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
using	swsh_derivative_tags
using	integration_independent_tags
using	temporary_tags
using	return_tags
using	argument_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 [21] [93] 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

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

.

Member Typedef Documentation

argument_tags

using Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiQ > >::argument_tags

Initial value:

 
      tmpl::append<pre_swsh_derivative_tags, swsh_derivative_tags,
                   integration_independent_tags>

integration_independent_tags

using Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiQ > >::integration_independent_tags

Initial value:

 
      tmpl::list<Tags::EthRDividedByR, Tags::BondiK>

pre_swsh_derivative_tags

using Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiQ > >::pre_swsh_derivative_tags

Initial value:

 
      tmpl::list<Tags::Dy<Tags::BondiBeta>, Tags::Dy<Tags::BondiJ>,
                 Tags::BondiJ>

return_tags

using Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiQ > >::return_tags

Initial value:

 
      tmpl::append<tmpl::list<Tags::RegularIntegrand<Tags::BondiQ>>,
                   temporary_tags>

swsh_derivative_tags

using Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiQ > >::swsh_derivative_tags

Initial value:

 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>>

temporary_tags

using Cce::ComputeBondiIntegrand< Tags::RegularIntegrand< Tags::BondiQ > >::temporary_tags

Initial value:

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

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

• src/Evolution/Systems/Cce/Equations.hpp