From General relativity we know that the sun's mass causes the spacetime to dip, so can we theoretically calculate the exact the time period and speed of earth's revolution around the sun using a mathematical model of curved spacetime and known masses of earth and sun?
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$\begingroup$ Related question for Mercury: physics.stackexchange.com/q/814/2451 , physics.stackexchange.com/q/26408/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Apr 17, 2018 at 17:48
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1$\begingroup$ This idea isn't very practical because the solar system isn't a 2 body system, so the Newtonian perturbations caused by various other bodies in the system (notably the Moon, Venus, and Jupiter) are larger than the relativistic corrections. So to do this properly, you need to include those bodies. $\endgroup$– PM 2RingCommented Apr 17, 2018 at 18:12
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$\begingroup$ Are you asking whether GR has a nonrelativistic limit that is the same as Newtonian gravity (yes), whether the calculation is easy (not too hard), how the calculation is done, or whether the resulting calculation is exact? $\endgroup$– user4552Commented Apr 17, 2018 at 21:31
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$\begingroup$ I wanted to know how the calculation is done and how accurate it is? $\endgroup$– cat's eyeCommented Apr 19, 2018 at 19:44
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2 Answers
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In theory one certainly can calculate the orbital parameter's of Earth around the Sun using General Relativity. The framework provides a mathematically consistent way to describe how the Earth orbits the Sun. Practically speaking though, it's much more difficult to work with general relativity than it is with Newtonian mechanics. So what we usually do is something like use Newtonian mechanics to get a rough idea of the orbits, and then we include some perturbations from General Relativity to get corrections to those orbits to higher and higher accuracy.
Take as a concrete example, the 2 body problem. We consider the Sun and the Earth as 2 bodies which mutually attract each other. This is one approximation step better than the 1 body problem where we make the simplification that the Sun is so much more massive than the Earth that we treat the Earth as a negligible mass. In Newtonian mechanics, the 2 body problem is exactly solvable. We find that the 2 bodies orbit each other in conic sections (closed orbits = ellipses) and we can get exact parametric solutions for the orbits of both bodies. In general relativity this 2 body problem has no known analytical solution. This doesn't mean a solution doesn't exist in general relativity, only that the problem is so complex and intractable that we haven't been able to analytically solve it. So if we want to use GR to solve this problem, we can really only do so perturbatively.
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Yes. One simply takes two independent orbital parameters of Earth's orbit (say, the distance to the sun, and the speed relative to the sun), and then uses them as initial conditions for Earth's geodesic equation, and solves the latter. It's the same procedure one would use for Newtonian gravity, just a different set of equations.
answered Apr 17, 2018 at 17:39
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$\begingroup$ can you provide a link for such a calculation? $\endgroup$ Commented Apr 17, 2018 at 18:01
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$\begingroup$ @cat'seye this is the first thing worked out in nearly every relativity book when they do the Schwarzschild solution. $\endgroup$ Commented Apr 17, 2018 at 18:13
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$\begingroup$ The full two body problem in GR has no closed solution and N-body problems are best described as "murder". So I'm not sure "simply" is a good choice of words here. And for galaxies there's the addition issue of Dark Matter to consider. $\endgroup$ Commented Apr 17, 2018 at 19:40
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$\begingroup$ @StephenG: yes, but approximating the Earth as a test particle is well beyond good enough especially given the scope of this question. $\endgroup$ Commented Apr 17, 2018 at 19:50