tivity, relativistic statistical physics, off-shell Boltzmann-equation. 1 Introduction. The Boltzmann .... If one uses t
General Relativistic Boltzmann Equation II. Manifestly 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 In a preceding article we presented a general relativistic treatment of the derivation of Boltzmann-equation. The four-momenta occurring in this formalism were all on-shell four-momenta, verifying the mass-shell restriction p2 = m2 c2 . Due to this restriction, the resulting Boltzmann-equation, although covariant, turned out to be not manifestly covariant. In the present article we switch from mass-shell momenta to off-shell momenta, and thereby arrive at a Boltzmann equation that is manifestly covariant. PACS numbers : 04.20.-q, 05.20.Dd, 02.40.-k, 51.10.+y Keywords : off-shell relativistic distribution function, general relativity, relativistic statistical physics, off-shell Boltzmann-equation
1
Introduction
The Boltzmann equation is an equation describing the evolution in time of the one-particle-density in position-momentum space. In non-relativistic physics, the one-particle phase space appears as the direct product of the 3-dimensional physical space and the 3-dimensional momentum space. A point in this phase space is usually represented by a set of six coordinates (x1, x2 , x3, p1 , p2 , p3 ). The general relativistic situation is more complex, since the 3-dimensional physical space has to be replaced by the 4-dimensional spacetime M with metric gµν (x). A first possibility, developed in the preceding article, is to 1
use a 7-dimensional phase-space. A typical point in this space has seven coordinates, namely the four coordinates x := (x0 = ct, x1, x2, x3) ∈ M and the three momentum coordinates (p1 , p2 , p3 ) on the mass-shell gµν (x)pµ pµ = m2c2 , (µ = 0, 1, 2, 3), at x ∈ M. Notwithstanding this complication, one may extend the usual concept of a one-particle distribution function to the realm of general relativity by defining a one-particle distribution function on a seven-dimensional manifold, and show that this function is a general relativistic scalar. Moreover, we could prove [2] that this general relativistic distribution function obeys a transport equation which may be considered as the general relativistic equivalent of the non-relativistic Boltzmann equation.
Although our approach was covariant, it was not manifestly so, a fact which is related to the use of a momentum space that is 3-dimensional, whereas a truly covariant treatment would require the use of all four components of the four-momentum as a variables; working on-shell restricts the 4-dimensional momentum-space to a 3-dimensional submanifold. Our approach so far used either the three spatial contravariant components (p1 , p2 , p3 ) or the three spatial covariant momentum components (p1 , p2 , p3 ) as coordinates on the mass-shell. An equation satisfied by a function which, apart from the 4 space-time coordinates, depends on only three momentum components — and not on four— obviously cannot be manifestly covariant. In order to relate our results [3] to earlier ones [5, 6, 8, 9] it is necessary to develop a manifestly covariant formalism, and this can only be achieved by using a 4-dimensional momentum-space, i.e., by working off-shell. The present article is devoted to this task. Our final results, will turn out to be closely related to, but slightly different from those found in the existing literature. In earlier publications, the Boltzmann-equation had been obtained via an educated guess, which resulted in some ambiguities related to the presence or non-presence of the zero-component of the momentum in the final equations. The material of this article is organized as follows. In section 2, we switch from the on-shell momentum variables pµ to offshell momentum variables pµ . We introduce an off-shell distribution function F∗(t, xi, p0 , p1 , p2 , p3 ) replacing on-shell distribution f∗(t, xi , p1 , p2 , p3 ) of the preceding article in case covariant momentum variables are used. Fur2
thermore, we derive the manifestly covariant Boltzmann-equation (53) for F∗(t, xi, p0 , p1 , p2 , p3 ). In section 3 we derive the results for the contravariant case from those for the covariant case. An off-shell distribution function F (t, xi, p0 , p1 , p2 , p3 ) takes over the role of f(t, xi, p1 , p2 , p3 ). The resulting equation is the manifestly covariant Boltzmann-equation (76) for F (t, xi, p0 , p1 , p2 , p3 ). In section 4 the Boltzmann-equations (53) and (76) for the off-shell functions F∗ and F are rewritten as the equations (91) and (111) for the on-shell functions f∗ and f. In the existing literature, one encounters Boltzmann-equations containing a distribution function called f, which closely resemble one of the four equations (53), (76), (91) or (111) derived in this article. It is not always clear, however, whether the authors used co- or contravariant momentum variables. Moreover, it is not always made explicit whether momentum variables are on-shell or off-shell. In other words, it is not always clear whether the f encountered in the literature equals F , F∗, f or f∗. This makes it difficult —and sometimes even impossible— to rightly interpret these equations or to compare them to our results. The appendices A–C give details to calculations of the main text.
2 2.1
Covariant momentum Off-shell distribution function F∗
Let us denote a covariant momentum four-vector, like pµ , which has lower indices, by p∗ , and a contravariant vector pµ by p. In general, a lower asterisk is used to indicate that lower indices are used. Let us now work in an eight dimensional phase-space, and let us choose the eight coordinates (x, p∗ ) as independent coordinates in this phase space. The choice of (x, p) as independent coordinates in an 8-dimensional phase-space will be envisaged later. If one uses the covariant momentum components pµ , the contravariant momentum components pµ are to be considered as functions of (x, p∗ ). One has 3
pµ (x, p∗) = g µν (x)pν
(1)
pµ (x, p) = gµν (x)pν
(2)
Inversely, one has
We will often suppress the explicit (t, xi)-dependence. In the preceding article [3] we used these relations to express the zeroth components p0 and p0 , given in terms of the three momenta pi and pi by q 1 2 i {−g0i(x)p + (g0i (x)pi ) − g00 (x)(gij (x)pi pj − m2 c2)} (3) p (p ) = g00 (x) q 1 p0 (pi ) = 00 {−g 0i(x)pi + (g 0i (x)pi )2 − g 00(x)(g ij (x)pi pj − m2 c2)}(4) g (x) 0
i
[see (I.??) and (I.??)]1 in terms of the components of the three-momentum pi and pi . Using (1) and (2) we obtain
0
p (pi ) = p0 (pi ) =
q
q
(g 0i (x)pi )2 − g 00(x)(g ij (x)pi pj − m2 c2),
(5)
(g0i (x)pi )2 − g00 (x)(gij (x)pi pj − m2c2 )
(6)
[see (I.??) and (I.??)]. In what follows it is important to distinguish between these four functions (3), (4), (5) and (6), and arbitrary integration variables, which we will name —for obvious reasons, but somewhat confusingly at first sight— p0 and p0 . In case that there can be confusion, we will write, when referring two one of the four functions (3)–(6), an explicit argument, e.g., p0 (pi ), p0 (pi ) or p0 (pi ), p0 (pi ) rather than p0 or p0 . 1
References to formulae of the preceding article [3] are preceeded the Roman numeral
I.
