Timeline for Periods of non-circular Schwarzschild orbits
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2018 at 3:29 | comment | added | Chiral Anomaly | @Gordon By free-fall equations, I mean the equations that a world-line must satisfy in order to be a geodesic (to represent an object in free-fall, such as an orbit). For the Schwarzschild metric, I wrote them out explicitly in $t,x,y,z$ coordinates at the end of this post: physics.stackexchange.com/a/436873/206691. I mention this because most people write them in spherical coordinates, which (despite the name) obscures their spherical symmetry. Writing them in $x,y,z$ coordinates makes the spherical symmetry obvious. But for non-circular orbits, I don't know if this helps much. | |
| Dec 9, 2018 at 1:00 | comment | added | Gordon | Thank you. Just so we are on the same page, what do you mean by free-fall equations? I presume the equations of motion for the metric. As to which world-lines correspond to orbits, I am considering non-circular geodesics: these lie in a plane, and precess between two radii (the perigee and the apogee). | |
| Dec 9, 2018 at 0:54 | comment | added | Chiral Anomaly | @Gordon Yes, you are thinking about the orbits correctly. Thanks to the symmetry of the Schwarzschild metric, these definitions that at first seem to be coordinate-dependent can actually be turned into (less concise) coordinate-independent definitions. So they are legitimate. As far as how one would calculate them, the hard part is determining which worldlines correspond to orbits (free-fall). I don't have any good hints to offer. The farthest I've ever gone with non-circular orbits is to write down the free-fall eqn's for $r,\phi,t$ that need to be solved, but I haven't actually solved them. | |
| Dec 9, 2018 at 0:48 | comment | added | Gordon | No worries, your answer helps a lot. I was asking 1. if this is a legitimate way of thinking about periods of Schwarzschild orbits, 2. how one would go about calculating them. | |
| Dec 9, 2018 at 0:44 | comment | added | Chiral Anomaly | @Gordon Okay, I guess I misunderstood the purpose of the question. Were you asking if there should be a third period, one corresponding to the function $t(\tau)$? | |
| Dec 9, 2018 at 0:42 | comment | added | Gordon | I realise that each of these defines a period for the given coordinate and that these are not equivalent. I don't really like the formalism of this paper, I might try to re-derive the periods in terms of other quantities. | |
| Dec 9, 2018 at 0:38 | history | answered | Chiral Anomaly | CC BY-SA 4.0 |