Schwarzschild Black Hole

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[edit] Schwarzschild coordinates

The spacetime metric is

ds^2 = - \left(1 - \frac{2M}{r}\right)dt^2 + \frac{1}{1-2M/r} dr^2 + r^2 d\Omega^2

where d\Omega^2 \equiv d\theta^2 + \sin^2\theta\, d\phi^2 is the metric for the unit two-sphere. The horizon is r = 2M and the singularity is r=0. The lapse function, shift vector, and extrinsic curvature defined by the t = {\rm const} slices and time flow vector field \partial_t:

\begin{array}{rcl} \alpha & = & \sqrt{1 - 2M/r} \\ \beta^a & = & 0 \\ K_{ab} & = & 0 \end{array}

[edit] Isotropic coordinates

The spacetime metric is

ds^2 = -\left( \frac{1 - M/(2r)}{1 + M/(2r)} \right)^2 dt^2 + \left( 1 + \frac{M}{2r} \right)^4 ( dr^2 + r^2 d\Omega^2 )

The horizon is r=M/2. The lapse function, shift vector, and extrinsic curvature defined by the t = {\rm const} slices and the time flow vector field \partial_t:

\begin{array}{rcl} \alpha & = & \left( \frac{1 - M/(2r)}{1 + M/(2r)} \right)\\ \beta^a & = & 0 \\ K_{ab} & = & 0 \end{array}

[edit] Kerr-Schild coordinates

The spacetime metric is

ds^2 = -\left( 1 - 2M/r \right) dt^2 + (4M/r)dt\,dr + \left( 1 + 2M/r\right) dr^2 + r^2 d\Omega^2

The lapse function, shift vector, and extrinsic curvature defined by the t = {\rm const} slices and the time flow vector field \partial_t:

\begin{array}{rcl} \alpha & = & 1/\sqrt{1 + 2M/r} \\ \beta^r & = & (2M/r)/(1 + 2M/r) \\ \beta^\theta & = & \beta^\phi \ = \ 0 \\ K_{rr} & = & -2M(M+r)/\sqrt{r^5(2M+r)} \\ K_{\theta\theta} & = & 2M\sqrt{r/(2M+r)} \\ K_{\phi\phi} & = & K_{\theta\theta} \sin^2\theta \\ K_{r\theta} & = & K_{r\phi} \ = \ K_{\theta\phi} \ = \ 0 \end{array}

[edit] Kruskal coordinates

The spacetime metric is

ds^2 = \frac{32M^3}{R} e^{-R/(2M)} (-dv^2 + du^2) + R^2 d\Omega^2

where the areal radius R is defined by

u^2 - v^2 = \left( \frac{R}{2M} - 1 \right) e^{R/(2M)}

The lapse function, shift vector, and extrinsic curvature defined by the v = {\rm const} slices and the time flow vector field \partial_v:

\begin{array}{rcl} \alpha & = & \sqrt{32M^3/R} e^{-R/(2M)} \\ \beta^u & = & \beta^\theta \ = \ \beta^\phi \ = \ 0 \\ K_{uu} & = & -2v(2M+R) \left( 2M/R\right)^{5/2} e^{-3R/(4M)} \\ K_{\theta\theta} & = & v\sqrt{2MR} e^{-R/(4M)} \\ K_{\phi\phi} & = & K_{\theta\theta} \sin^2\theta \\ K_{u\theta} & = & K_{u\phi} \ = \ K_{\theta\phi} \ = \ 0 \end{array}
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