Line data    Source code

       1           0 : // Distributed under the MIT License.
       2             : // See LICENSE.txt for details.
       3             : 
       4             : #pragma once
       5             : 
       6             : #include <array>
       7             : 
       8             : #include "DataStructures/Tensor/TypeAliases.hpp"
       9             : 
      10             : namespace gr {
      11             : namespace Solutions {
      12             : 
      13             : /*!
      14             :  * \brief Radius of Kerr horizon in Kerr-Schild coordinates.
      15             :  *
      16             :  * \details Computes the radius of a Kerr black hole as a function of
      17             :  * angles.  The input argument `theta_phi` is typically the output of
      18             :  * the `theta_phi_points()` method of a `YlmSpherepack` object; i.e.,
      19             :  * a std::array of two DataVectors containing the values of theta and
      20             :  * phi at each point on a Strahlkorper.
      21             :  *
      22             :  * \note If the spin is nearly extremal, this function has accuracy
      23             :  *       limited to roughly \f$10^{-8}\f$, because of roundoff amplification
      24             :  *       from computing \f$M + \sqrt{M^2-a^2}\f$.
      25             :  *
      26             :  * Derivation:
      27             :  *
      28             :  * Define spherical coordinates \f$(r,\theta,\phi)\f$ in the usual way
      29             :  * from the Cartesian Kerr-Schild coordinates \f$(x,y,z)\f$
      30             :  * (i.e. \f$x = r \sin\theta \cos\phi\f$ and so on).
      31             :  * Then the relationship between \f$r\f$ and the radial
      32             :  * Boyer-Lindquist coordinate \f$r_{BL}\f$ is
      33             :  * \f[
      34             :  * r_{BL}^2 = \frac{1}{2}(r^2 - a^2)
      35             :  *     + \left(\frac{1}{4}(r^2-a^2)^2 +
      36             :  *             r^2(\vec{a}\cdot \hat{x})^2\right)^{1/2},
      37             :  * \f]
      38             :  * where \f$\vec{a}\f$ is the Kerr spin vector (with units of mass),
      39             :  * \f$\hat{x}\f$ means \f$(x/r,y/r,z/r)\f$, and the dot product is
      40             :  * taken as in flat space.
      41             :  *
      42             :  * The horizon is a surface of constant \f$r_{BL}\f$. Therefore
      43             :  * we can solve the above equation for \f$r^2\f$ as a function of angles,
      44             :  * yielding
      45             :  * \f[
      46             :  *     r^2 = \frac{r_{BL}^2 (r_{BL}^2 + a^2)}
      47             :                   {r_{BL}^2+(\vec{a}\cdot \hat{x})^2},
      48             :  * \f]
      49             :  * where the angles are encoded in \f$\hat x\f$ and everything else on the
      50             :  * right-hand side is constant.
      51             :  *
      52             :  * `kerr_horizon_radius` evaluates \f$r\f$ using the above equation, and
      53             :  * using the standard expression for the Boyer-Lindquist radius of the
      54             :  * Kerr horizon:
      55             :  * \f[
      56             :  *   r_{BL} = r_+ = M + \sqrt{M^2-a^2}.
      57             :  * \f]
      58             :  *
      59             :  */
      60             : template <typename DataType>
      61             : Scalar<DataType> kerr_horizon_radius(
      62             :     const std::array<DataType, 2>& theta_phi, double mass,
      63             :     const std::array<double, 3>& dimensionless_spin) noexcept;
      64             : 
      65             : }  // namespace Solutions
      66             : }  // namespace gr