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    Light signals and time differences in Schwarzchild geometry

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    An observer at r = r1 in Schwarzchild geometry sends a light signal in the radial direction toward r = r2 where r2 > r1.
    (a) Determine the instantaneous coordinate velocity dr/dt of the signal.
    (b) Suppose the signal is reflected at r = r2 and returns to r1. Determine how long, as measured in coordinate time t, it takes the signal to return.
    (c) Determine how long it takes as measured by the clock of the observer at r1.

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    Solution Preview

    The line element is:

    ds^2 = c^2(1-2m/r)dt^2 - (1-2m/r)^(-1)dr^2 - r^2 d theta^2 - r^2sin^2(theta)d phi^2

    where m = MG/c^2

    You know that in a flat space-time the trajectory of a photon is such that ds^2 = 0 on the trajectory (such a trajectory is called a null geodesic). This is also true in general relativity. It follows form the usual reasoning that you can always define a local "free falling" coordinate system in which the metric looks ...

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