# On the Dynamical Instability of Monatomic Fluid Spheres in (N+1)-dimensional
Spacetime

Wei-Xiang Feng<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of California, Riverside, CA 92521

###### Abstract

In this note, I derive the Chandrasekhar instability of a fluid sphere in
($N$+1)-dimensional Schwarzschild-Tangherlini spacetime and take the
homogeneous (uniform energy density) solution for illustration. Qualitatively,
the effect of positive (negative) cosmological constant tends to destabilize
(stabilize) the sphere. In the absence of cosmological constant, the
privileged position of (3+1)-dimensional space is manifest in its own right.
As it is the _marginal dimensionality_ in which a monatomic ideal fluid sphere
is _stable but not too stable_ to trigger the onset of gravitational collapse.
Furthermore, it is the _unique_ dimensionality that can accommodate stable
hydrostatic equilibrium with _positive cosmological constant._ However, given
the current cosmological constant observed no stable configuration can be
larger than $10^{20}{\rm M}_{\odot}$. On the other hand, in (2+1) dimensions
it is _too stable_ to collapse either in the context of Newtonian Gravity (NG)
or Einstein’s General Relativity (GR). In GR, the role of _negative
cosmological constant_ is crucial not only to guarantee fluid equilibrium
(decreasing monotonicity of pressure) but also to have the Bañados-Teitelboim-
Zanelli (BTZ) solution. Owing to the negativeness of the cosmological
constant, there is no unstable configuration for a homogeneous fluid disk with
mass $0<\mathcal{M}\leq 0.5$ to collapse into a naked singularity, which
supports the Cosmic Censorship Conjecture. However, the relativistic
instability can be triggered for a homogeneous disk with mass
$0.5<\mathcal{M}\lesssim 0.518$ under causal limit, which implies that BTZ
holes of mass $\mathcal{M}_{\rm BTZ}>0$ could emerge from collapsing fluid
disks under proper conditions. The implicit assumptions and implications are
also discussed.

##  I introduction

Dimensionality of spacetime is a question with a long history Ehrenfest:1918 ;
Ehrenfest:1920 ; Whitrow:1955 ; Tangherlini:1963bw ; Barrow:1983 ;
Barrow:1988yia ; Caruso:1986de ; Tegmark:1997jg . Starting from Ehrenfest
Ehrenfest:1918 ; Ehrenfest:1920 , who argued based on the “stability
postulate” of the two-body problem, the fundamental laws of physics favors
(3+1) dimensions. His approach is still valid in the framework of general
relativity (GR) as well as hydrogen atom, as was shown by Tangherlini
Tangherlini:1963bw , however, see Ref. Caruso:1986de  for an alternative
procedure and references therein. Tegmark Tegmark:1997jg  argued that the
existence of only one temporal dimension by requiring hyperbolic equations of
motion, and hence _predictability_ , leaves the question why three
(macroscopic) spatial dimensions are favorable. Various arguments were
proposed for the reasoning Momen:2011jc ; Gonzalez-Ayala:2015xda ;
Brandenberger:1988aj ; Greene:2012sa ; Durrer:2005nz ; Nielsen:1993fd ;
Deser:2019oqc , although the dimensionality could be dynamical and scale-
dependent, as a physical observable Carlip:2017eud .

Presumably, the “non-compact” (3+1) dimensions might be just an illusion due
to our human perception. However, constraint from gravitational waves
indicates we certainly live in a universe of non-compact (3+1) dimensions
Pardo:2018ipy , as described by GR. On the other hand, although it was usually
cited that structures in (2+1) are not complex enough to accommodate life, as
opposed to common view, life could exist in (2+1) was discussed more recently
Scargill:2019rse .

Stellar equilibrium in different dimensions also has been explored previously
PoncedeLeon:2000pj ; Paul:2004nh ; Zarro:2009gd . One important feature on
stellar stability in GR is the Buchdahl stability bound Buchdahl:1959zz . It
states that the mass of a spherical compact object must exceed the $9/4$ of
its gravitational radius in (3+1) dimensions. Otherwise, there is no stable
stellar equilibrium and it would definitely collapse into a black hole.
Buchdahl bound in higher dimensions PoncedeLeon:2000pj  and with cosmological
constant Zarro:2009gd  were also investigated previously. Nonetheless, the
dynamical instability of a star might have set in well before the Buchdahl
bound.

The dynamical instability of a self-gravitating sphere in the context of GR
has been explored long ago by Chandrasekhar (1964) Chandrasekhar:1964zza ;
Chandrasekhar:1964zz  and Zel’dovich & Podurets (1966) Zeldovich:1966 , via
the study of pulsation equation and binding energy, respectively. It was found
that the turning point of fractional binding energy Ipser:1980 ;
Gunther:2021pbf  is very close to the result using the pulsation equation
Feng:2021rst . Chandrasekhar’s criteron Chandrasekhar:1964zza ;
Chandrasekhar:1964zz  provides a sufficient condition triggering the black
hole formation. In hydrostatic equilibrium, the pressure of a star balances
its self-gravity. As a gravitationally bound system, a star behaves as if it
has negative specific heat: The more it loses energy, the hotter it gets
LyndenBell:1968yw ; Spitzer:1987aa . Therefore it undergoes the gravothermal
evolution due to the heat dissipations. At each evolution stage the
instability might set in depending on the fluid’s _stiffness_ , which is
characterized by its adiabatic index Feng:2021rst . For example, the adiabatic
index of a (3+1) monatomic ideal fluid transits from $5/3$ (stiff) toward
$4/3$ (soft) when the particles become more relativistic through the
conversion of gravitational energy. This method has also been extended to
address the stellar instability in (3+1) spacetime with a non-zero
cosmological constant Boehmer:2005kk , the extra dimension influence
Arbanil:2019mae  on the strange quark stars, the stability of supermassive
stars Haemmerle:2020wwz , and the self-interacting dark halo core collapse
Feng:2020kxv ; Feng:2021rst .

In (2+1) spacetime, a negative cosmological constant is to guarantee not only
hydrostatic equilibrium Cruz:1994ar  (the pressure is monotonically
decreasing) but also permit a black hole solution of Bañados-Teitelboim-
Zanelli (BTZ) Banados:1992wn . The dynamical process of dust collapse
Ross:1992ba , critical collapse of scalar field Pretorius:2000yu ;
Husain:2000vm ; Jalmuzna:2015hoa  and ultra-relativistic fluid Bourg:2021ewu ;
Bourg:2021vpv  into a BTZ hole have been shown possible. Nevertheless, in this
note we are more interested in the condition triggering the dynamical
instability of a monatomic fluid disk in the _hydrodynamic_ limit. The
pressure in a fluid can not be ignored because it could prevent the fluid from
further collapse. On the other hand, as the random motion of the particles in
the fluid influences the gravitational potential in the macroscopic picture
through the pressure effect, (2+1) static stars of perfect fluid qualitatively
differ in their behavior from that of dust Giddings:1983es . As a result,
rather than counteract the gravitational attraction, it could further
destabilize the fluid disk at some point, just as the cases of (3+1). However,
the negative cosmological constant introduced tends to stabilize the fluid.
Thus, the competition between the pressure (relativistic) effect and the
cosmological constant is crucial to trigger the black hole formation.

In this note, I examine the _space dimensionality_ $N$ from the viewpoint of
the Chandrasekhar instability of an ideal monatomic fluid sphere in ($N$+1)
dimensions. The paper is organized as follows: We first derive the ($N$+1)
pulsation equation of a perfect fluid sphere with and without cosmological
constant and the corresponding Chandrasekhar’s criterion in Sec. II. As an
illustration, we then present the ($N$+1) homogeneous (uniform energy density)
solution in Sec. III and numerically determine the condition at the onset of
instabilities in Sec. IV. We briefly summarize the results with the implicit
assumptions, and discuss the physical implications in Sec. V. Geometric unit
($G_{N}=c=1$) is used throughout the text, where $G_{N}$ is the Newton’s
constant in ($N$+1) dimensions.

##  II (N+1)-dimensional spacetime of spherical symmetry

We consider a spherically symmetric and static spacetime in ($N$+1)
dimensions,

${\rm d}s^{2}=-e^{2\Phi(t,r)}{\rm d}t^{2}+e^{2\Lambda(t,r)}{\rm
d}r^{2}+r^{2}{\rm d}\Omega_{N-1}^{2},$ (1)

where

${\rm d}\Omega^{2}_{N-1}={\rm d}\theta_{1}^{2}+\sin^{2}\theta_{1}{\rm
d}\theta_{2}^{2}+...+\prod_{j=1}^{N-2}\sin^{2}\theta_{j}{\rm
d}\theta_{N-1}^{2}.$ (2)

After the standard calculations, the field equations
$G^{\mu}{}_{\nu}=\kappa_{N}T^{\mu}{}_{\nu}$ give

$-\frac{(N-1)\Lambda^{\prime}}{r}e^{-2\Lambda}-\frac{(N-1)(N-2)}{2r^{2}}\left[1-e^{-2\Lambda}\right]=\kappa_{N}T^{t}{}_{t},$
(3a)
$\frac{(N-1)\Phi{{}^{\prime}}}{r}e^{-2\Lambda}-\frac{(N-1)(N-2)}{2r^{2}}\left[1-e^{-2\Lambda}\right]=\kappa_{N}T^{r}{}_{r},$
(3b) $\frac{N-1}{r}\dot{\Lambda}e^{-2\Lambda}=\kappa_{N}T^{r}{}_{t},$ (3c)
$\displaystyle\left[\Phi{{}^{\prime\prime}}+\Phi{{}^{\prime}}^{2}-\Phi{{}^{\prime}}\Lambda{{}^{\prime}}+\frac{(N-2)}{r}(\Phi{{}^{\prime}}-\Lambda{{}^{\prime}})\right]e^{-2\Lambda}$
(3d)
$\displaystyle-\frac{(N-2)(N-3)}{2r^{2}}\left[1-e^{-2\Lambda}\right]-(\ddot{\Lambda}+\dot{\Lambda}^{2}-\dot{\Phi}\dot{\Lambda})e^{-2\Phi}=\kappa_{N}T^{\theta_{1}}{}_{\theta_{1}},$

where the “prime” and “dot” denote the “radial” and “time” derivatives,
respectively, and
$T^{\theta_{1}}{}_{\theta_{1}}=T^{\theta_{2}}{}_{\theta_{2}}=...=T^{\theta_{N-1}}{}_{\theta_{N-1}}\equiv
p$ because of spherical symmetry. On the other hand, combining Eq. (3a) and
(3b), we get

$\frac{(N-1)}{r}(\Phi{{}^{\prime}}+\Lambda{{}^{\prime}})e^{-2\Lambda}=\kappa_{N}(T^{r}{}_{r}-T^{t}{}_{t}).$
(4)

The conservation of energy-momentum tensor $\nabla_{\mu}T^{\mu}{}_{\nu}=0$
leads to

$\partial_{t}T^{t}{}_{t}+\partial_{r}T^{r}{}_{t}+(T^{t}{}_{t}-T^{r}{}_{r})\dot{\Lambda}+T^{r}{}_{t}\left(\Phi{{}^{\prime}}+\Lambda{{}^{\prime}}+\frac{N-1}{r}\right)=0$
(5a) and
$\partial_{t}T^{t}{}_{r}+\partial_{r}T^{r}{}_{r}+T^{t}{}_{r}(\dot{\Phi}+\dot{\Lambda})+(T^{r}{}_{r}-T^{t}{}_{t})\Phi{{}^{\prime}}+\frac{N-1}{r}(T^{r}{}_{r}-p)=0.$
(5b)

If we define the Schwarzschild mass function $M(r)$ through

$1-e^{-2\Lambda}\equiv 2M/r^{N-2},$ (6)

then $G^{t}{}_{t}=\kappa_{N}T^{t}{}_{t}=-\kappa_{N}\rho$ leads to
$M^{\prime}(r)=[\kappa_{N}/(N-1)]\rho r^{N-1}$. Accordingly the mass function

$M(r)=\omega_{N}\int_{0}^{r}\rho\bar{r}^{N-1}d\bar{r},$ (7)

where

$\omega_{N}=\frac{2\pi^{N/2}}{\Gamma(N/2)}$ (8)

is the area of the unit sphere in $N$-dimensional space. Therefore, by
construction the Einstein coupling constant $\kappa_{N}=(N-1)\omega_{N}$
automatically imposes the vacuum identity $G^{\mu}{}_{\nu}=0$ of Einsteinian
gravity in (1+1) dimensions.

