General Relativity

Einstein's geometric theory of gravitation.

Jun 12, 2026··Fork (sign in)Last verified: May 28, 2026

Einstein's field equations and the curvature of spacetime.

Einstein Field Equations

The curvature of spacetime is related to the energy and momentum of matter:

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

4

8πG

Tμν

where $G_{\mu\nu}$ is the Einstein tensor, $\Lambda$ the cosmological constant, and $T_{\mu\nu}$ the stress-energy tensor.

Free-falling particles follow geodesics — paths of extremal proper time:

\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0

For a spherically symmetric, non-rotating mass $M$ :

ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2\,dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1}dr^2 + r^2 d\Omega^2

where $r_s = 2GM/c^2$ is the Schwarzschild radius.

Back to classical mechanics: article-newtons-laws.

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