SpECTRE  v2021.11.01
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 [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 \(U\) 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: