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Is there an equation the can calculate gravity around black holes but is less time consuming than EFE? I want to find an equation that is simpler/faster than Einstein's Field Equations but can still accurately predict gravitation in and around black holes. Does an equation like this exist?

Chance Evans
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    Since you specify accuracy, it sounds like rather than an alternative theory of gravity, you just want expressions derived from general relativity that simplify calculations for your specific use case. But what is that use case? There is not enough detail here. – Sten Aug 05 '23 at 14:05
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    The dirty secret of physical theory is that as it evolves in a way that models are based on ever more abstract "simple" principles, the mathematics becomes more intractable. This, I believe, is evidence that mathematics cannot be the substance of reality. – John Doty Aug 05 '23 at 14:09
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    @JohnDoty thanks for making my (current) favourite pair of observations in one brief comment! Also, I'd like to add, in the particular case of numerical analysis, nature does not "iterate over multidimensional grids". – m4r35n357 Aug 05 '23 at 15:05
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    If there was a simpler theory with the same predictions, don't you think we would already be using it? – Javier Aug 05 '23 at 15:19
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    @m4r35n357 Yep. Numerical methods scale very badly, but Nature has no trouble making a whole universe go. – John Doty Aug 05 '23 at 16:28
  • For a Schwarzschild black hole that equation is already simple, but for rotating ones the equations are a little bit more complicated, so no, such an equation simpler than GR does not exist. – Yukterez Aug 05 '23 at 18:51
  • A realistic black hole rotates but has negligible electric charge, so should be Kerr. At least that's simpler than this. – J.G. Aug 05 '23 at 18:54
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    Regarding the new question, you don't need the Einstein field equations to evaluate the effect of gravity around black holes (unless you're considering the evolution of objects of comparable mass to the black holes). Just use an already-known solution to the Einstein field equations (Schwarzschild or Kerr) and use the geodesic equation to evaluate particle trajectories. – Sten Aug 05 '23 at 23:05
  • "Gravity" is gravitational acceleration. This page describes what we observe around a black hole: https://www.mathpages.com/rr/s6-07/6-07.htm – safesphere Aug 06 '23 at 04:30
  • Further to my previous comment, the Kerr metric's gravitational potential is Eq. (15) here. – J.G. Aug 06 '23 at 09:39

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Despite its fearsome reputation general relativity is a remarkably simple theory. One of the ways a theory can be defined is by a property called its action, and the action for GR is surprisingly simple. No simpler theory has ever been found that is consistent with experimental observations. Whether it has been proved that no simpler theory is possible I don't know.

So your quest is doomed to failure unless you would be happy with an approximation that holds only in limited conditions. For example there are many simplifications of GR that apply only when the gravitational fields are weak - in practice this means anywhere that isn't close to a black hole. A widely used example is Parameterised post-Newtonian formalism. However by definition these theories cannot be used in the most interesting situations e.g. black holes merging or observers falling into a black hole.

John Rennie
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    Except that it has the common defect of "remarkably simple" theories: the math is fantastically intractable in real situations. – John Doty Aug 05 '23 at 16:31
  • Maybe the fearsome reputations comes from the non-linearity of the theory. – Pipe Aug 05 '23 at 19:16
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    @Pipe Nonlinearity makes it harder, but even linear models scale badly in real life. – John Doty Aug 05 '23 at 20:18
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    Fortunately, the extremely simple example of a body moving near to a spherical, effectively spinless, mass distribution is extremely common in the universe . Accurate GR calcs. for a black hole that depends only on mass and spin are remarkably simple compared with say accurate satellite motion around the Earth using Newtonian gravity. – ProfRob Aug 06 '23 at 10:05