Xcts Namespace Reference

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 >, ::Tags::deriv<::Tags::deriv< Tags::ConformalFactor< DataVector >, tmpl::size_t< 3 >, Frame::Inertial >, 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::SpatialRicci< 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...
void	adm_mass_surface_integrand (gsl::not_null< tnsr::I< DataVector, 3 > * > result, const tnsr::i< DataVector, 3 > &deriv_conformal_factor, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const tnsr::Ijj< DataVector, 3 > &conformal_christoffel_second_kind, const tnsr::i< DataVector, 3 > &conformal_christoffel_contracted)
	Surface integrand for the ADM mass calculation. More...
tnsr::I< DataVector, 3 >	adm_mass_surface_integrand (const tnsr::i< DataVector, 3 > &deriv_conformal_factor, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const tnsr::Ijj< DataVector, 3 > &conformal_christoffel_second_kind, const tnsr::i< DataVector, 3 > &conformal_christoffel_contracted)
	Return-by-value overload.
void	adm_mass_volume_integrand (gsl::not_null< Scalar< DataVector > * > result, const Scalar< DataVector > &conformal_factor, const Scalar< DataVector > &conformal_ricci_scalar, const Scalar< DataVector > &trace_extrinsic_curvature, const Scalar< DataVector > &longitudinal_shift_minus_dt_conformal_metric_over_lapse_square, const Scalar< DataVector > &energy_density, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const tnsr::iJK< DataVector, 3 > &deriv_inv_conformal_metric, const tnsr::Ijj< DataVector, 3 > &conformal_christoffel_second_kind, const tnsr::i< DataVector, 3 > &conformal_christoffel_contracted, const tnsr::iJkk< DataVector, 3 > &deriv_conformal_christoffel_second_kind)
	Volume integrand for the ADM mass calculation. More...
Scalar< DataVector >	adm_mass_volume_integrand (const Scalar< DataVector > &conformal_factor, const Scalar< DataVector > &conformal_ricci_scalar, const Scalar< DataVector > &trace_extrinsic_curvature, const Scalar< DataVector > &longitudinal_shift_minus_dt_conformal_metric_over_lapse_square, const Scalar< DataVector > &energy_density, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const tnsr::iJK< DataVector, 3 > &deriv_inv_conformal_metric, const tnsr::Ijj< DataVector, 3 > &conformal_christoffel_second_kind, const tnsr::i< DataVector, 3 > &conformal_christoffel_contracted, const tnsr::iJkk< DataVector, 3 > &deriv_conformal_christoffel_second_kind)
	Return-by-value overload.
void	adm_linear_momentum_surface_integrand (gsl::not_null< tnsr::II< DataVector, 3 > * > result, const Scalar< DataVector > &conformal_factor, const tnsr::II< DataVector, 3 > &inv_spatial_metric, const tnsr::II< DataVector, 3 > &inv_extrinsic_curvature, const Scalar< DataVector > &trace_extrinsic_curvature)
	Surface integrand for the ADM linear momentum calculation. More...
tnsr::II< DataVector, 3 >	adm_linear_momentum_surface_integrand (const Scalar< DataVector > &conformal_factor, const tnsr::II< DataVector, 3 > &inv_spatial_metric, const tnsr::II< DataVector, 3 > &inv_extrinsic_curvature, const Scalar< DataVector > &trace_extrinsic_curvature)
	Return-by-value overload.
void	adm_linear_momentum_volume_integrand (gsl::not_null< tnsr::I< DataVector, 3 > * > result, const tnsr::II< DataVector, 3 > &surface_integrand, const Scalar< DataVector > &conformal_factor, const tnsr::i< DataVector, 3 > &deriv_conformal_factor, const tnsr::ii< DataVector, 3 > &conformal_metric, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const tnsr::Ijj< DataVector, 3 > &conformal_christoffel_second_kind, const tnsr::i< DataVector, 3 > &conformal_christoffel_contracted)
	Volume integrand for ADM linear momentum calculation defined as (see Eq. 20 in [153]): More...
tnsr::I< DataVector, 3 >	adm_linear_momentum_volume_integrand (const tnsr::II< DataVector, 3 > &surface_integrand, const Scalar< DataVector > &conformal_factor, const tnsr::i< DataVector, 3 > &deriv_conformal_factor, const tnsr::ii< DataVector, 3 > &conformal_metric, const tnsr::II< DataVector, 3 > &inv_conformal_metric, const tnsr::Ijj< DataVector, 3 > &conformal_christoffel_second_kind, const tnsr::i< DataVector, 3 > &conformal_christoffel_contracted)
	Return-by-value overload.
void	adm_angular_momentum_z_surface_integrand (gsl::not_null< tnsr::I< DataVector, 3 > * > result, const tnsr::II< DataVector, 3 > &linear_momentum_surface_integrand, const tnsr::I< DataVector, 3 > &coords)
	Surface integrand for the z-component of the ADM angular momentum. More...
tnsr::I< DataVector, 3 >	adm_angular_momentum_z_surface_integrand (const tnsr::II< DataVector, 3 > &linear_momentum_surface_integrand, const tnsr::I< DataVector, 3 > &coords)
	Return-by-value overload.
void	adm_angular_momentum_z_volume_integrand (gsl::not_null< Scalar< DataVector > * > result, const tnsr::I< DataVector, 3 > &linear_momentum_volume_integrand, const tnsr::I< DataVector, 3 > &coords)
	Volume integrand for the z-component of the ADM angular momentum. More...
Scalar< DataVector >	adm_angular_momentum_z_volume_integrand (const tnsr::I< DataVector, 3 > &linear_momentum_volume_integrand, const tnsr::I< DataVector, 3 > &coords)
	Return-by-value overload.
void	center_of_mass_surface_integrand (gsl::not_null< tnsr::I< DataVector, 3 > * > result, const Scalar< DataVector > &conformal_factor, const tnsr::I< DataVector, 3 > &coords)
	Surface integrand for the center of mass calculation. More...
tnsr::I< DataVector, 3 >	center_of_mass_surface_integrand (const Scalar< DataVector > &conformal_factor, const tnsr::I< DataVector, 3 > &coords)
	Return-by-value overload.
void	center_of_mass_volume_integrand (gsl::not_null< tnsr::I< DataVector, 3 > * > result, const Scalar< DataVector > &conformal_factor, const tnsr::i< DataVector, 3, Frame::Inertial > &deriv_conformal_factor, const tnsr::I< DataVector, 3 > &coords)
	Volume integrand for the center of mass calculation. More...
tnsr::I< DataVector, 3 >	center_of_mass_volume_integrand (const Scalar< DataVector > &conformal_factor, const tnsr::i< DataVector, 3, Frame::Inertial > &deriv_conformal_factor, const tnsr::I< DataVector, 3 > &coords)
	Return-by-value overload.
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. [14], 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.

