Anisotropic solutions of the Einstein-Boltzmann equations: I. General formalism

ANNALS OF PHYSICS 150, 455-486 (1983)

Anisotropic Solutions of the Einstein-Boltzmann Equations:

I. General Formalism

G. F. R. ELLIS, D. R. MATRAVERS, AND R. TRECIOKAS

Department of Applied Mathematics, Universit.v of Cape Town, Rondebosch 7700. South Africa

Received August 11, 1980; revised April 22, 1983

Given a choice of a timelike vector field. a particle distribution function in a general curved space-time can be analysed into spherical harmonics: the Liouville and Boltzmann equations can then be written as a set of equations relating these spherical harmonic components. We obtain these equations and the resulting equations for the spherical harmonic moments of the distribution function. An orthonormal tetrad formalism is used as an aid in our calculations; the set of moment equations used can be completed by giving Einstein’s field equations as equations for the rotation coefficients of this tetrad. We discuss time and space reversal symmetry properties of the Boltzmann equation, but leave applications of the set of equations obtained to further papers.

1. INTRODUCTION

A convenient way of describing the particles in a given region of space-time is to specify a one-particle distribution function f(x’, p’) which determines the number of particles at event xi with 4-momentum $. Between collisions, the particles are assumed to move freely, so their paths xi(r) are geodesics, i.e., pi = dx’/dr satisfies,

I &$ = 0 e z + rijktip" = 0.

The rate of change off along the particle world-lines is

The Boltzmann equation states that this is equal to C(x’,#,f), the rate of change off due to collisions. Equation (1.1) can be used to write this in the form

Lf =pi 5 + rjikPiPk af= c w

(1.2)

which determines Lf for all allowed 4-momenta pi (see, e.g., Refs. [ 14, 21, 37, 381

455 0003.4916/83 $7.50

Copyright ‘E 1983 by Academic Press. Inc. All rights of reproduction in any form reserved.

456 ELLIS, MATRAVERS, AND TRECIOKAS

for reviews of relativistic kinetic theory). The propagation of the particles may take place in a given space-time (e.g., propagation of photons or cosmic rays which takes place in a galaxy or universe model without appreciably affecting the space-time) or the system may be a self-gravitating system; in this case, the Ricci tensor R, of the space-time is given by Einstein’s field equations

R, - $Rg, + Ag, = Tij o R, = Tij - ;Tgij + Ag,, (1.3)

where the stress-tensor T, is wholly or partially that due to the particles described by f, T- T’,, R = Rii and /i is the cosmological constant.

Properties of solutions of the Einstein-Boltzmann equations in which the distribution function f is isotropic about some 4-velocity vector field ui are well understood (see [ 6-81); in these solutions, the 4-velocity vector field ui has no shear, is either non-expanding or non-rotating, and has an acceleration potential. Exact solutions of the coupled equations with isotropic distribution functions are known in the case of homogeneous, isotropic universe models (see, e.g., 17)) and in the case of static, spherically symmetric models of star clusters (see, e.g., [9]).

Some exact solutions are known where the distribution function is anisotropic, (see, e.g., [9-l 1, 40, 41 I). It is the aim of this paper to obtain exact equations describing detailed properties of the Boltzmann equation in a general curved space-time in which the distribution function is anisotropic; these equations are used in further papers to obtain some exact and approximate properties and solutions of the Einstein- Boltzmann equations.

The essential feature of our analysis is that we make a spherical harmonic analysis of the distribution function. This is done in two stages: first, we use the 3 + 1 splitting of space-time determined by an (arbitrary) timelike vector field to obtain a 3 + 1 splitting of the moments of the distribution function. We then carry out a spherical harmonic analysis of the distribution function in the rest-frame determined at each point by the timelike vector field, obtaining equations for a set of moments of the spherical harmonic components off, which are the quantities of physical interest and which we represent as trace-free symmetric spacelike tensors. To carry out the calculations, we introduce an orthonormal tetrad system and use a notation for the Ricci rotation coefficents similar to that used in classification of homogeneous cosmological models [ 121; the field equations (1.3) can readily be written out in terms of these rotation coefficients, so we obtain a complete set of equations for either the Boltzmann equation in a given space-time, or the coupled Einstein- Boltzmann equations in a self-gravitating system.

The idea of using such a harmonic analysis was proposed by Tauber and Weinberg [6], and aspects of our formalism have been developed in various papers (see, e.g., [ 1. 13-16, 351). As far as we are aware, the full set of equations given here has not been obtained before (although Thorne [ 351 goes a long way towards it in the case of photon gas). Our equations are related to those obtained by Grad’s method of moments (see, e.g., [ 16-191) but are, we believe, more convenient in many applications.

EINSTEIN-BOLTZMANN SOLUTION, 1 457

For generality our description is not restricted to particles of one rest-mass; the distribution function f will in general represent a mixture of particles of arbitrary rest- mass. The restriction to particles of particular (zero or non-zero) rest-mass can easily be made by specializing J:

A covariant description of the Einstein-Boltzmann equations is given in Section 2. The 3 + 1 decomposition is carried out in Section 3, and the spherical harmonic analysis in Section 4. Certain symmetry properties of the equations are discussed in Section 5; the collision operator is briefly considered in Section 6, and a summary of the procedures is given in Section 7. As we introduce an interrelated series of moments representing the distribution function f and collision term C, we give a summary of these relations in Tables I and II.

1.1 Notation

We assume that the space-time is pseudo-Riemannian with a symmetric torsion- free connection and that a global time orientation can be introduced, and we use the signature (-, +, +, +). Partial derivatives are denoted by a comma and covariant derivatives by a semicolon. A general tetrad basis (E,} is used in most of the paper: in some sections it is specialized to an orthonormal basis. The formulae required are collected below. For further details on these techniques readers may consult Refs. [ 12, 22-24, 361. Indices a, 6, c ,... are used for the tetrad basis and indices Lj, k,... for the coordinate basis. The tetrad components of Z (coordinate components Z’) are Z” = ETZ’, where ET E', = &, Eq EL = 8:. Tetrad indices are raised and lowered using the tetrad components of the metric tensor g,, = EkEj,g, and gab, where gabgb, = 8:. We define the derivative operator

the Ricci rotation coefficients

r,,, = E'aEcj. kEk , b’ (1.5)

and the “object of anholonomity” [5],

y;, = r;, - 1-;, . (1.6)

These quantities are related by the equations

ab gac = &bc + &ba? (1.7a)

rabc = tcab gca + a‘. gob - aa gbc) + thbc + Scab - Ybco)- (1.7b)

The curvature tensor is given by

Rbacd=acradb - adracb + pcepdb - raderecb f jV’ebfdc (1.8)

458 ELLIS,MATRAVERS, AND TRECIOKAS

and the Einstein field equations for the Ricci curvature are

The curvature tensor satisfies the identity:

Ra [bed] = o - &y’bc, + f,bcyad,e = ’ (1.10)

(square brackets denote skew symmetrization and round brackets denote symmetrization [5]).

