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Does anyone know if the Einstein Field equations have ever been written as Difference Equations, and if so does that simplify anything or produce solutions not available in the usual Differential Equation formulation.

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As I'm sure you must realize, it's unlikely that there exists a difference equation whose solutions are precisely the same as for a given differential equation. So I presume that by a difference equation you mean a numerical scheme to approximate solutions to Einstein Field Equations. There is an entire subfield of Relativity concerned with that: Numerical Relativity. It's already 50+ years old, has spawned hundreds of articles, also special conferences, textbooks and computer codes. Certainly, the numerical schemes that arise are not particularly simple, but they have been used to simulate situations like black hole collisions, that are not amenable to other approximation schemes. The linked Wikipedia article is a decent place to start reading.

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Higher order difference expression in several variables are notoriously difficult to handle in such a way that they converge suitably to iterated partial differentials. See

  • MR1245559 (94k:26024) Reviewed Frölicher, Alfred(CH-GENV-SM); Kriegl, Andreas(A-WIEN) Differentiable extensions of functions. (English summary) Differential Geom. Appl. 3 (1993), no. 1, 71–90.

for an impression about his. Difference equations are highly more complicated to handle than differential equations. Because of this, among others, the invention of differential calculus was such a huge leap forward. So trying to write Einstein's equations as difference equations would hugely complicate matters.

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There are analogues of the Einstein equations for piecewise flat manifolds, such as the Regge calculus. However, these cannot replace the Einstein equations, if your primary concern is the Einstein equations (Of course if you assume that the Regge calculus is the real thing, then you can forget about the Einstein equations). The "Regge equations" can approximate the Einstein equations, and its solutions may converge to the solutions of the Einstein equations in a suitable sense. As far as I know, the convergence issue is an open problem though.

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