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SpECTRE
v2024.09.16
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Computes the regular part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(W\). 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_w, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 0 > > * > script_av, const Args &... args) |
Computes the regular part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(W\).
The quantity \(W\) 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 \(q^A\):
\[ J \equiv h_{A B} q^A q^B, K \equiv h_{A B} q^A \bar{q}^B,\]
See [20] [82] 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 \(W\) on a surface of constant \(u\) given \(J\), \(\beta\), \(Q\), \(U\) on the same surface is written as
\[(1 - y) \partial_y W + 2 W = A_W + (1 - y) B_W. \]
We refer to \(A_W\) as the "pole part" of the integrand and \(B_W\) as the "regular part". The regular part is computed by this function, and has the expression
\[ B_W = \tfrac{1}{4} \partial_y (\eth (\bar{U})) + \tfrac{1}{4} \partial_y (\bar{\eth} (U)) - \frac{1}{2 R} + \frac{e^{2 \beta} (\mathcal{A}_W + \bar{\mathcal{A}_W})}{4 R}, \]
where
\begin{align*} \mathcal{A}_W =& - \eth (\beta) \eth (\bar{J}) + \tfrac{1}{2} \bar{\eth} (\bar{\eth} (J)) + 2 \bar{\eth} (\beta) \bar{\eth} (J) + (\bar{\eth} (\beta))^2 J + \bar{\eth} (\bar{\eth} (\beta)) J + \frac{\eth (J \bar{J}) \bar{\eth} (J \bar{J})}{8 K^3} + \frac{1}{2 K} - \frac{\eth (\bar{\eth} (J \bar{J}))}{8 K} - \frac{\eth (J \bar{J}) \bar{\eth} (\beta)}{2 K} \nonumber \\ &- \frac{\eth (\bar{J}) \bar{\eth} (J)}{4 K} - \frac{\eth (\bar{\eth} (J)) \bar{J}}{4 K} + \tfrac{1}{2} K - \eth (\bar{\eth} (\beta)) K - \eth (\beta) \bar{\eth} (\beta) K + \tfrac{1}{4} (- K Q \bar{Q} + J \bar{Q}^2). \end{align*}