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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*=2*M*. These have zero angular momentum (). 4-velocity .

More generally, $u^\mu=

Tetrad: Frolov §2.11.2 citing Luminet and Marck (1985) http://adsabs.harvard.edu/abs/1985MNRAS.212…57L