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SpECTRE
v2024.04.12
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Computes the linear factor which multiplies \(\bar{H}\) in the equation which determines the radial (y) dependence of the Bondi quantity \(H\). More...
#include <Equations.hpp>
Public Types | |
| using | pre_swsh_derivative_tags = tmpl::list< Tags::Dy< Tags::BondiJ >, Tags::BondiJ > |
| using | swsh_derivative_tags = tmpl::list<> |
| using | integration_independent_tags = tmpl::list< Tags::OneMinusY > |
| using | temporary_tags = tmpl::list<::Tags::SpinWeighted<::Tags::TempScalar< 0, ComplexDataVector >, std::integral_constant< int, 2 > > > |
| using | return_tags = tmpl::append< tmpl::list< Tags::LinearFactorForConjugate< Tags::BondiH > >, temporary_tags > |
| using | argument_tags = tmpl::append< pre_swsh_derivative_tags, swsh_derivative_tags, integration_independent_tags > |
Static Public Member Functions | |
| template<typename... Args> | |
| static void | apply (const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 4 > > * > linear_factor_for_conjugate_h, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 2 > > * > script_djbar, const Args &... args) |
Computes the linear factor which multiplies \(\bar{H}\) in the equation which determines the radial (y) dependence of the Bondi quantity \(H\).
The quantity \(H \equiv \partial_u J\) (evaluated at constant y) is defined via the Bondi form of the metric:
\[ds^2 = - \left(e^{2 \beta} (1 + r W) - r^2 h_{AB} U^A U^B\right) du^2 - 2 e^{2 \beta} du dr - 2 r^2 h_{AB} U^B du dx^A + r^2 h_{A B} dx^A dx^B. \]
Additional quantities \(J\) and \(K\) are defined using a spherical angular dyad \(q^A\):
\[ J \equiv h_{A B} q^A q^B, K \equiv h_{A B} q^A \bar{q}^B.\]
See [20] [81] for full details.
We write the equations of motion in the compactified coordinate \( y \equiv 1 - 2 R/ r\), where \(r(u, \theta, \phi)\) is the Bondi radius of the \(y=\) constant surface and \(R(u,\theta,\phi)\) is the Bondi radius of the worldtube. The equation which determines \(H\) on a surface of constant \(u\) given \(J\), \(\beta\), \(Q\), \(U\), and \(W\) on the same surface is written as
\[(1 - y) \partial_y H + H + (1 - y) J (\mathcal{D}_J H + \bar{\mathcal{D}}_J \bar{H}) = A_J + (1 - y) B_J.\]
The quantity \( (1 - y) J \bar{\mathcal{D}}_J\) is the linear factor for the non-conjugated \(H\), and is computed from the equation:
\[\mathcal{D}_J = \frac{1}{4}(-2 \partial_y \bar{J} + \frac{\bar{J} \partial_y (J \bar{J})}{K^2})\]