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Let's say I want to create a plot similar to the image below, but with two masses located at e.g. $x_0, y_0, z_0, 0$ and $x_1, y_1, z_1, 0$. How do I compute the curvature $g_{}$ at specific coordinates, e.g. $x, y, z, t = 17, 17, 17, 0$ given a mass e.g. at origo with mass $m$ (or $1$ if that's easier). If this doesn't make sense, consider something that does make sense, e.g. a uniform mass distribution within a radius $r$ around origo. Likewise, if the choice of coordinates ($x, y, z, t$) does not make sense, please feel free to suggest something that does make sense.

Note: A symbolic/exact solution is not necessary. It's enough if there is some equation-system, partial differential equation system or similar that can be solved numerically.

Curvature plot example

Qmechanic
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vidstige
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    The symbol $g_{\mu\nu}$ is traditionally reserved for the components of the metric tensor, not one of the curvature tensors. Do you mean the Einstein tensor $G_{\mu\nu}$, or perhaps the Ricci scalar $R$? – J. Murray Nov 26 '21 at 18:05
  • This might be relevant: https://physics.stackexchange.com/q/295814/168783 – Níckolas Alves Nov 26 '21 at 18:47
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    By the way, the Ricci tensor vanishes for all points where there are no masses, only the Weyl tensor will be non-vanishing. To be fair, I am not really sure whether these sorts of images are plotting anything at all, and I guess they might just be artistic illustrations – Níckolas Alves Nov 26 '21 at 18:49
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    First you have to define what exactly you are plotting. How do the grid lines relate to the metric and curvature? – Javier Nov 26 '21 at 20:13

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This image can be generated using ordinary Newtonian gravity. There is no need to do anything specifically relativistic here.

To generate this image simply start with a large number of test masses, arranged along the lines of this grid. Connect them to form a regular unbent grid. Then, allow the test masses to all free fall for a time. Connect them to form the bent grid in the picture.

Dale
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