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?
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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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