Добрый вечер!
Помогите, пожалуйста, решить такую вот задачку:
Имеется тензор энергии-импульса (ТЭИ) электромагнитного поля
![$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa
% aaleaacqaH8oqBaeaacqaH9oGBaaaaaa!3A61!
\[
T_\mu ^\nu
\]
$ $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaDa
% aaleaacqaH8oqBaeaacqaH9oGBaaaaaa!3A61!
\[
T_\mu ^\nu
\]
$](https://dxdy-03.korotkov.co.uk/f/e/4/2/e427ed36ba3e8303fc40136a8d23d20982.png)
в пространстве с метрикой
![$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaado
% hadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcqGHsislcaWGKbGaamiD
% amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadggadaahaaWcbeqaai
% aaikdaaaGccaGGOaGaamiDaiaacMcacaGGOaGaamizaiaadIhadaah
% aaWcbeqaaiaaikdaaaGccqGHRaWkcaWGKbGaamyEamaaCaaaleqaba
% GaaGOmaaaakiabgUcaRiaadsgacaWG6bWaaWbaaSqabeaacaaIYaaa
% aOGaaiykaaaa!4E41!
\[
ds^2 = - dt^2 + a^2 (t)(dx^2 + dy^2 + dz^2 )
\]
$ $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaado
% hadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcqGHsislcaWGKbGaamiD
% amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadggadaahaaWcbeqaai
% aaikdaaaGccaGGOaGaamiDaiaacMcacaGGOaGaamizaiaadIhadaah
% aaWcbeqaaiaaikdaaaGccqGHRaWkcaWGKbGaamyEamaaCaaaleqaba
% GaaGOmaaaakiabgUcaRiaadsgacaWG6bWaaWbaaSqabeaacaaIYaaa
% aOGaaiykaaaa!4E41!
\[
ds^2 = - dt^2 + a^2 (t)(dx^2 + dy^2 + dz^2 )
\]
$](https://dxdy-02.korotkov.co.uk/f/9/9/2/9926356ff1c7a147bfc6b0dbf1fc0cc182.png)
(1)
имеется, также, ТЭИ того же поля
![$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmivayaara
% Waa0baaSqaaiabeY7aTbqaaiabe27aUbaaaaa!3A79!
\[
\bar T_\mu ^\nu
\]
$ $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmivayaara
% Waa0baaSqaaiabeY7aTbqaaiabe27aUbaaaaa!3A79!
\[
\bar T_\mu ^\nu
\]
$](https://dxdy-03.korotkov.co.uk/f/6/6/e/66eb7f2be0fcf1b4579f9d173c1e2cc882.png)
в пространстве с метрикой
![% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGKb
% Gabm4CayaaraWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaeyOeI0Ia
% amizaiabeE7aOnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadsgaca
% WG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamizaiaadMhadaah
% aaWcbeqaaiaaikdaaaGccqGHRaWkcaWGKbGaamOEamaaCaaaleqaba
% GaaGOmaaaaaOqaaiaadsgacaWGZbWaaWbaaSqabeaacaaIYaaaaOGa
% eyypa0JaamyyamaaCaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaai
% ykaiaadsgaceWGZbGbaebadaahaaWcbeqaaiaaikdaaaaaaaa!5476!
\[
\begin{array}{l}
d\bar s^2 = - d\eta ^2 + dx^2 + dy^2 + dz^2 \\
ds^2 = a^2 (t)d\bar s^2 \\
\end{array}
\]
$ % MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGKb
% Gabm4CayaaraWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaeyOeI0Ia
% amizaiabeE7aOnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadsgaca
% WG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamizaiaadMhadaah
% aaWcbeqaaiaaikdaaaGccqGHRaWkcaWGKbGaamOEamaaCaaaleqaba
% GaaGOmaaaaaOqaaiaadsgacaWGZbWaaWbaaSqabeaacaaIYaaaaOGa
% eyypa0JaamyyamaaCaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaai
% ykaiaadsgaceWGZbGbaebadaahaaWcbeqaaiaaikdaaaaaaaa!5476!
\[
\begin{array}{l}
d\bar s^2 = - d\eta ^2 + dx^2 + dy^2 + dz^2 \\
ds^2 = a^2 (t)d\bar s^2 \\
\end{array}
\]
$](https://dxdy-04.korotkov.co.uk/f/3/2/e/32ecf3e3c0d0bb31344ff5117ae6900982.png)
(2)
Требуется показать, что, исходя из ковариатного закона сохранения ТЭИ
![$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaS
% baaSqaaiabe27aUbqabaGccaWGubWaa0baaSqaaiabeY7aTbqaaiab
% e27aUbaakiabg2da9iaaicdaaaa!3F9F!
\[
\nabla _\nu T_\mu ^\nu = 0
\]
$ $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaS
% baaSqaaiabe27aUbqabaGccaWGubWaa0baaSqaaiabeY7aTbqaaiab
% e27aUbaakiabg2da9iaaicdaaaa!3F9F!
\[
\nabla _\nu T_\mu ^\nu = 0
\]
$](https://dxdy-03.korotkov.co.uk/f/e/0/5/e05be8c7313d5eec2995f23e794c936182.png)
, следует
![$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaS
% baaSqaaiabe27aUbqabaGcceWGubGbaebadaqhaaWcbaGaeqiVd0ga
% baGaeqyVd4gaaOGaeyypa0JaaGimaaaa!3F97!
\[
\partial _\nu \bar T_\mu ^\nu = 0
\]
$ $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaS
% baaSqaaiabe27aUbqabaGcceWGubGbaebadaqhaaWcbaGaeqiVd0ga
% baGaeqyVd4gaaOGaeyypa0JaaGimaaaa!3F97!
\[
\partial _\nu \bar T_\mu ^\nu = 0
\]
$](https://dxdy-01.korotkov.co.uk/f/0/2/c/02c993a2fa42c4f36fe0e9c850e55c6182.png)
На сколько я понимаю необходимо понять как связаны векторный потенциал, тензор э-м поля и ТЭИ э-м поля в пространстве с метрикой (1) с аналогичными величинами, живущими в пространстве с метрикой (2).
Я исходил из того что
![$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa
% aaleaacqaH8oqBaeqaaOGaeyypa0JaamyyaiaacIcacaWG0bGaaiyk
% aiqadgeagaqeamaaBaaaleaacqaH8oqBaeqaaaaa!3F9D!
\[
A_\mu = a(t)\bar A_\mu
\]
$ $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa
% aaleaacqaH8oqBaeqaaOGaeyypa0JaamyyaiaacIcacaWG0bGaaiyk
% aiqadgeagaqeamaaBaaaleaacqaH8oqBaeqaaaaa!3F9D!
\[
A_\mu = a(t)\bar A_\mu
\]
$](https://dxdy-01.korotkov.co.uk/f/c/2/0/c202699373ea4a71087b3ad1b92a0ff982.png)
, поскольку компоненты 4-вектора преобразуется также как и компоненты 4-радиус вектора. Используя эту связь я получил тензор э-м поля и ТЭИ э-м, выражения оказались громоздкими и, вообще, я сомневаюсь что это верный ход решения.
У кого-нибудь есть какие-н. соображения на этот счёт?
Благодарю за помощь!