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Anisotropic solutions of the Einstein-Boltzmann equations. II. Some exact properties of the equations
ANNALS
OF PHYSICS
150,
Anisotropic II.
487-503
(1983)
Solutions of the Einstein-Boltzmann Equations. Some Exact Properties of the Equations
G. F. R. ELLIS, R. TRECIOKAS, AND D. R. MATRAVERS Department
of Applied
Mathematics,
Rondebosch
Received
August
7700.
Universifj, South
of
Cape
Town.
Africa
11. 1980: revised
April
22. 1983
A covariant harmonic analysis is used to determine exact properties of the Einstein Boltzmann equations. In particular. it is shown that if there are a finite number of harmonic components, or if the first and second harmonic components are zero. then the solutions are kinematically very restricted in many circumstances. Implications for the understanding of the microscopic foundations of perfect fluids and of transport coefficients are discussed.
1. INTRODUCTION When examining the Einstein-Liouville or Einstein-Boltzmann equations, a convenient way to represent anisotropic distribution functions is through a covariant spherical harmonic analysis ([ 11; see also [2]). In a previous paper, this method was used to obtain some exact solutions of the Einstein-Liouville equations with unexpected properties ([ 31). In this paper, the spherical harmonic decomposition of the Boltzmann equation ([ 1 ]) is used to obtain some exact properties of this equation in a curved space-time. These properties are summarised in Table I, which the reader may find useful when reading the paper. Part of the motivation for obtaining these results, is to examine the microscopic foundations of the concept of a “perfect fluid” (cf., e.g. ]4,5 I); we consider this briefly at the end of the paper.
2. THE HARMONIC DECOMPOSITION The harmonic decomposition used depends on first making a 3 + 1 decomposition of the physical quantities of interest, splitting these quantities into their spacelike and timelike parts relative to some specific 4-velocity field u~(u’u, = -l), and thereafter making a harmonic decomposition of these parts. We consider in turn the physical quantities of relevance (for a fuller discussion see [ 11). The particle distribution function f(x’, p”) determines the distribution of particles at the point xi with 4-momenta p”, Now the particle 4-momenta can be written pa
=
Eu” + lea,
e”e, = 1, e%, = 0 487
(E > 0, A > O), 0003.4916/83
(1) $7.50
Copyright % 1983 by Academic Press, Inc. All rights of reproduction m any form reserved.
488
ELLIS,
TRECIOKAS,
AND
TABLE Einstein-Liouville
MATRAVERS
1
Solutions
(Exact
Properties) Implications
Harmonic
coefficients Perfect
Conditions
F .AI F.42
Finite number of harmonic coefficients First three harmonics zero o=o f=f(orthogonal Killing tensors) J=f(orthogonal Killing tensors) Homogeneous orthogonal Bianchi I or V solutions
.
hj
.
F..l, (I > 3)
fluid
Ii”
finite no. #O
u
cm9
B
0" O"."
.
Stationary or FRW
0"
000 00.
0" ()O.d
potp
5:
0. 0
0
0.
FRW”
0‘
0”
0”
FRW”.”
Equilibriumf
potlh
Ob
Oh
4 Isotropicf
potl’
0’
0”
J
potl?
O?
O?
J?
00. 00.
d
Oh stationary
4 Perfect
fluidf
’ True with b True with ’ True with d Requires
generalised Krook collision term. Boltzmann collision term. Krook or Boltzmann collision term. Einstein equation. Note. ., arbitrary; 0, zero; ?. perhaps: J. yes; potl. acceleration Friedmann-Robertson-Walker space-time; ??, perhaps true almost always.
potential
exists:
FRW
=
is the particle energy relative to u“ and Le” is the particle where E = -p’u, relativistic momentum relative to u’. Then we can write f in the form f=f(x’, m, E, e’), where the particle mass is related to the quantities 1, E by the relation p“p,=-m2(>O)s-~2=E2-m2(>0). The harmonic analysis is then carried out (cf. [l-3])
(2) by writing’
f = F + F,e’ + Fabeaeb+ F,bceaebec+ “. = ‘? FA,tTA’, IT
(3)
I We adopt the notation of Thorne [ 191 for strings of sub- and superscripts M,4g = M,,,, .aq, M’* = Ma”* “q, where a capital sub- or superscript denotes a string of lower case indices; the actual number is denoted by a subscript on the capital sub- or superscript. Further we abbreviate the tensor product of vectors by a similar device but use a tilde to indicate the product Pdy = P,, P,,P,, Poe, P”q = palpa2po3 pw.
EINSTEIN-BOLTZMANN
where the spherical harmonic tensors orthogonal to u: FA,=%,.~..
SOLUTIONS
coefficients
.a,,.’
489
II
F,4,(xi, m, E) are symmetric,
F,d,m,bub = 0,
F,4,mzbchbc= 0.
trace-free
(4)
This is completely equivalent to carrying out a standard spherical harmonic analysis off in terms of the usual functions Y,“(B, 4). Clearly expansion (3) must converge, must be non-negative for all (xi, m, E, ea}, and must have suitably convergent integrals (e.g. (9) (lo), (13), (14) below). Now various moments off can be defined by integrating J weighted suitably, over the tangent space TX. The volume element z+ in and on the future light cone C+(x) E {p’ E TX: p” > 0, pap, > 0) in TX can be written in the form 7I A =mdmldEd.Cl,
(5)
where dLl is the volume element spanned by two independent de”, and one should note the integrals of the e’: if
r is odd
h(alQ
where S is the unit 2-sphere. Then the number-flux are defined by
N” =
The decomposition
if
. . . hQ’r-l+)
I T, v%
r is even,
vector na and mass-flux
+.
(6)
vector N”
Vb)
of these quantities relative to u’ is then
na = 3~” + j", jaU, = 0,
(84
Na = pU" + Ja, .h,
WI
= 0,
where ti is the particle number density, p the mass density, j” the number-flux-vector, and J” the mass-flux vector relative to ua. Using relations (lt(7), the quantities in (8) can be expressed in the form ’ Round
brackets
denote
symmetrization
and square
brackets
skew
symmetrization.
490
ELLIS,
TRECIOKAS,
p=47t
AND
MATRAVERS
Onm’dm .= EAF dE, i I -0 -In
J,=~I”m2dmjV~‘F,d~ 0
W)
(lob)
m
- showing how n, p are determined as integrals of F(x’, m, E) and j,, J, as integrals of Fa(xi, m, E). It is clear that if there are any particles present at all at xi, then F > 0 at xi for some m, E. Similarly, the matter stress tensor T,, is defined by
Tab= .T, PaPbfn+. J The decomposition
(11)
of T,, relative to ZP is given by T,,
=
pu,
ub
+
q(a
ub)
+
=,b
+
Pha,,
(12)
where qaua = 0, rcabub = 0, Pa = 0, and h,, = g,, + u,ub projects orthogonal to u. Then ,L is the total energy density, q, the energy flux vector, nz,b the trace-free stress tensor, and p the isotropic pressure, relative to u. Using (1 t(6) and (1 l), the quantities in (12) can be expressed in the form
(13b) (14a)
showing how ,u,p are determined as integrals of F(x’, m, E), q, is determined as an integral of Fa(xi, m, E), and scab as an integral of Fab(xi, m, E).
EINSTEIN-BOLTZMANN
3. THE TETRAD
FORMALISM
SOLUTIONS
491
I1
AND THE EINSTEIN-LIOUVILLE
EQUATIONS
In order to write down detailed versions of the Einstein and Liouville equations, it is convenient to choose as a vector basis an orthonormal tetrad3 (E,) = (u, E,,},
g,, s E,‘E,‘gij = E, . E, = diag(-1, +l, +I, +l).
(15a)
The rotation coefficients of the tetrad are Tabc_ E, . V,~E, = E,‘E,i:jEd
(* ra*c = -r&o).
(15b)
These may be characterised in the following way ([ 1, 6, 71):
where ti” is the acceleration (zPu, = 0), uab is the shear (u,~u~ = 0, crab= u,~~). un, = 0), IX,* is the vorticity tensor (w,*u* = 0, w,~ = wtDbl), and 0 is the expansion of the congruence generated by u, (see [4, 51 for further discussion of these quantities). Further, the quantity flu represents the rate of rotation of the basis vectors (E,“} as seen by an observer moving with 4-velocity ua (err0 is the usual permutation symbol: cur,,,= E,~,,,, , sIZ3= +l). The remaining rotation coefficients are given by
where no,, = ntvn,. Thus the rotation coefficients are represented in terms of 3dimensional quantities {li,,, uUu,IX,,, , 6’.Q”, n,, , a,), the first four of which are components of 4-dimensional quantities (because of the choice (15a) of tetrad, one automatically has ti,, = 0, uOa= 0, UJ,,~= 0). For some purposes it is useful to further split nPointo its trace II and its trace-free part tip,.: nuL’= $S,,. + ji,,,. fir,, = fi,rcr,j. ti”, = 0.
