Elliptical Orbits In The Schwarzschild Metric

[dman12]

I was just wondering how you would go about calculating the proper time for an observer following a freely falling elliptical orbit in a Schwarzschild metric.

I am happy with how to calculate the proper time for a circular orbit and was wondering whether if you had two observers start and end at the same spacetime point, for whom would more proper time elapse- one that followed a circular orbit or one that followed an elliptical orbit?

 

[George Jones]

I far as I know, there aren't geodesic (i.e., freely falling) elliptical orbits. There are elliptical orbits that are non-geodesics, but, to calculate the time required for one orbit, more knowledge of the orbit is required.

Also, ven circular orbits do not "start and end at the same spacetime point", as time elapses.

 

[PAllen]

To add a bit to George Jones answer, non-circular orbits in SC metric never quite close (due to perihelion advance), so they are not ellipses.

You certainly can arrange for a non-circular (near elliptic) free faller to meet a circular orbiter at two events. Then, the general rule is that the if the non-circular trajectory is outside of circular orbit between meetings, the non-circular free faller will age more. Conversely, if you arrange it so non-circular trajectory is inside the circular orbit between meetings, the circular orbiter will age more.

 

[George Jones]

In very, very special circumstances, there are (freely falling) closed "spirograph" orbits.

A condition for a closed orbit is that the precession angle divides evenly into an integral multiple of 360 degrees, i.e., n*360/(precession angle) = m, where n and m are integers. If this is true, then the total precession after m aphelia is n times 360 degrees, hence the repetition.

 

[Mentz114]

Here are a couple of references that might help with the calculation

Uros Kostic, Analytical time-like geodesics in Schwarzschild
space-time. General Relativity and Gravitation, 2012.
Preprint :http://arxiv.org/pdf/1201.5611v1.pdf

G. V. Kraniotis, S. B. Whitehouse,
Precession of Mercury in General Relativity, the Cosmolog-
ical Constant and Jacobi’s Inversion problem.
Preprint http://128.84.158.119/abs/astro-ph/0305181v3