adm_angular_momentum_z_surface_integrand()

void Xcts::adm_angular_momentum_z_surface_integrand	(	gsl::not_null< tnsr::I< DataVector, 3 > * >	result,
		const tnsr::II< DataVector, 3 > &	linear_momentum_surface_integrand,
		const tnsr::I< DataVector, 3 > &	coords
	)

Surface integrand for the z-component of the ADM angular momentum.

We define the ADM angular momentum surface integral as (see Eq. 23 in [153]):

\begin{equation} J_\text{ADM}^z = \frac{1}{8\pi} \oint_{S_\infty} \Big( x P^{yj} - y P^{xj} \Big) \, dS_j, \end{equation}

where \(1/(8\pi) P^{jk}\) is the result from adm_linear_momentum_surface_integrand.

Note

For consistency with adm_angular_momentum_z_volume_integrand, this integrand needs to be contracted with the Euclidean face normal and integrated with the Euclidean area element.

Parameters

result	output pointer
linear_momentum_surface_integrand	the quantity \(1/(8\pi) P^{ij}\) (result of adm_linear_momentum_surface_integrand)
coords	the inertial coordinates \(x^i\)

adm_angular_momentum_z_volume_integrand()

void Xcts::adm_angular_momentum_z_volume_integrand	(	gsl::not_null< Scalar< DataVector > * >	result,
		const tnsr::I< DataVector, 3 > &	linear_momentum_volume_integrand,
		const tnsr::I< DataVector, 3 > &	coords
	)

Volume integrand for the z-component of the ADM angular momentum.

We define the ADM angular momentum volume integral as (see Eq. 23 in [153]):

\begin{equation} J_\text{ADM}^z = - \frac{1}{8\pi} \int_{V_\infty} \Big( x G^y - y G^x \Big) \, dV, \end{equation}

where \(-1/(8\pi) G^i\) is the result from adm_linear_momentum_volume_integrand.

Note

For consistency with adm_angular_momentum_z_surface_integrand, this integrand needs to be integrated with the Euclidean volume element.