4
In order to rewrite the Boltzmann equation in a general covariant form, we introduce a function F∗ (t, xi, p0 , pi ) which is such that i
f∗ (t, x , pi ) =
Z
+∞
−∞
F∗(t, xi, p0 , pi )δ p0 − p0 (t, xi, pi ) dp0
(7)
or, equivalently, i
f∗(t, x , pi ) =
Z
+∞
F∗(t, xi, p0 , pi )dδ p0
(8)
−∞
where we introduced the abbreviation dδ p0 := δ p0 − p0 (t, xi, pi ) dp0
(9)
for an integration with respect to p0 that includes a delta-function restricting the integration to the mass shell. The function p0 (t, xi, pi ) is the on-shell value of the component p0 . It is given by the expression (4). Hence, the function F∗(t, xi , p0 , pi ) is a function defined in such a way that, if the variable p0 equals p0 (t, xi, pi ), it coincides with to the one-particle distribution function f∗ (t, xi, pi ) and, consequently, on the mass-shell, F∗ en f∗ may be interchanged. Another way of expressing all this is to say that f∗ is the restriction (in the mathematical sense of the word restriction) of the function F∗ to the mass-shell. Conversely, F∗ may be called the off-shell extension of f∗. At fixed t, xi and pi , the integration variables p2∗ and p0 are related according to
d(p2∗ ) := = = =
d(g µν pµ pν ) d g 00 p20 + 2g 0i p0 pi + g ij pi pj 2 g 00 p0 + g 0i pi dp0 2p0 (pi )dp0
Combining (I. ??) and (10), we find: 5
(10)
δ(p0 − p0 (t, xi , pi ))dp0 = 2θ(p0 )δ(p2∗ − m2c2 )p0 (pi )dp0 = θ(p0 )δ(p2∗ − m2 c2)d(p2∗ )
(11)
Since the right-hand side of this expression is a scalar, the left-hand side must also be a scalar. Now, let us go back to equation (7). Since f∗ in the left-hand side of (7) is a scalar [3], it is natural in view of (11) —and not inconsistent— to impose the further condition on F∗ that it has the same transformation properties as f∗ . Since f∗ is a general relativistic scalar, we will therefore demand the function F∗ to be a relativistic scalar too. Note, that the expression (7) does not define a unique function F∗, since to a single function f∗ correspond an infinity of off-shell functions F∗, the only requirement being that the function F∗ equals the function f∗ for values of the momentum which belong to the mass shell; for the derivatives, see (114) and (115). Hence, this does not present any problem, because physical quantities always are to be calculated on-shell, and, on-shell, all possible F∗’s coincide with f∗ . In what follows, F∗ will be any one of the functions obeying (7). The general relativistic Boltzmann equation for f∗ , eq (I.??), may be written s∗ (f∗ ) = c∗ (f∗ , f∗)
(12)
where s∗(f∗ ) is the streaming term of the Boltzmann equation, defined by ∂ dxi ∂ dpi p0 (pi ) ∂ s∗ (f∗ ) := f∗ + i f∗ + f∗ mc ∂t ∂x dt ∂pi dt
(13)
and where c∗ (f∗, f∗ ) is the collision term defined in (I.??). The covariant Boltzmann equation sought-for will be an equation obeyed by the offshell distribution function F∗, not an equation for the distribution function f∗ itself. It will be obtained by observing that a sufficient condition for f∗ (t, x1, x2, x3, p1 , p2 , p3 ) to satisfy the general relativistic Boltzmann equation (13) will be that F∗(t, x1, x2 , x3, p0 , p1 , p2 , p3 ) is a solution of a certain, manifestly covariant equation for F∗. In this article, one of our principle aims is to arrive at such an equation for F∗.
6
In the equations (12)–(13), the quantities p0 , 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∗ itself. Consequently, we will have to switch from the set of seven variables t, x1 , x2 , x3, p1 , p2 , p3 to the set of eight variables t, x1 , x2 , x3, p0 , p1 , p2 , p3 . This will be the subject of the next section, in which we will consider the left-hand side of the Boltzmann-equation, the streaming term.
2.2
Off-shell streaming term S∗
In appendix A, it is shown that the streaming term s∗ (f∗ ), depending on the on-shell function f∗ (p1 , p2 , p3 ) can be written as an integral depending on the off-shell function F∗ (p0 , p1 , p2 , p3 ): s∗(f∗ ) =
Z
∞
−∞
∂ 1 ∂ (pµ F∗ ) + µ m ∂x ∂pµ
dpµ F∗ dτ
dδ p0
(14)
where τ is the proper time. This leads us to the definition of the off-shell streaming term 1 ∂ ∂ S∗(F∗ ) := (pµ F∗) + µ m ∂x ∂pµ
dpµ F∗ dτ
(15)
Hence, s∗ (f∗ ) =
Z
∞
S∗ (F∗) dδ p0
(16)
−∞
We recall that dδ p0 has been defined in eq. (9). This concludes the discussion of the streaming term. In the literature, one may encounter different forms for the streaming terms. We now give some of them. 2.2.1
Six alternative forms of the streaming term
As a kind of side remark we give six alternative forms for the off-shell streaming term S∗(F∗ ). From eq. (15) we find
7
1 ∂pµ ∂ dpµ pµ ∂F∗ dpµ ∂F∗ S∗ (F∗) = + + F∗ + m ∂xµ dτ ∂pµ m ∂xµ ∂pµ dτ
(17)
The second term, the term between the square brackets, vanishes. This can be seen as follows. Between two collisions, the particles of the fluid are streaming freely, and their movement may be described by the geodesic equation for the covariant momentum 1 ∂gαβ α β dpµ = p p dτ 2m ∂xµ
(18)
(See [7, section 87, p. 245].) Now, let us define a Hamiltonian H according to H(x, p∗ ) =
1 µν g (x)pµ pν 2m
(19)
and consider the Hamiltonian equations dxµ ∂H = dτ ∂pµ dpµ ∂H =− dτ ∂xµ
(20) (21)
together with the relation between four-momentum and four-velocity pµ dxµ = m dτ
(22)
where τ is the proper time. It is easily seen that (20) implies (22). In oder to show that (21) is equivalent to (18), we first note that gαβ (x)pα (x, p∗ )pβ (x, p∗) = g αβ (x)pα pβ
8
(23)
where the notation p∗ = (p0 , p1 , p2 , p3 ) has been used. Differentiating this equation with respect to xµ at fixed p∗ leads to (∂µ gαβ )pα pβ + gαβ (∂µ pα )pβ + gαβ pα (∂µ pβ ) = (∂µ g αβ )pα pβ
(24)
One has ∂µ pα = (∂µ g αδ )pδ ,
∂µ pβ = (∂µ g βδ )pδ
(25)
Inserting these relations into (24) leads to (∂µ gαβ )pα pβ + 2(∂µ g αβ )pα pβ = (∂µ g αβ )pα pβ ,
(26)
(∂µ gαβ )pα pβ = −(∂µ gαβ )pα pβ
(27)
or
which imlpies, together with (19) that (21) is equivalent to the geodesic equation (18). Hence, the term in square brackets of (17) reduces to zero: ∂ dpµ ∂ 2H ∂ 2H ∂ pµ + = − ∂xµ m ∂pµ dτ ∂xµ ∂pµ ∂pµ ∂xµ =0
(28)
implying that the off-shell streaming term (17) for the off-shell function F∗ can be written: S∗ (F∗) =
pµ ∂F∗ dpµ ∂F∗ + m ∂xµ dτ ∂pµ
(29)
This is a second form for the off-shell streaming term. It is the form which one often encounters in the special theory of relativity. 9
Note that this equation is equivalent to S∗(F∗ ) =
pµ dpµ ∂F∗ ∇µ F∗ + m dτ ∂pµ
(30)
where ∇∗µ stands for the covariant derivative, which, when acting on a scalar field, is identical to the partial derivative ∂/∂xµ . It is a third form for the offshell streaming term, a form one sometimes encountered in general relativistic treatments of the Boltzmann equation. The geodesic equation for covariant components may also be written [7, section 87, p. 245], dpµ 1 = Γνλµ pλ pν dτ m
(31)
where the Γ’s are the coefficients of the affine connection. Inserting this expression into (30) we obtain 1 ∂F∗ λ λ ν S∗ (F∗) = p ∇λF∗ + p Γλµ pν m ∂pµ
(32)
a fourth form, encountered in general relativistic treatments of the Boltzmann equation. An equivalent way to write this is S∗ (F∗) =
pλ D∗λ F∗ m
(33)
where one makes use of a derivative D∗λ , defined according to: D∗λ := ∇λ + Γκλµ pκ
∂ ∂pµ
(34)
where ∇λ stands, as usual, for the ordinary covariant derivative. The operator D∗λ transforms like a covariant vector: for an explanation, see the footnote on p. 206 of Israel’s article [6]. For a vector field Aρ (x, p∗), one has, for example,
10
∂Aρ D∗λAρ = ∇λAρ + Γκλµ pκ ∂pµ ρ ∂A ∂Aρ ρ κ κ = + Γλκ A (x, p∗ ) + Γλµ pκ ∂xλ (x,p∗) ∂pµ (x,p∗)
Hence, it is appropriate to call the derivative D∗λ, defined by (34), a covariant derivative: it transforms like a covariant vector. Note that D∗λpρ = 0
(35)
since ∂λ(pρ ) = 0. This is in agreement with the fact that D∗λ is the derivative at constant covariant momentum. In eq. (33) pλ is to be understood as a function of xµ and pµ . It is therefore more appropriate to write (33) in the form g µλ pµ D∗λ F∗ S∗(F∗ ) = m
(36)
Since ∇λ g αβ = 0 and the metric g αβ is independent of pµ , we find using (35), S∗ (F∗) =
1 D∗λ(g λµ pµ F∗) m
(37)
This equation constitutes a sixth form of the off-shell streaming term.
2.3
Off-shell collision term
Similar to the way in which we could relate the on-shell streaming term s∗ (f∗) to an off-shell streaming term S∗(F∗ ) to obtain a manifestly covariant quantity, the on-shell collision term c∗ (f∗ , f∗), given by eq. (I.??) can be related to an off-shell collision term C∗(F∗ , F∗), and become a manifestly covariant quantity. In order to determine C∗(F∗ , F∗)), we introduce the offshell transition probability, W∗ (t, xi, p∗ , q∗ | p0∗ , q∗0 ), via 11
i
w∗ (t, x ; pi , qi |
p0i , qi0 )
=
Z
W∗ (x; p∗ , q∗ | p0∗ , q∗0 )dδ p00 dδ q00 .
(38)
where w∗ is defined in (I. ??). Furthermore, we introduced the abbreviations
dδ p00 := δ(p00 − p00 (t, xi , p0i ))dp00 dδ q00 := δ(q00 − q00 (t, xi, qi0 ))dq00
(39) (40)
with p00 (t, xi, p0i ) and q00 (t, xi , qi0) given in terms of three-momenta by the analogues of eq. (4) for p0i and qi0. Note, that the integrand W∗ in the right-hand side of (38) is defined by the left-hand side, but not in a unique way. Interchanging primed and unprimed variables in (38) we find w∗ (t, x
i
; p0i , qi0
| pi , qi ) =
Z
W∗ (x; p0∗ , q∗0 | p∗ , q∗)dδ p0 dδ q0.
(41)
where dδ p0 is defined in eq. (9), and where dδ q0 := δ(q0 − q0 (t, xi, qi))dq0
(42)
with p0 and q0 defined by (4) and is analogue for q, respectively. Thus, by definition, on the mass-shell, the off-shell transition probability W∗ equals the on-shell transition probability w∗ of eq. (I. ??). With the help of the off-shell transition probability W∗, and the off-shell distribution function (7) the on-shell collision term c∗(f∗ , f∗ ), given by (I. ??), can be written
c∗(f∗ , f∗ ) =
Z
[F∗(x, p0∗ )F∗(x, q∗0 )W∗0 − F∗ (x, p∗)F∗ (x, q∗)W∗ ]
dδ p0 dδ q0 d3 Vq∗ dδ p00 d3 Vp0∗ dδ q00 d3 Vq∗0
12
(43)
where we abbreviated
W∗0 ≡ W∗(x; p0∗ , q∗0 | p∗ , q∗) W∗ ≡ W∗(x; p∗ , q∗ | p0 ∗ , q∗0 )
(44) (45)
In appendix C we show that
dδ q0 d3 Vq∗ = d4δ Vq∗
(46)
where d4δ Vq∗ is a defined by d4δ Vq∗ := 2mc θ(q0) δ(q∗2 − m2c2 ) d4 Vq∗
(47)
In the latter expression, d4 Vq∗ is the usual volume element in four-momentum space, defined by eq. (I.??). Evidently, since d4 Vq∗ is a scalar quantity, d4 Vq∗ is a scalar integration element on the 4-dimensional momentum-space. The volume element (47) reduces the integration with respect to off-shell momenta to an integration over the mass-shell. Using (46) and its obvious analogues for p0∗ and q∗0 , we can rewrite eq. (43) in the form
c∗ (f∗, f∗ ) =
Z
[F∗(x, p0∗ )F∗(x, q∗0 )W∗0 − F∗ (x, p∗)F∗ (x, q∗)W∗ ]
dδ p0 d4δ Vq∗ d4δ Vp0 ∗ d4δ Vq0 ∗
(48)
or, equivalently, c∗(f∗ , f∗ ) =
Z
C∗ (F∗, F∗ )dδ p0
where 13
(49)
C∗ (F∗, F∗) :=
2.4
Z
[F∗(x, p0∗)F∗ (x, q∗0 )W∗0 −F∗(x, p∗ )F∗(x, q∗)W∗ ]d4δ Vq∗ d4δ Vp0 ∗ d4δ Vq0 ∗ (50)
Off-shell Boltzmann equation
In the preceding two sections 2.2 and 2.3, we obtained manifestly covariant expressions for the streaming term and the collision term of the Boltzmann equation. When we now insert (16) and (49) into (12) we find Z
S∗ (F∗)dδ p0 =
Z
C∗ (F∗, F∗)dδ p0
(51)
This equality is certainly obeyed if S∗(F∗ ) = C∗ (F∗, F∗)
(52)
which, using (33), may alternatively be written pµ D∗µ F∗ = C∗ (F∗, F∗) m
(53)
Since all quantities occurring in this equation transform like tensors, this is a manifestly covariant equation. It is the Boltzmann-equation for the off-shell manifestly covariant distribution function F∗ (x, p∗), a function depending on the spacetime point x and the covariant momentum p∗ . It constitutes one of the main results of this article. The definition of the derivative D∗µ is given in (34); the collision term C∗ (F∗, F∗) is given by the expression (50). The Boltzmann-equation (53) for F∗ (x, p∗) constitutes one of the main results of this article. 14
The off-shell function F∗ is related to the on-shell function f∗ . The latter is defined according to X f∗ (t, xi, pi ) := h δ (3)(xi − xir (t))δ (3)(pi − pri (t))iav ,
(54)
r
[See equation (I. ??).] If the on-shell function f∗ (t, xi, pi ) itself is used rather than the off-shell function F∗(t, xi, p0 , pi ), one finds the equation (91) below instead of the equation (53).
