General Relativistic Boltzmann Equation I. Covariant Treatment F. Debbasch Universit´e Paris 6 - CNRS, L.E.R.M.A. (E.R.G.A.), Tour 22-12, 4e`me ´etage, boˆıte 142, 4 place Jussieu, 75252 Paris Cedex 05, France

W. A. van Leeuwen Instituut voor Theoretische Fysica, Valckenierstraat 65, 1018 XE Amsterdam, the Netherlands Abstract A many-particle system, of particles numbered 1, 2, . . . , r, . . ., is considered. The particles have trajectories xir (t) in spacetime, where i = 1, 2, 3 labels a position coordinate. Using these trajectories, two equivalent one-particle distribution functions, f (t, xi, pi) and f∗ (t, xi , pi), which depend on the time t and position xi are defined: one with contravariant momentum variables pi and one with covariant momentum variables pi . Both f and f∗ can be considered as natural counterparts of the nonrelativistic and special relativistic one-particle distribution functions. In this article it is explicitly proved that the distribution functions f and f∗ so defined are scalars under general relativistic coordinate transformations. The general relativistic Boltzmann-equations to be obeyed by f and f∗ are derived in a way which parallels the usual derivation of the non-relativistic Boltzmann-equation as close as possible. The lefthand sides of these equations, the so-called streaming terms, turn out to be different for f and f∗ . The results of this article are covariant, but not manifestly covariant. The corresponding manifestly covariant results will be presented in a sequel to this article. PACS numbers: 04.20.-q, 05.20.Dd, 02.40.-k, 51.10.+y Keywords: distribution function, general relativity, relativistic statistical physics, Boltzmann-equation, covariant volume elements, induced metric.

1

1

Introduction

Non-equilibrium systems are systems in which some kind of transport takes place, that is, transfer of energy, mass, or any other particle property, from one place in a system to another. Central in the theoretical description of dilute non-equilibrium systems stands the so-called one-particle distribution function, usually denoted by the symbol f. All macroscopic properties of a dilute non-equilibrium system, such as its pressure, its energy density, its temperature, its electrical conductivity, its viscosity, etc., can be expressed in terms of this function f. The function f, in turn, can be determined by solving an equation — often called a transport equation— that describes the behavior of the non-equilibrium system. The most famous of these equations is the non-relativistic equation derived by Ludwig Boltzmann at the end of the 19th century. In the forties, Lichnerowich and Marrot generalized this equation to the realm of special relativity [9]. Israel [5] was one of the first to calculate relativistic transport coefficients on the basis this equation. He also was one of the first authors that wrote down a general relativistic version of the Boltzmann-equation [6]. It is the aim of this article to transfer the special relativistic Boltzmann theory to the realm of general relativity. To that end, both the definition of f and the equation for f have to be generalized. The existing literature on the Boltzmann-equations usually starts directly from manifestly covariant equations [6, 4, 10]. This makes it difficult to see what is actually happening: the steps of the derivation are obscured by the manifestly covariant formalism. Therefore, we opted for an approach which is covariant but not manifestly covariant. We will perform the transition to manifestly covariant equations in a the sequel to the present article [3]. The material is organized as follows. Section 2 reviews the non-relativistic and special relativistic distribution functions fnr and fsr . In general relativity, it is not clear, a priori, whether one should use the contravariant momentum pi or the contravariant momentum pi as a variable in the distribution function. We therefore define two distribution functions, one in the seven dimensional (t, xi, pi )-space, and one in the seven dimensional (t, xi , pi ) -space. We call them f and f∗ , respectively. Both f, given by eq. (27) below, and f∗ , given by eq. (4) below, are proven

2

to be general relativistic scalars. Although the choices for f or for f∗ are physically totally equivalent, the two sets of equations they lead to are sufficiently different to make mandatory a separate and full treatment for each choice. It turns out, however, that a treatment around f∗ is substantially simpler, and, therefore, highly preferable. In Section 3, the general relativistic counterparts of the Boltzmann equation, to be obeyed by the two afore-defined distribution functions f and f∗ , are derived. For simplicity’s sake, we only deal with non-quantal particles, and do not discuss Bose-Einstein or Fermi-Dirac statistics. The final equations for f and f∗ are (55) and (80), respectively. The present article is concerned with equations which are covariant, but not always manifestly covariant. In the next article, we will reformulate our results in a manifestly covariant way. In the main text, one needs two-, three- and four-dimensional surface and volume elements in spacetime and momentum space. These volume elements are defined and related to one another in appendix A. Appendix B recalls Stokes’s theorem, while appendix C is devoted to induced metrics. The appendices A, B and C have a pedagogical character. An explicit proof of the scalar character of f∗ (t, xi, pi ) has been given in section 2.3 below. The corresponding explicit proof of the scalar character of f(t, xi , pi ) is given in appendix D. We use a metric with signature −2.

2

One-particle distribution function

In this section we will be concerned with a general relativistic generalization of the special relativistic one-particle position-momentum distribution function, fsr (t, xi, pi ), (i = 1, 2, 3). It will turn out that in general relativity there are two possible generalizations, which we will denote f(t, xi , pi ) and f∗ (t, xi, pi ), both of which are general relativistic scalars. Here and elsewhere, a lower asterisk at some quantity indicates that covariant momentum components (like pi or qi ) are used as variables rather than contravariant ones (like pi or q i ).

3

At the one hand, the distribution function f(t, xi , pi ) depends on variables that are directly measurable, just as in special relativity, namely the timeposition (t, xi) and the contravariant momentum pi . In general relativity, however, this choice of variables leads to a quite intricate, and, hence, undesirable definition of f, namely the definition (27). It contains, next to the usual Dirac delta-functions in position and three-momentum space, factors containing the energy and the determinant of the metric. The distribution function f∗ (t, xi, pi ), on the other hand, contains covariant momentum variables, variables that depend on the metric via pi = giµ (t, xi)pµ (µ = 0, 1, 2, 3), which, therefore, are not directly measurable. However, the definition of the distribution function f∗ (t, xi, pi ) in terms of Dirac deltafunctions, given in eq. (4), is simpler than the corresponding definition (27) for f(t, xi, pi ). Now, there are no extra factors like the energy or the determinant of the metric. As a consequence, the Boltzmann equation (80) for f∗ takes a simpler form than the Boltzmann equation (55) for f. All this entails the question: which one to choose? Since both seem have their intrinsic advantages, we treat both f and f∗ and derive the Boltzmannequations they should obey. As a running start, we take the non-relativistic and the special relativistic distribution functions, fnr and fsr , respectively.

2.1

Non-relativistic distribution function

Let us start by recalling the classical, non-relativistic definition of the oneparticle position-momentum distribution function fnr (t, xi , pi ). It is a function of the time t, a position vector xi ≡ (x1, x2, x3 ) and a momentum vector pi ≡ (p1 , p2 , p3 ). By definition, the combination fnr (t, xi, pi ) ∆3x∆3p yields the average number of particles which, in a non-equilibrium fluid characterized by a certain set of macroscopic variables, will be found in a small but finite volume element ∆3 x around the point xi with momenta in a small but finite volume element ∆3 p around pi . The spatial volume elements ∆3x are supposed to be large enough to contain many particles, but small enough in order to make it possible to treat the distribution function as a constant all over these volume-elements. Consider a large, macroscopic system of identical particles numbered r = 1, 2, 3, . . .. Let xir (t) describe the trajectory of the r-th particle of the system. Let pir (t) be the momentum of the r-the particle at time t. Now, let us 4

consider the expression X r

δ (3)(xi − xir (t))δ (3)(pi − pir (t))

(1)

where the symbol δ (3) denotes a three-dimensional Dirac distribution, i.e., a product of three Dirac delta-functions, each of which contains a component of the vector xi or pi (i = 1, 2, 3). Integration of this expression with respect to the space and momentum volumes ∆3 x and ∆3p yields the number of particles which, at time t, are found within the volume element ∆3x around the point xi with momenta within the volume element ∆3 p around the point pi . Hence, the expression (1) gives the particle density in (xi, pi )-space. Two fluids are said to be macroscopically equal when their macroscopic properties are equal. Two fluids that are macroscopically equal will in general differ at a microscopic level: only averages are equal, not the precise position and momentum of each of the particles that constitute the two fluids. Let us consider a large set of macroscopically equal non-equilibrium systems, a so-called ensemble of non-equilibrium systems. One may associate, with each system of the ensemble, an expression like (1). Adding all these expressions, and dividing the sum so obtained by the number of systems of the ensemble, one gets the average particle density (averaged over the ensemble). Let us denote this average by h iav . Hence, the one-particle distribution function is given by the expression X fnr (t, xi, pi ) = h δ (3)(xi − xir (t))δ (3)(pi − pir (t))iav

(2)

r

In this way we obtained a mathematical expression for the non-relativistic distribution function fnr (t, xi, pi ).

2.2

Special relativistic distribution function

It can be shown that the non-relativistice definition (2) for the one-particle position-momentum distribution function happens to be valid also in special relativity and that

5

X fsr (t, xi , pi ) := h δ (3)(xi − xir (t))δ (3)(pi − pir (t))iav

(3)

r

is a scalar under Lorentz transformations (see [2] for a proof of these statements). So far for the non-relativistic and special relativistic one-particle distribution functions. We now want to generalize the special relativistic density fsr (t, xi, pi ) in position-momentum space given by eq. (3) to general relativity. This is less simple than the generalization from the non-relativistic to the special relativistic situation: in case contravariant momentum variables pi are used, an extra factor has to be incorporated in the expression which generalizes (3): see eq. (27). In case covariant momentum variables pi are used, no such extra factor is needed. For this reason, we begin our expos´e in the next section, section 2.3.1, by using pi ’s and the function f∗ (t, xi, pi ) rather than by using pi ’s and the function f(t, xi, pi ). Thereafter, in section 2.3.2 we continue with pi ’s and consider f(t, xi, pi ).

2.3 2.3.1

General relativistic distribution function Covariant components

Let M be the spacetime manifold endowed with a metric g(x) ≡ (gµν ). (Here and elsewhere, Greek indices run from 0 to 3.) Let x = (ct, xi) be local coordinates on M and let det g(x) be the determinant of g(x) at the point x ∈ M. Let us consider a fluid of identical particles, numbered 1, 2, . . ., in a neighborhood of x. Let xir (t) and pri (t) be the position and momentum of a particle r of the fluid at time t. Now, define a quantity f∗ (t, xi, pi ) by: X f∗ (t, xi, pi ) := h δ (3)(xi − xir (t))δ (3)(pi − pri (t))iav ,

(4)

r

in analogy to (3). In order to prove that f∗ is a scalar, we introduce the auxiliary function I∗(x, p∗ ) := 2θ(p0 )δ(p2∗ − m2c2 )f∗ (t, xi, pi ). 6

(5)

In the left-hand of eq. (5), x and p∗ are short-hand notations for the four contravariant spacetime coordinates xµ = (ct, xi) and the covariant momentum 4-vector pµ = (p0 , pi ). In the right-hand, θ stands for the unit step function and p2∗ := g µν pµ pµ , while m is the mass of a particle of the gas. Since in this equation the functions θ(p0 ) and δ(p2∗ − m2c2 ) are scalars, proving that the auxiliary function I∗(x, p∗ ) is a scalar is tantamount to proving that the distribution function f∗ (x, pi ) is a scalar. In order to show that I∗(x, p∗) is a scalar, we recall the identity δ(H(z)) =

1

X

|H 0 (z)|

k

δ(z − zk )

(6)

where the sum is extended over all roots zk of the equation H(z) = 0. The prime at H in the right-hand side stands for the derivative with respect to the argument z of the function H. Applying this equation to the function H(p0 ) := g µν pµ pν − m2 c2 one obtains 2θ(p0 )δ(p2∗ − m2c2 ) =

1 δ(p0 − p0 (x, pi )), p0 (x, pi )

(7)

where 1 p0 (x, pi ) = 00 {−g 0i (x)pi + g (x) p0 (x, pi ) =

q

q

(g 0i (x)pi )2 − g 00(x)(g ij (x)pi pj − m2c2 )} (8)

(g 0i (x)pi )2 − g 00(x)(g ij (x)pi pj − m2c2 ),

(9)

The first of these two expressions, eq. (8), is easily found by solving the following quadratic expression for p0 : g µν pµ pν = m2c2

(10)

The second expression, eq. (9), then follows immediately with p0 = g 0ν pν . Inserting (7) and (4) into (5) we find 7

X I∗ (x, p∗) = h r

1 p0r (pri (t))

δ (3)(xi − xir (t))δ (4)(p∗ − p∗r (t))iav

(11)

where

p0r (pri (t)) := p0 (t, xir (t), pri (t)) pr0 (pri (t)) := p0 (t, xir (t), pri (t))

(12) (13)

The functions p0 (t, xi , pi ) and p0 (t, xi, pi ) are given by (9) and (8). As a next step, we introduce an additional Dirac distribution δ(t − tr ) and, at the same time, add an integration with respect to tr . This yields:

I (x, p ) = Z ∗X ∗ 1 h δ(t − tr )δ (3)(xi − xir (tr ))δ (4)(p∗ − p∗r (tr ))dtr iav 0 (t ) p r r r

(14)

The zero component of the momentum of particle r is given by p0r = mdx0r /dτr , where τr is the proper time along the trajectory of particle r. Hence, dτr =

mc p0r (tr )

dtr

(15)

where tr := x0r /c. Changing the integration variable in (208) from tr to τr , and using (15), we find the expression: 1 I∗ (x, p∗) = m

Z X h δ (4)(x − xr (tr (τr )))δ (4)(p∗ − p∗r (tr (τr )))iav dτr r

8

(16)

√ √ In appendix A it is shown that δ 4(x − xr )/ − det g and − det g δ 4 (p∗ − p∗r ) are scalars [see eqs. (122) and (155)]. Hence, the product of the two deltafunctions occurring in (16) is a scalar. Consequently, the auxiliary function I∗(x, p∗ ) is a scalar, implying, finally, in view of (5), that f∗(x, pi ) is a scalar. Because of its interpretation in a local Lorentz system as a density, f∗ (x, pi ) as defined by eq. (4) is a general relativistic scalar that may be interpreted as the one-particle position-momentum distribution function in the sevendimensional (x, pi )-space. 2.3.2

Contravariant components

Let us now switch from the covariant momentum variables pi to the contravariant momentum variables pi (i = 1, 2, 3). From eq. (182) we have d3 p =

1 p0 (pi ) 3 d p∗ − det g p0 (pi )

(17)

The function p0 (pi ) occurring in this expression is given by (9). The co- and contravariant components p0 (pi ) and p0 (pi ) of the four-momentum pµ are given by 1 {−g0i (x)pi + p (x, p ) = g00(x) 0

i

q

(g0i (x)pi )2 − g00(x)(gij (x)pi pj − m2 c2 )} (18)

q p0 (x, pi ) = (g0i (x)pi )2 − g00 (x)(gij (x)pi pj − m2c2 )

(19)

The first of these two expressions, eq. (18), is easily found by solving, for p0 , the equation gµν pµ pν = m2c2

(20)

an equation which is quadratic in p0 . The second expression, eq. (19), then follows immediately with p0 = g0ν pν . Note, that in eq. (17), the function 9

p0 (pi ) is to be expressed in terms of the variables pi via the relation pi = giµ pµ , with p0 (pi ) given by (9). Let h(pi ) and h∗ (pi ) be arbitrary functions that are equal when pi = giµ pµ : h∗ (pi ) = h(pi )

(21)

Now, let p∗ri = (pr1 , pr2 , pr3 ) and pir = (p1r , p2r , p3r ) be fixed vectors related via pri = giµ pµr . Then (21) implies h∗(pri ) = h(pir ). The latter equality can be rewritten Z

3

3

h∗(pi )δ (pi − pri )d p∗ =

Z

h(pi )δ 3(pi − pir )d3 p

(22)

as follows by integrating. Using now the relation (17) between the differentials d3 p and d3 p∗ we get Z

3

3

h∗ (pi )δ (pi − pri )d p∗ =

Z

h(pi )δ 3(pi − pir )

1 p0 (pi ) 3 d p∗ − det g p0 (pi )

(23)

which implies, since h∗ (pi ) = h(pi ) is an arbitrary function, 3

3

i

δ (pi − pri ) = δ (p −

pir )

1 p0 (pi ) − det g p0 (pi )

(24)

With this expression, the one-particle distribution function f∗ , eq. (4) can be written

X f∗(t, xi , pi ) := h δ (3)(xi − xir (t))δ (3)(pi − pir (t) r

p0 (pi ) 1 iav , − det g p0 (pi )

(25)

Now, let us express p0 (pi ), given by (8), as a function of pi via pi = giµ pµ , with p0 given by (18), and, thereupon, define a new function f(t, xi, pi ) by f(t, xi, pi ) := f∗(t, xi , pi ) 10

(26)

or, explicitly,

f(t, xi , pi ) := h

X r

δ (3)(xi − xir (t))δ (3)(pi − pir (t))

1 p0 (pi ) iav , − det g p0 (pi )

(27)

where p0 (pi ) has been expressed as a function of pi , i.e., p0 (pi ) = p0 (giµ pµ ), with the p0 of the argument gµ0 p0 given in terms of pi by (18). Since f∗ is a scalar, f must be a scalar. If no factor p0 (pi )/(− det g p0 (pi )) would have been included in the expression (27) for f in terms of delta functions [like it was done in the corresponding expression (4) for f∗ ], f would not have been a scalar. See also appendix D, where it is shown explicitly that (27) is a scalar. In a local Lorentz system at the spacetime point x, the metric tensor gµν = diag(1, −1, −1, −1) so that p0 = p0 and det g = −1 , implying that f, given by eq. (27), reduces to the special relativistic expression (3), which shows that (27) is indeed the particle density in position-momentum space, since it is so in a local Lorentz system. Note that, although p0 (pi ) and p0 (pi ) do occur (27), f is not a function of these variables; in this expression they only are abbreviations for the right-hand sides of eqs. (19) and (9). Since f∗ is a scalar, f is a scalar, i.e., independent of the choice of coordinates. An explicit proof of this statement, independent of the relation (17) between the differential d3 p and d3 p∗ , and independent of the scalar character of f∗ is given in appendix D. In the next section we will derive the Boltzmann-equations for the distribution functions f and f∗ , respectively. We start with f, since the distribution function f is more closely related to the classical distribution function than f∗ . This is not essential point, however, it only is a choice of presentation.

3

General Relativistic Boltzmann equation

A transport equation is an equation for the spacetime behavior of the oneparticle momentum distribution function. The form of this equation depends on a number of assumptions. 11

In case of the Boltzmann-equation —a particular transport equation— there are three assumptions: i. The considerations are limited to binary collisions. ii. Furthermore, the possibility of a macroscopic description (i.e., averaged description) requires that changes of the distribution function on a microscopic scale of length and time (i.e., atomic or particle scale of length and time) are negligible small. iii. Finally, an essential assumption of statistical nature enters the description, namely the hypothesis of molecular chaos (or stosszahlansatz ), originally due to Boltzmann. The hypothesis of molecular chaos is the assumption that the probability that a collision with certain initial and certain final momenta will take place, is independent of the situation of other particles, particles in the neighborhood of the colliding particles. In fact, this number is supposed to be proportional to the product of the distribution functions of the colliding particles and to a so-called transition rate w(pi , q i | p0i , q 0i), which is a measure for the probability that a collision between two particles with initial momenta pi and q i and final momenta p0i and q 0i will take place. In this article, we generalize w by allowing w to be dependent of time t and position xi (i = 1, 2, 3): w = w(t, xi, pi , q i | p0i , q 0i). In this section, the transport equation is obtained in a way that is similar to Boltzmann’s original reasoning, but now placed within the framework of general relativity. This entails a number of technical difficulties, mainly concerning volume elements in spacetime or momentum-three and four-space. These are treated in appendix A.

For didactical purposes, we start in sections 3.1 and 3.2 by considering the case that is as close as possible to the original derivation of the Boltzmannequation, namely the case with contravariant momentum variables pi , variables corresponding to momenta that are directly measurable. As noted above, a description based on f∗ rather than f is preferable. It will be presented in sections 3.3 and 3.4.

3.1

Streaming term — contravariant momentum

The number density four-flow vector nµ (t, xi), defined according to

12

pµ f(t, xi , pi ) mc

nµ (t, xi) :=

(28)

In the present article, we will use only its zero component p0 f(t, xi , pi ) mc

n0 (t, xi ) :=

(29)

Since f has the dimension of a number density in momentum-position space, n0 also has the dimension of a number-density in momentum-position space. Let ∆3 x ⊂ Σµ (t) be volume elements of spacetime lying entirely in a leafs of times t of a foliation of spacetime [see appendix A.3.2]. Furthermore, let us choose the contravariant components pi (i = 1, 2, 3) as momentum variables on the mass shell S [see appendix A.5], and let pi ∈ ∆3 p, where ∆3p ⊂ S is some three-dimensional momentum interval of momenta belonging to the mass shell S [see eq. (85)]. Let N∆3 x∆3 p(t) be the total number of the particles, which, at a time t, have positions xi and momenta pi in the ranges ∆3x and ∆3p, respectively. With respect to coordinates adapted to the foliation of spacetime as defined in appendix A.3.2, the total number N∆3 x∆3 p (t) can be found by integrating the number-density (29) with respect to volume element d3 Σ0 of the threedimensional hypersurface Σ(t) of spacetime, and with respect to the volume element d3 Vp of the three-dimensional mass shell S. Thus we get N∆3 x∆3 p(t) =

Z

∆3 x×∆3 p

p0 f(t, xi , pi ) d3 Σ0 d3 Vp mc

(30)

We have to use d3 Σ0 , since we integrate a volume-element with constant time x0 = ct; see (125). We have to use d3 Vp since we integrate a volume element on the mass shell S: see (157). Using (128) and (169) we so find the expression N∆3 x∆3 p (t) =

Z

f(t, xi, pi ) r(t, xi , pi ) d3 xd3 p ∆3 x×∆3 p

with the function r(t, xi , pi ) is given by 13

(31)

p0 (x, pi ) r(t, x , p ) := − det g(x) p0 (x, pi ) i

i

(32)

