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  Numerical reconstruction of Einstein's field equations

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A few analytic solutions are known to the Einstein field equations:

$$ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R - kT_{\mu\nu} = 0$$

Taking a preexisting analytic solution such as Schwarzchild's solution:

$$g_{\mu\nu} = \begin{bmatrix} (1 - \frac{2m}{r}) & 0 & 0 & 0 \\ 0 & -\frac{1}{(1 - \frac{2m}{r})} & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -r^2\sin^2\theta \end{bmatrix}$$

Suppose we now delete a single arbitrary 2x2x2x2 grid locations' known $g_{\mu\nu}$ value. 

By what systematic method can we numerically recover the correct identical 10 numeric values of $g_{\mu\nu}$ in this grid, knowing only all of their neighboring points' analytic values except this 2x2x2x2 hole, forbidding any use of the analytic recipe that generates this numerical field?

asked Feb 3, 2023 in Open problems by user287658762 [ no revision ]

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