3
Contravariant momentum
One might expect, at this point, that we would derive the manifestly covariant Boltzmann-equation for f(t, xi, pi ) from the Boltzmann-equation (I. ??) derived in the preceding article [3]. This would require a lengthy rewriting of the streaming term s(f), comparable to what has been done for s∗ (f∗ ) in appendix A. The following shortcut, however, circumvents this problem. This approach has as a consequence, that the transition-probability W (t, xi; pi , q i | p0i , q 0i), which we will come across in the Boltzmann-equation (76) for the contravariant momenta, is related to W∗ and, thus, to w∗ rather than to w [see eq. (68)]. In other words, the transition rate w(pi , pj | p0i , p0j ), which played a role when we derived the expression (I.??) for the collision term c(f, f) will disappear from the stage altogether. The relation of (76) to the original Boltzmann-equation (I. ??), containing w is, therefore, not given explicitly.
3.1
Off shell distribution function F
In the preceding section we used the covariant momentum variables p∗ = (p0 , pi ). In this section we will derive the corresponding expressions for the streaming and collision terms for contravariant components p = (p0 , pi ). In order to do so, we define the off-shell distribution function F (x, p) which is such that
15
i
i
f(t, x , p ) =
Z
+∞
F (t, xi, p0 , pi ) dδ p0
(55)
−∞
where dδ p0 := δ p0 − p0 (t, xi, pi ) dp0
(56)
Hence, the function F (t, xi, p0 , pi ) is a function defined in such a way that, if the variable p0 equals p0 (t, xi, pi ), its value coincides with to the one-particle distribution function f(t, xi , pi ), which has been proved to be a general relativistic scalar. On the mass-shell, F (t, xi, p0 , pi ) equals f(t, xi , pi ): f is the restriction of the function F to the mass-shell. Conversely, F may be called the off-shell extension of f. On the mass-shell we thus have, by definition F (p0 , pi ) = f(pi )
(57)
In article I, [3], we defined, in equation (I. ??), f(pi ) = f∗ (pi )
(58)
In equation (8) we defined F∗ as the off-shell extension of f∗ . Hence, on the mass shell we have f∗ (pi ) = F∗ (p0 , pi )
(59)
Combining the three equations (57)–(59), we conclude that on the mass-shell, F (p0 , pi ) = F∗(p0 , pi )
(60)
We now do not start from the Boltzmann-equation (I. ??), but use the shortcut referred to above, and start from the Boltzmann-equation for F∗ , eq. (53). 16
Let us define, in analogy to (29), the off-shell streaming term by S(F ) :=
pµ ∂F dpµ ∂F + m ∂xµ dτ ∂pµ
(61)
then, on the mass-shell, S(F ) = S∗ (F∗)
(62)
as follows from (60) and (29). The geodesic equation for contravariant components may be written [7, section 87, p. 245] dpµ 1 = − Γµλν pλ pν dτ m
(63)
Inserting this expression into (29) we obtain S(F ) =
pµ ∂F 1 ∂F − Γµλν pλ pν µ µ m ∂x m ∂p
(64)
pλ Dλ F m
(65)
or, equivalently, S(F ) = where Dλ := ∇λ − Γκλν pν
∂ ∂pκ
(66)
The operator Dλ transforms like a covariant vector [6, see footnote p.206]. In particular one has, for a contravariant vector field Aρ :
17
∂Aρ Dλ Aρ = ∇λ Aρ − Γκλν pν κ ∂p ρ ρ ∂A ρ κ κ ν ∂A = + Γ A (x, p) − Γ p λν λκ ∂xλ (x,p∗) ∂pκ
Since ∂λ pρ = 0, we have
Dλ pρ = 0
(67)
in agreement with the fact that that Dλ is the derivative at constant contravariant momentum. The expression (65), with Dλ given by (66), is to be compared with the expression (36), with D∗λ given by (34). In view of (62) these two equations are equivalent, but they are expressed with respect to different momentum variables, namely contravariant or covariant momenta, respectively.
3.2
Off-shell collision term
We started from the streaming term S∗(F∗ ) to obtain a streaming term S(F ). Likewise, we now start from the collision term C∗(F∗, F∗ ) in order to find an expression for a collision term C(F, F ). Firstly, let us define a transition rate W (x; p, q | p0 , q 0), related to the transition rate W∗ (x; p∗ , q∗ | p0∗ defined by equation (38), in the following way W (x; p, q | p0 , q 0) := W∗ (x; p∗ , q∗ | p0∗ , q∗0 )
(68)
Furthermore, we note that d4δ Vq∗ = d4δ Vq
18
(69)
where d4δ Vq := 2mc θ(q 0) δ(q 2 − m2c2 ) d4 Vq
(70)
is the analogue of (47). The equality (69) follows by using that q 2 = q∗2 and d4 Vq = d4 Vq∗ [see eq. (I. ??)]. With the help of (60), (68) and (69), we find from (29) C∗ (F∗, F∗) = C(F, F )
(71)
where
C(F, F ) :=
Z
[F (x; p0)F (x, q 0)W 0 − F (x, p)F (x, q)W ]d4δVq d4δ Vp0 d4δ Vq0
(72)
In this equation we abbreviated
W ≡ W (x; p, q | p0 , q 0) W 0 ≡ W (x; p0, q 0 | p, q)
(73) (74)
where, in turn, W (x; p, q | p0 , q 0) is defined in terms of W∗ by (68).
