Schwarzschild metric

Example

Fix a mass parameter $m > 0$, in units where the gravitational constant and the speed of light are $1$. On $M = \R \times (2m, \infty) \times S^2$ with coordinates $(t, r, \theta, \phi)$, the Schwarzschild metric is $$g = -\left(1 - \frac{2m}{r}\right) dt^2 + \left(1 - \frac{2m}{r}\right)^{-1} dr^2 + r^2 \left(d\theta^2 + \sin^2\theta \, d\phi^2\right).$$ Check that $(M, g)$ is a Lorentzian manifold: for $r > 2m$ the number $1 - 2m/r$ is positive. So the coefficient of $dt^2$ is negative and the coefficient of $dr^2$ is positive. The last term is $r^2$ times the round metric of $S^2$, which is positive definite. Hence $g$ has signature $(-, +, +, +)$ at every point. A direct computation of the Christoffel symbols shows that the Ricci curvature of $g$ is zero, so $g$ solves the vacuum Einstein equations. It is static and spherically symmetric, and $g$ tends to the Minkowski metric as $r \to \infty$.

The slice $t = 0$ carries the Riemannian metric $(1 - 2m/r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2)$, the spatial Schwarzschild metric. It has zero scalar curvature, and in the coordinate $r$ extended to $r \ge 2m$ its boundary sphere $r = 2m$ is a minimal surface of area $16\pi m^2$.