(16~)
Given the 3 + 1 splitting (12) of the stress tensor T,,, the Einstein field equations now determine the Ricci tensor R,, by the relations RabUnUb= +tJd+ 3JI),
RobUahbc
= -q,,
Robhachbd=
+@ -P)
h,,
+ n,d.
(17)
One can explicitly write down these equations in terms of the rotation coefficients (16), see (82~(84) in [6), together with the Jacobi identities that are integrability conditions for Eq. (16), see (77)-(81) in [6]. The source terms in (17) are determined from the distribution function f by Eqs. 3 Here
and in the sequel, Greek
letters
run from
1 to 3.
492
ELLIS,
TRECIOKAS,
(13) and (14); and the evolution the Boltzmann equation L(f)
AND
MATRAVERS
of the distribution
= P~J-
- cob P”P~
function
itself is determined by
aflw = c(f ),
(18)
where L is the Liouville operator giving the derivative off along the particle paths” and C is the collision term giving the rate of change of S due to particle collisions. Given suitable prescriptions for this collision term, the evolution of the EinsteinBoltzmann equations is determined by Eqs. (17), (13), (14) and (18) where f is related to the harmonic coefftcients F,4, by (3) and (4). While the collision term C will generally be taken to be the n-particle Boltzmann interaction term (see, e.g. 18, 9]), for some applications it is appropriate to consider the collision-free case, when f satisfies the Liouville equation C=OeL(f)=O. In other circumstances
it may be appropriate c = -(f
(19) to use the Krook
equation
-f)/r
(20)
representing relaxation off towards a distribution functionf, where r is the relaxation time (cf. [ 10, 111); usually f is taken to be isotropic. In order to express (18) (with some suitable choice of collision term C, e.g., (19) or (20)) as a set of equations for the harmonic coefficients F,,,, one must harmonically analyse this equation. This somewhat lengthy process leads to a complicated (18) is expression (cf. [ 1, Eq. (4.12) I), which may be represented as follows: equivalent to the set of equations D, = + ,d, + + ,d, + ,d, +
,d, + - zd, = B,
(13 0).
(21)
Here f is expressed in terms of the harmonic coefficients F,q, by Eqs. (3) and (4). The d, are operators acting on the harmonic coefficients as follows: +&, is
(22a) (an operation only on the (I + 2) harmonic +,d! E ):Tiz
(&JP~
components
off ); + ,d, is
Wb)
- E~-‘-2a(~‘i2ZidFd.,,)/aE)
(an operation only on the (I + 1) harmonic components off), where as usual 1 denotes projection orthogonal to u on all indices by the projection tensor h,, ; ,d, is ’ ?,f
.
IS the derivative
of!
in the direction
of the basis vector
E,,, so ?,,J”=
ilfibx’
E,,’
EINSTEIN-BOLTZMANN
od, = E,(F,,)’
SOLUTIONS
- (n2/3)@aF..,,IaE
493
II
-EIFd(a,_,&d,,,~~b
where o” = 0, W’ = icK”‘~,,,, (an operation only on the I-harmonic f): _ ,d, = 0 if I = 0 (remember that I > 0) and if I> 1 is given by
-EL’
components
of
a(~-‘+‘F,,~,~,zi,,,)/aE 1
(an operation only on the (1- 1) harmonic components off): and if I > 2, is given by
-,d, = 0 if I = 0 or 1,
(an operation only on the (I - 2) harmonics off). Note that in all these expressions, the harmonic coefficients F4, are functions of xi, m, and E, while the rotation coefficients are functions only of the xi (and so are independent of m, E). Finally the B, are the harmonic components of the collision operator C; specifically, B, = 64,
CJI where C(j-) = \’ b,,,?, PO
(23)
where the coefficients b,,,, are trace-free, symmetric tensors orthonormal to u (that is, they obey Eq. (4)) and are functionals of the coefftcients FA, which determinef; note that in general the coefficient b,, for a particular value of I may depend on the coefficients F,,,, for all values I’ (i.e., harmonic mixing may take place through the collision term). It should be noted that a significant part of the complexity of these expressions results from the fact that each one must by itself be a symmetric, trace-free tensor
494
ELLIS,TRECIOKAS,AND
MATRAVERS
orthogonal to u; and in fact in each of the curly brackets in (22), only the first term is the real operational term; all the other terms are needed to make the resultant expression trace-free. The covariant derivatives in (22) may be explicitly written out in terms of the rotation coefficients (16); one finds that’ i(F,4,,;dhcd)
=
#d&4,)
hcd
+
21Fbc,A,&q)bdnCd
- 2(1 + 1) UbFb,,,
+ 21Fb bU-f%) i&I)’ &m,;a,J
=
~<4,Fa,)
=
&a,Fo;o2.. -(I-
-
(244 (24b)
1Fd(.4-,Edq)cQC, .a,))
+
(I
-
1)
&dc(&&y.
l)ad&(,&a,-,o,)
+
(I+
.a,~ l)F~~,-&)~
(24~)
The form of the collision harmonics 6,, will depend on the collision term C; in the case of the Liouville form (19), b/,,=O
(254
while in the case of the Krook equation (20), one will find b=r-‘(F-F),
b,, = -7-‘FA,
(12 1)
G-1
assumingthe function J‘ is isotropic. Equations (2 1E(24) enable one to write out the Boltzmann equation explicitly as a seriesof harmonic equations for 12 0 (in terms of the rotation coefficients (16)); the first three of these equations (for 1= 0, 1, 2) are written out explicitly in the Appendix.
4. MOMENT
CONDITIONS
In looking for exact solutions of the Einstein-Boltzmann equations, an obvious procedure is to place restrictions on the harmonic components FA, off by setting particular harmonics to zero; for example, one might truncate expansion (3) after some finite value L of 1 so that f has just a finite number of harmonic terms. More generally, we consider the case when any set of four consecutive harmonics vanish (which includes the case of a truncated expansion as a special case). As at least one of the FA, must be non-zero if there is any particle distribution present at all (in particular, F must be non-zero), there will then be some non-negative value L of 1 such that the FA, are zero for I = L + 1, L + 2, L + 3, and L + 4, but non-zero for 1 = L. Now let I = L + 2 in Eq. (21); the only non-zero term on the left will be -*d,, so the Liouville equation for this case can be written (from (21), (22e), (25a))
’ Note that these equations are independent extend the solutions to k = + 1).
of the value
of n = n’,
(which
fact was used in 13 1 to
EINSTEIN-BOLTZMANN
SOLUTIONS
II
495
where PSTF means “the projected symmetric trace-free part of’ ([2]; given by the operators T, S of [I], see Lemmas 2 and 3 of Section 4.2). Integrating, one finds = a(x’), and we must have a(xi) = 0 or else F,4, does not E+oOFA, = 0. Therefore we obtain (26) Now because each of FAL, oab are trace-free and FA, is non-zero, this implies that 0 ab = 0. To show this, choose the tetrad basis E, = {u. E,,} so that the {E,.} are shear eigenvectors, and let Y,’ = (2 ~ “‘)(Ela + iE,O), Ypa E (2 “‘)(Ela - iE,‘). Let A4 be the largest number such that when FA, is contracted with A4 vectors Y,“‘, a nonzero tensor results, but contracting with one more vector Y,’ results in a zero tensor: then
M factors
L-M
factors
is non-zero. Now multiplying (26) by M + 2 factors Y,” and (L -M - 2) factors Eja shows (s,~Y+‘Y+ b = 0; multiplying by factors Y-O instead of Y+a shows oa,,Y-aY-b = 0; together these show that cr,, = u12 = 0. Finally multiplying by M factors Y, a and (L -M) factors E,“ (and remembering that the E,.O are shear eigenvectors) shows that also uj3 = 0, so cob = 0. Now consider the case when the first three sets of coefficients (I = 1. 2, 3) are zero. Then the I = 2 Boltzmann equation is
Integrate with respect to E from m to co ; by the convergence properties of the harmonics the contribution from the term F,*,, vanishes, so we obtain the equation u ab
s
cc A’(aF/aE) m
dE = 0
(27)
in the Liouville case (when b,, = 0). Integrating by parts shows that (because F > 0) the integral is negative so uQb = 0. Finally, we may note that the above results will remain true in the case where the Krook equation (25b) holds instead of the Liouville equation (25a), because the source terms will vanish by virtue of there being no harmonic mixing. In fact, the results will remain true if we consider solutions where one has a generalised Krook equation given by b=z,-‘(F-F), representing a situation where the relaxation is different. Thus we have proved:
595/150/2-14
b,, = -r,-‘FA,
(28)
time for each of the spherical harmonics
496
ELLIS,TRECIOKAS,AND
MATRAVERS
THEOREM 1. If the distribution function F satisfies the Liouville equation, or the Boltzmann equation with generalised Krook collision term (28) and if there is a velocity field u such that relative to u,
(i) f has a finite number of harmonic components,or
f hasjust four consecutive harmonic componentszero, or (iii) f has its first, second and third harmonic componentszero, (ii)
the shear of u vanishes.
