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SpECTRE
v2023.05.16
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Calculates the time derivative of Psi0, the constant coefficient of the expansion of Psi.
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#include <TimeDerivative.hpp>
Public Types | |
| using | variables_tag = ::Tags::Variables< tmpl::list< Tags::Psi0, Tags::dtPsi0 > > |
| using | dt_variables_tag = db::add_tag_prefix<::Tags::dt, variables_tag > |
| using | return_tags = tmpl::list< dt_variables_tag > |
| using | argument_tags = tmpl::list< variables_tag, Stf::Tags::StfTensor< Tags::PsiWorldtube, 0, Dim, Frame::Grid >, Stf::Tags::StfTensor< Tags::PsiWorldtube, 1, Dim, Frame::Grid >, Stf::Tags::StfTensor< Tags::PsiWorldtube, 2, Dim, Frame::Grid >, Stf::Tags::StfTensor<::Tags::dt< Tags::PsiWorldtube >, 1, Dim, Frame::Grid >, gr::Tags::InverseSpacetimeMetric< double, Dim, Frame::Grid >, gr::Tags::TraceSpacetimeChristoffelSecondKind< double, Dim, Frame::Grid >, Tags::ExcisionSphere< Dim > > |
Static Public Member Functions | |
| static void | apply (const gsl::not_null< Variables< tmpl::list<::Tags::dt< Tags::Psi0 >, ::Tags::dt< Tags::dtPsi0 > > > * > dt_evolved_vars, const Variables< tmpl::list< Tags::Psi0, Tags::dtPsi0 > > &evolved_vars, const Scalar< double > &psi_monopole, const tnsr::i< double, Dim, Frame::Grid > &psi_dipole, const tnsr::ii< double, Dim, Frame::Grid > &psi_quadrupole, const tnsr::i< double, Dim, Frame::Grid > &dt_psi_dipole, const tnsr::AA< double, Dim, Frame::Grid > &inverse_spacetime_metric, const tnsr::A< double, Dim, Frame::Grid > &trace_spacetime_christoffel, const ExcisionSphere< Dim > &excision_sphere) |
Static Public Attributes | |
| static constexpr size_t | Dim = 3 |
Calculates the time derivative of Psi0, the constant coefficient of the expansion of Psi.
The derivation comes from expanding the scalar wave equation to second order and reads
\begin{equation} g^{00}_0 \ddot{\Psi}^R_0(t_s) + 2 g_0^{0i} \dot{\Psi}^N_i(t_s) + 2 g_0^{ij} \Psi^N_{\langle ij \rangle}(t_s) + \frac{2 \delta_{ij} g_0^{ij}}{R^2} \left(\Psi^N_0(t_s) - \Psi^R_0(t_s) \right) - \Gamma_0^0\dot{\Psi}R_0(t_s) - \Gamma_0^i \Psi_i^N(t_s) = 0. *\end{equation}
Here, \( \Gamma^\mu_0 \) and \( g^{\mu \nu}_0 \) are the trace of the spacetime Christoffel symbol and the inverse spacetime metric, respectively, evaluated at the position of the particle; \(\Psi^N_0\), \(\Psi^N_i\), \(\Psi^N_\langle i j \rangle\) are the monopole, dipole and quadrupole of the regular field on the worldtube boundary transformed to symmetric trace-free tensors and \( R\) is the worldtube radius.