Recall that p0 (x, pi ), given by (8), should be expressed as a function of pi via pi = giµ pµ , using eq. (18) for p0 in the right-hand side of the latter expression. One way to find dN∆3 x∆3 p (t)/dt, i.e., the increase of the total number of particles (per unit of time) contained in ∆3x × ∆3p, is to differentiate (31) with respect to t. Since ∆3 x is a volume element lying totally in Σ(t), where t is a constant, all spacetime points of the volume element ∆3x have the same time coordinate. Moreover, ∆3p is independent of t. Hence, the derivative may be taken under the integral: dN∆3 x∆3 p (t) = dt

Z

∆3 x×∆3 p

 ∂  f(t, xi, pi )r(t, xi, pi ) d3 x d3 p ∂t

(33)

This is a first way to calculate the gain of the total number of particles. If there are no collisions, there is, however, an alternative way, since then no other possibility for a particle to get out of the position and momenta intervals ∆3 x and ∆3p exists than streaming out of them. Consequently, a second way to obtain dN∆3 x∆3 p (t)/dt is to calculate what flows out of the intervals ∆3 x and ∆3 p, respectively, and to add these contributions. This is what will be done now. The loss of particles per unit of time, J∆3 x (t), due to particles leaving the position interval ∆3x, is obtained by integrating the expression (29) —multiplied by the particle velocity dxi /dt— with respect to the two-dimensional covariant surface element d2 σ0i of the boundary of ∆3x, and with respect to the three-dimensional covariant momentum interval d3 Vp J∆3 x (t) =

Z

∂(∆3 x)×∆3 p

p0 dxi 2 f(t, xi, pi ) d σ0i d3 Vp mc dt

(34)

where ∂(∆3x) is the boundary (i.e., the two-dimensional surface) of ∆3x. Similarly, the loss of particles per unit of time, J∆3 p (t), due to particles leaving the momentum interval ∆3 p is 14

J∆3 p(t) =

Z

∆3 x×∂(∆3 p)

i p0 i i dp f(t, x , p ) d3 Σ0 d2 si mc dt

(35)

where d3 Σ0 is a three-dimensional element of the three-dimensional surface Σ(t) of constant time given by (128), and d2 si is a two-dimensional element of the surface of ∆3p, lying entirely on the contravariant mass-shell (20), and is given by (185). If there are no collisions, the increase (33) equals minus the total decrease (34)-(35), so that we have dN∆3 x∆3 p (t) = −J∆3 x (t) − J∆3 p (t) dt

(36)

The integrals J∆3 x (t) and J∆3 p(t) may both be rewritten. For the first of these, J∆3 x (t), we have, using (138) and (169), J∆3 x (t) =

Z

∂(∆3 x)×∆3 p

p0 p0

√ − det g dxi 2 p f(t, xi , pi ) d Ai d3 p dt g 00

(37)

Using Stokes’s theorem, eq. (194), we get

J∆3 x (t) =

Z

" # √ i ∂ p p0 − det g dx p d3 x d3 p − det gΣ f(t, xi , pi ) 00 ∂xi p0 dt g

∆3 x×∆3 p

(38)

where gΣ (t, x1, x2, x3 ) is the determinant of the metric of the three-dimensional surface Σ(t) at the point (t, x1, x2 , x3) around which the interval of ∆3 x is situated. With the help of (132) we so find for J∆3 x (t): J∆3 x (t) =

Z

∆3 x×∆3 p

  i ∂ i i i i dx f(t, x , p )r(t, x , p ) d3 x d3 p ∂xi dt

where r is given by (32). For the second integral, J∆3 p (t), we find, with (128), 15

(39)

J∆3 p (t) =

Z

∆3 x×∂(∆3 p)

i p p0 i i dp − det g d2 si d3 x f(t, x , p ) mc dt

(40)

where d2 si is the element of surface on the contravariant mass-shell (85), given by eq. (183). Using again Stokes’s theorem, now in momentum space, we get   p0 p dpi ∂ p − det gS J∆3 p (t) = f − det g d3 pd3 x i ∂p mc dt 3 3 ∆ x×∆ p √ √ With the help of − det gS = mc − det g/p0 [see eq. (170)] we get Z

J∆3 p (t) =

Z

∆3 x×∆3 p

  ∂ i i i i dpi f(t, x , p )r(t, x , p ) d3 p d3 x i ∂p dt

(41)

We now substitute (33), (39) and (41) into (36) to obtain Z

3.2

∆3 x×∆3 p



∂ ∂ (fr) + i ∂t ∂x

    dxi ∂ dpi fr + i fr d3 xd3p = 0 dt ∂p dt

(42)

Boltzmann equation — contravariant momentum

In case ∆3 x and ∆3p are small but finite volumes, defined by (xi , xi + ∆xi ) and (pi , pi + ∆pi ) in position- and momentum-space, the equation (42) can be approximated by 

 ∂ ∂ dxi ∂ dpi (fr) + i (fr ) + i (fr ) ∆3 x∆3p = 0 ∂t ∂x dt ∂p dt

(43)

where ∆3 x = ∆x1∆x2∆x3 and ∆3p = ∆p1 ∆p2 ∆p3 . This equation gives, in the case that there are no collisions, the increase of the number of particles per unit time, due to particles with momenta in the range (pi , pi + ∆pi ) that arrive in (xi , xi + ∆xi ). 16

If we want to take collisions into account, we may replace this equation by 

∂ ∂ (fr) + i ∂t ∂x



dxi fr dt



∂ + i ∂p



dpi fr dt



∆3 x∆3p = ∆c(f, f)

(44)

The equation (44) is to be considered as the defining equation for ∆c(f, f). The combination ∆c(f, f) gives the change of the number of particles in ∆3x at the spacetime point (t, x1, x2 , x3) due to particles that leave the momentum interval ∆3 p due to collisions. In order to find an expression for ∆c(f, f), we shall use the hypothesis of ‘molecular chaos’ mentioned in the beginning of this article. Let us consider a collision pi + q i → p0i + q 0i between two particles with initial contravariant three-momenta pi and q i and final contravariant threemomenta p0i and q 0i. According to the hypothesis of molecular chaos, the average number per unit of time of such collisions in a volume ∆3x of the foliation at the leaf Σ(t) characterized by a constant value of t, around the point (x1 , x2, x3) ∈ Σ, is proportional to: i. the density (i.e., average number of particles per unit volume) of particles with three-momenta pi in the range (pi , pi + ∆pi ): f(t, xi , pi )∆3 p, ii. the density of particles with three-momenta in the range (q i, q i + ∆q i): f(t, xi , q i)∆3q, iii. the interval ∆3 x in position space and the two final intervals ∆3 p0 and ∆3q 0 in momentum space. We so find that the average number of particles with momentum pi in the volume ∆3x which disappear per unit of time from the interval ∆3 p because of a collision with a particle with momentum q i, to become a particle with momentum p0i is proportional to the product of i, ii and iii : [f(t, xi , pi )∆3p] [f(t, xi, q i)∆3q] [∆3x ∆3p0 ∆3 q 0]

(45)

Let us write the coefficient of proportionality in the form (mc)4 p ( − det g)3 w(t, xi; pi , q i|p0i , q 0i) p0 q0p00 q00 17

(46)

This expression defines the function w(t, xi ; pi , q i | p0i q 0i), usually referred to as the transition rate. Adding this proportionality factor to (45), we obtain the following expression for the total number of particles lost, due to collisions, of particles with coordinates within the volume element ∆3x mc f(pi )f(q i )w(t, xi; pi , q i | p0i q 0i )∆3Vq ∆3 Vp0 ∆3 Vq0 ∆3 x∆3p p0

(47)

where ∆3 Vq := mc

p

− det g

∆3 q q0

(48)

and ∆3Vp0 := mc

p

− det g

∆3p0 , p00

∆3Vq0 := mc

p

− det g

∆3 q 0 , q00

(49)

are invariant volume elements in momentum space [cf. (169)]. In passing, we note that the factor mc/p0 (pi ), which results in the factor p0 (pi )/mc in the left-hand side of the Boltzmann-equation (55) with contravariant momenta, has been added here to make the analogy with the corresponding Boltzmann-equation (80) with covariant momenta as close as possible. In the latter equation a factor p0 (pi )/mc is added, and it is precisely this factor which makes that the streaming term of the Boltzmann-equation, i.e., the left-hand side of (80), becomes a scalar [see article II [3], eqs. (??) and (??)]. This explains the addition of the otherwise mysterious factor mc/p0 in the eq. (47). In an arbitrary spacetime, the transition rate w(t, xi ; pi , q i | p0i q 0i) depends not only on the initial momenta (pi , q i ) and the final momenta (p0i , q 0i) of the colliding particles, but also on the location (t, xi) of the collision in spacetime. Only in a flat spacetime and with respect to an inertial system, the proportionality constant w would be independent of t and xi . The total number of particles lost is obtained by integrating this number with respect to all possible values of q i, p0i and q 0i. In this way we find 18

∆clost

mc 3 = ∆ x ∆3 p p0

Z

f(pi )f(q i )w(t, xi; pi , q i | p0i , q 0i) d3 Vq d3 Vp0 d3 Vq0 (50)

Similarly, the total number of particles gained in the range ∆3x is given by mc 3 ∆cgained = ∆ x ∆3 p p0

Z

f(p0i )f(q 0i)w(t, xi ; p0i, q 0i | pi q i) d3 Vq d3 Vp0 d3 Vq0

(51)

Hence, the net gain ∆c(f, f) = ∆cgained − ∆clost

(52)

occurring in the right-hand side (44) is ∆c(f, f) =

mc 3 ∆ x ∆3p c(f, f) p0

(53)

where

c(f, f) :=

Z

[f(t, xi, p0i )f(t, xi , q 0i)w(t, xi; p0i , q 0i|pi , q i )

−f(t, xi, pi )f(t, xi, q i )w(t, xi; pi , q i |p0i , q 0i)] d3 Vq d3 Vp0 d3 Vq0 .

(54)

Inserting (53) into (44) we obtain      p0 ∂ ∂ dxi ∂ dpi (fr) + i rf + i rf = c(f, f) mc ∂t ∂x dt ∂p dt

(55)

where f = f(t, xi, pi ) is a density in (xi, pi )-space and p0 (pi ) is the function (19) of the contravariant momentum pi . The function r(t, xi , pi ) is given by (32). 19

Equation (55) is the general relativistic Boltzmann-equation for the distribution function when the contravariant momentum is used as a variable. Usually, the left-hand side of the Boltzmann-equation is referred to as the streaming term, while the right-hand side is called the collision term. Disappointingly, the streaming term does not have its usual appearance: it contains the combination f(t, xi , pi )r(t, xi , pi ), with the function r(t, xi, pi ) given by (32), instead of f(t, xi , pi ) only, as one would expect. This leads us to abandon the formalism with pi altogether, and to switch to a formalism for f∗ (t, xi, pi ), hoping for a closer analogy with the special relativistic and non-relativistic results. This hope will be fulfilled, as we will see in the next section. Surprisingly, however, at the very end of article II, the sequel to this article, we will find, as a by-product of the formalism based on f∗ (t, xi, pi ) developed in the present article, an equation for f(t, xi , pi ) with a streaming term in its usual form: see (II.??). From now on, we will often write f∗(pi ) and f(pi ) rather than f∗ (t, xi, pi ) and f(t, xi, pi ), respectively.

3.3

Streaming term — covariant momentum

With the help of (26), the zeroth component of the number density four-flow (29) can be written n0 (t, xi) :=

p0 f∗ (t, xi, pi ) mc

(56)

Since f∗ has the dimension of a number density in momentum-position space, n0 has the dimension of a number-density in momentum-position space. Let ∆3 x ⊂ Σµ (t) be volume elements of spacetime lying entirely in a leafs of time t of a foliation of spacetime. Furthermore, let us now choose the covariant components pi (i = 1, 2, 3) as momentum variables on the mass shell S∗, and let pi ∈ ∆3p∗ , where ∆3 p∗ ⊂ S∗ is some three-dimensional momentum interval of momenta belonging to the mass shell S∗ . Let N∆3 x∆3 p∗ (t) be the total number of particles, which, at time t, have positions xi and momenta pi in the ranges ∆3 x and ∆3 p∗ , respectively. With 20

respect to coordinates adapted to the foliation of spacetime as defined in appendix A.3.2, the total number N∆3 x∆3 p∗ (t) can be found by integrating the number-density (56) with respect to volume element d3 Σ0 of the threedimensional hypersurface Σ(t) of spacetime, and with respect to the volume element d3 Vp∗ of the three-dimensional mass shell S∗ defined by g µν pµ pν = mc2. Thus we get N∆3 x∆3 p∗ (t) =

Z

∆3 x×∆3 p∗

p0 f∗(t, xi , pi ) d3 Σ0 d3 Vp∗ mc

(57)

Using (128) and (180) we so find the expression N∆3 x∆3 p∗ (t) =

Z

f∗(t, xi , pi )d3 xd3 p∗ ∆3 x×∆3 p

(58)

∗

This expression is to be compared with (31). no factor r, since, √ Now, there is √ going from (57) to (58) the square roots − det g and 1/ − det g and the zero components p0 and 1/p0 have canceled. One way to find the increase of the number of particles lost (per unit of time) contained in ∆3x × ∆3p∗ , dE∆3 x∆3 p (t)/dt, is to differentiate (58) with respect to t. Since ∆3x is a volume element lying totally in Σ(t), where t is a constant, all spacetime points of ∆3x have the same time coordinate. Moreover, ∆3 p∗ is independent of t. Hence, dN∆3 x∆3 p∗ (t) = dt

Z

∆3 x×∆3 p∗

∂f∗(t, xi, pi ) 3 3 d x d p∗ ∂t

(59)

This is a first way to calculate the gain of number of particles lost due to the particle movements. If there are no collisions, there is a second way to calculate this gain of parricles. In case there are no collisions, there is no other possibility for a particle to get out of the position and momenta intervals ∆3x and ∆3 p∗ than by streaming out of them. Consequently, an alternative way to obtain dN∆3 x∆3 p∗ (t)/dt is to calculate what flows out and what flows in ∆3x and ∆3p∗ , respectively, and to add these two contributions. The total loss of particles per unit of time, J∆3 x (t), due to particles leaving 21

the position interval ∆3x is obtained by integrating the expression (56) — multiplied by the particle velocity dxi /dt— with respect to the two-dimensional covariant surface element d2 σ0i of the boundary of ∆3 x, and with respect to the three-dimensional covariant momentum interval d3 Vp∗ J∆3 x (t) =

Z

∂(∆3 x)×∆3 p∗

dxi 2 p0 f∗ (t, xi, pi ) d σ0i d3 Vp∗ mc dt

(60)

where ∂(∆3x) is the boundary (i.e., the two-dimensional surface) of ∆3x and d2 σ0i is given by (135). Similarly, the loss of particles per unit of time, J∆3 p∗ (t) due to particles leaving the momentum interval ∆3 p∗ is J∆3 p∗ (t) =

Z

∆3 x×∂(∆3 p∗ )

p0 dpi 2 i 3 f∗ (t, xi, pi ) d s∗ d Σ0 mc dt

(61)

where d3 Σ0 is a three-dimensional element of the three-dimensional surface Σ(t) of constant time given by (133), and d2 si∗ is a two-dimensional element of the surface of ∆3p∗ on the mass-shell (86) of covariant-momentum, given by (186). If there are no collisions, the increase equals minus the total decrease, so that we have dN∆3 x∆3 p∗ (t) = −J∆3 x (t) − J∆3 p∗ (t) dt

(62)

The integrals J∆3 x (t) and J∆3 p∗ (t) may both be rewritten. For the first of these, J∆3 x (t), we have, using (138) and (180), J∆3 x (t) =

Z

∂(∆3 x)×∆3 p∗

√

1 − det g

p

g 00

f∗ (t, xi, pi )

dxi 2 d Ai d3 p∗ dt

(63)

Using Stokes’s theorem, eq. (194), we get

J∆3 x (t) =

Z

∆3 x×∆3 p∗

" √

i

∂ dx − det gΣ p f∗ (t, xi, pi ) √ i 00 ∂x dt − det g g 22

#

d3 x d3 p∗ (64)

where gΣ (t, x1, x2, x3 ) is the metric of the three-dimensional surface Σ(t) at the point (t, x1, x2 .x3) around which the interval of ∆3x is situated. With the help of eq. (132) we so find for J∆3 x(t): J∆3 x (t) =

Z

∆3 x×∆3 p∗

  ∂ dxi i f∗(t, x , pi ) d3 x d3 p∗ i ∂x dt

(65)

For the second integral, J∆3 p∗ (t), eq. (61), we find, with (128), J∆3 p∗ (t) =

Z

∆3 x×∂(∆3 p∗ )

dpi p p0 f∗ (t, xi, pi ) − det g d2 si∗ d3 x mc dt

(66)

where d2 si∗ is the surface element on the covariant mass-shell (10), and is given by (186). Using Stokes’s theorem, eq. (194), in pi -space, we get

J∆3 p∗ (t) =

Z

∆3 x×∆3 p∗

or, using the relation

  p ∂ p p0 dpi − det gS∗ f∗ − det g d3 p∗ d3 x (67) ∂pi mc dt

p √ − det gS∗ = mc/p0 − det g, [eq. (181)],

J∆3 p∗ (t) =

Z

∆3 x×∆3 p∗

  dpi ∂ i f∗(t, x , pi ) d3 p∗ d3 x ∂pi dt

(68)

We now substitute (58), (65) and (68) into (62) to obtain Z

∆3 x×∆3 p∗

3.4



∂f∗ ∂ + i ∂t ∂x



dxi f∗ dt



∂ + ∂pi

  dpi f∗ d3 xd3 p∗ = 0 dt

(69)

Boltzmann equation — covariant momentum

In case ∆3x and ∆3p∗ are small but finite volumes, defined by (xi, xi + ∆xi ) and (pi , pi + ∆pi ) in position- and momentum-space, the equation (69) can be approximated by 23



 ∂f∗ ∂ dxi ∂ dpi + i (f∗ )+ (f∗ ) ∆3 x∆3p∗ = 0 ∂t ∂x dt ∂pi dt

(70)

where ∆3x = ∆x1∆x2∆x3 and ∆3p∗ = ∆p1 ∆p2 ∆p3. This equation gives, in the case that there are no collisions, the increase of the number of particles, per unit time, due to particles with momenta in the range (pi , pi + ∆pi ) that arrive in (xi , xi + ∆xi ). If we want to take collisions into account, we may replace this equation by 

∂ ∂f∗ + i ∂t ∂x



dxi f∗ dt



∂ + ∂pi



dpi f∗ dt



∆3x∆3p = ∆c∗(f∗ , f∗ )

(71)

The equation (71) is to be considered as the defining equation for ∆c∗(f∗ , f∗). The expression for ∆c∗(f∗ , f∗ ) corresponds to the increase of the number of particles due to collisions taking place in ∆3 x that change the number of particles with momenta in ∆3p∗ . To obtain an expression for ∆c∗ (f∗ , f∗), let us consider a collision pi + qi → p0i + qi0 between two particles with initial covariant four-momenta pi and qi and final covariant four-momenta p0i and qi0 . According to the hypothesis of molecular chaos, the average number per unit of time of such collisions in a volume ∆3 x of the foliation at a leaf Σ of spacetime around the point (x1, x2 , x3) is proportional to small contravariant volume intervals ∆3p, not proportional to covariant volume elements ∆3 p∗ . However, covariant volume elements ∆3p = dp1 dp2 dp3 , which are integrals over small but finite volume elements are approximately proportional to the corresponding contravariant volume elements ∆3 p∗ = dp1 dp2 dp3 . We have

3

∆ p∗ =

Z

d3 p∗ ∆3 p

∗

p0 det(−g)d3 p ∆3 p p0 Z p0 ≈ det(−g) d3 p p0 3 ∆ p p0 = det(−g)∆3 p p0 =

Z

24

(72)

where we used (182). Consequently, ∆3p∗ is proportional to ∆3p and we can proceed in the same way as in the case of the contravariant momentum, and use the postulate of molecular chaos with covariant rather than contravariant momentum intervals. We so find that the net average loss of particles per unit time due to particles with momentum pi in the volume ∆3 x which disappear to become a particle with momentum p0i because of a collision with a particle with momentum qi is proportional to [∆3p∗ f∗ (t, xi, pi )] [∆3q∗f∗ (t, xi, qi )] [∆3x ∆3p0∗ ∆3q∗0 ]

(73)

Let us write the proportionality coefficient in the form (mc)4 1 √ w∗ (t, xi; pi , qi | p0i qi0 ) p0 q 0p00 q 00 ( − det g)3

(74)

The loss term then can be written mc 3 ∆ p∗ ∆3xf∗ (pi )f∗ (qi )w∗(t, xi; pi , qi | p0i , qi0 )∆3Vq∗ ∆3 Vp0 ∗ ∆3Vq0 ∗ 0 p

(75)

In deriving this expression we used the analogues of (180) for q0, p00 and q00 . The total number of particles lost is found by integrating with respect to qi, p0i and qi0 mc 3 ∆ p∗ ∆3 x 0 p

Z

f∗ (pi )f∗ (qi )w∗(t, xi; pi , qi | p0i , qi0)d3 Vq∗ d3 Vp0 ∗d3 Vq0 ∗

(76)

Similarly, the total number of particles gained is mc 3 ∆ p∗ ∆3 x 0 p

Z

f∗ (p0i )f∗ (qi0 )w∗(t, xi; p0i , qi0 | pi , qi)d3 Vq∗ d3 Vp0 ∗d3 Vq0 ∗

Hence, the net number of particles gained is 25

(77)

∆c∗(f∗ , f∗ ) =

mc 3 3 ∆ x∆ p∗ c∗ (f∗, f∗ ) p0

(78)

where

c∗ (f∗, f∗ ) :=

Z

(f∗ (t, xi, p0i )f∗ (t, xi, qi0 )w∗(t, xi ; p0i , qi0 |pi , qi )

−f∗ (t, xi, pi )f∗ (t, xi, qi )w∗(t, xi ; pi , qi |p0i , q 0i )d3 Vq∗ d3 Vp0∗ d3 Vq∗0

(79)

Inserting (78) into (71) we obtain, after dividing by (mc/p0 )∆3x∆3 p∗ ,      p0 ∂ ∂ dxi ∂ dpi f∗ + i f∗ + f∗ = c∗ (f∗, f∗ ) mc ∂t ∂x dt ∂pi dt

(80)

Let us stress that in this equation p0 = p0 (pi ), dxi /dt and dpi /dt are all to be considered as functions of t, x1, x2, x3 , p1 , p2 , p3 , like the distribution function f∗ .