Now we consider isotropic pressure ($T^{r}{}_{r}=p$) in static situation, Eq.
(5b) leads to $p^{\prime}=-(\rho+p)\Phi^{\prime}$, and $\Phi^{\prime}$ can be
replaced via Eq. (3b), resulting in the Tolman-Oppenheimer-Volkov (TOV)
equation in ($N$+1) dimensions:

$\left(1-\frac{2M}{r^{N-2}}\right)p^{\prime}=-(\rho+p)\left(\frac{(N-2)M}{r^{N-1}}+\frac{\kappa_{N}}{N-1}pr\right).$
(9)

Equation-of-state (EoS) $p=p(\rho)$ and boundary conditions $M(0)=0$, $p(R)=0$
must be imposed in order to determine the total fluid mass $\mathcal{M}\equiv
M(R)$.

###  II.1 Linear radial perturbation & the adiabatic index

The perfect fluid description
$T^{\mu}{}_{\nu}=(\rho+p)u^{\mu}u_{\nu}+p\delta^{\mu}{}_{\nu}$ implies
$T^{r}{}_{t}=0$ in the static case. However, under the _linear radial
perturbation_ $\partial_{t}\xi=\dot{\xi}\equiv dr/dt$,

$T^{r}{}_{t}=-(\rho+p)\dot{\xi},$ (10)

where $u^{\mu}$ is the 4-velocity of the fluid element and we have introduced
the “Lagrangian displacement” $\xi$ as well as $u_{t}=-e^{\Phi}$,
$u^{r}=e^{-\Phi}\dot{\xi}$.

Now we denote all the variables $X$’s in equilibrium by $X_{0}(r)$, which is
independent of time. After perturbation $X(t,r)=X_{0}(r)+\delta X(t,r)$, only
the perturbed quantities have the time dependence, where $\delta$ denotes the
“Eulerian change” of the perturbation. Keeping only the terms of _first-order
corrections_ from Eqs. (3a), (3b), (3c), (5a), and (5b), we have the
linearized equations governing the perturbation

$\left[r^{N-2}e^{-2\Lambda_{0}}2\delta\Lambda\right]^{\prime}=\frac{2\kappa_{N}}{N-1}r^{N-1}\delta\rho,$
(11a)
$\frac{N-1}{r}e^{-2\Lambda_{0}}(\delta\Phi{{}^{\prime}}-2\Phi_{0}{{}^{\prime}}\delta\Lambda)-\frac{(N-1)(N-2)}{r^{2}}e^{-2\Lambda_{0}}\delta\Lambda=\kappa_{N}\delta
p,$ (11b)
$\frac{N-1}{r}e^{-2\Lambda_{0}}\delta\dot{\Lambda}=-\kappa_{N}(\rho_{0}+p_{0})\dot{\xi}=-\frac{N-1}{r}e^{-2\Lambda_{0}}(\Phi_{0}{{}^{\prime}}+\Lambda_{0}{{}^{\prime}})\dot{\xi},$
(11c)
$\delta\dot{\rho}+\left[(\rho_{0}+p_{0})\dot{\xi}\right]^{\prime}+(\rho_{0}+p_{0})\left[\delta\dot{\Lambda}+\dot{\xi}\left(\Phi_{0}^{\prime}+\Lambda_{0}^{\prime}+\frac{N-1}{r}\right)\right]=0,$
(11d) $e^{2(\Lambda_{0}-\Phi_{0})}(\rho_{0}+p_{0})\ddot{\xi}+\delta
p^{\prime}+(\rho_{0}+p_{0})\delta\Phi{{}^{\prime}}+(\delta\rho+\delta
p)\Phi_{0}{{}^{\prime}}=0,$ (11e)

respectively. Besides, Eq. (11c) also leads to

$\delta\Lambda=-\xi(\Phi_{0}{{}^{\prime}}+\Lambda_{0}{{}^{\prime}}),$ (12)

thus Eq. (11a) and (11d) (after performing time integration) identically lead
to

$\delta\rho=-\frac{1}{r^{N-1}}\left[r^{N-1}(\rho_{0}+p_{0})\xi\right]^{\prime}=-\xi\frac{{\rm
d}\rho_{0}}{{\rm d}r}-\xi\frac{{\rm d}p_{0}}{{\rm
d}r}-(\rho_{0}+p_{0})\frac{1}{r^{N-1}}\left(r^{N-1}\xi\right)^{\prime}$ (13a)
or with the substitution
$p_{0}^{\prime}=-(\rho_{0}+p_{0})\Phi_{0}{{}^{\prime}}$, we get
$\Delta\rho=\delta\rho+\xi\frac{{\rm d}\rho_{0}}{{\rm
d}r}=-(\rho_{0}+p_{0})\frac{e^{\Phi_{0}}}{r^{N-1}}\left(r^{N-1}e^{-\Phi_{0}}\xi\right)^{\prime},$
(13b)

where $\Delta=\delta+\xi\partial_{r}$ denotes the “Lagrangian change” of the
perturbation. To express $\delta{p}$ in terms of $\xi$, we assume the
conservation of the baryon number $\nabla_{\alpha}(nu^{\alpha})=0$, that is

$\partial_{t}(ne^{-\Phi})+\partial_{r}(n\dot{\xi}e^{-\Phi})+ne^{-\Phi}(\dot{\Phi}+\dot{\Lambda})+ne^{-\Phi}\dot{\xi}\left(\Phi{{}^{\prime}}+\Lambda{{}^{\prime}}+\frac{N-1}{r}\right)=0.$
(14)

Keeping only the first-order terms of perturbation, one gets

$e^{-\Phi_{0}}\delta{\dot{n}}+\frac{1}{r^{N-1}}\left(n_{0}r^{N-1}\dot{\xi}e^{-\Phi_{0}}\right)^{\prime}+n_{0}e^{-\Phi_{0}}\delta{\dot{\Lambda}}+n_{0}e^{-\Phi_{0}}\dot{\xi}(\Phi_{0}{{}^{\prime}}+\Lambda_{0}{{}^{\prime}})=0.$
(15)

After integration over time and using Eq. (12) reduce to

$\delta{n}=-\frac{e^{\Phi_{0}}}{r^{N-1}}\left(n_{0}r^{N-1}{\xi}e^{-\Phi_{0}}\right)^{\prime}=-\xi\frac{{\rm
d}n_{0}}{{\rm
d}r}-n_{0}\frac{e^{\Phi_{0}}}{r^{N-1}}\left(r^{N-1}e^{-\Phi_{0}}\xi\right)^{\prime}$
(16a) or $\Delta n\equiv\delta{n}+\xi\frac{{\rm d}n_{0}}{{\rm
d}r}=-n_{0}\frac{e^{\Phi_{0}}}{r^{N-1}}\left(r^{N-1}e^{-\Phi_{0}}\xi\right)^{\prime}.$
(16b)

The adiabatic perturbation of the pressure is related to that of the number
density through adiabatic index if the EoS $n=n(\rho,p)$ is given, then

$\delta{n}=\left(\frac{\partial{n}}{\partial{\rho}}\right)_{p}\delta{\rho}+\left(\frac{\partial{n}}{\partial{p}}\right)_{\rho}\delta{p}$

together with Eq. (13) and (16) lead to

$-\xi\frac{{\rm d}n_{0}}{{\rm
d}r}-n_{0}\frac{e^{\Phi_{0}}}{r^{N-1}}\left(r^{N-1}e^{-\Phi_{0}}\xi\right)^{\prime}=\left(\frac{\partial{n}}{\partial{\rho}}\right)_{p}\left[-\xi\frac{{\rm
d}\rho_{0}}{{\rm
d}r}-(\rho_{0}+p_{0})\frac{e^{\Phi_{0}}}{r^{N-1}}\left(r^{N-1}e^{-\Phi_{0}}\xi\right)^{\prime}\right]+\left(\frac{\partial{n}}{\partial{p}}\right)_{\rho}\delta{p}.$

After the substitution

$\frac{{\rm d}n_{0}}{{\rm
d}r}=\left(\frac{\partial{n}}{\partial\rho}\right)_{p}\frac{{\rm
d}\rho_{0}}{{\rm
d}r}+\left(\frac{\partial{n}}{\partial{p}}\right)_{\rho}\frac{{\rm
d}p_{0}}{{\rm d}r},$

one finds

$\delta{p}=-\xi\frac{{\rm d}p_{0}}{{\rm
d}r}-\gamma{p_{0}}\frac{e^{\Phi_{0}}}{r^{N-1}}\left(r^{N-1}e^{-\Phi_{0}}\xi\right)^{\prime}$
(17a) or $\Delta p\equiv\delta{p}+\xi\frac{{\rm d}p_{0}}{{\rm
d}r}=-\gamma{p_{0}}\frac{e^{\Phi_{0}}}{r^{N-1}}\left(r^{N-1}e^{-\Phi_{0}}\xi\right)^{\prime},$
(17b)

and the adiabatic index $\gamma$ of the fluid is defined by 111 In the
literature, the symbol of adiabatic index is conventionally denoted as
$\Gamma_{1}$ and $\gamma\equiv{c_{p}}/{c_{n}}$, where $c_{p}$, $c_{n}$ denote
isobaric, isochoric specific heat capacities, respectively. The two indices
are related by $\Gamma_{1}=\chi_{T}\gamma$, where
$\chi_{T}\equiv\left({\partial\ln p}/{\partial\ln n}\right)_{T}$ evaluated at
constant temperature $T$. However, for an ideal gas without radiation pressure
$\chi_{T}=1$ and so $\Gamma_{1}=\gamma$ Ogilvie:2016 .

$\gamma\equiv\frac{1}{p_{0}\left(\partial{n}/\partial{p}\right)_{\rho}}\left[n_{0}-(\rho_{0}+p_{0})\left(\frac{\partial{n}}{\partial\rho}\right)_{p}\right]=\frac{\rho_{0}+p_{0}}{p_{0}}\left(\frac{\partial{p}}{\partial\rho}\right)_{s}=\frac{n_{0}}{p_{0}}\left(\frac{\partial
p}{\partial n}\right)_{s},$ (18a) where Eqs. (43), (44) in App. A have been
used to reach the last equality, and the subscript $s$ denotes the change is
adiabatic. This also implies the Lagrangian change is equivalent to the
adiabatic change in view of Eqs. (13), (16), (17), one can write
$\gamma=\left(\frac{\partial\ln p}{\partial\ln
n}\right)_{s}\equiv\frac{\Delta\ln p}{\Delta\ln n}.$ (18b)

###  II.2 The adiabatic index of an ideal monatomic fluid

For an ideal monatomic fluid, the adiabatic index depends on the _degrees of
freedom_ of spatial dimensions. Given a distribution function
$f(\mathbf{x},\mathbf{p})$ of monatomic particles with phase space measure
${\rm d}^{N}\mathbf{x}~{}{\rm d}^{N}\mathbf{p}$, the EoS can be determined by

$n(\mathbf{x})=\int f(\mathbf{x},\mathbf{p})~{}{\rm d}^{N}\mathbf{p},$ (19a)
$\rho(\mathbf{x})=\int Ef(\mathbf{x},\mathbf{p})~{}{\rm d}^{N}\mathbf{p},$
(19b) $p(\mathbf{x})=\frac{1}{N}\int\mathbf{p}\cdot\frac{\partial
E}{\partial\mathbf{p}}f(\mathbf{x},\mathbf{p})~{}{\rm
d}^{N}\mathbf{p}=\frac{1}{N}\int\frac{\mathbf{p}^{2}}{E}f(\mathbf{x},\mathbf{p})~{}{\rm
d}^{N}\mathbf{p},$ (19c)

where $E=\sqrt{\mathbf{p}^{2}+m^{2}}$ the energy of the particle with rest
mass $m$, and $N$ in the denominator of the pressure expression is due to
_equipartition theorem_.

