I know that gravitational time dilation near a single body is:
$$T_2=T_1\sqrt{1-\frac{2GM}{rc^2}} $$
Can you give gravitational time dilation formula when in proximity to multiple bodies?
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$\begingroup$ You're going to need to clarify this question. $\endgroup$– blademan9999Commented Jul 1, 2023 at 14:31
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$\begingroup$ In the weak field you can multiply the factors each of the bodies give at the respective distance to them, but in the strong field you need to do it numerically since there is no exact solution for that. If you keep the two masses at a fixed distance relative to each other you could use the Kastor Traschen metric but if the orbit each other dynamically that won't work. $\endgroup$– YukterezCommented Jul 1, 2023 at 14:49
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$\begingroup$ @Yukterez if they orbit each other maintaining the proper distance fixed it should not be a problem, you can choose a co-rotating reference frame so that you treat them as fixed, if i remember correctly $\endgroup$– LolloBoldoCommented Jul 1, 2023 at 15:00
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$\begingroup$ You are highlighting the issue that, while in Newtonian gravity any system with more than 2 bodies has no general solution, in General Relativity it's even worse – no system with more than 1 body has a general solution. $\endgroup$– RC_23Commented Jul 1, 2023 at 15:14
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1$\begingroup$ In weak field limit the Newtonian potential gives you the factor. That also offers a useful rough approximation at intermediate fields. $\endgroup$– Andrew SteaneCommented Jul 1, 2023 at 15:37
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1 Answer
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A general formula doesn't exist, you need to calculate it (or evaluate it numerically) yourself depending in the specific parameters of your system. To evaluate it you first need to solve the Einstein equations and find the metric associated to your specific problem configuration. Given that you can proceed to evaluate the time dilation ad you would do with a single Schwartzschild body but using the metric tensor you have just found.
