Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <limits>
7 :
8 : #include "DataStructures/Tensor/TypeAliases.hpp"
9 :
10 : /// \cond
11 : namespace PUP {
12 : class er;
13 : } // namespace PUP
14 : /// \endcond
15 :
16 : namespace gr {
17 : /*!
18 : * \brief Contains helper functions for transforming tensors in Kerr
19 : * spacetime to Kerr-Schild coordinates.
20 : *
21 : * Consider the Kerr-Schild form of the Kerr metric in Cartesian
22 : * coordinates \f$ (t, x, y, z)\f$,
23 : *
24 : * \f{align*}
25 : * ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 + \dfrac{2Mr^3}{r^4 + a^2 z^2}
26 : * \left[dt + \dfrac{r(x\,dx + y\,dy)}{r^2 + a^2} +
27 : * \dfrac{a(y\,dx - x\,dy)}{r^2 + a^2} + \dfrac{z}{r}dz\right]^2,
28 : * \f}
29 : *
30 : * where \f$r\f$ is the function defined through
31 : *
32 : * \f{align*}
33 : * r^2(x, y, z) = \frac{x^2 + y^2 + z^2 - a^2}{2} +
34 : * \sqrt{\left(\frac{x^2 + y^2 + z^2 - a^2}{2}\right)^2 + a^2 z^2}.
35 : * \f}
36 : *
37 : * Depending on the physical context, the following coordinate transformations
38 : * are usually adopted:
39 : *
40 : * - \em Spherical \em coordinates, defined by the usual transformation
41 : *
42 : * \f{align*}
43 : * x &= r_\text{sph}\sin\theta\cos\phi,\\
44 : * y &= r_\text{sph}\sin\theta\sin\phi,\\
45 : * z &= r_\text{sph}\cos\theta.
46 : * \f}
47 : *
48 : * Note that \f$r_\text{sph} \neq r\f$, so the
49 : * horizon of a Kerr black hole is \em not at constant \f$r_\text{sph}\f$.
50 : *
51 : * - \em Oblate \em spheroidal \em coordinates,
52 : *
53 : * \f{align*}
54 : * x &= \sqrt{r^2 + a^2}\sin\vartheta\cos\phi,\\
55 : * y &= \sqrt{r^2 + a^2}\sin\vartheta\sin\phi,\\
56 : * z &= r\cos\vartheta.
57 : * \f}
58 : *
59 : * Notice that \f$r\f$ is that defined in the Kerr-Schild form of the Kerr
60 : * metric, so the horizon is at constant \f$r\f$. (As it will be noted below,
61 : * \f$r\f$ corresponds to the Boyer-Lindquist radial coordinate.) Also, note
62 : * that the azimuthal angle \f$\phi\f$ is the same as the corresponding
63 : * angle in spherical coordinates, whereas the zenithal angle
64 : * \f$\vartheta\f$ differs from the corresponding angle \f$\theta\f$.
65 : *
66 : * - \em Kerr \em coordinates (sometimes also referred to as
67 : * "Kerr-Schild coordinates"), defined by
68 : *
69 : * \f{align*}
70 : * x &= (r \cos\varphi - a \sin\varphi)\sin\vartheta =
71 : * \sqrt{r^2 + a^2}\sin\vartheta\cos\left(\varphi +
72 : * \tan^{-1}(a/ r)\right),\\
73 : * y &= (r \sin\varphi + a \cos\varphi)\sin\vartheta =
74 : * \sqrt{r^2 + a^2}\sin\vartheta\sin\left(\varphi +
75 : * \tan^{-1}(a/ r)\right),\\
76 : * z &= r \cos\vartheta.
77 : * \f}
78 : *
79 : * These are the coordinates used in the gr::KerrSchildCoords class.
80 : * Notice that \f$r\f$ and \f$\vartheta\f$ are the same as those in
81 : * spheroidal coordinates, whereas the azimuthal angle
82 : * \f$\varphi \neq \phi\f$.
83 : *
84 : * \note Kerr's original "advanced Eddington-Finkelstein" form of his metric
85 : * is retrieved by performing the additional transformation \f$t = u - r\f$.
86 : * Note that Kerr's choice of the letter \f$u\f$ for the time coordinate
87 : * is not consistent with the convention of denoting the advanced time
88 : * \f$v = t + r\f$ (in flat space), and the retarded time \f$u = t - r\f$
89 : * (in flat space). Finally, note that Kerr's original metric has the sign
90 : * of \f$a\f$ flipped.
91 : *
92 : * The Kerr coordinates have been used in the hydro community to evolve the
93 : * GRMHD equations. They are the intermediate step in getting the Kerr metric
94 : * in Boyer-Lindquist coordinates. The relation between both is
95 : *
96 : * \f{align*}
97 : * dt_\text{BL} &=
98 : * dt - \frac{2M r\,dr}{r^2 - 2Mr + a^2}, \\
99 : * dr_\text{BL} &= dr,\\
100 : * d\theta_\text{BL} &= d\vartheta,\\
101 : * d\phi_\text{BL} &= d\varphi - \frac{a\,dr}{r^2-2Mr + a^2}.
102 : * \f}
103 : *
104 : * The above transformation makes explicit that \f$r\f$ is the Boyer-Lindquist
105 : * radial coodinate, \f$ r = r_\text{BL}\f$. Likewise,
106 : * \f$\vartheta = \theta_\text{BL}\f$.
