relativistic kinetic theory

8/10/2019 Relativistic Kinetic Theory

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arXiv:1303.2899v1[gr-qc]12Mar2013

Relativistic Kinetic Theory: An Introduction

Olivier Sarbach and Thomas Zannias

Instituto de Fsica y Matemticas, Universidad Michoacana de San Nicols de HidalgoEdificio C-3, Ciudad Universitaria, 58040 Morelia, Michoacn, Mxico.

Abstract. We present a brief introduction to the relativistic kinetic theory of gases with emphasison the underlying geometric and Hamiltonian structure of the theory. Our formalism starts witha discussion on the tangent bundle of a Lorentzian manifold of arbitrary dimension. Next, weintroduce the Poincar one-form on this bundle, from which the symplectic form and a volume formare constructed. Then, we define an appropriate Hamiltonian on the bundle which, together with thesymplectic form yields the Liouville vector field. The corresponding flow, when projected ontothe base manifold, generates geodesic motion. Whenever the flow is restricted to energy surfacescorresponding to a negative value of the Hamiltonian, its projection describes a family of future-

directed timelike geodesics. A collisionless gas is described by a distribution function on suchan energy surface, satisfying the Liouville equation. Fibre integrals of the distribution functiondetermine the particle current density and the stress-energy tensor. We show that the stress-energytensor satisfies the familiar energy conditions and that both the current and stress-energy tensor aredivergence-free.

Our discussion also includes the generalization to charged gases, a summary of the Einstein-Maxwell-Vlasov system in any dimensions, as well as a brief introduction to the general relativisticBoltzmann equation for a simple gas.

Keywords: general relativity, kinetic theory, Einstein-Maxwell-Vlasov systemPACS: 04.20.-q,04.25.-g,04.40.-b

INTRODUCTION

Kinetic theory of gases is an old subject with rich history. Its acceptance as a scien-

tific theory with potential predictive power marked the revival of the atomic theory of

nature proposed by Democritus and Leucippus in the ancient time. As early as 1738,

Daniel Bernoulli proposed that gases consist of a great number of molecules movingin all directions and the notion of pressure is a manifestation of their kinetic energy of

motion. Gradually, the idea of Bernoulli developed further culminating in the formula-

tion of the Maxwellian distribution of molecular velocities and the Maxwell-Boltzmann

distribution at the end of the 19thcentury.1

The arrival of the 20thcentury marks a new era for kinetic theory. Einsteins funda-

mental 1905 paper on Brownian motion establishes the atomic structure of matter, and

moreover the birth of special relativity set new challenges in re-formulating kinetic the-ory and its close relative thermodynamics, so that they are Poincar covariant theories.

As early as 1911, Jttner treated the equilibrium state of a special relativistic gas [2, 3, 4]

while early formulations of (special) relativistic kinetic theory and relativistic thermo-

1 For historical facts regarding the early development of kinetic theory see Ref. [1].
http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1http://arxiv.org/abs/1303.2899v1


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dynamics, successes, failures as well as early references can be found in the books by

Pauli [5] and Tolman [6].

Synge in 1934 [7] (see also [8]) introduced the notion of the world lines of gas

particles as the most fundamental ingredient for the description of a relativistic gas,

and this idea led to the development of the modern generally covariant formulationof relativistic kinetic theory. Synges idea led naturally to the notion of the invariant

distribution function and a statistical description of a relativistic gas which is fully

relativistic.

The period after 1960 characterizes the modern development of relativistic kinetic

theory based on the relativistic Boltzmann equation, and an early treatment can be foundin the paper by Tauber and Weinberg [9]. An important contribution to the subject

was the work by Israel [10] where conservation laws and the relativistic version of

the H-theorem is presented and the important notion of the state of thermodynamicalequilibrium in a gravitational field is clarified. Further it has been recognized in [10]

that a perfect gas has a bulk viscosity, a purely relativistic effect. Gradually, with the

discovery of the cosmic microwave background radiation, pulsars and quasars it has

become evident that relativistic flows of matter are not any longer just mathematical

curiosities but they are of relevance to astrophysics and early cosmology as well. These

new discoveries led to further studies of relativistic kinetic theory and the formulation

of the transient thermodynamics [11, 12, 13, 14, 15, 16]. The development of black hole

physics and the realization that their interaction with the rest of the universe requires afully general relativistic treatment, promoted relativistic kinetic theory to an important

branch of relativistic astrophysics and cosmology. For an overview, see the recent book

by Cercignani and Kremer [17].

The formulation of the Cosmic Censorship Hypothesis (CSH) led to the search of

matter models that go beyond the traditional fluids or magneto-fluids, and here studies

of the Einstein-Liouville, Einstein-Maxwell-Vlasov and Einstein-Boltzmann equations

are very relevant and on the frontier of studies in mathematical relativity, see for instanceRefs. [18, 19, 20, 21] and Ref. [22] for a recent review.

The strong cosmic censorship hypothesis affirms that the maximal Cauchy develop-

ment of generic initial data for the Einsteins equations should be an inextendible space-

time. Since this development is the largest region of the spacetime which is uniquelydetermined by the initial data, CSH affirms that the time evolution of a spacetime can

be generically fixed by giving initial data. In any study of the CSH the choice of the

matter model is very important. For instance, dust and perfect fluid models may lead to

the formation of shell crossing singularities and shocks even in the absence of gravity.

Their formation obscures a clear understanding of the global spacetime structure associ-

ated to the gravitational singularities. Kinetic theory offers a good candidate for a matter

model that avoids these problems. Studies of solutions of the Einstein-Liouville systemare characterized by a number of encouraging properties, see for instance [23] and [22].

Motivated by the above considerations, in this work we present a modern introduction

to relativistic kinetic theory. Our work is mainly based on Synges ideas and early work

by Ehlers [24], however, in contrast to their work, we derive the relevant ingredientsof the theory using a Hamiltonian formulation. As will become evident further ahead,

our approach exhibits transparently the basic ingredients of the theory and leads to

generalizations.


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In this work, (M,g) denotes a C-differentiable Lorentzian manifold of dimensionn = 1 + d, with the signature convention(,+,+, . . . ,+)for the metricg. Greek indices,,, . . .run from 0 todwhile Latin indicesi,j,k, . . .run from 1 to d, and we use theEinstein summation convention. X(N) and k(N) denote the class ofC vector and

k-form fields on a differentiable manifold N, whileiXandXdenote the interior productand the Lie derivative with respect to the vector field X.

THE TANGENT BUNDLE

In this section we begin with the definition of the tangent bundle and summarize some

of its most important properties. Let TxMdenote the vector space of all tangent vectorspat some eventx M. The tangent bundle ofMis defined as

T M:= {(x,p):x M,p TxM},

with the associated projection map :T M M,(x,p) x. The fibre at x M is thespace1(x) = (x,TxM), and thus, it is isomorphic to TxM.

Lemma 1. TM is an orientable,2n-dimensional C-differentiable manifold.

Remark: Notice thatT Mis orientable regardless whetherMis oriented or not.