5. THE PERFECT FLUID
CONDITION
One can define an average 4-velocity U” (the “kinematic mean velocity,” the particle distribution by the prescription N” = pi?,
uaua = -1,
IS]) of
(29a)
where Na is defined by (7b); then p is the rest-massdensity measuredby an observer moving with 4-velocity U”. The matter distribution (considered macroscopically) is defined to be a perfectfluid’ if U” is a Ricci eigenvector and the pressuresrelative to U” are isotropic. In this case, supposewe choose u’ = U”; then (from (8) and (12)) we will find J” = 0, qa = 0, 7r,b= 0. (29b) Therefore we have a perfect fluid zf and only tf there is a 4-velocity ua such that for this velocity, ua = U” and (29b) is true (cf [4]). If there is a 4-velocity field ua such that thefirst and secondharmonic coefficients off vanish (that is, F, = 0 = Fat,) then the particle distribution is a perfect /7uid (with U = u). LEMMA
1.
This follows immediately from (10) and (14). While the vanishing of these coefficients is sufficient to show the distribution is a perfect fluid, there are particular examples (see, e.g. [3]) showing one can sometimeshave a perfect fluid even if one or both of these coefficients are non-zero. However, the nature of these examples suggest these cases are exceptional: in general we expect to find a perfect fluid distribution only if F, = Fab = 0. We consider in this paper various examples in which these restrictions are (exactly) true. 6 This definition is more restricted than is sometimes exist some 4-velocity ua for which the stress-energy additionally that u’ is the mean velocity V. If this is not take the usual form, and the matter described will associate with the concept of a perfect fluid. Note that zero (e.g., there may be a non-zero bulk viscosity).
used, as rather than simply requiring that there tensor takes the perfect fluid form, it demands not so. the fluid thermodynamics will in general not have all the properties we would normally (29b) does not imply that entropy production is
EINSTEIN-BOLTZMANN
SOLUTIONS II
491
First, supposethese coefficients vanish and in addition the fluid shear is zero. Then the l= 0 and I= 1 Boltzmann equations are
Ah,b(F),b - AE ~F/cYE ti, = 6,.
E(F)’ - (A2/3)t’ aF/aE = b,
However, these equations are the basic equations from which the Ehlers-GerenSachs theorem [ 121 and its generalisation [ 131 are derived. Therefore we can extend those results to this case also: THEOREM 2. If f satisfies the Liouville, Krook, or generalised Krook equations, and there is a velocity field u with vanishing shear and such that F, = F,, = 0, then the corresponding Einstein-Boltzmann solution has perfect fluid energy-momentum tensor, and (1) Bw = 0 and there exists an acceleration potential r, and (2) the solution is either stationary, or shear free and irrotational; in the latter case the solution is necessarily a Robertson- Walker solution if the collision term is zero.
Combining this with Theorem 1, we obtain COROLLARY. If f satisfies the Einstein-Liouville field u such that either
(1)
equations and there is a velocity
f has vanishing first, second, and third harmonic coefficients, or
(2) the first and second harmonic coefficients are zero and there are a finite number of moments, then the space time is either stationary or Robertson-Walker. This implies that virtually the only perfect-fluid Einstein-Liouville spacesin which one can actually construct f in harmonic form, and in which the fluid is expanding, are the Robertson-Walker space-times. Some of these results can be extended to the case when the moment conditions are only satisfied on an initial surface. An examination of the proof of Theorem 1 shows (see [ 141 for further details), THEOREM 3. Iff satisfies the Liouville, Krook, or generalised Krook equation in a space-time admitting a hypersurface-orthogonal velocity field u such that
(1)
(2)
Fab = 0 everywhere, and in addition F, = Fabc = 0 on an initial suflace S orthogonal to u,
then the shear of u vanishes on the initial
surface S.
An example where this is applicable is the study of perfect-fluid cosmologies in hypersurface-orthogonal Bianchi spaces. At decoupling one frequently assumes equilibrium, so condition (2) is satisfied; one also postulates a non-zero value for the shear, to study the evolution of the anisotropy. In the cases where the perfect fluid requirement is equivalent to F, = F,, = 0, this is inconsistent with a Liouville equation or a generalised Krook collision term.
498
ELLIS,TRECIOKAS,AND
6. KILLING
MATRAVERS
TENSOR CASES
If a space-time admits some Killing vectors Ku, or more generally a set of Killing tensors of which a general one may be written KAr (if it is a rank r tensor), then by virtue of Killing’s equations (KCaibj= 0, KC,,:,, = 0) a solution to the Liouville equations is given by any function g(Ki) of the constants of motion K, = K,p’, i?, = KA,jAA, (see, e.g. [20] or p. 65 in [8]). If the Killing tensors concerned are orthogonal to u, then the solutions so generated will be time-inversion invariant f(x’, m, E, e) =f(x’,
m, -E, e).
Pa)
It follows (see Section 5 of [ 1)) that the Liouville equation (21) separatesout into two sets of equations:
3, = +Zd,+ ,d1+- ,d,= 0
(1z O>?
Wb)
+D, = + ,dl + - ,d, = 0
(12 0).
(3Oc)
Suppose now that in addition F, = F,, = 0. Then the Liouville equation for I= 2 becomes an equation for FA, and FA, (which vanishes), and an equation for F,A,r F.qz (which vanishes), and F; this second equation will be identical in form to (26a) (with vanishing right-hand side), so the same argument may be applied as in Theorem 1. On the other hand, whether or not FA, and FA, vanish, if one has a Killing tensor orthogonal to u then the Killing equation contracted twice with u shows that K a,m,ctiC= 0. This implies (see [ 141 for further details) that the terms in the Liouville equation (30~) involving ti, vanish on their own. In particular in the case I= 1 we obtain the equation (2/5) 1 -‘a(A3ticFC’,,)/8E + ti, 3F/BE = 0. Multiplying by A3, integrating with respect to E from rn to co and using the convergence properties of the moments shows
which, by an argument similar to that in Theorem 1, shows zi” = 0. Thus we see: THEOREM 4. If f is a function of constants of the motion generated by Killing tensors orthogonal to a 4-velocity field u, and iff satisfies the Liouville equation, then ti” = 0. If in addition the first and second harmonic coefficients off vanish, then the shear of u is zero. If u is hypersurface orthogonal, the space is locally RobertsonWalker or static.
This result may be applicable for example in considering the relativistic star clusters in spherically symmetric space-time (cf. [ 151) or matter after decoupling in
EINSTEIN-BOLTZMANN
spatially invariant have
SOLUTIONS
II
499
homogeneous solutions. One should note in these cases that should u be under the Killing vectors concerned, then for each Killing vector Ka we
Km) = 0, K”;b ub = so strong restrictions equations.
IP;~K~, K%, = 0) =a K”ti, = 0 = Kacoob
on the acceleration and vorticity
7.
BIANCHI
will follow directly from these
SPACES
Finally, we consider spatially homogeneous Liouville solutions in Bianchi spaces (see, e.g. 17, 6, 211). Choosing a tetrad as in [6] or [7], the derivatives L(kYdF,A,)= 0. Choice of u normal to the homogeneous hypersurfaces (“non-tilted models”) necessarily implies that the fluid acceleration and vorticity vanish. We assume further that F, = F,, = 0. Consider first Class A universes; that is, models in which the rotation coefficients a, = 0. The I= 1, 1= 2 Liouville equations become u~~c~,(~~F,~,)/~E = 0, - ~.jL-3~(~58fF,fob)/~E
+ jAFfc,a~b,fd ndc - 2'0,~ aF/aE = 0
(3la) (3lb)
(on using (24)). Integrating (31a) with respect to E, the arbitrary functions of integration, which are functions of the xi, are seen to vanish on putting E = m. The resultant equations are integrated again with respect to E from m to oo, as are Eq. (3 lb) after multiplying by 13. Then defining Bnbc= i‘O /14FabcdE, -WI
G = 1.m A’(i?F/aE) dE -??I
we obtain CffB en = 0, Bfc~aEb,%ldC - iGo,, = 0. Now multiplying (32b) by nab shows that nabuab= 0. In the particular case when n = 0 (which is the case of a Bianchi I universe), Guab= 0; but G # 0 because F? 0, so Uob= 0. In the case of Class B universes, the equations corresponding to (32) again give a series of constraints on the shear; in the case of a Type V universe, this condition is
500
ELLIS,TRECIOKAS,AND
MATRAVERS
again that the shear vanish. These conclusions are unchanged if we replace the Liouville by the generalised Krook collision term. Thus we have THEOREM 5. If f is spatially homogeneous, has vanishing first and second harmonic coefficients, and satisfies the Liouville or generalised Krook equation in a Class A Bianchi orthogonal universe, then uahnah = 0. In Bianchi I or V universes. under these conditions, the shear must vanish.
8. DISCUSSION We have used a spherical harmonic analysis to prove a number of exact results on the Liouville and generalised Krook solutions in a curved space-time, which are of interest in their own right; for example, they generalise our knowledge of ellipsoidal distribution functions (see [ 16]), for these are examples of truncated expansions and so are subject to Theorem 1. A summary of the results obtained is given in Table I, which also gives results from (4, 12, 131. These results also relate to both the microscopic foundations of perfect fluids and of transport coefficients. We consider these in turn. It is known (see, e.g. [4]) that an equilibrium distribution function cannot be a solution of the Boltzmann equation in an expanding universe. Similarly it is known that an isotropic distribution function cannot be a solution of the Boltzmann equation in an anisotropically expanding universe ([ 12, 13I). We have conjectured ([ 13]) that these results may be extended to the statement, a solution of the Einstein-Boltzmann or Einstein-Liouville equations can only take the perfect fluid form if the fluid shear is zero and there is an acceleration potential r (i.e., li” = -h”‘(log r),J; and that further, we = 0. If true, this conjecture gives a cautionary note on interpreting strictly exact results obtained for anisotropically expanding perfect fluid solutions of the Einstein field equations. It is supported by the new results presented in this paper. which also give an indication of the difficulty involved in constructing a counterexample-this would have to involve a distribution function with an infinite number of moments. One may query this conclusion because the results given here are based on the Liouville or generalised Krook equation rather than the full Boltzmann equation. However, two points are relevant: First, that use of the full Boltzmann equation will probably result in harmonic mixing which will strengthen the argument, for then the demand of zero first and second harmonic coefficients (almost always needed for a perfect fluid form) will feed up to the higher harmonics, forcing them to remain zero if these conditions are to be fulfilled. This suggestionis strengthened by the linearised or truncated analyses of the Boltzmann equation carried out up to this time (cf., e.g. 19, 17], updated by [ 181) which lead to transport coefficient results which are consistent with the conjecture given here. Second, while it is known that nonequilibrium solutions tend to increase entropy, and the only solutions in which entropy does not increase are the equilibrium solutions which are isotropic,
EINSTEIN-BOLTZMANN
SOLUTIONS
501
II
nevertheless no proper proof has been given that collisions in fact tend to isotropise the distribution function. However, suppose this is true; then the actual effect of the Boltzmann collision term must be to tend to reduce the higher order harmonic components to zero; but this is precisely what is represented by the Krook equation (25b) or generalized Krook equation (28), so it is plausible that if indeed collisions do tend (through the Boltzmann collision integral) to isotropise f, then the actual effect of these collisions is well represented by one of these collision terms (cf. 110, 111). Thus it may well be that the set of self-consistent perfect fluid solutions based on the Boltzmann equation may only include non-shearing, stationary or nonrotating solutions. As regards the transport coefficients, the calculations carried out to this time have all been based on truncated harmonic expansions. It is certainly plausible that the higher order harmonics will have much smaller magnitude than the lower order harmonics, particularly as this condition is the only sure way to guarantee that the expansion (3) converges to a positive number for all directions e. Nevertheless it is clear that the exact solutions corresponding to truncated harmonic distributions are very restricted-see Theorem 1 for the Liouville, Krook, and generalised Krook cases, and note the above remarks related to the Boltzmann collision term. Indeed, if the Boltzmann collision term tends to isotropise f one may conjecture that it only causes harmonic drift-down-that is, for any value of I, the collision term b,,, depends on the F,4,, only for 1’ > 1 (so we do not find lower order harmonics generating higher order harmonics through the collisions). If this is so, then cases (i) and (ii) of Theorem 1 will remain exactly true for the Boltzmann collision term. These results do not necessarily mean that the usual formulae for the transport coefficients are incorrect; but it must clearly be a cause for concern if an approximation procedure used to calculate them (say, use of a truncated distribution function in an expanding, highly shearing universe) corresponds to exact solutions which can be shown to be inconsistent. It may wel! be possible to use bounding procedures in the full Boltzmann equation (21)-(24) to show why such exact results do not cause problems for the methods now in use for calculating transport coefficients in the corresponding situations. This would give a firm footing to physical arguments (relating collision times to expansion times) on which the present methods of calculation are based.
APPENDIX:
THE FIRST THREE
HARMONIC
EQUATIONS
We give here the first three equations (2 I)-(23). I=0 - +
- ‘a@ “aefF,)/dE + $(F&zef) - jEk’a(A2tidFd)/aE
+ E(F)’ - (A2/3)0BF/aE = b.
ELLIS,TRECIOKAS,AND
502 I=
MATRAVERS
1
- &-‘a(~“aefFefa)/aE
+ fL(habFb,:,hcd)
- $-2a(~32idFd,)/aE
+ Ehad(Fd)’ - (A2/3)8 aFa/aE
- f~“28(d3’2Fdad,)/aE - EFdmd, + dh,b(F)qb - AE aF/aE ti, = b, . 1=2 -g/l
p3a(/wfF’f,b
)/aE + +l(h,ehbfFefc:dhrd)
- $ELp3a(/14tidFd,b)/aE + Eh,‘hbf(Fef)’ - (A2/3)8 aF,,/aE - 41 “2a(L 3’2 { Fd(audb)- faefFefh,b})/aE - 2EFd,,odb,
+ ~haChbd(Fcc;dj- fF,;,hefh,,} - EkZa(lp’(F&b,
- +F,ti’h,,))/aE
- /I2 aF/aE oab= b,,.
ACKNOWLEDGMENT This paper
is largely
based on the Ph.D.
thesis
of one of the authors
(see j 14 1).
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17.
G. F. R. ELLIS. D. R. MATRAVERS. AND R. TRECIOKAS, ,4nn. Physics I50 (1983). 455. K. S. THORNE, Mon. NOI. Roy. Asf. SOC. 194 (1982), 439. G. F. R. ELLIS, D. R. MATRAVERS, AND R. TRECIOKAS, &r-f. Rel. Grac., in press. J. EHLERS, Abhand. Math. Nat. KI. Akad. Wiss. Lit. Maim I I (1961). G. F. R. ELLIS. in “General Relativity and Cosmology” (R. K. Sachs, Ed.). Varenna School, Course XLVII, Academic Press, New York. 1971. M. A. H. MACCALLUM, in “Cargese Lectures in Physics, Volume 6” (E. Schatzmann, Ed.), & Breach, 1973. G. F. R. ELLIS AND M. A. H. MACCALLUM. Commurz. Math. Phys. 12 (1969). 108. J. EHLERS. in “General Relativity and Cosmology” (R. K. Sachs, Ed.), Varenna Summer Course XLVII. Academic Press, New York. 1971. W. ISRAEL, in “General Relativity” (L. O’Raiffeartaigh, Ed.), Oxford Univ. Press, Oxford, P. L. BHATNAGAR, E. P. GROSS, AND M. KROOK, Phw. Rev. 94 (1954). 5 1 I. C. MARLE. Ann. Inst. Henri PoincarP X (1969), 67. J. EHLERS, P. GEREN. AND R. K. SACHS, J. Math. Phw. 8 (1968). 1344. R. TRECIOKAS AND G. F. R. ELLIS. Commun. Math. Phys. 23 (1971). I. R. TRECIOKAS, Ph.D. thesis, Department of Applied Mathematics and Theoretical University of Cambridge, August, 1972. J. IPSER AND K. S. THORNE, Astrophys. J. 154 (1968), 251. M. TRUMPER. Gen. Rel. Grav. 5 (1974). 1. New York. 197 I. J. M. STEWART, in “Lecture Notes in Physics No. IO,” Springer-Verlag,
Summer Gordon
School. 1972.
