|
SpECTRE
v2024.04.12
|
Compute the Weyl scalar \(\Psi_0\) in the volume according to a standard set of Newman-Penrose vectors. More...
#include <NewmanPenrose.hpp>
Public Types | |
| using | return_tags = tmpl::list< Tags::Psi0 > |
| using | argument_tags = tmpl::list< Tags::BondiJ, Tags::Dy< Tags::BondiJ >, Tags::Dy< Tags::Dy< Tags::BondiJ > >, Tags::BondiK, Tags::BondiR, Tags::OneMinusY > |
Static Public Member Functions | |
| static void | apply (gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 2 > > * > psi_0, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &bondi_j, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &dy_j, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &dy_dy_j, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &bondi_k, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &bondi_r, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &one_minus_y) |
Compute the Weyl scalar \(\Psi_0\) in the volume according to a standard set of Newman-Penrose vectors.
The Bondi forms of the Newman-Penrose vectors that are needed for \(\Psi_0\) are:
\begin{align} \mathbf{l} &= \partial_r / \sqrt{2}\\ \mathbf{m} &= \frac{-1}{2 r} \left(\sqrt{1 + K} q^A \partial_A - \frac{J}{\sqrt{1 + K}}\bar{q}^A \partial_A \right) \end{align}
Then, we may compute \(\Psi_0 = l^\alpha m^\beta l^\mu m^\nu C_{\alpha \beta \mu \nu}\) from the Bondi system, giving
\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*}