Proof. The proof is based on the observation that a local chart(U,)ofMdefines in anatural way a local chart(V,)as follows. LetV :=1(U)and define

: V (U)Rn R2n,

(x,p) x0,x1, . . . ,xd,p0,p1, . . . ,pd := ((x,p)),dx0x(p),dx1x (p), . . . ,dxdx (p) .By taking an atlas (U,) ofM, the corresponding local charts (V,) cover T M.Furthermore, one can verify that the transition functions are C-differentiable and that

their Jacobian matrix have positive determinant, yielding an oriented atlas ofT M.

We call the local coordinates(x0,x1, . . . ,xd,p0,p1, . . . ,pd)adapted local coordinates,

and

x0

(x,p)

, . . . , pd

(x,p)

and

dx0

(x,p), . . . ,d pd(x,p)

are the corresponding basis of

the tangent and cotangent spaces ofT Mat(x,p). Any tangent vectorL T(x,p)(T M)canthen be expanded as

L=X

x

(x,p)

+ P

p

(x,p)

, X =dx

(x,p)

(L), P =d p

(x,p)

(L).

Likewise, a cotangent vector T(x,p)(T M)can be expanded as

= dx|(x,p)+d p

|(x,p) , =

x

(x,p)

, =

p

(x,p)

.


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The projection map :T M M induces a projection (x,p) :T(x,p)(T M) TxMthrough the push-forward of, defined as(x,p)(L)[g]:=L[g]for a tangent vectorLinT(x,p)(T M)and a functiong:MR which is differentiable at x. It is a simple matterto verify that

(x,p)

x

(x,p)

=

x

x

, (x,p)

p

(x,p)

=0, (1)

and thus, the projection of an arbitrary vector fieldL X(T M)on T Mis given by

(x,p)(L(x,p)) =X(x,p)

x

x

, L(x,p)=X(x,p)

x

(x,p)

+ P(x,p)

p

(x,p)

,

in adapted local coordinates.

For the following, we consider a differentiable curve :I M, () on M. Itinduces a parameter-dependentliftwhich is defined in the following way:

: I T M, ():=

(),

d

d()

.

Since = it follows immediately that the tangent vectors X and X of and arerelated to each other by (X) =X. In adapted local coordinates(x

,p)we can expand

X()=X()

x

()

, X()=X()

x

()

+ P()

p

()

,

where the coefficients are given by

X() = x(), P() = x(),

wherex()parametrizes the curve in the local chart(U,)of the base manifold, anda dot denotes differentiation with respect to.

As a particular example of this lift, consider the trajectory:I Mof a particle ofmassm> 0 and chargeqin an external electromagnetic fieldF 2(M). The equationsof motion are

pp=q F(p), p:= d

d(), (2)

where F: X(M) X(M)is defined by g(X, F(Y)) =F(X,Y)for allX,YX(M),andis an affine parameter, normalized2 such thatg(p,p) =m2. Consider the tangent

vectorL to the associated lift :I T M. Since in adapted local coordinates x

= p

and p = pp+ qFp

, we find

L(x,p)= p

x

(x,p)

+

qF(x)p(x)p

p p

(x,p)

, (3)

2 Notice that Eq. (2) implies thatg(p,p)is constant along the trajectories, due to the antisymmetry ofF.


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and thisL defines a vector field on . By extending this construction to arbitrary curvesone obtains a vector fieldL on T M, called theLiouville vector field. In the next section,

we shall provide an alternative definition of the Liouville vector field based on Hamilto-

nian mechanics.

HAMILTONIAN DYNAMICS ON THE TANGENT BUNDLE

The purpose of this section is twofold. At first, we introduce a symplectic structureon the tangent bundle which in turn is the backbone of our formulation of kinetic

theory. Secondly, we introduce a natural Hamiltonian function onT Mwhose associated

Hamiltonian vector field Lcoincides with the Liouville vector field defined in Eq. (3).

Moreover, the symplectic structure defines a natural volume form on T Mwhich will behelpful to set up integration on T M.

In order to define the symplectic structure, we note that the spacetime metric ginduces

a natural one-form 1(T M) on the tangent bundle, called the Poincar or the

Lioville one-form. It is defined as

(x,p)(X):=gx(p,(x,p)(X)), X T(x,p)(T M), (4)

at an arbitrary point(x,p) T M. In terms of adapted local coordinates (x,p)we mayexpand

X= X

x

(x,p)

+Y

p

(x,p)

, (x,p)(X) =X

x

x

, p= p

x

x

,

and obtain

(x,p)=g(x)p dx|(x,p) , (5)

which shows that is C-differentiable. The symplectic form s on T Mis defined as

the two-forms:=d, (6)

which is closed. In adapted local coordinates we obtain from Eq. (5),

s=g(x)d pdx|(x,p)+

g

x (x)p dxdx|(x,p) . (7)

The following proposition shows that s induces a natural volume form on T M, and

thus it is non-degenerated.

Proposition 1. The n-fold product3

:=(1)

n(n1)2

n! ss . . .s

2n(T M) (8)

satisfies (x,p)=0 for all(x,p) T M, and thus defines a volume form on T M.

3 The choice for the normalization of will become clear later.


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Proof. Using Eq. (7) we find, in adapted local coordinates,

= (1)

n(n1)2

n!g00 g11 . . .gddd p

0 dx0 d p1 dx1 . . .d pddxd

= 1n!

g00 g11 . . .gddd p0 d p1 . . .d pddx0 dx1 . . .dxd

= 1

n!g00 g11 . . .gdd

01...d01...dd p0 . . .d pddx0 . . .dxd

= det(g)d p0 . . .d pddx0 . . .dxd, (9)

which is different from zero since the metric is non-degenerated.

Finally, we introduce the Hamiltonian function

H: T MR,(x,p) 1

2gx(p,p). (10)

This Hamiltonian and the symplectic structure s define the associated Hamiltonianvector fieldL X(T M)on the tangent bundle by

dH= s(,L) = iLs, (11)

which is well-defined since s is non-degenerated. In order to make contact with the

results from the previous section we work out the components ofL in an adapted local

coordinate system(x,p). For this, we expand

H(x,p) =1

2g(x)p

p, L=X

x

(x,p)

+Y

p

(x,p)

,

from which we obtain

dH=1

2

g

x ppdx+ gp

d p. (12)

On the other hand, using Eq. (7), we find

iLs = g[d p(L)dxdx(L)d p] +

g

x p[dx(L)dxdx(L)dx]

=

gY

+

g

x

g

x

pX

dxgX

d p. (13)

Comparing Eqs. (12,13) we concludeX =p and

gY =

1

2

g

x 2

g

x

pp = g

p

p,

and thus it follows that

L= p

xp

p

p. (14)


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In the absence of an external electromagnetic field this Hamiltonian vector field co-

incides with the Liouville vector field on T Mdefined in Eq. (3). Therefore, we have

shown:

Theorem 1. Let be an integral curve of the Hamiltonian vector field L defined in

Eq. (11), and consider its projection := onto M. Then,is necessarily an affinelyparametrized geodesics: its tangent vector, p satisfies pp= 0, with denoting theLevi-Civita connection belonging to g.