Physics.
EINSTEIN-BOLTZMANN
SOLUTIONS
W. ISRAEL AND J. M. STEWART, in “General Relativity Plenum, New York, 1980. 19. K. S. THORNE, Rev. Mod. Phys. 52, 2(l) (1980), 285. 20. J. R. RAY, Astrophys. J. 257 (1982), 578. 18.
21. J. R. RAY
AND
J. C. ZIMMERMAN,
Nuouo
Cimento
B 42
503
II
and
Gravitation”
(1977).
183.
(A.
Held,
Ed.).
Vol.
I.
OF PHYSICS
150,
Anisotropic II.
487-503
(1983)
Solutions of the Einstein-Boltzmann Equations. Some Exact Properties of the Equations
G. F. R. ELLIS, R. TRECIOKAS, AND D. R. MATRAVERS Department
of Applied
Mathematics,
Rondebosch
Received
August
7700.
Universifj, South
of
Cape
Town.
Africa
11. 1980: revised
April
22. 1983
A covariant harmonic analysis is used to determine exact properties of the Einstein Boltzmann equations. In particular. it is shown that if there are a finite number of harmonic components, or if the first and second harmonic components are zero. then the solutions are kinematically very restricted in many circumstances. Implications for the understanding of the microscopic foundations of perfect fluids and of transport coefficients are discussed.
1. INTRODUCTION When examining the Einstein-Liouville or Einstein-Boltzmann equations, a convenient way to represent anisotropic distribution functions is through a covariant spherical harmonic analysis ([ 11; see also [2]). In a previous paper, this method was used to obtain some exact solutions of the Einstein-Liouville equations with unexpected properties ([ 31). In this paper, the spherical harmonic decomposition of the Boltzmann equation ([ 1 ]) is used to obtain some exact properties of this equation in a curved space-time. These properties are summarised in Table I, which the reader may find useful when reading the paper. Part of the motivation for obtaining these results, is to examine the microscopic foundations of the concept of a “perfect fluid” (cf., e.g. ]4,5 I); we consider this briefly at the end of the paper.
2. THE HARMONIC DECOMPOSITION The harmonic decomposition used depends on first making a 3 + 1 decomposition of the physical quantities of interest, splitting these quantities into their spacelike and timelike parts relative to some specific 4-velocity field u~(u’u, = -l), and thereafter making a harmonic decomposition of these parts. We consider in turn the physical quantities of relevance (for a fuller discussion see [ 11). The particle distribution function f(x’, p”) determines the distribution of particles at the point xi with 4-momenta p”, Now the particle 4-momenta can be written pa
=
Eu” + lea,
e”e, = 1, e%, = 0 487
(E > 0, A > O), 0003.4916/83
(1) $7.50
Copyright % 1983 by Academic Press, Inc. All rights of reproduction m any form reserved.
488
ELLIS,
TRECIOKAS,
AND
TABLE Einstein-Liouville
MATRAVERS
1
Solutions
(Exact
Properties) Implications
Harmonic
coefficients Perfect
Conditions
F .AI F.42
Finite number of harmonic coefficients First three harmonics zero o=o f=f(orthogonal Killing tensors) J=f(orthogonal Killing tensors) Homogeneous orthogonal Bianchi I or V solutions
.
hj
.
F..l, (I > 3)
fluid
Ii”
finite no. #O
u
cm9
B
0" O"."
.
Stationary or FRW
0"
000 00.
0" ()O.d
potp
5:
0. 0
0
0.
FRW”
0‘
0”
0”
FRW”.”
Equilibriumf
potlh
Ob
Oh
4 Isotropicf
potl’
0’
0”
J
potl?
O?
O?
J?
00. 00.
d
Oh stationary
4 Perfect
fluidf
’ True with b True with ’ True with d Requires
generalised Krook collision term. Boltzmann collision term. Krook or Boltzmann collision term. Einstein equation. Note. ., arbitrary; 0, zero; ?. perhaps: J. yes; potl. acceleration Friedmann-Robertson-Walker space-time; ??, perhaps true almost always.
potential
exists:
FRW
=
is the particle energy relative to u“ and Le” is the particle where E = -p’u, relativistic momentum relative to u’. Then we can write f in the form f=f(x’, m, E, e’), where the particle mass is related to the quantities 1, E by the relation p“p,=-m2(>O)s-~2=E2-m2(>0). The harmonic analysis is then carried out (cf. [l-3])
(2) by writing’
f = F + F,e’ + Fabeaeb+ F,bceaebec+ “. = ‘? FA,tTA’, IT
(3)
I We adopt the notation of Thorne [ 191 for strings of sub- and superscripts M,4g = M,,,, .aq, M’* = Ma”* “q, where a capital sub- or superscript denotes a string of lower case indices; the actual number is denoted by a subscript on the capital sub- or superscript. Further we abbreviate the tensor product of vectors by a similar device but use a tilde to indicate the product Pdy = P,, P,,P,, Poe, P”q = palpa2po3 pw.
EINSTEIN-BOLTZMANN
where the spherical harmonic tensors orthogonal to u: FA,=%,.~..
SOLUTIONS
coefficients
.a,,.’
489
II
F,4,(xi, m, E) are symmetric,
F,d,m,bub = 0,
F,4,mzbchbc= 0.
trace-free
(4)
This is completely equivalent to carrying out a standard spherical harmonic analysis off in terms of the usual functions Y,“(B, 4). Clearly expansion (3) must converge, must be non-negative for all (xi, m, E, ea}, and must have suitably convergent integrals (e.g. (9) (lo), (13), (14) below). Now various moments off can be defined by integrating J weighted suitably, over the tangent space TX. The volume element z+ in and on the future light cone C+(x) E {p’ E TX: p” > 0, pap, > 0) in TX can be written in the form 7I A =mdmldEd.Cl,
(5)
where dLl is the volume element spanned by two independent de”, and one should note the integrals of the e’: if
r is odd
h(alQ
where S is the unit 2-sphere. Then the number-flux are defined by
N” =
The decomposition
if
. . . hQ’r-l+)
I T, v%
r is even,
vector na and mass-flux
+.
(6)
vector N”
Vb)
of these quantities relative to u’ is then
na = 3~” + j", jaU, = 0,
(84
Na = pU" + Ja, .h,
WI
= 0,
where ti is the particle number density, p the mass density, j” the number-flux-vector, and J” the mass-flux vector relative to ua. Using relations (lt(7), the quantities in (8) can be expressed in the form ’ Round
brackets
denote
symmetrization
and square
brackets
skew
symmetrization.
490
ELLIS,
TRECIOKAS,
p=47t
AND
MATRAVERS
Onm’dm .= EAF dE, i I -0 -In
J,=~I”m2dmjV~‘F,d~ 0
W)
(lob)
m
- showing how n, p are determined as integrals of F(x’, m, E) and j,, J, as integrals of Fa(xi, m, E). It is clear that if there are any particles present at all at xi, then F > 0 at xi for some m, E. Similarly, the matter stress tensor T,, is defined by
Tab= .T, PaPbfn+. J The decomposition
(11)
of T,, relative to ZP is given by T,,
=
pu,
ub
+
q(a
ub)
+
=,b
+
Pha,,
(12)
where qaua = 0, rcabub = 0, Pa = 0, and h,, = g,, + u,ub projects orthogonal to u. Then ,L is the total energy density, q, the energy flux vector, nz,b the trace-free stress tensor, and p the isotropic pressure, relative to u. Using (1 t(6) and (1 l), the quantities in (12) can be expressed in the form
(13b) (14a)
showing how ,u,p are determined as integrals of F(x’, m, E), q, is determined as an integral of Fa(xi, m, E), and scab as an integral of Fab(xi, m, E).
