Gravitational Radius
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Gravitational Radius
The formula for the gravitational radius of an object is
\[
r_{g} \equiv \frac{GM}{c^{2}}
\]
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:
\[
r_{g} = 1.4822 \left(\frac{M}{M_{\odot}}\right) \ {\rm km}
\]
or simply,
\[
r_{g} \sim 1.5 \left(\frac{M}{M_{\odot}}\right) \ {\rm km}
\]
where $M_{\odot}$ is the mass of the sun. For much larger black-hole masses it is more convenient to write:
\[
r_{g} = 1.4822 \times 10^{13} \ M_{8} \ {\rm cm},
\]
where $M_{8}$ is the mass of the black hole in units of $10^{8}$ solar masses.
The following is also useful:
\[
r_{g} \sim \left(\frac{M}{10^{8}M_{\odot}}\right) \ {\rm AU} \ = \ M_{8} \ \ {\rm AU},
\]
where AU is the astronomical unit.
Note that the event horizon of a non-spinning black hole is at $2r_{g}$, which is also equal to the Schwarzschild radius.
