To the method of the kinetic equation in general relativity

• 1.    Explicitly covariant ways of writing the kinetic equation

The kinetic equation in general relativity is (see [1])

K.f _a ( x,p) = I_a(T,p),                                         (1.1)

where -    8       id

K = P — P P .’ 8хг          8pa can be written in an explicitly covariant form by introducing invariant momentum components

p(m) = ^^т_;/, р^г = ^P^.

Here ©") is an arbitrary quadruple of vectors linearly independent at each point, and л^т) is a quadruple with ©") vectors (^j™)^^) = ^(т)))- Choosing p(a) as variables in the equation (1.1), we reduce the operator К to the form к — п(т), d +п(т)п(п)o' F(a) d                            (12)

К = P   8(т) ужг +P P ^(т)^(„)^г;, QpCt , where semicolon stands for covariant derivative. If the space admits several linearly independent Killing vectors at each point, then the operator К is simplified by choosing Killing vectors as ©"). If we choose an orthoreper as ^г(т), then we get an orthoreper notation of the kinetic equation (1.1).

Note one more property of the operator К, which is verified by direct computation

К а г 1 г ₂ ...г „ P 1 P 2 "• P "  = P^P 1 P 2 •• • P "  ^1 г ₂ ...г „ ;к .                            (1-3)

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This equality can be easily generalized if we bear in mind that the scalar function / ( х,р) can depend on the vector р^г only as a complex function /(х, д(х, р)), wliere д ( х,р) conventionally denotes the set of all scalars of the form а_г1_г2..._г„ р^г 1 р^г 2 • • • р^г "

К /(х,р) = р V г / ( х,р )

(1.4)

Here Vj means the derivative, which is calculated as ifpг is a covariantly constant vector. We will call such a. derivative the covariant partial derivative.2

Using (1.4), we can write (1.1) as pгVг f а(х,р^ = То(х,р),

(1.5)

and the generalization of the kinetic equation to the case of electromagnetic forces [3] in the form pгVг + -^ F“г рг — с      ора

^ а ⁽ Х^,р ⁾ = 1_а^(х,р⁾.

(1-6)

п what follows, we will need a. linearized kinetic equation in the collisionless approximation for the case when the electromagnetic field is absent in the equilibrium state, and in the nonequilibrium state it is “on” weak electromagnetic and gravitational disturbance. As shown in [2,3], the distribution function in a. nonequilibrium state should be sought in the form

/а(^х,р) = ^ а ^{( х,}р)^{{ 1+} фа^р ) }.

Then 1р_а ( х,р) will satisfy the equation:

р^ г Ч> _а ( х,р) = — р^гЕ,^кF k - р^гр^ _k 5 Г ^к с                          ^J

(1.7)

(in the corresponding equations obtained in [2,3], we replaced the operator К bу р^гV_г).

In (1.4) introduces the notion of a. partial covariant derivative. Note one property of its j Дррг1

• • р

I ‘ " J ( х,р)   = У Дрр^г

. г1

• р¹ " V j J ( х,р )

(1.8)

Неге Др = ^-^ ^р is an invariant volume element in momentum space, J ( х,р ) is a scalar or tensor function.

If we choose geodesic coordinates at the point Q as the coordinate system, then the equality (1.8) turns into a. trivial statement about the possibility of differentiating an integral with respect to a. parameter. Since the terms of this equality are components of the tensor, it takes place in any coordinate system and any point.

As an application of the property (1.8), we obtain the equations for the moments of the distribution function

/ ^г 1 ^г 2 ... ^г " = у Др^ ( х,р ) р^г1 ••• р^г "

by the following transformations of the kinetic equation:

рг Vг /а(х,р) = 1а(х,р),

У dpp^J1 • • • p^Jг " р^г V г / _а) х,р ) = У Дрр³¹ • • • p^Jг " 1 _а ( х,р ) ,

V г f J ¹ ... ^J " ^г (х)= Ц"³ " ( х ) .

