I Orbital Period In General Relativity

dsaun777
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What is the orbital period in General Relativity using the Schwarzschild metric? In classical mechanics, it is something like
T=2pi(GnM/a3). Where a is the semi-major axis, this is for a small body orbiting a larger one. I think I have an idea but I am not 100% sure. I am interested in an outside observer far away viewing a small particle m in orbit of some mass M.
 
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dsaun777 said:
What is the orbital period in General Relativity using the Schwarzschild metric?
For a circular orbit, it's the Kepler's Third Law expression with the Schwarzschild ##r## plugged in as the orbital radius. Note that this is the case even though ##r## is not the same as the physical distance from the center of mass of the central body.
 
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PeterDonis said:
For a circular orbit, it's the Kepler's Third Law expression with the Schwarzschild ##r## plugged in as the orbital radius. Note that this is the case even though ##r## is not the same as the physical distance from the center of mass of the central body.
Yeah, its the areal radius found by integrating over the radial coordinate from r to rs dr using the metric components related to radial coordinates.
 
dsaun777 said:
Yeah, its the areal radius
Yes, but...

dsaun777 said:
found by integrating over the radial coordinate from r to rs dr using the metric components related to radial coordinates.
...no, that's not what the areal radius is. The areal radius is ##r = \sqrt{A / 4 \pi}##, where ##A## is the surface area of the 2-sphere labeled by ##r## that is centered on the central mass.
 
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