XCTS quantities for a solution of the Einstein equations. More...

Detailed Description

template<typename GrSolution, bool HasMhd = false, typename GrSolutionOptions = typename GrSolution::options>
class Xcts::Solutions::WrappedGr< GrSolution, HasMhd, GrSolutionOptions >

XCTS quantities for a solution of the Einstein equations.

This class computes all XCTS quantities from the GrSolution. To do so, it chooses the conformal factor

\begin{equation} \psi = 1 \text{,} \end{equation}

so the spatial metric of the GrSolution is used as conformal metric, \(\bar{\gamma}_{ij = \gamma_{ij}\). This is particularly useful for superpositions, because it means that the superposed conformal metric of two WrappedGr solutions is probably a good conformal background to solve for a binary solution (see Xcts::AnalyticData::Binary).

For example, when the GrSolution is gr::Solutions::KerrSchild, the conformal metric is the spatial Kerr metric in Kerr-Schild coordinates and \(\psi = 1\). It is also possible to choose a different \(\psi\) so the solution is non-trivial in this variable, though that is probably only useful for testing and currently not implemented. It should be noted, however, that the combination of \(\psi=1\) and apparent-horizon boundary conditions poses a hard problem to the nonlinear solver when starting at a flat initial guess. This is because the strongly-nonlinear boundary-conditions couple the variables in such a way that the solution is initially corrected away from \(\psi=1\) and is then unable to recover. A conformal-factor profile such as \(\psi=1 + \frac{M}{2r}\) (resembling isotropic coordinates) resolves this issue. In production solves this is not an issue because we choose a much better initial guess than flatness, such as a superposition of Kerr solutions for black-hole binary initial data.

Warning: The computation of the XCTS matter source terms (energy density \(\rho\), momentum density \(S^i\), stress trace \(S\)) uses GR quantities (lapse \(\alpha\), shift \(\beta^i\), spatial metric \(\gamma_{ij}\)), which means these GR quantities are not treated dynamically in the source terms when solving the XCTS equations. If the GR quantities satisfy the Einstein constraints (as is the case if the GrSolution is actually a solution to the Einstein equations), then the XCTS solve will reproduce the GR quantities given the fixed sources computed here. However, if the GR quantities don't satisfy the Einstein constraints (e.g. because a magnetic field was added to the solution but ignored in the gravity sector, or because it is a hydrodynamic solution on a fixed background metric) then the XCTS solution will depend on our treatment of the source terms: fixing the source terms (the simple approach taken here) means we're making a choice of \(W\) and \(u^i\). This is what initial data codes usually do when they iterate back and forth between a hydro solve and an XCTS solve (e.g. see [177]). Alternatively, we could fix \(v^i\) and compute \(W\) and \(u^i\) from \(v^i\) and the dynamic metric variables at every step in the XCTS solver algorithm. This requires adding the source terms and their linearization to the XCTS equations, and could be interesting to explore.

Template Parameters

GrSolution	Any solution to the Einstein constraint equations
HasMhd	Enable to compute matter source terms. Disable to set matter source terms to zero.

The documentation for this class was generated from the following file:

src/PointwiseFunctions/AnalyticSolutions/Xcts/WrappedGr.hpp