EinsteinVlasov system with equal-angular momenta in AdS 5 Hiroki Asami1 Chul-Moon Yoo1 Ryo Kitaku1 and Keiya Uemichi1

2025-08-25 1 0 980.41KB 21 页 10玖币

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Einstein–Vlasov system with equal-angular momenta in AdS5

Hiroki Asami∗1, Chul-Moon Yoo†1, Ryo Kitaku‡1, and Keiya Uemichi§1

1Division of Particle and Astrophysical Science, Graduate School of Science, Nagoya University,

Nagoya 464-8602, Japan

Abstract

We investigate solutions of the 5–dimensional rotating Einstein-Vlasov system with

an R×SU(2) ×U(1) isometry group. In a ﬁve-dimensional spacetime, there are two

independent planes of rotation, thus, considering U(1) symmetry on each rotation plane,

we may impose an R×U(1) ×U(1) isometry to a stationary spacetime. Furthermore,

when the values of the two angular momenta are equal to each other, the spatial symmetry

gets enhanced to R×SU(2) ×U(1) symmetry, and the spacetime has a cohomogeneity-1

structure. Imposing the same symmetry to the distribution function of the particles of

which the Vlasov system consists, the distribution function can be dependent on three

mutually independent and commutative conserved charges for particle motion (energy,

total angular momentum on SU(2) and U(1) angular momentum). We consider the

distribution function which exponentially depends on the U(1) angular momentum and

reduces to the thermal equilibrium state in spherical symmetry. Then, in this paper, we

numerically construct solutions of the asymptotically AdS Einstein–Vlasov system.

1 Introduction

General relativistic self-gravitating collisionless many-particle systems, which are often called

Einstein-Vlasov systems, have long been investigated in astrophysics. In particular, spheri-

cally symmetric systems have been studied in detail, and a great deal of research has been

done on the existence of solutions [1–4] and the stability of the systems [5–10] (see also a

review [11]). To realize a speciﬁc conﬁguration of the Einstein-Vlasov system, it is necessary

to impose an appropriate ansatz on the distribution function. For example, to give a static

conﬁguration surrounding a black hole, we have to assume a distribution with a lower cutoﬀ

of the angular momentum [12]. If we assume a Maxwell–J¨uttner distribution, we can real-

ize a thermal equilibrium state of the self-gravitating system. However, there are no thermal

equilibrium states with ﬁnite mass in asymptotically ﬂat spacetimes because gravity is a long-

range interaction. Antonov investigated the non-relativistic many-particle systems with ﬁnite

mass by introducing an adiabatic wall conﬁning the particles [13], and the properties of those

systems have been generalized and investigated in detail by Lynden-Bell and Wood [14].

∗Email:asami.hiroki.b3@s.mail.nagoya-u.ac.jp

†Email:yoo.chulmoon.k6@f.mail.nagoya-u.ac.jp

‡Email:kitaku.ryo.f4@s.mail.nagoya-u.ac.jp

§Email:uemichi.keiya.j4@s.mail.nagoya-u.ac.jp

arXiv:2210.17112v2 [gr-qc] 13 Mar 2023

The thermodynamical instability of self-gravitating systems is often called the gravothermal

catastrophe which also applies to relativistic cases. It should be noted that, however, since

the thermal equilibrium states have inﬁnite mass with a vanishing cosmological constant,

the analyses of the gravothermal catastrophe in asymptotically ﬂat spacetime always require

an artiﬁcial wall to conﬁne the system. On the other hand, in a system with a negative

cosmological constant, the AdS barrier conﬁnes the particle system, allowing it to naturally

be in a thermal equilibrium state. Some of the authors have constructed a thermal equilibrium

state of such a system conﬁned by the AdS barrier under the assumption of static spherical

symmetry and analyzed its stability [15, 16].

Properties of the Einstein-Vlasov system in static and spherically symmetric cases have

been intensively studied, but there are still few studies on systems with rotation [17–21].

The properties of a system, such as stability, generally depend on the presence of angular

momentum because the total angular momentum may prevent the system from collapsing.

In the case of a self-gravitating many-particle system, it is expected that the instability

associated with gravitational collapse may be inhibited by the angular momentum of the

system. However, the existence of the non-zero angular momentum inevitably reduces the

spacetime symmetry, and the analyses become much more diﬃcult. To avoid this technical

diﬃculty, we focus on 5–dimensional spacetimes because it is known that the spacetime can

have a cohomogeneity-1 structure even with non-zero angular momentum in 5–dimensional

spacetimes. That is, we can construct a spacetime with ﬁnite angular momentum solving a set

of ordinary diﬀerential equations for unknown variables depending only on a radial coordinate.

More speciﬁcally, in a 5–dimensional spacetime, if the values of the angular momenta on the

two independent rotation planes are equal to each other, the spacetime symmetry can be

enhanced to R×SU (2) ×U(1). The corresponding black hole solution is called the Myers-

Perry (AdS) black hole with equal angular momenta [22–26]. Due to its high symmetry, this

spacetime is often used in the analyses of gravitational perturbations and phenomena speciﬁc

to rotating systems.

