Items related to solving the Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein constraint equations. More...

Namespaces
namespace	AnalyticData
	Analytic data for the XCTS equations, i.e. field configurations that do not solve the equations but are used as background or initial guess.

namespace	Solutions
	Analytic solutions of the XCTS equations.

namespace	Tags
	Tags related to the XCTS equations.

Classes
struct	FirstOrderSystem
	The Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein constraint equations, formulated as a set of coupled first-order partial differential equations. More...

struct	Fluxes
	The fluxes \(F^i\) for the first-order formulation of the XCTS equations. More...

struct	LinearizedSources
	The linearization of the sources \(S\) for the first-order formulation of the XCTS equations. More...

struct	Sources
	The sources \(S\) for the first-order formulation of the XCTS equations. More...

struct	SpacetimeQuantitiesComputer
	`CachedTempBuffer` computer class for 3+1 quantities from XCTS variables. See `Xcts::SpacetimeQuantities`. More...

Typedefs
using	SpacetimeQuantities = CachedTempBuffer< Tags::ConformalFactor< DataVector >, Tags::LapseTimesConformalFactor< DataVector >, ::Tags::deriv< Tags::ConformalFactor< DataVector >, tmpl::size_t< 3 >, Frame::Inertial >, detail::ConformalLaplacianOfConformalFactor< DataVector >, ::Tags::deriv< Tags::LapseTimesConformalFactor< DataVector >, tmpl::size_t< 3 >, Frame::Inertial >, ::Tags::deriv< Tags::ShiftExcess< DataVector, 3, Frame::Inertial >, tmpl::size_t< 3 >, Frame::Inertial >, Xcts::Tags::ShiftStrain< DataVector, 3, Frame::Inertial >, Xcts::Tags::LongitudinalShiftExcess< DataVector, 3, Frame::Inertial >, ::Tags::div< Tags::LongitudinalShiftExcess< DataVector, 3, Frame::Inertial > >, detail::LongitudinalShiftMinusDtConformalMetric< DataVector >, gr::Tags::SpatialMetric< DataVector, 3 >, gr::Tags::InverseSpatialMetric< DataVector, 3 >, gr::Tags::Lapse< DataVector >, gr::Tags::Shift< DataVector, 3 >, gr::Tags::ExtrinsicCurvature< DataVector, 3 >, gr::Tags::SpatialChristoffelSecondKind< DataVector, 3 >, gr::Tags::HamiltonianConstraint< DataVector >, gr::Tags::MomentumConstraint< DataVector, 3 > >
	General-relativistic 3+1 quantities computed from XCTS variables.

Enumerations
enum class	Equations { Hamiltonian , HamiltonianAndLapse , HamiltonianLapseAndShift }
	Indicates a subset of the XCTS equations. More...

enum class	Geometry { FlatCartesian , Curved }
	Types of conformal geometries for the XCTS equations. More...

Functions
template<int ConformalMatterScale>
void	add_hamiltonian_sources (gsl::not_null< Scalar< DataVector > * > hamiltonian_constraint, const Scalar< DataVector > &conformal_energy_density, const Scalar< DataVector > &extrinsic_curvature_trace, const Scalar< DataVector > &conformal_factor_minus_one)
	Add the nonlinear source to the Hamiltonian constraint on a flat conformal background in Cartesian coordinates and with \(\bar{u}_{ij}=0=\beta^i\). More...

template<int ConformalMatterScale>
void	add_linearized_hamiltonian_sources (gsl::not_null< Scalar< DataVector > * > linearized_hamiltonian_constraint, const Scalar< DataVector > &conformal_energy_density, const Scalar< DataVector > &extrinsic_curvature_trace, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &conformal_factor_correction)
	The linearization of `add_hamiltonian_sources`

void	add_distortion_hamiltonian_sources (gsl::not_null< Scalar< DataVector > * > hamiltonian_constraint, const Scalar< DataVector > &longitudinal_shift_minus_dt_conformal_metric_over_lapse_square, const Scalar< DataVector > &conformal_factor_minus_one)
	Add the "distortion" source term to the Hamiltonian constraint. More...

void	add_linearized_distortion_hamiltonian_sources (gsl::not_null< Scalar< DataVector > * > linearized_hamiltonian_constraint, const Scalar< DataVector > &longitudinal_shift_minus_dt_conformal_metric_over_lapse_square, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &conformal_factor_correction)
	The linearization of `add_distortion_hamiltonian_sources` More...