EINSTEIN-BOLTZMANN
3. THE TETRAD
FORMALISM
SOLUTIONS
491
I1
AND THE EINSTEIN-LIOUVILLE
EQUATIONS
In order to write down detailed versions of the Einstein and Liouville equations, it is convenient to choose as a vector basis an orthonormal tetrad3 (E,) = (u, E,,},
g,, s E,‘E,‘gij = E, . E, = diag(-1, +l, +I, +l).
(15a)
The rotation coefficients of the tetrad are Tabc_ E, . V,~E, = E,‘E,i:jEd
(* ra*c = -r&o).
(15b)
These may be characterised in the following way ([ 1, 6, 71):
where ti” is the acceleration (zPu, = 0), uab is the shear (u,~u~ = 0, crab= u,~~). un, = 0), IX,* is the vorticity tensor (w,*u* = 0, w,~ = wtDbl), and 0 is the expansion of the congruence generated by u, (see [4, 51 for further discussion of these quantities). Further, the quantity flu represents the rate of rotation of the basis vectors (E,“} as seen by an observer moving with 4-velocity ua (err0 is the usual permutation symbol: cur,,,= E,~,,,, , sIZ3= +l). The remaining rotation coefficients are given by
where no,, = ntvn,. Thus the rotation coefficients are represented in terms of 3dimensional quantities {li,,, uUu,IX,,, , 6’.Q”, n,, , a,), the first four of which are components of 4-dimensional quantities (because of the choice (15a) of tetrad, one automatically has ti,, = 0, uOa= 0, UJ,,~= 0). For some purposes it is useful to further split nPointo its trace II and its trace-free part tip,.: nuL’= $S,,. + ji,,,. fir,, = fi,rcr,j. ti”, = 0.
(16~)
Given the 3 + 1 splitting (12) of the stress tensor T,,, the Einstein field equations now determine the Ricci tensor R,, by the relations RabUnUb= +tJd+ 3JI),
RobUahbc
= -q,,
Robhachbd=
+@ -P)
h,,
+ n,d.
(17)
One can explicitly write down these equations in terms of the rotation coefficients (16), see (82~(84) in [6), together with the Jacobi identities that are integrability conditions for Eq. (16), see (77)-(81) in [6]. The source terms in (17) are determined from the distribution function f by Eqs. 3 Here
and in the sequel, Greek
letters
run from
1 to 3.
492
ELLIS,
TRECIOKAS,
(13) and (14); and the evolution the Boltzmann equation L(f)
AND
MATRAVERS
of the distribution
= P~J-
- cob P”P~
function
itself is determined by
aflw = c(f ),
(18)
where L is the Liouville operator giving the derivative off along the particle paths” and C is the collision term giving the rate of change of S due to particle collisions. Given suitable prescriptions for this collision term, the evolution of the EinsteinBoltzmann equations is determined by Eqs. (17), (13), (14) and (18) where f is related to the harmonic coefftcients F,4, by (3) and (4). While the collision term C will generally be taken to be the n-particle Boltzmann interaction term (see, e.g. 18, 9]), for some applications it is appropriate to consider the collision-free case, when f satisfies the Liouville equation C=OeL(f)=O. In other circumstances
it may be appropriate c = -(f
(19) to use the Krook
equation
-f)/r
(20)
representing relaxation off towards a distribution functionf, where r is the relaxation time (cf. [ 10, 111); usually f is taken to be isotropic. In order to express (18) (with some suitable choice of collision term C, e.g., (19) or (20)) as a set of equations for the harmonic coefficients F,,,, one must harmonically analyse this equation. This somewhat lengthy process leads to a complicated (18) is expression (cf. [ 1, Eq. (4.12) I), which may be represented as follows: equivalent to the set of equations D, = + ,d, + + ,d, + ,d, +
,d, + - zd, = B,
(13 0).
(21)
Here f is expressed in terms of the harmonic coefficients F,q, by Eqs. (3) and (4). The d, are operators acting on the harmonic coefficients as follows: +&, is
(22a) (an operation only on the (I + 2) harmonic +,d! E ):Tiz
(&JP~
components
off ); + ,d, is
Wb)
- E~-‘-2a(~‘i2ZidFd.,,)/aE)
(an operation only on the (I + 1) harmonic components off), where as usual 1 denotes projection orthogonal to u on all indices by the projection tensor h,, ; ,d, is ’ ?,f
.
IS the derivative
of!
in the direction
of the basis vector
E,,, so ?,,J”=
ilfibx’
E,,’
EINSTEIN-BOLTZMANN
od, = E,(F,,)’
SOLUTIONS
- (n2/3)@aF..,,IaE
493
II
-EIFd(a,_,&d,,,~~b
where o” = 0, W’ = icK”‘~,,,, (an operation only on the I-harmonic f): _ ,d, = 0 if I = 0 (remember that I > 0) and if I> 1 is given by
-EL’
components
of
a(~-‘+‘F,,~,~,zi,,,)/aE 1
(an operation only on the (1- 1) harmonic components off): and if I > 2, is given by
-,d, = 0 if I = 0 or 1,
(an operation only on the (I - 2) harmonics off). Note that in all these expressions, the harmonic coefficients F4, are functions of xi, m, and E, while the rotation coefficients are functions only of the xi (and so are independent of m, E). Finally the B, are the harmonic components of the collision operator C; specifically, B, = 64,
CJI where C(j-) = \’ b,,,?, PO
(23)
where the coefficients b,,,, are trace-free, symmetric tensors orthonormal to u (that is, they obey Eq. (4)) and are functionals of the coefftcients FA, which determinef; note that in general the coefficient b,, for a particular value of I may depend on the coefficients F,,,, for all values I’ (i.e., harmonic mixing may take place through the collision term). It should be noted that a significant part of the complexity of these expressions results from the fact that each one must by itself be a symmetric, trace-free tensor
494
ELLIS,TRECIOKAS,AND
MATRAVERS
orthogonal to u; and in fact in each of the curly brackets in (22), only the first term is the real operational term; all the other terms are needed to make the resultant expression trace-free. The covariant derivatives in (22) may be explicitly written out in terms of the rotation coefficients (16); one finds that’ i(F,4,,;dhcd)
=
#d&4,)
hcd
+
21Fbc,A,&q)bdnCd
- 2(1 + 1) UbFb,,,
+ 21Fb bU-f%) i&I)’ &m,;a,J
=
~<4,Fa,)
=
&a,Fo;o2.. -(I-
-
(244 (24b)
1Fd(.4-,Edq)cQC, .a,))
+
(I
-
1)
&dc(&&y.
l)ad&(,&a,-,o,)
+
(I+
.a,~ l)F~~,-&)~
(24~)
The form of the collision harmonics 6,, will depend on the collision term C; in the case of the Liouville form (19), b/,,=O
(254
while in the case of the Krook equation (20), one will find b=r-‘(F-F),
b,, = -7-‘FA,
(12 1)
G-1
assumingthe function J‘ is isotropic. Equations (2 1E(24) enable one to write out the Boltzmann equation explicitly as a seriesof harmonic equations for 12 0 (in terms of the rotation coefficients (16)); the first three of these equations (for 1= 0, 1, 2) are written out explicitly in the Appendix.
