SpECTRE Documentation Coverage Report
 Current view: top level - PointwiseFunctions/AnalyticSolutions/GeneralRelativity - KerrHorizon.hpp Hit Total Coverage Commit: 2db722c93a8e9b106e406b439b79c8e05c2057fb Lines: 0 1 0.0 % Date: 2021-03-03 22:01:00 Legend: Lines: hit not hit
  Line data Source code  1 0 : // Distributed under the MIT License. 2 : // See LICENSE.txt for details. 3 : 4 : #pragma once 5 : 6 : #include 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 61 : Scalar kerr_horizon_radius( 62 : const std::array& theta_phi, double mass, 63 : const std::array& dimensionless_spin) noexcept; 64 : 65 : } // namespace Solutions 66 : } // namespace gr 

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