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Sagittarius A* has an estimated mass of 4.154 million solar masses. A number of stars have been found to orbit near Sagittarius A*.

What is the maximum time dilation factor for any of these orbiting stars (taking into account the mass of Sagittarius A*, but also the star itself, if possible)?

Does the amount of time dilation depend upon where in the elliptical orbit the star is, i.e. is time dilation maximized at the nearest point (the periapsis)?

Luke Hutchison
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  • The gravitational time dilation depends on the distance to the horizon in the units of the Schwarzschild radius. This is an easy calculation. +1 – safesphere Oct 03 '22 at 02:13
  • @safesphere Isn't that using the Schwarzschild solution? Wouldn't the Kerr metric be the relevant choice? – Alex Nelson Oct 03 '22 at 02:23
  • @AlexNelson You are correct, but in this case the difference is insignificant. The closest any star gets to Sgr A* is about 150 times its Schwarzschild radius. The time dilation there is minuscule. – safesphere Oct 03 '22 at 02:59
  • This sounds a bit like a homework question. – ProfRob Oct 03 '22 at 06:23
  • @safesphere miniscule, but measured – ProfRob Oct 03 '22 at 06:25
  • @ProfRob Has it been measured? – safesphere Oct 03 '22 at 08:43
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    @safesphere yes. By measuring/inferring the velocity of closely orbiting stars and measuring a redshift one can uncover the "components" of the redshift to test the GR prediction. https://arxiv.org/pdf/1807.09409.pdf – ProfRob Oct 03 '22 at 09:20
  • @ProfRob It's not a homework question. It's something I was wondering about for my own interest. I already tried to figure out how to calculate this, but I was out of my depth. It's not really fair that this was closed as a homework question when the question should in fact have general interest value. A lot of people became interested in gravity well time dilation after the movie Interstellar, and the case I asked about pertains to our own galaxy. – Luke Hutchison Oct 04 '22 at 03:53
  • Rewrite the question so it isn't a homework-type question. The issue is not whether it is actually homework. e.g. Given the orbital parameters of a star around a supermassive black hole, how would I go about calculating the maximum level of time dilation expected? – ProfRob Oct 04 '22 at 06:36
  • @ProfRob I am bewildered that you don't see that that's basically exactly what I already wrote. – Luke Hutchison Oct 05 '22 at 07:05
  • @ProfRob Thanks for the link, pretty cool! There is a clear trend in physics for the list of authors soon to become the largest part of the paper ;) – safesphere Oct 06 '22 at 07:22
  • See https://physics.stackexchange.com/questions/730794/time-dilation-in-an-elliptical-orbit – ProfRob Oct 06 '22 at 16:38

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