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Gravitational radiation from linearly accelerating point masses 5/20/04

The total quadrupole moment of a mass density, T x , is defined by:00 00 2( ) ( ) /m T x c =

3 00

( ) ( ) 3

i ji j i j k

mQ t d x T x x x x x

k 1,2,3, i = (1)

Consider the general term: 3 00( ) ( )i j i j mq t d x x x T x=

(2)

Landau & Lifschitz (Class. Theory of Fields, eq. 104.6) show:

2 3( ) 2 ( )i j i jq t c d x T x =

(3)

where ./i iq q t

Eqn. (1) becomes:

2 3

( ) 2 ( ) ( )3

i ji j i j k

m m kQ t c d x T x T x

(4)

Assuming a perfect fluid form , T , for the momentum density:i j i jm m v= v

2 3( ) 2 ( ) ( ) ( ) ( ) ( )3

i ji j i j k

mQ t c d x x x s x s x s x s

k

=

(5)

Now consider a point fluid mass along a path ( )z s=

, then, for flat space-time

( ) ( ( )m ) x m x z s =

i ii dx dxvds c dt

g= = (6)

Eq. (5) becomes, for this case:

2 3( ) 2 ( ( )) ( ) ( ) ( ) ( )3

i ji j i j k

kmc d x x z s x s x s x s x s

Q t =

2( ) 2 ( ) ( ) ( ) ( )3

i ji j i j k

kQ s mc z s z s z s z s =

(7)

We write this in terms ofidz

dt:

2( ) 2 ( ) ( ) ( ) ( )3

i ji j i j k

kQ t m z t z t z t z t

=

(8)

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Taking one more time derivative:

( )2 4 2 2( ) 43 3

i j i ji j i j i jQ t mc = +

(9)

where we have converted to /v c = notation.

From Landau & Lifschitz (Class. Theory of Fields, eq. 104.12) the radiated power is,

using the Q definition, eq. (1):

515

i j

i j

dE GQ Q

dt c

= (10)

which becomes, squaring (9) and taking linear motion (

), and taking advantage of

the definition 2 21 2 = + yields exactly:

28 2 2

3

64

45

dE G ma

dt c = (11)

in terms of the acceleration a c= .

This form differs from the usual electromagnetic result by an additional factor.