In a curved space the volume element dΩ ≡ d4x has to be
substituted with a covariant expression, √{−g}dΩ, where g
is the determinant of the metric tensor gab (g < 0). Indeed the
metric tensor transforms as
gab=
∂x′c
∂xa
∂x′d
∂xb
g′cd.
(1)
Taking determinant of both sides gives g=J′2g′, or
√
−g
=J′
√
−g′
(2)
where J′=| [(∂x′a)/(∂xb)] |
is the Jacobian
of the transformation.
The 4-volume transforms as
dΩ =
∂xa
∂x′b
dΩ′ =
1
J′
dΩ′.
(3)
Apparently the combination √{−g}dΩ transforms as
√
−g
dΩ = J′
√
−g′
1
J′
dΩ′=
√
−g′
dΩ′,
(4)
and is thus a covariant volume element.
Action for matter
The action for matter in general relativity is the good old action from
special relativity, only rewritten, if needed, in a generally covariant
way. Particularly, dΩ→√{−g}dΩ,
∂aϕ→ gab∂bϕ, and
∂aAb→ DaAb. For example,
⌠ ⌡
(−Aaja)dΩ→
⌠ ⌡
(−Aaja)
√
−g
dΩ ,
(5)
⌠ ⌡
−1
2
∂aϕ∂aϕdΩ→
⌠ ⌡
−1
2
gab∂aϕ∂bϕ
√
−g
dΩ
(6)
⌠ ⌡
−1
16π
FabFabdΩ→
⌠ ⌡
−1
16π
FabFab
√
−g
dΩ
(7)
Energy-momentum tensor of matter
The variation of the matter action, Sm=∫L√{−g}dΩ,
due to variation δgab can be written in terms of a symmetric
tensor Tab,
δSm =
1
2
⌠ ⌡
Tab δgab
√
−g
dΩ = −
1
2
⌠ ⌡
Tab δgab
√
−g
dΩ,
(8)
where
1
2
√
−g
Tab =
δ(
√
−g
L)
δgab
.
(9)
This tensor satisfies the equation Tab;b=0 which in a flat space
turns into the energy-momentum conservation equation Tab,b=0
and we thus assume that it is the energy-momentum tensor (see also the
exercises next week).
Hilbert's action and Einstein's equation
The Hilbert action for the gravitation,
Sg = −
1
2κ
⌠ ⌡
R
√
−g
dΩ,
(10)
where κ is a (coupling) constant,
is the one which leads to Einsteins field equations. Its
variation with respect to δgab is
δ
⌠ ⌡
R
√
−g
dΩ = δ
⌠ ⌡
gabRab
√
−g
dΩ =
=
⌠ ⌡
Rab
√
−g
δgab+Rabgabδ
√
−g
+
+gab
√
−g
δRab
dΩ.
(11)
The last term can be proved (see t'Hooft) to not contribute; in the second
term we have1
δ
√
−g
=−
1
2
1
√
−g
ggabδgab = −
1
2
√
−g
gabδgab.
(12)
Thus the variation of the Hilbert action is
δSg = −
1
2κ
⌠ ⌡
Rab−
1
2
Rgab
δgab
√
−g
dΩ.
(13)
Combining (8) and (13) into δ(Sm+Sg)=0
gives the famous Einstein's equation,
In a curved space the electromagnetic field strength tensor Fab
is defined as Fab=Ab;a−Aa;b and the first Maxwell equation is
Fab;c+Fbc;a+Fca;b=0. Show that in the torsion free space of
general relativity, Γabc=Γacb, these equations
can still be written as in Minkowski space, Fab=Ab,a−Aa,b
and Fab,c+Fbc,a+Fca,b=0.
Derive the second Maxwell equation in a curved space,
√
−g
Fab
,a
=4π
√
−g
jb ,
from the action
⌠ ⌡
−
1
16π
FabFab −Aaja
√
−g
dΩ.
Show that the equation can also be written as
Fab;a=4πjb,
Hints:
show that Γaba=[1/(2g)]g,b=(ln√{−g}),b
show that Fab;a=[1/(√{−g})](√{−g}Fab),a
Footnotes:
1dg=ggabdgab=−ggabdgab.
2 notation:
;a ≡ Da ≡ [(D)/(dxa)] and
,a ≡ ∂a ≡ [(∂)/(dxa)]
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