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A Four velocity with the Schwarzchild metric

  1. Mar 16, 2017 #1
    I am trying to solve the following problem but have gotten stuck.
    Consider a massive particle moving in the radial direction above the Earth, not necessarily on a geodesic, with instantaneous velocity
    v = dr/dt
    Both θ and φ can be taken as constant. Calculate the components of the four-velocity Uu in terms of v using the normalization condition for Uu. Do not make any approximations yet.

    We are using the metric:

    ds2=-(1+2Φ)dt2+(1+2Φ)-1dr2+r22

    I have determined that U0 is going to be (1+2Φ)(-1/2) by using the equation

    guvUuUv=-1

    However when I try to solve for U1 I get

    g11*(U1)2=-1

    Which is clearly not correct as this would yield

    U1=(-(1+2Φ))(-1/2).

    What am I doing wrong?

    Apologies for formatting.
     
  2. jcsd
  3. Mar 16, 2017 #2

    PeterDonis

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    2016 Award

    Staff: Mentor

    Do you realize that this equation contains both ##U^0## and ##U^1##? You can't get an equation for just ##U^0## from it. The notation ##g_{uv} U^u U^v## means you have to sum over all possible values for the indexes ##u## and ##v##.
     
  4. Mar 16, 2017 #3
    This clears up the problem thank you. That slipped my mind
     
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