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SpECTRE
v2026.04.01
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The Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein constraint equations, formulated as a set of coupled first-order partial differential equations. More...
#include <FirstOrderSystem.hpp>
Public Types | |
| using | primal_fields |
| using | primal_fluxes |
| using | background_fields |
| using | inv_metric_tag |
| using | fluxes_computer = Fluxes<EnabledEquations, ConformalGeometry> |
| using | sources_computer |
| using | sources_computer_linearized |
| using | boundary_conditions_base |
| using | modify_boundary_data = void |
Static Public Attributes | |
| static constexpr Equations | enabled_equations = EnabledEquations |
| static constexpr Geometry | conformal_geometry = ConformalGeometry |
| static constexpr int | conformal_matter_scale = ConformalMatterScale |
| static constexpr size_t | volume_dim = 3 |
The Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein constraint equations, formulated as a set of coupled first-order partial differential equations.
See Xcts for details on the XCTS equations. The system can be formulated in terms of these fluxes and sources (see elliptic::protocols::FirstOrderSystem):
\begin{align}F^i_\psi ={} &\bar{\gamma}^{ij} \partial_j \psi \\ S_\psi ={} &-\bar{\Gamma}^i_{ij} F^j_\psi + \frac{1}{8}\psi\bar{R} + \frac{1}{12}\psi^5 K^2 - \frac{1}{8}\psi^{-7}\bar{A}^2 - 2\pi\psi^5\rho \end{align}
for the Hamiltonian constraint,
\begin{align}F^i_{\alpha\psi} ={} &\bar{\gamma}^{ij} \partial_j \alpha\psi \\ S_{\alpha\psi} ={} &-\bar{\Gamma}^i_{ij} F^j_{\alpha\psi} + \alpha\psi \left(\frac{7}{8}\psi^{-8} \bar{A}^2 + \frac{5}{12} \psi^4 K^2 + \frac{1}{8}\bar{R} + 2\pi\psi^4\left(\rho + 2S\right) \right) \\ &- \psi^5\partial_t K + \psi^5\left(\beta^i\bar{D}_i K + \beta_\mathrm{background}^i\bar{D}_i K\right) \end{align}
for the lapse equation, and
\begin{align}F^{ij}_\beta ={} &\left(\bar{L}\beta\right)^{ij} = \bar{\nabla}^i \beta^j + \bar{\nabla}^j \beta^i - \frac{2}{3} \bar{\gamma}^{ij} \bar{\nabla}_k \beta^k \\ S^i_\beta ={} &-\bar{\Gamma}^j_{jk} F^{ik}_\beta - \bar{\Gamma}^i_{jk} F^{jk}_\beta + \left(F^{ij}_\beta + \left(\bar{L}\beta_\mathrm{background}\right)^{ij} - \bar{u}^{ij}\right) \bar{\gamma}_{jk} \left(\frac{F^k_{\alpha\psi}}{\alpha\psi} - 7 \frac{F^k_\psi}{\psi}\right) \\ &- \bar{D}_j\left(\left(\bar{L}\beta_\mathrm{background}\right)^{ij} - \bar{u}^{ij}\right) + \frac{4}{3}\frac{\alpha\psi}{\psi}\bar{D}^i K + 16\pi\left(\alpha\psi\right)\psi^3 S^i \end{align}
for the momentum constraint, with
\begin{align}\bar{A}^{ij} ={} &\frac{\psi^7}{2\alpha\psi}\left( \left(\bar{L}\beta\right)^{ij} + \left(\bar{L}\beta_\mathrm{background}\right)^{ij} - \bar{u}^{ij} \right) \end{align}
and all \(f_\alpha=0\).
Note that the symbol \(\beta\) in the equations above means \(\beta_\mathrm{excess}\). The full shift is \(\beta_\mathrm{excess} + \beta_\mathrm{background}\). See Xcts::Tags::ShiftBackground and Xcts::Tags::ShiftExcess for details on this split. Also note that the background shift is degenerate with \(\bar{u}\) so we treat the quantity \(\left(\bar{L}\beta_\mathrm{background}\right)^{ij} - \bar{u}^{ij}\) as a single background field (see Xcts::Tags::LongitudinalShiftBackgroundMinusDtConformalMetric). The covariant divergence of this quantity w.r.t. the conformal metric is also a background field.
| using Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale >::boundary_conditions_base |
| using Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale >::inv_metric_tag |
| using Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale >::primal_fields |
| using Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale >::primal_fluxes |
| using Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale >::sources_computer |
| using Xcts::FirstOrderSystem< EnabledEquations, ConformalGeometry, ConformalMatterScale >::sources_computer_linearized |