void	add_curved_hamiltonian_or_lapse_sources (gsl::not_null< Scalar< DataVector > * > hamiltonian_or_lapse_equation, const Scalar< DataVector > &conformal_ricci_scalar, const Scalar< DataVector > &field, double add_constant=0.)
	Add the contributions from a curved background geometry to the Hamiltonian constraint or lapse equation. More...

template<int ConformalMatterScale>
void	add_lapse_sources (gsl::not_null< Scalar< DataVector > * > lapse_equation, const Scalar< DataVector > &conformal_energy_density, const Scalar< DataVector > &conformal_stress_trace, const Scalar< DataVector > &extrinsic_curvature_trace, const Scalar< DataVector > &dt_extrinsic_curvature_trace, const Scalar< DataVector > &shift_dot_deriv_extrinsic_curvature_trace, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one)
	Add the nonlinear source to the lapse equation on a flat conformal background in Cartesian coordinates and with \(\bar{u}_{ij}=0=\beta^i\). More...

template<int ConformalMatterScale>
void	add_linearized_lapse_sources (gsl::not_null< Scalar< DataVector > * > linearized_lapse_equation, const Scalar< DataVector > &conformal_energy_density, const Scalar< DataVector > &conformal_stress_trace, const Scalar< DataVector > &extrinsic_curvature_trace, const Scalar< DataVector > &dt_extrinsic_curvature_trace, const Scalar< DataVector > &shift_dot_deriv_extrinsic_curvature_trace, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one, const Scalar< DataVector > &conformal_factor_correction, const Scalar< DataVector > &lapse_times_conformal_factor_correction)
	The linearization of `add_lapse_sources` More...

void	add_distortion_hamiltonian_and_lapse_sources (gsl::not_null< Scalar< DataVector > * > hamiltonian_constraint, gsl::not_null< Scalar< DataVector > * > lapse_equation, const Scalar< DataVector > &longitudinal_shift_minus_dt_conformal_metric_square, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one)
	Add the "distortion" source term to the Hamiltonian constraint and the lapse equation. More...

void	add_linearized_distortion_hamiltonian_and_lapse_sources (gsl::not_null< Scalar< DataVector > * > hamiltonian_constraint, gsl::not_null< Scalar< DataVector > * > lapse_equation, const Scalar< DataVector > &longitudinal_shift_minus_dt_conformal_metric_square, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one, const Scalar< DataVector > &conformal_factor_correction, const Scalar< DataVector > &lapse_times_conformal_factor_correction)
	The linearization of `add_distortion_hamiltonian_and_lapse_sources` More...

template<int ConformalMatterScale>
void	add_curved_linearized_momentum_sources (gsl::not_null< Scalar< DataVector > * > linearized_hamiltonian_constraint, gsl::not_null< Scalar< DataVector > * > linearized_lapse_equation, gsl::not_null< tnsr::I< DataVector, 3 > * > linearized_momentum_constraint, const tnsr::I< DataVector, 3 > &conformal_momentum_density, const tnsr::i< DataVector, 3 > &extrinsic_curvature_trace_gradient, const tnsr::ii< DataVector, 3 > &conformal_metric, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one, const tnsr::I< DataVector, 3 > &conformal_factor_flux, const tnsr::I< DataVector, 3 > &lapse_times_conformal_factor_flux, const tnsr::II< DataVector, 3 > &longitudinal_shift_minus_dt_conformal_metric, const Scalar< DataVector > &conformal_factor_correction, const Scalar< DataVector > &lapse_times_conformal_factor_correction, const tnsr::I< DataVector, 3 > &shift_correction, const tnsr::I< DataVector, 3 > &conformal_factor_flux_correction, const tnsr::I< DataVector, 3 > &lapse_times_conformal_factor_flux_correction, const tnsr::II< DataVector, 3 > &longitudinal_shift_correction)
	The linearization of `add_curved_momentum_sources`

template<int ConformalMatterScale>
void	add_flat_cartesian_linearized_momentum_sources (gsl::not_null< Scalar< DataVector > * > linearized_hamiltonian_constraint, gsl::not_null< Scalar< DataVector > * > linearized_lapse_equation, gsl::not_null< tnsr::I< DataVector, 3 > * > linearized_momentum_constraint, const tnsr::I< DataVector, 3 > &conformal_momentum_density, const tnsr::i< DataVector, 3 > &extrinsic_curvature_trace_gradient, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one, const tnsr::I< DataVector, 3 > &conformal_factor_flux, const tnsr::I< DataVector, 3 > &lapse_times_conformal_factor_flux, const tnsr::II< DataVector, 3 > &longitudinal_shift_minus_dt_conformal_metric, const Scalar< DataVector > &conformal_factor_correction, const Scalar< DataVector > &lapse_times_conformal_factor_correction, const tnsr::I< DataVector, 3 > &shift_correction, const tnsr::I< DataVector, 3 > &conformal_factor_flux_correction, const tnsr::I< DataVector, 3 > &lapse_times_conformal_factor_flux_correction, const tnsr::II< DataVector, 3 > &longitudinal_shift_correction)
	The linearization of `add_flat_cartesian_momentum_sources`


