Locally Anisotropic Kinetic Processes and Thermodynamics in Curved Spaces

Annals of Physics290, 83–123 (2001)doi:10.1006/aphy.2000.6121, available online at http://www.idealibrary.com on

Locally Anisotropic Kinetic Processes and Thermodynamicsin Curved Spaces

Sergiu I. Vacaru1

Fachbereich Physik, Universitat Konstanz, Postfach M 638, D-78457, Konstanz, Germany; and Institute of Applied Physics,Academy of Sciences, 5 Academy Str., Chis¸inau MD2028, Republic of Moldova

Received August 19, 2000

The kinetic theory is formulated with respect to anholonomic frames of reference on curved space-times. By using the concept of nonlinear connection we develop an approach to modelling locallyanisotropic kinetic processes and, in corresponding limits, the relativistic nonequilibrium thermody-namics with local anisotropy. This leads to a unified formulation of the kinetic equations on (pseudo)Riemannian spaces and in various higher dimensional models of Kaluza–Klein type and/or generalizedLagrange and Finsler spaces. The transition rate considered for the locally anisotropic transport equa-tions is related to the differential cross section and spacetime parameters of anisotropy. The equations ofstates for pressure and energy in locally anisotropic thermodynamics are derived. The obtained generalexpressions for heat conductivity, shear, and volume viscosity coefficients are applied to determinethe transport coefficients of cosmic fluids in spacetimes with generic local anisotropy. We emphasizethat such local anisotropic structures are induced also in general relativity if we are modelling physicalprocesses with respect to frames with mixed sets of holonomic and anholonomic basis vectors whichnaturally admits an associated nonlinear connection structure.C© 2001 Academic Press

1. INTRODUCTION

The experimental data on anisotropies in the microwave background radiation (see, for instance,Refs. [13, 27]) and modern physical theories support the idea that in the very beginning the universehas to be described as an anisotropic and higher dimension spacetime. The investigation of higherdimension generalized Kaluza–Klein and string anisotropic cosmologies is of interest. Therefore,in order to understand the initial dynamical behavior of an anisotropic universe, in particular, tostudy possible mechanisms of anisotropic inflation connected with higher dimensions [32] we haveto know how we can compute the parameters (transport coefficients, damping terms, and viscositycoefficients) that characterize the cosmological fluids in spacetimes with generic anisotropy.

The first relativistic macroscopic thermodynamic theories were proposed in Refs. [9, 22]. Furtherdevelopments and applications in gravitational physics, astrophysics, and cosmology are connectedwith papers [16, 24, 28, 37]. Israel’s approach to a microscopic Boltzmann like kinetic theory forrelativistic gases [18] makes it possible to express the transport coefficients via the differential crosssections of the fluid’s particles. Here we also note Chernikov’s results on Boltzmann equations with acollision integral on (pseudo) Riemannian spaces [7]. A complete formulation of relativistic kineticsis contained in the monograph [15] by de Grootet al. We emphasize Vlasov’s monograph [36] wherean attempt to establish the statistical motivation of kinetic and thermodynamic theory on phase spacesenabled with Finsler-like metrics and connection structures was proposed. Recent developments andapplications of kinetics and nonequilibrium thermodynamics can be found in [38].

1 E-mail: [email protected].

830003-4916/01 $35.00

Copyright C© 2001 by Academic PressAll rights of reproduction in any form reserved.

84 SERGIU I. VACARU

The extension of the four-dimensional considerations to higher dimensions is due to [3, 29]. Thegeneralization of kinetic and thermodynamic equations and formulas to curved spaces is not a trivialtask. In order to consider flows of particles with noninteger spins, interactions with gauge fields, andvarious anisotropic processes of a microscopic or macroscopic nature it is necessary to reformulatethe kinetic theory in curved spacetimes by using Cartan’s moving frame method [6]. A generalapproach with respect to anholonomic frames contains the possibility of taking into account genericspacetime anisotropies which play an important role in the vicinity of cosmological or astrophysicalsingularities and for nontrivial reductions of higher dimensional theories to lower dimensional ones.In our works [30–32] we proposed to describe such locally anisotropic spacetimes and interactionsby applying the concept of a nonlinear connection (in brief, N-connection) [2] field which models thelocal splitting of spacetime into horizontal (isotropic) and vertical (anisotropic) subspaces. For somevalues of components the N-connection could parametrize, for instance, toroidal compactificationsof higher dimensions but, in general, its dynamics is to be determined, in different approaches, bysome field equations of constraints in some generalized Kaluza–Klein, gauge gravity, or (super)stringtheories [30–32, 35].

The concept of N-connection was first applied in the framework of Finsler and Lagrange geometryand gravity and their higher order extensions (see Refs. [11, 23, 30, 31] for details). Here it isemphasized that every generalized Finsler like geometry could be modelled equivalently on higherdimensions (pseudo) Riemannian spacetime (for some models by introducing additional torsionand/or nonmetricity structures) with correspondingly adapted anholonomic frame structures. If werestrict our considerations only to the higher dimensional Einstein gravity, the induced N-connectionstructure becomes a “pure” anholonomic frame effect which points to the fact that the set of dynamicalgravitational field variables given by the metric’s components was redefined by introducing localframe variables. The components of a fixed basis define a system (equivalently, frame) of reference(in four dimensions one uses, for instance, the terms of vierbien, or tetradic field) with respect towhich, in its turn, one states the coefficients of curved spacetime’s metric and of fundamental physicalvalues and field equations. It should be noted that the procedure of choosing (establishing) a systemof reference must be also physically motivated and that in general relativity this task is not consideredas a dynamical one following from the field equations. For trivial models of physical interactions oncurved spacetimes one can restrict consideration to only holonomic frames which locally are linearlyequivalent to some coordinate basis. Extending the class of physical fields and interactions (even inthe framework of Einstein’s gravity), for example, by introducing spinor fields and statistical andfluid models with spinning particles, we have to apply frame bundle methods and deal with generalanholonomic frames and phase spaces provided with Cartan’s Finsler-like connections (induced bythe Levi Civita connection on (pseudo) Riemannian spacetime) [5].

The modelling of kinetic processes with respect to anholonomic frames (which induces corre-sponding N-connection structures) is very useful with the aim of elucidating flows of fluids ofparticles not being in local equilibrium. More exactly, such generic anisotropic fluids are consideredto be in a thermodynamic equilibrium with respect to some frames which are locally adapted both tothe spacetime metric and N-connection structures, but in general (for instance, with respect to localcoordinate frames) the conditions of local equilibrium are not satisfied.

We developed a proper concept of locally anisotropic spacetime (in brief, la-spacetime) [30, 31,33] in order to provide a unified (super) geometric background for generalized Kaluza–Klein (super)gravities and low energy (super) string models when the higher dimensional spacetime is character-ized by generic local anisotropies and compatible metric, N-connection, and correspondingly adaptedlinear connection structures. It should be noted that the term la-spacetime can be used even for Ein-stein spaces if they are provided with anholonomic frame structures. Our treatment of local anisotropyis more general than that in Ref. [4] which is used for a subclass of Finsler-like metrics being con-formally equivalent (with conformal factors depending both on spacetime coordinates and tangentvectors) to the flat (pseudo) Euclidean, equivalently, Minkovschi metric. In the works of Bogoslovsky,

ANISOTROPIC KINETICS AND CURVED SPACES 85

Goenner, and Asanov [4, 11] on Finsler-like spacetimes, possible effects of the violation of localLorentz and/or Poincar´e symmetries are investigated. The background of our approach to locallyanisotropic strings, field interactions, and stochastics [30, 31, 33] is based on the fact that suchmodels could be constructed to be locally Lorentz invariant with respect to frames locally adaptedto N-connection structures.

The purpose of the present article is to formulate a Boltzmann-type kinetic theory and nonequilib-rium thermodynamics on spacetimes of higher dimension with respect to anholonomic frames whichmodel local anisotropies of both Einstein type and generalized Finsler–Kaluza–Klein theories. Weshall also discuss possible applications in modern cosmology and astrophysics.

Regarding the kinetic and thermodynamic aspects, our notations and derivations are often inspiredby Refs. [3, 15]. A number of formulas will be similar for both locally isotropic and anisotropicspacetimes if in the last case we consider the equations and vector, tensor, spinor, and connectionobjects with respect to anholonomic rest frames locally adapted to the N-connection structures. As aconsequence, some tedious proofs and intermediary formulas will be omitted by referring the readerto the corresponding works.

To keep the article self-contained we begin Section 2 with an overview of the so-called locallyanisotropic spacetime geometry and gravity. In Sections 3 and 4 we present the basic definitions forlocally anisotropic distribution functions, particle flow, and energy-momentum tensors and generalizethe kinetic equations for la-spacetimes. The equilibrium state and derivation of expressions for theparticle density and the entropy density for systems that are close to a locally anisotropic state ofchemical equilibrium are considered in Section 5. Section 6 is devoted to the linearized locallyanisotropic transport theory. There we discuss the problem of solution of kinetic equations, prove thelinear laws for locally anisotropic nonequilibrium thermodynamics, and obtain the explicit formulasfor transport coefficients. As an example, in Section 7, we examine the transport theory in curvedspaces with rotation ellipsoidal horizons. Concluding remarks are contained in Section 8.

2. SPACETIMES WITH LOCAL ANISOTROPY

We outline the necessary background on anholonomic frames and nonlinear connections(N-connections) modelling local anisotropies (la) in curved spaces and on locally anisotropic grav-ity [30, 31] (see Refs. [17, 23] for details on spacetime differential geometry and N-connectionstructures). We shall prove that Cartan’s moving frame method [6] allows a geometric treatment ofboth locally isotropic (for simplicity, we shall consider (pseudo) Riemannian spaces) and anisotropicspaces (the so-called generalized Finsler–Kaluza–Klein spaces).

2.1. Anholonomic Frames and Einstein Equations

In this paper spacetimes are modelled as smooth (i.e., classC∞) manifoldsV(d) of finite integerdimensiond ≥ 3, 4, . . . , being Hausdorff, paracompact, and connected. We denote the local coor-dinates onV(d) by variablesuα, where Greek indicesα, β, . . . = 3, 4, . . . could be either coordinateor abstract (Penrose) indices. A spacetime is provided with corresponding geometric structures ofsymmetric metricgαβ and of linear, in general nonsymmetric, connection0αβγ defining the covariantderivation∇α satisfying the metricity conditions∇αgβγ = 0. We shall underline indices,α

¯, β

¯, . . . ,

if we want to emphasize them as abstract.Let a set of basis (frame) vectorseα¯ = {e

αα = gαβeα¯

β} on V(d) be numbered by an underlinedindex. We shall consider only frames associated to symmetric metric structures via relations of type

eαα¯eβ¯α = ηα

¯,β

¯

86 SERGIU I. VACARU

and

eααeβ

βηα¯,β

¯= gαβ,

where the Einstein summation rule is accepted andηα¯,β

¯is a given constant symmetric matrix, for

simplicity a pseudo-Euclidean metric of signature (−,+ · · · ·+) (the sign minus is used in this workfor the time-like coordinate of spacetime). Operations with underlined and nonunderlined indicesare correspondingly performed by using the matrixηα

¯,β

¯, its inverseηα¯

β , and the metricgαβ and its

inversegαβ . A frame (local basis) structureeα onV(d) is characterized by its anholonomy coefficientswαβγ defined from the relations

eαeβ − eβeα = wγαβeγ . (1)

With respect to a fixed basiseα and its dualeβ we can decompose tensors and write down theircomponents; for instance,

T = Tγ

αβeγ ⊗ eα ⊗ eβ,

where by⊗ we denote the tensor product.A spacetimeV(d) is holonomic (locally integrable)if it admits a frame structure for which the

anholonomy coefficients from (1) vanishes; i.e.,wγαβ = 0. In this case we can introduce localcoordinate bases,

∂α = ∂/∂uα, (2)

and their duals,

dα = duα, (3)

and consider components of geometrical objects with respect to such frames.We note that the general relativity theory was formally defined on holonomic pseudo-Riemannian

manifolds. Even on holonomic spacetimes, for various (geometrical, computational, and physicallymotivated) purposes, it is convenient to use anholonomic frameseα

¯, but we emphasize that for such

spacetimes one can always define some linear transforms of frames to a coordinate basis,eα¯= a

α¯′α¯∂α¯′ .

