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Linearising the Geodesic Deviation Equation

  1. Nov 24, 2014 #1
    1. The problem statement, all variables and given/known data
    Write down the Newtonian approximation to the geodesic deviation equation for a family of geodesics. ##V^\mu## is the particle 4-velocity and ##Y^\mu## is the deviation vector.

    2. Relevant equations
    [tex]
    D = V^\mu\nabla_\mu \\
    V^\mu\approx(1,0,0,0) \\
    g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}
    [/tex]

    The geodesic deviation equation and the desired linearisation are
    [tex]
    D^2 Y^\mu = R^\mu{}_{\rho\lambda\nu} \\
    \partial_t^2 Y^i = -\partial_i\partial_j\phi Y^j
    [/tex]

    3. The attempt at a solution
    Apologies if this is not the correct way to post in this subforum, but I'm not going to write up all of what I have so far as I don't think it's relevant to my question. My issue is with the left hand side, which with the approximation for the 4-velocity becomes
    [itex]
    D^2 Y^\mu = V^\mu\nabla_\mu V^\nu\nabla_\nu Y^\mu = \nabla_t^2 Y^\mu
    [/itex]
    Everywhere I have looked, the next step is to replace this by ##\partial_t^2 Y^\mu##, but I don't follow this logic. As far as I can see, the covariant and partial derivatives differ by Christoffel symbols containing terms O(h), which should enter into the linearisation. What am I missing? Is Y also O(h)?
     
  2. jcsd
  3. Nov 27, 2014 #2

    stevendaryl

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    Staff Emeritus
    Science Advisor

    That equation [itex]D^2 Y^\mu = R^\mu{}_{\rho\lambda\nu}[/itex] can't possibly be right, because the left-hand side has only one index, [itex]\mu[/itex], while the right side has indices [itex]\mu, \rho, \lambda, \nu[/itex]. I think it's supposed to be something like:

    [itex]D^2 Y^\mu = R^\mu{}_{\rho\lambda\nu} V^\rho V^\lambda Y^\nu[/itex]
     
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