The EoS can be prescribed by the $\gamma$-law form $p=K(mn)^{\gamma}$
Tooper:1965 , which satisfies the definition of adiabatic index provided that
$K$, $\gamma$ are _not explicit_ functions of $n$ under adiabatic
perturbation. The first law of thermodynamics ${\rm
d(internal~{}energy)}=-p{\rm~{}d(volume)}$, under adiabatic change and
particle number conservation, results in

${\rm d}\left(\frac{u}{n}\right)=-p~{}{\rm
d}\left(\frac{1}{n}\right)=\frac{p}{n^{2}}{\rm d}n=Km^{\gamma}n^{\gamma-2}{\rm
d}n.$

Direct integration gives $u/n=Km^{\gamma}(\gamma-1)^{-1}n^{\gamma-1}$, thus
the internal energy density

$u\equiv\rho-mn=(\gamma-1)^{-1}p.$ (20)

It turns out the adiabatic index

$\gamma=1+\frac{p}{\rho-mn}$ (21)

depends on the spatial dimensions $N$ and the relativistic extent of the
particles. In non-relativistic limit
$\mathinner{\\!\left\lvert\mathbf{p}\right\rvert}\ll m$, $\gamma\rightarrow
1+2/N$; ultra-relativistic limit
$\mathinner{\\!\left\lvert\mathbf{p}\right\rvert}\gg m$, $\gamma\rightarrow
1+1/N$. It actually depends the velocity dispersion

$v\equiv\sqrt{\frac{Np}{\rho}}$ (22)

via (see App. B for an explicit example)

$\gamma(v)=1+\frac{1+\sqrt{1-v^{2}}}{N}.$ (23)

As an aside, we note that this only applies for ideal fluids. If the
_microscopic interaction_ between particles is significant, the internal
energy density $u$ will contain interacting energy between particles, and $K$,
$\gamma$ might depend explicitly on $n$.

###  II.3 The pulsation equation & the critical adiabatic index

Using Eqs. (11b), (11c) and (12) one can derive

$(\rho_{0}+p_{0})\delta\Phi{{}^{\prime}}=\left[\delta
p-(\rho_{0}+p_{0})\left(2\Phi_{0}{{}^{\prime}}+\frac{N-2}{r}\right)\xi\right](\Phi_{0}{{}^{\prime}}+\Lambda_{0}{{}^{\prime}}).$

Substitution of $\delta\Phi^{\prime}$ for Eq. (11e) and assuming all the
perturbed quantities have the time-dependence of the form $e^{i\omega{t}}$
with the eigen frequency $\omega$, one can show that

$\omega^{2}e^{2(\Lambda-\Phi)}(\rho+p){\xi}=\delta p^{\prime}+\delta
p(2\Phi{{}^{\prime}}+\Lambda{{}^{\prime}})+\delta\rho\Phi{{}^{\prime}}-(\rho+p)\left(2\Phi{{}^{\prime}}+\frac{N-2}{r}\right)(\Phi{{}^{\prime}}+\Lambda{{}^{\prime}})\xi,$

in which we drop the subscript “zero” for simplicity hereafter.

Now we can further simplify the result by replacing the perturbed quantities
(except $\xi$) with the unperturbed ones in equilibrium. With the proper
substitutions via Eqs. (3d), (9), (13), and (17), one can derive the
“pulsation equation”

$\displaystyle\omega^{2}e^{2(\Lambda-\Phi)}(\rho+p){\xi}=$
$\displaystyle\frac{2(N-1)}{r}\frac{{\rm d}p}{{\rm
d}r}\xi-e^{-(2\Phi+\Lambda)}\left[e^{3\Phi+\Lambda}\frac{\gamma
p}{r^{N-1}}\left(r^{N-1}e^{-\Phi}\xi\right)^{\prime}\right]^{\prime}$
$\displaystyle+\frac{2\kappa_{N}}{N-1}e^{2\Lambda}p(\rho+p)\xi-\frac{1}{\rho+p}\left(\frac{{\rm
d}p}{{\rm d}r}\right)^{2}\xi,$ (24)

governing the linear instability at the first-order with boundary conditions
$\xi=0$ at $r=0$ and $\delta{p}=0$ at $r=R$ (radius of the sphere). Clearly,
$N=3$ reduces to the result derived by Chandrasekhar (1964)
Chandrasekhar:1964zza ; Chandrasekhar:1964zz .

Before we proceed further, we observe that in the Newtonian limit ($p\ll\rho$
and $\Phi,\Lambda\ll 1$) it reduces to 222 This actually reduces to the
pulsation equation in (3+1)-dimensional spacetime in Ref. Shapiro:1983du  by
perturbing the Euler’s equation in Newtonian Gravity (NG).

$\omega^{2}\rho\xi=\frac{2(N-1)}{r}\frac{{\rm d}p}{{\rm
d}r}\xi-\left[\frac{\gamma
p}{r^{N-1}}\left(r^{N-1}\xi\right)^{\prime}\right]^{\prime}.$ (25)

This implies the _critical adiabatic index_ (see App. C for derivation)

$\boxed{\gamma_{\rm cr}=2\left(1-\frac{1}{N}\right).}$ (26)

To have a stable configuration, it is required that the pressure-averaged
$\langle\gamma\rangle>\gamma_{\rm cr}$. For relativistic (non-relativistic)
ideal fluids, this implies the spatial dimensions must be $N<3~{}(N<4)$ in
order to have a stable sphere. From this viewpoint, the priviledge of (3+1)
dimensions is manifest because the fluid sphere _is stable but not too
stable._ However, in (2+1) dimensions it is too stable because $\gamma_{\rm
cr}=1<1.5~{}(2)=\gamma$ as always for an ultra-relativistic (non-relativistic)
fluid. Nevertheless, the “pressure effect” is crucial in GR because the whole
energy-momentum should be taken as a single entity, _not only the energy
density but also the pressure is sourcing gravity_.

To determine the critical adiabatic index in the full relativistic context, we
perform the integration over $r$ with $\xi$ over the sphere with proper
measure $r^{N-1}e^{\Phi+\Lambda}$, and integration by parts for the term with
$\gamma$ in the integrand, resulting in

$\displaystyle\omega^{2}\int e^{3\Lambda-\Phi}(\rho+p)r^{N-1}{\xi}^{2}{\rm
d}r=2(N-1)\int e^{\Phi+\Lambda}r^{N-2}\frac{{\rm d}p}{{\rm d}r}{\xi}^{2}{\rm
d}r$ $\displaystyle+\int e^{3\Phi+\Lambda}\frac{\gamma
p}{r^{N-1}}\left[(r^{N-1}e^{-\Phi}\xi)^{\prime}\right]^{2}{\rm d}r-\int
e^{\Phi+\Lambda}\left(\frac{{\rm d}p}{{\rm
d}r}\right)^{2}\frac{r^{N-1}\xi^{2}}{\rho+p}{\rm d}r$
$\displaystyle+\frac{2\kappa_{N}}{N-1}\int
e^{3\Lambda+\Phi}p(\rho+p)r^{N-1}\xi^{2}{\rm d}r.$ (27)

By Rayleigh-Ritz principle (see App. D), $\omega^{2}\leq 0$ signals the
instability of the given configuration, and thus determines the critical
adiabatic index $\gamma_{\rm cr}$ at $\omega^{2}=0$. To see the relativistic
corrections to NG, on the RHS of Eq. (II.3) we perform (i) integration by
parts for the first term and with Eq. (4) to replace
$\Phi^{\prime}+\Lambda^{\prime}$; (ii) replacement of $p^{\prime}$ by Eq. (9)
in the third term; (iii) use of Eq. (3b) to replace $\Phi^{\prime}$ after
choosing the trial function $\xi(r)=re^{\Phi}$ Chandrasekhar:1964zz . After
arranging all the terms with care and setting RHS of Eq. (II.3) equals to
zero, one gets

$\displaystyle\gamma_{\text{cr}}$ $\displaystyle\equiv
2\left(1-\frac{1}{N}\right)+\frac{\int
e^{3\Phi+\Lambda}[8(N-1)p+(N-2)(e^{2\Lambda}-1)(\rho+p)](N-2)(e^{2\Lambda}-1)r^{N-1}{\rm
d}r}{4N^{2}\int e^{3\Phi+\Lambda}pr^{N-1}{\rm d}r}$ $\displaystyle+$
$\displaystyle\frac{\kappa_{N}\int
e^{3(\Phi+\Lambda)}[4(N-1)p+(N-2)(e^{2\Lambda}+1)(\rho+p)]pr^{N+1}{\rm
d}r}{N^{2}(N-1)\int e^{3\Phi+\Lambda}pr^{N-1}{\rm
d}r}+\frac{\kappa_{N}^{2}\int e^{3\Phi+5\Lambda}(\rho+p)p^{2}r^{N+3}{\rm
d}r}{N^{2}(N-1)^{2}\int e^{3\Phi+\Lambda}pr^{N-1}{\rm d}r},$ (28)

and

$\langle\gamma\rangle\equiv\frac{\int e^{3\Phi+\Lambda}\gamma pr^{N-1}{\rm
d}r}{\int e^{3\Phi+\Lambda}pr^{N-1}{\rm d}r}$ (29)

the “effective” (pressure-averaged) adiabatic index of the fluid sphere.

###  II.4  The effect of cosmological constant

Furthermore, if the _cosmological constant_ $\lambda$ is included in the
previous derivation, it turns out to be the Schwarzschild-Tangherlini
spacetime Tangherlini:1963bw , and the pulsation equation becomes

$\displaystyle\omega^{2}e^{2(\Lambda-\Phi)}(\rho+p){\xi}=$
$\displaystyle\frac{2(N-1)}{r}\frac{{\rm d}p}{{\rm
d}r}\xi-e^{-(2\Phi+\Lambda)}\left[e^{3\Phi+\Lambda}\frac{\gamma
p}{r^{N-1}}\left(r^{N-1}e^{-\Phi}\xi\right)^{\prime}\right]^{\prime}$
$\displaystyle+\frac{2\kappa_{N}}{N-1}e^{2\Lambda}(p+p_{\lambda})(\rho+p)\xi-\frac{1}{\rho+p}\left(\frac{{\rm
d}p}{{\rm d}r}\right)^{2}\xi.$ (30)

As we will see, the extra term $p_{\lambda}$ from the cosmological constant is
significant to the stability condition. And we follow the same procedure to
obtain

$\displaystyle\gamma_{\rm cr}\equiv 2\left(1-\frac{1}{N}\right)+\frac{\int
e^{3\Phi+\Lambda}[8(N-1)p+(N-2)(e^{2\Lambda}-1)(\rho+p)](N-2)(e^{2\Lambda}-1)r^{N-1}{\rm
d}r}{4N^{2}\int e^{3\Phi+\Lambda}pr^{N-1}{\rm d}r}$
$\displaystyle+\frac{\kappa_{N}\int
e^{3(\Phi+\Lambda)}\left\\{[4(N-1)p+(N-2)(e^{2\Lambda}+1)(\rho+p)](p+p_{\lambda})-2(N-1)p_{\lambda}(\rho+p)\right\\}r^{N+1}{\rm
d}r}{N^{2}(N-1)\int e^{3\Phi+\Lambda}pr^{N-1}{\rm d}r}$
$\displaystyle+\frac{\kappa_{N}^{2}\int
e^{3\Phi+5\Lambda}(\rho+p)(p+p_{\lambda})^{2}r^{N+3}{\rm
d}r}{N^{2}(N-1)^{2}\int e^{3\Phi+\Lambda}pr^{N-1}{\rm d}r}.$ (31)

The second term on RHS tends to destabilize the fluid sphere due the “pressure
effect” of the fluid; while the third and fourth terms depend on its
_competition_ with cosmological constant $\lambda$. _Qualitatively_ ,
$\lambda<0$ tends to stabilize the sphere; and $\lambda>0$ does the opposite
333We note that the impact of cosmological constant on the Chandrasekhar
instability is opposite to the Antonov instability (gravothermal catastrophe),
see Ref. Axenides:2012bf .. The expression is fully relativistic in its own
right, though it would be indicative to see the post-Newtonian expansion in a
particular model. Therefore, we apply this result to the homogeneous model in
the next section.