By applying both holonomic and anholonomic frames and their mutual transforms on holonomicpseudo-Riemannian spaces, different variants of tetradic and spinor gravity and extensions to linear,affine, and de Sitter gauge group gravity models [10, 12, 26] were developed.

A spacetime is genericallyanholonomic (locally nonintegrable)if it does not admit a framestructure for which the anholonomy coefficients from (1) vanish; i.e.,w

γ

αβ 6= 0. In this case theanholonomy becomes a proper spacetime characteristic. For instance, a generic anholonomy couldbe obtained if we consider nontrivial reductions from higher dimensional spaces to lower dimensionalones. It induces nonvanishing additional terms into the torsion,

T(δγ , δβ) = Tαβγ δα,

and curvature,

R(δτ , δγ )δβ = Rαβ γ τ δα,


tensors of a linear connection0αβγ , with coefficients defined respectively as

Tαβγ = 0αβγ − 0αγβ + wαβγ (4)

and

Rαβ γ τ = δτ0αβγ − δγ 0αβδ + 0ϕβγ 0αϕτ − 0ϕβτ0αϕγ + 0αβϕwϕγ τ . (5)

The Ricci tensor is defined as

Rβγ = Rαβ γα (6)

and the scalar curvature is

R= gβγRβγ . (7)

The Einstein equations on an anholonomic spacetime are introduced in a standard manner,

Rβγ − 1

2gβγR= kϒβγ , (8)

where the energy–momentum d-tensorϒβγ includes the cosmological constant terms and possi-ble contributions of torsion (4) and matter andk is the coupling constant. For a symmetric linearconnection the torsion field can be considered as induced by some anholonomy (or equivalently,by some imposed constraints) conditions. This way locally anisotropic structures could be inducedby anholonomic frames on (pseudo) Riemannian spaces and in Einstein gravity. For more generalgravitational theories with dynamical torsion there are necessary additional field equations; see, forinstance, the case of locally anisotropic gauge-like theories [35].

The usual locally isotropic Einstein gravity is obtained on the supposition that for every anholo-nomic frame, corresponding linear transforms to a coordinate frame basis can be defined.

It is a topic of further theoretical and experimental investigations to establish if the present-dayexperimental data on the anisotropic structure of the universe are a consequence of matter andquantum fluctuation induced anisotropies, and for some scales the anisotropy is a consequence ofanholonomy of the observer’s frame. The spacetime anisotropy could be also a generic propertyfollowing, for instance, from string theory, and from a more general self-consistent gravitationaltheory when both the left (geometric) and right (matter energy–momentum tensor) parts of Einsteinequations depend on anisotropic parameters.

2.2. The Local Anisotropy and Nonlinear Connection

A subclass of anholonomic spacetimes consists of those with local anisotropy modelled by anonlinear connection structure. In this section we briefly outline the geometry of anholonomicframes with induced nonlinear connection structure.

The la-spacetime dimension is split locally into two components,n for isotropic coordinates andmfor anisotropic coordinates, whenn(a) = n+mwith n ≥ 2 andm≥ 1. We shall use local coordinatesuα = (xi, ya), where Greek indicesα, β, . . . take values 1, 2, . . . ,n+m and Latin indicesi anda arecorrespondinglyn- andm-dimensional, i.e.,i, j, k . . . = 1, 2, . . . ,n anda, b, c, . . . = 1, 2, . . . ,m.

Now, we consider an invariant geometric definition of spacetime’s splitting into isotropic andanisotropic components. For modelling la-spacetimes one uses a vector bundleE = (En+m, p,M(n),

F(m),Gr ) provided withnonlinear connection(in brief, N-connection) structureN = {Naj (uα)},

where theNaj (uα) are its coefficients [23]. We denoteEm+n as an (n+m)-dimensional total space

88 SERGIU I. VACARU

of a vector bundle,M(n) as then-dimensional base manifold,F(m) as the typical fiber being am-dimensional real vector space,Gr as the group of automorphisms ofF(m), and p as a surjectivemap. For simplicity, we shall consider only local constructions on vector bundles.

The N-connection is a new geometric object which generalizes that of linear connection. Thisconcept came from Finsler geometry (see Cartan’s monograph [5]); its global formulation is due toBarthel [2], and it is studied in detail in the works of Miron and Anastasiei [23]. We have extendedthe geometric constructions for spinor bundles and superbundles with further applications in locallyanisotropic field theory and strings and modern cosmology and astrophysics [30–32]. We have alsoillustrated [33] that the N-connection could be introduced on (pseudo) Riemannian spacetimes and inEinstein gravity if we consider anholonomic frames consisting of sets of basis vectors, some of thembeing holonomic and the rest anholonomic. In this case the N-connection coefficients are associatedwith the frame structure and they transform into some metric components if the considerations aretransferred with respect to a coordinate basis.

The rigorous mathematical definition of N-connection is based on the formalism of horizontaland vertical subbundles and on exact sequences of vector bundles. Here, for simplicity, we define anN-connection as a distribution which for every pointu = (x, y) ∈ E defines a local decomposition ofthe tangent space of our vector bundle,TuE, into horizontal,HuE, and vertical (anisotropic),VuE,subspaces; i.e.,

TuE = HuE ⊕ VuE.

If an N-connection with coefficientsNaj (uα) is introduced on the vector bundleE the modelled

spacetime possesses a generic local anisotropy and in this case we cannot apply the operators ofpartial derivatives and their duals, differentials, in the usual manner. Instead of coordinate bases (2)and (3) we must consider some bases adapted to the N-connection structure:

δα = (δi , ∂a) = δ

∂uα=(δi = δ

∂xi= ∂

∂xi− Nb

i (x j, y)∂

∂yb, ∂a = ∂

∂ya

)(9)

and

δβ = (di , δa) = δuβ = (di = dxi, δa = δya = dya + Nak (x j, yb) dxk

). (10)

A nonlinear connection (N-connection) is characterized by its curvature,

Äai j =

∂Nai

∂x j− ∂Na

j

∂xi+ Nb

i

∂Naj

∂yb− Nb

j

∂Nai

∂yb. (11)

Here we note that the class of usual linear connections can be considered as a particular case when

Naj (x, y) = 0a

bj (x)yb.

The elongation (by N-connection) of partial derivatives, in the partial derivatives adapted N-connection (9) or the locally adapted basis (la-basis)δβ , reflects the fact that the spacetimeE is locallyanisotropic and generically anholonomic because there are satisfied anholonomy relations (1),

δαδβ − δβδα = wγαβδγ ,


where the anholonomy coefficients are

wki j = 0, wk

aj = 0, wkia = 0, wk

ab = 0, wcab = 0,

wai j = −Äa

i j , wbaj = −∂aNb

i , wbia = ∂aNb

i .

On a la-spacetime the geometrical objects one distinguished (by N-connection) into horizontaland vertical components. They are briefly called d-tensors, d-metrics, and/or d-connections. Theircomponents are defined with respect to a la-basis of type (9), its dual (10), or their tensor products(d-linear or d-affine transforms of such frames could also be considered). For instance a covariantand contravariant d-tensorZ is expressed as

Z = Zαβδα ⊗ δβ = Zij δi ⊗ d j + Zi

aδi ⊗ δa + Zbj ∂b ⊗ d j + Zb

a∂b ⊗ δa.

A symmetric d-metric on la-spaceE is written as

δs2 = gαβ(u)δα ⊗ δβ = gi j (x, y)dxi dxj + hab(x, y)δyaδyb. (12)

A linear d-connectionD on la-spaceE ,

Dδγ δβ = 0αβγ (xk, y)δα,

is parametrized by nontrivial h–v-components,

0αβγ =(Li

jk, Labk,C

ijc,C

abc

). (13)

Some d-connection and d-metric structures are compatible if the following conditions are satisfied:

Dαgβγ = 0.

For instance, a canonical compatible d-connection

c0αβγ =(

cLijk,

cLabk,

cCijc,

cCabc

)is defined by the coefficients of the d-metric (12),gi j (x, y) andhab(x, y), and by the coefficients ofN-connection,

cLijk =

1

2gin(δkgnj + δ j gnk − δngjk),

cLabk = ∂bNa

k +1

2hac(δkhbc− hdc∂bNd

i − hdb∂cNdi

),

(14)cCi

jc =1

2gik∂cgjk,

cCabc =

1

2had(∂chdb+ ∂bhdc− ∂dhbc).

The coefficients of the canonical d-connection generalize for la-spacetimes the well-known Cristoffelsymbols.

90 SERGIU I. VACARU

For a d-connection (13) we can compute the components of, in our case, d-torsion (4),

Ti. jk = Ti

jk = Lijk − Li

k j , Tija = Ci

. ja, Tiaj = −Ci

ja,

Ti. ja = 0, Ta

.bc = Sa.bc = Ca

bc− Cacb,

Ta.i j = −Äa

i j , Ta.bi = ∂bNa

i − La.bj , Ta

.ib = −Ta.bi .

In a similar manner, putting nonvanishing coefficients (13) into the formula for curvature (5), wecan compute the nontrivial components of a d-curvature,

R.ih. jk = δkLi.hj − δ j L

i.hk + Lm

.hj Limk− Lm

.hkLimj − Ci

.haÄa. jk,

R.ab. jk = δkLa.bj − δ j L

a.bk + Lc

.bj La.ck − Lc

.bkLa.cj − Ca

.bcÄc. jk,

P.ij .ka = ∂kLi

. jk + Ci. jbTb

.ka−(∂kCi

. ja + Li.lkCl

. ja − Ll. jkCi

.la − Lc.akC

i. jc

),

P.cb.ka = ∂aLc

.bk + Cc.bdTd

.ka−(∂kCc

.ba+ Lc.dkC

d.ba− Ld

.bkCc.da− Ld

.akCc.bd

)S.ij .bc = ∂cC

i. jb − ∂bCi

. jc + Ch. jbCi

.hc− Ch. jcCi

hb,

S.ab.cd = ∂dCa.bc− ∂cC

a.bd + Ce

.bcCa.ed− Ce

.bdCaec.

The components of the Ricci tensor (15) with respect to locally adapted frames (9) and (10) (inthis case, the d-tensor) are

Ri j = R.ki . jk, Ria = −2Pia = −P.ki .ka,

(15)Rai = 1Pai = P.b

a.ib, Rab = S.ca.bc.

We point out that because, in general,1Pai 6= 2Pia the Ricci d-tensor is nonsymmetric.Having defined a d-metric of type (12) inE we can compute the scalar curvature (7) of a

d-connectionD,

←R= GαβRαβ = R+ S, (16)

whereR= gi j Ri j andS= habSab.Now, by introducing the values (15) and (16) into anholonomic gravity field equations (8) we

can write down the system of Einstein equations for la-gravity with prescribed N-connection struc-ture [23],

Ri j − 1

2(R+ S)gi j = kϒi j , (17)

Sab− 1

2(R+ S)hab = kϒab,

1Pai = kϒai ,

2Pia = −kϒia,

whereϒi j , ϒab, ϒai , andϒia are the components of the energy–momentum d-tensor field. We notethat such decompositions into h- and v-components of gravitational field equations have to be consid-ered even in general relativity if physical interactions are examined with respect to an anholomomicframe of reference with associated N-connection structure.


There are variants of la-gravitational field equations derived in the low-energy limits of the theoryof locally anisotropic (super)strings [31] or in the framework of gauge-like la-gravity [32, 35] whenthe N-connection and torsions are dynamical fields and satisfy some additional field equations.

