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

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1. Mar 27, 2017

I am trying to derive weak-field Schwarzschild metric using Linearized Einstein's field equations of gravity:

[]hμν – 1/2 ημν []hγγ = -16πG/ c4 Tμν

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

Tμν = diag { ρc2 , 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 ημν= { -1 , 1, 1, 1 }:

hμν = diag { -htt , hxx , hyy , hzz }

Trace of the metric:

h = hγγ = - htt + hxx + hyy + hzz

The four equations:

1) []htt – 1/2 ηtt []hγγ = -16πG/ c4 Ttt

=> []htt + 1/2 []( - htt + hxx + hyy + hzz )= -16πGρ/ c2

=> 1/2 []( htt + hxx + hyy + hzz )= -16πGρ/ c2

2) []hxx – 1/2 ηxx []hγγ = -16πG/ c4 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/ c2

Solving the equations using:

[] ≈ ▼2 ≈ 1/R2 d/ dR ( R2 d/ dR ) for static spherically symmetric case; we get:

( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= -8πGρR2 / 3c2 – 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:

- 8πGρr2 / 3c2 ≈ - 2Gm/ rc2

The remaining two constants must be:

K1 = 0 & K1' = 2Gm/ Rc2

Therefore, the complete solution becomes:

( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= - 2Gm/ Rc2

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

( htt + hxx ) = ( htt + hyy ) = ( htt + hzz )= + 4Gm/ Rc2

Thanks.

2. Mar 27, 2017

### dextercioby

It really helps you more to learn about LaTex scientific writing software than how to derive a solution to a differential equation...

3. Mar 27, 2017

### Staff: Mentor

To compare with the literature, I think you need a + sign on the RHS.