4. MOMENT
CONDITIONS
In looking for exact solutions of the Einstein-Boltzmann equations, an obvious procedure is to place restrictions on the harmonic components FA, off by setting particular harmonics to zero; for example, one might truncate expansion (3) after some finite value L of 1 so that f has just a finite number of harmonic terms. More generally, we consider the case when any set of four consecutive harmonics vanish (which includes the case of a truncated expansion as a special case). As at least one of the FA, must be non-zero if there is any particle distribution present at all (in particular, F must be non-zero), there will then be some non-negative value L of 1 such that the FA, are zero for I = L + 1, L + 2, L + 3, and L + 4, but non-zero for 1 = L. Now let I = L + 2 in Eq. (21); the only non-zero term on the left will be -*d,, so the Liouville equation for this case can be written (from (21), (22e), (25a))
’ Note that these equations are independent extend the solutions to k = + 1).
of the value
of n = n’,
(which
fact was used in 13 1 to
EINSTEIN-BOLTZMANN
SOLUTIONS
II
495
where PSTF means “the projected symmetric trace-free part of’ ([2]; given by the operators T, S of [I], see Lemmas 2 and 3 of Section 4.2). Integrating, one finds = a(x’), and we must have a(xi) = 0 or else F,4, does not E+oOFA, = 0. Therefore we obtain (26) Now because each of FAL, oab are trace-free and FA, is non-zero, this implies that 0 ab = 0. To show this, choose the tetrad basis E, = {u. E,,} so that the {E,.} are shear eigenvectors, and let Y,’ = (2 ~ “‘)(Ela + iE,O), Ypa E (2 “‘)(Ela - iE,‘). Let A4 be the largest number such that when FA, is contracted with A4 vectors Y,“‘, a nonzero tensor results, but contracting with one more vector Y,’ results in a zero tensor: then
M factors
L-M
factors
is non-zero. Now multiplying (26) by M + 2 factors Y,” and (L -M - 2) factors Eja shows (s,~Y+‘Y+ b = 0; multiplying by factors Y-O instead of Y+a shows oa,,Y-aY-b = 0; together these show that cr,, = u12 = 0. Finally multiplying by M factors Y, a and (L -M) factors E,“ (and remembering that the E,.O are shear eigenvectors) shows that also uj3 = 0, so cob = 0. Now consider the case when the first three sets of coefficients (I = 1. 2, 3) are zero. Then the I = 2 Boltzmann equation is
Integrate with respect to E from m to co ; by the convergence properties of the harmonics the contribution from the term F,*,, vanishes, so we obtain the equation u ab
s
cc A’(aF/aE) m
dE = 0
(27)
in the Liouville case (when b,, = 0). Integrating by parts shows that (because F > 0) the integral is negative so uQb = 0. Finally, we may note that the above results will remain true in the case where the Krook equation (25b) holds instead of the Liouville equation (25a), because the source terms will vanish by virtue of there being no harmonic mixing. In fact, the results will remain true if we consider solutions where one has a generalised Krook equation given by b=z,-‘(F-F), representing a situation where the relaxation is different. Thus we have proved:
595/150/2-14
b,, = -r,-‘FA,
(28)
time for each of the spherical harmonics
496
ELLIS,TRECIOKAS,AND
MATRAVERS
THEOREM 1. If the distribution function F satisfies the Liouville equation, or the Boltzmann equation with generalised Krook collision term (28) and if there is a velocity field u such that relative to u,
(i) f has a finite number of harmonic components,or
f hasjust four consecutive harmonic componentszero, or (iii) f has its first, second and third harmonic componentszero, (ii)
the shear of u vanishes.
5. THE PERFECT FLUID
CONDITION
One can define an average 4-velocity U” (the “kinematic mean velocity,” the particle distribution by the prescription N” = pi?,
uaua = -1,
IS]) of
(29a)
where Na is defined by (7b); then p is the rest-massdensity measuredby an observer moving with 4-velocity U”. The matter distribution (considered macroscopically) is defined to be a perfectfluid’ if U” is a Ricci eigenvector and the pressuresrelative to U” are isotropic. In this case, supposewe choose u’ = U”; then (from (8) and (12)) we will find J” = 0, qa = 0, 7r,b= 0. (29b) Therefore we have a perfect fluid zf and only tf there is a 4-velocity ua such that for this velocity, ua = U” and (29b) is true (cf [4]). If there is a 4-velocity field ua such that thefirst and secondharmonic coefficients off vanish (that is, F, = 0 = Fat,) then the particle distribution is a perfect /7uid (with U = u). LEMMA
1.
This follows immediately from (10) and (14). While the vanishing of these coefficients is sufficient to show the distribution is a perfect fluid, there are particular examples (see, e.g. [3]) showing one can sometimeshave a perfect fluid even if one or both of these coefficients are non-zero. However, the nature of these examples suggest these cases are exceptional: in general we expect to find a perfect fluid distribution only if F, = Fab = 0. We consider in this paper various examples in which these restrictions are (exactly) true. 6 This definition is more restricted than is sometimes exist some 4-velocity ua for which the stress-energy additionally that u’ is the mean velocity V. If this is not take the usual form, and the matter described will associate with the concept of a perfect fluid. Note that zero (e.g., there may be a non-zero bulk viscosity).
used, as rather than simply requiring that there tensor takes the perfect fluid form, it demands not so. the fluid thermodynamics will in general not have all the properties we would normally (29b) does not imply that entropy production is
EINSTEIN-BOLTZMANN
SOLUTIONS II
491
First, supposethese coefficients vanish and in addition the fluid shear is zero. Then the l= 0 and I= 1 Boltzmann equations are
Ah,b(F),b - AE ~F/cYE ti, = 6,.
E(F)’ - (A2/3)t’ aF/aE = b,
However, these equations are the basic equations from which the Ehlers-GerenSachs theorem [ 121 and its generalisation [ 131 are derived. Therefore we can extend those results to this case also: THEOREM 2. If f satisfies the Liouville, Krook, or generalised Krook equations, and there is a velocity field u with vanishing shear and such that F, = F,, = 0, then the corresponding Einstein-Boltzmann solution has perfect fluid energy-momentum tensor, and (1) Bw = 0 and there exists an acceleration potential r, and (2) the solution is either stationary, or shear free and irrotational; in the latter case the solution is necessarily a Robertson- Walker solution if the collision term is zero.
Combining this with Theorem 1, we obtain COROLLARY. If f satisfies the Einstein-Liouville field u such that either
(1)
equations and there is a velocity
f has vanishing first, second, and third harmonic coefficients, or
(2) the first and second harmonic coefficients are zero and there are a finite number of moments, then the space time is either stationary or Robertson-Walker. This implies that virtually the only perfect-fluid Einstein-Liouville spacesin which one can actually construct f in harmonic form, and in which the fluid is expanding, are the Robertson-Walker space-times. Some of these results can be extended to the case when the moment conditions are only satisfied on an initial surface. An examination of the proof of Theorem 1 shows (see [ 141 for further details), THEOREM 3. Iff satisfies the Liouville, Krook, or generalised Krook equation in a space-time admitting a hypersurface-orthogonal velocity field u such that
(1)
(2)
Fab = 0 everywhere, and in addition F, = Fabc = 0 on an initial suflace S orthogonal to u,
then the shear of u vanishes on the initial
surface S.
An example where this is applicable is the study of perfect-fluid cosmologies in hypersurface-orthogonal Bianchi spaces. At decoupling one frequently assumes equilibrium, so condition (2) is satisfied; one also postulates a non-zero value for the shear, to study the evolution of the anisotropy. In the cases where the perfect fluid requirement is equivalent to F, = F,, = 0, this is inconsistent with a Liouville equation or a generalised Krook collision term.
498
ELLIS,TRECIOKAS,AND
6. KILLING
MATRAVERS
TENSOR CASES
If a space-time admits some Killing vectors Ku, or more generally a set of Killing tensors of which a general one may be written KAr (if it is a rank r tensor), then by virtue of Killing’s equations (KCaibj= 0, KC,,:,, = 0) a solution to the Liouville equations is given by any function g(Ki) of the constants of motion K, = K,p’, i?, = KA,jAA, (see, e.g. [20] or p. 65 in [8]). If the Killing tensors concerned are orthogonal to u, then the solutions so generated will be time-inversion invariant f(x’, m, E, e) =f(x’,
m, -E, e).
Pa)
It follows (see Section 5 of [ 1)) that the Liouville equation (21) separatesout into two sets of equations:
3, = +Zd,+ ,d1+- ,d,= 0
(1z O>?
Wb)
+D, = + ,dl + - ,d, = 0
(12 0).
(3Oc)
Suppose now that in addition F, = F,, = 0. Then the Liouville equation for I= 2 becomes an equation for FA, and FA, (which vanishes), and an equation for F,A,r F.qz (which vanishes), and F; this second equation will be identical in form to (26a) (with vanishing right-hand side), so the same argument may be applied as in Theorem 1. On the other hand, whether or not FA, and FA, vanish, if one has a Killing tensor orthogonal to u then the Killing equation contracted twice with u shows that K a,m,ctiC= 0. This implies (see [ 141 for further details) that the terms in the Liouville equation (30~) involving ti, vanish on their own. In particular in the case I= 1 we obtain the equation (2/5) 1 -‘a(A3ticFC’,,)/8E + ti, 3F/BE = 0. Multiplying by A3, integrating with respect to E from rn to co and using the convergence properties of the moments shows
which, by an argument similar to that in Theorem 1, shows zi” = 0. Thus we see: THEOREM 4. If f is a function of constants of the motion generated by Killing tensors orthogonal to a 4-velocity field u, and iff satisfies the Liouville equation, then ti” = 0. If in addition the first and second harmonic coefficients off vanish, then the shear of u is zero. If u is hypersurface orthogonal, the space is locally RobertsonWalker or static.