We adopt the notation of Thorne [39] for strings of subscripts and superscripts:

MAqa4 ala*. .a43 (1.11)

&fAq E ipfalaz”.G,, (1.12)

where a capital subscript or superscript denotes a string of lowercase indices; their number is denoted by a subscript on the capital subscript or superscript. Further, we abbreviate the tensor product of vectors by a similar device, using a tilde to indicate the product:

(1.13)

(1.14)

2. ~-DIMEN~IONAL FORMULATION

2.1. The Distribution Function

If a particle at a point x has 4-momentum pa, then its rest-mass m is related to p”

by

p* = +gab(X’)papb = -m* > 0, (2.1)

where pa = Eai dx’ldv and v = t/m. If ua is a future directed timelike vector field, then future directed 4-momenta satisfy

-p%, > 0. (2.2)

One can represent the particle 4-momenta at x by points in the tangent space T, ; then the 4-momenta of particles at x with rest-mass m lie on the future mass shell P,(x): the set of points in TX which obey (2.1), (2.2). Since ap*/ap” = -2gabpb, the vector pa is normal to P,(x) at each point pa in TX. The volume element at x spanned by displacements d,xi, d,x’, d3xk, d,x’ is r = qijk,dlxi d,x’ d,xk d,x’, where rijkl is the totally skew pseudo-tensor defined by qijk, = v,~~~,,, q0123 = 6,

EINSTEIN-BOLTZMANN SOLUTION, I 459

where g = det(gJ < 0. The volume element in 7’, spanned by displacements d, pa, d2pb, d, P’, da pd is

713 u,bcd d, Pa d, pb d, PC d, pd. (2.3)

The volume element 71 induces a natural volume element on P,(x); denote this volume element by (mn,) (the factor m allows one to include particles of zero-rest mass, see [ 3 ]), then

(2.4a)

as dm measures the increase in magnitude of p” normal to P,(x). Conversely, one can express rt, in the form [3]

7cm = *H(-u,p”) 6(p2 + m’)n, (2.4b)

where the step-function H(z) = 1 if z > 0, H(z) = 0 if z < 0. The distribution function f(x’,p”) represents the number of particles at x with 4-momentum pa. More precisely, if an observer with 4velocity zP(u’u, = -1) observes dN particles with 4-momenta in the range 7c about pa in a volume element dV in his rest-frame, then

dN =f(x’,p”) . (-u,p’) . rz dV. (2.5)

By Synge’s projection theorem [20], the number (-uapa) dV is independent of the observer’s 4-velocity u*, sof(xi,pa) is a scalar. We wish to specify that only future- directed timelike and null 4-momenta be considered. We do so by allowing f(x’,p”) to include all momenta, but restricting (2.5) to vectors pa obeying (2.1), (2.2) and using in all integrals the volume element,

?r+ = H(--u,p=‘) H(-p2)n (2.3a)

which selects from all 4-momenta, future-directed non-spacelike vectors. Here and later we use the volume element rc+ rather than x,,,, and allow the distribution function f to refer to particles of any rest-mass. Iffrefers to particles of rest-mass m only, then (2.5) takes the form dN =f,(x’,p”) . (-ucpc) . 71, dV for some function f,(x’,pa). In this way any of the following formulae can be specialised to particles of a particular (zero or non-zero) rest-mass.

By definition off, f(x’,p”) > 0 for all xi and allowed p”. It is a distribution when considered as a function of m, and it is at least C2 as a function of pa, for any fixed value of m. The assumptions involved in the existence off have been examined in detail by Ehlers (31; in particular the smoothness off depends on the existence of sufficient particles, and obviously this will be unsatisfactory for some regime as E -+ co on each mass shell because there are no particles ultimately, as one takes this limit.


2.2. Moments of the Distribution Function

One can, at any point x, define moments of the distribution function j-(x’. p”) and collision term C(x’,p”,f) by multiplying by t powers of m and q powers of pa and integrating over TX. We denote the moments by

(2.6)

lyAqZ . J m& cn+ 1 t > 0, q > 0. (2.7) T.7

Clearly these quantities are totally symmetric. From (2.1), it follows that the moments M satisfy the recursion relation

and the N’s satisfy the corresponding relations. In the case of a distribution of particles of rest-mass m, this relation can be written

M&-d d = -mz M”s-2. t t

(2.8a)

The most important moments off in many applications are

na s p”fn+ = $fa, Na E mp”fn, = “(‘“,

(2.9)

The vector na is the number flux vector, which is used to defined the average (number flux) velocity vector va and proper density n (measured by an observer with 4- velocity ~1’) by

n” = nva, u,ua = -1.

Similarly, N* is the mass flux vector, which defines the average (mass flux) velocity vector Vu and proper rest-mass density p by

N” = pV”, v-v, = -1

(see [21 I). The energy-momentum tensor of the particles with distribution function f is Tab and it represents the contribution of the particles to Eq. (1.3). This tensor also determines a unique average velocity, namely, its timelike eigenvector (see, e.g., (31).

Because of the Boltzmann equation, the moments M and N are related by the divergence relations

$lAqb;b = NAq (2.10) I

EINSTEIN-BOLTZMANN SOLUTION.1 461

which states that the divergences of the moments M are given by the moments N. To obtain this relation, substitute (1.2) in (2.7); the equality follows easily on noting that

L(m2) = gn*:&fp*pc = 0

follows from (1.2), (2.1) and that

a,72 = [a,(logfi)]rc = l-*,*71

is a consequence of definition (2.3). The divergence relations (2.10) for the quantities (2.9) are

na;Cl =NsE, 0

N”;, = N SE i?, I

(2.11)

Here, G is the particle number generation density, which vanishes if particle numbers are conserved in collisions, and E” is the particle rest-mass generation density which vanishes if rest-mass is conserved. The vector Ja represents the generation-density of energy and momentum of the particles due to collisions (31. In many applications, t: = E’= J” = 0; this holds, for example, if there are only binary collisiqns in which particle rest-mass is conserved [3, 211. In a self-consistent solution of the field equations where Tab in (1.3) is given entirely by the particle stress tensor (2.9), J” must vanish because of the contracted Bianchi identities Rub;* = +R:“. If collisions occur with some other system (e.g., in the case of photons moving through a medium), J” will represent the transfer of energy and momentum between the two systems. When l? and J” vanish, Vab is the first non-zero (I = 0) moment of the collision term, and so is the important term in determining transport properties of a simple gas (see, e.g., [4, 18, 191).

3. 3 + 1 DECOMPOSITION

3.1. The Velocity Vector Field

In any physical situation, there are various preferred 4-velocity vector fields in a given region of space-time. For example, a well-defined average vector exists in most reasonable cosmological models (see [25]); and we have seen in Section 2.2 that there are at least three different ways in which a distribution function may define a preferred 4-velocity at a point x. In this section we simply assume that there exists some chosen 4-velocity vector field u’I, and use it to give a 3 + 1 splitting of space- time, without enquiring further about the optimal choice of such a 4-velocity.

We take zP to be a unit vector field:

ff%, = -1. (3.1)


It defines a projection tensor {habj into the rest-space of an observer moving with 4- velocity ua (see, e.g., [21, 251):

h,, = g,, + uaub, hobhbc = h,‘, hobub = 0.

We denote the projection of any tensor by a 1 sign, so

(3.2)

lT&. .d = h,chbe’-hdfTc,. .* * UaiTab.. .d = 0;

and use a superscript dot to denote the covariant derivative in the direction u’, so

V’a.. .b)’ = T,. . .b;cUC.

The covariant derivative of u, can be written in the form j21, 251

U a;b = w,b + Dab + @h,, - i, ub, (3.3)

where the vorticity tensor oab is spacelike, wab = ~~~~~ and o ub = 0; the shear tensor cob is spacelike, oab = uCobJ, uaa = 0 and cabub = 0; the zipansion t? = ua;, ; and the acceleration vector zi, = u,:~ ub so (3.1) implies Pu, = 0. The vorticity vector d z f ?pbCdUb Wed ; it is completely equivalent to the vorticity tensor wab, as w ab =v abcdWcUds

Given a e 4-velocity vector field ua, there clearly exists a preferred family of orthonormal tetrads associated with u’, namely, those in which the timelike vector E, of the tetrad is chosen to be parallel to u’. For such a tetrad basis, one has

h,, = 0, h,, = J,,

(Greek indices y, v, u,... range over the values 1,2,3 only). From (3.3), (3.4) and definition (1.5), one finds that

RJr7 = L r,,, = o,,, + wua + ;es,., .

