50
3 Mathematical Framework
where the last term corresponds to the contribution of the electromagnetic field given
in Eq. (3.2.55). The conservation equation (3.4.48), along with the Maxwell equation
(3.2.67), then imply the usual conservation of T
ab
∇
a T ab ≡
s
∇
a T
s
ab + ∇
a T
EM
ab
=
s
q s F
a
b T
s
a + ∇ a F
ac F bc +
3
2
F
ac
∇ [a F bc]
=
s
q s F
a
b T
s
a +
s
J
c
s F bc +
3
2
F
ac
∇ [a F bc] ≡ 0 ,
(3.4.52)
where the first two terms cancel each other out and the last term vanishes through
the Bianchi identity in the tetrad basis.
References
1. R. Durrer, A. Neronov, Cosmological magnetic fields: their generation, evolution and observation. Astron. Astrophys. Rev. 21, 62 (2013). arxiv:1303.7121
2. K. Becker, M. Becker, J.H. Schwarz, String theory and M-theory: A Modern Introduction (Cambridge University Press, 2006)
3 Mathematical Framework
where the last term corresponds to the contribution of the electromagnetic field given
in Eq. (3.2.55). The conservation equation (3.4.48), along with the Maxwell equation
(3.2.67), then imply the usual conservation of T
ab
∇
a T ab ≡
s
∇
a T
s
ab + ∇
a T
EM
ab
=
s
q s F
a
b T
s
a + ∇ a F
ac F bc +
3
2
F
ac
∇ [a F bc]
=
s
q s F
a
b T
s
a +
s
J
c
s F bc +
3
2
F
ac
∇ [a F bc] ≡ 0 ,
(3.4.52)
where the first two terms cancel each other out and the last term vanishes through
the Bianchi identity in the tetrad basis.
References
1. R. Durrer, A. Neronov, Cosmological magnetic fields: their generation, evolution and observation. Astron. Astrophys. Rev. 21, 62 (2013). arxiv:1303.7121
2. K. Becker, M. Becker, J.H. Schwarz, String theory and M-theory: A Modern Introduction (Cambridge University Press, 2006)