SQUARE45

For a static, spherically symmetric mass

M

in vacuum, the Schwarzschild metric

g_{\mu\nu}

in coordinates

(t, r, \theta, \phi)

is given by:\n\n

ds^2 = g_{\mu\nu} dx^\mu dx^\nu = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \frac{1}{\left(1 - \frac{2GM}{c^2 r}\right)} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)

\n\nWhere

G

is the gravitational constant,

M

is the mass, and

c

is the speed of light. The components are

g_{tt} = -\left(1 - \frac{2GM}{c^2 r}\right)

g_{rr} = \left(1 - \frac{2GM}{c^2 r}\right)^{-1}

, and

g_{\theta\theta} = r^2

Schwarzschild Metric