template<int ConformalMatterScale>
void	add_curved_momentum_sources (gsl::not_null< Scalar< DataVector > * > hamiltonian_constraint, gsl::not_null< Scalar< DataVector > * > lapse_equation, gsl::not_null< tnsr::I< DataVector, 3 > * > momentum_constraint, const tnsr::I< DataVector, 3 > &conformal_momentum_density, const tnsr::i< DataVector, 3 > &extrinsic_curvature_trace_gradient, const tnsr::ii< DataVector, 3 > &conformal_metric, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const tnsr::I< DataVector, 3 > &minus_div_dt_conformal_metric, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one, const tnsr::I< DataVector, 3 > &conformal_factor_flux, const tnsr::I< DataVector, 3 > &lapse_times_conformal_factor_flux, const tnsr::II< DataVector, 3 > &longitudinal_shift_minus_dt_conformal_metric)
	Add the nonlinear source to the momentum constraint and add the "distortion" source term to the Hamiltonian constraint and lapse equation. More...

template<int ConformalMatterScale>
void	add_flat_cartesian_momentum_sources (gsl::not_null< Scalar< DataVector > * > hamiltonian_constraint, gsl::not_null< Scalar< DataVector > * > lapse_equation, gsl::not_null< tnsr::I< DataVector, 3 > * > momentum_constraint, const tnsr::I< DataVector, 3 > &conformal_momentum_density, const tnsr::i< DataVector, 3 > &extrinsic_curvature_trace_gradient, const tnsr::I< DataVector, 3 > &minus_div_dt_conformal_metric, const Scalar< DataVector > &conformal_factor_minus_one, const Scalar< DataVector > &lapse_times_conformal_factor_minus_one, const tnsr::I< DataVector, 3 > &conformal_factor_flux, const tnsr::I< DataVector, 3 > &lapse_times_conformal_factor_flux, const tnsr::II< DataVector, 3 > &longitudinal_shift_minus_dt_conformal_metric)
	Add the nonlinear source to the momentum constraint and add the "distortion" source term to the Hamiltonian constraint and lapse equation. More...


template<typename DataType >
void	extrinsic_curvature (const gsl::not_null< tnsr::ii< DataType, 3 > * > result, const Scalar< DataType > &conformal_factor, const Scalar< DataType > &lapse, const tnsr::ii< DataType, 3 > &conformal_metric, const tnsr::II< DataType, 3 > &longitudinal_shift_minus_dt_conformal_metric, const Scalar< DataType > &trace_extrinsic_curvature)
	Extrinsic curvature computed from the conformal decomposition used in the XCTS system. More...

template<typename DataType >
tnsr::ii< DataType, 3 >	extrinsic_curvature (const Scalar< DataType > &conformal_factor, const Scalar< DataType > &lapse, const tnsr::ii< DataType, 3 > &conformal_metric, const tnsr::II< DataType, 3 > &longitudinal_shift_minus_dt_conformal_metric, const Scalar< DataType > &trace_extrinsic_curvature)
	Return-by-value overload.


template<typename DataType >
void	longitudinal_operator (gsl::not_null< tnsr::II< DataType, 3 > * > result, const tnsr::ii< DataType, 3 > &strain, const tnsr::II< DataType, 3 > &inv_metric)
	The longitudinal operator, or vector gradient, \((L\beta)^{ij}\). More...

template<typename DataType >
void	longitudinal_operator (gsl::not_null< tnsr::II< DataType, 3 > * > result, const tnsr::I< DataType, 3 > &shift, const tnsr::iJ< DataType, 3 > &deriv_shift, const tnsr::II< DataType, 3 > &inv_metric, const tnsr::Ijj< DataType, 3 > &christoffel_second_kind)
	The longitudinal operator, or vector gradient, \((L\beta)^{ij}\). More...


template<typename DataType >
void	longitudinal_operator_flat_cartesian (gsl::not_null< tnsr::II< DataType, 3 > * > result, const tnsr::ii< DataType, 3 > &strain)
	The conformal longitudinal operator \((L\beta)^{ij}\) on a flat conformal metric in Cartesian coordinates \(\gamma_{ij}=\delta_{ij}\). More...

