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SpECTRE
v2024.09.29
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Compute the Weyl scalar \(\Psi_0\) in the volume for the purpose of CCM, the quantity is in the Cauchy coordinates. More...
#include <NewmanPenrose.hpp>
Public Types | |
| using | return_tags = tmpl::list< Tags::Psi0Match > |
| using | argument_tags = tmpl::list< Tags::BondiJCauchyView, Tags::Dy< Tags::BondiJCauchyView >, Tags::Dy< Tags::Dy< Tags::BondiJCauchyView > >, Tags::BoundaryValue< Tags::BondiR >, Tags::OneMinusY, Tags::LMax > |
Static Public Member Functions | |
| static void | apply (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 2 > > * > psi_0, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &bondi_j_cauchy, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &dy_j_cauchy, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &dy_dy_j_cauchy, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &bondi_r_cauchy, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &one_minus_y, const size_t l_max) |
Compute the Weyl scalar \(\Psi_0\) in the volume for the purpose of CCM, the quantity is in the Cauchy coordinates.
The Weyl scalar \(\Psi_0\) is given by:
\begin{align*} \Psi_0 = \frac{(1 - y)^4}{16 r^2 K} \bigg[& \partial_y \beta \left((1 + K) (\partial_y J) - \frac{J^2 \partial_y \bar J}{1 + K}\right) - \frac{1}{2} (1 + K) (\partial_y^2 J) + \frac{J^2 \partial_y^2 \bar J}{2(K + 1)}\\ & + \frac{1}{K^2} \left(- \frac{1}{4} J \left(\bar{J}^2 \left(\partial_y J\right)^2 + J^2 \left(\partial_y \bar J\right)^2\right) + \frac{1 + K^2}{2} J (\partial_y J) (\partial_y \bar J) \right)\bigg]. \end{align*}
The quantities above are all in the Cauchy coordinates, where \(K\) is updated from \(J\) and \(\bar J\), \((1-y)\) is invariant under the coordinate transformation. \(r\) transforms as
\begin{align*} r = \omega \hat r \end{align*}