##  III Homogeneous Fluid Solutions

The total fluid mass of homogeneous density is

$\mathcal{M}=\frac{\kappa_{N}}{N(N-1)}\rho R^{N}$ (32)

thus we can write

$\kappa_{N}\rho=\frac{N(N-1)\mathcal{M}}{R^{N}}.$ (33)

On the other hand, the definition of cosmological constant is ambiguous up to
some factor depending on the space dimensionality. We adopt the convention
based on
$R_{\lambda\mu\nu\rho}=\lambda(g_{\lambda\nu}g_{\mu\rho}-g_{\lambda\rho}g_{\mu\nu})$
for a space of constant curvature to define the cosmological constant in
($N$+1) dimensions from the vacuum Einstein equations
$G_{\mu\nu}=-\frac{N(N-1)}{2}\lambda g_{\mu\nu}$, thus

$\kappa_{N}\rho_{\lambda}=\frac{N(N-1)}{2}\lambda=-\kappa_{N}p_{\lambda},$
(34)

where $\lambda=\pm 1/\ell^{2}$, with $\ell$ the curvature of radius, depends
on the positiveness of the scalar curvature. Then the TOV equation Eq. (9) can
be solved analytically:

$\frac{p(r)}{\rho}=\frac{[(N-2)(\mathcal{M}/R^{N-2})-\lambda
R^{2})]\left[e^{-\Lambda(r)}-e^{-\Lambda(R)}\right]}{N(\mathcal{M}/R^{N-2})~{}e^{-\Lambda(R)}-[(N-2)(\mathcal{M}/R^{N-2})-\lambda
R^{2})]e^{-\Lambda(r)}}$ (35a) and
$e^{\Phi(r)}=\frac{N(\mathcal{M}/R^{N-2})~{}e^{-\Lambda(R)}-[(N-2)(\mathcal{M}/R^{N-2})-\lambda
R^{2})]e^{-\Lambda(r)}}{2\mathcal{M}/R^{N-2}+\lambda R^{2}},$ (35b) where
$e^{-\Lambda(r)}=\sqrt{1-\left(\frac{2\mathcal{M}}{R^{N-2}}+\lambda
R^{2}\right)\frac{r^{2}}{R^{2}}}.$ (35c) The solution is parameterized by the
_compactness parameter_ $\mathcal{C}_{N}\equiv\mathcal{M}/R^{N-2}$, the
_curvature parameter_ $\lambda R^{2}$ and the space dimensionality $N\geq 2$.
The parametrization makes sense only if $\rho\neq 0$ or the compactness
$\mathcal{M}/R^{N-2}\neq 0$.

Even though the homogeneous model is not so realistic, it captures the essence
of some underlying physics. For example, the Buchdahl stability bound can be
shown simply by demanding $p(0)<\infty$ and $e^{\Phi(0)}>0$:

$\frac{N-1}{N^{2}}\left(1-\sqrt{1-\frac{N^{2}}{(N-1)^{2}}\lambda
R^{2}}\right)<\frac{\mathcal{M}}{R^{N-2}}<\frac{N-1}{N^{2}}\left(1+\sqrt{1-\frac{N^{2}}{(N-1)^{2}}\lambda
R^{2}}\right).$ (36)

Clearly, for $N=3,~{}\lambda=0$, it reduces to the familiar Buchdahl bound
$0<9\mathcal{M}/4<R$ in (3+1) dimensions without cosmological constant. We
also note that the lower bound is larger than zero if $\lambda>0$. For more
realistic models, assuming the decreasing monotonicity of density is
sufficient to prove the Buchdahl bound, see App. E for a rigorous proof.
Nonetheless, when it comes to real stellar equilibrium the instability might
already trigger well before the Buchdahl bound.

In the post-Newtonian expansion with background curvature, Eq. (II.4) leads to

$\gamma_{\rm cr}=\frac{\lambda R^{2}}{(N-2)\mathcal{M}/R^{N-2}-\lambda
R^{2}}+\sum_{j,k=0,1,...}f_{jk}^{(N)}\left(\frac{\mathcal{M}}{R^{N-2}}\right)^{j}\left(\lambda
R^{2}\right)^{k},$ (37a) where the second terms on the RHS are post-Newtonian
corrections with coefficients $f_{jk}^{(N)}$ depending on the density
distribution and spatial dimensions $N$, except
$f_{00}^{(N)}=2\left(1-1/N\right)$ the exact Newtonian result; the first term
is a _stabilizer/destabilizer_ characterizing the _competition_ between the
compactness and the background curvature, and it can be expanded as
$\frac{\lambda R^{2}}{(N-2)\mathcal{M}/R^{N-2}-\lambda
R^{2}}=\begin{cases}\sum_{n=1}^{\infty}\left(\frac{\lambda
R^{2}}{(N-2)\mathcal{M}/R^{N-2}}\right)^{n}\quad{\rm
if}~{}\mathinner{\\!\left\lvert\lambda
R^{2}\right\rvert}<(N-2)\mathcal{M}/R^{N-2}\\\
-1-\sum_{n=1}^{\infty}\left(\frac{(N-2)\mathcal{M}/R^{N-2}}{\lambda
R^{2}}\right)^{n}\quad{\rm if}~{}\mathinner{\\!\left\lvert\lambda
R^{2}\right\rvert}>(N-2)\mathcal{M}/R^{N-2}.\end{cases}$ (37b)

We note that this term is always $-1$ for $N=2$ if $\lambda\neq 0$, and the
two limits ($\lambda\rightarrow 0\Leftrightarrow N\rightarrow 2$) do not
commute

$0=\lim_{N\rightarrow 2}\lim_{\lambda\rightarrow 0}\frac{\lambda
R^{2}}{(N-2)\mathcal{M}/R^{N-2}-\lambda R^{2}}\neq\lim_{\lambda\rightarrow
0}\lim_{N\rightarrow 2}\frac{\lambda R^{2}}{(N-2)\mathcal{M}/R^{N-2}-\lambda
R^{2}}=-1$

reflecting the fact that GR has no Newtonian limit in (2+1) dimensions.
Therefore, _Einsteinian stars are even stabler than Newtonian stars in (2+1)
as the critical adiabatic index is reduced by one unit in GR compared to NG._

##  IV Numerical Results

The effect of cosmological constant cannot be neglected when it comes to
stability, as it could stabilize the fluid sphere in higher dimensions ($N>3$)
with $\lambda<0$ or destabilize it in lower dimensions ($N<3$) with
$\lambda>0$. However, GR in lower dimensions, (1+1) has no dynamics (vacuum)
and (2+1) has no Newtonian limit. In particular, $\lambda<0$ is required to
have the Bañados-Teitelboim-Zanelli (BTZ) BH solution Banados:1992wn  and
stellar equilibrium Cruz:1994ar , it would then be interesting to see if the
instability condition can be triggered in (2+1) dimensions. In this section we
_numerically_ solve the marginal stable configurations and determine the
critical compactness of homogeneous spheres in ($N$+1) dimensions. We first
discuss (3+1) and higher, then (2+1) dimensions, respectively.

###  IV.1 Fluid spheres in (3+1) and higher-dimensional spacetime

Without cosmological constant, a (3+1)-dimensional homogeneous ideal fluid
becomes unstable as $\gamma_{\rm cr}=\langle\gamma\rangle=1.6219$, and the
critical central velocity dispersion $v_{c}\equiv v(0)=0.681433$ with the
critical compactness $\mathcal{C}_{3}=\mathcal{M}/R=0.189$, see Fig. 1
(_top_). Positive cosmological constant tends to destabilize the sphere owing
to the extra energy density and reduced pressure. Interestingly, in Fig. 1
(_middle_) there are _two critical points_ as $\lambda>0$ is turned on. In
reality, if the configuration is not sufficiently compact, the fluid is
unstable and it tends to further contract until it transits into a stable
configuration Feng:2021dibh . However, the stable region shrinks as $\lambda
R^{2}$ increases, and it could directly form a BH if the stable region
vanishes, see Tab. 1.

The stable configurations are bounded by the _two critical points_ up to
$\lambda R^{2}=0.01786$, above which there is no stable configuration, see
Fig. 1 (_bottom_). At this _degenerate_ critical point,
$\mathcal{C}_{3}=\mathcal{M}/R=0.1164$, we can eliminate the dependence of $R$
to get $\mathcal{M}\sqrt{\lambda}=0.015556$, or $\mathcal{M}=0.015556\ell
c^{2}/G_{3}$ with $G_{3}$ and $c$ restored. Now given the cosmological
constant observed Planck:2015fie  $\ell\sim 10^{60}\ell_{\rm Pl}$, where
$\ell_{\rm Pl}$ is the Planck length in (3+1) dimensions, there is no stable
stellar equilibrium for $\mathcal{M}\gtrsim 0.015556\ell c^{2}/G_{3}\sim
10^{20}{\rm M}_{\odot}$. Therefore, a virialized mass sphere should be much
smaller $10^{20}{\rm M}_{\odot}$ in order to have long-lived hydrostatic
equilibrium before it can trigger the BH formation. Curiously, this number is
close to the maximal Jeans mass just prior to the recombination of hydrogen
$M_{\rm J}\simeq 1.2\times 10^{16}\left(\Omega_{b,0}h^{2}\right)^{-2}{\rm
M}_{\odot}\simeq 10^{19}{\rm M}_{\odot}$ with the current baryon abundance
$\Omega_{b,0}h^{2}\simeq 0.0224$ Mo:2010ga . As the Jeans mass is the
threshold that a gas cloud can clump into gravitationally bound states,
structures can form only after recombination as the horizon mass of hydrogen
is always less than the Jeans mass before recombination Kolb:1990vq .
Therefore the stable upper bound $10^{20}{\rm M}_{\odot}$ seems delicately
protected in our universe.

Figure 1: $N=3$: Pressure-averaged & critical adiabatic indices v.s.
compactness $\mathcal{C}_{3}$ given $\lambda\geq 0$. The configurations are
unstable if $\langle\gamma\rangle<\gamma_{\rm cr}$, and the instabilities will
set in at critical points $\langle\gamma\rangle=\gamma_{\rm cr}$. In the case
of zero cosmological constant $\lambda R^{2}=0$ (_top_), the arrows on the
lines of adiabatic indices exhibit the directions when the fluid sphere is
being compressed while keeping $\mathcal{M}$ fixed. There is only _one_
critical point for the instability to be triggered at $\mathcal{C}_{3}=0.189$.
On the other hand, if $\lambda>0$ is turned on (no matter how small it is),
_two_ critical points will present. It is shown that for $\lambda R^{2}=0.01$
(_middle_), the stable region is bounded between $\mathcal{C}_{3}=0.0449$ and
$0.1685$ and shrinks as $\lambda R^{2}$ increases until $\lambda
R^{2}=0.01786$ (_bottom_), at which $\mathcal{C}_{3}=0.1164$ the two critical
points become degenerate. In Tab. 1, we list the corresponding compactness
$\mathcal{C}_{3}$ and central velocity dispersion $v_{c}$ of stable regions
with various $\lambda R^{2}\geq 0$.