The ideas outlined so far can be extended to diverse physical systems. As an example,

here we discuss the case of particles with charge q interacting with an external electro-

magnetic and gravitational field. Remarkably, a minimal modification of the Poincar

one-form is sufficient to generalize the previous result. The modified Poincar one-form

is

(x,p)(X):=gx(p,(x,p)(X)) + qAx((x,p)(X)), X T(x,p)(T M), (15)

whereA 1(M)is the electromagnetic potential. In adapted local coordinates Eq. (5)

is replaced by(x,p)=

g(x)p

+ qA(x)

dx|(x,p) . (16)

This modification is natural in view of the fact that in the presence of an external

electromagnetic field the canonical momentum is related to the physical momentumpby = p+ qA. The symplectic form s=d now reads

s=g(x)d pdx|(x,p)+

gx

(x)p+q

2F(x)

dxdx|(x,p) , (17)

whereF= dAis the electromagnetic field strength. Note that this s is independent ofthe gauge choice. Furthermore, Eq. (17) shows that the volume form defined in Eq. (8)

is unaltered. Choosing the Hamiltonian function as in Eq. (10), the resulting Hamiltonianvector fieldL coincides with the one defined in Eq. (3).

We end this section with a simple but useful result:

Proposition 2. We have LH = 0, Ls = 0 and L = 0, which implies that thequantities H , s and are invariant with respect to the flow generated by L.

Proof. Using the Cartan identity we first find LH= iLdH=i2

Ls= 0 and Ls=diLs+ iLds= d

2H= 0. With this,L =0 follows directly from the definition inEq. (8).

Remark: Since L = (divL) it follows from this proposition that the Liouvillevector field is divergence-free, divL=0. Therefore, relative to the volume form , the

flow inT Mgenerated byL is volume-preserving.

THE MASS SHELL

We now consider a simple gas, that is, a collection of neutral or charged, spinless

classical particles of the same rest mass m> 0 and the same charge q moving in atime-oriented background spacetime(M,g)and an external electromagnetic field F. We


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assume that the particles interact only via binary elastic collisions idealized as a point-

like interaction.4 Therefore, between collisions, for the uncharged case, the particles

move along future-directed timelike geodesics of(M,g)while for the charged case theymove along the classical trajectories determined by Eq. (2). From the tangent bundle

point of view, the gas particles follow segments of integral curves of the Liouville vectorfieldL. Since all gas particles have the same rest mass, these segments are restricted to

a particular subset mofT Mreferred to as the mass shell. m is defined as

m:=H1

m2

2

=

(x,p) T M: 2H(x,p) =gx(p,p) = m2, (18)

whereH: T M R is the Hamiltonian defined in Eq. (10). In this section we discuss afew relevant properties of the mass shell. The first important property is described in the

following lemma.

Lemma 2. m is a(2n1)-dimensional C-differentiable manifold.

Proof. We consider an arbitrary point (x,p) m. Sincegx(p,p) = m2 0. Then,(M,g) is time-orientable ifand only ifm is disconnected, in which case it is the disjoint union of two connected

components +m andm .

4 For the uncharged case, the self-gravity of the gas particles will be incorporated in a self-consistentmanner in the Einstein-Liouville system. For the charged case, the self-gravity and self-electromagneticfield of the gas particles will be taken into account by imposing the Einstein-Maxwell-Vlasov equationsdiscussed further ahead.


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Proof. See appendix A.

In Lemma 2 we proved that m is a submanifold ofT M. In the next lemma we show

that the volume form on T Mdefined in Eq. (8) induces a (2n 1)-form on themass shell mwhich is non-vanishing at every point. In particular, m is oriented by the

volume form

.Lemma 3. (i) Consider the open subset V := {(x,p) T M: H(x,p)< 0} TM of

the tangent bundle. There exists a(2n1)-formon V such that for all(x,p) V

dH(x,p)(x,p)= (x,p). (21)

(ii) The(2n1)-form on m, defined by the pull-back := ofwith respect to

the inclusion map : m V TM, is independent of the choice for in (i) anddefines a volume form on m.

Remark: Notice that the (2n 1)-form is not unique since we can add to it anyfield of the form d Hwith a(2n2)-form. However, the pull-back of to m is

unique.Proof. (i) As a first step, we show the existence of a vector field N X(V) with

the property that dH(N) = 1 onV. For this, consider the one-parameter group ofdiffeomorphisms : VV, (x,p) (x,ep), R, which induces the rescalingby the factor e in each fibre. LetXX(V) be the corresponding generating vectorfield,

X(x,p):= d

d(x,p)

=0

.

Then, it follows for each(x,p) V that

dH(x,p)(X) =X(x,p)[H] = d

dH(

(x,p))=0

= d

d

1

2gx(e

p,e

p)=0

= 2H(x,p).

Therefore,N:=X/(2H)yields the desired vector field.Next, define :=iN = (N, , , . . . , ). We claim that this satisfies Eq. (21).In order to verify this claim, we note that it is sufficient to show that Eq. (21)

holds at each (x,p) V when both sides are evaluated on a particular basis ofT(x,p)(T M). A convenient basis is constructed as follows: consider the (2n 1)-dimensional submanifold H=const. through (x,p). Since N(x,p) is transverse tothis submanifold, it can be completed to a basis {X1 := N(x,p),X2, . . . ,X2n} ofT(x,p)(T M)such that eachXi,i = 2, . . . ,2n, is tangent to the submanifoldH= const,that is,dH(x,p)(Xi) =0 fori=2, . . . ,2n. The claim follows by noting that

dH(x,p)(x,p)(X1,X2, . . . ,X2n) = (x,p)(X2, . . . ,X2n) = (x,p)(N(x,p),X2, . . . ,X2n).

(ii) Let(x,p) m Vand let the basis {X1=N(x,p),X2, . . . ,X2n} at(x,p)be definedas above. Then,

(x,p)(X2, . . . ,X2n) = ()(x,p)(X2, . . . ,X2n) = (x,p)(X2, . . . ,X2n)

= (x,p)(X1,X2, . . . ,X2n),


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where we have used Eq. (21) in the last step. This shows that (x,p) is uniquely

determined by (x,p). Furthermore, as a consequence of Proposition 1, the right-

hand side is different from zero which proves that (x,p) =0.

For the formulation of the next result, it is important to note that the Liouvillevector field L X(T M) is tangent to m. This property follows from the fact thatdH(L) =iLdH=LH=0, see Proposition 2. Therefore, we may also regard L as avector field on m.