107 : *
108 : * \note Sometimes (especially in the GRMHD community), the Kerr coordinates
109 : * as defined above are referred to as "spherical Kerr-Schild" coordinates.
110 : * These should not be confused with the true spherical coordinates defined
111 : * in this documentation.
112 : *
113 : * The Kerr metric in Kerr coodinates is sometimes used to write analytic
114 : * initial data for hydro simulations. In this coordinate system, the
115 : * metric takes the form
116 : *
117 : * \f{align*}
118 : * ds^2 &= -(1 - B)\,dt^2 + (1 + B)\,dr^2 + \Sigma\,d\vartheta^2 +
119 : * \left(r^2 + a^2 + B a^2\sin^2\vartheta\right)\sin^2\vartheta\,d\varphi^2\\
120 : * &\quad + 2B\,dt\,dr - 2aB\sin^2\vartheta\,dt\,d\varphi -
121 : * 2a\left(1 + B\right)\sin^2\vartheta\,dr\,d\varphi,
122 : * \f}
123 : *
124 : * where \f$B = 2M r/ \Sigma\f$ and \f$\Sigma = r^2 + a^2\cos^2\vartheta\f$.
125 : * Using the additional relations
126 : *
127 : * \f{align*}
128 : * \sin\vartheta &= \sqrt{\frac{x^2 + y^2}{r^2 + a^2}},\quad
129 : * \cos\vartheta = \frac{z}{r},\\
130 : * \sin\varphi &= \frac{y r - x a}{\sqrt{(x^2 + y^2)(r^2 + a^2)}},\quad
131 : * \cos\varphi = \frac{x r + y a}{\sqrt{(x^2 + y^2)(r^2 + a^2)}},
132 : * \f}
133 : *
134 : * the Jacobian of the transformation to Cartesian Kerr-Schild coordinates is
135 : *
136 : * \f{align*}
137 : * \dfrac{\partial(x, y, z)}{\partial(r, \vartheta, \varphi)} &=
138 : * \left(\begin{array}{ccc}
139 : * \dfrac{xr + ya}{r^2 + a^2} &
140 : * \dfrac{xz}{r}\sqrt{\dfrac{r^2 + a^2}{x^2 + y^2}} & -y\\
141 : * \dfrac{yr - ax}{r^2 + a^2} &
142 : * \dfrac{yz}{r}\sqrt{\dfrac{r^2 + a^2}{x^2 + y^2}} & x\\
143 : * \dfrac{z}{r} & - r\sqrt{\dfrac{x^2 + y^2}{r^2 + a^2}} & 0
144 : * \end{array}\right),
145 : * \f}
146 : *
147 : * which can be used to transform tensors between both coordinate systems.
148 : */
149 1 : class KerrSchildCoords {
150 : public:
151 0 : KerrSchildCoords() = default;
152 0 : KerrSchildCoords(const KerrSchildCoords& /*rhs*/) = default;
153 0 : KerrSchildCoords& operator=(const KerrSchildCoords& /*rhs*/) = default;
154 0 : KerrSchildCoords(KerrSchildCoords&& /*rhs*/) = default;
155 0 : KerrSchildCoords& operator=(KerrSchildCoords&& /*rhs*/) = default;
156 0 : ~KerrSchildCoords() = default;
157 :
158 0 : KerrSchildCoords(double bh_mass, double bh_dimless_spin);
159 :
160 : /// Transforms a spatial vector from Kerr (or "spherical Kerr-Schild")
161 : /// coordinates to Cartesian Kerr-Schild coordinates. If applied on points
162 : /// on the z-axis, the vector to transform must have a vanishing
163 : /// \f$v^\vartheta\f$ in order for the transformation to be single-valued.
164 : template <typename DataType>
165 1 : tnsr::I<DataType, 3, Frame::Inertial> cartesian_from_spherical_ks(
166 : const tnsr::I<DataType, 3, Frame::NoFrame>& spatial_vector,
167 : const tnsr::I<DataType, 3, Frame::Inertial>& cartesian_coords) const;
168 :
169 : /// Kerr-Schild \f$r^2\f$ in terms of the Cartesian coordinates.
170 : template <typename DataType>
171 1 : Scalar<DataType> r_coord_squared(
172 : const tnsr::I<DataType, 3, Frame::Inertial>& cartesian_coords) const;
173 :
174 : // NOLINTNEXTLINE(google-runtime-references)
175 0 : void pup(PUP::er& /*p*/);
176 :
177 : private:
178 0 : friend bool operator==(const KerrSchildCoords& lhs,
179 : const KerrSchildCoords& rhs);
180 :
181 : // The spatial components of the Jacobian of the transformation from
182 : // Kerr coordinates to Cartesian coordinates (x, y, z).
183 : template <typename DataType>
184 0 : tnsr::Ij<DataType, 3, Frame::NoFrame> jacobian_matrix(
185 : const DataType& x, const DataType& y, const DataType& z) const;
186 :
187 0 : double spin_a_ = std::numeric_limits<double>::signaling_NaN();
188 : };
189 :
190 0 : bool operator!=(const KerrSchildCoords& lhs, const KerrSchildCoords& rhs);
191 :
192 : } // namespace gr
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