Further, one can define a 3-vector P’ by the relation

(3.5a)

rvl),, = Eb,,,fla; (3Sb)

then LP’ represents the rate of rotation of the basis vectors {E,} as seen by an observer with 4-velocity ua (eLIDO is the permutation symbol spvo = sluvo,, s,23 = +l). The rotation coefficients I-,,, can be expressed in terms of quantities’ a, and n = n,,,, as follows [ 121: split yrrKO.sUKU into parts which orthe indices ,LL, v. The symmetric part is defined to be n”;

are skew and symmetric the skew part is written

as elrvOaO. Then

(3.5c)

Note that ti,, aUv and urcr, are the projections of 4-tensors, and so transform as 3- tensors when transformations of the vectors {E,} are made; but Q,, n,,,, and a, do


not transform as tensors because the Tabc do not transform as a tensor on the index c and the yabc do not transform as a tensor on either index b or c under a general space-time dependent transformation.

Because the rotation coefficients rabc are skew in a and c when the basis is orthonormal (see (1.7)), one can obtain them all from the quantitites zi,, a,,,., 0, a,, n I and crK. Thus, when dealing with equations involving generalised Christoffel s&nbols (1.5) for an orthonormal basis, one can write out the equations explicitly in terms of these quantities.

3.2. 4-Momenta and the Boltzmann Equation

Given a 4-velocity vector field u’, any particle 4-momentum pa can be written in terms of its components parallel and perpendicular to P:

pa = Eu” + hea, e’e, = 1, eau, = 0, A > 0. (3.6)

E is the energy of the particle for an observer moving with 4-velocity ua and ea is the direction of the particle motion in the observer’s rest-space. From (3.6) (2.1) (2.2) and (3.1)

E = -p%, > 0, A2=E2-m2+E2>m2, (3.7)

where A is the magnitude of the particles’ (relativistic) 3-momentum. From (3.6). rr, and TC, (see (2.3a), (2.4)) can be written

(3.8)

where dQ is the solid angle spanned by two independent de”. The distribution function can now be written as a function of xi, m, E and ea; i.e.,

as f(x’, m, E, e“). Substitution into the Boltzmann equation shows that the partial derivatives in af/am cancel as expected because L(m2) = 0 and the equation becomes

Lf = Eu” a, f t lea a, f + u,Pbc(Eub t Aeb)(Euc + M) -f&

E2 1 (3.9a)

- u,rabc-ed + - hdarabc ,I A

(Eub + leb)(Euc t Ae’) g = C;

when the orthonormal basis with ua = Sg is used, one obtains

Eu”a,f tle"a,f + -~~~Ei.e’li,.-l’o,,.e”e” g i i

I 2

-I c (-ti” + e’e%,) - E(a”, + wv, + &‘,,fi~) e”

+ EeKe”u,,ev - ,lEv,,nTUeaeP - Aa” + Ae”a,eu ! af -= C. a8

595/150/2-I2


By specializing the rotation coefficients, one can obtain the equations in a space with particular symmetries, e.g., Lindquist’s form for the case of spherical symmetry ] I] and Dautcourt and Wallis’ form for the case of a Bianchi I universe [ 141.

3.3. Moment Equations

One can decompose any tensor field into parts perpendicular to and parallel to the vector field U” [25]. In particular

n” = rid + ja, N” =/‘iua + I”, j%, = 0 = Pu,. (3. IO)

Then H is the number density, p the rest-mass density, y the number flux vector and P the rest-mass flux vector observed by zP. Similarly, the energy-momentum tensor Tab can be split into the energy density ,u, energy flux vector q, and pressure tensor II,, observed by ua, where

T,, = iu”, ud + 2q,d ub) + r;r,d 3 qa Ua = 0,

n,b = =Cab, ’ II,, Ub = 0.

(3.1 la)

One can further split the pressure tensor ZZ,, into its trace 3p (p is the isotropic pressure) and its trace-free part rcGb:

IIab =ph,, + irob, zabub = 0, ra, = 0. (3.1 lb)

The vector J” (see (2.11)) can be written

J” = Joua + J,’ where Jlau, = 0. (3.12)

Then J, is the rate of change of energy, and J,, the rate of change of 3-momentum of the particles. In this section we consider the projection of the general moments M and N off and C perpendicular to and parallel to ua to obtain a representation of these tensors in terms of totally symmetric tensors orthogonal to u” as in (3.10), (3.1 la), (3.12). (We consider the further splitting of these tensors into their trace-free parts, as in (3.11b), in Section 4.1).

To obtain this representation for M, substitute (3.6) into the definition (2.6)‘; one finds

M,,= + q! Pu pyo p!(q -p)! I !?(“P UP+1 ... u%)l I

where

’ Round brackets (A,) are used to denote symmetrization over the index string.

(3.13)

(3.14)


which implies these moments of f are totally symmetric, and “spacelike” (i.e.. orthogonal to u”); that is,

p.4p = P,A$ and PA p~la ZP = 0. 1.q t-4 1.9

Similarly the moments N can be expressed in terms of the totally symmetric spacelike moments

of the collision term. Substituting into (2.8), one obtains the recursion relations

f+~~= PAP 2 _ pAp-> t>o+q>p>.2, t.q

(3.16) t*q tts.q-2

for the moments P; the moments Q obey a similar set of recursion relations. One can obtain a useful interpretation of the moments P by noting that 1e, = ED,, where ~1, is the particle’s 3-velocity relative to ua (cf. Section 3.2) and that by (2.5) one can write f7r+ = dN/E dV. Thus if the particle * has a rest-mass m,, energy E, and 3- velocity u* relative to u’, (3.14) can be written

where the sum is over all the particles in the 3-volume dV. As particular examples, the quantities in (3.10) are %= 04” = (I/dV) x (No. of particles in dV), ja = “7” =

(l/W C ve”, p= o( = (l/dV) 2 m,,

stress-te;fsor T,, ’

and I” = ,<’ = (l/dV) C m, v.+‘. If the

in (3.11) is tha*t of the particles represented b*y the distribution functionf (see (2.9)), then p = ,< = (l/dV) C E, , q’ = ,<” = (l/dV) C E, L:*~, and

ZKab = o<nb = (I/‘dV) C E,t~,“b,~; taking the* trace of the’ last express& shows that

the isotropic pressure*p is given by p = f P,” = (l/dV) C fE, t’*‘. An alternative 0.2 *

interpretation is obtained on noting that if a surface element dS in U”‘S rest-frame is represented by the normal vector dS, (dS,u” = 0, dS” dS, = (dS)‘), then the flux of the quantity m’Eq~‘C,.,,,~, across the surface is dS” PO,,,- ,, i.e..

1-q

where the sum is taken over the particles crossing the surface dS, in the + direction in time dt. Thus dS” n, is the flux of particles across dS,, dS”N, is the flux of rest- mass across dS,, and dS”T,, is the flux of (relativistic) 3-momentum Ae, across dS”.

The moments P and Q are related by the divergence relations (2.10). Substituting (3.13) into (2.10) and separating into parts perpendicular to and parallel to u,. one

466 ELLIS,MATRAVERS, AND TRECIOKAS

can equate the (symmetrized) coefficients of u,, . . . u,~ ~,. As a result one obtains the equation

Ic4,r;dhcd + IVA,)’ + (9 -P - 1) PAp*c(ubc + @hbC) t.q 1.P t.q

+ 0 PA, + P PC, _ uaP) = P I Q+,t t>O,q>p++>l, t.q t.q t-q-- I

for the moments P. One could obtain this directly from Eq. (3.9), by multiplying by ,,+E4-“Ape’ Aplc+, and integrating over TX. Examples of these equations are the usual low-order moment equations of radiation transfer theory (t = 0, q = 2; p = 0 and 1).

One can obtain another set of moment equations by integrating Eqs. (3.9), after multiplying by ml’ ‘Eq-“Apt ’ dm dE, over h4 and E but not over angles, obtaining equations for the moments

m ‘+’ dmj” Eq-PAP+ tf&

m

(cf. f 1, Eq. (3.19); 16, Eq. (25)j). Clearly we would obtain (3.17) from these equations by multiplying by dJ2 and integrating over the unit 2sphere S.