Detailed Description

Items related to solving the Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein constraint equations.

The XCTS equations

\begin{align} \bar{D}^2 \psi - \frac{1}{8}\psi\bar{R} - \frac{1}{12}\psi^5 K^2 + \frac{1}{8}\psi^{-7}\bar{A}_{ij}\bar{A}^{ij} &= -2\pi\psi^5\rho \\ \bar{D}_i(\bar{L}\beta)^{ij} - (\bar{L}\beta)^{ij}\bar{D}_i \ln(\bar{\alpha}) &= \bar{\alpha}\bar{D}_i\left(\bar{\alpha}^{-1}\bar{u}^{ij} \right) + \frac{4}{3}\bar{\alpha}\psi^6\bar{D}^j K + 16\pi\bar{\alpha} \psi^{10}S^j \\ \bar{D}^2\left(\alpha\psi\right) &= \alpha\psi\left(\frac{7}{8}\psi^{-8}\bar{A}_{ij}\bar{A}^{ij} + \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\beta^i\bar{D}_i K \\ \text{with} \quad \bar{A} &= \frac{1}{2\bar{\alpha}} \left((\bar{L}\beta)^{ij} - \bar{u}^{ij}\right) \\ \quad \text{and} \quad \bar{\alpha} &= \alpha \psi^{-6} \end{align}

are a set of nonlinear elliptic equations that the spacetime metric in general relativity must satisfy at all times. For an introduction see e.g. [13], in particular Box 3.3 which is largely mirrored here. We solve the XCTS equations for the conformal factor \(\psi\), the product of lapse times conformal factor \(\alpha\psi\) and the shift vector \(\beta^j\). The remaining quantities in the equations, i.e. the conformal metric \(\bar{\gamma}_{ij}\), the trace of the extrinsic curvature \(K\), their respective time derivatives \(\bar{u}_{ij}\) and \(\partial_t K\), the energy density \(\rho\), the stress-energy trace \(S\) and the momentum density \(S^i\), are freely specifyable fields that define the physical scenario at hand. Of particular importance is the conformal metric, which defines the background geometry, the covariant derivative \(\bar{D}\), the Ricci scalar \(\bar{R}\) and the longitudinal operator

\begin{equation} \left(\bar{L}\beta\right)^{ij} = \bar{D}^i\beta^j + \bar{D}^j\beta^i - \frac{2}{3}\bar{\gamma}^{ij}\bar{D}_k\beta^k \text{.} \end{equation}

Note that the XCTS equations are essentially two Poisson equations and one Elasticity equation with nonlinear sources on a curved geometry. In this analogy, the longitudinal operator plays the role of the elastic constitutive relation that connects the symmetric "shift strain" \(\bar{D}_{(i}\beta_{j)}\) with the "stress" \((\bar{L}\beta)^{ij}\) of which we take the divergence in the momentum constraint. This particular constitutive relation is equivalent to an isotropic and homogeneous material with bulk modulus \(K=0\) (not to be confused with the extrinsic curvature trace \(K\) in this context) and shear modulus \(\mu=1\) (see Elasticity::ConstitutiveRelations::IsotropicHomogeneous).

Once the XCTS equations are solved we can construct the spatial metric and extrinsic curvature as

\begin{align} \gamma_{ij} &= \psi^4\bar{\gamma}_{ij} \\ K_{ij} &= \psi^{-2}\bar{A}_{ij} + \frac{1}{3}\gamma_{ij} K \end{align}

from which we can compose the full spacetime metric.

Enumeration Type Documentation

◆ Equations

enum class Xcts::Equations

strong

Indicates a subset of the XCTS equations.

Enumerator
Hamiltonian	Only the Hamiltonian constraint, solved for \(\psi\).
HamiltonianAndLapse	Both the Hamiltonian constraint and the lapse equation, solved for \(\psi\) and \(\alpha\psi\).
HamiltonianLapseAndShift	The full XCTS equations, solved for \(\psi\), \(\alpha\psi\) and \(\beta_\mathrm{excess}\).

◆ Geometry

enum class Xcts::Geometry

strong

Types of conformal geometries for the XCTS equations.

Enumerator
FlatCartesian	Euclidean (flat) manifold with Cartesian coordinates, i.e. the conformal metric has components \(\bar{\gamma}_{ij} = \delta_{ij}\) in these coordinates and thus all Christoffel symbols vanish: \(\bar{\Gamma}^i_{jk}=0\).
Curved	The conformal geometry is either curved or employs curved coordinates, so non-vanishing Christoffel symbols must be taken into account.

