KerrHorizon.hpp
1 // Distributed under the MIT License.
2 // See LICENSE.txt for details.
3 
4 #pragma once
5 
6 #include <array>
7 
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>
62  const std::array<DataType, 2>& theta_phi, const double& mass,
63  const std::array<double, 3>& dimensionless_spin) noexcept;
64 
65 } // namespace Solutions
66 } // namespace gr
Holds functions related to general relativity.
Definition: KerrHorizon.cpp:14
Defines a list of useful type aliases for tensors.
Scalar< DataType > kerr_horizon_radius(const std::array< DataType, 2 > &theta_phi, const double &mass, const std::array< double, 3 > &dimensionless_spin) noexcept
Radius of Kerr horizon in Kerr-Schild coordinates.
Definition: KerrHorizon.cpp:18
Tensor< T, Symmetry<>, index_list<> > Scalar
Scalar type.
Definition: TypeAliases.hpp:21