On the other hand, negative cosmological constant tends to stabilize the
sphere due to the extra pressure and reduced energy density. Compared to zero
and positive cosmological constant, the critical compactness gets larger in
order to trigger the collapse. There is _only one critical point_ for a given
compactness down to $\lambda R^{2}=-0.094853$, below which there is no
physical solution due to causality ($v_{c}\leq 1$). At this critical point,
$\mathcal{C}_{3}=0.248$, see Fig. 2.

Figure 2: $N=3$: Pressure-averaged & critical adiabatic indices (top panels),
and central velocity dispersion $v_{c}$ (bottom panels) v.s. compactness
$\mathcal{C}_{3}$ given $\lambda<0$. The configurations are unstable if
$\langle\gamma\rangle<\gamma_{\rm cr}$, and the instabilities will set in at
critical points $\langle\gamma\rangle=\gamma_{\rm cr}$. There is only one
critical point for $\lambda<0$. For $\lambda R^{2}=-0.01$ (_left_), the
instability is to be triggered at $\mathcal{C}_{3}=0.202$ with $v_{c}=0.737$;
while for $\lambda R^{2}=-0.02$ (_middle_), it occurs at
$\mathcal{C}_{3}=0.212$ with $v_{c}=0.782$. Compared to $\lambda R^{2}=0$ in
Fig. 1 (_top_), it becomes harder to trigger the instabilities as higher
$\mathcal{C}_{3}$ (thus $v_{c}$) is required if $\lambda$ is more negative
until $\lambda R^{2}=-0.094853$ (_right_), at which $\mathcal{C}_{3}=0.248$
with the causal limit $v_{c}=1$. Beyond this point, no physical configuration
can trigger the instability on the grounds of causality. In Tab. 2, we list
the critical points at causal limits for $N=3,4,5,6,$ and $7$.

For $N>3$ the fluid is genuinely unstable in NG. In GR, however, the fluid
gets stabilized if $\lambda<0$ is turned on. For $N=4,5,6$, and $7$, we also
identify the lower bound of $\lambda R^{2}$ in Tab. 2 that the fluid can still
collapse into a black hole under the causal limit $v_{c}=1$.

Table 1: $N=3$: Stable regions $\langle\gamma\rangle\geq\gamma_{\rm cr}$ with
various $\lambda R^{2}>0$. There is no stable configuration for $\lambda
R^{2}\gtrsim 0.01786$.

$\lambda R^{2}$ | $\mathcal{C}_{3}=\mathcal{M}/R$ | $v_{c}=v(0)$ | $\langle\gamma\rangle$
---|---|---|---
$0$ | $0\textup{--}0.189057$ | $0\textup{--}0.681433$ | $1.66667\textup{--}1.62190$
$0.010$ | $0.044912\textup{--}0.16846$ | $0.240284\textup{--}0.604073$ | $1.66116\textup{--}1.63163$
$0.015$ | $0.075802\textup{--}0.149753$ | $0.328969\textup{--}0.541498$ | $1.65634\textup{--}1.63857$
$0.016$ | $0.084228\textup{--}0.14377$ | $0.352167\textup{--}0.522649$ | $1.65483\textup{--}1.64051$
$0.017$ | $0.09501\textup{--}0.13543$ | $0.3818\textup{--}0.497134$ | $1.65275\textup{--}1.64302$
$0.0175$ | $0.10278\textup{--}0.12889$ | $0.403279\textup{--}0.47766$ | $1.65113\textup{--}1.64485$
$0.0176$ | $0.10485\textup{--}0.12705$ | $0.409033\textup{--}0.472255$ | $1.65069\textup{--}1.64534$
$0.0177$ | $0.10738\textup{--}0.12475$ | $0.416092\textup{--}0.465544$ | $1.65013\textup{--}1.64594$
$0.0178$ | $0.1109\textup{--}0.1215$ | $0.425964\textup{--}0.456132$ | $1.64933\textup{--}1.64678$
$0.01785$ | $0.1142\textup{--}0.1183$ | $0.435279\textup{--}0.446946$ | $1.64856\textup{--}1.64757$
$0.01786$ | $0.1164$ | $0.441523$ | $1.64804$

Table 2: Critical points for $N=3,4,5,6,$ and $7$ with $\lambda<0$ at causal
limit $v_{c}=1$.

$N$ |  $\mathcal{C}_{N}$ |  $\lambda R^{2}$ |  $\langle\gamma\rangle=\gamma_{{\rm cr}}$
---|---|---|---
$3$ |  $0.248179$ |  $-0.094853$ |  $1.56387$
$4$ |  $0.117505$ |  $-0.134605$ |  $1.43352$
$5$ |  $0.062846$ |  $-0.151149$ |  $1.35328$
$6$ |  $0.037099$ |  $-0.154395$ |  $1.29861$
$7$ |  $0.023595$ |  $-0.151406$ |  $1.25884$

###  IV.2 Fluid disks in (2+1)-dimensional spacetime

It is well known that (2+1) GR has no local degrees of freedom (locally flat),
thus no gravitational wave (or graviton) can propagate. This means that
particles do not gravitate if they are static in the (2+1)-dimensional
spacetime Giddings:1983es ; Deser:1983tn ; Deser:1983nh . On the other hand,
the collective behavior of particles demands the fluid description under the
influence of gravity Giddings:1983es . To have hydrostatic equilibrium, a
negative cosmological constant $\lambda=-1/\ell^{2}$ is to guarantee not only
hydrostatic equilibrium Cruz:1994ar  (the pressure is monotonically
decreasing) but also permit a black hole solution (BTZ) in (2+1) dimensions.
The metric interior to the fluid disk turns out to be 444In (2+1) dimensions,
we note that the integration constant in $e^{-2\Lambda(r)}$ here is arbitrary,
but it can always be normalized to “unity” by adjusting the natural mass scale
$G_{2}$.

${\rm d}s_{\rm disk}^{2}=-e^{2\Phi(r)}{\rm d}t^{2}+e^{2\Lambda(r)}{\rm
d}r^{2}+r^{2}{\rm d}\phi^{2}\quad{\rm with}\quad
e^{-2\Lambda(r)}=1-2M(r)+\frac{r^{2}}{\ell^{2}}.$ (38)

On the other hand, the non-rotating BTZ metric reads

${\rm d}s_{\rm BTZ}^{2}=-f(r){\rm d}t^{2}+\frac{{\rm d}r^{2}}{f(r)}+r^{2}{\rm
d}\phi^{2}\quad{\rm with}\quad
f(r)=-\mathcal{M}_{\text{BTZ}}+\frac{r^{2}}{\ell^{2}}.$ (39)

To match spacetime of the fluid interior to the BTZ exterior, the junction
condition at the fluid radius $R$ leads to the relation of the BTZ mass and
the fluid mass

$\boxed{\mathcal{M}_{\rm BTZ}=2\mathcal{M}-1.}$ (40)

We note that the ADM mass in (2+1) is the BTZ mass Banados:1992gq ;
Brown:1986nw ; Brown:1994gs , rather than the fluid mass. The phase diagrams
of homogeneous fluid configurations with $0<\mathcal{M}\leq 0.5$ and
$\mathcal{M}>0.5$ are shown in Figs. 3(a) and 3(b), respectively. The absence
of fluid $\mathcal{M}=0$ corresponds to the BTZ bound state (anti-de Sitter
space) $\mathcal{M}_{\rm BTZ}=-1$, which is separated from the mass spectrum.
The fluid mass $\mathcal{M}=0.5$ is the BTZ ground (vacuum) state
$\mathcal{M}_{\rm BTZ}=0$. For $0<\mathcal{M}<0.5$, they are a sequence of
states of naked conical singularity $-1<\mathcal{M}_{\rm BTZ}<0$ if the fluid
were to collapse. Further, the threshold to have BTZ excited state
$\mathcal{M}_{\rm BTZ}>0$ is $\mathcal{M}>0.5$. It is interesting to note that
the central velocity dispersion $v_{c}$ is always monotonically increasing in
$\mathinner{\\!\left\lvert\lambda R^{2}\right\rvert}$ for $0<\mathcal{M}\leq
0.5$; while there are minima of $v_{c}\neq 0$ for $\mathcal{M}>0.5$, and no
causal solution if $\mathcal{M}\gtrsim 0.5208$. This manifests that the two
phase diagrams are separated by the _phase boundary_ $\mathcal{M}=0.5$ (or
$\mathcal{M}_{\rm BTZ}=0$).

(a) $0<\mathcal{M}\leq 0.5$ ($-1<\mathcal{M}_{\rm BTZ}\leq 0$)

(b) $\mathcal{M}>0.5$ ($\mathcal{M}_{\rm BTZ}>0$)

Figure 3: $v_{c}-\lambda R^{2}$ phase diagrams of homogeneous fluid disks in
(2+1) dimensions with the total fluid mass (a) $0<\mathcal{M}\leq 0.5$; and
(b) $\mathcal{M}>0.5$. The BTZ bound state (AdS spacetime) $\mathcal{M}_{\rm
BTZ}=-1$ ($\mathcal{M}=0$ only if $R=0$ as $\rho\neq 0$) is separated from the
mass spectrum. The fluid solution for $\mathcal{M}\gtrsim 0.5208$ is forbidden
regarding causality. In App. F, we list the corresponding $\lambda R^{2}$ at
causal limit $v_{c}=1$ for various $0<\mathcal{M}\leq 0.5$ in Tab. 3 and
$\mathcal{M}>0.5$ in Tab. 4, respectively. The minima of $v_{c}$ for
$\mathcal{M}>0.5$ are shown in Tab. 5.

By Chandrasekhar’s criterion at the critical point
$\langle\gamma\rangle=\gamma_{\rm cr}$, we can examine if the BTZ excited
states and the BTZ bound states (naked singularities) can result from a
collapsing fluid. For a homogeneous disk the critical adiabatic index, Eq.
(37), reduces to

$\displaystyle\gamma_{\rm cr}=-1$
$\displaystyle+\sum_{j,k=0,1,...}f_{jk}^{(2)}\mathcal{M}^{j}\left(\lambda
R^{2}\right)^{k}=\left(-\frac{1}{3}\left(\lambda
R^{2}\right)-\frac{11}{72}\left(\lambda
R^{2}\right)^{2}-\frac{37}{432}\left(\lambda R^{2}\right)^{3}+...\right)$
$\displaystyle+\left(-\frac{3}{4}\left(\lambda
R^{2}\right)-\frac{3}{4}\left(\lambda
R^{2}\right)^{2}-\frac{125}{192}\left(\lambda
R^{2}\right)^{3}+...\right)\mathcal{M}$
$\displaystyle+\left(-\frac{19}{12}\left(\lambda
R^{2}\right)-\frac{733}{288}\left(\lambda
R^{2}\right)^{2}-\frac{439}{144}\left(\lambda
R^{2}\right)^{3}+...\right)\mathcal{M}^{2}$
$\displaystyle+\left(-\frac{157}{48}\left(\lambda
R^{2}\right)-\frac{133}{18}\left(\lambda
R^{2}\right)^{2}-\frac{821}{72}\left(\lambda
R^{2}\right)^{3}+...\right)\mathcal{M}^{3}+\mathcal{O}\left(\mathcal{M}^{4}\right).$
(41)

It ostensibly starts from “zero” rather than “one” in NG as $\lambda
R^{2}\rightarrow 0$. However, Eq. (IV.2) is _not necessarily_ convergent to
zero as $\lambda R^{2}\rightarrow 0$ and the convergence really relies on the
fluid mass $\mathcal{M}$. In fact, $\gamma_{\rm cr}\rightarrow 0$ as $\lambda
R^{2}\rightarrow 0$ _only if_ $\mathcal{M}\leq 0.5$ when the fluid is being
compressed. Thus the presence of the negative cosmological constant makes the
relativistic instability hardly be triggered for $\mathcal{M}\leq 0.5$. In
Fig. 4 (_left_) we take $\mathcal{M}=0.4$ for example that it is always
$\gamma_{\rm cr}<\langle\gamma\rangle$, thus stable, under causal region
$v_{c}\leq 1$. On the other hand, the instability could set in if the fluid
disk exceeds, _no matter how tiny amount_ , the threshold $\mathcal{M}=0.5$
(or $\mathcal{M}_{\rm BTZ}=0$). Nevertheless, some external agent must
compress the fluid to make it unstable if $\lambda$ is fixed. In Fig. 4
(_middle_), for $\mathcal{M}=0.508$ the instability can set in at $\lambda
R^{2}=-0.0247$ with central velocity dispersion $v_{c}=0.799$. For higher
$\mathcal{M}$ both $\mathinner{\\!\left\lvert\lambda R^{2}\right\rvert}$ and
$v_{c}$ increase at the critical point of instability until
$\mathcal{M}=0.518$ at $\lambda R^{2}=-0.0609$ with $v_{c}=1$. Beyond this
mass there is no unstable configuration under causal range $v_{c}\leq 1$. In
App. F, we also list the critical points ($\lambda R^{2},v_{c}$) of
instability for various masses $0.5<\mathcal{M}\lesssim 0.518$ in Tab. 6.

