SpECTRE  v2023.01.13
Cce::ComputeBondiIntegrand< Tags::Integrand< Tags::BondiU > > Struct Reference

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)

Detailed Description

Computes the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity $$U$$.

Details

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 [15] [68] 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

$\partial_y U = \frac{e^{2\beta}}{2 R} (K Q - J \bar{Q}).$

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