- #1
Kyrios
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Homework Statement
An observer is orbiting at a radius r = 3GM, [itex] \theta = \frac{\pi}{2} [/itex] and [itex] \phi = \omega t [/itex] where w is constant.
The observer sends a photon around the circular orbit in the positive [itex] \phi [/itex] direction. What is the proper time [itex] \Delta \tau [/itex] for the photon to complete one orbit and return to the observer?
Homework Equations
Schwarzschild line element where dr =0 and [itex]d\theta[/itex] =0.
The Attempt at a Solution
From the line element we have
[tex] \left(\frac{d\tau}{dt}\right)^2 = 1 - \frac{2GM}{r} - r^2 \left(\frac{d\phi}{dt}\right)^2 [/tex]
[tex] \left(\frac{d\tau}{dt}\right)^2 = 1 - \frac{2GM}{r} - r^2 \omega^2 [/tex]
I was trying to use [itex] \omega^2 = \frac{GM}{r^3} [/itex] but that just gives
[tex] \left(\frac{d\tau}{dt}\right)^2 = 1 - \frac{2GM}{r} - r^2 \frac{GM}{r^3} [/tex]
[tex] \left(\frac{d\tau}{dt}\right)^2 = 1 - \frac{2GM}{r} - \frac{GM}{r} [/tex]
[tex] \left(\frac{d\tau}{dt}\right)^2 = 1 - \frac{3GM}{3GM} [/tex]
as r = 3GM. This gives zero and I'm not really sure what to do with it. Have I gone wrong somewhere? What should I do with the w?