Theorem 2 (Liouvilles theorem). The volume form on m defined in the previous

lemma satisfies

L = (divL) =0. (22)

Proof. Let 2n1(V)be as in Lemma 3(i). Taking the Lie-derivative with respect toLon both sides of Eq. (21) we obtain, taking into account the results from Proposition 2,

dH(x,p) (L)(x,p)=0 (23)

for all(x,p) m. LetX2,X3, . . . ,X2nbe vector fields on m, and letNX(V)be suchthatdH(N) =1. Evaluating both sides of Eq. (23) on(N,X2,X3, . . . ,X2n), we obtain

0 = (L)(X2,X3, . . . ,X2n)

= L [(X2,X3, . . . ,X2n)]([L,X2],X3, . . . ,X2n) . . .(X2,X3, . . . , [L,X2n])

= L [(X2,X3, . . . ,X2n)]([L,X2],X3, . . . ,X2n) . . .(X2,X3, . . . , [L,X2n])

= (L)(X2,X3, . . . ,X2n),

where we have used the properties of the Lie derivative in the second and fourth step

and the definition of plus the fact that L is tangent to m in the third step. Therefore,L =0 and the lemma follows.

We close this section by introducing local coordinate charts on the mass shell m.For that, let (U,) be a local chart of (M,g) with corresponding local coordinates(x0,x1, . . . ,xd), such that

x0

x

,

x1

x

, . . . ,

xd

x

, x U,

is a basis ofTxM, with the property that for each x U, x0

x

is timelike and xi

x

,i =

1,2, . . . ,dare spacelike. Let(V,)denote the local chart ofT Mwith the correspondingadapted local coordinates (x,p) constructed in the proof of Lemma 1. Relative tothese local coordinates, the mass shell is determined by

m2 =g(x)pp =g00(x)(p

0)2 + 2g0j(x)p0pj + gi j(x)p

ipj. (24)

Therefore, the mass shell m can be locally represented as those(x,p0,pi) (V)

R2n for which p0 = p0(x

,pi)with

p0(x,pi):=

g0j(x)pj

[g0j(x)pj]2 + [g00(x)]

m2 + gi j(x)pipj

g00(x)

. (25)


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withKMa compact subset ofMwhich we assume to be contained inside a local chart(U,)ofM. Relative to such a local chart and the induced local coordinates(x,pi)of

+m introduced in the previous section, the integral of a C

-function f : +m R ofcompact support over Vtakes the form

V

f =

(K)

Rd

f( x, p)det(g( x))

p+0(x,pi)ddpdnx

=

(K)

Rd

f( x, p)

det(g( x))

p+0(x,pi) ddp

det(g( x))dnx (29)

where in these expressions,

x:= 1(x), p:=p0+(x,pi)

x0

x

+pj

xj

x

. (30)

Provided(M,g)is oriented, the integral over p can be interpreted as a fibre integraloverP+x . This follows from the observation that P

+x is a submanifold ofTxM, and this

linear space carries the natural volume formx=det(g(x))dx

0 dx1 . . .dxd.Proceeding as in Lemma 3, the volume form x induces a volume form x on P

+x . In

terms of coordinates(p0,pj)ofTxMchosen such that p0

is timelike, pj

are spacelike

for j=1, . . . ,d, and such that p0 >0 onP+x ,xtakes the form

x=

det(g(x))

|p+0(x,pi)| d p1 . . .d pd, (31)

with the function p+0(x,pi)given in Eq. (26). In particular, if the coordinates(p0,pj)

are chosen such that they determine an inertial frame in TxM, then we have g(x) =, and it follows that

x=d p1 . . .d pd

m2 +i jpipj,

from which we recognize the special-relativistic Lorentz-invariant volume-form on the

future mass hyperboloid.With the definition in Eq. (31) we can rewrite the integral over p in Eq. (29) as a fibre

integral overP+x , and thus

V

f =

(K)

P+x

f( x,p)x

det(g( x))dnx=

K

P+x

f(x,p)x

, (32)

where we have used again the fact that (M,g)is oriented and the definition of the naturalvolume form . We summarize this important result in the following lemma.


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Lemma 4(Local splitting I). Suppose(M,g)is oriented and time-oriented. Let V +mbe a subset of the future mass shell which is of the form given in Eq. (28), where K Mis a compact subset of M , contained in a coordinate neighborhood. Suppose f : +m Ris C-differentiable and has compact support. Then,

V

f =

K

P+x

f(x,p)x

, (33)

where x is the fibre volume element defined in Eq. (31) and is the natural volumeelement of(M,g).

We now consider the integration of functions f : +m R on suitable submanifolds of+m . The submanifolds we are considering are 2d-dimensional and oriented. In order

to define the integral of f on we also need a 2d-form. Given the volume form and

the Liouville vector field L on ma natural definition for such a 2d-form is

:=iL 2d(m). (34)

Denoting by : +m the inclusion map, it follows that for each C-function f : +m

R with compact support the integral

(f), (35)

is well-defined. Note that for the particular case that L is transverse to at every point

of, it follows that the pull-back of=iL is nowhere vanishing so that it defines avolume form on , and in this case is automatically oriented. We will come back to

this case at the end of this section.Relative to the local coordinates(x,pi)of+m introduced in the previous section, we

find

= det(g(x))

p+0(x,pi)

1

(n1)!p1...ddx

1 . . .dxdd p1 . . .d pd

+ 1

(d1)!

qFi(x)p

i(x)pp

i j2...jdd p

j2 . . .d pjddx0 . . .dxd.

where we have used Eqs. (3) and (27). In terms of the quantities

:=

1

(n1)!det(g(x))1...ddx

1

. . .dxd

,

i := 1

(d1)!

det(g(x))

|p+0(x,xi)| i j2...jdd p

j2 . . .d pjd.

we can rewrite the right-hand side in more compact form,

= p+

qFi(x)pi(x)p

pi . (36)


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For the particular case of hypersurfaces of the form

:= {(x,p):x S,p P+x }, (37)

withSMa d-dimensional, spacelike hypersurface in Mwhich is contained inU,5 wehave

(f) =

S

P+x

f(x,p)px

, (38)

where : S M is the inclusion map of S in M. As a preparation of what followsit is worth noticing that the fibre integral yields a well-defined vector field J on the

base manifold(M,g), see Eq. (39) below. Observing that J=iJ we arrive at thefollowing lemma.

Lemma 5(Local splitting II). Suppose(M,g)is oriented and time-oriented. Let +mbe a2d-dimensional submanifold of the mass shell which is of the form given in Eq. (37),where SM is a d-dimensional, spacelike hypersurface of M, contained in a coordinateneighborhood. Suppose f :+mR is C

-differentiable and has compact support. Then,

(f) =S

(iJ), J :=

P+x

f(x,p)px, (39)

where x is the fibre volume element defined in Eq. (31) and is the natural volumeelement of(M,g).

