5.2 ADM energy

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We cannot de_ne energy in the same way because this is associated with

a conserved symmetric tensor T__ , rather than a vector. This is not unexpected

because a locally conserved energy can exist only in a spacetime

admitting a timelike Killing vector _eld.

[Unlike photons, which do not carry charge, gravitons do carry energy

) possibility of energy exchange between matter and its gravitational _eld.]

We can still de_ne a total energy in asymptotically at spacetimes as

a surface integral at in_nity because @=@t is asymptotically Killing in such

spacetimes. In this case

g__ ! ___ as r ! 1 (___ Minkowski metric) (5.11)

We shall assume that, in Cartesian coordinates,

h__ = g__ 􀀀 ___ = O

_

1

r

_

(5.12)

which will justify a linearization of Einstein's equations near 1.

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Exercise Show that G__ = 8_GT__ becomes the Pauli-Fierz equation

􀀀 h__ + h;__ 􀀀 2h(_;_) = 􀀀16_G

_

T__ 􀀀

1

2

___T

_

(5.13)

where

􀀀 = ___@_@_ (5.14)

h = ___h__ (5.15)

h_ = ___h__;_ = h_

_;_ (5.16)

T = ___T__ (5.17)

Take the trace to get

􀀀 h 􀀀 h_

;_ = 8_GT (5.18)

We shall _rst consider a weak static dust source

T__ =

0

BB@

_ 0

0 0

1

CCA

zero pressure for `dust' (5.19)

__ = 0 for static

4_G_ _ 1

T0i = 0

_

for weak

Since source is static we may assume static h__ , i.e. _h__ = 0. Then _ = _ =

0 component of (5.13) becomes

r2h00 = 􀀀8_GT00 (5.20)

while (5.18) becomes

􀀀 r2h00 + r2hjj 􀀀 hij;ij | {z }

@i (@ihjj 􀀀 @jhij )

= 􀀀8_GT00 (5.21)

Add (5.20) and (5.21) to get

T00 =

1

16_G

@i (@jhij 􀀀 @ihjj ) (Cartesian coordinates) (5.22)

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Since the source is weak we can assume that the spacetime is almost Minkowski,

i.e. we treat h__ as a _eld on Minkowski spacetime. The total energy is now

found by integrating T00 over all space.

E =

Z

t = constant

all space

d3x T00 (5.23)

Using Gauss' law we can rewrite result as the surface integral

E =

1

16_G

I

1

dSi (@jhij 􀀀 @ihjj) (Cartesian coordinates) (5.24)

But this depends only on the asymptotic data, so we may now change the

source in any way we wish in the interior without changing E, provided that

the asymptotic metric is unchanged. So formula for E is valid in general.

This is the ADM formula for the energy of asymptotically at spacetimes.

5.2.1 Alternative Formula for ADM Energy

Subtract (5.21) from (5.20) to get

@i (@jhij 􀀀 @ihjj ) = 􀀀2r2h00 (5.25)

This allows us to rewrite ADM formula as

E = 􀀀

1

8_G

I

1

dSi @ih00 (5.26)

But (Exercise)

gij􀀀 0

0j = 􀀀

1

2

@ih00 + O

_

1

r3

_

(􀀀 = a_ne connection) (5.27)

and hence

E =

1

4_G

I

1

dSi gij􀀀 0

0j (5.28)

=

1

4_G

I

1

dS0iDik0 where k =

@

@t

; dSi _ dS0i (5.29)

But k is asymptotically Killing, i.e.

D_k_ + D_k_ = O

_

1

r3

_

(5.30)

so

E = 􀀀

1

8_G

I

1

dS__D_k_ (5.31)

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