General Relativity

Einstein Field Equations

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

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

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

Geodesics

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

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

Schwarzschild Metric

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.