So far, we have assumed the function f to beC-smooth and have compact support

on +m . Although convenient from a mathematical point of view since the requirement

of compact support avoids convergence problems, for the physical interpretation givenin the next section this assumption on fmight be too strong. In the following we give

a brief explanation of what we believe is the correct function space for f within the

context of kinetic theory.For this, we assume spacetime (M,g) to be globally hyperbolic, so that it can be

foliated by spacelike Cauchy surfacesSt,tR. There is a corresponding foliation of thefuture mass shell +m given by

t:= {(x,p):x St,p P+

x }, t R. (40)

SinceStis spacelike,Lis everywhere transverse to t, and as discussed above it follows

that the pull-back of=iL to tdefines a volume form on t. Therefore, we candefine for each real-valued,C-function f : t R of compact support the integral

t

f. (41)

5 Notice thatL is transverse to at each point of sinceSis spacelike. Therefore, is oriented.


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In fact, the integral is also well-defined for continuous functions f:tR with compactsupport. In this case, it follows from the Riesz representation theorem (see, for instance

chapter 7 in Ref. [25]) that there exists a unique measure ton the space of Borel setsoftsuch that

t

f dt=t

f

for all such f. With this measure at hand, one can immediately define the spaceL1(t,dt). We claim that this is the appropriate function space for kinetic theory.Primary, it turns out that the space L1(t,dt) is, in some sense, independent of theCauchy surface St. The precise statement is contained in the next lemma, whose proof

is sketched in Appendix B.

Proposition 4. Let1 and2 be 2d-dimensional submanifolds of+m with the property

that each integral curve of L intersects 1and2exactly at one point. Then, there exists

a diffeomorphism: 1 2 such that2

f d2=

1

(f)d1

for all C-functions f : 2 R of compact support.

Secondly, the physical interpretation that the total number of particles in the gas is

finite is accommodated in our assumption that the integral of f over t is finite, as we

will see below.

After these remarks concerning the integration of real-valued functions on the mass

shell we are now ready to discuss physical applications of the above mathematical

formalism.

DISTRIBUTION FUNCTION AND LIOUVILLE EQUATION

As we have mentioned earlier on, for a simple gas, the world lines of the gas particles

between collisions are described by integral curves of the Liouville vector field Lon themass shell m. This section is devoted to the definition and properties of the one-particle

distribution function and its associated physical observables.

Following Ehlers [24] we consider a Gibbs ensemble of the gas on a fixed spacetime

(M,g). The central assumption of relativistic kinetic theory is that the averaged prop-erties of the gas are described by a one-particle distribution function. This function is

defined as a nonnegative function f :

+m

Ron the future mass shell such that for any(2d)-dimensional, oriented hypersurface +m , the quantity

N[]:=

(f), (42)

gives the ensemble average of occupied trajectories that pass through . In particular,if = V is the boundary of an open set V +m in

+m with (piecewise) smooth


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boundaries, then the quantity

N[V] =V

(f) (43)

gives the ensemble average of the net change in number of occupied trajectories due to

collisions inV.

As explained at the end of the last section, the distribution function f lies in the space

of integrable functionsL1(,d)with respect to a hypersurface of+m to which L is

transverse. However, for mathematical convenience, we assume in the following that f isC-smooth and is compactly supported on the future mass shell +m . This automatically

guarantees the existence of all the integrals we write below.

As a consequence of the interpretation of the integrals (42,43), we show in this sectionthat for a collisionless gas the distribution function fmust obey theLiouville equation

Lf=0. (44)

The derivation of the Liouville equation (44) is based on the following result.

Proposition 5. The 2d-form = iL is closed. Morevover, for any real-valued C-

function f on mwe have

d(f) = (Lf). (45)

Proof. The proposition is a consequence of Liouvilles theorem (Theorem 2) and the

Cartan identity. Using these results one finds

(Lf) =L(f) = (diL+ iLd)(f) =d(f iL) =d(f).

In particular, for f=1 it follows thatd= 0.

By means of Stokes theorem and this proposition we can rewrite the integral inEq. (43) as

N[V] =V

d(f) =V

(Lf). (46)

In particular, let V=

0tTtbe the tubular region obtained by letting flow a(2d)-dimensional hypersurface 0 in

+m to which L is transverse along the integral curves

ofL. Since by definition L is tangent to the cylindrical piece, T :=

0tTt, of theboundary, it follows that

T

f= 0. Therefore, we obtain from Eqs. (42,46)

N[T]N[0] = V

(Lf),

that is, the averaged number of occupied trajectories at T is equal to the averaged

number of occupied trajectories at 0 plus the change in occupation number due to

collisions in the region V. For a collisionless gas the integral in Eq. (46) must be zerofor all volumesV, and in this case the distribution function f must satisfy the Liouville

equationLf=0. (47)


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Using Eq. (3) we find

(Lf)(x,p) = p f

x(x,p) +

qF(x)p

(x)pp

fp

(x,p) =0 (48)

in adapted local coordinates ofT M. The above Liouville equation can also be rewritten

as an equation for the function f(x,pi):= f( x, p), depending on the local coordinates(x,pi)of the future mass shell +m introduced earlier, where we recall the definitions ofx and pin Eq. (30). By differentiating both sides of Eq. (24) one finds

p0+(x,pj)

x =

1

2p+0(x,pj)p p

g

x ,

p0+(x,pj)

pi =

1

p+0(x,pj)gi p

,

where here p0 = p0+(x,pj) and pj = pj. Using this and

f

x =

f

x+

f

p0p0+x =

f

x

p p

2p+0

g

xf

p0 ,

f

pi =

f

pi+

f

p0p0+pi

= f

pi

gi p

p+0

f

p0,

Eq. (48) can be rewritten in terms of the local coordinates(x,pj)on the mass shell as

p f

x(x,pj) +

qFi(x)p

i(x)p p

fpi

(x,pj) =0. (49)

CURRENT DENSITY AND STRESS-ENERGY TENSOR

In this section, using the distribution function f, we construct important tensor fields

on the spacetime manifold(M,g)by integrating suitable geometric quantities involvingfover the future mass hyperboloidalP+x . At this point, we remind the reader that(M,g)is assumed to be oriented and time-oriented, and thusP+x carries the natural volume formxdefined in Eq. (31).

We restrict our consideration to the quantities defined by:

Jx():=

P+x

f(x,p)(p)x (current density), (50)

Tx(,):=

P+x

f(x,p)(p)(p)x (stress-energy tensor), (51)

where here ,TxM. Since f : +m R is assumed to be smooth and compactly

supported, these quantities are well-defined. Moreover, by construction,Jxis linear inand Tx bilinear in(,), and thus Jand Tdefine a vector field and a symmetric, con-travariant tensor field onM, respectively. Their components relative to local coordinates


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(x)ofMare obtained fromJ(x) =Jx(dx),T(x) =Tx(dx

,dx), which yields

J(x) =

P+x

f(x,p)px=Rd

f( x, p) p

|p0+(x,pi)|

det(g( x))d

dp, (52)

T(x) =

P+x

f(x,p)ppx=Rd

f( x, p) pp

|p0+(x,pi)|

det(g( x))d

dp,(53)

where here(x,pi)are the adapted local coordinates of+m with the functionp0+(x,pi)

defined in Eq. (26). It follows from these expressions thatJand Tare C-smooth tensor

fields onM.The physical significance of these tensor fields can be understood by considering

a future-directed observer in (M,g) whose four-velocity is u. At any event x alongthe observers world line, we choose an orthonormal frame so that g(x) = and

u =

0 . Relative to this rest frame we have the usual relations from special relativity,

(p) = (E,pi) =m(1,vi), = 11i jvivj

,

withvi the three-velocity of a gas particle measured by the observer at x. Therefore, in

the observers rest frame, we have the following expressions which are familiar from the

non-relativistic kinetic theory of gases:

J0(x) =Rd

f(x,p)ddp=n(x), (particle density) (54)

Ji(x) = Rd

f(x,p)viddp=n(x)x, (particle current density) (55)

T00(x) =

Rd

f(x,p)Eddp=n(x)x, (energy density) (56)

T0j(x) =Rd

f(x,p)pjddp=n(x)< pj >x, (momentum density) (57)

Ti j(x) =

Rd

f(x,p)vipjddp=n(x)x, (kinetic pressure tensor),(58)

where the averagex atx Mof a functionA : m R on phase space is defined

as

x:=

Rd

A(x,p)f(x,p)ddp

Rd

f(x,p)ddp=

1

n(x)

Rd

A(x,p)f(x,p)ddp.