APPENDICES In these appendices we make explicit some elements of differential geometry that are indispensable for a good understanding of this article. Moreover, we formulate Stokes theorem in a form useful in general relativity, and we give some formulae useful when working with spaces embedded in higher dimensional spaces. As general references we mention [13] and [11].

A

Volume and surface elements

In this appendix we will treat, as far as possible, spacetime and four-momentumspace on an equal footing . In particular, M will denote either the fourdimensional manifold of spacetime or the four-dimensional manifold of fourmomenta at a given point of spacetime. Let X µ (µ = 0, 1, 2, 3) be a coordinate system on M. Thus, the X µ ’s (with upper index) denote either the usual 26

spacetime coordinates xµ (i.e., X µ := xµ ) or four-momentum components with upper or lower indices, i.e., X µ := pµ or X µ := pµ (no error here!). In what follows, we will need to introduce various three-dimensional submanifolds of M. Let N be such a submanifold. In practice, N will be either the ‘three-space’ of spacetime M defined, in a given coordinate system, by the equation: x0 = a

(81)

where a is a constant, or the three-dimensional mass-shell in momentumspace at a given spacetime point x = (ct, x1, x2, x3 ), or an equation relating the four momentum variables. With respect to the contravariant momentum variables pµ , the mass-shell N in the four-dimensional space M of momenta p = (p0 , p1 , p2 , p3 ) is defined by the equation: gµν (x)pµ pν = m2 c2

(82)

with gµν (x) the metric of spacetime at point x. Sometimes, however, it is necessary or convenient to use the covariant pµ rather than the contravariant pµ . The relation (82) defining the mass-shell N of the four-dimensional covariant momentum space p∗ = (p0 , p1 , p2 , p3 ) at the spacetime point x then reads: g µν (x)pµ pν = m2c2 .

(83)

Hence, in this appendix, the points of the subspace N are related by one of the equations

x0 − a = 0 gµν (x)pµ pν − m2 c2 = 0 g µν (x)pµ pν − m2 c2 = 0

27

(84) (85) (86)

depending on whether the X µ ’s stand for xµ , pµ or pµ . In all three cases, one may use the X 1 , X 2 and X 3 as the free coordinates on N, and consider the coordinate X 0 as a given function F of the coordinates on N, i.e., X 0 = F (X 1 , X 2 , X 3 ).

(87)

In the cases (85) and (86), where M is the momentum four-space, the function F will depend, through the metric g, on the point x in spacetime.

A.1

Tangent vectors

Let Q be an arbitrary point of the manifold M introduced in the preceding section, and let v(Q) be defined according to: ∂ v(Q) := v (Q) ∂X µ Q µ

(88)

This newly defined quantity is called a tangent vector (or vector, for short) to the manifold M at the point Q. The four functions v µ(Q) are called the components of v(Q) with respect to the basis formed by the four derivatives ∂/∂X µ at Q. Note that a tangent vector is an operator acting on functions defined on M. Now, let Xvµ (λ) be a curve in M which passes through the point Q. We choose the parameterization on the curve in such a way the coordinates of the point Q are found by setting λ = 0: Q = (Xv0 (0), Xv1 (0), Xv2 (0), Xv3 (0)). Suppose that the curve is such that: dXvµ (λ) v (Q) = dλ λ=0 µ

(89)

The quantity (89) then is a tangent vector to the curve Xvµ (λ) in the classical sense: it is not an operator. Upon substitution of (89) into (88), we get: dXvµ (λ) ∂ v(Q) = dλ ∂X µ λ=0 28

(90)

This shows that, in any coordinate system (X 0 , X 1 , X 2 , X 3 ), the components of the tangent vector (88) can be expressed as derivatives of the coordinates along any curve which passes through Q and which is tangent, in the usual sense of the word, to v µ (Q) at Q. The square of a tangent vector v(Q) with components v µ (Q) is defined by: v 2(Q) = gµν (Q)v µ(Q)v ν (Q),

(91)

where gµν is a metric on the manifold M. Let N be, as before, a three-dimensional submanifold of M and suppose Q ∈ N. Suppose also that the curve X µ (λ) lies entirely in N. In view of (87) one has: Xv0 (λ) = F (Xv1(λ), Xv2 (λ), Xv3 (λ)) (92) i.e., for each value of λ, the zeroth coordinate Xv0 (λ) on the curve Xvµ (λ) is completely determined by its three spatial coordinates Xv1 (λ), Xv2 (λ) and Xv3 (λ). By inserting (92) into (89) with µ = 0 we find: ∂F j v (Q) = v (Q), ∂X j Q 0

(93)

where we first used the chain rule for differentiation of composite functions, and thereupon replaced (∂X j /∂λ)|λ=0 by v j (Q). From (87) we find, with the help of the implicit function theorem, X  ∂F −1 ∂ ∂ = . ∂X 0 ∂X i ∂X i i

(94)

Thus a vector v(Q), tangent at point Q to a curve which lies entirely in the submanifold N, may be rewritten as a linear combination of the three operators ∂/∂X i only. Inserting (93) and (94) into (88) we find v(Q) =

X

i vN (Q)

i

29

∂ ∂X i

(95)

where we abbreviated: i vN (Q)

i

:= v (Q) +



∂F ∂X i

−1 X j

∂F j v (Q) (i = 1, 2, 3) ∂X j

(96)

The above results are valid for an arbitrary three-dimensional submanifold N of a four-dimensional manifold M. In this article, we have used these results in two different situations. Firstly, have taken M to be the spacetime manifold, in which case N has been the submanifold of constant time-coordinate (84), and, secondly, we have taken M to be a four-dimensional momentumspace, in which case N is a three-dimensional mass-shell. In the latter case, N = Nx is x-dependent since the mass-shell is defined by the x-dependent equation (85) or (86). However, we will not indicate the x-dependence of the mass-shell in what follows, in order to make it possible to denote by N both a constant time-coordinate surface in spacetime, or a mass-shell in four-momentum space. The three relations (96) can be inverted to give: 1 2 3 , vN ) (i = 1, 2, 3) , vN v i = v i(vN

(97)

Using the expressions (93) and (97), one can express the four components i of v µ in terms of the three components vN . Inserting the result into the 2 expression (91) for v (Q), one can express the square of v µ(Q) entirely in terms of v N (Q) and obtain an expression of the form: j i v 2(Q) = (gN )ij (Q)vN (Q)vN (Q)

(98)

As noted above, if N is a constant time-coordinate submanifold of spacetime, Q represents a point x in spacetime. If, however, N is the mass-shell, Q represents p and x, since, then, the equation defining N depends on x via the metric g. The quantities (gN )ij define a 3 × 3 matrix, which we will denote by gN . The matrix gN defines a metric on the submanifold N of M. This metric is called the metric induced by N on M. This induced metric plays a crucial role in the derivation of Boltzmann equation. In conclusion, N can be considered both as a submanifold of M and as a 30

manifold in its own right, endowed with the metric gN . We will say more about this in appendix C.

A.2

Forms and exterior product

To derive properly the general relativistic Boltzmann equation, one needs several types of volume elements, and their relations. Volume elements can be expressed in terms of exterior products of one-forms, which explains why forms and exterior products are interesting in the context of this article. One-forms Let M be an arbitrary manifold, and let X µ , µ = (0, 1, 2, 3) be coordinates on (a part of) M. Let us define an operator, dX µ , that acts linearly on an arbitrary tangent vector v, the action being determined by the effect of dX µ on the basis formed by the four vectors (derivatives) ∂/∂X ν : dX

µ



∂ ∂X ν



= δ µν .

(99)

Hence we have, using the linearity of dX µ :

µ

dX (v) = dX

µ

= v µ.



∂ v ∂X ν ν



(100)

In other words, the linear operator dX µ extracts the component v µ from the tangent vector v = v ν ∂/∂X ν . Let now a(X) be an arbitrary tangent vector field on M. This field can be represented by its contravariant components aµ (X) with respect to the coordinates X. From the contravariant components aµ (X) one finds the covariant components of a in the usual way: aµ (X) = gµν (X)aν (X)

31

(101)

These components can be used to define the scalar operator field aµ (X)dX µ . Such operators are called one-forms or, alternatively, cotangent vectors. By construction, the one-forms dX µ form a basis of the vector-space of oneforms. Two-forms The next step is to introduce two-forms. Let us define a bilinear form, denoted by dX µ ∧ dX ν , acting on an ordered set (v, w) of two arbitrary tangent vectors v and w by:

dX µ ∧ dX ν (v, w) = dX µ (v)dX ν (w) − dX ν (v)dX µ (w) = v µ wν − v ν wµ X = (−1)σ(P ) v P (µ)wP (ν) ,

(102)

P

where the summation is over the two possible permutations of the two indices µ and ν. The symbol σ(P ) stands for the signature of the permutation: σ(P ) is 0 if the permutation is even and +1 if the permutation is odd. The bilinear operator dX µ ∧ dX ν acting on the two tangent vectors v and w yields an antisymmetric tensor, the components of which are related to the area of the surface spanned by v and w. For example, let v µ = (0, a, 0, 0) and wµ = (0, 0, b, 0), where a and b are (strictly) positive real numbers. The antisymmetric tensor dX µ ∧ dX ν (v, w) then has essentially only one nonvanishing component, namely dX 1 ∧ dX 2 (v, w) = ab − 0

(103)

Clearly, the non-zero component of dX µ ∧ dX ν (v, w) is equal to the area of the surface spanned by v and w. The other nonzero component is found by permutation of 1 and 2: dX 2 ∧ dX 1 (v, w) = 0 − ab.

The bilinear operator dX µ ∧ dX ν is called the wedge product or exterior product of the two one-forms dX µ and dX ν . Given the purely contravariant components aµν (X) of an arbitrary tensor field, one can construct the purely covariant components of the same field: aµν (X) = gµα (X)gνβ (X)aαβ (X) 32

(104)

and define the scalar two-form corresponding to a as the linear combination aµν (X)dX µ ∧dX ν . The set of all two-forms is a vector space and the collection {(dX µ ∧ dX ν )|µ < ν; µ, ν = 0, 1, 2, 3} is, by construction, a basis of this vector-space. Three-forms The exterior product dX µ ∧ dX ν ∧ dX ρ is a multi-linear operator acting on triplets of tangent vectors and is defined by: X σ(P ) dX µ ∧ dX ν ∧ dX ρ (v, w, u) = (−1) dX P (µ) (v)dX P (ν) (w)dX P (ρ) (u) P

=

X

(−1)σ(P ) v P (µ)wP (ν) uP (ρ),

(105)

P

where the summation is over all 3! = 6 permutations of the three indices µ, ν and ρ. The multi-linear operator dX µ ∧ dX ν ∧ dX ρ applied to three four-vectors v, w and u yields a completely antisymmetric tensor that is related to the volume of the parallelepiped spanned by v, w and u. For example, with v and w chosen as above and uµ = (0, 0, 0, c), we have: dX 1 ∧ dX 2 ∧ dX 3 (v, w, u) = abc.