Function Documentation

◆ add_curved_hamiltonian_or_lapse_sources()

void Xcts::add_curved_hamiltonian_or_lapse_sources	(	gsl::not_null< Scalar< DataVector > * >	hamiltonian_or_lapse_equation,
		const Scalar< DataVector > &	conformal_ricci_scalar,
		const Scalar< DataVector > &	field,
		double	add_constant = `0.`
	)

Add the contributions from a curved background geometry to the Hamiltonian constraint or lapse equation.

Adds \(\frac{1}{8}\psi\bar{R}\). This term appears both in the Hamiltonian constraint and the lapse equation (where in the latter \(\psi\) is replaced by \(\alpha\psi\)).

This term is linear.

See also: Xcts

◆ add_curved_momentum_sources()

template<int ConformalMatterScale>

void Xcts::add_curved_momentum_sources	(	gsl::not_null< Scalar< DataVector > * >	hamiltonian_constraint,
		gsl::not_null< Scalar< DataVector > * >	lapse_equation,
		gsl::not_null< tnsr::I< DataVector, 3 > * >	momentum_constraint,
		const tnsr::I< DataVector, 3 > &	conformal_momentum_density,
		const tnsr::i< DataVector, 3 > &	extrinsic_curvature_trace_gradient,
		const tnsr::ii< DataVector, 3 > &	conformal_metric,
		const tnsr::II< DataVector, 3 > &	inv_conformal_metric,
		const tnsr::I< DataVector, 3 > &	minus_div_dt_conformal_metric,
		const Scalar< DataVector > &	conformal_factor_minus_one,
		const Scalar< DataVector > &	lapse_times_conformal_factor_minus_one,
		const tnsr::I< DataVector, 3 > &	conformal_factor_flux,
		const tnsr::I< DataVector, 3 > &	lapse_times_conformal_factor_flux,
		const tnsr::II< DataVector, 3 > &	longitudinal_shift_minus_dt_conformal_metric
	)

Add the nonlinear source to the momentum constraint and add the "distortion" source term to the Hamiltonian constraint and lapse equation.

Adds \(\left((\bar{L}\beta)^{ij} - \bar{u}^{ij}\right) \left(\frac{\partial_j (\alpha\psi)}{\alpha\psi} - 7 \frac{\partial_j \psi}{\psi}\right) + \partial_j\bar{u}^{ij} + \frac{4}{3}\frac{\alpha\psi}{\psi}\bar{\gamma}^{ij}\partial_j K + 16\pi\left(\alpha\psi\right)\psi^{3-n} \bar{S}^i\) to the momentum constraint, where \(n\) is the ConformalMatterScale and \(\bar{S}^i=\psi^n S^i\) is the conformally-scaled momentum density.

Note that the \(\partial_j\bar{u}^{ij}\) term is not the full covariant divergence, but only the partial-derivatives part of it. The curved contribution to this term can be added together with the curved contribution to the flux divergence of the dynamic shift variable with the Elasticity::add_curved_sources function.

Also adds \(-\frac{1}{8}\psi^{-7}\bar{A}^{ij}\bar{A}_{ij}\) to the Hamiltonian constraint and \(\frac{7}{8}\alpha\psi^{-7}\bar{A}^{ij}\bar{A}_{ij}\) to the lapse equation.

See also: Xcts

◆ add_distortion_hamiltonian_and_lapse_sources()

void Xcts::add_distortion_hamiltonian_and_lapse_sources	(	gsl::not_null< Scalar< DataVector > * >	hamiltonian_constraint,
		gsl::not_null< Scalar< DataVector > * >	lapse_equation,
		const Scalar< DataVector > &	longitudinal_shift_minus_dt_conformal_metric_square,
		const Scalar< DataVector > &	conformal_factor_minus_one,
		const Scalar< DataVector > &	lapse_times_conformal_factor_minus_one
	)

Add the "distortion" source term to the Hamiltonian constraint and the lapse equation.

Adds \(-\frac{1}{8}\psi^{-7}\bar{A}^{ij}\bar{A}_{ij}\) to the Hamiltonian constraint and \(\frac{7}{8}\alpha\psi^{-7}\bar{A}^{ij}\bar{A}_{ij}\) to the lapse equation.

See also: Xcts; Xcts::Tags::LongitudinalShiftMinusDtConformalMetricSquare

◆ add_distortion_hamiltonian_sources()

void Xcts::add_distortion_hamiltonian_sources	(	gsl::not_null< Scalar< DataVector > * >	hamiltonian_constraint,
		const Scalar< DataVector > &	longitudinal_shift_minus_dt_conformal_metric_over_lapse_square,
		const Scalar< DataVector > &	conformal_factor_minus_one
	)

Add the "distortion" source term to the Hamiltonian constraint.