3.4. The Field Equations

One can write out the field equations (1.9) in terms of the rotation coefficients (3.5) of a preferred orthonormal tetrad; in doing so, it is convenient to use the decomposition (3.11) for the stress tensor TbC. Similarly, the Jacobi identity and contracted Bianchi identities can be written out in this way. The resulting equations are given in [36, Eqs. 77-861. It is easy to substitute into these equations for any particular geometry; see, e.g., [I, 241, for discussions of the spherically symmetric case, and [ 12, 33, 361 for a discussion of some homogeneous cosmological models.

4. SPHERICAL HARMONIC ANALYSIS

4.1. The Distribution Function and Its Moments

We have seen how choice of a timelike vector field u4 lets one write the distribution function f as a function f (xi, m, E, e*). Now for any fixed values of xi, m, and E, f is a function only of the direction e* in u”s rest-frame, i.e., of angles 8, 4 defined by some orthonormal triad of vectors orthogonal to u’. Thus one can carry out a spherical harmonic analysis off, writing it as


where the Y,“’ are the usual surface spherical harmonics (see, e.g., [26]), and rewrite the Boltzmann equation as a set of equations for the coefficients A,m (see 129)). An alternative way to carry out this harmonic analysis is to write f in the form

j-z G QsA 170

e I= F + F,e” + Fobeaeb + F,,,e”ebec + **a, (4.2)

where the spherical harmonic coefficients FA,(xi, m, E) are symmetric, trace-free tensors orthogonal to u’:

F,,=Fg.+ FA,-zcdhcd=O> F,imiaua=O. (4.3)

Similarly, one can write out the collision term C(x’, m, E, e”,f) in terms of its harmonic components, i.e., C = C;“),JI, either in the form

c= ? tr I?) mLir

B,“(f, xi, m, E) Y,m(8, (p)

or in the form

C = ? bA,zA’, 1-Z

(4.4)

where the coefficients b,,(f, xi, m, E) satisfy conditions (4.3). In working out consequences of definitions (4.2)-(4.4), the following result is

useful :

LEMMA 1. Let ea be a unit vector orthogonal to u’. Then

li CArda= if r is odd i s

4n =--

r+ 1 ,j(Qloz haI” . . . h%lQ’r) if r is even, r > 0, X4.5)

where S is the unit 2-sphere.

There is I-I mapping between all symmetric trace-free tensors of rank 1 and the spherical harmonics of order 1 [28]; we shall use the representations (4.2), (4.4). As in any other harmonic decomposition, one can invert (4.2) to express the coefficients FA, as integrals off: The easiest way to do this is to form the trace-free, symmetric part of the tensor c”, by following Pirani’s methods [28], obtaining the quantities’.’

[l/21 ()A’= \’ (1!)2 (21- 2k)!

-k=o (-l) k!(l- k)! (I- 2k)!(21)! h(alG

... hQ-lQeazkt, ,.. e”/’

’ These quantities obey the symmetry relations (4.3~ that js. 0 ‘1 = ()‘?I’. 011 zh,, = 0. 0 l/-lhuh = 0, ’ (k-1 denotes the integer part of k.


and the orthogonality relations

il OAqpmdfi=(j ’ 2’(1!)2

s m 2z4; 1 * (2z)! /zQbl ... ha/*,

(obtained by calculating J‘[,y &I). From these properties, it is clear that we can write (4.2) in terms of the d’l instead of the e’s; and then the inversion follows:

m

f = x F,,O*b FAa, = I=0

(4.2a)

Similarly one can invert the expression (4.4) to find the b’s as integrals of the collision term C.

Given the spherical harmonic decomposition (4.2) of the distribution function, the spherical harmonic moments lI offare defined by

Using expression (4.2a) for the Fs, one can reexpress this relation in the form

=A, = (21 -t 1) (2z)! . (,t~q-~~~o*/)fj+,

t3q.p 4?t 2’02 7, J (4.7)

We shall assume that the function f is such that the harmonic components F,,!, are bounded, well-behaved functions for which lim,_, F,,r,= 0 in such a way that the moments Il exist for all powers q >p > 0, t > 0. One can similarly define spherical harmonic moments of the collision term C, using (4.4) by

t>o, I>O.q>p>O, (4.8)

which can be reexpressed as an integral over the collision term C exactly analogous to (4.7). (Replace II by K and f by C in (4.7).) The moments ll and K are trace-free, symmetric, and orthogonal to ua because the coefftcients F and b obey (4.3).

Substituting (4.2) into Eq. (3.14) and using Lemma 1, one obtains a relation giving the splitting of the symmetric spacelike tensor P into its trace-free symmetric spacelike parts ll:

P

PAP= f,q

l;o 47cp! ((1 +p)/2)!

(z +p + l)! ((p _ z),2)! 21 n’“, hnl+iaf+2 .f. hap-l@, t47.p

(I+p) even

t>o, q>p>l>O. (4.9)

EINSTEIN-BOLTZMANN SOLUTION. 1 469

Similar relations relate the moments Q and K of C. The recursion relations (3.16) for the P’s imply the relations

nA/ = nA/ - LlA’ , t3q.p 1,9,P-2

9>P>LP-221>0, ttz.q-2,p-2

(for the II’s; these follow from (4.7) and (2.1). The K’s satisfy similar relations. The quantities defined in (3.10), (3.11) are examples of the II’s;

n=4n n 0.1.0 ’

p=47l II ) l.130

p=% (4.10)

3 n 3 7[ab=?!k naba

0.2.2 15 0.2.2

The recursion relation shows that .D - 3p = 47~ I7 . Certain inequalities follow 2.0.0

from f > 0 for all x, m, E, since this implies jI, f dl2 > 0, and so (using (4.2) and Lemma 1)

F(x’. m, E) > 0 for all X, m, E.

If some matter is present the equality cannot hold for all m, E at any given x. The inequalities

n >,,a+, >o, 1,q.p 1 1

,,F,, > n > 0. , * It 1.q.p

follow directly, and the generalized Taub inequality

n+ 17 2 n + 1.Q.P t.q+1.p+2 1.4-t 1.p

follows from Schwarz’ inequality. These inequalities show that for any distribution of particles,

3P+P>Pu3P>O, jf>>>/.-3p>o.

One obtains the stronger inequalities 1271

3P, + P, > iu, > s P, + v75tnG,)2 > 3Prn > 0

for the contribution of particles of mass m to the pressure, density and energy density of the gas, and so in particular these inequalities are fulfilled by a gas consisting of particles of a single mass. The quantities defined in (2.11). (3.12) are expressible in terms of the first few moments K; in fact,

&=4x K 0.0.0 ’

l7=4n K 1.0.0 ’

Jo = 4~ K , and 0.1.0

Ja, = To:;. . 3


4.2. The Boltzmann Equation and the Moment Equations

The Boltzmann equation (3.9) will give a set of equations for the spherical harmonic components FA, off when (4.2) is substituted into it. Before giving these equations, we state two further lemmas.

LEMMA 2. Let aA, be a symmetric, trace-free tensor orthogonal to ua; and let b, be a vector orthogonal to u,. Define the operator T by

if r > 0, and T(a,,d,b,,) = 0 if r < 0. Then T(a,,, b,,) is a symmetric, trace-free tensor

orthogonal to ua.

LEMMA 3. Let aA, be a symmetric trace-free tensor orthogonal to u’, and let b,, be a symmetric tensor orthogonal to ua. Define the operator S by

1 W,A,Ld = acA,b - - acA,hab, hcdbcd 3 + 2r

2r - ---ac(A,~,b,d,a,hob)hcd + (2r ;;,1), 3) h’nghbRh

3 + 2r

and

S(a&cd = 0 if r-CO.

Then S(a(,rb,b,) is a symmetric, trace-free 3-tensor orthogonal to u’.

