SpECTRE  v2021.12.06
Cce::ComputeBondiIntegrand< Tags::LinearFactor< Tags::BondiH > > Struct Reference

Computes the linear factor which multiplies \(H\) 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::LinearFactor< 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, 0 > > * > linear_factor_for_h, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 2 > > * > script_djbar, const Args &... args)
 

Detailed Description

Computes the linear factor which multiplies \(H\) in the equation which determines the radial (y) dependence of the Bondi quantity \(H\).

Details

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 [13] [54] 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 \(1 +(1 - y) J \mathcal{D}_J\) is the linear factor for the non-conjugated \(H\), and is computed from the equation:

\[\mathcal{D}_J = \frac{1}{4}(-2 \partial_y \bar{J} + \frac{\bar{J} \partial_y (J \bar{J})}{K^2})\]


The documentation for this struct was generated from the following file: