Action for matter and gravitation

Covariant volume element

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=xc

xa
xd

xb
gcd.
(1)
Taking determinant of both sides gives g=J2g′, or


 

g
 
=J

 

g
 
(2)
where J′=| [(∂xa)/(∂xb)] | is the Jacobian of the transformation. The 4-volume transforms as
dΩ =
xa

xb

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→ gabbϕ, and ∂aAbDaAb. For example,


(−Aaja)dΩ→
(−Aaja)

 

g
 
dΩ  ,
(5)


−1

2
aϕ∂aϕdΩ→
−1

2
gabaϕ∂bϕ

 

g
 
d
(6)


−1

16π
FabFabdΩ→
−1

16π
FabFab

 

g
 
d
(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



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




Rab 1

2
Rgab
δgab

 

g
 
dΩ.
(13)
Combining (8) and (13) into δ(Sm+Sg)=0 gives the famous Einstein's equation,
Rab 1

2
RgabTab  .
(14)

Exercises2

  1. In a curved space the electromagnetic field strength tensor Fab is defined as Fab=Ab;aAa;b and the first Maxwell equation is Fab;c+Fbc;a+Fca;b=0. Show that in the torsion free space of general relativity, Γabcacb, these equations can still be written as in Minkowski space, Fab=Ab,aAa,b and Fab,c+Fbc,a+Fca,b=0.
  2. Derive the second Maxwell equation in a curved space,



     

    g
     
    Fab

    ,a 
    =4π

     

    g
     
    jb  ,
    from the action


    1

    16π
    FabFabAaja


     

    g
     
    dΩ.
    Show that the equation can also be written as Fab;a=4πjb, Hints:
    1. show that Γaba=[1/(2g)]g,b=(ln√{−g}),b
    2. show that Fab;a=[1/(√{−g})](√{−g}Fab),a

Footnotes:

1 dg=ggabdgab=−ggabdgab.
2 notation: ;aDa ≡ [(D)/(dxa)] and ,a ≡ ∂a ≡ [(∂)/(dxa)]


File translated from TEX by TTH, version 3.77.
On 6 Oct 2007, 00:26.