---
tag: 6Y50T68WKY
kind: example
name: Schwarzschild metric
msc: ["83C15", "83C57", "53C50"]
instance_of: [BY4QYYS788]
uses: [4GTNM0WHDH]
sources:
  - url: https://en.wikipedia.org/wiki/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|BY4QYYS788]]: 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|4GTNM0WHDH]] 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 $16\pi m^2$.

## Explanations

### L1 human

The Schwarzschild metric describes space and time outside a single round, non-rotating, uncharged mass, such as a star or a black hole.
Far away it looks like flat spacetime, and near the mass time runs slower and space is stretched.
The sphere $r = 2m$ is the event horizon of the black hole: the coordinates above break down there, although the spacetime itself does not.
It was the first exact solution of Einstein's equations, found by Karl Schwarzschild in 1916.