# How can I calculate curvature in space which can be used in formula of general theory of relativity? [closed]

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

• "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. Commented May 4, 2020 at 18:23
• You can calculate the curvature of the Earth in space using differential geometry. To calulate the curvature of space-time, you need general relativty. Commented May 4, 2020 at 18:24

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.