Figure 4: $N=2$: Pressure-averaged & critical adiabatic indices (top panels),
and central velocity dispersion $v_{c}$ (bottom panels) v.s. curvature
parameter $\lambda R^{2}$. The configurations are unstable if
$\langle\gamma\rangle<\gamma_{\rm cr}$, and the instabilities will set in at
critical points $\langle\gamma\rangle=\gamma_{\rm cr}$. The arrows on the
lines of adiabatic indices exhibit the directions when the fluid sphere is
being compressed while keeping $\mathcal{C}_{2}=\mathcal{M}$ and $\lambda$
fixed. For $\mathcal{M}\leq 0.5$, there is no crossing of the two indices,
thus no instability will be triggered. For example, for $\mathcal{M}=0.4$
(_left_), $\gamma_{\rm cr}$ ($\langle\gamma\rangle$) is maximal (minimal) at
$v_{c}=1$ and decreases down (increases up) to zero (one) as $R$ decreases and
the two indices never cross. On the contrary, instabilities can occur for
$\mathcal{M}>0.5$. For $\mathcal{M}=0.508$ (_middle_) the instability can set
in at $\lambda R^{2}=-0.0247$ with $v_{c}=0.799$ ; while for
$\mathcal{M}=0.518$ (_right_) it is at $\lambda R^{2}=-0.0609$ with $v_{c}=1$,
no instability can be triggered if $\mathcal{M}\gtrsim 0.518$ regarding
causality.

##  V discussions and implications

We have derived Chandrasekhar’s criterion in ($N$+1) spacetime with and
without cosmological constant. As an illustration, we take the homogeneous
solution to determine the instability provided that the sphere is composed of
an ideal monatomic fluid.

In (3+1) spacetime, the privileged position is manifest as it is the marginal
dimensionality in which the fluid sphere is stable but not too stable to
trigger the onset of gravitational collapse. In particular, it is the _unique_
dimensionality that allows stable hydrostatic equilibrium with positive
cosmological constant. For higher spatial dimensions, the fluid sphere is
genuinely unstable either in the context of NG or GR. However, a negative
cosmological constant can stabilize it.

In (2+1) spacetime the effect of negative cosmological constant wins the
relativistic effect so that it is too stable for a fluid disk of mass
$0<\mathcal{M}\leq 0.5$ to collapse into a naked singularity. This somewhat
supports the Cosmic Censorship Conjecture Penrose:1969pc . However, the BTZ
hole emergence is possible from a collapsing fluid under proper conditions.
This is reasonable because if it were not the case, the anti-de Sitter space
$\mathcal{M}_{\rm BTZ}=-1$ (bound state) could be deformed _continuously_ into
a BTZ black hole of mass $\mathcal{M}_{\rm BTZ}\geq 0$ (vacuum or excited
state) by growing the mass of the fluid disk from $\mathcal{M}=0$ to
$\mathcal{M}\geq 0.5$.

We now summarize the assumptions made implicitly in the results and the
implications.

#### Assumptions:

  * •

First law of thermodynamics and equipartition theorem hold.

  * •

Mass-energy dispersion relation $E=\sqrt{\mathbf{p}^{2}+m^{2}}$ is valid.

  * •

Einstein equations hold in ($N$+1) dimensions.

The particles in a fluid sphere (or disk) are well thermalized at each local
point in the spacetime such that the equipartition theorem holds, thus the
pressure is isotropic macroscopically. Microscopically, every particle in the
fluid follows the mass-energy relation if the Lorentz symmetry is locally
preserved 555The mass-energy relation implied by local Lorentz invariance is
based on the assumption that particle states can be described by vectors in
some irreducible representation of the Poincaré group, however, Lorentz
symmetry is not necessary if the vectors describing particles can be generated
from a vacuum vector with the help of local field operators Borchers:1984yv ..
In the fluid description, its adiabatic index can vary from $1+2/N$ to $1+1/N$
as particles go from non-relativistic to ultra-relativistic regimes as more
and more gravitational energy is converted into the fluid. On the gravity
side, we restrict to only “one time dimension” since the hyperbolicity of
field equations (in this note GR or its Newtonian limit) determines the causal
structure Tegmark:1997jg , hence the “predictivity” allows physicists to do
theoretical physics. Moreover, in view of the Ostrogradsky theorem
Woodard:2015zca , the field equations are at most second derivative because
equations of motion with higher-order derivatives are in principle unstable or
non-local. Both GR and NG are free of Ostrogradsky instability.

Although gravitational attraction of a fluid is getting weaker as space
dimension $N$ increases, its pressure is even weaker when being compressed
because the _randomly_ moving particles inside the fluid will have _more
directions_ to go, thus easier to collapse. That is the reason why an ideal
monatomic fluid is genuinely unstable in higher dimensions.

#### Implications:

  * •

Baby universe emerges from collapsing matter in a BH.

  * •

Spacetime dimensionality reshuffles in the reign of quantum gravity.

We can reexamine the idea that the observable universe is the interior of a
black hole existing as one of possibly many inside a larger parent universe,
or multiverse Pathria:1972 ; Good:1972 ; Easson:2001qf . Since singularity is
generic Penrose:1964wq ; Hawking:1965mf ; Geroch:1966ur ; Ellis:1968vy ;
Garfinkle:2003bb  in GR, some limiting curvature must exist Frolov:1988vj ;
Frolov:1989pf  to avoid the singularity formation and transit to a baby
universe. Several mechanisms, including extended GR with torsion
Poplawski:2010kb , invoking stringy Hagedorn matter Dubovsky:2012wk ;
Feng:2018jrh  to produce a bouncing solution Biswas:2005qr ; Biswas:2006bs ,
braneworld scenario Dvali:2000hr ; Pourhasan:2013mqa , or transition through
an S-brane Brandenberger:2021ken , have been introduced to realize the idea.

Suppose the space dimension reshuffling is a random process as collapsing
matter into a BH in a universe of arbitrary space dimension $N$. When the
collapsing matter is squeezed down to the Planck scale (near the classical BH
singularity), it would emerge to a new universe with different space
dimensions. The change of spacetime dimensions through phase transition near
Planck scale might require some unknown theory of quantum gravity or string
theory Brandenberger:1988aj ; Greene:2012sa . Some inflation models that
predict parts of exponentially large size having different dimensionalities
Linde:1988yp  might provide an alternative mechanism. This process will repeat
again and again until the new-born universe is just (3+1)-dimensional, which
is stable but not too stable for the pristine gas in it to form complicated
structures, including both BHs and stars. In any higher dimensions $N>3$, it
is to too easy for matter to collapse into BHs; in (2+1), there is no way to
form BHs from self-gravitating fluid without external agents— _Both are too
barren to have complex structures._

The above discussion reverberates the _anthropic principle_ Barrow:1983 ;
Barrow:1988yia ; Smolin:1999 ; Susskind:2003kw ; Linde:2002gj  in one way or
another, though, it only relies on the _assumptions_ regardless of the
anthropic reasoning. However, it tells _more_ than that because gravitational
bound states of monatomic fluid are genuinely unstable and bound to BHs
immediately in space dimensions higher than three. Although negative
cosmological constant could stabilize the fluid spheres, it decelerates the
expansion of the whole universe on a large scale. This leads to another issue
on the anthropic bound of cosmological constant Barrow:1988yia ; Linde:2002gj
; Bousso:2004 , even though the falsifiability has been criticized Ball:2004 ;
Smolin:2004yv . In (3+1), the anthropic bound on the positive cosmological
constant Weinberg:1987dv  argues that it could not be very large, or the
universe would expand too fast for galaxies, stars (or us) to form. While once
these local structures are detached from the background expansion, its
_smallness_ also prevents the gravitational bound state (typically of mass
much less than $10^{20}{\rm M}_{\odot}$) from forming a BH immediately given
the current cosmological constant observed.

Why $N=3$? Perhaps the more sensible question is not, what makes (3+1) the
preferred dimensionality Momen:2011jc ; Gonzalez-Ayala:2015xda ;
Brandenberger:1988aj ; Greene:2012sa ? Rather, _why the physical principles
(thermodynamics, Lorentz symmetry, GR,…) allow fecund universes to exist only
in (3+1), which is the unique dimensionality that permits stable hydrostatic
equilibrium with positive cosmological constant?_

## Acknowledgements

The author acknowledges the Institute of Physics, Academia Sinica, for the
hospitality during the completion of this work, and many colleagues there for
discussions. The author is also grateful for helpful correspondence with Steve
Carlip and Stanley Deser. This work is supported in part by the U.S.
Department of Energy under grant No. DE-SC0008541.

##  Appendix A Thermodynamic identity

Assuming the EoS $n=n(\rho,p)$, we have

$\left(\frac{\partial{n}}{\partial\rho}\right)_{s}=\left(\frac{\partial{n}}{\partial\rho}\right)_{p}+\left(\frac{\partial{n}}{\partial{p}}\right)_{\rho}\left(\frac{\partial{p}}{\partial\rho}\right)_{s}$
(42)

and the first law of thermodynamics under adiabatic process leads to

$\left(\frac{\partial{n}}{\partial\rho}\right)_{s}=\frac{n_{0}}{\rho_{0}+p_{0}}$
(43)

combining the above results, we get

$\left(\frac{\partial{n}}{\partial\rho}\right)_{p}=\frac{n_{0}}{\rho_{0}+p_{0}}-\left(\frac{\partial{n}}{\partial{p}}\right)_{\rho}\left(\frac{\partial{p}}{\partial\rho}\right)_{s}$
(44)

where the subscript $s$ denotes an isentropic (adiabatic) process.

