SpECTRE  v2021.11.01
Cce::ComputeBondiIntegrand< Tags::LinearFactorForConjugate< Tags::BondiH > > Struct Reference

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)

## Detailed Description

Computes the linear factor which multiplies $$\bar{H}$$ 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 $$(1 - y) J \bar{\mathcal{D}}_J$$ is the linear factor for the non-conjugated $$H$$, and is computed from the equation:

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

The documentation for this struct was generated from the following file:
• src/Evolution/Systems/Cce/Equations.hpp