2.3. Modelling of Generalized Finsler Geometries in (Pseudo) Riemannian Spaces

The present-day trend is to consider the Finsler-like geometries and their generalizations as quitesophisticated for straightforward applications in quantum and classical field theory. The aim of thissection is to prove that, as a matter of principle, such geometries could be equivalently modelledon corresponding (pseudo) Riemannian manifolds (tangent or vector bundles) by using Cartan’smoving frame method and from this viewpoint a wide class of Finsler-like metrics could be treatedas solutions of the usual Einstein field equations.

2.3.1. Almost Hermitian Models of Lagrange and Finsler Spaces.This topic was originallyinvestigated by Miron and Anastasiei [23]; here we outline some basic results.

Let us model a la-spacetime not on a vector bundleE but on a manifoldT M = T M\{0} associatedto a tangent bundleT M of ann-dimensional base spaceM (when the dimensions of the typical fiberand base are equal,n = m and\{0} means that the null cross-section of the bundle projectionτ : T M→ M is eliminated) and consider d-metrics of type

δs2 = gαβ(u)δα ⊗ δβ = gi j (x, y)dxi dxj + gi j (x, y)δyi δyi . (18)

On T M we can define a natural almost complex structureC(a) as

C(a)(δi ) = −∂/∂yi

and

C(a)(∂/∂yi ) = δi ,

where the la-derivativeδi = ∂/∂xi − Nki ∂/∂yk (9) and la-differentialδi = dyi + Ni

kdxk (10) acton T M, being adapted to a nontrivial N-connection structureN = {Nk

j (x, y)} in T M. It is obviousthatC2

(a) = −I . The pair (δs2,C(a)) defines an almost Hermitian structure onT M with an associate2-form

θ = gi j (x, y)δi3 dxj

and the triadK 2n = (T M, δs2,C(a)) is an almost K¨ahlerian space. By straightforward calculationswe can verify that the canonical d-connection (13) satisfies the conditions

cDX(δs2) = 0, cDX(C(a)) = 0

for any d-vectorX on T M and has zerohhh- andvvv-torsions.The notion ofLagrange space[20, 23] was introduced as a generalization of Finsler geometry in

order to geometrize the fundamental concepts in mechanics. A regular LagrangianL(xi , yi ) on T Mwas introduced as a continuity classC∞ functionL : T M→ R for which the matrix

gi j (x, y) = 1

2

∂2L

∂yi ∂y j(19)

92 SERGIU I. VACARU

has rankn and is of constant signature onT M. A d-metric (18) with coefficients of form (19), acorresponding canonical d-connection (13), and an almost complex structureC(a) define an almostHermitian model of Lagrange geometry.

For arbitrary metricsgi j (x, y) of rankn and constant signature onT M, which cannot be determinedas a second derivative of a Lagrangian, one defines the so-called generalized Lagrange geometry onT M (see details in [23]).

A particular subclass of metrics of type (19) consists of those where instead of a regular Lagrangianwe consider a Finsler metric functionF on M defined asF : T M → R having the followingproperties. It is of classC∞ on T M and only continuous on the image of the null cross-section inT M; the restriction ofF on T M is a positive function homogeneous of degree 1 with respect to thevariablesyi ; i.e.,

F(x, λy) = λF(x, y), λ ∈ Rn;

and the quadratic form onRn with coefficients

gi j (x, y) = 1

2

∂2F2

∂yi ∂y j, (20)

defined onT M, is positive definite. Different approaches to Finsler geometry, its generalizations,and applications are examined in a number of monographs [5, 11, 23] and as a rule they are basedon the assertion that in this type of geometries the usual (pseudo) Riemannian metric interval

ds=√

gi j (x)dxi dxj

on a manifoldM is changed into a nonlinear one defined by the Finsler metricF (fundamentalfunction) onT M (we note an ambiguity in terminology used in monographs on Finsler geometryand on gravity theories with respect to such terms as Minkowschi space, metric function, and so on)

ds= F(xi, dxj ). (21)

Geometric spaces with a “cumbersome” variational calculus and a number of curvatures, torsions,and invariants connected with nonlinear metric intervals of type (19) are considered less suitable forpurposes of modern field and particle physics.

In our investigations of generalized Finsler geometries in (super) string, gravity, and gauge theories[30, 31] we advocated the idea that instead of the usual geometric constructions based on straightfor-ward applications of derivatives of (20) following from a nonlinear interval (21) one should considerd-metrics (12) with coefficients of necessity determined via an almost Hermitian model of a Lagrange(19) or Finsler geometry (20) and/or their extended variants. In this way, by a synthesis of the movingframe method with the geometry of N-connection, we can investigate various classes of higher andlower dimensional gravitational models with generic or induced anisotropies in a unified manner onsome anholonomic and/or Kaluza–Klein spacetimes.

As a matter of principle, having a physical model with a d-metric and geometrical objects associatedto la-frames, we can redefine the physical values with respect to a local coordinate base on a (pseudo)Riemannian space. The coefficientsgi j (x, y) andhab(x, y) of a d-metric written for a la-basisδuα =(dxi , δya) transform into a usual (pseudo) Riemannian metric if we rearrange the components with


respect to a local coordinate basisduα = (dxi , dya),

gαβ =gi j + Na

iNb

jhab Ne

jhae

Neihbe hab

, (22)

where “hats” on indices emphasize that coefficients of the metric are given with respect to a coordinate(holonomic) basis on a spacetimeVn+m. Parametrizations (ansatzs) of metrics of type (20) are largelyapplied in Kaluza–Klein gravity and its generalizations [25]. In our works [30–32], following thegeometric constructions from [23], we proved that the physical model of interactions is substantiallysimplified. In addition we can correctly elucidate anisotropic effects if we work with diagonal blocksof d-metrics (12) with respect to anholonomic frames determined by a N-connection structure.

2.3.2. Finsler Like Metrics in Einstein’s Gravity.Some classes of locally anisotropic cosmo-logical and black hole-like solutions (in three, four, and higher dimensions) which can be treatedas generalized Finsler metrics of nonspheric symmetry (with rotation ellipsoid, torus, and cylin-drical event horizons, or with elliptical oscillations of horizons) have been obtained [32, 34] (seethe Appendix). Under corresponding conditions such metrics could be solutions of field equationsin general relativity or its lower or higher dimension variants. Here we shall formulate the generalcriteria for when a Finsler-like metric could be a solution of gravitational field equations in Einsteingravity.

Let us consider onT M an ansatz of type (22) whengi j = hi j = 12∂

2F2/∂yi ∂y j (for simplicity,we omit hats on indices), i.e.,

gαβ =1

2

∂2F2

∂yi ∂y j + Nki Nl

j∂2F2

∂yk∂yl Nlj∂2F2

∂yk∂yl

Nki∂2F2

∂yk∂yl∂2F2

∂yi ∂y j

. (23)

A metric (23), induced by a Finsler quadratic form (20), could be treated in the framework of a Kaluza–Klein model if for some values of the Finsler metricF(x, y) and N-connection coefficientsNk

i (x, y)this metric were a solution of the Einstein equations (8) written with respect to a holonomic frame. Forthe dimensionn = 2, when the valuesF andN are chosen to induce locally a (pseudo) Riemannianmetricgαβ of signature (−,+,+,+) and with coefficients satisfying the four-dimensional Einsteinequations, we define a subclass of Finsler metrics in the framework of general relativity. Here wenote that, in general, an N-connection, on a Finsler space, subjected to the condition that the induced(pseudo) Riemannian metric is a solution of the usual Einstein equations does not coincide with thewell known Cartan’s N-connection [5, 11]. We have to examine possible compatible deformationsof N-connection structures [23].

Instead of Finsler like quadratic forms we can consider ansatzes of type (22) withgi j andhi j

induced by a Lagrange quadratic form (19). A general approach to the geometry of spacetimes withgeneric local anisotropy can be developed on embeddings into corresponding Kaluza–Klein theoriesand adequate modelling of la-interactions with respect to anholonomic or holonomic frames andassociated N-connection structures.

3. COLLISIONLESS RELATIVISTIC KINETIC EQUATION

As argued in Section 2, the spacetimes could be of generic local anisotropy after nontrivial reduc-tions from some higher dimension theories or possess a local anisotropy induced by anholonomicframe structures even when we restrict our considerations to the general relativity theory. In this

94 SERGIU I. VACARU

line of particular interest is the formulation of relativistic kinetic theory with respect to generalanholonomic frames and elucidation of locally anisotropic kinetic and thermodynamic processes.

3.1. The Distribution Function and Its Moments

We use the relativistic approach to kinetic theory (we refer readers to monographs [15, 36] forhistory and complete treatment). Let us consider a simple system consisting of$ point particlesof massm in a la-spacetime with a d-metricgαβ . Every particle is characterized by its coordinatesuα(l ) = (xi

(l ), ya(l )) (u1 = x1 = ct is considered the time-like coordinate; for simplicity we put

hereafter the light velocityc = 1); x2(l ), x3

(l ), . . . , xn(l ) andy1

(l ), y2(l ), . . . , ym

(l ) are respectively space-likeand anisotropic coordinates, where the index (l ) enumerates the particles in the system. The particles’momenta are denoted bypα(l ) = gαβ pβ(l ),. We shall use the distribution function8(uα, pβ), given onthe space of supporting elements (uα, pβ), as a general characteristic of a particle system.

The system (of particles) is defined by using the random function

φ(uα, pβ) =$∑

l=1

∫δsδ(n(a))(uα − uα(l )(s)

)× δ(n(a))(pβ − pβ(l )(s)), (24)

where the sum is taken on all system’s particles,

δs=√

gαβδuαδuβ

is the interval element along the particle trajectoryuα(l )(s) parametrized by a natural parameters,andδn(a) (uα) is then(a)-dimensional delta function. The functionsuα(l )(s) and pβ(l )(s) describing thepropagation of thel -particle are found from the motion equations on la-spacetime

mδuα(l )ds= pα(l )

(25)

mDδpα(l )

ds= m

δpα(l )

ds− c0ταβ

(uγ(l )(s)

)pτ (l ) pβ(l ) = Fα(l ),

wherec0αβγ is the canonical d-connection with coefficients (14) and byFα(l ) we denote an exteriorforce (electromagnetic or another type) acting on thel th particle.

The distribution function8(u, p) is defined by averaging on paths of the random function

8(u, p) = 〈〈φ(u, p)〉〉,

where brackets〈〈. . .〉〉 denote path averaging.Let us consider a space-like hypersurfaceF(uα) = const with elementsδ6α = nαδ6, where

nα = DαT /|DT |, d6 =√|g|δn(a)u/nαδuα, |DT | =

√gαβDαDβT , and

√|g|δn(a)u is the invari-ant space–time volume. A local system of refernce in a pointuα(0), with the metricg(0)

i j = ηi j =diag(−1, 1, 1, . . . ,1), is obtained ifF(uα(0)) = x1

(0). In this caseδ6α = δ1αδ

n(a)−1u, δn(a)−1u =dx2dx3 . . .dxnδy1δy2 . . . δym, andδαβ is the Kronecker symbol. The value

8(u, p)vαd6αδn(a) p√|g| ,

wherevα = pα/m andδP = δn(a) p/√|g| = δp1δp2 · · · δpn(a)/

√|g|, is the invariant volume in themomentum space and defines the quantity of particles intersecting the hypersurface elementδ6α


with momentapα included in the elementδP in the vicinity of the pointuα. The first〈pα〉 and secondmoments〈pα pβ〉 of the distribution function8(x, p) give respectively the flux of particlesnα andthe energy–momentumTαβ ,

〈pα〉 =∫8(u, p)pαδP = mnα (26)

and

〈pα pβ〉 =∫8(u, p)pα pβδP = mϒαβ. (27)

We emphasize that the motion equations (25) have the first integralgαβ p(l )α p(l )

β = m2 = const foreveryl -particle, so the functions (24) and, in consequence,8(u, p), are nonzero only on the masshypersurface

gαβ pα pβ = m2, (28)

which when distinguished by an N-connection form (see the dual to the d-metric (12)) is written

gi j pi pj + gabpa pb = m2.

