Schwarzschild spacetime

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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

ds^2=-\Schw dt^2+\Schw^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)

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. u^\mu=\left(\Schw^{-1/2},0,0,0\right).

Geodesic motion. Worldlines parametrised [well, mostly…] by invariants e, the “energy per unit mass”, and \ell, 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 (\ell=0). 4-velocity u^\mu=\left(e\Schw^{-1},\pm\eroot,0,0\right).

More generally, $u^\mu=

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

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