3.3
Off-shell Boltzmann equation
Using the equalities (62) and (71), the Boltzmann-equation (52) for the offshell distribution function F∗ can now be rewritten as an equation for the off-shell function F . We find, using the expression (64) for S(F ), pµ ∂F 1 ∂F − Γµλν pλ pν µ = C(F, F ) µ m ∂x m ∂p where C(F, F ) is given by (72). 19
(75)
An equivalent way of writing (75) is pµ Dµ F = C(F, F ) m
(76)
with Dµ defined in eq. (66). It now follows that (76) is a manifestly covariant equation, since the left- and right-hand sides are manifestly covariant. We thus obtained a manifestly covariant, off-shell Boltzmann equation for the manifestly covariant off-shell distribution function F (x, p). The Boltzmann-equation (76) for F (t, xi, p0 , pi ) is one of the main results of this article. Its form is well-known in the literature: see, e.g., the 1972 article of Israel [6, p. 206], the 1972 article of Ehlers [4, p. 66] or the 1999 textbook of Peacock [8, p. 303]. In these treatments, however, it is not explicitly stated that F (t, xi, p0 , pi ) is the off-shell function (55), with f(t, xi, pi ) defined according to
f(t, xi , pi ) := h
X r
δ (3)(xi − xir (t))δ (3)(pi − pir (t))
p0 (pi ) 1 iav , − det g p0 (pi )
(77)
[See the equations (I. ??) and (I. ??) of article I.] If the on-shell function f(t, xi , pi ) itself is used rather than the off-shell function F (t, xi, p0 , pi ), one finds the equation (111) below instead of the equation (75).
4
On-shell Boltzmann-equation
The off-shell Boltzmann equation for F∗, given by eq. (53), and the off-shell equation for F , given by eq. (76), can be rewritten as on-shell Boltzmann equations for f∗ and f: see eqs. (91) and (111), respectively. The aim of the present section is to elaborate this point, which, in the end, will give us the equations which are generally encountered in the literature.
20
4.1
Covariant momentum
In this section we will rewrite the manifestly covariant off-shell Boltzmann equation for F∗ as an equation for f∗, by integrating the off-shell Boltzmann equation with respect to the measure D4 Vp∗ , thus restricting the off-shell momentum p0 to on-shell values. By construction, the resulting equation will be equivalent to the original Boltzmann equation (I. ??), but it will turn out not to be identical with it! Let us start by integrating the streaming term S∗ (f∗ with respect to the measure dδ p0 . With the help of the sixth expression for the streaming term, given by (36), we find Z
S∗ (F∗)dδ p0 =
Z
g µλ pµ D∗λ F∗ dδ p0 m
(78)
Using the definition (34) of D∗λ we obtain Z
pλ ∂F∗ pλ ν ∂F∗ S∗ (F∗) dδ p0 = + Γλα pν m ∂xλ m ∂pα = K∗ + L∗ + M∗ Z
(79)
where we introduced the abbreviations Z 1 ∂F∗ pλ λ dδ p0 K∗ := m ∂x Z 1 ∂F∗ L∗ := Γνλ0pλ pν dδ p0 m ∂p0 Z 1 ∂F∗ M∗ := Γνλi pλ pν dδ p0 m ∂pi
(80) (81) (82)
Using now (114) with Gµ1 ,...,µr and g µ1 ,...,µr equal to F∗ and f∗ , respectively, we find pλ ∂f∗ 1 K∗ = − m ∂xλ m
Z 21
pλ
∂p0 ∂F∗ dδ p0 ∂xλ ∂p0
(83)
where p0 is to be interpreted as g 0µ pµ , and the pi are given by pi = g iµ pµ , while p0 = p0 (pi ) is on the mass-shell. In a similar manner, using (115), M can be rewritten 1 ∂f∗ 1 M∗ = Γνλi pλ pν − m ∂pi m
Z
Γνλi pλ pν
∂p0 ∂F∗ dδ p0 ∂pi ∂p0
(84)
Inserting (81), (83) and (84) and into (79) we find Z
1 S∗(F∗ )dδ p0 = m Z
∂ ∂ + Γνλi pµ pν p λ ∂x ∂pi ∂F∗ Z(x, p∗ ) dδ p0 ∂p0 λ
f∗ + (85)
where Z∗ is the abbreviation given by Z∗ (x, p∗) :=
1 ν λ pµ ∂p0 1 ∂p0 Γλ0 p pν − − Γνλi pλ pν µ m m ∂x m ∂pi
(86)
Using the geodesic equation pµ 1 = Γνλµ pλ pν dτ m
(87)
[7, section 87, p. 245], we have Z∗ =
dp0 pµ ∂p0 dpi ∂p0 − − µ dτ m ∂x dτ ∂pi
(88)
Using now the chain rule to calculate dp0 (xµ , pi )/dτ , we find dp0 ∂p0 dxµ ∂p0 dpi = + dτ ∂xµ dτ ∂pi dτ 22
(89)
Hence, Z∗ = 0
(90)
From (51), using the expression (49) for the on-shell collision term, and (85) with (90) for the on-shell streaming term, we so obtain the on-shell Boltzmann-equation for f∗ : 1 m
∂ ∂ p + Γνµi pµ pν µ ∂x ∂pi µ
f∗ = c∗ (f∗, f∗ ),
(91)
In this equation, p0 and the four components of pµ are to be understood as functions of (t, x1, x2, x3 , p1 , p2 , p3 ). Notice, that the second lower index of Γ runs over i = 1, 2, 3 only. We thus have given a solid basis to (91), the general relativistic Boltzmannequation, an equation which can be found at many places in the literature [4, 5, 8, 9]. We finally note that (91) is obtained starting with the Boltzmann equation in a quite different form, namely the form (I. ??).