For computations it is convenient to use a new distribution functionf (u, p) on ann(a) − 1dimensional la-hyperspace (28)

8(uα, pβ) = f (uα, pβ)δ(√

gαβ pα pβ − m)θ (p1),

where

θ (p1) ={

1, p1 ≥ 0

0, p1 < 0

and the indexβ runs values inn(a) − 1 dimensional la-space. The flux of particles (26) and ofenergy–momentum (27) is computed by usingf (u, p) as

nα =∫δςpα f (uγ , pβ),

with respective baseni and fiberna components,

ni =∫δςpi f (uγ , pβ) and na =

∫δςpa f (uγ , pβ),

and

ϒαβ =∫δςpα pβ f (uγ , pβ),

where p1 is expressed viap2, p3, . . . , pn(a) by using Eq. (28). The abbreviationδς = δn(a)−1 p/(√|g|p1) is used here. The energy–momentum can also be split into base–fiber (horizontal–vertical,

96 SERGIU I. VACARU

in brief, h- and v-, or hv-components) by a corresponding distinguishing of momenta,ϒαβ ={ϒ i j , ϒaj , ϒ ib, ϒab}.

The one-particle distribution functionf (u, p) characterizes the number of particles (with massm, or massless ifm = 0) at a pointuα of a la-spacetime of dimensionn(a) being distinguished byan N-connection momentum vector (in brief, momentum d-vector)pα = (pi, pa) = (p1, Ep). Onestates that

f (u, p)pβδσβδς = f (u, p)pi δσi δς + f (u, p)paδσaδς

gives the number of world-lines of particles with momentum d-vectorpα in an intervalδpα aroundp crossing a space-like hypersurfaceδσβ at a pointu. Whenδσβ is taken to be time-like one hasδσβ = (δn(a)u, 0, . . . ,0).

The velocity d-field of a fluidUµ in la-spacetime can be defined in some ways as in relativistickinetic theory [15]. To obtain a particularly simple form for the energy–momentum d-tensor oneshould follow the Landau–Lifshitz approach [22] where the locally anisotropic fluid velocityUµ(u) =(Ui (u),Ua(u)) and the energy densityε(u) are defined respectively as the eigenvector and eigenvalueof the eigenvalue equation

ϒαβUβ = εUα. (29)

A unique value ofUα can be found from the conditions being time-like and normalized to a unityd-vector,

UαUα = 1.

Having so fixed the fluid d-velocity we can define correspondingly the particle density

n(u) = nαUα, (30)

the energy density

ε(u) = UαϒαβUβ, (31)

and the average energy per particle

e= ε/n. (32)

We can also introduce the pressure d-tensor

Pαβ(u) = 1ατϒτν1νβ (33)

by applying the projector

1αβ = gαβ −UαUβ (34)

with properties

1αβ1βµ = 1αµ,1

αβUβ = 0


and

1µµ = n(a) − 1.

With respect to a la-frame (9) we can introduce a particular Lorentz system, called the locallyanisotropic rest frame of the fluid, whenUµ = (1, 0, 0, . . . ,0) and1α

µ = diag(0, 1, 1, . . . ,1). Sothe pressure d-tensor was defined as coinciding with the la-space part of the energy–momentumd-tensor with respect to a locally anisotropic rest frame.

3.2. Collisionless Kinetic Equation

We start our proof by applying the identity∫ds

d

dsδn(a)

(uα − uα(l )(s)

)δn(a)

(pβ − pβ(l )(s)

) = 0.

Differentiating under this integral and taking into account Eq. (28) we get the relation

δ(pαφ)

∂uα+ ∂

∂pβ

(c0

µβε pµpεφ

) = 0. (35)

In our further considerations we neglect the interactions between the particles and assume thatthe metric of the background gravitational fieldgαβ does not depend on motion of particles. Afteraveraging (35) on paths we define the equation for the one-particle distribution function

δ(pα8)

∂uα+ ∂

∂pβ

(c0

µβε pµpε8

) = 0. (36)

As a matter of principle we can generalize the problem [36] when particles are considered to bealso sources of a gravitational field which is self-consistently defined from the Einstein equationswith the energy–momentum tensor defined by the formula (27).

Taking into account the identity

δpα

∂uα+ ∂

∂pβ

(c0

µβε pµpε

) = 0

we get from (36) the collisionless kinetic equation for the distribution function8(uα, pβ)

pβ Dβ8 = 0, (37)

where

Dβ = δ

∂uβ− c0εβα pα

∂

∂pε(38)

generalizes Cartan’s covariant derivation [5] for a space with higher order anisotropy (locallyparametrized by supporting elements (uα, pβ)), provided with an (extended to momenta coordi-nates) higher order nonlinear connection

Nεβ = δεaδi

βNai − c0εβα pα (39)

98 SERGIU I. VACARU

(on the geometry of higher order anisotropic spaces and superspaces and possible applications inphysics see [23, 30, 31]).

Equation (37) can be written in equivalent form for the distribution functionf (uα, pβ) as

pλ Dλ f = 0, (40)

with Cartan’s operator defined on (n(a) − 1)-dimensional momentum space.The kinetic equations (37) and, equivalently, (40) reflect the conservation law of the quantity of

particles in every volume of the space of supporting elements which holds in the absence of collisions.Finally, in this section, we note that the kinetic equation (37) is formulated on the space of

supporting elements (uα, pβ), which is characterized by coordinate transforms

xi ′ = xi ′ (xk), ya′ = yaK a′a (xi ) (41)

and

pj ′ = pi∂xi

∂x j ′ , pa′ = paK aa′ (x

i ),

whereK a′a (xi ) and K a

a′ (xi ) takes values correspondingly in the set of matrices parametrizing the

group of linear transformsGL(m,Rm), whereR denotes the set of real numbers. In a particular case,for dimensionsn = m, we can parametrize

K i ′i (xi ) = ∂xi ′

∂xiand K i

i ′ (xi ) = ∂xi

∂xi ′

∣∣∣∣xi=xi (xi ′ )

.

A tensorQα1α2···αrβ1β2···βq

(uε, pτ ) distinguished the higher order nonlinear connection (39) being con-travariant of rankr and covariant of rankq satisfies the next transformation laws by a change ofcoordinates of type (41):

Qα′1α

′2···α′r

β ′1β′2···β ′q (uε

′, pτ ′ ) = ∂uα

′1

∂uα1· · · ∂uα

′r

∂uαr

∂uβ

∂uβ′1· · · ∂uβq

∂uβ′q

Qα1α2···αrβ1β2···βq

(uε, pτ ). (42)

Operators of type (39) and transformations (41) and (42), on the first order of anisotropy and on thetangent bundleT M on ann-dimensional manifoldM , were considered by Cartan in his approachto Finsler geometry [5] and by Vlasov [36] in order to formulate the statistical kinetic theory ona Finsler geometry background. It should be emphasized that the Cartan–Vlasov approach is tobe applied even in general relativity because the kinetic processes are to be examined in a phasespace provided with local coordinates (uε, pτ ). Our recent generalizations [30, 31] to higher orderanisotropy (including spinor and supersymmetric spaces) are to applied in the case of models withnontrivial reductions (modelled by N-connections) from higher to lower dimensions.

4. KINETIC EQUATION WITH PAIR COLLISIONS

In this section we shall prove the relativistic kinetic equations for a one-particle distributionfunction f (uα, pβ) with pair collisions [7]. We summarize the related results and generalize theconstructions for Minkowski spaces [15].


4.1. Integral of Collisions, Differential Cross Section, and Velocity of Transitions

If collisions of particles are taken into account (for simplicity we shall consider only pair collisions),the quantity of particles from a volume in the space of supporting elements is not constant. We haveto introduce a source of particlesC(x, p), for instance, in the kinetic equations (40),

pα Dα f = C( f ) = C(u, p). (43)

The scalar functionC(u, p) is called the integral of collisions (in the space of supporting elements).The value

1n(a)u1n(a)−1 p

p1C(u, p)

is the change of the quantity of particles under pair collisions in a region1n(a)u1n(a)−1 p.Let us denote by

W(p, p[1] | p′, p′[1]

)p1 p1

[1] p′1 p′1[1]

(44)

the probability of transition for two particles which before scattering have the momentapi and p1iand after scattering the momentap′

iand p′

1iwith respective inaccuracies1n(a)−1 p′ and1n(a)−1 p′1.

The functionW(p, p[1] | p′, p′[1] ) (the so-called collision rate) is symmetric on argumentsp, p[1] andp′, p′[1] and describes the velocity of transitions with conservation of momenta of type

pα + pα[1] = p′α + p′α[1]

when the conditions (28) hold.The number of binary collisions within a la-spacetime intervalδu around a pointu between

particles with initial momenta in the ranges

( Ep, Ep+ δ Ep) and( Ep[1], Ep[1] + δ Ep[1]

)and final momenta in the ranges

( Ep′, Ep′ + δ Ep′) and( Ep′[1], Ep′[1] + δ Ep′[1]

)is given by

f (u, p) f(u, p[1]

)W(p, p[1] ; p′, p′[1]

)δςpδςp[1]δςp′δςp′[1]

δu, (45)

where, for instance, we abbreviatedδςp = δn(a)/p(√|g|p1).

The collision integral of the Boltzmann equation in la-spacetime (43) is expressed in terms of thecollision rate (44)

C( f ) =∫

f (u, p) f(u, p[1]

)×W(p, p[1] ; p′, p′[1]

)δςp[1]δςp′δςp′[1]

. (46)

The collision integral is related to the differential cross section (see below).

100 SERGIU I. VACARU

4.2. The Cross Section in La-Spacetime

The numbern(bin) of binary collisions per unit time and unit volume when the initial momenta ofthe colliding particles lie in the ranges

( Ep, Ep+ δ Ep) and( Ep[1], Ep[1] + δ Ep[1]

)and the final momenta are in some intervalς in the space of variablesEp′ and Ep′[1] is to be obtainedfrom (45) by dividing onδu = δtδx2 · · · δxndy1 · · ·dym and integrating with respect to the primedvariables,

n(bin) = f (u, p) f(u, p[1]

)× δςpδςp[1]

∫ς

W(p, p[1] ; p′, p′[1]

)δςp′δςp′[1]

. (47)

Let us introduce an auxiliary velocity

v = F/(p1 p1

[1]

), (48)

where the so-called M¨oller flux factor is defined by

F = [(pα p[1]α)2− m4

]1/2. (49)

It may be verified that the speedv reduces to the relative speed of particles in a frame in whichone of the particles initially is at rest. We also consider the product of the number density of targetparticles with momenta in the range (Ep[1], Ep[1] + δ Ep[1] ) (given by f (u, p[1] )δn(a)−1 p[1] ) and the fluxof incoming particles with momenta in the range (Ep, Ep+ δ Ep) (given by f (u, p)δn(a)−1 pv, wherevis the speed (48)). This product can be written

F f (u, p) f(u, p[1]

)δςpδςp[1] . (50)

By definition (as usual in the relativistic Boltzmann theory [3, 7, 15]) the cross section ˜σ is thedivision of the number (47) into the number (50),

σ = 1

F

∫ς

W(p, p[1] ; p′, p′[1]

)δςp′δςp′[1]

. (51)

The condition that collisions are local implies that the collision rate must containn(a) delta-functions

W(p, p[1] ; p′, p′[1]

) = w(p, p[1] ; p′, p′[1]

)× δn(a)(p+ p[1] − p′ − p′[1]

). (52)

Let En(a)−1 be an (n(a) − 1)-dimensional Euclidean space enabled with an orthonormal basis(e1, e2, . . . ,en(a)−1) and denote by (u1, u2, . . . ,un(a)−1) the Cartesian coordinates with respect to thisbasis. For calculations on scattering particles it is useful to apply the system of spherical coordinates(rn(a) = r, θn(a)−1, θn(a)−2, . . . , θ1) associated with the Cartesian coordinates (u1, u2, . . . ,). The volumeelement can be expressed as

dn(a)−1u = (r )n(a)−2dr dn(a)−2Äθ,


where the element of solid angle, the (n(a) − 1)-spherical element, is given by

dn+m−2Äθ = sinn+m−3 θn+m−2 · sinn+m−4 θn+m−3

· · · · sin2 θ3 · sinθ2× dθn+m−2 · · · · · dθ2 · dθ1. (53)

In our further considerations we shall consider that the Cartesian and spherical (n(a)−1)-dimensionalcoordinates are given with respect to an la-frame of type (9).

