SpECTRE
v2023.01.13

Computes the pole part of the integrand (righthand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(Q\). More...
#include <Equations.hpp>
Public Types  
using  pre_swsh_derivative_tags = tmpl::list<> 
using  swsh_derivative_tags = tmpl::list< Spectral::Swsh::Tags::Derivative< Tags::BondiBeta, Spectral::Swsh::Tags::Eth > > 
using  integration_independent_tags = tmpl::list<> 
using  temporary_tags = tmpl::list<> 
using  return_tags = tmpl::append< tmpl::list< Tags::PoleOfIntegrand< Tags::BondiQ > >, 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 > > * > pole_of_integrand_for_q, const Args &... args) 
Computes the pole part of the integrand (righthand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(Q\).
The quantity \(Q\) 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 \(Q = Q_A q^A\). See [15] [66] 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 \(Q\) on a surface of constant \(u\) given \(J\) and \(\beta\) on the same surface is written as
\[(1  y) \partial_y Q + 2 Q = A_Q + (1  y) B_Q.\]
We refer to \(A_Q\) as the "pole part" of the integrand and \(B_Q\) as the "regular part". The pole part is computed by this function, and has the expression
\[A_Q = 4 \eth \beta.\]