- #1
Fek
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Homework Statement
Show Kepler's Third Law holds for circular Schwarzschild orbits.
Homework Equations
The Attempt at a Solution
Setting [itex] r' = 0 , \theta' = 0 [/itex] and [itex] \theta = \pi / 2 [/itex] , where the derivatives are with respect to the variable [itex] \lambda [/itex] and setting c = 1 the Lagrangian is:
[itex] L = (1 - r_s / r) \dot t^2 - r^2\dot\phi^2 [/itex]
E-L equation for r:
$$\frac{r_s} {r^2} \dot t^2 - 2r \dot\phi^2 = 0$$
Therefore $$d\phi / dt = GM/r $$ , the solution.
However I'm struggling to understand why t here corresponds to the proper time at infinity, but r corresponds to the radius of the ciruclar orbit?