In order to eliminate the delta functions from (52) put into (51), we fix as a reference frame thecenter of mass frame for the collision between two particles with initial momentaEp and Ep[1] . Thequantities defined with respect to the center of mass frame will be enabled with the subindexC M.One denotes byEpC M the polar axis and characterizes the directionsEp′C M of the outgoing particleswith respect to the polar axis by means of generalized spherical coordinates. In this case

δn(a)−1 p′C M = | Ep′C M|n(a)−2d| Ep′C M|dÄC M, (54)

wheredÄC M is given by the formula (53).The total (n(a) + 1)-momenta before and after collision in la-spacetime are given by d-vectors

Pα = pα + pα[1] P′α = p′α + p′α[1] . (55)

Following from

P′1C M = 2p′1C M = 2√| Ep′C M|2+ m2

we have

| Ep′C M|2d| Ep′C M| =1

4P′1C Md P′1C M.

As a result the formula for the volume element (54) transforms into

δn(a)−1 p′C M =1

4| Ep′C M|n(a)−2P′1C Md P′1C MdÄC M.

Inserting this collision rate (52) into (51) we obtain an integral which can be rewritten (see theAppendixes 1 and 7 to the paper [3] for details on transition to the center of mass variables; inla-spacetimes one holds similar considerations with that difference that must work with respect tola-frames (9) and (10) and d-metric (12))

σ = Fn(a)−4

En(a)−2

∫ς

w dÄC M, (56)

whereE = √PαPα is the total energy (divided byc = 1).The differential section in the center of mass system is found from (56)

δσ

dÄ

∣∣∣∣C M

= Fn(a)−4

En(a)−2w


which allows us to express the collision rate (52) as

W(p, p[1] ; p′, p′[1]

) = En(a)−2

Fn(a)−4× δσ

dÄ

∣∣∣∣C M

· δn(a)(p+ p[1] − p′ − p′[1]

). (57)

We shall use the formula (57) in order to compute the transport coefficients.

5. EQUILIBRIUM STATES IN LA-SPACETIMES

When the system is in equilibrium we can derive an expression for the particle distribution functionf (u, p) = f(eq)(u, p) in a similar way as for locally isotropic spaces and write

f(eq)(u, p) = 1

(2πh–)n(a)−1exp

(µ− pi Ui − paUa

kBT

), (58)

wherekB is Boltzmann’s constant andh– is Planck’s constant divided by 2π andµ = µ(u) andT =T(u) are respectively the thermodynamic potential and temperature. For simplicity, in this sectionwe shall omit explicit dependencies ofµ andT on la-spacetime coordinatesu. Our thermodynamicsystems will be considered in local equilibrium in a vicinity of a pointu[0] with respect to a restframe locally adapted to a N-connection structure, like (9) and (10). In the simplest case then+msplitting is trivially given by an N-connection with vanishing curvature (11). For such conditions oftrivial la-spacetimeµ andT can be considered as constant values.

5.1. Particle Density

The formula relating the particle densityn, temperatureT , and thermodynamic potentialµ isobtained by inserting (58) into (30), with (26). Choosing the value locally adapted to the N-connection(9) to be the rest frame, whenUµ = (1, 0, . . . ,0) the calculus is performed as for isotropicn(a)

dimensional spaces [3]. The integral for the number of particles per unit volume in the rest la-frameis

n(µ, T) = 1

(2πh–)n(a)−1exp

(µ

kBT

)×∫

p1 exp

(− p1

kBT

)dn(a)−1 p

p1, (59)

with

p1 = (| Ep|2+ m2)1/2 (60)

and depends only on the length of then(a) − 1 dimensional d-vectorEp. By applying sphericalcoordinates, when

dn(a)−1 p = | Ep|n(a)−2d| Ep| dÄ

anddÄ is given by the expression (53), and differentiating on radiusρ the well known formula forthe volumeVn(a)−1 in an (n(a) − 1)-dimensional Euclidean space

Vn(a)−1 =(n(a) − 1

)π (n(a)−1)/2

0[(

n(a) + 1)/2] ρ, (61)


where0 is the Euler gamma function, then puttingρ = 1 we get

∫dÄ =

(n(a) − 1

)π (n(a)−1)/2

0[(

n(a) + 1)/2] . (62)

We note that from (60) one follows| Ep|d| Ep| = p1dp1 and we can transform (59) into an integral withrespect to the angular variables through the replacement

dn(a)−1 p

p1→

(n(a) − 1

)π (n(a)−1)/2

0[(

n(a) + 1)/2] | Ep|n(a)−3dp1.

Introducing the dimensionless quantities

ξ = p1/kBT and ϑ = m/kBT, (63)

for which, respectively,

dp1 = kBT dξ and | Ep| = kBT√ξ2− ϑ2, (64)

the integral (59) is computed as

n(µ, T) = 2n(a)/2

(πm

2πh–

)n(a)−1(kBT

m

)(n(a)−2)/2× exp

(µ

kBT

)K(n(a)+1)/2

(m

kBT

), (65)

where the modified Bessel function of the second kind of ordern(a)/2 has the integral representation

Ks(η) =√π

(2η)s0

(s+ 1

2

) ∞∫η

e−q(q2− η2)s−1/2 dq. (66)

We note the dependence of (65) onm anisotropic parameters (coordinates). The formulas provedin this section transform into locally anisotropic ones [3] if the N-connection is fixed to be trivial(with vanishing N-connection curvature (11)) and the d-metric (12) transforms into a usual (pseudo)Riemannian one.

5.2. Average Energy and Pressure

The average energy per particle, for a system in equilibrium, can be calculated by introducingthe distribution function (58) into (32) and applying the formulas (27), (31), and (65). In terms ofdimensionless variables (63) we have

ε(µ, T) = (kBT)n(a)

(2πh–)n(a)−1

(n(a) − 1

)π (n(a)−1)/2

0[ (

n(a) + 1)/2] × exp

(µ

kBT

)∫ ∞ϑ

(ξ2− ϑ2)(n(a)−3)/2ξ2 dξ.

After carrying out partial integrations together with applications of the formula

(q2− ζ 2)(s−2)/2 = 1

s

d

dq

[(q2− ζ 2)s/2

]


we obtain

ε(ϑ) = π (n(a)−2)/2 mn(a)

(2πh–)n(a)−1

(2

ϑ

)n(a)/2

× exp

(µ

kBT

)[ϑK(n(a)+2)/2(ϑ)− Kn(a)/2(ϑ)

]. (67)

Dividing the energy density (67) to the particle density (59) and substituting the inverse dimen-sionless temperatureϑ = ϑ(T) (63) we get the energy per particle (32)

e(T) = kBT

(m

kBT

K(n(a)+2)/2(m/kBT)

Kn(a)/2(m/kBT)− 1

). (68)

This formula is the thermal equation of state of a equilibrium system havingmanisotropy parameters(we remember thatn(a) = n+m) with respect to a la-frame (9).

The pressure d-tensor is computed by substituting (58) into (33), applying also the integral (27).We get

Pαβ = −kBTn(µ, T)1αβ, (69)

where, by definition, the coefficients before1αβ determine the pressure

P = kBTn(µ, T). (70)

So, for a system ofn(a) − 1= n+m− 1 dimensions the expression (70) defines the equation ofstate of ideal gas of particles with respect to a la-frame, havingm anisotropic parameters.

5.3. Enthalpy, Specific Heats, and Entropy

The average enthalpy per particle

h(T) = e(T)+ P(µ, T)/n(µ, T) (71)

is computed directly by substituting in this formula the values (68), (70), and (65). The result is

h(T) = mK(n(a)+2)/2(m/kBT)

Kn(a)/2(m/kBT). (72)

Using (68) and (72) we can compute respectively the specific heats at constant pressure and atconstant volume

cp =(∂h

∂T

)p

and cp =(∂e

∂T

)p

.

Substractingh ande, after carrying out the differentiation with respect toT , one finds Mayer’srelation,

cp − cv = kB,


and, introducing the adiabatic constantγ = cp/cv,

γ

γ − 1= ϑ2+ (n(a) + 1

) h

kBT−(

h

kBT

)2

. (73)

The relation (73) can be proven by straightforward calculations by using the properties of Bessel’sfunction (66) (see a similar proof for isotropic spacetimes in [3]).

The entropy per particle is introduced as in the isotropic case

s= h− µT

. (74)

We have to insert the values ofµ (found from (65))

µ(n, T) = kBT × ln(2πh–)n(a)−1n

(2m)n(a)/2(πkBT)(n(a)−2)/2Kn(a)/2(m/kBT)(75)

and ofh (see (72) in order to obtain the dependences= s(T)) (for simplicity, we omit this cumber-some formula).

5.4. Low and High Energy Limits

Let us investigate the nonrelativistic locally anistropic limit by applying the asymptotic formula [1]

Ks(ϑ) ∼√π

2ϑe−ϑ

valid for largeϑ . The formula (68) gives

e∼ m+ n+m− 1

2kBT.

Thus, each isotropic and anisotropic dimension contributes withkBT/2 to the average energy perparticle.

Now, we consider high temperatures. Forϑ → 0 one holds the asymptotic formula [1]

Ks(ϑ) ∼ 2s−10(s)ϑ−s

and the particle number density (65), the energy (68), enthalpy (72), and entropy (74) can be approx-imated respectively (we state explicitly the dimensionsn+m) by

n(µ, T) = 2n+m−1πn+m−20

(n+m

2

)×(

kBT

2πh–

)n+m−1

exp

(µ

kBT

),

e= (n+m− 1)kBT, h = (n+m)kBT, (76)

s = (n+m)kB.

As a consequence, the corresponding specific heats and adiabatic constant are

cp = (n+m)kB, cv = (n+m− 1)kB, and γ = n+m

n+m− 1.


These formulas imply that for largeT the spacetime anisotropy (very possible at the beginning ofour universe) could modify substantially the thermodynamic parameters.

Finally we emphasize that in the ultrarelativistic limit

s= 2n+m−1πn+m−2(n+m)× 0(

n+m

2

)(kBT

2πh–

)n+m−1 kB

n. (77)

We also note that our treatment was based on Maxwell–Boltzmann instead of Bose–Einstein statistics.

6. LINEARIZED LOCALLY ANISOTROPIC TRANSPORT THEORY

For systems with local anisotropy outside equilibrium the distribution functionf(eq)(u, p) (58) mustbe generalized to another one,f (u, p), solving the kinetic la-equation (43). We follow a standardprocedure of linearization and construction of solutions of kinetic equations by generalizing to la-spacetimes the Chapman–Enskog approach (we shall extend to locally anisotropic backgrounds theresults presented in [15]).

We write

f (u, p) = f[0] (u, p)[1+ ϕ(u, p)] (78)

with the lowest order of approximation tof taken similarly to (58),

f[0] (u, p) = 1

(2πh–)n(a)−1× exp

(µ(u)− pi Ui (u)− paUa(u)

kBT(u)

), (79)

where the constant variables of a trivial equilibriumµ, T , andUα are changed by some local con-terpartsµ(u), T(u), andUα(u). Outside equilibrium the first two dependenciesµ(u) andT(u) aredefined respectively from the relations (75) and (68).

6.1. Linearized Transport Equations

The so-called deviation functionϕ(u, p) from (78) describes the deformation by nonequilibriumflows of f[0] (u, p) into f (u, p). It is considered that in equilibriumϕ vanishes and not too far fromequilibrium states it must be small. The Chapman–Enskog method states that after substitutingf[0]

into the left side andf[0] (1+ ϕ) into the right hand of (43) we shall neglect the quadratic and higherterms inϕ. As a result we obtain a linearized equation for the deviation function (for simplicity,hereafter we shall not point to the explicit dependence of functions and kinetic and thermodynamicvalues on spacetime coordinates)

−pα Dα f[0] = f[0] L[ϕ],

where, having introduced (79) into (46), we write for the right side

L[ϕ] = −pα Dα

(µ− pi Ui − paUa

kBT

).

