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General Relativity- geodesics, killing vector, Conserved quantity Schwarzschild metric

  1. Jun 25, 2017 #1
    1. The problem statement, all variables and given/known data

    Conserved quantity Schwarzschild metric.

    2. Relevant equations


    3. The attempt at a solution

    ##\partial_u=\delta^u_i=k^u## is the KVF ##i=1,2,3##

    We have that along a geodesic ##K=k^uV_u## is constant , where ##V^u ## is the tangent vector to some affinely parameterised geodesic.

    For example if we take the Schwarzschild metric, ##K^u=(1,0,0,0)## , ##V^u=(\dot{t},\dot{r},\dot{\theta},\dot{\phi})## so we get ##K= (1-\frac{2GM}{r})\dot{t}## is conserved, for example.

    where dot denotes a derivative with respect to some affine parameter ##s##

    QUESTION:

    What here is to say that ##s## is an affine parameter?
    I.e- why is ##V^u=(\dot{t},\dot{r},\dot{\theta},\dot{\phi})## the tangent vector to an affinely parameterised geodesic?
     
    Last edited by a moderator: Jun 25, 2017
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  3. Jun 25, 2017 #2

    Orodruin

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    You assumed it to be an affine parameter and then wonder why it is affine?

    If it is not an affine parameter ##k^u V_u## will not be a conserved quantity.
     
  4. Jun 25, 2017 #3

    PeterDonis

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    Moderator's note: moved to homework.
     
  5. Jun 25, 2017 #4
    ahh apologies got it yes, it is in the proof showing that ##k^u V_u## will not be a conserved quantity is a conserved quantity.
     
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