##  Appendix B Ideal monatomic fluid in $N$-dimensional space

If we assume a fluid element of volume $\mathcal{V}$, and composed of
$\mathcal{N}$ structureless point particles with the same mass $m$ in
$N$-dimensional space, then

$n(\mathbf{x})=\sum_{i=1}^{\mathcal{N}}\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i})=\frac{\mathcal{N}}{\mathcal{V}},$
(45)
$\rho(\mathbf{x})=\sum_{i=1}^{\mathcal{N}}E_{i}\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i}),$
(46)
$p(\mathbf{x})=\frac{1}{N}\sum_{i=1}^{\mathcal{N}}\frac{\mathbf{p}_{i}^{2}}{E_{i}}\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i}),$
(47)

with the velocity of the $i-$th particle being
$\mathbf{v}_{i}=\mathbf{p}_{i}/E_{i}$. We have

$\mathbf{p}_{i}=\frac{m\mathbf{v}_{i}}{\sqrt{1-\mathbf{v}_{i}^{2}}},$ (48)
$E_{i}=\frac{\mathinner{\\!\left\lvert\mathbf{p}_{i}\right\rvert}}{\mathinner{\\!\left\lvert\mathbf{v}_{i}\right\rvert}}=\frac{m}{\sqrt{1-\mathbf{v}_{i}^{2}}}.$
(49)

Given a distribution of these particles, we may take the root-mean-square
speed as velocity dispersion by

$v\equiv
v_{\text{rms}}=\sqrt{\frac{\mathcal{V}}{\mathcal{N}}\sum_{i}^{\mathcal{N}}\mathbf{v}_{i}^{2}\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i})}.$
(50)

First, we express (root-mean-square speed averaged) energy density
$\rho=\rho(p)$, by

$p=\frac{1}{N}\sum_{i}^{\mathcal{N}}\frac{\mathbf{p}_{i}^{2}}{E_{i}}\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i})=\frac{1}{N}\sum_{i}^{\mathcal{N}}E_{i}\mathbf{v}_{i}^{2}\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i})=\frac{1}{N}\rho
v^{2},\quad{\rm thus}\quad\rho=\frac{Np}{v^{2}}$ (51)

Second, we express $mn=mn(p)$ by

$p=\frac{1}{N}\sum_{i}^{\mathcal{N}}\bigg{(}\frac{m\mathbf{v}_{i}}{\sqrt{1-\mathbf{v}_{i}^{2}}}\bigg{)}^{2}\frac{\sqrt{1-\mathbf{v}_{i}^{2}}}{m}\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i})=\frac{1}{N}\sum_{i}^{\mathcal{N}}\frac{\mathbf{v}_{i}^{2}}{\sqrt{1-\mathbf{v}_{i}^{2}}}m\delta^{(N)}(\mathbf{x}-\mathbf{x}_{i})=\frac{1}{N}\frac{v^{2}}{\sqrt{1-\tilde{v}^{2}}}mn,$
(52)

whence we can identify

$\frac{1}{\sqrt{1-\tilde{v}^{2}}}mn=\rho,\quad{\rm thus}\quad
mn=\frac{N\sqrt{1-\tilde{v}^{2}}}{v^{2}}p.$ (53)

The internal energy density

$u\equiv\rho-
mn=\frac{N(1-\sqrt{1-\tilde{v}^{2}})}{v^{2}}p\equiv(\gamma-1)^{-1}p,$ (54)

as $\tilde{v}\simeq v$, we find

$\gamma(v)=1+\frac{v^{2}/N}{1-\sqrt{1-v^{2}}}=1+\frac{1+\sqrt{1-v^{2}}}{N}.$
(55)

As expected, in nonrelativistic limit ($v\rightarrow 0$), $\gamma\rightarrow
1+2/N$; in extremely relativistic limit ($v\rightarrow 1$), $\gamma\rightarrow
1+1/N$.

##  Appendix C The critical adiabatic index in Newtonian Gravity

Starting from the pulsation equation in NG

$\omega^{2}\rho\xi=\frac{2(N-1)}{r}\frac{{\rm d}p}{{\rm
d}r}\xi-\left[\frac{\gamma
p}{r^{N-1}}\left(r^{N-1}\xi\right)^{\prime}\right]^{\prime},$ (56)

we perform the integration over $r$ with $\xi$ over the sphere with proper
measure $r^{N-1}$ (see App. D in the Newtonian limit $p\ll\rho$ and
$\Phi,\Lambda\ll 1$), and after integration by parts one obtains

$\omega^{2}\int\rho\xi^{2}r^{N-1}{\rm d}r=\int\frac{\gamma
p}{r^{N-1}}\left(r^{N-1}\xi\right)^{\prime 2}{\rm d}r-2(N-1)\int
p\left(r^{N-2}\xi^{2}\right)^{\prime}{\rm d}r.$

Once the trial function $\xi=r$ is chosen, we get

$\omega^{2}\int\rho r^{N+1}{\rm
d}r=\int\left[N^{2}\gamma-2N(N-1)\right]pr^{N-1}{\rm d}r.$

By Rayleigh-Ritz principle, this implies the critical adiabatic index

$\gamma_{\text{cr}}=2\left(1-\frac{1}{N}\right)$ (57)

with the pressure-averaged adiabatic index

$\langle\gamma\rangle=\frac{\int\gamma pr^{N-1}{\rm d}r}{\int pr^{N-1}{\rm
d}r}.$ (58)

As a result, $\langle\gamma\rangle>2(1-1/N)$ corresponds to stable
oscillations ($\omega^{2}>0$); $\langle\gamma\rangle<2(1-1/N)$ corresponds to
unstable collapse or explosion ($\omega^{2}<0$); and
$\langle\gamma\rangle=2(1-1/N)$ is marginal stable ($\omega^{2}=0$).

##  Appendix D The orthogonality relation & Rayleigh-Ritz principle

The orthogonality relation:

$\int e^{2(\Lambda-\Phi)}(\rho+p)\xi^{(i)}\xi^{(j)}\sqrt{-g}{\rm d}r=\int
e^{3\Lambda-\Phi}(\rho+p)r^{N-1}\xi^{(i)}\xi^{(j)}{\rm d}r=0\quad(i\neq j),$
(59)

where $\xi^{(i)}$ and $\xi^{(j)}$ are the proper (eigen) solutions belonging
to different characteristic values of $\omega^{2}$. For Eq. (II.3) can be
written as

$-\frac{\partial^{2}\xi}{\partial{t}^{2}}=\omega^{2}\xi=A\xi,$ (60)

where $A$ is a linear differential operator which is self-adjoint, _i.e._

$\langle\xi^{(i)},A\xi^{(j)}\rangle=\langle A\xi^{(i)},\xi^{(j)}\rangle,$ (61)

with the inner product $\langle\cdot,\cdot\rangle$ taken with a weight factor,
$e^{3\Lambda-\Phi}(\rho+p)r^{N-1}$ in our case. By _Rayleigh-Ritz principle_ ,
we have

$\omega^{2}\leq\frac{\langle\xi,A\xi\rangle}{\langle\xi,\xi\rangle},$ (62)

for some chosen “trial function” $\xi$ (need not be an eigen solution). So if
$\langle\xi,A\xi\rangle=0$, we get $\omega^{2}\leq 0$, then $\xi$ grows
without bound and the perturbation is unstable. Therefore, a “sufficient”
condition for the onset of dynamical instability is that the RHS of Eq. (II.3)
vanishes for the chosen $\xi$ which satisfies the required boundary
conditions.

##  Appendix E A rigorous proof on the Buchdahl bound

The basic assumptions in the Buchdahl stability bound are:

  * •

The energy density $\rho$ is finite and monotonically non-increasing, _i.e._
${\rm d}\rho/{\rm d}r\leq 0$.

  * •

$e^{2\Phi}$ and $e^{2\Lambda}$ are positive definite, thus no horizon is
present inside the fluid sphere.

Including the cosmological constant $\lambda$, $\rho_{\lambda}$ and
$p_{\lambda}$ are given by
$\kappa_{N}\rho_{\lambda}=\frac{N(N-1)}{2}\lambda=-\kappa_{N}p_{\lambda}$, and
the TOV equation Eq. (9) along with the conservation law
$p^{\prime}=-(\rho+p)\Phi^{\prime}$ becomes

$\frac{\kappa_{N}}{N-1}p=-(N-2)\frac{M}{r^{N}}+\lambda+\frac{1}{r}\left(1-\frac{2M}{r^{N-2}}-\lambda
r^{2}\right)\frac{{\rm d}\Phi}{{\rm d}r},\quad{\rm with}\quad
M(r)=\frac{\kappa_{N}}{N-1}\int_{0}^{r}\rho(\bar{r})\bar{r}^{N-1}{\rm
d}\bar{r}.$

Now we consider taking derivative with respect to $r$ of the above equation,

$\frac{\kappa_{N}}{N-1}\frac{{\rm d}p}{{\rm d}r}=-(N-2)\frac{\rm d}{{\rm
d}r}\left(\frac{M}{r^{N}}\right)+\frac{\rm d}{{\rm
d}r}\left[\frac{1}{r}\left(1-\frac{2M}{r^{N-2}}-\lambda r^{2}\right)\frac{{\rm
d}\Phi}{{\rm d}r}\right].$ (63)

In addition, note that if ${\rm d}\rho/{\rm d}r\leq 0$, the first term on the
RHS of Eq. (63)

$\frac{\rm d}{{\rm d}r}\left(\frac{M}{r^{N}}\right)=\left(\frac{{\rm d}M}{{\rm
d}r}\right)\frac{1}{r^{N}}-\frac{NM}{r^{N+1}}=\frac{\kappa_{N}}{(N-1)r}\left(\rho-\bar{\rho}\right)<0,$
(64)

where $\bar{\rho}$ is defined through the mean value theorem for $0\leq
r_{0}\leq r$,

$M(r)=\frac{\kappa_{N}}{N(N-1)}\rho(r_{0})r^{N}\equiv\frac{\kappa_{N}}{N(N-1)}\bar{\rho}(r)r^{N}$

and $\bar{\rho}(r)\geq\rho(r)$ if ${\rm d}\rho/{\rm d}r\leq 0$. Furthermore,
the second term on the RHS of Eq. (63) can be written as

$\frac{\rm d}{{\rm
d}r}\left[\left(\frac{\Phi^{\prime}}{r}e^{\Phi}\sqrt{1-\frac{2M}{r^{N-2}}-\lambda
r^{2}}\right)\left(e^{-\Phi}\sqrt{1-\frac{2M}{r^{N-2}}-\lambda
r^{2}}\right)\right]=\frac{\rm d}{{\rm
d}r}\left[\left(\frac{\Phi^{\prime}}{r}e^{\Phi}e^{-\Lambda}\right)\left(e^{-\Phi}e^{-\Lambda}\right)\right]$
$=e^{-\Phi}e^{-\Lambda}\frac{\rm d}{{\rm
d}r}\left(\frac{\Phi^{\prime}}{r}e^{\Phi}e^{-\Lambda}\right)+\frac{\Phi^{\prime}}{r}e^{\Phi}e^{-\Lambda}\frac{\rm
d}{{\rm d}r}\left(e^{-\Phi}e^{-\Lambda}\right)\quad{\rm with}\quad
e^{2\Lambda}=\left[1-\frac{2M}{r^{N-2}}-\lambda r^{2}\right]^{-1}$ (65)

and

$\frac{\rm d}{{\rm d}r}\left(e^{-\Lambda}\right)=-e^{\Lambda}\left[\frac{\rm
d}{{\rm d}r}\left(\frac{M}{r^{N-2}}\right)+\lambda r\right].$

Therefore,

$e^{-\Lambda}\frac{\rm d}{{\rm
d}r}\left(e^{-\Phi}e^{-\Lambda}\right)=e^{-\Lambda}\left[-e^{-\Phi}e^{\Lambda}\left(\frac{\rm
d}{{\rm d}r}\left(\frac{M}{r^{N-2}}\right)+\lambda
r\right)-e^{-\Phi}\Phi^{\prime}e^{-\Lambda}\right]=-e^{-\Phi}\left\\{\left[\frac{\rm
d}{{\rm d}r}\left(\frac{M}{r^{N-2}}\right)+\lambda
r\right]+\Phi^{\prime}e^{-2\Lambda}\right\\}$
$=-e^{-\Phi}\left\\{\left[\left(\frac{\kappa_{N}\rho}{N-1}-\frac{(N-2)M}{r^{N}}+\lambda\right)r\right]+\Phi^{\prime}e^{-2\Lambda}\right\\}=-e^{-\Phi}\left[\frac{\kappa_{N}(\rho+p)}{N-1}\right]r,$
(66)

where we have used the TOV equation in the last equality

$-\frac{(N-2)M}{r^{N}}+\lambda=\frac{\kappa_{N}}{N-1}p-\frac{1}{r}e^{-2\Lambda}\Phi^{\prime}.$

