Covariant transport equation and gravito-conductivity in generic stationary spacetimes

2025-05-06 1 0 566.68KB 17 页 10玖币

侵权投诉

Covariant transport equation and

gravito-conductivity

in generic stationary spacetimes

Song Liu1, Xin Hao2, Shaofan Liu3, and Liu Zhao1∗

1School of Physics, Nankai University, Tianjin 300071, China

2School of Physics, Hebei Normal University, Shijiazhuang 050024, China

3Department of Physics, University of Houston, Houston, TX 77204, USA

email: 2120190124@mail.nankai.edu.cn

xhao@hebtu.edu.cn,sliu74@cougarnet.uh.edu

and lzhao@nankai.edu.cn

Abstract

We ﬁnd a near detailed balance solution to the relativistic Boltzmann equation under

the relaxation time approximation with a collision term which diﬀers from the Anderson-

Witting model and is dependent on the stationary observer. Using this new solution,

we construct an explicit covariant transport equation for the particle ﬂux in response to

the generalized temperature and chemical potential gradients in generic stationary space-

times, with the transport tensors characterized by some integral functions in the chemical

potential and the relativistic coldness. To illustrate the application of the transport equa-

tion, we study probe systems in Rindler and Kerr spacetimes and analyze the asymptotic

properties of the gravito-conductivity tensor in the near horizon limit. It turns out that

both the longitudinal and lateral parts (if present) of the gravito-conductivity tend to be

divergent in the near horizon limit. In the weak ﬁeld limit, our results coincide with the

non-relativistic gravitational transport equation which follows from the direct application

of the Drude model.

1 Introduction

Transport equations for macroscopic systems under the inﬂuence of external ﬁelds are critical

in understanding non-equilibrium processes. For electron gas in solid state physics, the

external ﬁelds can be electromagnetic ﬁeld and/or temperature gradient. For compact stars

and accretions around black holes, the relativistic gravitational ﬁeld must be taken into

consideration. In the recent years, there has been an increasing interest in studying the

evolution of kinetic gases in the vicinity of black holes. On the one hand, the macroscopic

properties of near horizon systems are likely to be connected with thermodynamic eﬀects of

gravity. On the other hand, the rotating plasma surrounding supermassive black holes has

been revealed by Event Horizon Telescope Collaboration [1]. Motivated by these astrophysical

∗Correspondence author.

arXiv:2210.10907v3 [gr-qc] 7 Dec 2022

scenarios, it is necessary to dwell on some more details of the transport phenomena in the

strong gravity regime.

Kinetic theory has long been used for constructing transport equations. In the early

stages of the universe and in the near horizon region, gravity is so strong that the relativistic

eﬀect becomes important. In this situation, a relativistically covariant approach is needed.

Such a systematic approach is known as the relativistic kinetic theory (RKT) and could be

dated back to 1911 [2]. The modern geometrical description of specially relativistic kinetic

theory is formulated in [3], which is fundamental for the generalization to curved spacetime

backgrounds [4–6]. Renewed interests in this area arise following Refs. [7–9], where RKT has

been respectively applied to the study of the quark-gluon plasma and heavy ion collisions

[10–19], the early Universe [20–29] and astrophysics [30–33], particularly accretion around

black holes [9,33–35]. The covariant formalism also led to the development of relativistic

thermodynamics [36,37].

The aim of the present work is to provide a covariant formulation for transport phenomena

in curved spacetimes. There have been studies about the gravitational eﬀects on the transport

of conserved charge, heat and momentum [38–40]. The major diﬀerence between the present

work and the previous ones lies in that we emphasize the importance of the observer, and that

our collision model is independent of Landau frame. The reason for stressing the important

role of observers lies as follows. As we have pointed out in [41], in relativistic physics,

physical laws are independent of the choice of coordinate system, whilst the values of physical

observables are observer dependent. In the description of relativistic transport phenomena,

the transport equation encodes the underlying physical law which needs to be described in

a covariant formalism. On the other hand, the values of the relevant transport coeﬃcients

— although also need to be tensorial objects — describe the phenomenological properties of

concrete systems which depend on the choice of observer. As will be seen in the main texts

below, the proper velocity of the stationary observer is crucial when modeling the collision

term and interpreting the kinetic tensor.

Another diﬀerence of the present work with previous ones is that the lateral transport in

our framework can be clearly separated from the longitudinal transport. One may think that

the lateral eﬀects are negligibly small as compared to the longitudinal eﬀects. This may be a

truth in some cases, but not always. For instance, around rapidly spinning black holes, the

lateral eﬀects are at least equally as important as the longitudinal eﬀects.

For simplicity, we will work on the assumption that the gaseous (or ﬂuid) system is

consisted of massive neutral particles and is regarded as a probe system.

2 GEM ﬁeld and the gravitational Hall coeﬃcient under post-

Newtonian approximation

In this section, we work within the post-Newtonian limit, i.e. when the gravitational ﬁeld

is weak, and the motion of the source is slow but still taken into consideration. In this

limit, the gravitational ﬁeld can be described by small perturbations hµν around Minkowski

metric. When all the nonlinear terms in hµν are neglected, there will be an analogy between

Einstein equations under de Donder gauge and the Maxwell equation under Lorenz gauge. In

4-dimensional spacetimes, such an analogy is known as the linear gravito-electromagnetism

(GEM) [42–45]. The solution of the linearized Einstein equation suggests that, up to O(c−2),

we can write the metric as

ds2=−c21 + 2φ

c2dt2+4

cA·dxdt+1−2φ

c2δij dxidxj,(1)

where i, j = 1,2,3, φ=h00/2c2is the scalar potential and Ai=h0i/2c2is the vector

potential. For slowly varying source, the gravito-electric (GE) ﬁeld and the gravito-magnetic

(GM) ﬁeld become E(g)=−∇φand B(g)=∇×A, which satisfy a set of equations analogous

to the Maxwell equations in 3-vector formulation.