Adds \(-\frac{1}{8}\psi^{-7}\bar{A}^{ij}\bar{A}_{ij}\).

See also: Xcts; Xcts::Tags::LongitudinalShiftMinusDtConformalMetricOverLapseSquare

◆ add_flat_cartesian_momentum_sources()

template<int ConformalMatterScale>

void Xcts::add_flat_cartesian_momentum_sources	(	gsl::not_null< Scalar< DataVector > * >	hamiltonian_constraint,
		gsl::not_null< Scalar< DataVector > * >	lapse_equation,
		gsl::not_null< tnsr::I< DataVector, 3 > * >	momentum_constraint,
		const tnsr::I< DataVector, 3 > &	conformal_momentum_density,
		const tnsr::i< DataVector, 3 > &	extrinsic_curvature_trace_gradient,
		const tnsr::I< DataVector, 3 > &	minus_div_dt_conformal_metric,
		const Scalar< DataVector > &	conformal_factor_minus_one,
		const Scalar< DataVector > &	lapse_times_conformal_factor_minus_one,
		const tnsr::I< DataVector, 3 > &	conformal_factor_flux,
		const tnsr::I< DataVector, 3 > &	lapse_times_conformal_factor_flux,
		const tnsr::II< DataVector, 3 > &	longitudinal_shift_minus_dt_conformal_metric
	)

Add the nonlinear source to the momentum constraint and add the "distortion" source term to the Hamiltonian constraint and lapse equation.

Adds \(\left((\bar{L}\beta)^{ij} - \bar{u}^{ij}\right) \left(\frac{\partial_j (\alpha\psi)}{\alpha\psi} - 7 \frac{\partial_j \psi}{\psi}\right) + \partial_j\bar{u}^{ij} + \frac{4}{3}\frac{\alpha\psi}{\psi}\bar{\gamma}^{ij}\partial_j K + 16\pi\left(\alpha\psi\right)\psi^{3-n} \bar{S}^i\) to the momentum constraint, where \(n\) is the ConformalMatterScale and \(\bar{S}^i=\psi^n S^i\) is the conformally-scaled momentum density.

Note that the \(\partial_j\bar{u}^{ij}\) term is not the full covariant divergence, but only the partial-derivatives part of it. The curved contribution to this term can be added together with the curved contribution to the flux divergence of the dynamic shift variable with the Elasticity::add_curved_sources function.

Also adds \(-\frac{1}{8}\psi^{-7}\bar{A}^{ij}\bar{A}_{ij}\) to the Hamiltonian constraint and \(\frac{7}{8}\alpha\psi^{-7}\bar{A}^{ij}\bar{A}_{ij}\) to the lapse equation.

See also: Xcts

◆ add_hamiltonian_sources()

template<int ConformalMatterScale>

void Xcts::add_hamiltonian_sources	(	gsl::not_null< Scalar< DataVector > * >	hamiltonian_constraint,
		const Scalar< DataVector > &	conformal_energy_density,
		const Scalar< DataVector > &	extrinsic_curvature_trace,
		const Scalar< DataVector > &	conformal_factor_minus_one
	)

Add the nonlinear source to the Hamiltonian constraint on a flat conformal background in Cartesian coordinates and with \(\bar{u}_{ij}=0=\beta^i\).

Adds \(\frac{1}{12}\psi^5 K^2 - 2\pi\psi^{5-n}\bar{\rho}\) where \(n\) is the ConformalMatterScale and \(\bar{\rho}=\psi^n\rho\) is the conformally-scaled energy density. Additional sources can be added with add_distortion_hamiltonian_sources and add_curved_hamiltonian_or_lapse_sources.

See also: Xcts

◆ add_lapse_sources()

template<int ConformalMatterScale>

void Xcts::add_lapse_sources	(	gsl::not_null< Scalar< DataVector > * >	lapse_equation,
		const Scalar< DataVector > &	conformal_energy_density,
		const Scalar< DataVector > &	conformal_stress_trace,
		const Scalar< DataVector > &	extrinsic_curvature_trace,
		const Scalar< DataVector > &	dt_extrinsic_curvature_trace,
		const Scalar< DataVector > &	shift_dot_deriv_extrinsic_curvature_trace,
		const Scalar< DataVector > &	conformal_factor_minus_one,
		const Scalar< DataVector > &	lapse_times_conformal_factor_minus_one
	)

Add the nonlinear source to the lapse equation on a flat conformal background in Cartesian coordinates and with \(\bar{u}_{ij}=0=\beta^i\).