Parameters

result	output pointer
linear_momentum_volume_integrand	the quantity \(-1/(8\pi) G^i\) (result of adm_linear_momentum_volume_integrand)
coords	the inertial coordinates \(x^i\)

adm_linear_momentum_surface_integrand()

void Xcts::adm_linear_momentum_surface_integrand	(	gsl::not_null< tnsr::II< DataVector, 3 > * >	result,
		const Scalar< DataVector > &	conformal_factor,
		const tnsr::II< DataVector, 3 > &	inv_spatial_metric,
		const tnsr::II< DataVector, 3 > &	inv_extrinsic_curvature,
		const Scalar< DataVector > &	trace_extrinsic_curvature
	)

Surface integrand for the ADM linear momentum calculation.

We define the ADM linear momentum integral as (see Eqs. 19-20 in [153]):

\begin{equation} P_\text{ADM}^i = \frac{1}{8\pi} \oint_{S_\infty} \psi^{10} \Big( K^{ij} - K \gamma^{ij} \Big) \, dS_j. \end{equation}

Note

For consistency with adm_linear_momentum_volume_integrand, this integrand needs to be contracted with the Euclidean face normal and integrated with the Euclidean area element.

Parameters

result	output pointer
conformal_factor	the conformal factor \(\psi\)
inv_spatial_metric	the inverse spatial metric \(\gamma^{ij}\)
inv_extrinsic_curvature	the inverse extrinsic curvature \(K^{ij}\)
trace_extrinsic_curvature	the trace of the extrinsic curvature \(K\)

adm_linear_momentum_volume_integrand()

void Xcts::adm_linear_momentum_volume_integrand	(	gsl::not_null< tnsr::I< DataVector, 3 > * >	result,
		const tnsr::II< DataVector, 3 > &	surface_integrand,
		const Scalar< DataVector > &	conformal_factor,
		const tnsr::i< DataVector, 3 > &	deriv_conformal_factor,
		const tnsr::ii< DataVector, 3 > &	conformal_metric,
		const tnsr::II< DataVector, 3 > &	inv_conformal_metric,
		const tnsr::Ijj< DataVector, 3 > &	conformal_christoffel_second_kind,
		const tnsr::i< DataVector, 3 > &	conformal_christoffel_contracted
	)

Volume integrand for ADM linear momentum calculation defined as (see Eq. 20 in [153]):

\begin{equation} P_\text{ADM}^i = - \frac{1}{8\pi} \int_{V_\infty} \Big( \bar\Gamma^i_{jk} P^{jk} + \bar\Gamma^j_{jk} P^{jk} - 2 \bar\gamma_{jk} P^{jk} \bar\gamma^{il} \partial_l(\ln\psi) \Big) \, dV, \end{equation}

where \(1/(8\pi) P^{jk}\) is the result from adm_linear_momentum_surface_integrand.

Note

For consistency with adm_linear_momentum_surface_integrand, this integrand needs to be integrated with the Euclidean volume element.

Parameters

result	output pointer
surface_integrand	the quantity \(1/(8\pi) P^{ij}\) (result of adm_linear_momentum_surface_integrand)
conformal_factor	the conformal factor \(\psi\)
deriv_conformal_factor	the gradient of the conformal factor \(\partial_i\psi\)
conformal_metric	the conformal metric \(\bar\gamma_{ij}\)
inv_conformal_metric	the inverse conformal metric \(\bar\gamma^{ij}\)
conformal_christoffel_second_kind	the conformal christoffel symbol \(\bar\Gamma^i_{jk}\)
conformal_christoffel_contracted	the contracted conformal christoffel symbol \(\bar\Gamma_i\)

adm_mass_surface_integrand()

void Xcts::adm_mass_surface_integrand	(	gsl::not_null< tnsr::I< DataVector, 3 > * >	result,
		const tnsr::i< DataVector, 3 > &	deriv_conformal_factor,
		const tnsr::II< DataVector, 3 > &	inv_conformal_metric,
		const tnsr::Ijj< DataVector, 3 > &	conformal_christoffel_second_kind,
		const tnsr::i< DataVector, 3 > &	conformal_christoffel_contracted
	)

Surface integrand for the ADM mass calculation.

