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Wikipedia says:

According to Einstein's theory of general relativity, particles of negligible mass travel along geodesics in the space-time.

Does that "negligible mass" only refer to particles or objects like a spacecraft, whose mass are literally negligible to that of a black hole, are also taken into account?

If the mass of a spacecraft is not negligible, how has one to modify the geodesics equation corresponding to that mass?

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  • $\begingroup$ It really means photons, particles with no rest mass. Everything else might become questionable under some circumstances. E.g. neutrinos are very close to being massless, so they must move very close to geodesics, but in some rare cases pretty major deviations may happen. $\endgroup$
    – Florin Andrei
    May 6, 2019 at 1:15
  • $\begingroup$ @Florin But photons travel on null geodesics. I assume that Roboticist is also interested in timelike geodesics. $\endgroup$
    – PM 2Ring
    May 6, 2019 at 4:32

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Yes its means that you ignore the effect that an object such as a spacecraft has on the central mass that it is orbiting, that the orbiting body has infinitely less mass.

Unfortunately, I think there is no exact solution found of the geodesics equations when that approximation is no longer valid or when you try to devolop geodesics equation for a spacecraft under the influence of two (or more) spherically symmetric mass distributions such as the combination of planets and the Sun in our solar system.

In the solar system JPL who calculates orbits of planets and other celestial objects use what is called the "post Newtonian expansion". You can look at their official documentation, Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation, expression 4-26 on page 4-19. Some of the the terms may be zero if you only have two non-neglibly massive objects.

Note that this is not really a "geodesics equation" per se. Nasa/JPL uses what I believe is a first order expansion of the Schwarzshild solution in isotropic coordinate and then uses a scheme I do not quite follow to get to the relativistic acceleration terms of the post-Newtonian that they add to the classical Newtonian gravitational acceleration term to calculate the orbits.

I do not know if there is a simpler way to do it.

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  • $\begingroup$ Thus, one may conclude that (if approximated solutions are fine) the geodesics equation can be used to study the motion of a hypothetical spacecraft around "only one" super massive object, right? $\endgroup$
    – User
    May 5, 2019 at 7:17
  • $\begingroup$ The Schwarzschild solution is only applicable to the case of an "infinitely lighter" object moving around a spherically symmetric mass distribution. When you have more than one spherically symmetric mass distributions you cannot simply add the contributions like in Newtonian gravitation. $\endgroup$
    – Agerhell
    May 5, 2019 at 11:15
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    $\begingroup$ @Roboticist The idea is that the "particles of negligible mass" is light enough that the curvature it causes in spacetime is tiny relative to the curvature caused by the body(s) it's orbiting. So you can pretend that it's moving in a static spacetime, which is a lot easier to calculate than one where the orbiting body is drastically modifying the curvature as it moves. $\endgroup$
    – PM 2Ring
    May 6, 2019 at 4:38
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Something of a tautology. Objects of negligible mass have a mass that can be neglected. By definition.

To put it another way, if you have decided to not bother calculating in the mass of some object, because you have decided for whatever reason that it's small enough that it won't significantly affect the result of some calculation, then you would compute that that object travels along a geodesic.

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    $\begingroup$ while this is certainly a good first answer, it does not appear to answer the specifics of the question. $\endgroup$
    – dalearn
    May 6, 2019 at 1:02

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