We also define the mean kinetic pressure by

p(x):=1

dTii(x) =

1

dn(x)x,


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and write (x):=mn(x) for the rest mass density. We stress that these quantities areobserver-dependent; so far, only the tensor fieldsJand T defined in Eqs. (50,51) have a

covariant meaning. It is worth noting the following Lemma (cf. Eq. (111) in Ref. [24])

Lemma 6. The (observer-dependent) quantities (x):=n(x)< E>x, p(x) and(x)

satisfy the following inequalities:

0 d p(x) d

2p(x) +

d

2p(x)

2+(x)2 (x) (x) + d p(x). (59)

for all x M.

Proof. The first two inequalities are obvious since p 0. For the last two inequalities,we first observe that

(x)d p(x) =mRd

1f(x,p)ddp.

Since 1 1 the fourth inequality follows immediately. Next, using the Cauchy-Schwarz inequality:

(x)2 = m2

Rd

1/21/2f(x,p)ddp

2

Rd

mf(x,p)ddpRd

m1f(x,p)ddp= (x) [(x)d p(x)] ,

from which the third inequality follows.

After these estimates concerning observer-dependent quantities, we now turn to co-

variant considerations. For this, the following lemma is useful (cf. [26]):

Lemma 7. Consider a vector p and a covariant, symmetric tensor T on a finite-

dimensional vector space V with Lorentz metric g. Then, the following statements hold:

(i) p V is future-directed timelike if and only if g(p,k)< 0 for all non-vanishing,future-directed causal vectors kV .

(ii) Suppose that T(k,k) > 0for all non-vanishing null vectors kV . Then, there existsa timelike vector k such that

T(k, ) = g(k, )

withR. If, in addition T(k,k) 0for all vectors kV , then the timelike vectork is unique up to a rescaling.

Proof. See Appendix C.

Using this lemma, we establish that at any given event x Mwith f(x, ) not iden-tically zero, the current density Jx and the stress-energy tensor Tx satisfy the follow-

ing properties: By appealing to statement (i) of this Lemma and the definition of Jx


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in Eq. (50) we conclude that gx(Jx,k)


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of+m . We compute the averaged number of collisions inside Vin two different ways.

First, using Eq. (46) and the local splitting result in Lemma 4, we have

N(V) =V

(Lf) =K

P+x

Lf(x,p)x . (64)

On the other hand, using the definition ofN(V), the local splitting result in Lemma 5,Stokes theorem and Cartans identity, we find

N(V) =

V

f=

K

iJ=

K

diJ=

K

J=

K

(divJ). (65)

Comparing Eqs. (64,65), and taking into account that Kcan be chosen arbitrarily small,

yields the first claim, Eq. (62) of the lemma.

For the second identity, we fix a one-form on M, and replace the four-currentdensity J by J :=T(,) in Eq. (62), which is equivalent to formally replace f(x,p)by f(x,p)(p)in the calculation above. Then, we obtain

div Jx=

P+x

L [f(x,p)(p)]x=

P+x

(Lf(x,p))(p)x+

P+x

f(x,p) [L((p))]x.

Using local coordinates, we find on one hand div J = (T) = (divT)() +

T, and on the other hand

L((p)) =L[(x)p] = pp+ qF

(x)p

,

where we have used the coordinate expression (3) for the Liouville vector field L. These

observations, together with the definitions in Eqs. (52,53) of J and T yield the

desired result.

THE EINSTEIN-MAXWELL-VLASOV SYSTEM

In this section we consider a simple gas of charged particles and take into accountits self-gravitation and the self-electromagnetic field generated by the charges. This

system is described by an oriented and time-oriented spacetime manifold (M,g) withg

the gravitational field, an electromagnetic field tensorF

, and a distribution functionf : +m R describing the state of the gas. These fields obey the Einstein-Maxwell-Vlasov equations, given by

G = 8GN

Tem+ Tgas

, (66)

F = qJ, [F]=0, (67)

Lf = 0, (68)


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whereG denotes the Einstein tensor,GNNewtons constant, and where

Tem =FF

1

4gFF

(69)

is the stress-energy tensor associated to the electromagnetic field,

Tgas =

P+x

f(x,p)ppx (70)

the stress-energy tensor associated to the gas particles, and

J =

P+x

f(x,p)px (71)

is the particles current density. Notice that thetotalstress-energy tensor is divergence-

free, as a consequence of the identity (63), Maxwells equations (67), and the Vlasov

equation (68). Similarly, the divergence-free character of the current density followsfrom the identity (63) and Eq. (68). For recent work related to the above system, see

Refs. [28, 29, 30].

THE RELATIVISTIC BOLTZMANN EQUATION FOR A SIMPLE

GAS

In the previous sections we developed the theory describing a collisionless simple

gas. For the development of this theory, we postulated that the averaged properties of

the gas are described by a one-particle distribution function fdefined on the future massshell +m and by simple arguments concluded that this distribution function obeys the

Liouville equationLf=0.However, undoubtedly one of the most important equations in the kinetic theory of

gases is the famous Boltzmann equation, and in order to complete this work, in thissection we shall sketch the structure of this equation. Like the Liouville equation, the

central ingredient of the Boltzmann equation is again the one-particle distribution func-

tion f associated to the gas. However, and in sharp contrast to the previous equations, the

Boltzmann equation describes the time evolution of a system where collisions between

the gas particles can no longer be neglected. This occurs when the mean free path (mean

free time) is much shorter than the characteristic length scale (time) associated with the

system.In order to describe the Boltzmann equation, for simplicity we consider a simple

charged gas i.e. a collection of spinless, classical particles which are all of the same rest

mass m >0 and all have the same charge q, and which interact only via binary elasticcollisions. For the purpose of this section we shall neglect the self-gravity and the self-electromagnetic field of the gas, assuming a fixed background spacetime(M,g)which isoriented and time-oriented. For the uncharged case, the gas particles move along future-directed timelike geodesics of(M,g)except at binary collisions which are idealized as


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point-like interactions. If at an eventx Ma binary collision occurs, then two geodesicsegments representing the trajectories of the gas particles end and two new ones emerge.

