Computes the leading part of \(\Psi_2\) near \(\mathcal I^+\). More...

#include <ScriPlusValues.hpp>

Public Types
using	return_tags = tmpl::list<Tags::ScriPlus<Tags::Psi2>>
using	tensor_argument_tags
using	argument_tags

Static Public Member Functions

static void apply (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 0 > > * > psi_2, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &exp_2_beta, const Scalar< SpinWeighted< ComplexDataVector, 1 > > &dy_bondi_q, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &ethbar_dy_bondi_q, const Scalar< SpinWeighted< ComplexDataVector, 1 > > &dy_bondi_u, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &eth_dy_bondi_u, const Scalar< SpinWeighted< ComplexDataVector, 1 > > &dy_dy_bondi_u, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &ethbar_dy_dy_bondi_u, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &dy_dy_bondi_w, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &dy_bondi_j, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &dy_du_bondi_j, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &boundary_r, const Scalar< SpinWeighted< ComplexDataVector, 1 > > &eth_r_divided_by_r, size_t l_max, size_t number_of_radial_points)

Detailed Description

Computes the leading part of \(\Psi_2\) near \(\mathcal I^+\).

Details

The value \(\Psi_2\) scales asymptotically as \(r^{-3}\), and has the form (in the coordinates used for regularity preserving CCE)

\begin{align*}\Psi_2^{(3)} = -\frac{e^{-2 \beta^{(0)}}}{4} \left(e^{2 \beta^{(0)}} \eth \bar Q^{(1)} + \eth \bar U^{(2)} + \bar \eth U^{(2)} + J^{(1)} \bar \eth \bar U^{(1)} + J^{(1)} \bar \partial_u J^{(1)} - 2 W^{(2)}\right) \end{align*}

,

where \(A^{(n)}\) is the \(1/r^n\) part of \(A\) evaluated at \(\mathcal I^+\), so for any quantity \(A\),

\begin{align*}\eth A^{(1)} &= (-2 R \eth \partial_y A - 2 \eth R \partial_y A)|_{y = 1} \notag\\ \eth A^{(2)} &= (2 R^2 \eth \partial_y^2 A + 2 R \eth R \partial^2_y A)|_{y = 1}, \notag\\ A^{(1)} &= (- 2 R \partial_y A)|_{y = 1}, \notag\\ A^{(2)} &= (2 R^2 \partial_y^2 A)|_{y = 1}, \end{align*}

where the expansion is determined by the conversion between Bondi and numerical radii \(r = 2 R / (1 - y)\).

Member Typedef Documentation

◆ argument_tags

using Cce::CalculateScriPlusValue< Tags::ScriPlus< Tags::Psi2 > >::argument_tags

Initial value:

tmpl::push_back<tensor_argument_tags, Tags::LMax,

Tags::NumberOfRadialPoints>

◆ tensor_argument_tags

using Cce::CalculateScriPlusValue< Tags::ScriPlus< Tags::Psi2 > >::tensor_argument_tags