Adds \((\alpha\psi)\left(\frac{5}{12}\psi^4 K^2 + 2\pi\psi^{4-n} \left(\bar{\rho} + 2\bar{S}\right)\right) + \psi^5 \left(\beta^i\partial_i K - \partial_t K\right)\) where \(n\) is the ConformalMatterScale and matter quantities are conformally-scaled. Additional sources can be added with add_distortion_hamiltonian_and_lapse_sources and add_curved_hamiltonian_or_lapse_sources.

See also: Xcts

◆ add_linearized_distortion_hamiltonian_and_lapse_sources()

void Xcts::add_linearized_distortion_hamiltonian_and_lapse_sources	(	gsl::not_null< Scalar< DataVector > * >	hamiltonian_constraint,
		gsl::not_null< Scalar< DataVector > * >	lapse_equation,
		const Scalar< DataVector > &	longitudinal_shift_minus_dt_conformal_metric_square,
		const Scalar< DataVector > &	conformal_factor_minus_one,
		const Scalar< DataVector > &	lapse_times_conformal_factor_minus_one,
		const Scalar< DataVector > &	conformal_factor_correction,
		const Scalar< DataVector > &	lapse_times_conformal_factor_correction
	)

The linearization of add_distortion_hamiltonian_and_lapse_sources

Note that this linearization is only w.r.t. \(\psi\) and \(\alpha\psi\).

◆ add_linearized_distortion_hamiltonian_sources()

void Xcts::add_linearized_distortion_hamiltonian_sources	(	gsl::not_null< Scalar< DataVector > * >	linearized_hamiltonian_constraint,
		const Scalar< DataVector > &	longitudinal_shift_minus_dt_conformal_metric_over_lapse_square,
		const Scalar< DataVector > &	conformal_factor_minus_one,
		const Scalar< DataVector > &	conformal_factor_correction
	)

The linearization of add_distortion_hamiltonian_sources

Note that this linearization is only w.r.t. \(\psi\).

◆ add_linearized_lapse_sources()

template<int ConformalMatterScale>

void Xcts::add_linearized_lapse_sources	(	gsl::not_null< Scalar< DataVector > * >	linearized_lapse_equation,
		const Scalar< DataVector > &	conformal_energy_density,
		const Scalar< DataVector > &	conformal_stress_trace,
		const Scalar< DataVector > &	extrinsic_curvature_trace,
		const Scalar< DataVector > &	dt_extrinsic_curvature_trace,
		const Scalar< DataVector > &	shift_dot_deriv_extrinsic_curvature_trace,
		const Scalar< DataVector > &	conformal_factor_minus_one,
		const Scalar< DataVector > &	lapse_times_conformal_factor_minus_one,
		const Scalar< DataVector > &	conformal_factor_correction,
		const Scalar< DataVector > &	lapse_times_conformal_factor_correction
	)

The linearization of add_lapse_sources

Note that this linearization is only w.r.t. \(\psi\) and \(\alpha\psi\). The linearization w.r.t. \(\beta^i\) is added in add_curved_linearized_momentum_sources or add_flat_cartesian_linearized_momentum_sources.

◆ extrinsic_curvature()

template<typename DataType >

void Xcts::extrinsic_curvature	(	const gsl::not_null< tnsr::ii< DataType, 3 > * >	result,
		const Scalar< DataType > &	conformal_factor,
		const Scalar< DataType > &	lapse,
		const tnsr::ii< DataType, 3 > &	conformal_metric,
		const tnsr::II< DataType, 3 > &	longitudinal_shift_minus_dt_conformal_metric,
		const Scalar< DataType > &	trace_extrinsic_curvature
	)

Extrinsic curvature computed from the conformal decomposition used in the XCTS system.

The extrinsic curvature decomposition is (see Eq. 3.113 in [13]):

\begin{equation} K_{ij} = A_{ij} + \frac{1}{3}\gamma_{ij}K = \frac{\psi^4}{2\lapse}\left((\bar{L}\beta)_{ij} - \bar{u}_{ij}\right) + \frac{\psi^4}{3} \bar{\gamma}_{ij} K \end{equation}

Parameters

result	output buffer for the extrinsic curvature
conformal_factor	the conformal factor \(\psi\)
lapse	the lapse \(\alpha\)
conformal_metric	the conformal metric \(\bar{\gamma}_{ij}\)
longitudinal_shift_minus_dt_conformal_metric	the term \((\bar{L}\beta)^{ij} - \bar{u}^{ij}\). Note that \((\bar{L}\beta)^{ij}\) is the conformal longitudinal shift, and \((\bar{L}\beta)^{ij}=\psi^4(L\beta)^{ij}\) (Eq. 3.98 in [13]). See also Xcts::longitudinal_operator.
trace_extrinsic_curvature	the trace of the extrinsic curvature, \(K=\gamma^{ij}K_{ij}\). Note that it is a conformal invariant, \(K=\bar{K}\) (by choice).