We define the ADM mass integral as (see Eq. 3.139 in [14]):

\begin{equation} M_\text{ADM} = \frac{1}{16\pi} \oint_{S_\infty} \Big( \bar\gamma^{jk} \bar\Gamma^i_{jk} - \bar\gamma^{ij} \bar\Gamma_{j} - 8 \bar\gamma^{ij} \partial_j \psi \Big) d\bar{S}_i. \end{equation}

Note

We don't use the other versions presented in [14] of this integral because they make assumptions like \(\bar\gamma = 1\), \(\bar\Gamma^i_{ij} = 0\) and faster fall-off of the conformal metric.

For consistency with adm_mass_volume_integrand, this integrand needs to be contracted with the conformal face normal and integrated with the conformal area element.

Parameters

result	output pointer
deriv_conformal_factor	the gradient of the conformal factor \(\partial_i \psi\)
inv_conformal_metric	the inverse conformal metric \(\bar\gamma^{ij}\)
conformal_christoffel_second_kind	the conformal christoffel symbol \(\bar\Gamma^i_{jk}\)
conformal_christoffel_contracted	the conformal christoffel symbol contracted in its first two indices \(\bar\Gamma_{i} = \bar\Gamma^j_{ij}\)

adm_mass_volume_integrand()

void Xcts::adm_mass_volume_integrand	(	gsl::not_null< Scalar< DataVector > * >	result,
		const Scalar< DataVector > &	conformal_factor,
		const Scalar< DataVector > &	conformal_ricci_scalar,
		const Scalar< DataVector > &	trace_extrinsic_curvature,
		const Scalar< DataVector > &	longitudinal_shift_minus_dt_conformal_metric_over_lapse_square,
		const Scalar< DataVector > &	energy_density,
		const tnsr::II< DataVector, 3 > &	inv_conformal_metric,
		const tnsr::iJK< DataVector, 3 > &	deriv_inv_conformal_metric,
		const tnsr::Ijj< DataVector, 3 > &	conformal_christoffel_second_kind,
		const tnsr::i< DataVector, 3 > &	conformal_christoffel_contracted,
		const tnsr::iJkk< DataVector, 3 > &	deriv_conformal_christoffel_second_kind
	)

Volume integrand for the ADM mass calculation.

We cast the ADM mass as an infinite volume integral by applying Gauss' law on the surface integral defined in adm_mass_surface_integrand:

\begin{equation} M_\text{ADM} = \frac{1}{16\pi} \int_{V_\infty} \Big( \partial_i \bar\gamma^{jk} \bar\Gamma^i_{jk} + \bar\gamma^{jk} \partial_i \bar\Gamma^i_{jk} + \bar\Gamma_l \bar\gamma^{jk} \bar\Gamma^l_{jk} - \partial_i \bar\gamma^{ij} \bar\Gamma_j - \bar\gamma^{ij} \partial_i \bar\Gamma_j - \bar\Gamma_l \bar\gamma^{lj} \bar\Gamma_j - 8 \bar D^2 \psi \Big) d\bar{V}, \end{equation}

where we can use the Hamiltonian constraint (Eq. 3.37 in [14]) to replace \(8 \bar D^2 \psi\) with

\begin{equation} 8 \bar D^2 \psi = \psi \bar R + \frac{2}{3} \psi^5 K^2 - \frac{1}{4} \psi^5 \frac{1}{\alpha^2} \Big[ (\bar L \beta)_{ij} - \bar u_{ij} \Big] \Big[ (\bar L \beta)^{ij} - \bar u^{ij} \Big] - 16\pi \psi^5 \rho. \end{equation}

Note

This is similar to Eq. 3.149 in [14], except that here we don't assume \(\bar\gamma = 1\).

For consistency with adm_mass_surface_integrand, this integrand needs to be integrated with the conformal volume element.

Parameters

result	output pointer
conformal_factor	the conformal factor
conformal_ricci_scalar	the conformal Ricci scalar \(\bar R\)
trace_extrinsic_curvature	the extrinsic curvature trace \(K\)
longitudinal_shift_minus_dt_conformal_metric_over_lapse_square	the quantity computed in Xcts::Tags::LongitudinalShiftMinusDtConformalMetricOverLapseSquare
energy_density	the energy density \(\rho\)
inv_conformal_metric	the inverse conformal metric \(\bar\gamma^{ij}\)
deriv_inv_conformal_metric	the gradient of the inverse conformal metric \(\partial_i \bar\gamma^{jk}\)
conformal_christoffel_second_kind	the conformal christoffel symbol \(\bar\Gamma^i_{jk}\)
conformal_christoffel_contracted	the conformal christoffel symbol contracted in its first two indices \(\bar\Gamma_{i} = \bar\Gamma^j_{ij}\)
deriv_conformal_christoffel_second_kind	the gradient of the conformal christoffel symbol \(\partial_i \bar\Gamma^j_{kl}\)

center_of_mass_surface_integrand()

void Xcts::center_of_mass_surface_integrand	(	gsl::not_null< tnsr::I< DataVector, 3 > * >	result,
		const Scalar< DataVector > &	conformal_factor,
		const tnsr::I< DataVector, 3 > &	coords
	)

Surface integrand for the center of mass calculation.

