PeterDonis
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Huh? The stress-energy tensor is symmetric, so it only has 10 independent components, and spacetime is 4D, not 3D.James1238765 said:Starting with ##T_{ij}## we have 16 numbers for each 3D gridpoint
You don't "reverse engineer" anything. You write down the components of the Einstein Field Equation and solve them. But in a case like this, as has already been noted, you can't solve them analytically, you can only solve them numerically, and to write them down you need an ansatz for the metric in terms of functions of the coordinates.James1238765 said:The problem seems to be trying to reverse engineer what R and ##R_{ij}## will give rise to this result
Since you have isolated regions where the SET is nonzero, with the rest of the spacetime having the SET zero, you actually have to do three solutions, and then join them at boundaries.
The short version: this is a lot more complicated than you seem to imagine, and you would be much better served by trying to look at worked examples from the numerical relativity community, than by trying to guess for yourself.