SpECTRE  v2021.11.01
Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiQ > > 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 $$Q$$. More...

#include <Equations.hpp>

## Public Types

using pre_swsh_derivative_tags = tmpl::list< Tags::Dy< Tags::BondiBeta >, Tags::Dy< Tags::BondiJ >, Tags::BondiJ >

using swsh_derivative_tags = tmpl::list< Spectral::Swsh::Tags::Derivative< Tags::Dy< Tags::BondiBeta >, Spectral::Swsh::Tags::Eth >, Spectral::Swsh::Tags::Derivative< ::Tags::Multiplies< Tags::BondiJ, Tags::BondiJbar >, Spectral::Swsh::Tags::Eth >, Spectral::Swsh::Tags::Derivative< ::Tags::Multiplies< Tags::BondiJbar, Tags::Dy< Tags::BondiJ > >, Spectral::Swsh::Tags::Eth >, Spectral::Swsh::Tags::Derivative< Tags::Dy< Tags::BondiJ >, Spectral::Swsh::Tags::Ethbar >, Spectral::Swsh::Tags::Derivative< Tags::BondiJ, Spectral::Swsh::Tags::Ethbar > >

using integration_independent_tags = tmpl::list< Tags::EthRDividedByR, Tags::BondiK >

using temporary_tags = tmpl::list<::Tags::SpinWeighted<::Tags::TempScalar< 0, ComplexDataVector >, std::integral_constant< int, 1 > > >

using return_tags = tmpl::append< tmpl::list< Tags::RegularIntegrand< 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 > > * > regular_integrand_for_q, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 1 > > * > script_aq, 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 $$Q$$.

### Details

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 [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 $$Q$$ 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 regular part is computed by this function, and has the expression

$B_Q = - \left(2 \mathcal{A}_Q + \frac{2 \bar{\mathcal{A}_Q} J}{K} - 2 \partial_y (\eth (\beta)) + \frac{\partial_y (\bar{\eth} (J))}{K}\right),$

where

$\mathcal{A}_Q = - \tfrac{1}{4} \eth (\bar{J} \partial_y (J)) + \tfrac{1}{4} J \partial_y (\eth (\bar{J})) - \tfrac{1}{4} \eth (\bar{J}) \partial_y (J) + \frac{\eth (J \bar{J}) \partial_y (J \bar{J})}{8 K^2} - \frac{\bar{J} \eth (R) \partial_y (J)}{4 R}.$

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