SpECTRE  v2022.09.02
Cce::ComputeBondiIntegrand< Tags::PoleOfIntegrand< Tags::BondiH > > Struct Reference

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)

## Detailed Description

Computes the pole part of the integrand (right-hand side) of 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] [61] 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*}

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