The left side is computed by applying the generalized Cartan derivative (38).


We introduce a locally anisotropic generalization of the gradient operator by considering theoperator (in brief, la-gradient)

∇α = 1βαDβ (80)

with 1βα given by (34) (we use a d-covariant derivative defined by a d-connection (13) instead of

partial and/or isotropic derivatives for isotropic spaces [3, 15]. In a local rest frame of the isotropicfluid ∇α→ (0, ∂1, ∂2, . . .), i.e., in the flat isotropic space, the la-gradient reduces to the ordinaryspace gradient. By applying∇α we can eliminate the time la-derivatives

kBT L[ϕ] = pαpβ∇αUβ − T pα∇α(µ

T

)− pαpβUα

(∇βT

T− ∇β p

hn

)+[(γ − 1)pαUα + T2(γ − 1)

∂

∂T

(µ

T

)+ n

(∂µ

∂n

)]pεUε∇vUv. (81)

We can verify, using the expressionµ = µ(n, T) from (75), that one holds the equalities

∂

∂T

(µ

T

)= − e

T2,

∂µ

∂n= kBT

n

and

1

n∇α p = h

∇αT

T+ T∇α

(µ

T

).

Introducing these expressions into (81) we get

kBT L[ϕ] = 4X + (h− pαUα)1βε pβXβ +

(1βε1

αγ −

1

n+m− 11εγ1

βα

)pβ p(0)

α Xεγ ,

where

4 =(

n+m

n+m− 1− γ

)(pαUα

)2+ [(γ − 1)h− γ kBT ] pαUα − m2

n+m− 1(82)

and there were forces deriving the system towards equilibrium

X = −∇µUµ, Xα = ∇αT

T− ∇

α p

nh

and

(0)Xεγ = 1

2(∇εU γ +∇γU ε)− 1

n+m− 11εγ∇µUµ.

We note that the valueskBT L[ϕ], 4, and(0)Xεγ depend explicitly on the dimensions of the basesubspace,n, and of the fiber subspace,m. The anisotropy also modifies both the thermodynamicand kinetic values via operators∇µ and1β

ε which depend explicitly on d-metric and d-connectioncoefficients.


6.2. On the Solution of Locally Anisotropic Transport Equations

Let us suppose that the solution of these three equations is known,

kBT L[ A(p)] = 4,(83)

kBT L[B(p)1β

α pβ] = (h− pαUα)1β

α pβ,

and

kBT L

[C(p)

(1βα1

σε −

1

n+m− 11αε1

βσ

)pβ pσ

]=(1βα1

σε −

1

n+m− 11αε1

βσ

)pβ pσ ,

for some functionsA(p), B(p), andC(p). Because the integral operatorL is linear the linear com-bination of thermodynamic forces

ϕ = AX+ BαXα + C(0)αβ Xαβ,

where

Bα = B(p)1βα pβ

and

Cαβ = B(p)

(1εα1

σβ −

1

n+m− 11αβ1

εσ

)pε pσ ,

is a solution for the deviation functionϕ.It can also be verified that every function of the formϕ = a+bα pα with some parametersa andbα

not depending onpα is a solution of the homogeneous equationL[ϕ] = 0. As a consequence, it wasproved (see [3, 8]) that the scalar parts ofA(p) andB(p) are determined respectively up to functionsof the formsϕ = a+ bα pα andbα pα. We emphasize that solutions of the transport equations (83)(see the next sections) will be expressed in terms ofA, B, andC.

6.3. Linear Laws for Locally Anisotropic Nonequilibrium Thermodynamics

LetUβ(u) be a velocity field of Landau–Lifshitz type characterizing a locally anisotropic fluid flow.The heat flow and the viscous pressure d-tensor with respect to such a la-field are correspondinglydefined as in [15] but in terms of objects on la-spacetime (see formulas (72), (69), (34), (33), (27),(26), (13), (9), (10), (12)):

I α(heat)= (Uαϒαβ − hnβ)1ε

β,(84)

5αβ = Pαβ + p1αβ.

It should be noted that if the Landau–Lifshitz condition (29) is satisfied the first term inI ν(heat)vanishes and the heat flow is the enthalpy carried away by the particles. The pressureP(u) is definedby n(u)kBT(u); see (70). Inserting (78) and (79) into (84) we prove (see locally isotropic cases in[3, 15]) the linear laws of locally anisotropic nonequilibrium thermodynamics

I ν(heat)= λT Xν


and

5αβ = 2η(0)Xαβ − η(v) X1εσ

with the transport coefficients (the heat conductivityλ, the shear viscosityη, and the volume viscositycoefficientη(v); in order to compare with formulas from [3] we shall introduce explicitly the lightvelocity constantc) defined as

λ = − c

(n+m− 1)T

∫pσ Bε1

εσ (h− pνUν) f[0] dςp,

η(ν) = − c

n+m− 1

∫pε pσ1

εσ A f[0] dςp, (85)

η = c

(n+m)(n+m− 4)

∫pσ pεCνµ1

νµσε f[0] dςp,

where one uses the d-tensor

1νµσε =

1

2

(1vσ1

µε −1v

ε1µσ

)− 1

n+m− 11νµ1σε

with the properties

1νµρσ1

ρσφε = 1νµ

φε

and

1ρσρσ = −1+ (n+m)(n+m− 1)

2.

The main purpose of nonequilibrium thermodynamics is the calculation of transport coefficients.With respect to la-frames the formulas are quite similar with those for isotropic spaces with thatdifference that we have to consider the values as d-tensors and take into account the numberm ofanisotropic variables.

On both types of locally isotropic and anisotropic spacetimes one holds the so-called conditionsof fit (see, for instance, [3])∫

pεUεA f[0] dςp = 0 and

∫(pεU

ε)2A f[0] dςp = 0

which allow us to write the volume viscosity (see (82))

η(v) = c∫4A f[0] dςp.

For some d-tensorsHα1,...αq (p) andSα1,...αq (p) we define the symmetric bracket

{H, S} = 1

n2

∫Hα1,...αq (p)L[Sα1,...αq (p)] f[0] dςp,


wheren is the particle density andL is a linearized operator. In terms of such brackets the coefficients(85) can be rewritten in an equivalent form:

λ = − ckBn2

n+m− 1{Bα, Bα},

η = ckBTn2

(n+m− 1)2− 3

{Cαβ,Cαβ

}, (86)

η(v) = ckBTn2{A, A}.

The addition of invariants of typeϕ = a + bα pα to some solutions forA(p), Bα(p), or Cαβ(p)does not change the values of the transport coefficients.

6.4. Integral and Algebraic Equations

The solutions of transport equations (83) are approximated [3] by considering a power series onζ = pαUα/kBT for functions

A(ζ ) =∞∑

a=2

Aaζa, B(ζ ) =

∞∑b=1

Bbζb, C(ζ ) =

∞∑c=0

Ccζc.

The starting valuesa = 2 andb = 1 have been introduced with the aim or determining the scalarfunctionsA(ζ ) andB(ζ ), respectively, up to contributions of the formsa+bα pα andbα pα. Insertingthese power series into the integral equations (83), multiplying respectively onζ s f[0], ζ

s pα f[0] , andζ s pα pβ f[0] , after integrating ondςp we get

∞∑a=2

aa1a Aa = αa1,

∞∑b=1

bb1bBb = βb1,

∞∑c=0

cc1cCc = γc1,

wherea1 = 2, 3, . . . , b1 = 1, 2, . . . , c1 = 0, 1, . . . , and there are symmetric brackets

aa1a = {ζ a1, ζ a},bb1b =

{ζ b1 pα, ζ b1v

α pν}, (87)

cc1c ={ζ c1 pα pβ, ζ c1

vµαβ pν pµ

}and integrals

αa1 =∫ζ a14 f[0]

kBTn2dςp,

βb1=∫ζ b1(h− pαUα)1µν pµpν f[0]

kBTn2dςp, (88)

γc1 =∫ζ c11

vµαβ pν pµpα pβ f[0]

kBTn2dςp.


The lowest approximation is given by the coefficients

A2 = α2/a22, B1 = β1/b11, C0 = γ0/c00.

Introducing these values into (86) we obtain the first-order approximations to the transport co-efficients

λ = − ckBn2

n+m− 1

β21

a22,

η = ckBTn2

(n+m)(n+m− 1)− 2

γ 20

c00, (89)

η(v) = ckBTn2α2

2

a22.

The valuesα2, β1, andγ0 are (n+m−1)-fold integrals which can be expressed in terms of enthalpyh and temperatureT ,

α2 =n+m+ 1− (n+m− 1)γ

ckBTn− (n+m− 1)γ

cn→︸︷︷︸

T→∞0,

β1 =kBT

cn

(n+m− 1)γ

γ − 1→︸︷︷︸

T→∞

kBT

cn(n+m− 1)(n+m),

(90)

γ0 =kBT

c2n(n+m− 2)(n+m+ 1)h →︸︷︷︸

T→∞

1

n

(kBT

c

)2

[(n+m)2− 1](n+m).

In order to complete the calculus of the transport coefficients it is necessary to compute the backetsfrom the denominators of (89).

6.5. Brackets from Flat to la-Spacetimes

The calculation of brackets is a quite tedious task (see [3]) which should involve the curvedspacetime metric and connection. For simplicity, we consider a background flat spacetime withtrivial local anisotropy. In this case we can apply directly the formulas proved for the Boltzmanntheory inn(a) dimension but with respect to la-frames and by introducing the locally anisotropicd-connections and d-metrics instead of their isotropic analogues.

Let us consider a rest frame locally adapted to an N-connection,Ee being a unit vector, and denoteby (2n(a)−1,2n(a)−2, . . . , 21) the angles of spherical coordinates with respect to a Cartesian vector(X1, X2, . . . , Xn(a) ) chosen as theXn(a) -axis alongEpC M (where| EpC M| = F/P, P = |Pα|; see (49),(55), andEpC MEe in the plane ofXn(a) andXn(a)−1 axes). We assume that the scattering angle is2n(a)−1

and the differential cross section (dσ/dÄ)C M is a function only of the module of total energy and ofscattering angle2 = 2n(a)−1,

(dσ

dÄ

)C M

=(

dσ

dÄ

)C M

(P,2).

The results of the calculations of symmetric brackets will be expressed in terms of twofold integrals


for arbitrary integers,

J(i , j ,k)r s (ζ, t, u, v, w) = ν(ζ )

(rt

)(su

)(d/2v

)2b+v−d

√π× 0

[b+ v + 1

2

]F(t, u, v)

×∞∫

2ζ

dχ χ (n+m−1+d−u−b−v+i ) ×(χ2

4− ζ 2

)(n+m−2+t+u+ j )/2

Kb+v−h(ζ )

×[δ0w −

(uw

)sinw 2 cosu−w 2

] π∫2ζ

d2 sinn+m−12

(dσ

dÄ

)C M

(P,2),

where there are introduced the integers

d = r + s− t − u, b = (t + u+ n+m− 2)/2, h = [1− (−1)d]

and the variablesχ = c/(kBT) andζ = mc2/(kBT). The factors in (91) are defined as

ν(ζ ) = π (n+m−3)/2

20((n+m− 1)/2)0(n+m− 3)ζ 2[K n+m

2(ζ )]2 , (91)

and

F(t, u, w) = 1

4[1+ (−1)w][1 + (−1)t+u−w]

× B

(n+m− 3

2,w + 1

2

)B

(n+m− 2+ w

2,

t + u− w + 1

2

),

where the beta function is given by gamma functions,

B(x, y) = 0(x)0(y)

0(x + y),

and (rt ) denotes Newton’s binomial coefficient.