◆ longitudinal_operator() [1/2]

template<typename DataType >

void Xcts::longitudinal_operator	(	gsl::not_null< tnsr::II< DataType, 3 > * >	result,
		const tnsr::I< DataType, 3 > &	shift,
		const tnsr::iJ< DataType, 3 > &	deriv_shift,
		const tnsr::II< DataType, 3 > &	inv_metric,
		const tnsr::Ijj< DataType, 3 > &	christoffel_second_kind
	)

The longitudinal operator, or vector gradient, \((L\beta)^{ij}\).

Computes the longitudinal operator

\begin{equation} (L\beta)^{ij} = \nabla^i \beta^j + \nabla^j \beta^i - \frac{2}{3}\gamma^{ij}\nabla_k\beta^k \end{equation}

of a vector field \(\beta^i\), where \(\nabla\) denotes the covariant derivative w.r.t. the metric \(\gamma\) (see e.g. Eq. (3.50) in [13]). Note that in the XCTS equations the longitudinal operator is typically applied to conformal quantities and w.r.t. the conformal metric \(\bar{\gamma}\).

In terms of the symmetric "strain" quantity \(B_{ij}=\nabla_{(i}\gamma_{j)k}\beta^k\) the longitudinal operator is:

\begin{equation} (L\beta)^{ij} = 2\left(\gamma^{ik}\gamma^{jl} - \frac{1}{3} \gamma^{ij}\gamma^{kl}\right) B_{kl} \end{equation}

Note that the strain can be computed with Elasticity::strain.

◆ longitudinal_operator() [2/2]

template<typename DataType >

void Xcts::longitudinal_operator	(	gsl::not_null< tnsr::II< DataType, 3 > * >	result,
		const tnsr::ii< DataType, 3 > &	strain,
		const tnsr::II< DataType, 3 > &	inv_metric
	)

The longitudinal operator, or vector gradient, \((L\beta)^{ij}\).

Computes the longitudinal operator

\begin{equation} (L\beta)^{ij} = \nabla^i \beta^j + \nabla^j \beta^i - \frac{2}{3}\gamma^{ij}\nabla_k\beta^k \end{equation}

of a vector field \(\beta^i\), where \(\nabla\) denotes the covariant derivative w.r.t. the metric \(\gamma\) (see e.g. Eq. (3.50) in [13]). Note that in the XCTS equations the longitudinal operator is typically applied to conformal quantities and w.r.t. the conformal metric \(\bar{\gamma}\).

In terms of the symmetric "strain" quantity \(B_{ij}=\nabla_{(i}\gamma_{j)k}\beta^k\) the longitudinal operator is:

\begin{equation} (L\beta)^{ij} = 2\left(\gamma^{ik}\gamma^{jl} - \frac{1}{3} \gamma^{ij}\gamma^{kl}\right) B_{kl} \end{equation}

Note that the strain can be computed with Elasticity::strain.

◆ longitudinal_operator_flat_cartesian()

template<typename DataType >

void Xcts::longitudinal_operator_flat_cartesian	(	gsl::not_null< tnsr::II< DataType, 3 > * >	result,
		const tnsr::ii< DataType, 3 > &	strain
	)

The conformal longitudinal operator \((L\beta)^{ij}\) on a flat conformal metric in Cartesian coordinates \(\gamma_{ij}=\delta_{ij}\).

See also: Xcts::longitudinal_operator

Namespaces

Classes

Typedefs

Enumerations

Functions

Detailed Description

Enumeration Type Documentation

◆ Equations

◆ Geometry

Function Documentation

◆ add_curved_hamiltonian_or_lapse_sources()

◆ add_curved_momentum_sources()

◆ add_distortion_hamiltonian_and_lapse_sources()

◆ add_distortion_hamiltonian_sources()

◆ add_flat_cartesian_momentum_sources()

◆ add_hamiltonian_sources()

◆ add_lapse_sources()

◆ add_linearized_distortion_hamiltonian_and_lapse_sources()

◆ add_linearized_distortion_hamiltonian_sources()

◆ add_linearized_lapse_sources()

◆ extrinsic_curvature()

◆ longitudinal_operator() [1/2]

◆ longitudinal_operator() [2/2]

◆ longitudinal_operator_flat_cartesian()