Combining Eq. (63),(64),(65),(66), and the conservation law
$p^{\prime}=-(\rho+p)\Phi^{\prime}$, it turns out

$(N-2)e^{\Phi+\Lambda}\frac{\rm d}{{\rm
d}r}\left(\frac{M}{r^{N}}\right)=\frac{\rm d}{{\rm
d}r}\left(\frac{\Phi^{\prime}}{r}e^{\Phi}e^{-\Lambda}\right)\leq 0.$ (67)

Integration of Eq. (67) from $r$ to $R$ gives

$\frac{\Phi^{\prime}(R)}{R}e^{\Phi(R)-\Lambda(R)}-\frac{\Phi^{\prime}(r)}{r}e^{\Phi(r)-\Lambda(r)}\leq
0,$

also note that in vacuum

$e^{\Phi(r)}=\sqrt{1-\frac{2\mathcal{M}}{r^{N-2}}-\lambda
r^{2}}=e^{-\Lambda(r)}\quad{\rm for}\quad r\geq R,$

so

$\Phi^{\prime}(R)e^{\Phi(R)}=\frac{\rm d}{{\rm
d}r}\left(e^{\Phi(r)}\right)\bigg{|}_{r=R}=\frac{(N-2)\frac{\mathcal{M}}{R^{N-1}}-\lambda
R}{\sqrt{1-\frac{2\mathcal{M}}{R^{N-2}}-\lambda
R^{2}}}=\left[(N-2)\frac{\mathcal{M}}{R^{N-1}}-\lambda R\right]e^{\Lambda(R)}$
$\Rightarrow\frac{\Phi^{\prime}(R)}{R}e^{\Phi(R)-\Lambda(R)}=\left[(N-2)\left(\frac{\mathcal{M}}{R^{N}}\right)-\lambda\right]\Rightarrow\frac{\Phi^{\prime}(r)}{r}e^{\Phi(r)-\Lambda(r)}\geq\left[(N-2)\left(\frac{\mathcal{M}}{R^{N}}\right)-\lambda\right]$

or

$\Phi^{\prime}(r)e^{\Phi(r)}\geq
re^{\Lambda(r)}\left[(N-2)\left(\frac{\mathcal{M}}{R^{N}}\right)-\lambda\right].$
(68)

Integration of Eq. (68) from $0$ to $R$ yields

$e^{\Phi(R)}>e^{\Phi(R)}-e^{\Phi(0)}\geq\left[(N-2)\left(\frac{\mathcal{M}}{R^{N}}\right)-\lambda\right]\int_{0}^{R}\frac{r{\rm
d}r}{\sqrt{1-\frac{2M(r)}{r^{N-2}}-\lambda r^{2}}}$ (69)

and the non-increasing monotonicity of $\rho$ leads to the inequality

$M(r)=\frac{\kappa_{N}}{N(N-1)}r^{N}\bar{\rho}(r)=\left(\frac{r}{R}\right)^{N}\frac{\kappa_{N}}{N(N-1)}R^{N}\bar{\rho}(r)\geq\left(\frac{r}{R}\right)^{N}\frac{\kappa_{N}}{N(N-1)}R^{N}\bar{\rho}(R)=\left(\frac{r}{R}\right)^{N}\mathcal{M},$

hence

$\int_{0}^{R}\frac{r{\rm d}r}{\sqrt{1-\frac{2M(r)}{r^{N-2}}-\lambda
r^{2}}}\geq\int_{0}^{R}\frac{r{\rm
d}r}{\sqrt{1-\frac{2\mathcal{M}}{R^{N}}r^{2}-\lambda
r^{2}}}=\left[\frac{1-\sqrt{1-2\mathcal{M}/R^{N-2}-\lambda
R^{2}}}{2\mathcal{M}/R^{N}+\lambda}\right]>0$ (70)

either for $2\mathcal{M}/R^{N}+\lambda>0$ or $<0$ if there is no horizon
inside the sphere. Consequently Eq. (69) and (70) give rise to

$\sqrt{1-\frac{2\mathcal{M}}{R^{N-2}}-\lambda
R^{2}}=e^{\Phi(R)}>\left[\left(\frac{N-2}{2}\right)\left(\frac{2\mathcal{M}}{R^{N-2}}\right)-\lambda
R^{2}\right]\left[\frac{1-\sqrt{1-2\mathcal{M}/R^{N-2}-\lambda
R^{2}}}{2\mathcal{M}/R^{N-2}+\lambda R^{2}}\right]$

or

$\boxed{\sqrt{1-2\mathcal{C}_{N}-a}>\left[(N-2)\mathcal{C}_{N}-a\right]\left[\frac{1-\sqrt{1-2\mathcal{C}_{N}-a}}{2\mathcal{C}_{N}+a}\right]}$
(71)

with $\mathcal{C}_{N}\equiv\mathcal{M}/R^{N-2}>0$ and $a\equiv\lambda R^{2}$.
Note that if $2\mathcal{C}_{N}+a=0$, we have
$1>N\mathcal{C}_{N}\times\frac{1}{2}\rightarrow
2\mathcal{C}_{N}=2\mathcal{M}/R^{N-2}=-\lambda R^{2}=-a<4/N$. However, if
$2\mathcal{C}_{N}+a\neq 0$, one obtains

$N\mathcal{C}_{N}\sqrt{1-2\mathcal{C}_{N}-a}\gtrless\left[(N-2)\mathcal{C}_{N}-a\right]\quad{\rm
if}\quad 2\mathcal{C}_{N}+a\gtrless 0.$

Both lead to the same inequality:

$\mathcal{C}_{N}^{2}-\frac{2}{N}\left(\frac{N-1}{N}\right)\mathcal{C}_{N}+\frac{a}{N^{2}}<0$
(72)

and hence

$\boxed{\frac{N-1}{N^{2}}\left(1-\sqrt{1-\frac{N^{2}a}{(N-1)^{2}}}\right)<\mathcal{C}_{N}<\frac{N-1}{N^{2}}\left(1+\sqrt{1-\frac{N^{2}a}{(N-1)^{2}}}\right).}$
(73)

##  Appendix F Various tables for $N=2$

Table 3: End points of $\lambda R^{2}$ for $N=2$ with various fluid mass
$0<\mathcal{C}_{2}=\mathcal{M}\leq 0.5$ at causal limit $v_{c}=1$.

$\mathcal{C}_{2}=\mathcal{M}$ |  $\lambda R^{2}$ |  $\langle\gamma\rangle$ |  $\gamma_{{\rm cr}}$
---|---|---|---
$0.1$ |  $-1.08681$ |  $1.81892$ |  $0.289458$
$0.2$ |  $-0.91554$ |  $1.81892$ |  $0.312447$
$0.3$ |  $-0.73198$ |  $1.81892$ |  $0.347248$
$0.4$ |  $-0.52604$ |  $1.81893$ |  $0.410982$
$0.5$ |  $-0.25$ |  $1.81893$ |  $0.636364$

Table 4: The lower and upper causal limits ($v_{c}=1$) of $\lambda R^{2}$ with
various fluid mass $\mathcal{C}_{2}=\mathcal{M}>0.5$. The two causal limits
become degenerate at $\mathcal{M}=0.5208\bar{3}$.

$\mathcal{C}_{2}=\mathcal{M}$ |  $\lambda R^{2}$ (lower/upper) |  $\langle\gamma\rangle$ (lower/upper) |  $\gamma_{{\rm cr}}$ (lower/upper)
---|---|---|---
$0.501$ |  $-0.245968$ / $-0.0020368908$ |  $1.81892$ / $1.81893$ |  $0.643072$ / $44.9933$
$0.504$ |  $-0.233361$ / $-0.00863897$ |  $1.81893$ / $1.81892$ |  $0.665408$ / $10.8801$
$0.508$ |  $-0.215107$ / $-0.0188929$ |  $1.81892$ / $1.81892$ |  $0.702112$ / $5.16155$
$0.512$ |  $-0.194394$ / $-0.0316059$ |  $1.81893$ / $1.81893$ |  $0.751605$ / $3.21809$
$0.516$ |  $-0.169208$ / $-0.048792$ |  $1.81893$ / $1.81893$ |  $0.827182$ / $2.19555$
$0.520$ |  $-0.13$ / $-0.08$ |  $1.81893$ / $1.81893$ |  $1.00000$ / $1.45455$
$0.5208$ |  $-0.1092$ / $-0.0992$ |  $1.81893$ / $1.81893$ |  $1.13986$ / $1.22727$
$0.5208\bar{3}$ | $-0.1041\bar{6}$ | $1.81893$ | $1.18182$

Table 5: The minimal $v_{c}$ and the corresponding $\lambda R^{2}$ with
various fluid mass $\mathcal{C}_{2}=\mathcal{M}>0.5$. $\mathcal{M}\simeq
0.5208$ is the upper bound without violation of the causal limit $v_{c}=1$.

$\mathcal{C}_{2}=\mathcal{M}$ |  $\lambda R^{2}$ |  $v_{c}=v(0)$ |  $\langle\gamma\rangle$ |  $\gamma_{{\rm cr}}$
---|---|---|---|---
$0.50001$ |  $-0.0000401797$ |  $0.134047$ |  $1.99701$ |  $1.00447$
$0.5001$ |  $-0.000405737$ |  $0.239529$ |  $1.99045$ |  $1.01407$
$0.501$ |  $-0.00418706$ |  $0.432509$ |  $1.96890$ |  $1.04396$
$0.504$ |  $-0.0175648$ |  $0.625459$ |  $1.93483$ |  $1.08603$
$0.508$ |  $-0.0365920$ |  $0.757625$ |  $1.90366$ |  $1.11908$
$0.512$ |  $-0.0566768$ |  $0.850335$ |  $1.87713$ |  $1.14315$
$0.516$ |  $-0.0776784$ |  $0.924608$ |  $1.85193$ |  $1.16248$
$0.520$ |  $-0.0995169$ |  $0.987848$ |  $1.82538$ |  $1.17873$
$0.5208$ |  $-0.10398$ |  $0.999520$ |  $1.81921$ |  $1.18170$
$0.5208\bar{3}$ |  $-0.1041\bar{6}$ |  $1$ |  $1.81893$ |  $1.18182$

Table 6: The critical $\lambda R^{2}$ with various fluid mass
$\mathcal{C}_{2}=\mathcal{M}>0.5$ under causal range $v_{c}\leq 1$.
$\mathcal{M}\simeq 0.518$ is the upper bound that the instability can be
triggered without violation of causal limit $v_{c}=1$.

$\mathcal{C}_{2}=\mathcal{M}$ |  $\lambda R^{2}$ |  $v_{c}=v(0)$ |  $\langle\gamma\rangle=\gamma_{{\rm cr}}$
---|---|---|---
$0.501$ |  $-0.0027978$ |  $0.461740$ |  $1.96455$
$0.504$ |  $-0.0117926$ |  $0.663440$ |  $1.92657$
$0.508$ |  $-0.0247187$ |  $0.799338$ |  $1.89227$
$0.512$ |  $-0.0385385$ |  $0.893364$ |  $1.86307$
$0.516$ |  $-0.0532156$ |  $0.967696$ |  $1.83468$
$0.518$ |  $-0.0609066$ |  $0.999979$ |  $1.81894$
$0.5180013$ |  $-0.0609116$ |  $1$ |  $1.81893$

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