I am trying to derive weak-field Schwarzschild metric using Linearized Einstein's field equations of gravity:(adsbygoogle = window.adsbygoogle || []).push({});

[]hμν – 1/2 ημν []h^{γ}γ = -16πG/ c^{4}Tμν

For static, spherically symmetrical case, the Energy- momentum tensor:

Tμν = diag { ρc^{2}, 0, 0, 0 }

Corresponding metric perturbations for static ortho-normal coordinates:

hμν = diag { htt , hxx , hyy , hzz }

With one index rised using flat space-time Minquoskwi metric η^{μν}= { -1 , 1, 1, 1 }:

h^{μ}ν = diag { -htt , hxx , hyy , hzz }

Trace of the metric:

h = h^{γ}γ = - htt + hxx + hyy + hzz

The four equations:

1) []htt – 1/2 ηtt []h^{γ}γ = -16πG/ c^{4}Ttt

=> []htt + 1/2 []( - htt + hxx + hyy + hzz )= -16πGρ/ c^{2}

=> 1/2 []( htt + hxx + hyy + hzz )= -16πGρ/ c^{2}

2) []hxx – 1/2 ηxx []h^{γ}γ = -16πG/ c^{4}Txx

=> []hxx - 1/2 []( - htt + hxx + hyy + hzz )= 0

=> 1/2 []( htt + hxx - hyy - hzz )= 0

Similarly:

3) 1/2 []( htt - hxx + hyy - hzz )= 0

4) 1/2 []( htt - hxx - hyy + hzz )= 0

Adding equations 2), 3) & 4) to 1) respectively, yield:

[]( htt + hxx ) = []( htt + hyy ) = []( htt + hzz )= -16πGρ/ c^{2}

Solving the equations using:

[] ≈ ▼^{2}≈ 1/R^{2}d/ dR ( R^{2}d/ dR ) for static spherically symmetric case; we get:

( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= -8πGρR^{2}/ 3c^{2}– K1/ R + K2

Similar solutions for vacuum case, with Tμν= 0 would be:

( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= – K1'/ R + K2'

For the metric to be asymptotically flat:

K2 = K2' = 0

For the solutions to be continuous at boundary, R= r, the radius of spherically symmetric matter:

- 8πGρr^{2}/ 3c^{2}≈ - 2Gm/ rc^{2}

The remaining two constants must be:

K1 = 0 & K1' = 2Gm/ Rc^{2}

Therefore, the complete solution becomes:

( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= - 2Gm/ Rc^{2}

But, as per the literature, the weak field Schwarzschild metric must be:

( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= + 4Gm/ Rc^{2}

I am not able to make out where I am making mistake. Can anybody please help?

Thanks.

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# I How to derive weak-field Schwarzschild metric from LEFE's?

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