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I want to do some calculate and I need to know how can I calculate the curvature of space time so that I can plug in the digits in formula of general theory of relativity

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  • $\begingroup$ "Can be used in formula", which formula? Calculating curvature in general relativity is not as trivial as it sounds like you're expecting it to be. $\endgroup$
    – Charlie
    Commented May 4, 2020 at 18:23
  • $\begingroup$ You can calculate the curvature of the Earth in space using differential geometry. To calulate the curvature of space-time, you need general relativty. $\endgroup$ Commented May 4, 2020 at 18:24

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I don't know exactly what you're after but I'll try. The Schwarzschild metric describes the metric outside of a spherically symmetric body. Like a planet or black hole. $$g=-\left(1-\frac {r_s}r\right)c^2dt^2+\left(1-\frac {r_s}r\right)^{-1}dr^2+r^2\left(d\theta^2+\sin^2(\theta)\ d\phi^2\right)$$ You can loosely write this as a matrix: $$g_{\mu\nu}=\pmatrix{\left(1-\frac {r_s}r\right)c^2&&\\ &\left(1-\frac {r_s}r\right)^{-1}&\\ &&r^2&\\ &&&r^2\sin^2\phi\\ }$$ You would then have to calculate the Christoffel symbols. These tell you a lot about how vectors change when you move in your spacetime. $$\Gamma^\lambda_{\mu\nu}=\tfrac 1 2g^{\lambda\alpha}\left(\frac{\partial g_{\nu\alpha}}{\partial x^\mu}+\frac{\partial g_{\mu\alpha}}{\partial x^\nu}-\frac{\partial g_{\mu\nu}}{\partial x^\alpha}\right)$$ Here I used Einstein notation. This means any time you see an index twice you have to sum over all spacetime components. From the Christoffel symbols you can calculate the Riemann tensor. $$R^\rho_{\sigma\mu\nu}=\partial_\mu\Gamma^\rho_{\nu\sigma}-\partial_\nu\Gamma^\rho_{\mu\sigma}+\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$$ This tensor uses the information in the Christoffel symbols to tell you how the space is curved (in a nontrivial way).

The Riemann tensor allows you to calculate the Ricci tensor: $$R_{\mu\nu}=R^\lambda_{\mu\lambda\nu}$$ The Ricci tensor allows you to calculate the Ricci scalar: $$R=R^\mu_\mu$$ This is everything you need for the curvature part of the Einstein field equations. If you are confused by any of this you should. General relativity is notoriously hard. But this is the minimal amount of knowledge you need to calculate curvature. If you really want to do this I recommend this video playlist about tensors followed by this one about tensor calculus. After watching these playlists you should be able to understand every equation in my answer. You could probably also find the Ricci tensor of the Schwarzschild metric online if you want a shortcut.

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  • $\begingroup$ Thanks you so much, idk how much should I thank you $\endgroup$ Commented May 5, 2020 at 17:00
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Usually the problem is not calculating curvature. Einstein's equation gives the (Einstein) curvature as proportional to the stress energy tensor (plus cosmological constant term). The stress energy is known for a given situation. Of course that does not tell you Riemann curvature, but you are unlikely to want Riemann curvature.

The problem is to not to find curvature, but to find the metric which describes the geometry. A few solutions are known, such as Schwarzschild and Friedmann cosmologies, but you seem to be asking about arbitrary solutions, and they are not known. In any event, it is unlikely that you can find anything other than the standard analytic solutions which would enable you to "plug in the digits in formula of general theory of relativity".

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  • $\begingroup$ Typically, you also won’t know the stress energy in a given situation as it will itself depend on the metric. Instead you ended up with a coupled set of equations of motion for both. $\endgroup$
    – TimRias
    Commented May 5, 2020 at 9:03

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