To obtain the equations for the moments FA,, use these lemmas to convert the Boltzmann equation, after substitution of (4.2) into (3.9). into relations for coefficients of C4/ which are totally symmetric, orthogonal to U’ and trace-free. Since these expressions are then linearly independent, one can equate the coefficients of I?’ to zero for each I > 0, obtaining the spherical harmonic analysis of the Boltzmann equation. Having carried out the calculation in a general basis, one can finally write the result in a covariant form. An alternative way of obtaining the final equation is to use the fact that for any symmetric tensor K,dr the Liouville operator L acts in the following way:

L(KAr?) = KAr;,pArpa. (4.11)

Using this relation one can directly evaluate the effect of L on f. expressed in the form (4.2), without using the form (3.9) of the Boltzmann equation; and then again separate out the resultant equations into equations for symmetric, trace-free coefficients f?‘.


The final form of the equation is

(I + lx1 + 2, - (21+ 3)(21+ 5)

{L20’fa(-F,,,)/aE + (1 + 3) Eaef -Fef.A,l

(I+ 1) + (21+ 3) u+-F.4,c;dhcd - EAtid a( +Fd,.,,)/aE

+$(l+2)dd+Fd,~,/ + E,(-&I)’ - $1’ a(-&,>/aE

& i2~‘T(a(~F,,,,~,)/aEUd,,,) + 3ET(-Fd,4,-ladq,)l

- ~E(-Fdt.+wda,,) + T -IX a(+F,.,,m,W i,,

+ u- 1): +f,..4-,~a)) I + Tin +f’,,+ .,:a,,\ - SV2 a(LFuI JlaE ~a, ,a!)

- (I- 2)E -F(,4,JJa,-,a,) I = + b.‘r, for I>O. (4.12)

(For the present, ignore the f subscripts on the F’s and b’s, which will be explained later; Eq. (4.12) holds for coefficients F,.r,, b,d, in which all the f, - subscripts have simply been omitted.) It is clear from the form of this equation that it links at most’ moments I+ 2, I + 1, I, I - 1 and 1 - 2. These equations for the harmonic coefficients off are completely equivalent to the Boltzmann equation.

After multiplying by m ” ‘Eq-pApt ‘dm dE d.0 one can integrate these equations over T,. The limits F,4,-+ 0 as E -+ co, A= 0 when E = m, and the definitions (4.7), (4.8) of the harmonic moments enable one to obtain the equation

(I+ l)(l+ 2) (21+3)(21+5) (‘-‘) i

df 1 ,,,;F;y/ +(P-l)u”‘lT;f$\

Cl+ 1) + (21+3) 1

L(+ =.4,c) ;d hcd + I i

&++:‘, (9 -P + l) lid i&4, f.q+ I.P+z t.q+ I.ptl

+ (p - 1) lid

,.,:f,“-‘: I + I -‘-%, I

+$ (4-P)

i

-II,, tb+3) 44

t.q+ l,PiZ t.q+ I,:, I + 1 (WdL4,. , %,A

t.qi 1.P

2(67-P> -nd~.4,&q~d + t2p + 3, -ndcA, I %,,d f.q+l.P+2 t.q+ 1-P

’ Note that if the shear oah = 0. then these equations link only I + I to I - I.


+ T (4 -P + 1) +fl,,,_, k,,) + (P + I+ 1) +nc..,., &,I t.qt l.p+ 1 t,qtl,P-l

+ T I

+44-,x7,) I I

+ s (9-P) -*u,-2 (JtJ&,c7,) t.q+ I.P+ 1 t,qt1,pt2

+@+I+ 1) -44J+a,) =+h, I I>at>o,s>p>o (4.13) t,4+ 1.P IJ7.P

for the moments II. (Again, ignore the subscripts + for the present.) These are the detailed equations, derived by integrating the Boltzmann equation, determining the relations between the moments II and the source terms K.

In many applications it is the first three of equations (4.12), (4.13) which are most important. Particular examples of (4.13) are the matter conservation equations (I = 0, t=l,q=P=O)

6 + /itI + I,.:,hcd + lidId = ,6, (4.14)

which can be obtained directly by substituting (3.10) in (2.11); the energy conservation equation (I = 0, t = 0, q =p = 1)

fi + @ +p)e + qc;dhcd + 2tidq, + Cf%, = Jo

and the momentum conservation equation (/ = 1, t = 0, q = p = 1)

(4.15)

@’ +P> i, + hd(P,, + %c:ehce + (C&j) + %dZjd

+ ((iad + Uod + @had) qd = J,o, (4.16)

which can be obtained by substituting (3.1 l), (3.12) in (2.11); Eqs. (4.15), (4.16) correspond to the moment equations of radiation transfer theory. For the case of a gas in which particle collisions preserve number and rest mass, the first non-zero collision term is Vab (see (2.11)). Thus it is useful to have the next equations, those for Sabc (see (2.9)), explicitly; writing

(4.17)

and

where

S,=4n n , 093.0

s& II,, 0.3-L

s2=q n, 0.3.2

s,ab = ; &b, 0,3.2

%a=% =,, 0.3.3

g&,r=s nabc, 0.3.3

EINSTEIN-BOLTZMANN SOLUTION. I 473

and

5=4x K , 0.2,0

v,, =q K,, V=%K 2

0.2.1 3 ’ 0.2.2

and

v -8°K Zab - 15 ab,

0.2.2

then the moment equations are

v,,t + e(so + ZS,) + 2a’f& + S,c;dhcd + 3fidS,, = v, (4.19a)

(for I = 0, f = 0, q = 2, p = 0),

(s2)’ + SOS2 + $ddS,, + $(S1,;dhcd + ti”f+) + $??&= vz (4.19b)

(for I= 0. T = 0, q = 2, p = 2),

(‘0 + 2s2) &I + h,b((s2);b + 3,bc;dhcd + (s,,)‘) + 2S2,,tib

+ S3b(2u,b + $9hab) + S,,(a/ +W,b + ;&,")+ d!?,efrr= v,, (4.19C)

(for I= 1, t=O, q=2,p= l), and

h n ehbf(g3Pfc.dhcd)+ lid,!? >dob + hnchhd(&d)’

+ :@-&I + 2(S2d(c,wb,d + s2d(ou,,,d - hab+udr&fr)

+ 2(s,,,z& - ho, +idS,,) + 2(&,&, - th,,tidS,,,)

+ 2hnrhbd&r:d, - h,, $hc”sly;f) + 2s2 ~7,~ = p,,,, (4.19d)

(for I= 2, t = 0, p = q = 2). It is these equations which play a central role in determining the transport coefftcients of a gas [ 17-19, 38).

We have given Eqs. (3.17), (4.12), (4.13) in a covariant form; they can be written out in terms of any general basis. and in particular may be given in terms of an orthonormal basis. If this basis is used, the tetrad components of these equations can easily be obtained from the relations

LtF,I ,c:d hcd) = @dF.4,c) /cd + 24,,,.4, ,cu,,bdn”d

+ 21Fb bL4,. , ‘a,) - 2(1 + 1) ah&,.,, (4.20a)

#‘A,)* = MoF,,) - fFd(,4,_,Ed,,,cRC~ (4.20b)

L(F(A,-,:o,1)= L(~co,Fa:...a,l)- (I- 1x4, pa,&,CEdo,tc

- (I - 1) adFd(A,-Zh-IO1b + cl- I)F(.,~ ,a,,, (4.20~)


which hold for any symmetric tensor FA, which is orthogonal to zP. One can obtain many further relations from (4.12) and (4.13). For example one can contract (4.12) with FA, or ba,; one can contract (4.13) with I&, ; and one can contract (4.19d)

I’JI’.P with nab, gab, cPmb, nab, or aaab.