Another important aspect of the linear GEM framework lies in the equation of motion of a

test particle. In non-relativistic limit, the proper velocity of the test particle is approximately

uα≈(c, v), and up to O(c−1) the geodesic equation in terms of GE and GM reduces to

Lorentz-force-like equation

dt=E(g)+ 2 v

c×B(g),(2)

as a simple application of eq.(2), we can now follow the Drude model of traditional Hall eﬀect

and introduce a linear friction term −v/τ to the RHS, which is due to scattering between

particles, and the scattering time τis the average time between collisions. When the particle

becomes equilibrated, we have roughly dv/dt= 0. In this case, the particle ﬂow j=nvis

related to external force E(g)by

R(g)j− R(g)

Hj×B(g)=E(g),(3)

where R(g)= 1/nτ is the gravitational resistance and R(g)

H= 2/nc is the gravitational

Hall coeﬃcient for neutral systems. This analogy is clear and instructive, but the Drude

model is highly dependent on the Lorentz-force-like equation, therefore only applies to weak

gravitational ﬁeld.

In the rest of this work, we will generalize the transport equation to generic stationary

spacetime backgrounds in a fully covariant form and calculate the gravitational conductivity

for neutral systems as an example case for applications.

3 Relativistic Boltzmann equation and approximate solution

Before proceeding, let us mention that, in ordinary electrodynamics, diﬀerent observers should

observe diﬀerent electric (E) and magnetic (B) ﬁelds, which is known as the Faraday eﬀect.

Likewise, The GE and GM ﬁelds must also be observer dependent. The description presented

in the last section did not show this observer dependence, because a speciﬁc observer, i.e. the

static observer in the Minkowski background spacetime, is implicitly adopted.

In the following, we will try to establish a fully covariant formalism for the relativistic

transport equations. For this purpose, we need to bring back the explicit observer dependence.

It turns out that the stationary observer is best suited for our investigation. Therefore we

shall assume that the spacetime is stationary (i.e. there exists a timelike Killing vector ﬁeld).

From now on, we will be working in natural units with c=h=kB= 1. We introduce a

generic observer Owith proper velocity Zµ(with ZµZµ=−1) and the observer dependence

of the GE and GM ﬁelds are best illustrated by their expressions in terms of Zµ, i.e.

E(g)

µ=−Zν∇νZµ, B(g)

µν =1

2∇[αZβ]∆αµ∆βν,(4)

where ∆µν =gµν +ZµZµis the normal projection tensor associated with the observer O. It

should be mentioned that the covariant GE ﬁeld E(g)

µis proportional to the proper acceleration

of the observer O. The GM ﬁeld represented in vector form can be introduced as

B(g)µ=−∆µνεναβ B(g)

αβ ,

where εναβ is the Levi-Civita tensor.

Now let us proceed with a brief review of the relativistic kinetic theory for a system

composed of neutral particles moving in curved spacetime. The macroscopic state of the

system is described by the particle number ﬂow Nµ, the energy momentum tensor Tµν and

the entropy current Sµ. These quantities are primary in the ﬂuid description of macroscopic

systems [46]. In the absence of external ﬁeld besides gravity, one has

∇µNµ= 0,∇µTµν = 0,∇µSµ>0.(5)

The expressions for Nµ,Tµν and Sµin terms of other macroscopic observables such as

temperature T, chemical potential µand entropy density sare called constitutive relations. It

is important to keep in mind that T,µ,sand therefore the constitutive relations are observer

dependent, but the above primary macroscopic quantities are not, since they have microscopic

deﬁnitions. In standard relativistic kinetic theory, the primary macroscopic quantities are

expressed as integrations over one-particle distribution function (1PDF),

Nµ=Z$pµf, T µν =Z$pµpνf, (6)

Sµ=−Z$pµflog f−ς−1(1 + ςf) log (1 + ςf),(7)

where ς= 0,+1,−1 respectively corresponds to the non-degenerate, bosonic and fermionic

cases, $=g√g

p0(dp)dis the invariant volume element in the momentum space with gbeing the

intrinsic degeneracy factor for individual particles in the system, and the spacetime dimension

is chosen to be (d+ 1). The above deﬁnitions are valid for systems away from equilibrium.

Therefore, in kinetic theory, we can ask whether the equilibrium state can be achieved in

curved spacetime, and ﬁnd the constraints for the spacetime metric to embed an equilibrium

system. Since the primary macroscopic quantities are totally determined by the 1PDF, the

relativistic Boltzmann equation (RBE) is then of fundamental importance

LHf=C(f),(8)

where LH=pµ∂

∂xµ−Γµαβ pαpβ∂

∂pµis the Liouville vector ﬁeld on the tangent bundle of the

spacetime.

There are two-fold complexities in the RBE, which make it challenging to ﬁnd an exact

solution. On the left hand side, the spacetime geometry is encoded in the Liouville vector

ﬁeld, however the spacetime geometry itself can only be determined by solving the Einstein

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Covarianttransportequationandgravito-conductivityingenericstationaryspacetimesSongLiu1,XinHao2,ShaofanLiu3,andLiuZhao1*1SchoolofPhysics,NankaiUniversity,Tianjin300071,China2SchoolofPhysics,HebeiNormalUniversity,Shijiazhuang050024,China3DepartmentofPhysics,UniversityofHouston,Houston,TX77204,USAemail...

展开>> 收起<<

Covariant transport equation and gravito-conductivity in generic stationary spacetimes.pdf

共17页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Covariant transport equation and gravito-conductivity in generic stationary spacetimes

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: