Undergrad Orbital Period In General Relativity

[dsaun777]

What is the orbital period in General Relativity using the Schwarzschild metric? In classical mechanics, it is something like
T=2pi(G_nM/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.

 

[PeterDonis]

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.

 

[dsaun777]

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 r_s dr using the metric components related to radial coordinates.

 

[PeterDonis]

Yeah, its the areal radius

Yes, but...

found by integrating over the radial coordinate from r to r_s 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.