# Schwarzschild spacetime

Difficulty level:   ★ ★ ★

Schwarzschild spacetime

A Schwarzschild black hole is the simplest type of black hole: it does not rotate and has no electric charge. It is named after Karl Schwarzschild, discovered in * and published in 1916.

One choice of coordinates, and probably the most common one, is Schwarzschild-Droste coordinates (t,r,θ,φ), under which the metric takes form

in geometric units G=c=1. (Droste is not usually credited, but deserves to be. See *)

#### History

This was the first non-trivial exact solution found to Einstein’s field equations.

#### Symmetries

Schwarzschild spacetime does not change over time, and is spherically symmetric. Mathematically, these symmetries are described by the following Killing vectors:

\partial_t \partial_\phi

#### Curvature

Christoffel symbols, and curvature tensors. Some sources giving curvature quantities in various coordinates are: Hartle §B for Schwarzschild coordinates, Frolov

#### Coordinate systems

Schwarzschild

G-P

E-F

K-S

Orbits: velocities and frames

Static observer. .

Geodesic motion. Worldlines parametrised [well, mostly…] by invariants e, the “energy per unit mass”, and , the “angular momentum per unit mass”.

Radial motion: Taylor & Wheeler term “rain”, “hail”, “drips”. I add a 4th metaphor, “snow”, for e≤0 which is only allowed inside the event horizon r=2M. These have zero angular momentum (). 4-velocity .

More generally, \$u^\mu=