5. TIME AND SPACE INVERSIONS

5.1. Time Inversions

Consider a reversal of the time direction accompanied by a reversal of the vector field u’. Then (3.6) shows that

(5-l)

The arguments of the distribution function S(x, m, E, e) and collision term C(x, m, E, e) will change under this reversal of the time directions:

f(x, m, E, e) +f(x, m, -E, e), C(x, m, E, e) -+ C(x, m, --E, e). (5.2)

We shall call the transformation (5.1), (5.2) time inversion, and denote it by F. Our aim now is to separate the Boltzmann equation into parts which are invariant under K, and parts which change sign under K. To do this, define new functions

+ f(x, m, E, e) = i [ f(x, m, E, e) +fk m, -E, e)],

_ f(x, m, E, e) = f [ f(x, m, E, e) --j-(x, m 4, e) II (5.3a)

and

+ C(x, m, E, e) = i [ C(x, m, E, e) + C(x, m, -E. e)],

-C-(x, m, E, e) = i [ C(x, m, E, e) - C(c, m, -E, e)] (5.3b)

fromf, C. Then we have split f, C into eigenfunctions of the operator K:

f=,f+-L a(+./-) = + L g(-.f-) = -- f;

c=+c+-c, F(+C) = + c, B(-C) = -- c.

Now one can harmonically analyse + f, -f, +C, -C, just as we did f and C; so we define the coeffkients +I;a,, -F,.,, by Eq. (4.2) applied to +h -fi respectively, and coeffkients b + A!, -b,4, by Eq. (4.4) applied to +C, -C. Clearly those coeffkients have the same inversion properties as the functions from which they are derived:

U+F,,) = +Fe,,, a(-F.,,) = -- Fq,,

EINSTEIN-BOLTZMANN SOLUTION. 1 415

and similar relations for b,,. Further, one can define moments +IIA’, -DA’, +K and t3q.p 1,q.p

-K. These moments will not, in general, have the same a-invariance properties as the coefficients F * A,, *b,, from which they are obtained. However one has the useful result that if f is 8-invariant5, all the terms with a subscript (-) vanish:

a(f) =f e -f= 0 o { pF,+ = 0, I > O} * { -ZZA’ = 0) f4.P

(5.4)

A similar result will hold for C. We shall call the Boltzmann equation (3.9) for fk w E, e), the 0-Boltzmann equation. One can obtain an equation for f(x, M, -E, e) from this equation, by applying the transformation (5.1). Adding these two equations, one obtains

Eu” a,( -f) + de” a,(+f) + A’(-$9 - eL(el’uer) y

+ EAe”zi, 3(+f) z 7 + 1 $- (-2 + e’e?i,)

- A(.5”,,n’,e”eN - a” + eva,a”) I

K+f > T

+ E{eKe’uu,eV - (av, + co”, + E~~,Q~) eU I_ = Kf) + c, (5.5) de”

which we call the + Boltzmann equation. Subtracting, one obtains the same equation except that all the f indices are interchanged; we call this the - Boltzmann equation. Thus the 0-Boltzmann equation (3.9) is equivalent to the pair of f Boltzmann equations, which are a pair of equations linking the eigenfunctions of + f, _ f of the operator 6. The usefulness of this splitting results from (5.4): an exact solution of the Boltzmann or Liouville equation may be g-invariant (for example, any function g(c,,,p”) is. by Eq. (4.1 l), a solution of the Liouville equation if the vectors <,,U (A = I,..., r) are Killing vectors: g is g-invariant if these Killing vectors are orthogonal to u’, i.e., if rnnu“ = 0 [ 291). Then the _ f terms vanish, and the O- Boltzmann equation (3.9) splits into the pair of * Boltzmann equations which now have a simpler form: for if f = 0, the + Boltzmann equation is

Ae" a,,( + f) + EAe”zi, aE + $ (4” + e”e’$,) d(+f>

- A(E”,,nT,eKeU -a” + eOa,e”) I

3,f 1 r = + C, (5.6a)

’ This is not the same as saying that f is time-symmetric, i.e. thatf(x”, t, m, E, e) =f(xr, +, M. E, e) for some coordinates (x”, 1).


and the - Boltzmann equation is

at+.f- 1 Eu” a,(+f) + L’(-$9 - e’e”u,,) a~

Without further information, it is not clear if one would expect -C to vanish or not; however for many reasonable choices of the collision term (cf. Section 6) _ f = 0 * _ C = 0. The splitting of the 0- Boltzmann equation (3.9) into its parts ((5.5) and the corresponding equation with the +,- indices interchanged) results in a corresponding splitting of the harmonic form of the Boltzmann equation. We shall call the equation (4.12) for FA,(x, m, E) without the f the O-harmonic equation. Making substitution (5.1) in this equation, one obtains an equation for F.,,,(x, m, -E). Adding these equations, one obtains precisely (4.12), which we call the + harmonic equation. Subtracting, one obtains the same equation except that all the + indices are interchanged; we shall call this the -harmonic equation. Thus the O-harmonic equation (for each value of t) is equivalent to a pair of i harmonic equations. Similarly Eq. (4.13) without the f indices may be called the O-moment equation; Eq. (4.13) itself, the + moment equation, is obtained by integrating the + Boltzmann equation (4.12), and Eq. (4.13) with the f subscripts interchanged, the -moment equation, is obtained by integrating the -Boltzmann equation. If a solution of the Boltzmann or Liouville equations is a-invariant, the terms with the minus vanish (see (5.4)), and for each value of I, the O-equations (4.12), (4.13) split into a simpler pair of f equations (just as (5.5) split into the pair (5.6a), (5.6b)): the + equation links terms in I+ 1 and I - 1, while the - equation links terms in I + 2, I and I- 2.

5.2. Space Inversions

One can similarly consider the effect of a space inversion on f and C; i.e., one can consider the effect of the transformation 9’ where

{u”+u’,e”+-e”}d {E+E,A-+-A}. (5.7)

Now one can define new functions *S, *C by the relations

+f = t(f (x, m, E, e> +f (x, m, E, -e>>,

-f= f(f(x,m,E,e)--f(x,m,E,--e)),

+C = i(C(x, m, E, e) + C(x, m, E, -e>),

-C = f (C(x, m, E, e) - C(x, m, E. -e)).

(5.8)


Then we splitf, C into eigenfunctions of the operator ~7’:

f=+f+-.L W+f) = + f, 9(-f)=--f;

c=+c+-c, .i"'( +C) = + c, .4*'( -C) = - _ c.

Proceeding as before, one can define harmonic components J,.,,, *b,, and moments *II, *K of ,fand *C. One now finds from (5.8) and (4.2) that

+F.+ = 0 if 1 is odd,

&,=O if 1 is even, (5.9)

and similar relations hold for the b’s Considering the effect of .Y‘ on Eq. (3.9), one sees that this equation is invariant under F; so the functions ,fdefined by (5.8) each separately satisfy precisely Eq. (3.9). Considering Eqs. (4.12) and (4.13), one obtains effectively the same splitting of these equations as before. In fact from (5.9) we see that the /-harmonic Boltzmann equation (using notation (5.8)) has precisely the plus form (4.12) when 1 is odd, except that one must replace the source term +b,, by d,,; and will have the minus form except that the source term will be +b,4,, when 1 is even (i.e., the + and - subscripts will be interchanged on the left-hand side of (4.12)). The same results will clearly hold for (4.13). The splitting of the equations is useful, for example, in the study of stability of star clusters 1301. As in the previous case, invariance offwill result in vanishing of the terms with - coefficients: by (5.9),

and then Eqs. (4.12), (4.13) will each split into two simpler equations.