4.2
Contravariant momentum
A calculation similar to the one carried out in the preceding section shows that the general relativistic Boltzmann equation (I. ??) obeyed by f can also be put in an alternative form. By multiplying the streaming term S(F ), eq. (64), by the delta-function δ(p0 − p0 (x, p)) and integrating with respect to p0 , we obtain, Z
0
S(F ) dδ p =
Z
pλ ∂F 1 ∂F − Γµλν pλ pν µ λ m ∂x m ∂p
dδ p0
(92) (93)
23
where we abbreviated dδ p0 := δ(p0 − p0 (x, p)) dp0
(94)
Z
(95)
This can be written
S(F ) dδ p0 = K + L + M
where we introduced three abbreviations, namely
pλ ∂F dδ p0 m ∂xλ Z 1 ∂F Γ0λν pλ pν 0 dδ p0 L=− m ∂p Z 1 ∂F M =− Γiλν pλ pν i pλ dδ p0 m ∂p K=
Z
(96) (97) (98)
We now use the analogues of the relations (114) and (115) for p0 instead of p0 to rewrite K and M. We then get pλ ∂f − K= m ∂xλ
Z
1 ∂f 1 M = − Γiλν pλ pν i + m ∂p m
pλ ∂p0 ∂F dδ p0 m ∂xλ ∂p0 Z
Γiλν pλ pν
(99)
∂p0 ∂F dδ p0 i 0 ∂p ∂p
(100)
Hence, Z
pλ ∂f 1 ∂f S(F ) dδ p = − Γiλν pλ pν i + λ m ∂x m ∂p 0
where Z is the abbreviation 24
Z
Z
∂F dδ p0 ∂p0
(101)
Z=−
pλ ∂p0 1 0 λ ν 1 i λ ν ∂p0 − Γ p p + Γ p p λν m ∂xλ m m λν ∂pi
(102)
Using the equation of motion for the contravariant momentum, eq. (63), we find Z=−
∂p0 pλ dp0 dpi ∂p0 + − ∂xλ m dτ dτ ∂pi
(103)
Since p0 = p0 (xλ , pi ), we have, using the chain rule to calculate dp0 /dτ , ∂p0 dxµ ∂p0 dpi dp0 = + i dτ ∂xµ dτ ∂p dτ
(104)
Z =0
(105)
Hence,
4.3
Off-shell collision term
Multiplying both sides of (75) by δ(p0 − p0 (pi )) and integrating with respect to p0 we obtain, using (101) with (105) pλ ∂f 1 ∂f − Γiλν pλ pν i = λ m ∂x m ∂p
Z
C(F, F )dδ p0
(106)
From (72) we have Z
0
C(F, F ) dδ p =
Z
[F (p0)F (q 0)W (p0 , q 0 | p, q)
−F (p)F (q)W (p, q | p0 , q 0)]dδ p0 d4δ Vq d4δ Vp0 d4δ Vq0 25
(107)
Using (159) we find Z
0
C(F, F ) dδ p =
Z
[f(p0 )f(q 0)w(p ˆ 0 , q 0 | p, q)
−f(p)f(q)w(p, ˆ q | p0 , q 0)]d3Vq d3 Vp0 d3 Vq0
(108)
where
0 0
w(p, ˆ q | p q ) := 0
0
w(p ˆ , q | p, q) :=
Z
W (p, q | p0 q 0)dδ p0 dδ q 0
(109)
Z
W (p0 , q 0 | p, q)dδ p00 dδ q 00
(110)
We recall that W (x; p, q | p0 , q 0) is defined in terms of W∗ by eq. (68). We thus find the on-shell Boltzmann-equation for 1 m
∂ i µ ν ∂ − Γ p p p µν ∂xµ ∂pi µ
f = cˆ(f, f),
(111)
where cˆ(f, f) is the collision term defined by
cˆ(f, f) :=
Z
[f(p0 )f(q 0)w(p ˆ 0 , q 0 | p, q) − f(p)f(q)w(p, ˆ q | p0 , q 0)]d3 Vq d3 Vp0 d3 Vq0 (112)
In eq. (111), p0 and the four components of pµ are to be understood as functions of (t, x1, x2, x3, p1 , p2 , p3 ).
5
Summary
Let is summarize the reasoning of the present article (article II) and its predecessor (article I). The central, basic equation, is the equation (II. 53) 26
for the distribution function F∗(pµ ) defined by (II. 8). This equation has been shown to imply the equation (I. ??) for f∗ (pi ) given by (I. ??). This equation turned out to be equivalent to the equation (I. ??) for the the distribution function f(pi ) given by (I. ??). However, the central equation (II. 53) also implies the equation (II. 75) for F (pµ ) defined by (II. 55), which, in turn, implies the equation (II. 111) for f(pi ). Hence, the equations (I. ??) and (II. 111), both equations for f(pi ), finding their origin in the same equation (II. 53), are equivalent. Finally, we observe that the equation (II. 91) for f∗ (pi ) also comes from the basic equation (II. 53). Hence, the equations (I. ??) and (II. 91) are also equivalent. In conclusion, the equations for f(pi ), f∗ (pi ), F (pµ ) and F∗(pµ ) all are equivalent, despite of the fact that they have quite different forms.
A
Reduction of the streaming term
The streaming term s∗, given by (12) for f∗ contains derivatives with respect to time t, space xi and three-momentum pi . These seven derivatives have to be transformed to the eight derivatives of F∗ with respect to the four-vector spacetime xµ and the four-momentum pµ . This transformation requires some acrobatics, which will now be performed. In order to switch from mass-shell quantities to non-mass-shell quantities, we start with some helpful identities. Consider
g
µ1 ...µr
i
(t, x , pi ) =
Z
+∞
−∞
Gµ1 ...µr (t, xi , p0 , pi )δ(p0 − p0 (t, xi, pi ))dp0
(113)
the analogue of (7) for an arbitrary tensor Gµ1 ...µp (t, xi , p0 , pi ) of order r, i.e., the tensor g µ1 ...µr is the on-shell restriction of an off-shell tensor Gµ1 ...µr . Next, we derive two auxiliary identities, which will help us to transform the streaming term for f∗ to a streaming term for F∗. Differentiating (113) with respect to xµ we find the first identity
27
Z +∞ ∂g µ1 ,...,µr ∂ ∂p0 ∂ = + µ Gµ1 ,...,µr (t, xi, p0 , pi ) dδ p0 µ ∂xµ (t,xi,pi ) ∂x ∂x ∂p 0 −∞
(114)
where we used the abbreviation (9). Differentiating (113) with respect to pi we get the second identity Z +∞ ∂ ∂p0 ∂ ∂g µ1 ,...,µr = + Gµ1 ,...,µr (t, xi, p0 , pi ) dδ p0 ∂pi ∂pi ∂pi ∂p0 −∞ (t,xi ,pi)
(115)
In deriving (114) and (115) we used the definition of the derivative of a delta-function Z
+∞
−∞
0
h(y)δ (y0 − y)dy = −
Z
+∞
−∞
h0(y)δ(y0 − y)dy
and the fact that G(p0 ) has been chosen to be a function which is such that its derivatives vanish at p0 = ±∞.
