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General Relativity: Particle velocity travelling on Schwarzchild orbit

  1. Mar 6, 2012 #1
    1. The problem statement, all variables and given/known data

    What is the speed of a particle in the smallest possible circular orbit in the Schwarzschild
    geometry as measured by a stationary observer at that orbit? Note: The orbit in
    question happens to be unstable.


    2. Relevant equations

    Normalization condition:

    [itex]\textbf{u}_{obs}(r)[/itex][itex]\cdot[/itex][itex]\textbf{u}_{obs}(r)[/itex]=[itex]g_{αβ}[/itex][itex]u^{α}_{obs}(r)[/itex][itex]u^{β}_{obs}(r)[/itex]=-1

    Schwarzchild Metric...


    3. The attempt at a solution

    So from the Normalization condition and the Schwarzchild metric it is easy to find the four-velocity of the observer, because the spatial components are zero. My question relates to the observer's measurement of the particle's velocity. If his four-velocity is
    [itex]{u}^{a}[/itex]=[1/[itex]\sqrt{1-2m/r}[/itex],0,0,0]
    then, how does observe the particle's velocity?
    An answer I found somewhere said

    [itex]\frac{u^{2}}{1-u^{2}}[/itex]=[itex]u^{a}u^{b}(g_{ab}+u_{a}u_{b})=(u^{a}u_{a})^{2}-1[/itex]

    But I have no idea how to arrive at this relation, and why it relates to the observer's measurement of the particle. Is it possibly a derivation of a relation between t and the proper time?

    Also, this might seem very rudimentary but I am not sure why we are able to write:

    [itex]u_{a}u_{b} \ast u^{a}u^{b} = (u^{a}u_{a})^{2}[/itex]

    I'm very new to general relativity, and would really appreciate the help!
    Thanks a lot!
     
  2. jcsd
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