Kerr

Kerr initial data for an isolated rotating black hole is specified using the ``quasi-isotropic'' coordinates [33]:

$$ds^2 = \psi^4 (dr^2 + r^2(d\theta^2 + \chi^2\sin^2\theta d\phi^2)),$$ (part327)

where

$$\psi^4 = - 2\frac{a^2}{r^2}\cos\theta\sin\theta,$$ (part328)

$$\chi^2 = p^2 / \Sigma,$$ (part329)

$$p^2 = a^2 + {r_k}^2 - a B_\phi,$$ (part3210)

$$r_k = r + M + \frac{M^2 - a^2}{4r},$$ (part3211)

$$B_\phi = -2 M r_k a \sin^2\theta / \Sigma,$$ (part3212)

$$\Sigma = {r_k}^2 + a^2 \cos^2\theta.$$ (part3213)

The two free parameters are the Kerr mass, $M$, and angular momentum, $a$ (assumed to be aligned with the z-axis). These are specified using the parameters idanalyticbh::mass and idanalyticbh::a_kerr respectively. (Note that the default values for these parameters are $M=2$ and $a=0.1$.) The black hole is assumed to reside at the centre of the coordinate system, at $x=y=z=0$.

The admbase::metric_type parameter can be used to specify whether the metric should be conformal or not. If the metric is conformal, then $\psi$ is initialised as a separate grid function, and it's first and second derivatives are calculated analytically and also stored as grid functions. Otherwise, the conformal factor is multiplied through in the expression for the 3-metric before the values of the admbase::metric variables are set. The extrinsic curvature is also determined analytically.

The gauge can be set to the Kerr lapse and shift with the parameters

  idanalyticbh::initial_lapse = "kerr"
  idanalyticbh::initial_shift = "kerr"

in which case the formulas

$$\alpha = \sqrt{\frac{\Delta}{p^2}},$$ (part3214)

$$\beta^\phi = -2 m r_k a / p^2,$$ (part3215)

where

$$\sqrt{\Delta} = r - \frac{m^2 - a^2}{4r}.$$ (part3216)

A set of parameters which initialise an evolution to use the Kerr intial data with mass $M=1$ and angular momentum $a=0.3$ are:

  ActiveThorns = "... ADMBase StaticConformal IDAnalyticBH ..."

  admbase::metric_type = "static conformal"

  admbase::initial_data = "kerr"
  admbase::initial_lapse = "kerr"
  admbase::initial_shift = "kerr"

  idanalyticbh::mass = 1.0
  idanalyticbh::a_kerr = 0.3