(106)

Note, that this component together with those obtained by permutation of 1,2 and 3 in this expression, are the only non-vanishing components of the tensor under consideration. Four-forms A basis a four-forms is given by the single exterior product dX µ ∧ dX ν ∧ dX ρ ∧ dX σ , defined as that multi-linear form which acts on four arbitrary 4-vectors according to: dX µ ∧ dX ν ∧ dX ρ ∧ dX σ (v, w, u, s) = X = (−1)σ(P ) dX P (µ) (v) dX P (ν) (w)dX P (ρ) (u)dX P (σ) (s) P

=

X

σ(P ) P (µ)

(−1)

v

wP (ν) uP (ρ) sP (σ)

P

33

(107)

The linear operator dX µ ∧ dX ν ∧ dX ρ ∧ dX σ applied to the four 4-vectors v, w, u and s with v, w and u as given above and sµ = (d, 0, 0, 0) is related to the four-volume of the four dimensional parallelepiped spanned by these vectors in four-space: dX 1 ∧ dX 2 ∧ dX 3 ∧ dX 0 (v, w, u, s) = abcd.

(108)

The exterior product of five one-forms is a five-form. Like all five-forms, it is identically zero since it is anti-symmetric in any two of its indices and any component of a five-form has at least two equal indices, since they are picked from the set {0, 1, 2, 3} of only four elements. Similarly, all forms of higher order vanish. Up to now, we have used the Greek indices µ, ν, ... which run from 0 to 3; in other words, the manifolds which we considered were implicitly assumed to be four-dimensional. In what follows, we will also consider three-dimensional manifolds: the three-dimensional physical space, as defined above, and the three-dimensional mass-shell in momentum space. A similar formulation then applies. It should be kept in mind that an n-form vanishes when n is (strictly) larger than the dimension of the manifold on which it is defined.

A.3

Volume elements

Let xµ be four coordinates for spacetime, e.g., the time and spherical coordinates t, r, θ and φ. Choose a particular order for these coordinates, e.g., x0 = t, x1 = r, x2 = θ and x3 = φ. Let, in this particular coordinate system εµνρσ , related to the chosen coordinates (whichever they are) be the permutation symbol in four dimensions, and let ε0123 = +1. By definition, εµνρσ takes the values 0 and 1 in any coordinate system. As is well-known, in curved spacetime the permutation symbol εµνρσ is not a tensor, but the combination Eµνρσ :=

p

− det g εµνρσ

34

(109)

is. In (109), det g stands for the determinant of the 4 × 4-matrix of the covariant components gµν of the metric tensor on spacetime with respect to the coordinates xµ . In order to show that Eµνρσ transforms like a tensor, we must show that with respect to a different set of coordinates x0 one has 0 Eµνρσ =

p

− det g 0 εµνρσ

(110)

0 where g 0 is the determinant of the metric gµν . Note that in (109) and (110) the same permutation symbol appears, which takes, by definition, the same values 0 and 1 with respect to any coordinate system.

To prove (110), we first recall that, with respect to a different set of coordinates, x0µ , one has for the components of the metric: 0 gµν (x)

∂xρ ∂xσ gρσ (x) = ∂x0µ ∂x0ν

(111)

Taking the determinant at both sides, we find   2 ∂x det g = det det g ∂x0 0

(112)

where (∂x/∂x0) stands for the matrix with components ∂xµ/∂x0ν . The latter matrix is called the Jacobian matrix of the transformation. The Jacobian matrix is related to the transformation from the new coordinates x0 to the old ones x = x(x0). One has: det



∂x ∂x0



= εανρσ

∂xα ∂xβ ∂xγ ∂xδ ∂x00 ∂x01 ∂x02 ∂x03

(113)

Secondly, let us suppose that, as we claim, Eµνρσ , defined by eq. (109), transforms as a tensor with lower indices, and check whether this is consistent with its definition. If Eµνρσ would transform like a tensor, it would read, with respect to the new coordinates x0,

35

0 Eµνρσ :=

∂xα ∂xβ ∂xγ ∂xδ p − det g εαβγσ ∂x0µ ∂x0ν ∂x0ρ ∂x0σ

(114)

Taking in this expression µ = 0, ν = 1, ρ = 2, σ = 3, we find, using (113): 0 E0123



∂x = det ∂x0



p

− det g

(115)

or, with (112), 0 E0123 =

p

− det g 0

(116)

Since (114) is completely antisymmetric with respect to the indices µ,ν,ρ and σ, we indeed must have (110). Hence, the supposition that Eµνρσ transforms like a tensor with lower indices is justified. A.3.1

Four-dimensional invariant volume form on spacetime

With the help of the tensor Eµνρσ we may construct, on the basis consisting of the exterior products dxµ ∧ dxν ∧ dxρ ∧ dxσ , a new scalar form: d4 Vx :=

1 Eµνρσ dxµ ∧ dxν ∧ dxρ ∧ dxσ 4!

(117)

It is a form, since it is a sum of four-forms; it is a scalar since all tensor indices of the tensors Eµνρσ and dxµ ∧ dxν ∧ dxρ ∧ dxσ are contracted. By substituting (109) into this definition we find d4 Vx =

1 p − det g εµνρσ dxµ ∧ dxν ∧ dxρ ∧ dxσ 4!

(118)

Defining d4 x :=

1 εµνρσ dxµ ∧ dxν ∧ dxρ ∧ dxσ 4! 36

(119)

we find for the invariant volume element of spacetime d4 Vx =

p

− det g d4 x

(120)

The four-form d4 Vx is invariant, whereas the four-form is d4 x is not invariant under coordinate transformations. One therefore might call invariant volume elements like (120) that contain a square root of the determinant ‘covariant volume elements’, to distinguish them from volume elements like (119), the usual volume element in flat spaces. Now, let Ωx be some four-dimensional domain of spacetime, and let T (x, p) be a suitably smooth tensor field of arbitrary rank, defined on Ωx and on four-momentum space. It is then possible to define, in a mathematically rigorous manner, the integral over a four-form Z

T (x, p) d4 Vx

(121)

Ωx

where d4 Vx is given by eq. (118), or, equivalently, eq. (120). Let us calculate (121) in case Ωx contains a fixed point a of spacetime, and T is given by p G(x)δ 4 (x − a)/ − det g

(122)

where δ 4 is the four-dimensional Dirac-delta function and G(x) is an arbitrary scalar function of spacetime x. We then find Z

p δ 4(x − a) √ G(x) − det g d4 x = G(a) − det g

(123)

√ Since G(x) and − det g√d4 x are a scalars, and G(x) is arbitrary, the combination G(x)δ 4(x − a)/ − det g of the integrand is a scalar. Hence, since G(x) is a scalar δ 4(x − a) √ − det g is a scalar. This fact has been used in section 2. 37

(124)

Volume elements of three-dimensional subspaces Σ(t) of spacetime

A.3.2

A k-dimensional local foliation of an n-dimensional manifold M is a family of k-dimensional manifolds N with the property that there is, around each point p ∈ M, a coordinate neighborhood U with coordinates (x1 , x2 . . . , xn ), such that the coordinates of N∩M are of the form (x1, x2, . . . , xk , ak+1, ak+2 , . . . , an ) where ak+1 , ak+2 ,. . ., an are constants. These coordinates on U are called adapted to the foliation. Let M be a region of spacetime described with coordinates {x0, xi }. Consider the three-dimensional leaves Σ(t) defined by x0 = ct, with t ∈ R a constant. The union of these leaves constitutes a local foliation of spacetime. The coordinates {x0, xi } are adapted to the foliation.

We will now suppose that the spacetime has a global foliation. In other words, we will suppose that there exist a family of three-dimensional spacelike submanifolds Σ(t) of spacetime, each of which is characterized by a fixed value of t, such that every point (t, xi) of spacetime belongs to one and only one member Σ(t) of the family. Three-dimensional surface elements in three-space Σ(t)

A.3.3

Let us define a three-dimensional surface element on a leaf Σ(t) of this foliation in a way analogous to eq. (117): d3 Σµ :=

1 Eµνρσ dxν ∧ dxρ ∧ dxσ 3!

(125)

In a system of coordinates adapted to the foliation, one effectively has dx0 = 0, so that for a tensor T (x, p) one has Z

3

T (x, p)d Σµ = Vx

Z

T (x, p) Vx

1p − det g εµijk dxi ∧ dxj ∧ dxk (126) 3!

38

Since εµijk = δµ0 εijk , we have Z

Vx

3

T (x, p)d Σµ =

δµ0

Z

T (x, p) Vx

1p − det g εijk dxi ∧ dxj ∧ dxk (127) 3!

Hence, in a system adapted to the coordinates (125) reduces to d3 Σµ = δµ0 where d3 x :=

p

− det g d3 x

1 εijk dxi ∧ dxj ∧ dxk 3!

(128)

(129)

In a coordinate system adapted to the foliation one has (gΣ )ij = gij

(130)

where gΣ is the metric induced by g on Σ(t). The element g 00 is, by definition, the 00-component of the matrix inverse to the matrix formed by the components of gµν . Hence, g 00 =

det gij det gµν

(131)

g 00 =

det gΣ det g

(132)

or

Upon substituting this equation into (128) we find 1 p d3 Σµ = δµ0 p − det gΣ d3 x 00 g 39

(133)

If, on the other hand, one considers Σ(t) as a three-dimensional manifold in its own right, i.e., not as an embedded space, one can use the analogue of (118) in three dimensions: 1 p − det gΣ εijk dxi ∧ dxj ∧ dxk 3! p = − det gΣ d3 x

d3 Vx =

(134)

Note, that (133) with (129) is a form-valued vector, whereas (134) is a formvalued scalar. A.3.4

Two-dimensional surface elements in three-space Σ(t)

Let A be a two-dimensional surface totally contained in one of the threedimensional submanifolds constituting the preceding foliation, say Σ(t). The two-dimensional counterpart of (125), 1 Eµνρσ dxρ ∧ dxσ 2! 1 p − det g εµνρσ dxρ ∧ dxσ = 2!

d2 σµν :=

(135)

defines a first surface element on A. The only non-vanishing components of d2 σµν are the three components d2 σ0i and d2 σi0 . Using (132) we find from (135) d2 σ0i =

1 1 p p − det gΣ εijk dxj ∧ dxk 00 2! g

(136)

Using d2 σµν amounts to considering the surface A as a two-dimensional submanifold of the four-dimensional spacetime manifold. But A is also, by definition, a two-dimensional submanifold of a three-dimensional space-like manifold Σ(t). From that point of view, the covariant surface element is the two-dimensional version of (134): 40

d2 Ai =

1 p − det gΣ εijk dxj ∧ dxk 2!

(137)

Hence, 1 d2 Ai d2 σ0i = p 00 g

(138)

This result has been used when we derived the expression (37) from (34).

A.4

Volume elements of four-dimensional momentum space

In this section, we will distinguish between volume elements related to the contravariant components pµ and to the covariant components pµ of the momentum four-vector. A.4.1

Contravariant components

In the same way as we introduced a scalar volume form (117), we may define a scalar volume form in the space of the contravariant components of the four momenta: d4 Vp =

1 Eµνρσ dpµ ∧ dpν ∧ dpρ ∧ dpσ 4!

(139)

or, equivalently, using (109), d4 Vp =

p

− det g d4 p

(140)

where d4 p :=

1 εµνρσ dpµ ∧ dpν ∧ dpρ ∧ dpσ 4!

41

(141)

In this equation, we treat the momentum-space (p0 , p1 , p2 , p3 ) as a flat √ fourdimensional manifold M, together with a function gµν (x). The factor − det g ensures that d4 Vp is a scalar with respect to coordinate transformations in spacetime. The quantity (141) is a four-form, since it is the sum of four-forms. In view of (141) one has Z

4

T (x, p) d Vp = Ωp

Z

Ωp

p T (x, p) − det g d4 p

(142)

where Ωp is a domain in the space of (p0 , p1 , p2 , p3 ). Using the same argument as used to obtain (124), we now find that δ 4 (p − a) √ − det g

(143)

where a is some constant momentum four-vector, is a scalar. This fact has been used in appendix D, where we prove the scalar character of the distribution function f defined by (27). A.4.2

Covariant components

Similar to the definition of d4 Vp we may define d4 Vp∗ :=

1 µνρσ E dpµ ∧ dpν ∧ dpρ ∧ dpσ 4!

(144)

where the contravariant components of the tensor E are obtained by using expression (109) for the covariant components: E µνρσ = g µα g νβ g ργ g σδ

p

− det g εαβγδ

(145)

In particular we have E 0123 = det g −1

42


p

− det g

(146)

since the matrix formed from the contravariant components g µν of the metric is, by definition, the inverse of the matrix with covariant components gµν , i.e., g −1 is the matrix with components g µ . We note that E µνρσ is totally antisymmetric in all of its indices, and, hence, is proportional to the permutation symbol, which we will write, in this context, with upper indices: εµνρσ := +εµνρσ . We so find E µνρσ = det g −1


p

− det g εµνρσ

(147)

Furthermore, det (g −1 ) = 1/ det g. Hence, E µνρσ = √

1 εµνρσ − det g

(148)

Using this expression in (144) we find d4 Vp∗ = √

1 d4 p∗ − det g

(149)

where 1 µνρσ ε dpµ ∧ dpν ∧ dpρ ∧ dpσ 4!

(150)

1 µνρσ E dpµ ∧ dpν ∧ dpρ ∧ dpσ 4!

(151)

d4 p∗ := From (139) we find d4 Vp = or, equivalently,

d4 Vp = √

1 d4 p∗ − det g

(152)

Hence, comparing this equation and (149), we find d4 Vp∗ = d4 Vp 43

(153)

i.e., the invariant co- and contravariant volumes elements are equal. The counterpart of eq. (142) reads Z

4

T (x, p∗) d Vp∗ = Ωp

Z

Ωp

T (x, p∗) √

1 d4 p∗ − det g

(154)

In conclusion: in the invariant momentum volume-elements there should be √ incorporated a factor − det g if pµ ’s are used [see eq. (140)], and a factor √ 1/ − det g if pµ ’s are employed [see eq. (152]. Applying (154) to the case that T equals the delta-function δ 4(p∗ − a∗ ), we find that p − det gδ 4(p∗ − a∗)

(155)

is a scalar. This fact was used in section 2.3.1 in order to prove that the distribution function f∗ is a scalar.

A.5

Volumes on the mass shell

Let us consider again the four-space formed by the four contravariant components pµ of the momenta. Let S be a three-dimensional mass shell in this space, defined by (85). Note, that the mass shell depends on the spacetime point x. However, we will not indicate this fact explicitly in the notation for the mass shell S. A.5.1

Contravariant components

As was done in section A.3.2 for Σ(t), one may construct two different 3forms to be used in integrals with respect to the mass shell S, eq. (85). The first one is the analogue of (125) d3 Sµ :=

1 Eµνρσ dpν ∧ dpρ ∧ dpσ 3!

44

(156)

A second volume-element on the mass-shell can be obtained by treating S as a three-dimensional manifold in its own right. This can be done by using the metric g of spacetime as a metric in momentum-space and by considering the metric gS induced via g on the mass shell S as a metric in momentum-space. Let det gS be the determinant of the 3 × 3 matrix formed by the (covariant) components of gS . Then one may define, analogous to (134), a second 3-form by d3 Vp :=

1p − det gS εijk dpi ∧ dpj ∧ dpk 3!

(157)

On the mass shell S, the component p0 is a function of the component pi . Differentiating (85) with respect to pµ we have pµ dpµ = 0

(158)

or dp0 = −

pi i dp p0

(159)

Inserting (109) into (156) we find d3 Sµ =

pµ p − det g dp1 ∧ dp2 ∧ dp3 p0

(160)

as may easily be checked for µ = 0. In case µ = 1, 2, 3 one also must use (159) to check this equality. On the mass shell one may calculate

p

mc p − det g dp1 ∧ dp2 ∧ dp3 p0 1 mc p = − det g εijk dpi ∧ dpj ∧ dpk 3! p0

d3 Sµ d3 S µ =

45

(161)

Since the expression at the left-hand side is a scalar, the expression at the right-hand side must also be a scalar. Furthermore, we notice that the righthand sides of (157) and (161) have the same differential structure. Hence, they must be proportional, i.e., d3 Vp = a(pi )

p

d3 Sµ d3 S µ

(162)

where a(pi ) is a scalar and dimensionless function defined on the mass shell. The only scalar that can be formed on the mass shell out of the three components of pi is (gS )ij pi pj . But, according to (199) we have (gS )ij pi pj = gµν pµ pν

(163)

Since on the mass shell gµν pµ pν = m2c2 is a constant, the scalar a(pi ) must be a constant, a(pi ) = a say. By evaluating (161) in a local Lorentz frame, where gµν = diag(1, −1, −1, −1), we find d3 Vp = a mc where d3 p :=

d3 p p0

1 εijk dpi ∧ dpj ∧ dpk 3!

(164)

(165)

and where we used det g = −1 and p0 = p0 . The well-known invariant special relativistic expression for d3 Vp reads d3 Vp = mcd3 p/p0 . The latter expression and (164) coincide when a=1

(166)

p

(167)

Consequently, d3 Vp =

d3 Sµ d3 S µ

or, with (161), 46

d3 Vp =

1 mc p − det g εijk dpi ∧ dpj ∧ dpk 3! p0

(168)

which is the scalar volume-form that was used on the mass shell in the main part of this article. Note that p0 = g0µ pµ in the equations above; so, p0 depends on p1 , p2 , p3 and on the spacetime point x. Note that an alternative way of writing (168) is d3 Vp = mc

p

− det g

d3 p p0

(169)

where we used (165). With (157) this implies p

− det gS =

mc p − det g p0

(170)

the relation we used to obtain (41). A.5.2

Covariant components

The steps of the preceding section can be repeated to construct volume elements on the mass shell S∗ , eq. (86). One may define, analogous to (156), a second volume-element on the massshell by treating S∗ as a three-dimensional manifold in its own right. This can be done by using the metric g of the spacetime as a metric in momentumspace and by considering the metric gS∗ induced by g on the mass shell S∗ in momentum-space. Let det gS∗ be the determinant of the 3 × 3 matrix formed by the (covariant) components of gS∗ . Then one may define, analogous to (134), a second 3-form by d3 Vp∗ :=

1p − det gS∗ εijk dpi ∧ dpj ∧ dpk 3!

(171)

or d3 Vp∗ =

p

− det gS∗ d3 p∗ 47

(172)

where d3 p∗ :=

1 ijk ε dpi ∧ dpj ∧ dpk 3!

(173)

From (156) we obtain d3 S µ := g µν d3 Sν ;

(174)

Substituting (156) in this definition, we get 1 µνρσ E dpν ∧ dpρ ∧ dpσ 3!

(175)

pµ 1 √ dp1 ∧ dp2 ∧ dp3 0 p − det g

(176)

d3 S µ = Note that, similarly to (160), d3 S µ =

where we used (148) and dp0 = −(pi /p0 )dpi , the analogue of (159). Compared to the similar expression (160) all indices have interchanged their positions: down indices have become upper indices and, conversely, upper indices are now lower indices. Furthermore, that the square root now is in the denominator. Analogous to the way we arrived at the result (167) one can show that d3 Vp∗ =

p

d3 Sµ d3 S µ

(177)

Hence, d3 Vp∗ = d3 Vp

48

(178)

Substituting (176) into (177) we get d3 Vp∗ =

1 mc 1 √ εijk dpi ∧ dpj ∧ dpk 0 3! p − det g

(179)

An alternative way of writing this expression is 3

d Vp∗

1 d3 p∗ = mc √ − det g p0

(180)

with d3 p∗ defined in (173). This equation is the counterpart of the expression (169) when the covariant components of the momentum are used. Note that we encounter, in this expression, the contravariant component p0 rather then the covariant component p0 , and the covariant components pi instead of the contravariant components pi . From (172) and (180) we find the relation p 1 mc √ = − det gS∗ 0 p − det g

(181)

From (169), (178) and (179) we find the connection √

1 d3 p∗ p d3 p = − det g p0 − det g p0

(182)

This relation is used in section 2.3.2 to switch from the distribution function f∗ (pi ) to the distribution function f(pi ).

A.6 A.6.1

Surface element on the mass shell Contravariant momentum

Let A be a two-dimensional surface on the three-dimensional mass shell. The analogue of d2 Ai, eq. (137), reads: d2 si :=

1 p − det gA εijk dpj ∧ dpk 2! 49

(183)

Here, gA is the metric on S restricted to A. Hence, (gA )ij = (gS )ij

(184)

With the help this equation and (170), we find from (183) d2 si =

1 mc p − det g εijk dpj ∧ dpk 2! p0

(185)

Note that d2 si , eq. (183), is precisely the surface element figuring in Stokes theorem: compare eq. (192). The expression d2 si is the two-dimensional counterpart of d3 Vp , eq. (179). A.6.2

Covariant momentum

Let B be a two-dimensional surface on the three-dimensional mass shell S∗ of covariant momenta (86). The analogue of d2 Ai , eq. (137), now reads: d2 si∗ :=

1 p − det gB εijk dpj ∧ dpk 2!

(186)

Here, gB is the metric on S∗ restricted to B. Hence, (gB )ij = (gS∗ )ij

(187)

We so find d2 si∗ :=

1 p − det S∗ εijk dpj ∧ dpk 2!

(188)

or, using (181), d2 si∗ :=

1 mc 1 √ εijk dpj ∧ dpk 0 2! p − det g 50

(189)

Note that d2 si∗, eq. (186), is the surface element figuring in Stokes theorem for covariant momentum: compare eq. (192). The expression d2 si∗ is the two-dimensional counterpart of d3 Vp∗ , eq. (179). We used (183) and (186) when we applied Stokes’s theorem in momentum space to obtain (67) and (40), respectively. However, we did not need the explicit expressions (185) and (189). They are mentioned here for reasons of completeness only.