6.5.1. Scalar Type Brackets.The first type of brackets necessary for calculation of transportcoefficients (86) (see the series approximation (87)) is the so-called scalar brackets, decomposedinto a sum of two integrals

aa1a = a′a1a + a′′a1a, (92)

wherea1a = 2, 3, . . . ,

a′a1a = n−2∫

f[0] (p) f[0](p[1])(ζp)a1

[(ζp[1]

)a − (ζp′[1]

)a]W dςp dςp[1] dςp′ dςp′[1]

,

a′′a1a = n−2∫

f[0] (p) f[0](p[1])(ζp)a1[(ζp)a − (ζp′ )

a]W dςp dςp[1] dςp′ dςp′[1],


with the integration variables

ζp = pαUα/kBT, ζp[1] = pα[1]Uα/kBT,

ζp′ = p′αUα/kBT, ζp′[1]= p′α[1]Uα/kBT.

The explicit calculations of integrals (93) from (92) (see the method and basic intermediary formulasin [3]; in this paper we deal with la-values) give

a′a1a =a1∑

t=0

a∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,0,0)a1a (t, u, v, w),

a′′a1a =a1∑

t=0

a∑u=0

d/2∑v=0

u∑w=0

J(0,0,0)a1a (t, u, v, w).

In the lowest approximation, fora1 = a = 2, we obtain

a22 = 2[J(0,0,0)

22(2, 2, 0, 0)+ J(0,0,0)

22(2, 2, 0, 2)

].

So we have reduced the scalar brackets to twofold integrals which are expressed in terms of sphericalcoordinates with respect to a locally anisotropic rest frame.

6.5.2. Vector Type Brackets.The vector type brackets from (86) and (87) are also split into twotypes of integrals,

bb1b = b′b1b+ b′′

b1b,

whereb1, b = 1, 2, . . . ,

b′b1b= n−2

∫f[0] (p) f[0]

(p[1])1σε(ζp)b1 pσ × [pε[1]

(ζp[1]

)b − p′ε[1]

(ζp′ [1]

)b]W dςp dςp[1] dςp′ dςp′[1]

,

(93)

b′′b1b= n−2

∫f[0] (p) f[0]

(p[1])1σε(ζp)b1 pσ × [ pε(ζp)b − p′ε(ζp′ )

b]W dςp dςp[1] dςp′ dςp′[1].

A tedious calculus similar to that presented in [3] implies further decompositions of coefficients andtheir representation as

b′b1b=

3∑(i )=1

b′(i )b1b

and b′′b1b=

3∑(i )=1

b′′(i )b1b

,

with corresponding sums

b′(1)b1b= 1

4

(kBT

c

)2 b1∑t=0

b∑u=0

d/2∑v=0

u∑w=0

(−1)u J(2,0,0)a1a (t, u, v, w),

b′(2)b1b=(

kBT

c

)2 b1∑t=0

b∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,2,1)a1a (t, u, v, w),

b′(2)b1b= −

(kBT

c

)2 b1∑t=0

b∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,2,1)a1+1a+1(t, u, v, w),

with corresponding sum expressions forb′′(i )b1b

by omitting the factor (−1)u.


In the lowest approximation (forb1 = b = 1) we have

b11 = −2

(kBT

c

)2

× [J(0,2,1)11

/(1, 1, 0, 0)+ J(0,0,0)

22(2, 2, 0, 0)+ J(0,0,0)

22(2, 2, 0, 2)

].

Here it should be noted that in generalT = T(u) is a function on la-spacetime coordinates.

6.5.3. Tensor Type Brackets.In a fashion similar to that for scalar and vector type symmetricbrackets from (86) and (87) one holds the decomposition

cc1c = c′c1c + c′′c1c,

wherec1, c = 1, 2, . . . ,

c′c1c = n−2∫

f[0] (p) f[0](p[1])1νµσε (ζp)c1 pν pµ

× [pσ[1] pε[1]

(ζp[1]

)c − p′σ[1] p′ε[1]

(ζp′ [1]

)c]W dςp dςp[1] dςp′ dςp′ [1]

,(94)

c′′c1c = n−2∫

f[0] (p) f[0](p[1])1νµσε (ζp)c1 pν pµ

× [pσ pε(ζp[1]

)c − p′σ p′ε(ζp′ [1]

)c]W dςp dςp[1] dςp′ dςp′[1]

.

For tensor-like brackets we have to consider sums on nine terms

c′c1c =9∑

(s)=1

c′(s)c1c and c′′c1c =9∑

(s)=1

c′′(s)c1c.

These terms are [3] (with the exception that we have dependencies on the number of anisotropicvariables andT = T(u) is a function on la-spacetime coordinates)

c′(1)c1c =1

16

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(4,0,0)c1c (t, u, v, w),

c′(2)c1c =(

kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,4,2)c1c (t, u, v, w),

c′(3)c1c =n+m− 2

n+m− 1

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,0,0)c1+2c+2(t, u, v, w),

c′(4)c1c =1

2

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(2,2,1)c1c (t, u, v, w),

c′(5)c1c = −2

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,2,1)c1+1c+1(t, u, v, w),

c′(6)c1c = −1

2

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(2,0,0)c1+1c+1(t, u, v, w),


c′(7)c1c = −ζ 4

n+m− 1

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,0,0)c1c (t, u, v, w),

c′(8)c1c =ζ 4

n+m− 1

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,0,0)c1+2c (t, u, v, w),

c′(9)c1c =ζ 4

n+m− 1

(kBT

c

)4 c1∑t=0

c∑u=0

d/2∑v=0

u∑w=0

(−1)u J(0,0,0)c1c+2 (t, u, v, w).

The twice primed valuesc′′(s)c1c are given by similar sums by omitting the factor (−1)u and by changinginto c′′(4)c1c andc′′(5)c1c the overall signs.

In the lower approximation, forc1 = c = 0, one holds

c00 = 2

(kBT

c

)4{J(0,4,2)

00(0, 0, 0, 0)+ 2J(0,2,1)

11(1, 1, 0, 0)

+ n+m− 2

n+m− 1

[J(0,0,0)

22(2, 2, 0, 0)+ J(0,0,0)

22(2, 2, 0, 2)

]}.

6.6. Locally Anisotropic Transport Coefficients for a Class of Cross Sections

For astrophysical applications we can substitute

(dσ

dÄ

)C M

= ξPr , (95)

with some scalar factorξ (with or without dimension) andr being a positive or negative number into(91). In this subsection we putζ = mc2/(kBT) for high values ofT . The chosen type of differentialcross section (95) is used, for instance, for calculations of neutrino–neutrino scattering (whenr = 2and ξ is connected with the weak coupling constant). In the first approximation the symmetricbrackets (87) are

a22 = ξ(

2kBT

c

)rπ (n+m−1)/20[n+m+ 1+ r/2]0[(n+m+ r )/2]

0[n+m− 2]0[(n+m)/2]0[(n+m+ 1)/2],

b11 = −(

kBT

c

)2 n+m+ 2+ r

2a22,

c00 =(

kBT

c

)2[ (n+m+ r )(n+m+ 4+ r )

2+ n+m− 2

n+m− 1

]a22.

Putting these values into (90) we obtain the locally anisotropic variant of transport coefficients inthe first approximation, whenη(v) ' 0 but with nonzero

λ= 2kBc

ξ

(c

2kBT

)r (n+m)2(n+m− 1)

n+m+ 2+ r× 0[n+m− 2]0[(n+m)/2]0[(n+m+ 1)/2]

0[n+m+ 1+ r/2]0[(n+m+ r )/2], (96)


and

η = 1

ξ

(c

2kBT

)r−1 1

π (n+m−1)/2[(n+m)2− n−m− 2]

× [(n+m− 1)2− 1](n+m+ 1)2

[(n+m+ r )(n+m+ 4+ r )+ 2((n+m− 2)/(n+m− 1))]

× 0[n+m− 2]0[(n+m)/2]0[(n+m+ 1)/2]

0[n+m+ 1+ r/2]0[(n+m+ r )/2]. (97)

These formulas present a locally anisotropic generalization of the Boisseau and van Leeuwen [3]results.

7. TRANSPORT THEORY IN CURVED SPACES WITH ROTATIONELLIPSOIDAL HORIZONS

After having established in a general way the scheme for calculation of transport coefficients onla-spacetimes, we now specify an example for a four-dimensional static metric (being a solution ofEinstein equations in both general relativity and la-gravity) with the event horizon described by ahypersurface of rotation ellipsoid [34] (see the Appendix). We note that in this casen = 3,m = 1,andn(a) = 4.

The main formulas for kinetic and thermodynamic variables from the previous Sections 3–6 werebased on spherical symmetry of then(a) − 1 dimensional volume (61) and spherical integral (62).For the rotation ellipsoid we have to modify the volume’s formula by introducing the ellipsoidaldependence

Vn(a)−1 =(n(a) − 1

)π (n(a)−1)/2

0[(

n(a) − 1)/2] ρ(θ ) =︸︷︷︸

n(a)=4

3π3/2

0[3/2]ρ(θ ), (98)

where

ρ(θ ) = ρ(0)

1− ε cosθ

is the parametric formula of an ellipse with constant parameterρ(0), angle variableθ , and the ec-centricityε = 1/σ <1 is defined by the axes of rotation ellipsoid (see the formula (117) from theAppendix).

In the case of spherical symmetry 4∫ π/2

0 dθ = 2π , the ellipse deformation gives the result

4

π/2∫0

dθ

1− ε cosθ= 8√

1− ε2arctan

√1− ε1+ ε .

So, performing integrations on solid angles in spaces with rotational ellipsoid symmetry, we can usethe same formulas as for spherical symmetry but multiplied by

q(ε) = 4

π√

1− ε2arctan

√1− ε1+ ε .


For instance, the integral (62) transforms as

∫dÄ→

∫dÄ(ε) = q(ε)

(n(a) − 1

)π (n(a)−1)/2

0[(

n(a) − 1)/2] =︸︷︷︸

n(a)=4

q(ε)3π3/2

0[3/2].

The formulas for the particle density (59) and energy density (67) of point particles must bemultiplied byq(ε),

n→ n(ε) = q(ε)n and ε→ ε(ε) = q(ε)ε,

but the energy per particle will remain constant. We also have to modify the formula for pressure(70), having been proportional to the particle density, but consider unchanged the averaged enthalpy(71). The entropy per particle (74) and chemical potential (75) depends explicitly on theq(ε)-factorbecause their formulas were derived by using the particle densityn(ε). Here we note that all provedformulas depend onm (=1, in this section) anisotropic parameters and on the volume elementdetermined by a d-metric. In the first approximation of transport coefficients we could choose alocally isotropic background but introduce the factorq(ε) and take into account the dependence onanisotropic dimension.

Puttingε-corrections into (89) we obtain the first-order approximations to the transport coefficientsin a la-spacetime with the symmetry of rotation ellipsoid

λ = −ckBn2(ε)

3

β21

a22,

η = ckBTn2(ε)

10

γ 20

c00, (99)

η(v) = ckBTn2(ε)

α22

a22.

The valuesα2, β1, andγ0 are 3-fold integrals expressed in terms of enthalpyh and temperatureTand have the limits

α2 =5γ

ckBTn(ε)− 3γ

cn(ε)→︸︷︷︸

T→∞0,

β1 =kBT

cn(ε)

3γ

γ − 1→︸︷︷︸

T→∞12

kBT

cn(ε), (100)

γ0 =kBT

c2n(ε)10h →︸︷︷︸

T→∞60

1

n(ε)

(kBT

c

)2

.

In the next step we compute theq(ε)-deformations of brackets from the denominators of (99).Because the squares ofα, β, andγ coefficients from (88) are proportional to deformations ofn−2

(this conclusion follows from the formulas (96) in theT→∞ limit, see (100).) and the scalar(93), vector (93), and tensor (94) type brackets do not change underq(ε)-deformations (see (96)) weconclude that the transport coefficients (89) do not contain the factorq(ε) but depend only on thenumberm of anisotropic dimensions. This conclusion is true only in the first approximation and forlocally isotropic backgrounds. As a consequence, the final formulas for the transport coefficient, see


(96) and (97), in a (3+ 1) locally anisotropic spacetime with rotation ellipsoid symmetry are

η(v) ' 0

λ = 2kBc

ξ

(c

2kBT

)r 48

6+ r

0[2]20[5/2]

0[5+ r/2]0[(4+ r )/2],

η = 1

ξ

(c

2kBT

)r−1 20

π3/2[(4+ r )(8+ r )+ 4/3]

0[2]20[5/2]

0[5+ r/2]0[(4+ r )/2].

