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SpECTRE
v2024.09.29
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Update the inertial gauge cartesian coordinate derivative \(\partial_u \hat x(x)\). More...
#include <GaugeTransformBoundaryData.hpp>
Public Types | |
| using | return_tags = tmpl::list<::Tags::dt< Tags::PartiallyFlatCartesianCoords >, Tags::BondiUAtScri > |
| using | argument_tags = tmpl::list< Tags::PartiallyFlatCartesianCoords, Tags::CauchyGaugeC, Tags::PartiallyFlatGaugeOmega, Tags::CauchyGaugeD, Tags::LMax, Spectral::Swsh::Tags::SwshInterpolator< Tags::PartiallyFlatAngularCoords > > |
Static Public Member Functions | |
| static void | apply (gsl::not_null< tnsr::i< DataVector, 3 > * > cartesian_inertial_du_x, gsl::not_null< Scalar< SpinWeighted< ComplexDataVector, 1 > > * > evolution_gauge_u_at_scri, const tnsr::i< DataVector, 3 > &cartesian_inertial_coordinates, const Scalar< SpinWeighted< ComplexDataVector, 2 > > &gauge_cauchy_c, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &omega, const Scalar< SpinWeighted< ComplexDataVector, 0 > > &gauge_cauchy_d, size_t l_max, const Spectral::Swsh::SwshInterpolator &interpolator) |
Update the inertial gauge cartesian coordinate derivative \(\partial_u \hat x(x)\).
For the asymptotically inertial angular coordinates \(\hat{x}^{\hat{A}}\), we have:
\begin{align*} \partial_u \hat{x}^{\hat{A}} = -U^{(0)B}\partial_B \hat{x}^{\hat{A}} \end{align*}
and the Cartesian version reads
\begin{align*} \partial_u \hat{x}^{\hat{i}}= - \text{Re}(\bar{U}^{(0)} \eth \hat{x}^{\hat{i}}) \end{align*}
Note that \(U^{0}\) and \(\mathcal U^{(0)}\) are related by
\begin{align*} U^{(0)} &= \frac{1}{2\omega^2} \left( \bar{d} \mathcal U^{(0)} - c \bar{\mathcal U}^{(0)} \right) \\ &= \frac{\hat \omega^2}{2} \left( \bar{d} \mathcal U^{(0)} - c \bar{\mathcal U}^{(0)} \right) \end{align*}
see Eq. (79) of [136].