B

Stokes’s Theorem

Let N be an m-dimensional manifold on R, endowed with a metric g. A metric g defines a unique derivative, ∇, which, acting on a tensors T , yields another tensor, ∇T . In what follows, N will be either a three-dimensional submanifold Σ(t) of a foliation of the spacetime M with coordinates xµ , or the mass shell in the space of covariant momentum variables pµ or the space of contravariant momentum variables. Let us therefore restrict the discussion to the case m = 3, and denote the coordinates on N by X i . Hence, as usual by now, X i stands either for spatial coordinates xi on Σ(t) or for momenta variables pi or pi on N. In the first case, the g on N coincides with the three-dimensional metric gΣ induced on Σ(t) by the four-dimensional spacetime metric; in the second case, the metric g is the metric gS induced on the mass-shell by the spacetime metric, considered as a metric in four-dimensional momentum-space. Therefore, in this appendix, appendix B —and in this appendix only—, g does not stand for the spacetime metric, but for the metric that is relevant for the case which is considered. Let T i be a vector field defined on N. Let D be an arbitrary bounded domain in N, and ∂D its boundary (the surface of D). Then Stokes Theorem states that I Z i 2 T d Si = ∇i T i d3 V (190) ∂D

D

where d3 V is the covariant volume element d3 V =

1 p det g εijk dX i ∧ dX j ∧ dX k 3! 51

(191)

and where d2 Si =

1 p det g εijk dX j ∧ dX k 2!

(192)

An explicit expression for the covariant derivative of T i is [12] ∇i T i = √

1 ∂ p ( det g T i) det g ∂X i

(193)

Inserting (191) and (193) into (194) we find I

i

2

T d Si = ∂D

Z

D

∂ p ( det g T i ) d3 X ∂X i

(194)

where d3 X is given by d3 X :=

1 εijk dX i ∧ dX j ∧ dX k 3!

(195)

Equation (194) is Stokes’ theorem for a tensor that is a vector in a threedimensional space with metric g, in which coordinates X i have been chosen. The coordinates X i may be the space-coordinates xi , or the co- or contravariant coordinates of the momentum vector, pi or pi , respectively. Formula (194) has been used in order to obtain the final expressions for the streaming terms, in sections 3.1 and 3.3.

C

Induced metric

In appendix C.1 we discuss how a metric defined over some n-dimensional manifold M induces a metric on an (n − 1)-dimensional submanifold N of M. In the main part of this article, these considerations are applied to three-dimensional spacelike submanifolds Σ(t) of spacetime and to the mass shells S and S∗ , which are three-dimensional submanifolds of the momentum four-spaces of pµ and pµ , respectively. 52

C.1

Induced metric

Let us restrict our discussion to the case where the dimension n of the manifold M equals 4, which is the only case needed in the present article. Let X µ be a coordinate system on M and let gµν be the metric of M. Let N be a three-dimensional submanifold of M. In view of the applications considered in this article, we will suppose that, locally, N is defined by an equation of the form (87). We will choose the X i ’s as the coordinates on N. At any point Q(X) of N, one can consider two different tangent-spaces. The first is T M(Q), the four-dimensional tangent-space to M at Q, and the other is T N(Q), the three-dimensional tangent space to N at Q. Obviously, T N(Q) ⊂ T M(Q). Let V be a vector of T N(Q). Since this vector belongs also to T M(Q), on which the metric gµν is defined, one can calculate V 2 simply by: V 2 = gµν V µ V ν

(196)

But since V lies totally in T N(Q), its four components are not independent, and one may express one of them, V 0 say, in terms of the other components V i . One has V µ = dX µ /dt, hence, using (87) ∂X 0 dX i ∂X i dt ∂F Vi = ∂X i

V0 =

(197) (198)

The scalar V 2 can therefore be expressed as a quadratic form in the V i ’s and this defines on N a three-dimensional metric, gN : V 2 = gµν V µ V ν = (gN )ij V i V j

(199)

where (gN )ij := gij + gi0

∂X 0 ∂X 0 ∂X 0 ∂X 0 + g + g · 0j 00 ∂X j ∂X i ∂X i ∂X j 53

(200)

The metric (gN )ij is called the metric induced by the 4-dimensional metric gµν (which is defined on M) on the submanifold N. The induced metric gN can be used to lower the indices of the components of vectors tangent to the submanifold N. In particular, one can define a set of three quantities V˜i by: V˜i = (gN )ij V j

(201)

These quantities are not identical to the usual covariant components Vi of the 4-vector V . Indeed, the standard covariant components of V can be evaluated with the help of the original metric g and we have: Vi = gi0 V 0 + gij V j

(202)

or, with (198), Vi = (gij + gi0

∂F )V j · ∂X j

(203)

This makes it clear that Vi is in general different from V˜i . Let us remark that (203) cannot be used to define a three-dimensional metric on N, if only because of the fact that the quantity between brackets on the right-hand side is not symmetrical with respect to the indices i and j.

C.2

Spacelike subspaces of spacetime

Let us now relate the above to the main text of this article. In section A.3.2 we encountered three-dimensional spacelike submanifolds Σ(t) of the spacetime M. In the coordinate system used earlier, the equation of Σ(t) is simply x0 = ct. This means that if one chooses the X µ ’s to be coordinates in spacetime, the function X 0 then simply is a constant function, the constant being equal to ct. Any vector tangent to Σ(t) therefore has a vanishing timecomponent in these coordinates [see equation (198)]. Moreover, according to (200) the components (gN )ij of induced metric coincide with the components gij of the spacetime metric. Because of the choice of reference-frame (i.e. the 54

choice of coordinates), the quantities V˜i thus turn out to be identical to the natural covariant components Vi [see eq. (203)].

C.3

Mass shell

The three dimensional subspaces of momentum space defined by (85) or (86) are commonly called the mass shell. At a given point x of spacetime endowed with a metric gµν(x) , four-momentum space is nothing but the manifold R4. It is a flat manifold, since its metric does not depend on the point p in momentum-space. As a first possible choice of coordinates in momentum-space, let us consider the usual contravariant components pµ of p, which will now play the role of the X µ ’s of the main text (see section A). In contradistinction to the momentumspace itself, the mass-shell is not flat. The function F (X i ) figuring in eq. (87) is now given by p0 (pi ) (18), which makes it possible to calculate the three covariant components of a vector Vi on the mass-shell, given the three contravariant components V i on the mass shell, using (203) with X j replaced by pj .

D

Scalar character of the distribution function

In this appendix will now explicitly show that the density f(t, xi , pi ), defined by eq. (27) with respect to a certain system of coordinates (t, xi), is independent of this choice of coordinates. In other words, when one goes from one system of coordinates to another, the expression (27) with p0 and p0 given by (18) and (19) does not change, i.e., f(t, xi, pi ) is a general relativistic scalar. To carry out the proof, we introduce the auxiliary function: I(x, p) := 2θ(p0 )δ(p2 − m2 c2)f(t, xi , pi )

(204)

and we will show that is invariant under spacetime transformations, i.e., I is a general relativistic scalar. In the left-hand side of (204), x and p are 55

short-hand notations for the four spacetime coordinates xµ = (ct, xi) and the momentum 4-vector pµ = (p0 , pi ), θ stands for the unit step function and p2 := pµ pµ , while m is the mass of a particles of the gas. Since θ and δ are scalars, proving that I is a scalar is equivalent to showing that f is a scalar. In order to show that I is a scalar, we apply the identity (6) to the function H(p0 ) := gµν pµ pν − m2 c2, to obtain 2θ(p0 )δ(p2 − m2c2 ) =

1 δ(p0 − p0 (x, pi )), p0 (x, pi )

(205)

Inserting (27) and (205) into (204) we find: X I(x, p) = h r

1 1 δ (3)(xi − xir (t))δ (4)(p − pr (t))iav , − det g p0r (t)

(206)

where p0r (t) is defined by:

p0r (t) := p0 (t, xir (t), pri(t))

(207)

with the function p0 (t, xi , pi ) given by (9), in which the argument pi is given by pi = giµ pµ with p0 , in turn, given by (18). As a next step, we introduce an additional Dirac distribution δ(t − tr ) and, at the same time, add an integration with respect to tr . This yields:

I(x, p) = Z X 1 1 h δ(t − tr )δ (3)(xi − xir (tr ))δ (4)(p − pr (tr ))dtr iav 0 − det g p (t ) r r r

(208)

The zero component of the momentum of particle r is given by p0r = mdx0 /dτr , where τr is the proper time along the trajectory of particle r. Changing the integration variable in (208) from tr to τr , using (15), we find 56

1 I(x, p) = m

Z X (4) δ (x − xr (tr (τr ))) δ (4)(p − pr (tr (τr ))) √ √ h iav dτr − det g − det g r

(209)

where we used x0 = ct.

√ According to the√eqs. (122) and (143) the combinations δ 4(x − xr )/ − det g and δ 4(p − pr )/ − det g are scalars. Consequently, the auxiliary function I(x, p) is a scalar, implying, in view of (204), that f(t, xi , pi ) is a scalar. In a local Lorentz frame f(t, xi, pi ) given by the microscopic expression (27) can be interpreted as a macroscopic density in (xi, pi )-space, since then − deg g = 1 and p0 = p0 . Hence, the expression (27), which intuitively is not clear, is the general relativistic scalar that may be interpreted indeed as the one-particle distribution function f(pi ). In the main text, we have shown how this expression (27) for f(pi ) arises naturally from the distribution function for f∗(pi ): compare eq. (26).

References [1] S. Chapman and T. Cowling, in co-operation with D. Burnett, The mathematical Theory of non-uniform Gases, 3nd Ed., Cambridge University Press, 1970. [2] F. Debbasch, J.P. Rivet, W.A. van Leeuwen, Invariance of the relativistic one-particle distribution function, Physica A 301(2001)181. [3] F. Debbasch, W.A. van Leeuwen, General Relativistic Boltzmann Equation II — Manifestly Covariant Formalism, to be published in XXX. [4] J. Ehlers, Survey of general relativity theory, in: Relativity, Astrophysics and Cosmology, Proceedings of the summer school held, 14-16 August, 1972, at the Banff Center, Banff, Alberta, Reidel, Dordrecht, pp. 1–125, W. Israel, ed. [5] W. Israel, Relativistic Kinetic Theory of a Simple Gas, J. Math. Phys. 4(1963)1163–1181.

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[6] W. Israel, The relativistic Boltzmann equation, in: L. O’Raifeartaigh (Ed.), General Relativity, papers in honour of J.L. Synge, pp. 201–241 (Clarendon Press, Oxford, 1972). [7] W. Israel, in: Relativistic Fluid Dynamics, A. Anile and Y. ChoquetBruhat, eds., Springer Verlag, 1989. [8] J.A. Peacock, Cosmological Physics, Cambridge University Press, 1999. [9] A. Lichnerowicz and R. Marrot, Compt. Rend. Acad. Sc. Paris 210(1940)759–761. [10] R.K. Sachs, Survey of general relativity theory in: [4], pp. 197–236. [11] B.F. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, 1980. [12] Steven Weinberg, Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity, Wiley, New York, 1972. [13] R.M. Wald, General Relativity, University of Chicago Press, 1984. [14] W. Zimdahl, Relativistic stochastic Boltzmann equation and fluctuations in general relativity, Classical and Quantum Gravity 6(1989)1879– 1892.

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