Note that in the left-hand side of (114), the derivative ∂/∂xµ, (µ = 0, 1, 2, 3), acts on g at constant xν , (ν 6= µ), and constant pi , (i = 1, 2, 3), whereas, in the right-hand side of (114), ∂/∂xµ acts on G at constant xν , ν 6= µ and constant pα , (α = 0, 1, 2, 3). Similarly, in the left-hand sides of (115), the operator ∂/∂pi works on g at constant xµ , µ = 0, 1, 2, 3 and constant pj , j 6= i; but, in the right-hand side of (115), ∂/∂pi acts on G at constant xµ , µ = 0, 1, 2, 3 and constant pj , j 6= i and constant p0 . These auxiliary results may now be used to rewrite the left-hand side (13) of the Boltzmann equation for f∗ , eq. (12), as an integral over F∗. To that end we choose Gµ (x, p∗) :=
1 p0 (x, p∗) 28
pµ (x, p∗ )F∗(x, p∗ )
(116)
We then have, using (7),
µ
i
g (t, x , pi ) = =
Z
∞
1
0 −∞ p (x, p∗ )
1 p0 (pi )
pµ (x, p∗ )F∗(x, p∗ )δ (p0 − p0 (pi )) dp0
pµ f∗ (t, xi, pi )
(117)
In the last line of this equation pµ = (p0 (pi ), p1 , p2 , p3 ), as a result of the action of the delta-function in the first line: on the mass-shell the variable p0 takes the value p0 (pi ) given by eq. (4) while p0 takes the value p0 (pi ) given by eq. (5). As a next step, let us substitute g µ (t, xi, pi ), eq. (117), and Gµ (x, p∗), eq. (116), into the left- and right-hand sides of the first identity, eq. (114). The left-hand side is easy, we immediately find the left-hand side of (119). The right-hand side is a little bit tricky. First, we note that because of the deltafunction occurring in (116), the 1/p0 at the right-hand side may be replaced by the function 1/p0 (pi ). In other words, under the integral, the integrand Gµ can be considered as the product of the function 1/p0 (pi ) and the function pµ F∗(t, xi, p0 , pi ), i.e., µ
G =
1 0 p (pi )
pµ F∗(t, xi, p0 , pi )
(118)
Secondly, we use Leibniz’s rule to find the derivative of the product (118). We so obtain µ ∂ p i i f∗ (t, x , p ) = ∂xµ p0 (pi ) Z +∞ ∂ ∂p0(pi ) ∂ 1 + (pµ F∗ ) + = 0 (p ) µ µ ∂p p ∂x ∂x i 0 −∞ µ p F∗ ∂ ∂p0 (pi ) ∂ 0 − 0 + p (pi ) dδ p0 p (pi )2 ∂xµ ∂xµ ∂p0
29
(119)
In the left-hand side, we replace the derivative ∂/∂xµ by (c−1 ∂/∂t, ∂/∂xi). Moreover, we replace pµ /p0 (pi ) by (1, c−1 ∂xi/dt). In second term of the righthand side, we replace the function 1/p0 (pi )2 by 1/p0 p0 (pi ), which can be done thanks to the delta-function. Finally, we multiply both sides with p0 (pi )/m. We thus obtain the identity i ∂ dx p0 (pi ) ∂ f∗ + i f∗ = mc ∂t ∂x dt Z +∞ 1 ∂ ∂p0 (pi ) ∂ = + (pµ F∗) + µ µ ∂p m ∂x ∂x 0 −∞ 1 pµ F∗ ∂ ∂p0 (pi ) ∂ 0 − + p (pi ) dδ p0 m p0 ∂xµ ∂xµ ∂p0
(120)
In this way we have achieved that the first two terms of the streaming term s∗ , given by eq. (13), containing the on-shell distribution function f∗ are expressed as an integral over the off-shell distribution function F∗ . The third and last term of s∗ , eq. (13), may be rewritten is a similar way, applying the second identity, eq. (115) to
G :=
mc dpi F∗(x, p∗ ) p0 dτ
(121)
where cτ is the path length of a particle in four space. Firstly, by substituting (121) into (113) we find for the on-shell counterpart g of G ∞
1 dpi F∗ δ (p0 − p0 (pi )) dp0 0 −∞ p dτ mc dpi = 0 f∗ p (pi ) dτ dpi = f∗ dt
g = mc
Z
30
(122)
In the last step we used the well-known relation p0 (pi )dτ = mcdt between eigentime τ and time t. Secondly, by substituting this expression for g and the expression (121) for G into (115) we find
∂ ∂pi
dpi f∗ dt
+∞
mc ∂ ∂p0 ∂ dpi = + F∗ + p0 (pi ) ∂pi ∂pi ∂p0 dτ −∞ ∂ ∂p0 ∂ dpi mcF∗ + p0 dδ p0 − dτ (p0 )2 ∂pi ∂pi ∂p0 Z
(123)
implying p0 (pi ) ∂ mc ∂pi
Z +∞ dpi ∂ ∂p0 ∂ dpi f∗ + F∗ + = dt ∂pi ∂pi ∂p0 dτ −∞ dpi F∗ ∂ ∂p0 ∂ 0 + p (pi ) dδ p0 (124) − dτ p0 ∂pi ∂pi ∂p0
In this way we have rewritten the third term of s∗ (f∗) (13) as an integral containing the off-shell function F∗. By adding the identities (120) and (124) we find for the streaming term s∗ (f∗ ) the expression s∗ (f∗) =
Z
+∞
[A + B + C + D + E + F + G]dδ p0
(125)
−∞
where
1 ∂ 1 ∂p0 (pi ) ∂ µ (pµ F∗), B= (p F∗ ) µ m ∂x m ∂xµ ∂p0 1 pµ F∗ ∂p0 (pi ) 1 pµ F∗ ∂p0 (pi ) ∂p0 (pi ) C =− , D = − m p0 ∂xµ m p0 ∂xµ ∂p0 ∂ dpi ∂p0 ∂ dpi E = i F∗ , F = F∗ ∂p dτ ∂pi ∂p0 dτ dpi F∗ ∂p0(pi ) dpi F∗ ∂p0 ∂p0(pi ) G =− , H = − dτ p0 ∂pi dτ p0 ∂pi ∂p0 A =
31
(126) (127) (128) (129)
These expressions for A, B, . . . , H can immediately, without any calculation, be read off from the expressions (120) and (124). We shall now rewrite this expression for s∗ in a shorter form. As a first step in the process of rewriting (125), we take together the terms B and F : 1 ∂p0 ∂pµ 1 ∂p0 µ ∂F∗ ∂p0 dpi ∂F∗ ∂p0 ∂ B+F = F∗ + p + + F∗ µ µ m ∂x ∂p0 m ∂x ∂p0 ∂pi dτ ∂p0 ∂pi ∂p0
dpi dτ (130)
This expression can be reduced by noting p0 is a function of t, xi and pi . Using the chain rule of differentiation we get pµ ∂p0 ∂p0 dpi dp0 = + µ dτ m ∂x ∂pi dτ
(131)
where we used that dxµ /dτ = pµ /m. Hence, the second and third terms of (130) can be taken together, and we obtain dp0 ∂F∗ 1 ∂p0 ∂pµ ∂p0 ∂ dpi B+F = + F∗ + dτ ∂p0 m ∂xµ ∂p0 ∂pi ∂p0 dτ ∂ dp0 1 ∂p0 ∂pµ ∂p0 ∂ dpi ∂ ∂p0 = F∗ + F∗ + − ∂p0 dτ m ∂xµ ∂p0 ∂pi ∂p0 dτ ∂p0 ∂τ (132) As a next step in the reduction of (125) we consider the sum F∗ pµ ∂p0 (pi ) dpi ∂p0 (pi ) ( + ) p0 m ∂xµ dτ ∂pi F∗ ∂p0(pi ) pµ ∂p0 (pi ) dpi ∂p0 − 0 ( + ) p ∂p0 m ∂xµ dτ ∂pi
C +G+D+H = −
32
(133) (134)
The first term between parentheses can be rewritten by using the chain rule dp0 pµ ∂p0 dpµ ∂p0 = + dτ m ∂xµ dτ ∂pµ
(135)
whereas the second term between parentheses can be reduced with the help of (131). We so find F∗ ∂p0 dp0 F∗ dp0 dp0 ∂p0 C +G+D+H =− 0 − − 0 p dτ dτ ∂p0 p ∂p0 dτ 0 F∗ dp =− 0 p dτ
(136) (137)
Hence, from (132) and (137), ∂ B+F +C +G+D+H = ∂p0
dp0 F∗ dτ
+∆
(138)
where ∆(x, p∗) stands for the abbreviation
∆(x, p∗) := F∗
1 ∂p0 ∂pµ ∂p0 ∂ dpi ∂ ∂p0 1 dp0 + − − m ∂xµ ∂p0 ∂pi ∂p0 dτ ∂p0 ∂τ p0 dτ
(139)
which will be shown to vanish identically in appendix B . We so find ∂ E +B +F +C +G+D+H = ∂pµ
dpµ F∗ dτ
(140)
and, finally, 1 ∂ ∂ µ A+E +B +F +C +G+D+H = (p F ) + ∗ m ∂xµ ∂pµ
dpµ F∗ dτ
(141)
33
By substituting (141) into (125) we find the expression (14) for the streaming term s∗ . What still has to be shown ist that the term ∆(x, p∗ ) vanishes.
