# Schwarzschild Radius

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### The Schwarzschild Radius of a Black Hole

The Schwarzschild radius of a black hole is the location of the event horizon from the center of a non-spinning (stationary) black hole. In other words, light originating at radii less than the Schwarzschild radius cannot escape to infinity because it would be subject to an infinite redshift. The formula for the Schwarzschild radius is

\[

R_{s} \equiv \frac{2GM}{c^{2}}

\]

where

$G$ is the gravitational constant,

$M$ is the black-hole mass, and

$c$ is the speed of light.

We can conveniently express this in terms of solar masses:

\[

R_{s} = 2.9644 \left(\frac{M}{M_{\odot}}\right) \ {\rm km}

\]

or simply,

\[

R_{s} \sim 3 \left(\frac{M}{M_{\odot}}\right) \ {\rm km}

\]

where $M_{\odot}$ is the mass of the sun. For much larger black-hole masses it is more convenient to write:

\[

R_{s} = 2.9644 \times 10^{13} \ M_{8} \ {\rm cm},

\]

where $M_{8}$ is the mass of the black hole in units of $10^{8}$ solar masses.

It also quite handy to write

\[

R_{s} \sim 2 \left(\frac{M}{10^{8}M_{\odot}}\right) \ = \ 2 M_{8} \ \ {\rm AU},

\]

where AU is the astronomical unit.