The nature and properties of the emerging geodesics are described by probabilistic laws

incorporated in the transition probability through a Lorentz scalar which is the basic

ingredient of the collision integral. The history of the gas in the spacetime consists of acollection of broken future-directed geodesic segments describing the particles between

collisions. In the charged case, the same situation occurs except that the geodesic

segments are replaced by segments of the classical trajectories of Eq. (2).

If p1,p2 stand for the four-momenta of the incoming particles and p3,p4 for themomenta of the outgoing particles at the event x, then an elastic collision obeys thelocal conservation law:

p1+p2= p3+p4. (72)

In the tangent bundle description such a binary collision at x M involves four points(x,p1), (x,p2),(x,p3), (x,p4)belonging to the same mass shell

+m and additionally four

orbits of the Liouville vector field L. In particular, the orbits through (x,p1), (x,p2)

become unoccupied while the orbits through (x,p3),(x,p4) become occupied. Thisinterchange in the occupation of the orbits of L is the effect of a binary collision as

perceived from the mass shell +m .Like for the case of a collisionless system, it is postulated that its averaged properties

are described by a distribution function f : +m R on the mass shell +m , defined by

Eq. (42). However, the Liouville equation is replaced by the Boltzmann equation which

has the following form:

Lf(x,p) =

P+x

P+x

P+x

W(p3+p4p +p2)

[f(x,p4)f(x,p3)f(x,p)f(x,p2]x(p4)x(p3)x(p2), (73)

where the right-hand side is the collision integral describing the effects of binary col-

lisions. The quantity W(p3+p4 p +p2) is referred to as the transition probabilityscalar, and it obeys the following symmetries:

W(p3+p4p +p2) = W(p4+p3p2+p), (74)

W(p +p2p3+p4) = W(p3+p4p +p2). (75)

The first symmetry is trivial, while the second symmetry expresses microscopic re-

versibility or, as often called, the principle of detailed balancing [10].

Intuitively speaking the termP+x

P+x

P+x

W(p3+p4 p +p2)f(x,p4)f(x,p3)x(p4)x(p3)x(p2)

in the collision integral describes the averaged number of collisions taking place in an

infinitesimal volume element centered around x Mwhose net effect is to increase the


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averaged number of occupied orbits through (x,p). On the other hand the term

P+x

P+x

P+x

W(p3+p4p +p2)f(x,p)f(x,p2)x(p4)x(p3)x(p2)

describes the depletion of the occupied orbits through(x,p)due to the binary scattering.In the previous sections we have shown that the mere existence of the distribution

function f : +m R leads to the construction of the current densityJand stress-energytensor Tdescribing the gas, and naturally this property of fremains valid for a collision-

dominated gas. For a collisionless gas Jand Tsatisfy the conservation laws divJ= 0,divT=q F(J), as a consequence of the Liouville equation Lf=0, see Proposition 6.We shall show below that these properties ofJ and Tremain valid for the case where

the distribution function satisfies the Boltzmann equation, Eq. (73).

In order to prove this property, let : +m R :(x,p) (x,p)be an arbitraryC-

smooth real-valued function. Multiplying both sides of Boltzmann equation by and

integrating overP+

x yieldsP+x

(x,p)Lf(x,p)x(p)

= 1

4

P+x

P+x

P+x

P+x

W(p3+p4p +p2) [(x,p4) +(x,p3)(x,p)(x,p2)]

[f(x,p4)f(x,p3)f(x,p)f(x,p2]x(p4)x(p3)x(p2)x(p), (76)

where we have made use of the symmetry properties (74,75) for the transition probability

scalar. In particular, the right-hand side vanishes identically if is chosen to be a

collision-invariant quantity. The choice (x,p) =1 leads toP+x

Lf(x,p)x(p) =0,

which implies that Jis divergence-free, see Eq. (62), while the choice (x,p) =x(p)from some 1(M), together with the momentum conservation law, Eq. (72), yields

P+x

x(p)Lf(x,p)x(p) =0,

which implies that div Tx(x) =qx(Fx(J)), see Eq. (63).However, by far the most important implication of the Boltzmann equation is the

existence of a vector field S on (M,g) whose covariant divergence is semi-positivedefinite provided that micro-reversibility holds. In order to construct this field, weassume for the following that the distribution function is strictly positive, with suitable

fall-off conditions to guarantee the convergence of the integrals below. Since we abstainfrom specifying such fall-off conditions explicitly, the following arguments should be


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taken as formal. We choose (x,p) =1 + log(A f(x,p)), where A has been insertedto make the argument of the logarithm a dimensionless quantity. For this choice, the

identity (76) yields

P+x

[1 + log(A f(x,p))] Lf(x,p)x(p) = 1

4

P+x

P+x

P+x

P+x

W(p1+p2p3+p4)

[log(f1f2) log(f3f4)] [f1f2 f3f4]x(p1)x(p2)x(p3)x(p4),

where we have abbreviated fj := f(x,pj)for j= 1,2,3,4. The right-hand side is non-positive, due to the positivity ofW(p1+p2p3+p4)and the inequality

(logy logx)(yx) 0,

which is valid for all x,y > 0. On the other hand, noting that [1+log(A f)]Lf =L[log(A f)f] we obtain, by replacing the function f with the function log(A f)f in

Eq. (62),divSx=

P+x

[1 + log(A f(x,p))]Lf(x,p)x(p),

with the entropy flux vector fieldSdefined by

Sx():=

P+x

log(A f(x,p))f(x,p)(p)x, (77)

for Tx M. Therefore, the entropy flux vector fieldSsatisfies

divSx = 1

4

P+x

P+x

P+x

P+x

W(p1+p2 p3+p4)

[log(f1f2) log(f3f4)] [f1f2 f3f4]x(p1)x(p2)x(p3)x(p4) 0.

The above relation expresses the famous Boltzmann H-theorem. It is beyond the scope

of this article to provide a detailed discussion of the mathematical framework underlying

the structure of the collision integral and related properties of the Boltzmann equation.

The reader is referred to Refs. [31, 18, 17] for a more detailed discussion and properties

of the Boltzmann equation.

CONCLUSIONS

In this work, we have presented a mathematically-oriented introduction to the field

of relativistic kinetic theory. Our emphasis has been placed on the basic ingredients that

constitute the foundations of the theory. As it has become clear, of a prime importancefor the description of this theory are not any longer point particles but rather and

according to Synges fundamental idea, world lines of gas particles. From the tangentbundle point of view, these world lines appeared as the integral curves of a Hamiltonian


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vector field. It is worth stressing here the liberty and the flexibility that characterizes the

relativistic kinetic theory of gases. The Poincar one-form as well as the Hamiltonian

and thus the structure of the Hamiltonian vector field are at our disposal. In this work

we have made simple and natural choices for the Poincar one-form and Hamiltonian,

and we were led from first principles to the Liouville equation, while for the case wherethe self-gravity and self-electromagnetic field of the gas are accounted for we arrived

naturally to the Einstein-Liouville and the Einstein-Maxwell-Vlasov systems.

