SpECTRE  v2021.11.01
Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiW > > Struct Reference

Computes the regular part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(W\). More...

#include <Equations.hpp>

Public Types

using pre_swsh_derivative_tags = tmpl::list< Tags::Dy< Tags::BondiU >, Tags::Exp2Beta, Tags::BondiJ, Tags::BondiQ >
using swsh_derivative_tags = tmpl::list< Spectral::Swsh::Tags::Derivative< Tags::BondiBeta, Spectral::Swsh::Tags::Eth >, Spectral::Swsh::Tags::Derivative< Tags::BondiBeta, Spectral::Swsh::Tags::EthEth >, Spectral::Swsh::Tags::Derivative< Tags::BondiBeta, Spectral::Swsh::Tags::EthEthbar >, Spectral::Swsh::Tags::Derivative< Spectral::Swsh::Tags::Derivative< Tags::BondiJ, Spectral::Swsh::Tags::Ethbar >, Spectral::Swsh::Tags::Eth >, Spectral::Swsh::Tags::Derivative< ::Tags::Multiplies< Tags::BondiJ, Tags::BondiJbar >, Spectral::Swsh::Tags::EthEthbar >, Spectral::Swsh::Tags::Derivative< ::Tags::Multiplies< Tags::BondiJ, Tags::BondiJbar >, Spectral::Swsh::Tags::Eth >, Spectral::Swsh::Tags::Derivative< Tags::Dy< Tags::BondiU >, Spectral::Swsh::Tags::Ethbar >, Spectral::Swsh::Tags::Derivative< Tags::BondiJ, Spectral::Swsh::Tags::EthbarEthbar >, Spectral::Swsh::Tags::Derivative< Tags::BondiJ, Spectral::Swsh::Tags::Ethbar > >
using integration_independent_tags = tmpl::list< Tags::EthRDividedByR, Tags::BondiK, Tags::BondiR >
using temporary_tags = tmpl::list<::Tags::SpinWeighted<::Tags::TempScalar< 0, ComplexDataVector >, std::integral_constant< int, 0 > > >
using return_tags = tmpl::append< tmpl::list< Tags::RegularIntegrand< Tags::BondiW > >, 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, 0 > > * > regular_integrand_for_w, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 0 > > * > script_av, const Args &... args)

Detailed Description

Computes the regular part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(W\).


The quantity \(W\) 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 \(W\) on a surface of constant \(u\) given \(J\), \(\beta\), \(Q\), \(U\) on the same surface is written as

\[(1 - y) \partial_y W + 2 W = A_W + (1 - y) B_W. \]

We refer to \(A_W\) as the "pole part" of the integrand and \(B_W\) as the "regular part". The regular part is computed by this function, and has the expression

\[ B_W = \tfrac{1}{4} \partial_y (\eth (\bar{U})) + \tfrac{1}{4} \partial_y (\bar{\eth} (U)) - \frac{1}{2 R} + \frac{e^{2 \beta} (\mathcal{A}_W + \bar{\mathcal{A}_W})}{4 R}, \]


\begin{align*} \mathcal{A}_W =& - \eth (\beta) \eth (\bar{J}) + \tfrac{1}{2} \bar{\eth} (\bar{\eth} (J)) + 2 \bar{\eth} (\beta) \bar{\eth} (J) + (\bar{\eth} (\beta))^2 J + \bar{\eth} (\bar{\eth} (\beta)) J + \frac{\eth (J \bar{J}) \bar{\eth} (J \bar{J})}{8 K^3} + \frac{1}{2 K} - \frac{\eth (\bar{\eth} (J \bar{J}))}{8 K} - \frac{\eth (J \bar{J}) \bar{\eth} (\beta)}{2 K} \nonumber \\ &- \frac{\eth (\bar{J}) \bar{\eth} (J)}{4 K} - \frac{\eth (\bar{\eth} (J)) \bar{J}}{4 K} + \tfrac{1}{2} K - \eth (\bar{\eth} (\beta)) K - \eth (\beta) \bar{\eth} (\beta) K + \tfrac{1}{4} (- K Q \bar{Q} + J \bar{Q}^2). \end{align*}

The documentation for this struct was generated from the following file: