Schwarzschild Black Hole

Schwarzschild coordinates

The spacetime metric is

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

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 are:

(1)   \begin{eqnarray*}    \alpha & = & \sqrt{1 - 2M/r } \\    \beta^a & = & 0 \\    K_{ab} & = & 0 \end{eqnarray*}

The geodesic equations are:

Isotropic coordinates

Starting with Schwarzschild coordinates, the transformation

    \[ r=\rho \left(1+M/(2\rho) \right)^2  \]

gives the line element

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

The horizon is \rho =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 are:

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

The geodesic equations are:

Kerr-Schild coordinates

Starting with Schwarzschild coordinates, the transformation

    \[ T =t + 2M \ln\left| \frac{r}{2M}-1 \right| +k \]

gives the line element

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

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

(3)   \begin{eqnarray*}    \alpha & = &  \frac{1}{\sqrt{1 + 2M/r} } \\    \beta^r & = & \frac{2M/r}{\left(1 + 2M/r\right)} \\    \beta^\theta & = &  \beta^\phi \ = \ 0 \\    K_{rr} & = & \frac{-2M(M+r)}{\sqrt{r^5(2M+r)} } \\    K_{\theta\theta} & = & 2M\sqrt{\frac{r}{(2M+r)} } \\    K_{\phi\phi} & = & K_{\theta\theta} \sin^2\theta \\    K_{r\theta} & = & K_{r\phi} \  = \  K_{\theta\phi} \  = \  0 \end{eqnarray*}

Changing from spherical coordinates r, \theta, \phi to Cartesian coordinates x^i gives

    \[ ds^2 = -\left(1 - \frac{2M}{r}\right)dT^2 + \left(\frac{4Mx^i}{r^2}\right) dx^i\,dT + \left( \delta^{ij} + \frac{2Mx^i x^j}{r^3}\right) dx^i\, dx^j \]

This can also be written as

    \[ g_{\mu\nu} = \eta_{\mu\nu} + 2H\ell_\mu \ell_\nu  \]

where \eta_{\mu\nu} = diag(-1,1,1,1) is the Minkowski metric, \ell_\mu = \left(1,\frac{x}{r},\frac{y}{r},\frac{z}{r}\right) is a covector, and H = M/r. Also let \ell^\mu \equiv \eta^{\mu\nu}\ell_\nu = \left(-1,\frac{x}{r},\frac{y}{r},\frac{z}{r}\right). Then the inverse metric is g^{\mu\nu} = \eta^{\mu\nu} - 2H\ell^\mu \ell^\nu. Note that the vector \ell^\mu is null in both the Minkowski and physical metrics.

The geodesic equations:

Kruskal coordinates

Starting with Schwarzschild coordinates, let

(4)   \begin{eqnarray*}  u^2 - v^2 & = & \left( \frac{r}{2M} - 1 \right) e^{r/(2M)} \\ \frac{u}{v} & = & \tanh\left(\frac{t}{4M}\right) \end{eqnarray*}

The line element becomes

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

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

(5)   \begin{eqnarray*}    \alpha & = & \sqrt{\frac{32M^3}{r} } e^{-r/(2M)}  \\    \beta^u & = & \beta^\theta \  = \  \beta^\phi  \  = \   0 \\    K_{uu} & = & -2v(2M+r) \left( \frac{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{eqnarray*}

Geodesic equations: