|
SpECTRE
v2024.08.03
|
Computes the regular part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the scalar quantity \(\Pi\). More...
#include <Equations.hpp>
Static Public Member Functions | |
| template<typename... Args> | |
| static void | apply (const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 0 > > * > regular_integrand_for_kg_pi, const Args &... args) |
Computes the regular part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the scalar quantity \(\Pi\).
The evolution equation for \(\Pi\) is written as
\[(1 - y) \partial_y \Pi + \Pi = A_\Pi + (1 - y) B_\Pi.\]
We refer to \(A_\Pi\) as the "pole part" of the integrand and \(B_\Pi\) as the "regular part". The regular part is computed by this function, and has the expression:
\[ B_\Pi = \frac{1}{4R}e^{2\beta}\left(N_{\psi 1}-N_{\psi 2} +N_{\psi 3}\right) -\frac{R}{2}\frac{N_{\psi 4}}{(1-y)^2} +\tau+\frac{\partial_{u}R}{R}(1-y)\partial_y^2\psi, \]
where
\begin{align*} & N_{\psi 1}= K(2\Re \eth\beta\bar{\eth}\psi + \eth\bar{\eth}\psi) \\ & N_{\psi 2}=\Re \bar{J}\eth\eth\psi + 2\Re \bar{\eth}\beta \bar{\eth}\psi J + \Re \bar{\eth}J\bar{\eth}\psi \\ & N_{\psi 3}=\Re \eth K\bar{\eth}\psi \\ & N_{\psi 4}=2\Re (\bar{\eth}\psi ) \partial_rU + 2\Re \eth \bar{U}\partial_r\psi + 4\Re \bar{U}\eth\partial_r\psi \\ & \tau=\frac{1}{2}(1-y)\partial_y W\partial_y\psi + \frac{(1-y)^2}{4R}\partial_y^2\psi + \frac{1}{2}(1-y)W\partial_y^2\psi + \frac{1}{2}W\partial_y\psi \end{align*}