So the geodesics that point towards the Earth brings space-time towards the Earth and then back out again, but then the moon has its own geodesics so wouldn't it be kind of like geodesics affecting other geodesics because there is no other force in this situation other than gravity? Or do geodesics add? Also how are black holes affected by geodesics if all geodesics point into their singularity and how can a black hole the mass of the moon orbit the Earth at the same radius as the current moon? So aren't the geodesics that end at the singularity constantly changing due to the black hole moving?
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$\begingroup$ Note that geodesics are in 4D spacetime, not in 3D space. $\endgroup$– StenCommented Mar 12 at 1:52
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$\begingroup$ Apparently you confuse 4D geodesics with 3D trajectories. Your question makes no sense. $\endgroup$– safesphereCommented Apr 3 at 16:16
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Space time is a smooth manifold, in this sense we have one set of global geodesics for the cosmic whole, i.e. the geodesics at any one point in space time are truly determined by everything in the universe all at once, as the Einstein tensor is proportional to the stress energy. However, the practical matter is that General Relativity is a local theory where the geodesics at any point are more influenced by the matter/energy in the neighborhood of the point in question than by something on the other side of the universe from it. You can think of space time as somewhat like a bed spread, it is one unified piece, that can have ripples and indentions from say small masses placed here and there, however, at the end of the day, these imperfections all seamlessly flow into the whole.
Yes, space time might have singularities, and indeed, if it does, then at the singularity there is something non-smooth going on, however, that is a question for another more complete theory all together. It is also true that space time is not static, we have time changing gravitational fields that warp and morph the geodesics according to the specifics of the time varying parameters under consideration, e.g. orbits of massive bodies, rotation of black holes, etc. To know exactly how this occurs, one must solve Einstein's equation, you can get for example, gravitational waves in addition to more constant local deformities of space time. Just keep in mind that, as you get farther and farther away from the time dependent phenomena, the geodesics approach a limit more consistent with whatever is at the point distant from the disturbing mass/energy, i.e. space time far away from orbiting black holes, and from any other gravitating matter just becomes more and more like flat Minkowski space.
answered Mar 11 at 21:20
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$\begingroup$ How do black holes feel tidal forces from non-black holes?How would that work if the singularity can't be warped by anything other than rotation and charge? $\endgroup$ Commented Mar 11 at 21:33
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1$\begingroup$ @MiltonTheMeme Black holes are capable of "feeling" gravity from all other massive bodies. Any other gravitating body perturbs the field and alters the geodesics. To understand how black holes interact gravitationally with other massive bodies one has to solve Einstein's equations for the case at hand by sticking all the relevant mass/energy into the stress energy tensor and solving for $g^{\mu\nu}$ (no easy feat). Systems such as black holes orbiting stars and such have been analyzed, at least the gravitational waves emanating from the orbiting pairs has. $\endgroup$ Commented Mar 11 at 21:51