5.3. Symmetry Restrictions

We have seen that when f is invariant under either time inversion T or space inversion 9, Eqs. (4.12) and (4.13) split into two simpler equations, one (say, the (1, +) equation) linking If 1, I- 1, and the other (say, the (1, -) equation) linking It 2, 1 and I- 2. This has an interesting consequence: consider any particular value 1’ of 1. The (l’, -) equation will be a relation linking 1’ + 2, 1’ and 1’ - 2. The (1’ - 1, +) equation will link 1’ and 1’ - 2, and the (1’ + 1, +) equation will link 1’ to 1’ - 2, so if one combines these two equations one obtains a second relation between 1’ + 2, 1’ and 1’ - 2. These relations will be different (the first involves oab, 0 and w ob, while the second involves ti,) so the moments (1’ + 2, l’, 1’ - 2) will, for each value of 1’, be related by two separate equations, and so will be more restricted than one might have thought at first. The simplest examples of this when a(f) =f are the moment equations one obtains for low values of 1’. The (0, -) and (1, +) equations are then


-+ p&f-&

[

i?F +3EdfFQf +E(Fj+&b,

I

[

f3F +~(h,bFbc;dhcd)-+ A$$ + 3 $ zi’F,,

I

+ Ah,‘(F),, - ;1E $ ti, = + b,,

which both link F and F,,; and the (1, -) and (2, +) equations are

-& p&f-&L%

[

C?F + 4EaefF,,,

1 + hd,(F,)’ E

-;t’$12;~od,-jEFdod,-EFdwda=-b,,

+ 12h,‘hbf(Fe,c:dhcd) - -f [

aFdab 2 EAtid BE + 4% ZjdFdab 1 + hachbdA Fc‘vd, -; h,,Fefhef 1

2

- AE[dF,,/aE Lib, -+Eh,,tids ++[F&-h,,ticFc]=+boh. 1 which both link F, and Fabc. It is clear that if there are kinematic restrictions on the congruence ua, then these equations will change form again. For example, if the shear (3 ab vanishes, the highest moments drop out of the first equation of each pair. In particular problems with high symmetry, it may be useful to examine more detailed invariance properties of the equations, such as invariance under reflection about one preferred direction (cf. [9, 241) or under discrete isotropies which may occur (cf.

]311).

6. THE COLLISION TERM

The discussion so far holds for any form of the collision term C. This will usually be a function of f, so it can be regarded as an operator on the functions F,4,, producing the term b,, which acts as a source term in Eq. (4.12) for the harmonic coefftcients of $; or correspondingly as producing the moment K which acts as a source term in Eq. (4.13) for the harmonic moments lI of f: In any particular problem, one has to specify further properties of these operators. The usual way of choosing them are as follows.


6.1. Liouville Form

When there is either an absence of collision, or detailed balancing [3], the collision term vanishes:

c = 0 0 b,, = 0, I> 0. (6-l)

In this case, we refer to the Boltzmann equation as the Liouuille equation. Clearly in this case

KA’ = 0. t.p.9

6.2. BGK Form

This form represents relaxation of the distribution function f(xj m, E, e) towards a distribution functionf(x, m, E, e), with a relaxation time $x’, E) ([ 17, 321). Then one has

c = 4’(f-f) u b,, = r-‘(F*, - F,,), (6.2)

where the F,,(x, m, E) are the harmonic components of the functionc This is usually taken to be an equilibrium distribution function; in which case, on choosing u“ as the mean velocity defined by the distribution function 7 is isotropic about uQ and so FA, = 0 for I> 1, which implies b,, = t-IF,, for 12 1, and the moments

are proportional.

KA’ f.P.Q

and Il*/ 1.p.q

6.3. Transport Form

The transport form of the collision function represents the effect of the passage of the particles considered through some absorbing and emitting medium. Choosing ua to be the average velocity of the medium, which has proper density n(x’), the collision term takes the form [ 1 ]

C = n(xi)(Q(xi,pa)- ~(.d,p~)f+ .r, W(P,P’)./-@‘N’) ‘f,). (6.3) x

The first term represents spontaneous particle emission by the medium, the second represents loss of particles with 4-momentum pa due to absorption and scattering by the medium, and the third represents the gain of particles with 4-momentum pa by scattering from particles with other 4-momenta. The first two terms have the same form as the BGK terms (cf. (6.2)). The last term can be analysed harmonically: expressing p, p’ in terms of m, E, e and m’, E’, e’, one can write W in the form

w= F ,.;;‘=a

W(m, E, m’, El),,,,, e’A9’si,. (6.4)

595/150/Z 13


The coefficients W,,,B,, completely describe the harmonic properties of the scattering term. On using (4.2), one finds from (5.3), (5.4) that the contribution to b,) due to W, which is given by

n(x’> 1 W(p,p’) f(x’,p’“) 7c’+ = c b,,2’, T.Y I=0

is 2”(f’!)*

. (21’)! J . m, dm’ . w

i ,a,B,,FRf.IZ’ dE’ (6.5)

showing how the scattering of the initial distribution function f (represented by the Fs) leads to sources for the various harmonics (via the Boltzmann equation in the form (4.12)). When the scattering is elastic, m - m’ and E + E’ and so

W A+,,, = 2&m* - m’*) W -E’) +vq,H,, (6Sa)

where the coefficients W,,,,,, (xi, m, E) give the transformation by the collision from the V to the 1 harmonic: in this case

b,4,= n(x’) ,z, & . “;;;;ii’ ’ w,~,~, F”‘,. (6.5b)

Integrating suitably will give an expression for the moments K in terms of the coefficients W,,,,,, and FA, (in general there will be no simple relation between the moments II and K). If in addition the scattering is isotropic in the rest-frame, then

(see 111) b = n(x’) a&m, E) /iF(x’, m, E), 4, = 0 if I> 1, (6.6)

where u&m, E) is the scattering cross section. In this case, the expression (6.3) for the collision term reduces to the BGK form discussed above (cf. [2]).

6.4. Boftzmann Form

One can write down general expressions for C representing II’ particles of various species colliding and producing n particles as a result,” and including, for example, electromagnetic effects and quantum-mechanical statistical effects (see e.g. [ 3, 4. 37. 381). For simplicity, we consider here only binary collisions of uncharged classical particles.6 In that case (see e.g. [6], [ 211) the collision term can be written’

C(f) =JJj W(p,p’;p”,p”‘)(f’f” -ff’) 71’ + 7c” + n”’ + ) (6.7)

’ This description is valid when the mean free path is large compared with the effective range of individual interparticle forces (cf. (3.4 1).

b We omit indices of the arguments of W: e.g., p-p”. f.(p) - L”,ph. etc. and use the notation f’ -f(p’). etc.

’ For a discussion of further effects (cf. [SS]).


where W(p,p’;p”,p”‘) is the probability of a collision (p,p’) + (p”.p”‘). This will obey the symmetries

W(p,p’;p”,p”‘) = W(p’,p;p”,p”‘) = W(p,p’;p”‘,p”) = W(p’,p;p”‘,p”), (6.8a)

W(p,p’;p”,p”‘) = W(p”.p”‘;p,p’), (6.8b)

the first set being essentially relabellings, and the second expressing the microscopic reversibility of collisions. In addition, W will be Lorentz invariant: that is,

W(p,p’;p”,p”‘) = W(L(p), L(p’); L(p”), L(p”‘)) (6.8~)

for any Lorentz transformation L (because this probability will be unchanged if we apply such a Lorentz transformation to the entire collision). Finally 4-momentum will be conserved in collisions, so W will take the form

W(p, p’; p”, p”‘) = S(p + p’ - p” - p”‘)R(P, p’; p”. p”‘), (6.9)

where R has the symmetries (6.8). In addition, other microscopic conservation laws may hold in particular circumstances, for example in the case of elastic collisions. R will incorporate a factor 6(m - m”) 6(m’ - m”‘).

As in the previous cases, one can harmonically analyse the distribution functions and the collision term W in (6.7) or the term R in (6.9). Thus one can analyse the effect of the collision term on the spherical harmonic structure of the distribution function.

