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

  1. Mar 27, 2017 #1
    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

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

    Thanks.
     
  2. jcsd
  3. Mar 27, 2017 #2

    dextercioby

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    It really helps you more to learn about LaTex scientific writing software than how to derive a solution to a differential equation...
     
  4. Mar 27, 2017 #3

    PeterDonis

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    2016 Award

    Staff: Mentor

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