Schwarzschild
The Schwarzschild metric is a class of metrics describing 4 dimensional spherically symmetric vacuum spacetimes, parametrized by the Schwarzschild radius $r_s$, $r_s > 0$, or alternatively by the mass $M$,$M > 0$, with the relation $$r_s = \frac{2GM}{c^2}$$
1. History
The Schwarzschild metric originally comes from Karl Schwarzschild's 1916 paper "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie".
2. Topology
The maximally extended Schwarzschild spacetime has topology $\mathbb{R}^2 \times S^2$. All variants on the Schwarzschild spacetime are either subsets or quotients of this topology.
3. Metrics and coordinates
Schwarzschild coordinates
The original coordinates of the Schwarzschild metrics, corresponding to an observer at infinity.
$$ds^2 = -(1 - \frac{r_s}{r}) dt^2 + (1 - \frac{r_s}{r})^{-1} dr^2 + r^2 (d\theta + \sin^2 \theta d\varphi^2)$$Isotropic coordinates
$$ds^2 = - \left( \frac{1-\frac{r_s}{4r}}{1+\frac{r_s}{4r}} \right)^2 dt^2 + (1 + \frac{r_s}{4r})^4 (dx^2 + dy^2 + dz^2)$$Lorentz transformed coordinates
$$ds^2 = (1+A)^2 (-dt^2 + dx^2 + dy^2 + dz^2) - \left[ (1 + A)^4 - \left( \frac{1-A}{1+A} \right)^2 \right] \frac{dt - v dx}{1 - v^2} $$A = \frac{M}{2r}$$Tortoise coordinates
Tortoise coordinates are named after the Zeno paradox, as the radial distance becomes infinite as one approaches the horizon.
$$r^* = r + r_s \ln |\frac{r}{r_s} - 1|$$Eddington–Finkelstein coordinates
There are two Eddington-Finkelstein coordinates. First, the ingoing Eddington-Finkelstein coordinates, with $v = t + r^*$ :
$$ds^2 = - (1 - \frac{r_s}{r}) dv^2 + 2dv dr + r^2 (d\theta^2 + \sin^2 \theta d\varphi^2)$$And then the outgoing Eddington-Finkelstein coordinates, with $u = t - r^*$ :
$$ds^2 = - (1 - \frac{r_s}{r}) du^2 - 2du dr + r^2 (d\theta^2 + \sin^2 \theta d\varphi^2)$$Kruskal-Szekeres coordinates
$$ds^{2} = \frac{32M^3}{r}e^{-r/2M}(-dT^2 + dX^2) + r^2 d\Omega^2$$Gullstrand–Painlevé coordinates
$$ d\tau^{2} = \left(1-\frac{2M}{r} \right) dt^2 -\frac{dr^2}{ \left(1-\frac{2M}{r} \right)}- r^2 d\theta^2-r^2\sin^2\theta d\phi^2$$Lemaître coordinates
4. Tensor quantities
4.1. In Schwarzschild coordinates
Christoffel symbols
The non-zero Christoffel symbols are
| \begin{eqnarray} {\Gamma^r}_{tt} &=& \frac{r_s}{2r^3} (r - r_s)\\ {\Gamma^\theta}_{r\theta} &=& r^{-1}\\ {\Gamma^r}_{\varphi\varphi} &=& - (r - r_s) \sin^2 \theta \end{eqnarray} | \begin{eqnarray} {\Gamma^r}_{rr} &=& -\frac{r_s}{2r(r - r_s)}\\ {\Gamma^r}_{\theta\theta} &=& - (r - r_s)\\ {\Gamma^\theta}_{\varphi\varphi} &=& - \sin\theta \cos\theta \end{eqnarray} | \begin{eqnarray} {\Gamma^t}_{rt} &=& \frac{r_s}{2r(r - r_s)}\\ {\Gamma^\varphi}_{r\varphi} &=& r^{-1}\\ {\Gamma^\varphi}_{\theta\varphi} &=& \frac{\cos \theta}{\sin \theta} \end{eqnarray} |
Ricci tensor
$$R_{\mu\nu} = 0$$Ricci scalar
$$R = 0$$5. Symmetries
As a spherically symmetric static spacetime, the Schwarzschild spacetime has $3$ Killing vectors : $\partial_t$, $\partial_\theta$ and $\partial_\varphi$
6. Stress-energy tensor
As a vacuum spacetime, the stress-energy tensor is always $0$ :
$$T_{\mu\nu} = 0$$In distributional general relativity, the stress energy tensor can be defined at the singularity as well, with the form
$$T_{\mu\nu} = -M \delta(r) \delta_{\mu}^0 \delta_{\nu}^0\delta$$7. Curves
In Schwarzschild coordinates
The geodesic equation
\begin{eqnarray} \frac{d^2 t}{d\lambda^2} + \frac{r_s}{r(r - r_s)} \frac{dr}{d\lambda} \frac{dt}{d\lambda} &=& 0\\ \frac{d^2 r}{d\lambda^2} + \frac{r_s}{2r^3} (\frac{dt}{d\lambda})^2 - \frac{r_s}{2r(r - r_s)} (\frac{dr}{d\lambda})^2 - (r - r_s) \left[ (\frac{d\theta}{d\lambda})^2 + \sin^2 \theta (\frac{d\varphi}{d\lambda})^2 \right]&=& 0\\ \frac{d^2 \theta}{d\lambda^2} + \frac{2}{r} \frac{d\theta}{d\lambda} \frac{dr}{d\lambda} - \sin\theta\cos\theta (\frac{d\varphi}{d\lambda})^2 &=& 0\\ \frac{d^2 \varphi}{d\lambda^2} + \frac{2}{r} \frac{d\varphi}{d\lambda} \frac{dr}{d\lambda} + 2 \frac{\cos\theta}{\sin\theta}\frac{d\varphi}{d\lambda}\frac{d\theta}{d\lambda} &=& 0\\ \end{eqnarray}8. Equations
In Schwarzschild coordinates
The wave equation
As the metric is spherically symmetric, the angular part of the equation reduces to the Laplace equation on the sphere, with spherical harmonics as a solution. For a solution of the form $\Psi = r^{-1} f(r,t) Y_{lm}(\theta, \varphi)$, the equation is
$$\partial_t^2 f + \partial_{r^*}^2 f + (1 - \frac{r_s}{r}) \left[ \frac{l(l+1)}{r^2} + \frac{r_s}{r^3} \right] f = 0$$The Klein-Gordon equation
$$\partial_t^2 f + \partial_{r^*}^2 f + (1 - \frac{r_s}{r}) \left[ \frac{l(l+1)}{r^2} + \frac{r_s}{r^3} + m^2 \right] f = 0$$9. Causal structure
The Schwarzschild spacetime is globally hyperbolic. The maximally extended Schwarzchild spacetime has two singularities. It is not nakedly singular.
10. Asymptotic structure
Schwarzschild spacetime is asymptotically simple and flat, and has an event horizon (at the $2$-sphere $r = r_s$ for all $t$).
11. Energy conditions
As a vacuum spacetime, the Schwarzschild spacetime obeys most energy conditions.
12. Limits and related spacetimes
In the limit $M = 0$, the Schwarzschild metric reduces to Minkowski spacetime.
There exists a negative mass version of the Schwarzschild metric.