1. Databases
2. Spacetime database
3. Kerr

Kerr

1. History

2. Topology

3. Metrics and coordinates

Kerr coordinates

\begin{eqnarray} ds^2 &=& -\left[ 1 - \frac{2mr}{r^2 + a^2 \cos^2 \theta} \right] (du + a \sin^2 \theta d\varphi)^2 \\ &&+ 2(du + a \sin^2 \theta d\varphi) (dr + a \sin^2 \theta d\varphi)\\ &&+ (r^2 + a^2 \cos^2 \theta) (d\theta^2 + \sin^2 \theta d\varphi^2) \end{eqnarray}

Kerr-Schild "Cartesian" coordinates

\begin{eqnarray} ds^2 &=& -dt^2 + dx^2 + dy^2 + dz^2 \\ && + \frac{2mr^3}{r^4 + a^2 z^2} \left[ dt + \frac{r(x dx + y dy)}{a^2 + r^2} + \frac{a(y dx - x dy)}{a^2 + r^2} + \frac{z}{r} dz \right]^2 \end{eqnarray} $$x^2 + y^2 + z^2 = r^2 + a^2 \left[ 1 - \frac{z^2}r^2{} \right]$$

Boyer-Lindquist coordinates

\begin{eqnarray} ds^2 &=& -dt^2 + dr^2 + 2a \sin^2 \theta dr d\varphi + (r^2 + a^2 \cos^2 \theta) d\theta^2\\ &&+ (r^2 + a^2)\sin^2 \theta d\varphi^2 + \frac{2mr}{r^2 + a^2 \cos^2 \theta} (dt + dr + a \sin^2 \theta d\varphi)^2 \end{eqnarray}

Rational polynomial coordinates

\begin{eqnarray} ds^2 &=& - \left[ 1 - \frac{2mr}{r^2 + a^2 \chi^2} \right] dt^2 - \frac{4amr(1 - \chi^2)}{r^2 + a^2 \chi^2} d\varphi dt\\ && + \frac{r^2 + a^2 \chi^2}{r^2 - 2mr + a^2} dr^2 + (r^2 + a^2 \chi^2) \frac{d\chi^2}{1 - \chi^2}\\ && + (1 - \chi^2) \left[ r^2 + a^2 + \frac{2ma^2 r (1-\chi^2)}{r^2 + a^2 \chi^2} \right] d\varphi^2 \end{eqnarray}

Doran coordinates

\begin{eqnarray} ds^2 &=& - dt^2 + (r^2 + a^2 \cos^2 \theta) d\theta^2 + (r^2 + a^2) \sin^2 \theta d\varphi^2\\ && + \frac{r^2 + a^2 \cos^2 \theta}{r^2 + a^2} \left[ dr + \frac{\sqrt{2mr(r^2 + a^2)}}{r^2 + a^2 \cos^2 \theta} (dt - a \sin^2 \theta d\varphi) \right]^2 \end{eqnarray}

4. Tensor quantities

5. Symmetries

6. Stress-energy tensor

7. Curves

8. Equations

9. Causal structure

10. Asymptotic structure

11. Energy conditions

12. Limits and related spacetimes

13. Misc.

Bibliography