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SpECTRE
v2024.09.29
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Computes the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(U\). More...
#include <Equations.hpp>
Public Types | |
| using | pre_swsh_derivative_tags = tmpl::list< Tags::Exp2Beta, Tags::BondiJ, Tags::BondiQ > |
| using | swsh_derivative_tags = tmpl::list<> |
| using | integration_independent_tags = tmpl::list< Tags::BondiK, Tags::BondiR > |
| using | temporary_tags = tmpl::list<> |
| using | return_tags = tmpl::append< tmpl::list< Tags::Integrand< Tags::BondiU > >, 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, 1 > > * > regular_integrand_for_u, const Args &... args) |
Computes the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(U\).
The quantity \(U\) 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 \(q^A\):
\[ J \equiv h_{A B} q^A q^B, K \equiv h_{A B} q^A \bar{q}^B,\]
and \(Q\) is defined as a supplemental variable for radial integration of \(U\):
\[ Q_A = r^2 e^{-2\beta} h_{AB} \partial_r U^B\]
and \(U = U_A q^A\). See [20] [82] 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 \(U\) on a surface of constant \(u\) given \(J\), \(\beta\), and \(Q\) on the same surface is written as
\[\partial_y U = \frac{e^{2\beta}}{2 R} (K Q - J \bar{Q}). \]