In this paper, we consider rotating solutions for the Einstein-Vlasov system with an R×

SU (2) ×U(1) isometry group appropriately setting the distribution function of the Vlasov

ﬁeld. A negative cosmological constant is introduced for the realization of the stationary

solutions with ﬁnite total mass and angular momentum. In general, a resultant spacetime

has an asymptotically locally AdS structure, whose spatial geometry is given by the foliation

of squashed S3hyper-surfaces even at spatial inﬁnity. The squashing parameter at spatial

inﬁnity can be set to zero by tuning the boundary condition at the center, and the spacetime

can be asymptotically AdS without squashing of S3at spatial inﬁnity. We note that a similar

situation has been reported in the vacuum cases [27].

The purpose of introducing a negative cosmological constant is not only for the construc-

tion of a physical solution with ﬁnite mass. Recently, asymptotically AdS spacetimes have

attracted much attention in the context of the AdS/CFT correspondence [28–30] and the

gravitational turbulent instability [31]. The conditions for the onset of the instability have

not yet been clariﬁed [32–45], and there are still few clues to the ﬁnal state. The system

provided in this paper may be treated as a macroscopic model of the ﬁnal state of a system

complicated by turbulent phenomena, and we expect that our analyses will be helpful to get

useful insights into the ﬁnal state of turbulent instability. In addition, the instability in the

Einstein-Vlasov system is also discussed in Refs. [8, 46, 47], and the possible relation with the

Hawking-Page transition has been pointed out in Refs. [48,49]. Another related phenomenon

is the superradiant instability of rotating black holes in asymptotically AdS spacetime, for

which the existence of ﬁnite angular momentum is essential. The ﬁnal fate of the super-

radiant instability has not been also clariﬁed yet. In order to approach the superradiant

instability through the construction of a macroscopic model with the Einstein-Vlasov system,

the introduction of ﬁnite angular momentum is a necessary step to be performed.

This paper is organized as follows. In section 2, we provide the metric ansatz of the spacetime

with an R×SU (2) ×U(1) isometry group. We also deﬁne the conserved quantities along

the geodesic of a particle and list the conditions which we impose on the metric functions for

technical and practical reasons. In section 3, introducing a speciﬁc form of the distribution

function, we show the explicit forms for the energy-momentum tensors (detailed calculations

are shown in App. A). We write down the Einstein ﬁeld equations for our system in section 4

and numerically solve them in section 5. Section 6 is devoted to a summary and conclusion.

Throughout this paper, we use the geometrized units in which both the speed of light and

gravitational constant in 5–dimension are unity, c=G= 1.

2 Metric Ansatz and conserved quantities

2.1 Metric ansatz with an R×SU (2) ×U(1) isometry group

We start with the following form of the metric:

g:=−e2µ(r)dt2+ e2ν(r)dr2+r2

4hσ12+σ22+σ32i+h(r)dt−a(r)

2σ32

,(2.1)

where σiare one-forms deﬁned as

σ1:=−sin φdθ+ sin θcos φdψ , (2.2a)

σ2:=−cos φdθ−sin θsin φdψ , (2.2b)

σ3:= dφ+ cos θdψ , (2.2c)

satisfying the Maurer-Cartan equation dσi+1

2ijkσj∧σk= 0. The ranges of the coordinate

variables are given by t∈(−∞,∞), r∈[0,∞), θ∈[0, π], φ∈[0,4π) and ψ∈[0,2π). The

third term in the metric

γ:=1

4hσ12+σ22+σ32i

4dθ2+ dφ2+ dψ2+ 2 cos θdφdψ.(2.3)

describes the metric on S3.

The three-sphere S3has two sets of SU(2) generators {ξi}i∈{1,2,3}and {σi}i∈{1,2,3}written

in the form:

ξ1=−sin ψ∂θ+ csc θcos ψ∂φ−cot θcos ψ∂ψ,(2.4a)

ξ2=−cos ψ∂θ−csc θsin ψ∂φ+ cot θsin ψ∂ψ,(2.4b)

ξ3=∂ψ,(2.4c)

and

σ1=−sin φ∂θ−cot θcos φ∂φ+ csc θcos φ∂ψ,(2.5a)

σ2=−cos φ∂θ+ cot θsin φ∂φ−csc θsin φ∂ψ,(2.5b)

σ3=∂φ.(2.5c)

The SU (2) generators {ξi,σi}satisfy the following commutation relations:

[ξi,ξj] = ijkξk,[σi,σj] = ij kσk,[ξi,σj] = [σi,ξj]=0,(2.6)

or equivalently

Lξiξj=ijkξk,Lσiσj=ijkσk,Lξiσj= 0.(2.7)

The Killing vectors {ξi}and σ3on the three-sphere are also the Killing vectors on the

spacetime due to Eq. (2.7). On the other hand, neither σ1nor σ2is the Killing vector

on the spacetime due to the last term in Eq. (2.1) unless a(r) = 0 everywhere. Since the

vector σ3generates a U(1) isometry group and the metric has a time-like Killing vector

η=∂t, the spacetime has the Rt×SU (2)ξ×U(1)σisometry group. The black hole solutions

which have the same symmetry are known as the Myers–Perry black holes with equal angular

momenta [24, 25]. It would be worth noting that the angular coordinates θ,φand ψare

related to the Hopf coordinates ˜

θ,˜

φand ˜

ψthrough θ= 2˜

θ,ψ=−˜

φ+˜

ψand φ=˜

φ+˜

ψ. Then

the two equal angular momenta are the conserved charges associated with the Killing vectors

∂˜

φand ∂˜

ψ.