This is consistent with the fact that we chose as an example a static metric with a local anisotropy thatdoes not cause drastic changes in the structure of the transport coefficients. Nevertheless, there areq(ε)-deformations of such values as the density of particles, kinetic potential, and entropy which reflectmodifications of kinetic and thermodynamic processes even by static spacetime local anisotropies.

8. CONCLUDING REMARKS

The formulation of Einstein’s theory of relativity with respect to anholonomic frames raises anumber of questions concerning locally anisotropic field interactions and kinetic and thermodynamiceffects.

We argue that spacetime local anisotropy (la) can be modelled by applying Cartan’s moving framemethod [6] with associated nonlinear connection (N-connection) structures. A remarkable fact is thatthis approach allows a unified treatment of various types of theories with generic local anisotropy likegeneralized Finsler-like gravities, of standard Kaluza–Klein models with nontrivial compactifications(modelled by N-connection structures), of standard general relativity with anholonomic frames,and even of low-dimensional models with distinguished anisotropic parameters. We have showna relationship between a subclass of Finsler-like metrics with (pseudo) Riemannian ones beingsolutions of Einstein equations.

This paper has provided a generalization of relativistic kinetics and nonequilibrium thermody-namics in order to include possible spacetime locally anisotropies. It should not be considered asa work on the definitions of some sophisticated theories on Finser-like spaces but as an attemptto develope an approach the kinetics and thermodynamics in (pseudo) Riemannian spacetimes ofarbitrary dimension provided with anholonomic frame structures. We introduced into considerationthe generalized Finsler and Kaluza–Klein spaces because of the facts that have been recently con-structed (see the Appendix and [33] for solutions of the Einstein equations with generic anisotropy,and of Finsler and another type, like ellipsoidal static black holes, black tora, anisotropic solitonicbackgrounds, and so on) and because in the low energy limits of string theories, various classes ofgeneralized Finsler–Kaluza–Klein metrics could be obtained alternatively to the well known (pseudo)Riemannian ones [31]. By applying the moving frame method it is possible to elaborate a generalschema for defining physical values, basic equations, and approximated calculations of kinetic andthermodynamic values with respect to la-frames in all mentioned types.

The crucial ingredient in the definition of a collisionless relativistic locally anisotropic kineticequation was the extension of the moving frame method to the space of supporting elements (uα, pβ)provided with an induced higher order anisotropic structure. The former Cartan–Vlasov approach [5,36], proposing a variant of statistical kinetic theory on curved phase spaces provided with Finsler-like and Cartan N-connection structures, was self-consistently modified for both types of locallyisotropic (the Einstein theory) and anisotropic (generalized Finsler–Kaluza–Klein) spacetimes withN-connection structures induced by a local anholonomic frame or via reductions from higher di-mensions in (super) string or (super) gravity theories. The physics of pair collisions in la-spacetimeswas examined by introducing on the space of supporting elements of (correspondingly adapted


to the N-connection structure) the integral of collisions differential cross-sections and velocity oftransitions.

Despite all the complexities of a definition of equilibrium states with generic anisotropy, a rigorousdefinition of local equilibrium particle distribution functions is possible by fixing some anholonomicframes of reference adapted to the N-connection structure. The basic kinetic and thermodynamicvalues such as particle density, average energy and pressure, enthalpy, specific heats, and entropy arederived via integrations on volume elements determined by metric components given with respect tola-bases. In the low- and high-energy limits the formulas reflect explicit dependencies on the numberof anisotropic dimensions as well as on anisotropic deformations of spacetime metric and linearconnection.

One can linearize the transport equations and prove the linear laws for locally anisotropic nonequi-librium thermodynamics. A general scheme for calculation of transport coefficients (the heat conduc-tivity, the shear viscosity, and the volume viscosity) in la-spacetimes is also established. An explicitcomputation of such values was performed for a metric with a rotation ellipsoidal event horizon (anexample of spacetime with static local anisotropy), recently found as a new solution of the Einsteinequations.

Our overall conclusion is that in order to obtain a self-consistent formulation of the locallyanisotropic kinetic and thermodynamic theory in curved spacetimes and calculation of basic phys-ical values we must consider moving frames with correspondingly adapted nonlinear connectionstructures.

APPENDIX

A Locally Anisotropic Solution of Einstein’s Equations

Before presenting an explicit construction [34] of a four-dimensional solution with local anisotropyof the Einstein equations (8) we briefly review the properties of four-dimensional metrics whichtransform into (2+ 2) or (3+ 1) anisotropic d-metrics (we note that this is not a (space+ time) buta (isotropic+ anisotropic) decomposition of coordinates) by transitions to correspondingly definedanholonomic bases of tetrads (vierbeins)).

Let us consider a four-dimensional (in brief, 4D) spacetimeV (2+2) (with two isotropic plus twoanisotropic local coordinates) provided with a metric of signature (+,+,−,+) parametrized by asymmetric matrix of type

g1+ q2

1h3+ n21h4 q1q2h3+ n1n2h4 q1h3 n1h4

q1q2h3+ n1n2h4 g2+ q22h3+ n2

2h4 q2h3 n2h4

q1h3 q2h3 h3 0

n1h4 n2h4 0 h4

(101)

with components being some functions

gi = gi (xj ), qi = qi (x

j , z), ni = ni (xj , z), ha = ha(x j , z)

of necessary smooth class. With respect to a la-basis (10) this ansatz results in diagonal 2× 2 h- andv-metrics for a d-metric (10) (for simplicity, we shall consider only diagnoal 2D nondegeneratedmetrics because for such dimensions every symmetric matrix can be diagonalized).


An equivalent diagonal d-metric of type (12) is obtained for the associated N-connection withcoefficients being functions on three coordinates (xi , z),

N31 = q1(xi , z), N3

2 = q2(xi , z),(102)

N41 = n1(xi , z), N4

2 = n2(xi , z).

For simplicity, we briefly denote the partial derivatives asa= ∂a/∂x1,a′ = ∂a/∂x2,a∗ = ∂a/∂za′ =∂2a/∂x1∂x2,a∗∗ = ∂2a/∂z∂z.

The nontrivial components of the Ricci d-tensor (15) (for the ansatz (101)), whenR11 = R2

2 andS3

3 = S44, are computed as

R11 =

1

2g1g2

[−(g′′1 + g2)+ 1

2g2

(g2

2 + g′1g′2)+ 1

2g1

(g′1

2+ g1g2)], (103)

S33 =

1

h3h4

[−h∗∗4 +

1

2h4(h∗4)2+ 1

2h3h∗3h∗4

], (104)

P3i = qi

2

[(h∗3h3

)2

− h∗∗3h3+ h∗4

2h24

− h∗3h∗42h3h4

]+ 1

2h4

[h4

2h4h∗4 − h∗4 +

h3

2h3h∗4

], (105)

P4i = − h4

2h3n∗∗i . (106)

The curvature scalar←R (16) is defined by two nontrivial componentsR= 2R1

1 andS= 2S33.

The system of Einstein equations (17) transforms into

R11 = −κϒ3

3 = −κϒ44 , (107)

S33 = −κϒ1

1 = −κϒ22 , (108)

P3i = κϒ3i , (109)

P4i = κϒ4i , (110)

where the values ofR11, S3

3, Pai , are taken respectively from (103), (104), (105), (106).We note that we can define the N-coefficients (102),qi (xk, z), andni (xk, z), by solving Eqs. (109)

and (110) if the functionhi (xk, z) are known as solutions of Eqs. (108).An elongated rotation ellipsoid hypersurface is given by the formula [21]

x2+ y2

σ 2− 1+ z2

σ 2= ρ2, (111)

whereσ ≥ 1 andρ is similar to the radial coordinate in the spherical symmetric case.The space 3D coordinate system is defined by

x = ρ sinhu sinv cosϕ, y = ρ sinhu sinv sinϕ, z= ρ coshu cosv,

whereσ = coshu (0≤ u <∞, 0≤ v ≤ π, 0≤ ϕ < 2π ). The hypersurface metric is

guu = gvv = ρ2(sinh2 u+ sin2 v),(112)

gϕϕ = ρ2 sinh2 u sin2 v.


Let us introduce a d-metric

δs2 = g1(u, v)du2+ dv2+ h3(u, v, ϕ)(δt)2+ h4(u, v, ϕ)(δϕ)2, (113)

whereδt andδϕ are N-elongated differentials.As a particular solution of (107) for the h-metric, considering−κϒ3

3 = −κϒ44 , we choose the

coefficient

g1(u, v) = cos2 v. (114)

Theh3(u, v, ϕ) = h3(u, v, ρ(u, v, ϕ)) is considered as

h3(u, v, ρ) = 1

sinh2 u+ sin2 v

[1− rg/4ρ]2

[1+ rg/4ρ]6. (115)

In order to define theh4 coefficient solving the Einstein equations, for simplicity with a diagonalenergy–momentum d-tensor for vanishing pressure we must solve Eq. (108) which transforms intoa linear equation ifϒ1 = 0. In our cases(u, v, ϕ) = β−1(u, v, ϕ), whereβ = (∂h4/∂ϕ)/h4, mustbe a solution of

∂s

∂ϕ+ ∂ ln

√|h3|∂ϕ

s= 1

2.

After two integrations (see [19]) the general solution forh4(u, v, ϕ), is

h4(u, v, ϕ) = a4(u, v) exp

− ϕ∫0

F(u, v, z) dz

, (116)

where

F(u, v, z) = 1

/{√|h3(u, v, z)|

[s1(0)(u, v)+ 1

2

z∫z0(u,v)

√|h3(u, v, z)| dz

]},

ands1(0)(u, v) andz0(u, v) are some functions of necessary smooth class. We note that if we puth4 = a4(u, v), Eqs. (108) are satisfied for everyh3 = h3(u, v, ϕ). In this case the coefficientsqi = 0and our two-dimensional anisotropy degenerates the one-dimensional (functionsni do not vanish).

Every d-metric (113) with coefficients of type (114), (115), and (116) solves the Einstein equations(107)–(110) with the diagonal momentum d-tensor

ϒαβ = diag[0, 0,−ε = −m0, 0],

whenrg = 2κm0; we set the light constantc = 1. If we choose

a4(u, v) = sinh2 u sin2 v

sinh2 u+ sin2 v

our solution is conformally equivalent (if not considering the time–time component) to the hyper-surface metric (112). The condition of vanishing of the coefficient (115) parametrizes the rotation


ellipsoid for the horizon

x2+ y2

σ 2− 1+ z2

σ 2=(

rg

4

)2

, (117)

where the radial coordinate is redefined via relationr = ρ(1+ rg/4p)2. After multiplication on theconformal factor

(sinh2 u+ sin2 v)

[1+ rg

4ρ

]4

,

approximatingg1(u, v) = cos2 v ≈ 1, in the limit of locally isotropic spherical symmetry,

x2+ y2+ z2 = r 2g,

the d-metric (113) reduces to

ds2 =[1+ rg

4ρ

]4

(dx2+ dy2+ dz2)−[1− rg

4ρ

]2[1+ rg

4ρ

]2 dt2

which is just the Schwazschild solution with the redefined radial coordinate when the space compo-nent becomes conformally Euclidean.

So, the d-metric (113), the coefficients of N-connection being solutions of (109) and (110), de-scribes a static 4D solution of the Einstein equations when instead of a spherical symmetric horizonone considers a locally anisotropic deformation to the hypersurface of rotation elongated ellipsoid.

ACKNOWLEDGMENTS

The author thanks Professors G. Neugebauer and H. Dehnen for hospitality and support of his participation at JourneesRelativistes 99 at Weimar and his visit at Konstantz University, Germany, where the bulk of the results were communicated.

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