5.3 Komar Integrals

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Let V be a volume of spacetime on a spacelike hypersurface _, with boundary

@V . To every Killing vector _eld _ we can associate the Komar integral

Q_(V ) =

c

16_G

I

@V

dS__D___ (5.32)

for some constant c. Using Gauss' law

Q_(V ) =

c

8_G

Z

V

dS_D_D___ (5.33)

Lemma D_D___ = R____ for Killing vector _eld _.

Proof By contraction of previous `Killing vector Lemma.'

Using Lemma,

Q_(V ) =

c

8_G

Z

V

dS_R_

___ (5.34)

= c

Z

dS_

_

T_

_ __ 􀀀

1

2

T__

_

(by Einstein's eqs.) (5.35)

=

Z

dS_J_(_) (5.36)

where

J_(_) = c

_

T_

_ __ 􀀀

1

2

T__

_

(5.37)

Proposition @_J_(_) = 0.

Proof Using D_T__ = 0 we have

D_J_ = c

_

T__D___ 􀀀

1

2

TD___

_

| {z }

0 for Killing vector _

􀀀

c

2

_ _ @T (5.38)

=

c

2

_ _ @R (by Einstein's eqs.) (5.39)

= 0 for Killing vector _eld _ (5.40)

97

(In this last step, choose coordinates s.t. _ _ @ = @=@_, then the metric is

_-independent (@g__=@_ = 0), so R is too (@R=@_ = 0)).

Since J_(_) is a `conserved current', the chargeQ_(V ) is time-independent

provided J_(_) vanishes on @V , just as for electric charge.

Exercise _ = k (time-translation Killing vector _eld)

E(V ) = 􀀀

1

8_G

I

@V

dS__D_k_ (5.41)

i.e. c = 􀀀2, is _xed by comparison with previous formula derived for total

energy, i.e. by choosing V = 2-sphere at spatial 1.

Exercise Verify that E(V ) = M for Schwarzschild, for any V with @V in exterior (r >

2M) spacetime.

5.3.1 Angular Momentum in Axisymmetric Spacetimes

Return to Komar integral. Let _ = m = @=@_ and choose c = 1 to get

J(V ) =

1

16_G

I

@V

dS__D_m_ (5.42)

Note here factor of 􀀀1=2 relative to Komar integral for the energy.

To check coe_cient, use Gauss' law to write J(V ) =

R

V dS_J_(m) where

J_(m) = T_

_m_ 􀀀

1

2

Tm_ (5.43)

If we choose V to be on t = constant hypersurface, and m = @=@_, then

dS_m_ = 0, so

J(V ) =

Z

V

dV T0

_m_ =

Z

V

dV

􀀀

T0

2x1 􀀀 T0

1x2_

(5.44)

in Cartesian coordinates

_

xi; i = 1; 2; 3

           

where

m = x1 @

@x2 􀀀 x2 @

@x1 (5.45)

For a weak source, g _ _ and

J(V ) _ "3jk

Z

V

d3x xjTk0 (5.46)

98

which is result for 3rd component of angular momentum of _eld in Minkowski

spacetime with stress tensor T__.

So the total angular momentum of an asymptotically at spacetime is

found by taking @V to be a 2-sphere at spatial in_nity

J =

1

16_G

I

1

dS__D_m_ (5.47)

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