B
Proof that ∆(x, p∗ ) vanishes
Let us rewrite systematically the four terms of (139) in terms of the components of the metric gαβ and the Christoffel symbols Γµαβ . In order to do so, we will first express all elements occurring in ∆(x, p∗ ) in terms of these quantities. Firstly, from the equation which states that the physical momentum is on the mass-shell, g 00 p0 (pi )2 + 2g 0i p0 pi + g ij pi pj = m2c2
(142)
we find, by differentiation of this equation with respect to spacetime xµ , p0
∂p0 1 ∂g 00 2 ∂g 0i 1 ∂g ij = − p − p p − pi pj 0 i ∂xµ 2 ∂xµ 0 ∂xµ 2 ∂xµ
(143)
for the first factor of the first term of ∆, eq. (139). Secondly, from (1) we find the second factor of the first term of ∆: ∂pµ = g 0µ ∂p0
(144)
Thirdly, by differentiation of (142) with respect to pi we get p0
∂p0 = −pi . ∂pi
(145)
for the first factor of the second term of (139). Fourthly, the second factor of the second term, and the third term of ∆ both follow from the geodesic equation for the covariant momentum (18) after one has taken the derivative with respect to p0 : 34
∂ dp0 1 ∂gαβ 0α β = g p , ∂p0 dτ m ∂x0
∂ dpi 1 ∂gαβ 0α β = g p ∂p0 dτ m ∂xi
(146)
Fifthly and finally, the fourth term of ∆ follows from the geodesic equation for the contravariant momentum [7, section 87, p. 245] dpµ 1 = − Γµαβ pα pβ dτ m
(147)
We now substitute these results into (139). We then find 1 ∂g ij 1 ∂g 00 2 ∂g 0i p − p p − pi pj mp ∆ = g − 0 i 2 ∂xµ 0 ∂xµ 2 ∂xµ ∂gαβ 0α β g p −pi ∂xi gαβ −p0 0 g 0αpβ ∂x 0 +Γαβ pα pβ 0
0µ
(148)
for the four terms of ∆, respectively. The second and third terms can be taken together. We then get
mp0 ∆ = D − g 0αpβ pµ ∂µ gαβ + Γ0αβ pα pβ
(149)
where we introduced the abbreviation D := g
0µ
p2 1 − 0 ∂µ g 00 − p0 pi ∂µ g 0i − pi pj ∂µ g ij 2 2
(150)
In (149) and (150) we have written ∂µ instead of ∂/∂xµ. We now rewrite this expression for D using the relation [7, formula (86.8), p. 243] 35
∂µ g αβ = −Γαµρ g ρβ − Γβµρ g ρα
(151)
Inserting (151) into (150) we get 1 p20 (−2Γ0µρ )g ρ0 − p0 pi (−Γ0µρ g ρi − Γiµρ g 0ρ ) − pi pj (−Γiµρ g ρj − Γjµρ g ρi )] 2 2 0µ 0ρ iρ 0 0ρ i jρ i = g [p0 (p0 g + pi g )Γµρ + pi (p0 g Γµρ + pj g )Γµρ ]
D = g 0µ [−
= g 0µ [p0 pα g αρ Γ0µρ + pi pα g αρ Γiµρ ] = g 0µ pα pβ g αρ Γβµρ
(152)
With this result we find for (149): mp0 ∆ = g 0µ pβ pρ Γβµρ − g 0αpβ pµ ∂µ gαβ + Γ0αβ pα pβ
(153)
Using now the expression 1 Γβµρ = g βκ (∂µ gκρ + ∂ρ gκµ − ∂κ gµρ ) 2
(154)
we find 1 0µ ρ βκ g pβ p g (∂µ gκρ + ∂ρ gκµ − ∂κ gµρ ) − g 0αpβ pµ ∂µ gαβ 2 1 + pα pβ g 0κ (∂αgκβ + ∂β gκα − ∂κ gαβ ) 2 1 0µ κ ρ = g p p (∂µ gκρ + ∂ρ gκµ − ∂κ gµρ ) − g 0αpβ pµ ∂µ gαβ 2 1 +pα pβ g 0κ (∂αgκβ − ∂κ gαβ ) 2 1 0µ κ ρ = g p p ∂µ gκρ − g 0α pβ pµ ∂µ gαβ 2 1 +pα pβ g 0κ (∂αgκβ − ∂κ gαβ ) 2 = 0 (155)
mp0 ∆ =
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This was the result used in the preceding section.
C
C.1
Invariant volume elements in four-momentum space Covariant momentum
In the derivation of the expression for the off-shell collision term C∗ (F∗, F∗), we encounter the invariant volume element d4δ Vp∗ , given by (47), which brings off-shell covariant four-vectors pµ back to their mass-shell. In this appendix we will prove the equality (46), containing this invariant volume element. From (I.??) we find 1 mc 1 √ εijk dqi ∧ dqj ∧ dqk 3! q 0 − det g 1 mc 1 √ = εµνρσ dqµ ∧ dqν ∧ dqρ ∧ dqσ 4! q 0 − det g
dq0d3 Vq∗ = dq0
(156)
or dq0 d3 Vq∗ =
mc 4 d Vq∗ q0
(157)
where d4 Vq∗ is given by (I. ??) combined with (I. ??). Using now the identity (I. ??) we can rewrite this equality in the form
δ(q0 − q0(t, xi, qi ))dq0d3 Vq∗ = 2q 0θ(q0 )δ(q∗2 − m2c2 ) implying (46), the equation to be proved.
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mc 4 d Vq∗ q0
(158)
C.2
Contravariant momentum
A derivation similar to the one performed in the preceding subsection yields
dδ q 0 d3 Vq = d4δ Vq
(159)
References [1] S. Chapman and T. Cowling, The mathematical Theory of non-uniform Gases, 2nd Ed., Cambridge University Press, 1952. [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 I — Covariant Formalism to appear. [4] J. Ehlers, in: Relativity, Astrophysics and Cosmology, Proceedings of the summer school held, 14–26 August, 1972, at the Banff Centre, Banff, Alberta, W. Israel, ed. [5] W. Israel, in: Relativistic Fluid Dynamics, A. Anile and Y. ChoquetBruhat, eds., Springer Verlag, 1989. [6] W. Israel, The Relativistic Boltzmann Equation in: General Relativity, papers in honour of J.L. Synge, p. 201–241. [7] L.D. Landau and E.M. Lifshitz, The classical theory of fields, fourth revised edition, Pergamon Press, Oxford, 1975. [8] J.A. Peacock, Cosmological Physics, Cambridge University Press, 1999. [9] R.K. Sachs, Survey of general relativity theory, in: [4], pp. 197–236.
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