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Circular orbits in Schwarzschild geometry

  1. Dec 5, 2016 #1
    1. The problem statement, all variables and given/known data
    Hartle, Gravity, P9.8
    A spaceship is moving without power in a circular orbit about a black hole of mass M, with Schwarzschild radius 7M.
    (a) What is the period as measured by an observer at infinity?

    (b) What is the period as measured by a clock on the spaceship?

    2. Relevant equations
    Eqn 9.46 from Hartle:
    [tex]
    \omega =\sqrt{(M/r^3)}
    [/tex]

    Proper time:
    [tex]
    d\tau^2=-ds^2
    [/tex]
    and the Schwarzschild metric.

    3. The attempt at a solution
    (a) This part is fine. Using T=2*pi/omega and substituting r=7M, T=14*pi*sqrt(7)*M.

    However, I'm not sure how Eqn 9.46 is derived? Can it be derived using the Schwarzschild metric and noting that the coordinate r is constant?

    (b) The clock in the spaceship measures proper time:
    [tex]
    d\tau=\sqrt{(1-2M/r)dt^2+r^2d\phi^2}=\sqrt{(1-2M/r)+M/r}dt
    [/tex]
    Here, [tex]d\phi^2=M/r^3dt[/tex]
    [tex]
    T'=\sqrt{(1-2M/r)+M/r}T
    [/tex]
    Is my reasoning correct? Assuming this is correct, is all that remains is to substitute r=7M and the period T from part (a)?
     
  2. jcsd
  3. Dec 5, 2016 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Calling 7M "Schwarzschild radius" is confusing.
    Sure (but don't ask me how exactly).

    For (b), I would have expected an expression that becomes 0 at the photon sphere, but I'm not sure if that is right.
     
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