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GR algebra pretty much (weak limit thm)

  1. Jan 23, 2017 #1
    1. The problem statement, all variables and given/known data
    Hi

    I am stuck on a small algebra set in the weak limit theorem to recover Newtonian equations

    The text I am looking at:

    ##\frac{d^2x^i}{ds^2}+\Gamma^i_{tt}\frac{dt}{ds}\frac{dt}{ds}=0## (1)

    ##\Gamma^{i}_{tt}=-1/2 \eta^{ij}\partial_{j}h_{tt} ## (to first oder in the metric ##h_{uv}##) (2)

    ##dt/ds \approx 1##
    and so using (2), (1) becomes:

    ##\frac{d^2 x^i}{ds^2}=-1/2\partial_ih_{tt}## (3)

    MY QUESTION

    ##-1/2\eta^{ij}\partial_jh_{tt}## in (2)
    ##= -1/2 \partial^{i} h_tt ##

    So for (3) I am getting

    ##\frac{d^2 x^i}{ds^2}=1/2\partial^ih_{tt}##

    Im really confused how

    ##-1/2\eta^{ij}\partial_jh_{tt}=1/2\partial_{i}h_{tt}## , or at least that is what it looks like has been done.

    Many thanks

    2. Relevant equations

    see above



    3. The attempt at a solution
    see above
     
  2. jcsd
  3. Jan 23, 2017 #2

    stevendaryl

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    There are two conventions for the flat space metric tensor (in Cartesian coordinates):
    1. [itex]\eta^{tt} = +1, \eta^{xx} = \eta^{yy} = \eta^{zz} =-1[/itex] (all the other components zero)
    2. [itex]\eta^{tt} = -1, \eta^{xx} = \eta^{yy} = \eta^{zz} =+1[/itex] (all the other components zero)
    If they are using the first convention, then [itex]\partial_i = - \partial^i[/itex].
     
  4. Jan 24, 2017 #3
    I have ##\eta^{ij}\partial_{j}=-\partial^i ## , dont know how to show [itex]\partial_i = - \partial^i[/itex]. ( well since ##\eta## is diagonal I know I really have ##i=j## but to keep the index notation clear..)
     
  5. Jan 24, 2017 #4

    stevendaryl

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    In the usual tensor notation, [itex]\partial_\mu \equiv \sum_{\nu} \eta_{\mu \nu} \partial^\nu[/itex] where [itex]\eta_{\mu \nu}[/itex] is the metric tensor. So if [itex]\eta_{\mu \nu}[/itex] is diagonal with diagonal entries [itex](+1, -1, -1, -1)[/itex], then

    [itex]\partial_t = \partial^t[/itex]
    [itex]\partial_x = - \partial^x[/itex]
    [itex]\partial_y = - \partial^y[/itex]
    [itex]\partial_z = - \partial^z[/itex]
     
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