defined in sec 2. This
gives the following constraint on $\alpha$ and $\beta$:

$4(292-375\alpha+480\beta)-45(68+81\alpha+224\beta)\ln 3\overset{!}{=}0.$
(3.7)

If this relation is satisfied, then the shell can support itself at a critical
point. If we now use the Buchdahl constraint $\alpha=\beta+2/3$ obtained in
[15], we can uniquely solve for $\alpha$ and $\beta$. The result is remarkably
simple:

$\begin{split}\alpha&=\frac{4}{15},\\\ \beta&=-\frac{2}{5}.\end{split}$ (3.8)

What is even more remarkable is that this satisfies the constraint
$\alpha<4/9$ and exactly saturates the constraint $6\alpha-\beta>2$. So a
rotating bubble at a critical point is just barely compatible with the
stability of a non-rotating bubble that accretes such that it always sits at
the Buchdahl radius. This is a highly non-trivial result. It is worth noting
that this solution for $\alpha,\beta$ does not depend on the value of
$\left(m_{1}-n_{1}\right)$ that was chosen in order to pick the physically
most interesting solution with $c_{1}=c_{2}=0$. It appears to be a universal
feature of rotating black shells that are made of the same ingredients
allowing the total energy momentum tensor to be split in same way as in eq
(2.24).

To summarize, the source term compatible with stability of the non-rotating
case, as well as the existence of a critical point in the case of a rotating
black shell is given by

$j_{i}\equiv\frac{\rho_{g}}{5}\left(4\frac{\nabla_{i}a_{{}_{R}}}{a_{{}_{R}}}+3\frac{\nabla_{i}R^{(3)}}{R^{(3)}}\right).$
(3.9)

##  4 Conclusions

In summary, we have re-examined the rotating black shell alternatives for
black holes proposed in [14]. A black shell is made of a higher dimensional
brane from string theory, wrapping internal cycles and yielding a spherical
membrane in four dimensional space time. The membrane carries magnetic fluxes,
whose massless vibrations within the world volume of the membrane give rise to
a gas of radiation. This is heated up to the local Unruh temperature of the
shell, thus equilibrating the system and making it lie at a critical point of
the gravitational potential. Recently, [15] analyzed the stability of the non-
rotating black shells of [13] and examined a two parameter family of fluxes to
ensure that this critical point is stable. It was concluded that stability
against radial perturbations, with damping, requires $\alpha<4/9$ and
$6\alpha-\beta>2$. Furthermore, under an impulsive accretion event, for the
shell to settle back to its new Buchdahl radius, it is necessary to have
$\alpha=\beta+2/3$.

In this paper, we have extrapolated the stability criteria of [15] to slowly
rotating black shells, and re-examined the flux exchanged between the brane
and gas in this case. We found that the flux indeed supports the shell at a
critical point in the gravitational potential. Additionally, requiring that
the parameters are such that a non-rotating shell relaxes back to its new
Buchdahl radius under accretion, fixes the flux parameters to unique values.
Remarkably, despite the complicated numerical factors involved, this yields an
extremely simple result: $\alpha=4/15$, $\beta=-2/5$. These satisfy one of the
constraints: $\alpha<4/9$, and exactly saturate the other: $6\alpha-\beta>2$.
So the existence of rotating black shells forces the parameters to saturate
the stability criteria obtained in [15] for non-rotating shells.

This opens up several interesting avenues for further study. First, one needs
to make sure that rotating black shells accrete matter in a way consistent
with that proposed in [15] for the non-rotating black shells. Saturating the
constraint $6\alpha-\beta=2$ in eq (2.6) implies that there is a zero mode of
the perturbation. However, since eq (2.6) is valid in the large mass limit
$m\gg 1/k$, a careful analysis of the sub-leading terms is needed. This, along
with extending the analysis of [15] to rotating black shells, will require
care and careful consideration when approaching this margin of stability.
Second, given these remarkably simple and promising values, it is of
fundamental importance to find an argument based on first principles, for the
source term as given in eq (3.9).

Third, the analysis, including the suggested reduction of the quadrupole
moment by $7\%$ compared to Kerr, is only valid for slowly spinning black
shells. It is of great phenomenological interest to extend this analysis to
black shells where the rotation is not slow. Given that most supermassive
black holes, for example the one at the center of the Milky way in Sagittarius
A*, are expected to be spinning close to maximal [25], such an analysis would
be important for astrophysical tests that could become available in the near
future.

## Acknowledgement

UD wants to thank Luis Lehner and Frans Pretorius for discussions. SG is
partially supported by the INFN and the MIUR-PRIN contract 2017CC72MK003AZ.

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