This result may be applicable for example in considering the relativistic star clusters in spherically symmetric space-time (cf. [ 151) or matter after decoupling in
EINSTEIN-BOLTZMANN
spatially invariant have
SOLUTIONS
II
499
homogeneous solutions. One should note in these cases that should u be under the Killing vectors concerned, then for each Killing vector Ka we
Km) = 0, K”;b ub = so strong restrictions equations.
IP;~K~, K%, = 0) =a K”ti, = 0 = Kacoob
on the acceleration and vorticity
7.
BIANCHI
will follow directly from these
SPACES
Finally, we consider spatially homogeneous Liouville solutions in Bianchi spaces (see, e.g. 17, 6, 211). Choosing a tetrad as in [6] or [7], the derivatives L(kYdF,A,)= 0. Choice of u normal to the homogeneous hypersurfaces (“non-tilted models”) necessarily implies that the fluid acceleration and vorticity vanish. We assume further that F, = F,, = 0. Consider first Class A universes; that is, models in which the rotation coefficients a, = 0. The I= 1, 1= 2 Liouville equations become u~~c~,(~~F,~,)/~E = 0, - ~.jL-3~(~58fF,fob)/~E
+ jAFfc,a~b,fd ndc - 2'0,~ aF/aE = 0
(3la) (3lb)
(on using (24)). Integrating (31a) with respect to E, the arbitrary functions of integration, which are functions of the xi, are seen to vanish on putting E = m. The resultant equations are integrated again with respect to E from m to oo, as are Eq. (3 lb) after multiplying by 13. Then defining Bnbc= i‘O /14FabcdE, -WI
G = 1.m A’(i?F/aE) dE -??I
we obtain CffB en = 0, Bfc~aEb,%ldC - iGo,, = 0. Now multiplying (32b) by nab shows that nabuab= 0. In the particular case when n = 0 (which is the case of a Bianchi I universe), Guab= 0; but G # 0 because F? 0, so Uob= 0. In the case of Class B universes, the equations corresponding to (32) again give a series of constraints on the shear; in the case of a Type V universe, this condition is
500
ELLIS,TRECIOKAS,AND
MATRAVERS
again that the shear vanish. These conclusions are unchanged if we replace the Liouville by the generalised Krook collision term. Thus we have THEOREM 5. If f is spatially homogeneous, has vanishing first and second harmonic coefficients, and satisfies the Liouville or generalised Krook equation in a Class A Bianchi orthogonal universe, then uahnah = 0. In Bianchi I or V universes. under these conditions, the shear must vanish.
8. DISCUSSION We have used a spherical harmonic analysis to prove a number of exact results on the Liouville and generalised Krook solutions in a curved space-time, which are of interest in their own right; for example, they generalise our knowledge of ellipsoidal distribution functions (see [ 16]), for these are examples of truncated expansions and so are subject to Theorem 1. A summary of the results obtained is given in Table I, which also gives results from (4, 12, 131. These results also relate to both the microscopic foundations of perfect fluids and of transport coefficients. We consider these in turn. It is known (see, e.g. [4]) that an equilibrium distribution function cannot be a solution of the Boltzmann equation in an expanding universe. Similarly it is known that an isotropic distribution function cannot be a solution of the Boltzmann equation in an anisotropically expanding universe ([ 12, 13I). We have conjectured ([ 13]) that these results may be extended to the statement, a solution of the Einstein-Boltzmann or Einstein-Liouville equations can only take the perfect fluid form if the fluid shear is zero and there is an acceleration potential r (i.e., li” = -h”‘(log r),J; and that further, we = 0. If true, this conjecture gives a cautionary note on interpreting strictly exact results obtained for anisotropically expanding perfect fluid solutions of the Einstein field equations. It is supported by the new results presented in this paper. which also give an indication of the difficulty involved in constructing a counterexample-this would have to involve a distribution function with an infinite number of moments. One may query this conclusion because the results given here are based on the Liouville or generalised Krook equation rather than the full Boltzmann equation. However, two points are relevant: First, that use of the full Boltzmann equation will probably result in harmonic mixing which will strengthen the argument, for then the demand of zero first and second harmonic coefficients (almost always needed for a perfect fluid form) will feed up to the higher harmonics, forcing them to remain zero if these conditions are to be fulfilled. This suggestionis strengthened by the linearised or truncated analyses of the Boltzmann equation carried out up to this time (cf., e.g. 19, 17], updated by [ 181) which lead to transport coefficient results which are consistent with the conjecture given here. Second, while it is known that nonequilibrium solutions tend to increase entropy, and the only solutions in which entropy does not increase are the equilibrium solutions which are isotropic,
EINSTEIN-BOLTZMANN
SOLUTIONS
501
II
nevertheless no proper proof has been given that collisions in fact tend to isotropise the distribution function. However, suppose this is true; then the actual effect of the Boltzmann collision term must be to tend to reduce the higher order harmonic components to zero; but this is precisely what is represented by the Krook equation (25b) or generalized Krook equation (28), so it is plausible that if indeed collisions do tend (through the Boltzmann collision integral) to isotropise f, then the actual effect of these collisions is well represented by one of these collision terms (cf. 110, 111). Thus it may well be that the set of self-consistent perfect fluid solutions based on the Boltzmann equation may only include non-shearing, stationary or nonrotating solutions. As regards the transport coefficients, the calculations carried out to this time have all been based on truncated harmonic expansions. It is certainly plausible that the higher order harmonics will have much smaller magnitude than the lower order harmonics, particularly as this condition is the only sure way to guarantee that the expansion (3) converges to a positive number for all directions e. Nevertheless it is clear that the exact solutions corresponding to truncated harmonic distributions are very restricted-see Theorem 1 for the Liouville, Krook, and generalised Krook cases, and note the above remarks related to the Boltzmann collision term. Indeed, if the Boltzmann collision term tends to isotropise f one may conjecture that it only causes harmonic drift-down-that is, for any value of I, the collision term b,,, depends on the F,4,, only for 1’ > 1 (so we do not find lower order harmonics generating higher order harmonics through the collisions). If this is so, then cases (i) and (ii) of Theorem 1 will remain exactly true for the Boltzmann collision term. These results do not necessarily mean that the usual formulae for the transport coefficients are incorrect; but it must clearly be a cause for concern if an approximation procedure used to calculate them (say, use of a truncated distribution function in an expanding, highly shearing universe) corresponds to exact solutions which can be shown to be inconsistent. It may wel! be possible to use bounding procedures in the full Boltzmann equation (21)-(24) to show why such exact results do not cause problems for the methods now in use for calculating transport coefficients in the corresponding situations. This would give a firm footing to physical arguments (relating collision times to expansion times) on which the present methods of calculation are based.
APPENDIX:
THE FIRST THREE
HARMONIC
EQUATIONS
We give here the first three equations (2 I)-(23). I=0 - +
- ‘a@ “aefF,)/dE + $(F&zef) - jEk’a(A2tidFd)/aE
+ E(F)’ - (A2/3)0BF/aE = b.
ELLIS,TRECIOKAS,AND
502 I=
MATRAVERS
1
- &-‘a(~“aefFefa)/aE
+ fL(habFb,:,hcd)
- $-2a(~32idFd,)/aE
+ Ehad(Fd)’ - (A2/3)8 aFa/aE
- f~“28(d3’2Fdad,)/aE - EFdmd, + dh,b(F)qb - AE aF/aE ti, = b, . 1=2 -g/l
p3a(/wfF’f,b
)/aE + +l(h,ehbfFefc:dhrd)
- $ELp3a(/14tidFd,b)/aE + Eh,‘hbf(Fef)’ - (A2/3)8 aF,,/aE - 41 “2a(L 3’2 { Fd(audb)- faefFefh,b})/aE - 2EFd,,odb,
+ ~haChbd(Fcc;dj- fF,;,hefh,,} - EkZa(lp’(F&b,
- +F,ti’h,,))/aE
- /I2 aF/aE oab= b,,.
ACKNOWLEDGMENT This paper
is largely
based on the Ph.D.
thesis
of one of the authors
(see j 14 1).
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