The question arises as to whether this is a sensible thing to do, for the spherical harmonic decomposition is based on the choice of the velocity vector ua used in the 3 + 1 decomposition of Section 3, while the individual particles taking part in any collision will be unaware of this mean velocity. In fact, the preferred velocity defined in their collision is the centre-of-mass velocity Ua = Pa/(phPh)“*, where paFpa+pl”=p’lQ+pplllU is the total particle momentum; and U” will almost never be the same as uQ (we are indebted to Professor Israel for raising this point). However, there will in each case be a Lorentz transformation L relating U to u, that is

L(U) = u t> L”, u6 = ugl (6. IO)

and by (6.8~) the collision probability W is invariant under this transformation. Thus the harmonic structure of W in the centre-of-mass frame will determine a harmonic structure of W in the u-frame in a unique way. More specifically,

W(p,p’;p”.p”‘) = W(L(p), L(p’); L(p”). L(p”‘)) = L W(p,p’;p”p”‘) (6.11)

where the left-hand expression can be regarded as the collision probability evaluated in’the centre-of-mass rest frame, while the right-hand side is the collision probability evaluated in the u-frame but derived by Lorentz transformation from the expression in the centre-of-mass frame. Note that as L depends both on u and on p (that is. on p.


p’) the functional relation between the p’s in the function W and the function L W is not as simple as might appear at first glance. In fact, in the integrals (6.7), (3.15). (4.8) each collision will have a different centre-of-mass, and so the Lorentz transformation (6.10) will be different for each collision (it will depend on P”); but this does not affect the feature that there will be a well-determined harmonic structure (in the u-frame) for these integrals, which we can write either directly as integrals of W or equivalently as integrals of L W (via Eq. (6.11)). It is therefore reasonable to analyse in a spherical harmonic way (based on u’) the two integrals

c, = jjj*, W(P,P’;P”,P”‘)f(P”)f(p”‘)~+‘~+“~+”’,

c, = jjj W(P,P’;P”,p”‘)f(P)f(P’) ~+‘~+“n+“‘, G

(6.12a)

(6.12b)

which together make up (6.7):

c=c,-c,. (6.12~)

Given the spherical harmonic decomposition (4.2) off(p), and breaking up W into spherical harmonic parts,

W = z W(m, E, m’, E’, m”, E”, m”‘, E”‘)A,B,,C,,,D ,,,, OA~OEf’O’~“O”~“‘. (6.13) , ’ 1 II 111,

Equations (6.12) show the spherical harmonic effect of the collision term and can be analysed in a way similar to the way the transport form is analysed (see Eqs. (6.5),

(6.6)). In many cases, the distribution function will be very nearly isotropic in the chosen

velocity frame ua; that is, the termf, will dominate the expansion

In this case, it is convenient to note that one can rewrite the collision term C(f) (which is the r.h.s. of Eq. (1.2)) in the form

C(f)= c c(A)+ x L(fi,fi~)~ I>0 /'>I>0

(6.14)

where C(S) is defined by (6.7) and L( f, g) is defined by

L(f, ‘!?I = jjjTx (f “g”’ +f”‘g” -fg’ --f’g) W(p,p’;p”,p”‘) n’7c”z”‘. (6.15)

EINSTEIN-BOLTZMANN SOLUTION,1 483

Then the linear approximation to the Boltzmann collision term is given by ignoring the higher-order terms in (6.14); that is, by using the approximation

w-1 = C(fo> + \‘ w/Lfo>. ITI

(6.16)

Spherical harmonic analysis of this equation will be much more tractable than that of (6.12) because now the collision term is only multiplied by one anisotropic distribution function each time, instead of three. In particular, becausef, is isotropic, the dominant term C(f,) will not generate higher harmonics in (6.16); and the analysis of the L(f,,fo) terms will be very similar to that given in Eqs. (6.5), (6.6). In general, one will have to truncate the analysis after a finite value 1 of L, thereby obtaining a truncated linear analysis.

Alternatively, one can use the full equations (6.14) but truncate that expansion after a finite number of terms (say, include only terms for which I, I’ <L), resulting in an approximation similar to the Grad method of moments (see [ 17, 18, 191) for calculating the transport coefficients of a gas near equilibrium. However the method proposed here allows a more detailed examination of the energy dependence of the transport coefficients than in the usual Grad method.

It is worth noting that because we allow particles of all rest-masses to be included in our formalism, one can easily include the effects of chemical and nuclear reaction in the collision terms.

7. SUMMARY

The way in which we have carried out first a 3 + 1 decomposition and then a harmonic analysis, is summarised in Tables I and II.

Once one has chosen a 4-velocity vector field, one can obtain a harmonic analysis

TABLE I

Functions”

Form Distribution

function Collision

term Boltzmann

equation

Covariant C(X’,P”) Lf= c for j-(x’. p,“)

(1.2)

3+1 splitting

SC-u’. m. E, e) C(x’. m, E. e) for f(x’* m. E, e) (3.9)

Harmonic components

b, ,(x. m, E) (4.4)

for F,,(x. m. E) (4.12)

” Effect of 3 + 1 and harmonic splitting on the functionsf, C and the Boltzmann equation.


TABLE II

Moments”

Form

Distribution function

(Moments)

Collision term

(Moments)

Boltzmann equation

(Moments)

Covariant M.6 NJ4 div M = N i

(2.6) I

(2.7) (2.10)

3+1 splitting

P” for P ” Id7

(3.14) 13’“’

(3.15) I4 (3.17)

Harmonic nA’ KA’ for Il” components

I.P.il (4.7)

1.8.4 (4.8) ‘.p,4 (4.13)

’ Effect of 3 + 1 and harmonic splitting on the moments off. C and the Boltzmann equation. Note: Each cell is the moment form of the corresponding cell in Table I.

of any distribution function f, expressing it in terms of trace-free, symmetric, traceless 3-tensors F,I(x, m, E) (see Eq. (4.2)). The harmonic moments m”/ of the distribution

tsp.9

function, obtained by integrating the F,41(x, m, E) over mass and energy after multiplication by suitable factors of m, e and (E2 - m2)1'2. are the physically interesting macroscopic quantities defined by f (see (4.10)). They can be combined to give the spacelike parts PAP of the general moments M,,, of the distribution function. Y

The moments M are relfa:ed to the moments NA, of the collision term C by a set of

divergence relations (Eqs. (2.10)). When the dollision term is harmonically analysed with harmonic coefficients b,, and harmonic moments K"f, these divergence relations

l.P.9

become relations (Eqs. (4.13)) between the harmonic moments ll and K. On the other hand the Boltzmann equation can be written as a relation determining the harmonic coefficients F.q, from the b,g,, (Eqs. (4.12)); on integrating this equation, one again obtains the moment equations (4.13). Any specified form of the collision term will determine coefficients 6,,, from the FA,, and so gives one a complete set of equations determining the evolution of the distribution function in any space-time.

It is convenient to use an orthonormal tetrad in the process of carrying out the analysis; this has the advantage that Einstein’s field equations for the space-time can easily be written out in terms of the rotation coefficients of this tetrad (see [36]). Thus one can either solve the Boltzmann equation in a given space-time (when the rotation coefficients can easily be determined from the given geometry) or consider properties of the combined Einstein-Boltzmann equations. When combined with the tetrad form of the field equations, Jacobi identities and contracted Bianchi identities (see [36]), the Boltzmann equation (4.12) (or the moment equation (4.13)) form a complete set of equations equivalent to the full Einstein-Boltzmann equations, as long as the collision term C is fully specified.

EINSTEIN-BOLTZMANN SOLUTION,1 485

Solution of the Boltzmann equation is simplified if the distribution function is invariant under inversion either of the direction of time, or of spatial directions, at each point; in this case the equations split into pairs of rather simpler equations (Section 6).

The harmonic analysis of the Boltzmann equation proposed in this paper has one major advantage: one deals directly with the harmonic components of the distribution function which determine the quantities of physical interest. Thus F(x, m, E) determines the density, energy density and pressure of the particle distribution; FO(x, m, E) determines the mass flux and energy of the distribution; and F,,(x, m, E) determines the anisotropic pressures due to the particles. It has the disadvantage that when one has determined the set of coefficients F,,(x. m, E), it is in general diffkult to prove that the distribution defined by these coefficients through the infinite sum (4.2) is (a) convergent, (b) positive definite (i.e.,f(x, m, E, e) > 0 for all x, m, E, e). However other similar methods of analysis, such as the Grad method, suffer from similar problems. We aim to show the usefulness of this formalism by giving some applications in further papers [42.43 1.

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Anisotropic solutions of the Einstein-Boltzmann equations: I. General formalism