Deriving Weak-Field Schwarzschild Metric from LEFEs

[Nikhil Hadap]

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

[]hμν – 1/2 ημν []h^γγ = -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 η^μν= { -1 , 1, 1, 1 }:

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

Trace of the metric:

h = h^γγ = - htt + hxx + hyy + hzz

The four equations:

1) []htt – 1/2 ηtt []h^γγ = -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 []h^γγ = -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

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

Thanks.

 

[dextercioby]

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

 

[PeterDonis]

Linearized Einstein's field equations of gravity:

[]h_μν – 1/2 η_μν []h^γ_γ = -16πG/ c4 T_μν

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