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SpECTRE
v2024.04.12
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Computes the pole part of the integrand (right-hand side) of 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::BondiJ, Tags::BondiU, Tags::BondiW > |
| using | swsh_derivative_tags = tmpl::list< Spectral::Swsh::Tags::Derivative< Tags::BondiU, Spectral::Swsh::Tags::Eth >, Spectral::Swsh::Tags::Derivative< Tags::BondiJ, Spectral::Swsh::Tags::Ethbar >, Spectral::Swsh::Tags::Derivative< ::Tags::Multiplies< Tags::BondiJbar, Tags::BondiU >, Spectral::Swsh::Tags::Ethbar >, Spectral::Swsh::Tags::Derivative< Tags::BondiU, Spectral::Swsh::Tags::Ethbar > > |
| using | integration_independent_tags = tmpl::list< Tags::BondiK > |
| using | temporary_tags = tmpl::list<> |
| using | return_tags = tmpl::append< tmpl::list< Tags::PoleOfIntegrand< 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, 2 > > * > pole_of_integrand_for_h, const Args &... args) |
Computes the pole part of the integrand (right-hand side) of 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] [80] 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 \(W\) on a surface of constant \(u\) given \(J\), \(\beta\), \(Q\), \(U\), and \(W\) on the same surface is written as
\[(1 - y) \partial_y H + H + (1 - y)(\mathcal{D}_J H + \bar{\mathcal{D}}_J \bar{H}) = A_J + (1 - y) B_J.\]
We refer to \(A_J\) as the "pole part" of the integrand and \(B_J\) as the "regular part". The pole part is computed by this function, and has the expression
\begin{align*} A_J =& - \tfrac{1}{2} \eth (J \bar{U}) - \eth (\bar{U}) J - \tfrac{1}{2} \bar{\eth} (U) J - \eth (U) K - \tfrac{1}{2} (\bar{\eth} (J) U) + 2 J W \end{align*}