(1.9)

As you can see, the concept of a. partial covariant derivative is a. rather convenient mathematical trick that allows you to avoid sometimes cumbersome calculations (compare, for example, the derivation of the equation (1.9) in [1]).

2. Calculation of the 4-vector of the current of a nonequilibrium relativistic gas

The perturbation of the 4-vector of the current and the energy tensor — the momentum of a. relativistic gas in a. nonequilibrium state is determined by the formulas:

jk = ^eac / dppkf2(x,p)pa(x,p), a t^ = ^ c ( dpp'lpkfa0(x,p)pa(x,p). a

(2.1)

(2-2)

These expressions are valid if 5 r ^k j = 0. In general, J^k = j'^k + j^k, t^ik = t^ + t ^ik, wliere j'^k ,t'^l 1 ^k are calculated from (2.1) and (2.2), and j^ , t0^k are easily found from the equilibrium distribution function (see [2]).

In the expressions (2.1), (2.2) it is necessary to substitute the solution of the equation (1.7) in the collisionless approximation. We will write this solution in the form

∞

^a(x,p) ^ j dr e" -^'  [ pe^kF k, - ■p^jI k 5T^k ₃ ] .                  (2.3)

Substitution of (2.3) into (2.1) and (2.2) followed by formal integration over momenta, (considering that V₄ is a partial covariant derivative that does not act on p⁴) is possible only if the commutators [1^ , V j ] and [V₄, V j ] are equal to zero, that is, only in flat space. Obviously, in this case, when the unperturbed metric (background metric) admitting an equilibrium state of the gas is flat, the expressions obtained in this way will coincide after the Fourier transform with the corresponding expressions for a. homogeneous plasma.

In the case when the background metric is not flat, i.e., the indicated commutators do not vanish, we will proceed as follows. Substitute (2.3), for example, into (2.1) and expand in a. power series

ik =

J a

∞∞ eac / dre-8T ^ (-p [ dpfa0(x,p)pl

0          "=0 ^'

-^k p* 1 ...p¹ " V₄ ... V "    hF%p^j.         (2.4)

c J

After that, we count in some approximation each of the members of the series (2.4), and then again sum up the series. This can be done, for example, when the conditions

X T « X g ~ XW g ,

(2 . 5 a )

X ^L,

(2 . 5 b )

where X and L are the characteristic size of the disturbance and background inhomogeneity, respectively, and w _g is equal in order of magnitude to cd _i g _kl. Obviously, x _g is the speed acquired by a particle in the gravitational field during the period of wave oscillation (see [4]). This gives the following expression for the 4-current vector (for simplicity, we put 5 Г у = 0):

Ja = ea У dpfa(x,O)

(2.6)

where you need to integrate in a. formal way, and the result of integration is written in the form

Ja^k = AI Y ( I, V²) + д^кг<7 2 ( 1, V²) + a ₃ ( I, V ² ) VⁱV^k + I^I, V ² ) V^k + ^( I, V ² ) Vⁱ ] ^^lFn.

Here I = |ⁱV_i, V² = g^ij V ₄ V j are operators that can be considered commuting in our approximation, and о ₄( !, V²) — scalar operator expressions. In (2.6), the tensor operator T _mn is introduced, which is defined by the relations

VⁱT m = 0,      !4³ T ij = [ Iⁱ ( V i I^j ) V j ]= g.                         (2.7)

The derivation of the formula. (2.6) is given in Appendix I.

Appendix I. Derivation of formula (2.6)

We will start from the expression for the 4-current vector (2.4), which we rewrite in a. different way, using the equality

I dpf2 ( x,p)p^{1 1} • • • p

> гп

n

= f (^p¹-^^..^ ^, 1=0

(I.1)

where

(y, 9)^г1...^г£^|г£+1...^гп = €^(г1 • • • с'> 9 1+ ^г е+2 . . . ^n -

1 ^г п )

.

Substituting (LI) in (2.4), we get

+ = £ J dT ^{f      f} ^2 + gy v .^U^...^ V . 1 • • • V , „ € F _ej .