We define the center of mass integral as

\begin{equation} C_\text{CoM}^i = \frac{3}{8 \pi M_\text{ADM}} \oint_{S_\infty} (\psi^4 - 1) n^i \, dA, \end{equation}

where \(n^i = x^i / r\) and \(r = \sqrt{x^2 + y^2 + z^2}\).

Analytically, this is identical to the definition in Eq. (25) of [153] because

\begin{equation} \oint_{S_\infty} n^i \, dA = 0. \end{equation}

Numerically, we have found that subtracting \(1\) from \(\psi^4\) results in less round-off errors, leading to a more accurate center of mass.

One way to interpret this integral is that we are summing over the unit vectors \(n^i\), rescaled by \(\psi^4\), in all directions. If \(\psi(\vec r)\) is constant, no rescaling happens and all the unit vectors cancel out. If \(\psi(\vec r)\) is not constant, then \(\vec C_\text{CoM}\) will emerge from the difference of large numbers (sum of rescaled \(n^i\) in each subdomain). With larger and larger numbers being involved in this cancellation (i.e. with increasing radius of \(S_\infty\)), we loose numerical accuracy. In other words, we are seeking the subdominant terms. Since \(\psi^4 \to 1\) in conformal flatness, subtracting \(1\) from it in the integrand makes the numbers involved in this cancellation smaller, reducing this issue. We have also tried different variations of this integrand with the same motivation, but \((\psi^4 - 1)\) is the best one when taking simplicity and accuracy gain into consideration.

Note

We don't include the ADM mass \(M_{ADM}\) in this integrand. After integrating the result of this function, you have to divide by \(M_{ADM}\).

For consistency with center_of_mass_volume_integrand, this integrand needs to be integrated with the Euclidean area element.

See also

Xcts::adm_mass_surface_integrand

Warning

This integral assumes that the conformal metric falls off to flatness faster than \(1/r^2\). That means that it cannot be directly used with the Kerr-Schild metric, which falls off as \(1/r\). This is not a problem for XCTS with Superposed Kerr-Schild (SKS) because of the exponential fall-off terms.

Parameters

result	output pointer
conformal_factor	the conformal factor \(\psi\)
coords	the inertial coordinates \(x^i\)

center_of_mass_volume_integrand()

void Xcts::center_of_mass_volume_integrand	(	gsl::not_null< tnsr::I< DataVector, 3 > * >	result,
		const Scalar< DataVector > &	conformal_factor,
		const tnsr::i< DataVector, 3, Frame::Inertial > &	deriv_conformal_factor,
		const tnsr::I< DataVector, 3 > &	coords
	)

Volume integrand for the center of mass calculation.

We cast the center of mass as an infinite volume integral by applying Gauss' law on the surface integral defined in center_of_mass_surface_integrand:

\begin{equation} C_\text{CoM}^i = \frac{3}{8 \pi M_\text{ADM}} \int_{V_\infty} \Big( 4 \psi^3 \partial_j \psi n^i n^j + \frac{2}{r} \psi^4 n^i \Big) dV = \frac{3}{4 \pi M_\text{ADM}} \int_{V_\infty} \frac{1}{r^2} \Big( 2 \psi^3 \partial_j \psi x^i x^j + \psi^4 x^i \Big) dV, \end{equation}

where \(n^i = x^i / r\) and \(r = \sqrt{x^2 + y^2 + z^2}\).

Note

For consistency with center_of_mass_surface_integrand, this integrand needs to be integrated with the Euclidean volume element.

See also

center_of_mass_surface_integrand

Parameters

result	output pointer
conformal_factor	the conformal factor \(\psi\)
deriv_conformal_factor	the gradient of the conformal factor \(\partial_i \psi\)
coords	the inertial coordinates \(x^i\)

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 [14]):

\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 [14]). 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 [14]). 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 [14]). 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