In this paper, to avoid possible technical problems, we focus on the cases satisfying the

following four conditions:

1. Non-degeneracy: det g<0,

2. No-horizon: r·r>0,

3. Time-like Killing vector exists everywhere: η·η<0,

4. Time-like unit normal exists everywhere: n·n<0,

where r:=|grr|−1

2drand n:=−gtt

−1

2dtare the unit normal to the r= const.and the

t= const.hyper-surfaces, respectively. The condition 3 implies that the spacetime has no

ergo-region1. The conditions 1 - 4 yield

e2µ(r)−h(r),e2ν(r), F (r), G(r)>0,(2.8)

for any r > 0, where G(r):=r2+a2hand F(r):= e2µG−hr2.

2.2 Conserved quantities for the geodesic motion

The symmetry of the spacetime indicates the existence of conserved quantities for the geodesic

motion in the form of the inner product between the Killing vector and the momentum of a

particle. Then we can consider the following mutually independent conserved quantities for

the metric (2.1):

ε=−p·η=−pt,(2.9a)

Jξ1=p·ξ1=−pθsin ψ+pφcsc θcos ψ−pψcot θcos ψ, (2.9b)

Jξ2=p·ξ2=−pθcos ψ−pφcsc θsin ψ+pψcot θsin ψ, (2.9c)

Jξ3=p·ξ3=pψ,(2.9d)

jσ=p·σ3=pφ,(2.9e)

1In Ref. [20], the solution with an ergo-region is constructed in a 4–dimensional asymptotically ﬂat space-

time.

where pis the momentum of the particle and ‘·’ denotes the inner product concerning g. The

total angular momentum Jξ≥0 of the SU(2)ξsector can be deﬁned as

Jξ2:=

i=1

Jξi

2=1

4γµν pµpν,(2.10)

where we have deﬁned γµν as γµν := 4 Piσiµσiν. Then ε,jσand Jξare mutually commutative

conserved charges and can be used as independent coordinates in the phase space.

The momentum of the particle must satisfy the on-shell condition: p2+m2= 0, which can

be rewritten as follows:

e−2νG

Fε−2ah

Gjσ2

−e−2νm2+4

r2Jξ2−a2h

Gjσ2= (pr)2.(2.11)

We note that the left-hand side of Eq. (2.11) can be regarded as the eﬀective potential for

the geodesic motion. Here we impose that the momentum is future pointing n·p<0. Then

the positivity of the local energy of the particle is ensured:

ε−2ah

Gjσ>0.(2.12)

The allowed region in the momentum space for the particle is the subspace satisfying the

conditions (2.11) and (2.12).

3 Model and physical quantities

3.1 Distribution function and the energy-momentum tensor

Considering a collisionless many-particle system, the particles follow the geodesic motion.

Therefore the distribution function fsatisﬁes the Vlasov (collisionless Boltzmann) equation:

pµ∇µf=pµ∂f

∂xµ−Γiµν pµpν∂f

∂pi= 0,(3.1)

which implies the conservation of the distribution function along the geodesic. If the distri-

bution function is written in the form

f=f(ε, Jξ, jσ),(3.2)

the Vlasov equation (3.1) is automatically satisﬁed because ε,Jξand jσare conserved quan-

tities along the geodesic.

In this paper, as a simple speciﬁc model, we assume the following distribution function

f(ε, jσ) = exp [α−β(ε−Ωjσ)],(3.3)

with constants α∈R,β > 0 and Ω >0. The distribution function (3.3) reduces to the

Maxwell–J¨uttner distribution, which describes the relativistic thermal equilibrium states in

static cases, with Ω = 0. Thus the system with this distribution function can be regarded as

an extension of the thermal equilibrium state to the system with a ﬁnite angular momentum.

The Maxwell–J¨uttner distribution is derived by extremizing the entropy of the system ﬁxing

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Einstein{Vlasovsystemwithequal-angularmomentainAdS5HirokiAsami*1,Chul-MoonYoo1,RyoKitaku1,andKeiyaUemichi§11DivisionofParticleandAstrophysicalScience,GraduateSchoolofScience,NagoyaUniversity,Nagoya464-8602,JapanAbstractWeinvestigatesolutionsofthe5{dimensionalrotatingEinstein-VlasovsystemwithanRSU...

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EinsteinVlasov system with equal-angular momenta in AdS 5 Hiroki Asami1 Chul-Moon Yoo1 Ryo Kitaku1 and Keiya Uemichi1

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