(I.2)

0 n=0   "'   1=0

Obviously, 7^+2 = 0 for n — I + 2 = 2p + 1. Also the expression г 1 ...г1 \г1+1 ...гп kj y7 , . . . y7 . = ____1(11«рк       г1-г1:-2\У-1...г„ V . . .„V

^{( €,9 )              Vn Угп}   (n +1)(n + 2)^{€€ (€,9)              Vn   Угп}

+ (n +1)(n + 2)9 (€,9)                     ' + (n +1)(n + 2)€9 (€,9)             ^г1'

⁽ⁿ + ²¹⁾¹ ck_aj(_i₁(e yi...4ilii+i...4„⁾v₇.

(I.3)

+ (n + 1)(n + 2) €9   (^,9)'

⁽ⁿ + ^{2 - l)(n - l)} ^ l^j|(₄2/^ \гз...г_£+2|г_£+з...г„⁾у₇. ...у/.

+ (n + 1)(n + 2) 9   9    ^,9).

It is easy to get an approximate value for y^M • • • €'" V_i1 • • • V^:

A

+ ••• + V . 1 •••V₄„ «Г

n(n — 1) -fn—2 9 €

(I.4)

(members containing (V+)ⁿ и (V_г1 • • • V^€^k) при n ^ 2, are discarded).

In the same approximation, the commutator [Vг, Vk] in (1.3) should be set equal to zero. Therefore, (1.3), taking into account (1.4), takes the form

(€,9)

^г 1 ... ^г г ^{| г} г+1 ... ^г n ^кjv _г . • • V _г

+ — 1)    . .-

(n + 1)(n + 2)? ■

y?l-2 - ^{(l — 2)(l — 3)} ^y?l-4

V n — 1+2

^{(n + 2 — 1)1}  jk _c4 _ ^{1(1 —} 1)  4-2I _Vn-l,   ^{(n + 2 — 1)1} _£j  1-1 _ (£—J + —_2) _йс1-з1 _vn-l_vk

(n + 1)(n + 2) ^{9   €}       2     ^{€         +}(n + 1)(n + 2)^{€ €}            2       ^€

+

( n + 2 — 1)1 _€ k (n + 1)(n + 2)^

€ ^{- 1}

^{(l — 1)(l — 2)} ^y?i-3

V^{n - l}V^j

(n + 2 — 1)1 Г 4   1(1 — 1), 4-2I Vn-l-2vkv,

⁺ (fTf)F+4) ^{€ —}         J^{V VV} ’

This equality can be written as

(y, 9)г1...гl|гl+1...гnjkVг1 • • • Vгn = (€, 9)г1...г1|г1+1...гпЗк Vг1 • • • Vin

— n(n-l) (€,9)г1...г£|г£+1...гnjkVг1• ^^гп-2СТгп-1гп , (I.5)

where (€,9)^г1 ...^г1|^г1+1...^гп1"^кV_г1 • ^^_гп is formed from (у,9')^гг..-^г1\^г^+¹-.-^гпЗ^к V_г1 • • • V_гn as follows: we consider €^г and V_k commuting until then after commutations we get an expression in which in the first place there are V_k, not convoluted with the vector €^г, and on the second, the third and fourth places are the corresponding operators is formed similarly E,^p, V^2m and V^j. Similarly, it is formed and

(£, g^^-'^eV ^ +L .-tTijk v^ • ••V . ^ — 2 from (£, g)⁴1-^-4i|W:i. ■■■^г ™ з^к V . 1 •••V . _t 2. Now it remains to substitute

• (1.5)    in (2.4) and take into account (LI)

∞∞

^k = '2 / dr e ^{- ST} / dpf 0 ( x,p ) У ^(-^(p⁴V₄)"p^k p^jptc

(I.6)

o         J            n=o

∞

1У

2 ^

(_T\n+2                                     Г

⁽ ^  (p⁴V₄)Vp^mp^kp^j Tm^ = U у dpf 0 ( x,o )

ГpWFlj _ p^kp^jp^mp " T _mH E,^lF tj P^lV _t              (p»V₄)3

Estimates show that the above calculations are valid under conditions (2.5a)-(2.5b).