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SpECTRE
v2024.09.29
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Computes the pole part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(H\). More...
#include <Equations.hpp>
Static Public Member Functions | |
| template<typename... Args> | |
| static void | apply (const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 2 > > * > regular_integrand_for_h, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 0 > > * > script_aj, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 0 > > * > script_bj, const gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 2 > > * > script_cj, const Args &... args) |
Computes the pole part of the integrand (right-hand side) of the equation which determines the radial (y) dependence of the Bondi quantity \(H\).
The quantity \(H \equiv \partial_u J\) (evaluated at constant y) 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] [85] 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 \(H\) on a surface of constant \(u\) given \(J\), \(\beta\), \(Q\), \(U\), and \(W\) on the same surface is written as
\[(1 - y) \partial_y H + H + (1 - y)(\mathcal{D}_J H + \bar{\mathcal{D}}_J \bar{H}) = A_J + (1 - y) B_J.\]
We refer to \(A_J\) as the "pole part" of the integrand and \(B_J\) as the "regular part". The pole part is computed by this function, and has the expression
\begin{align*} B_J =& -\tfrac{1}{2} \left(\eth(\partial_y (J) \bar{U}) + \partial_y (\bar{\eth} (J)) U \right) + J (\mathcal{B}_J + \bar{\mathcal{B}}_J) \notag\\ &+ \frac{e^{2 \beta}}{2 R} \left(\mathcal{C}_J + \eth (\eth (\beta)) - \tfrac{1}{2} \eth (Q) - (\mathcal{A}_J + \bar{\mathcal{A}_J}) J + \frac{\bar{\mathcal{C}_J} J^2}{K^2} + \frac{\eth (J (-2 \bar{\eth} (\beta) + \bar{Q}))}{4 K} - \frac{\eth (\bar{Q}) J}{4 K} + (\eth (\beta) - \tfrac{1}{2} Q)^2\right) \notag\\ &- \partial_y (J) \left(\frac{\eth (U) \bar{J}}{2 K} - \tfrac{1}{2} \bar{\eth} (\bar{U}) J K + \tfrac{1}{4} (\eth (\bar{U}) - \bar{\eth} (U)) K^2 + \frac{1}{2} \frac{\eth (R) \bar{U}}{R} - \frac{1}{2} W\right)\notag\\ &+ \partial_y (\bar{J}) \left(- \tfrac{1}{4} (- \eth (\bar{U}) + \bar{\eth} (U)) J^2 + \eth (U) J \left(- \frac{1}{2 K} + \tfrac{1}{2} K\right)\right)\notag\\ &+ (1 - y) \bigg[\frac{1}{2} \left(- \frac{\partial_y (J)}{R} + \frac{2 \partial_{u} (R) \partial_y (\partial_y (J))}{R} + \partial_y (\partial_y (J)) W\right) + \partial_y (J) \left(\tfrac{1}{2} \partial_y (W) + \frac{1}{2 R}\right)\bigg]\notag\\ &+ (1 - y)^2 \bigg[ \frac{\partial_y (\partial_y (J)) }{4 R} \bigg], \end{align*}
where
\begin{align*} \mathcal{A}_J =& \tfrac{1}{4} \eth (\eth (\bar{J})) - \frac{1}{4 K^3} - \frac{\eth (\bar{\eth} (J \bar{J})) - (\eth (\bar{\eth} (\bar{J})) - 4 \bar{J}) J}{16 K^3} + \frac{3}{4 K} - \frac{\eth (\bar{\eth} (\beta))}{4 K} \notag\\ &- \frac{\eth (\bar{\eth} (J)) \bar{J} (1 - \frac{1}{4 K^2})}{4 K} + \tfrac{1}{2} \eth (\bar{J}) \left(\eth (\beta) + \frac{\bar{\eth} (J \bar{J}) J}{4 K^3} - \frac{\bar{\eth} (J) (-1 + 2 K^2)}{4 K^3} - \tfrac{1}{2} Q\right)\\ \mathcal{B}_J =& - \frac{\eth (U) \bar{J} \partial_y (J \bar{J})}{4 K} + \tfrac{1}{2} \partial_y (W) + \frac{1}{4 R} + \tfrac{1}{4} \bar{\eth} (J) \partial_y (\bar{J}) U - \frac{\bar{\eth} (J \bar{J}) \partial_y (J \bar{J}) U}{8 K^2} \notag\\&- \tfrac{1}{4} J \partial_y (\eth (\bar{J})) \bar{U} + \tfrac{1}{4} (\eth (J \partial_y (\bar{J})) + \frac{J \eth (R) \partial_y(\bar{J})}{R}) \bar{U} \\ &+ (1 - y) \bigg[ \frac{\mathcal{D}_J \partial_{u} (R) \partial_y (J)}{R} - \tfrac{1}{4} \partial_y (J) \partial_y (\bar{J}) W + \frac{(\partial_y (J \bar{J}))^2 W}{16 K^2} \bigg] \\ &+ (1 - y)^2 \bigg[ - \frac{\partial_y (J) \partial_y (\bar{J})}{8 R} + \frac{(\partial_y (J \bar{J}))^2}{32 K^2 R} \bigg]\\ \mathcal{C}_J =& \tfrac{1}{2} \bar{\eth} (J) K (\eth (\beta) - \tfrac{1}{2} Q)\\ \mathcal{D}_J =& \tfrac{1}{4} \left(-2 \partial_y (\bar{J}) + \frac{\bar{J} \partial_y (J \bar{J})}{K^2}\right) \end{align*}