Therefore, given the freedom in the Poincar one-form and the Hamiltonian it is worth

thinking of relativistic kinetic theory models describing dark matter or dark energy. This

could be interesting from an astrophysical and cosmological point of view.

ACKNOWLEDGMENTS

This work was supported in part by CONACyT Grant No. 101353 and by a CIC Grant

to Universidad Michoacana.

APPENDIX A: PROOF OF PROPOSITION 3

We prove that m is connected if and only if(M,g) is not time-orientable. Supposefirst that m is connected and let us show that (M,g) cannot be time-orientable. Forthis, choose an arbitrary point x M and two timelike tangent vectors k+ and kat x such that k P

x . Since m is connected by hypothesis, there exists a curve

:[0,1] m, t ((t),k(t))which connects the two points (x,k+) and(x,k). Theprojection of this curve yields a closed curve :[0,1] MinM throughx. Along thiscurve is defined the continuous timelike vector fieldk(t)which connectsk+and k(see

Fig. 1). Therefore,(M,g)is not time-orientable.Conversely, suppose (M,g) is not time-orientable. We shall prove that m is con-nected. In order to show this we note the hypothesis implies the following. There exists

an eventx M, a closed curve: [0,1]Mthroughxand a continuous timelike vectorfield k(t) T(t)Malongwhich we may normalize such that g(k(t),k(t)) = m

2 for

all t [0,1], with the property thatk:= k(0) P

x andk+:= k(1) P+

x . In this way weobtain a curve : [0,1] m, t ((t),k(t)) in mwhich connects(x,k)with(x,k+).Since the setsPx are connected, it follows that any two points(x,p),(x,p

) m in thesame fibre overx can be connected to each other by a curve in m.

Now let us extend this connectedness property to arbitrary points (x1,p1), (x2,p2) m. Since Mis connected there exist curves 1,2 inM which connect x with x1 and xwith x2, respectively. By parallel transporting p1 along 1 and p2 along 2 we obtaincurves 1 and 2 in m which connect (x1,p1) with (x,p) and (x2,p2) with (x,p)respectively.6 By the result in the previous paragraph,(x,p)and(x,p)can be connectedto each other by a curve in m. Therefore,(x1,p1)and(x2,p2)can also be connected toeach other and it follows that mis connected.

6 1 and 2 are the horizontal lifts of1and 2 through the points(x1,p1)and(x2,p2), respectively.


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M

k

k+

x11

PxTM

x

k

(x, k+)

(x, k)

FIGURE 1. An illustration of the curves T Mand M, together with the vector field kalong.

In order to conclude the proof we need to show that m consists of two connected

components if(M,g)is time-orientable. In order to prove this, choose x Mand a time-orientationP+x andP

x atx. Given any point y Mwe may connect it to x by means of

a curve in MsinceMis connected. We choose the time orientation Py aty such that

kx P+

x if and only ifky P+

y for all parallel transported timelike vector fieldskalong

. This choice is independent ofotherwise(M,g)would not be time-orientable. In thisway we obtain the two connected subsets

m := {(x,p) m: p P

x }

ofm which are disjoint and whose union is m.

APPENDIX B: SKETCH OF THE PROOF OF PROPOSITION 4

We define: 1 2 to be the map that associates to each point p1 1 the uniquepoint p2 2 that lies on the integral curve ofL through p1. By transporting the valuesof falong the integral curve ofL in the segment between 1 and 2, we can extend fto aC-function f : +m Rwith compact support, such that L f=0 in the regionVbetween 1 and 2. It follows by Stokes theorem that

2

2 (f)1

1 (f) =V

d(f) =V

(L f) =0,

where j : j +m denote the inclusion maps for j= 1,2, and where we have used

Proposition 5 in the second step. Since 2f= f and 1

f=fthe Proposition follows.


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APPENDIX C: PROOF OF LEMMA 7

Without loss of generality we can assume that(V,g)is Minkowski spacetime, so thatthe standard special relativistic notions of causality hold. We work in coordinates (x)

for which the metric has the standard diagonal form.(i) At first we note that any non-vanishing causal, future-directed vector k V has

the components(k) = (k0,ki)withk0 >0 andi jkikj (k0)2. Suppose first that

p V is future-directed timelike. Then, after a Lorentz transformation, (p) =(p0,0, . . . ,0) with p0 > 0 and it follows that g(p,k) =p0k0 < 0. Conversely,suppose that g(p,k)< 0 for all non-vanishing causal, future-directed k. After arotation the components of p are (p) = (p0,p1,0, . . . ,0) and the two choices(k

+) = (1,1,0, . . . ,0)and(k

) = (1,1,0, . . . ,0)lead to 0>g(p,k+) = p

0 +p1

and 0>g(p,k) = p0 p1 which implies |p1|0},

whose elements may be parametrized according to the stereographic map,

u0 = 1

1|v|2, ui =

vi1|v|2

, v B1(0),

withB1(0):= {vR3 : i jv

ivj 0, it follows that f(v) .Therefore, the function fhas a global minimum at some point v B1(0). Takingthe gradient on both sides of7

(1|v|2)f(v) =T00+ 2T0ivi + Ti jv

ivj

and evaluating atv=v, we obtain

f(v)vi =T0i+ Ti j(v)j,

7 Alternatively, one might also invoke the Lagrange multiplier method, applied to the restriction of thequadratic formT(u,u)on H+, in order to conclude that at the minimum u,T(u, ) = g(u, ).


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from which we also have f(v)|v|2 = T0i(v)i + Ti j(v

)i(v)j and so f(v) =T00+T0i(v

)i. Therefore, the timelike vector k := (1,v) satisfies T(k) =

kwith= f(v).

As for uniqueness, we first observe that after a rescaling and a Lorentz transfor-

mation we may assume that k

= (1,0,0, . . . ,0). Then, T00= , T0j =0, and byapplying a rotation if necessary, we can assume that Ti j is diagonal, such that(T) = diag(,p1,p2, . . . ,pd). It follows from the hypothesis that 0 andpj 0 for all j = 1,2, . . . ,d. Furthermore,= 0 impliespj> 0 for all j = 1,2, . . . ,dbecause of the hypothesis. Therefore, the eigenspace belonging to the eigenvalue

of the matrix(T) =diag(,p1,p2, . . . ,pd)must be one-dimensional since= pj for all j=1,2, . . . ,d.

Remark: Due to the fact that the scalar product defined by the Lorentz metric g is

not positive definite, it is not always the case that the stress-energy tensor T can be

diagonalized, despite of the fact that it is symmetric. In fact, the results above show thatTis diagonalizable if and only if it admits a timelike eigenvector. An explicit example

for which the condition in assumption (ii) in Lemma 7 is not satisfied is

T=kk

with a null vector k, corresponding to a stress-energy tensor for null dust. In this case,

the only eigenvalue ofTis zero, and its eigenspace consists of the vectors which are

orthogonal tok. In particular, this example shows that the imposition of the weak energy

condition is not sufficient to guarantee the diagonalizability of the stress-energy tensor.

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