ScalarTensor Namespace Reference

Items related to evolving the first-order scalar tensor system. More...

Namespaces
namespace	AnalyticData
	Holds classes implementing analytic data for the ScalarTensor system.
namespace	BoundaryConditions
	Boundary conditions for the combined Generalized Harmonic and CurvedScalarWave systems.
namespace	BoundaryCorrections
	Boundary corrections/numerical fluxes.

Classes
struct	AnalyticDataBase
	Base struct for properties common to all Scalar Tensor analytic initial data. More...
struct	System
	Scalar Tensor system obtained from combining the CurvedScalarWave and gh systems. More...
struct	TimeDerivative
	Compute the RHS terms of the evolution equations for the scalar tensor system. More...

Functions
void	add_scalar_source_to_dt_pi_scalar (gsl::not_null< Scalar< DataVector > * > dt_pi_scalar, const Scalar< DataVector > &scalar_source, const Scalar< DataVector > &lapse)
	Add in the source term to the \(\Pi\) evolved variable of the CurvedScalarWave system. More...
void	mass_source (gsl::not_null< Scalar< DataVector > * > scalar_source, const Scalar< DataVector > &psi, const double mass_psi)
	Computes the source term given by the mass of the scalar. More...
void	add_stress_energy_term_to_dt_pi (gsl::not_null< tnsr::aa< DataVector, 3_st > * > dt_pi, const tnsr::aa< DataVector, 3_st > &trace_reversed_stress_energy, const Scalar< DataVector > &lapse)
	Add in the trace-reversed stress-energy source term to the \(\Pi\) evolved variable of the gh system. More...
void	trace_reversed_stress_energy (gsl::not_null< tnsr::aa< DataVector, 3_st > * > stress_energy, const Scalar< DataVector > &pi_scalar, const tnsr::i< DataVector, 3_st > &phi_scalar, const Scalar< DataVector > &lapse)
	Compute the trace-reversed stress-energy tensor of the scalar field. More...
void	scalar_charge_integrand (const gsl::not_null< Scalar< DataVector > * > result, const tnsr::i< DataVector, 3 > &phi, const tnsr::I< DataVector, 3 > &unit_normal_vector)
	The scalar charge per unit solid angle. More...

Detailed Description

Items related to evolving the first-order scalar tensor system.

Tags for the scalar tensor system.

Function Documentation

add_scalar_source_to_dt_pi_scalar()

void ScalarTensor::add_scalar_source_to_dt_pi_scalar	(	gsl::not_null< Scalar< DataVector > * >	dt_pi_scalar,
		const Scalar< DataVector > &	scalar_source,
		const Scalar< DataVector > &	lapse
	)

Add in the source term to the \(\Pi\) evolved variable of the CurvedScalarWave system.

Details

The only source term in the wave equation

\[ \Box \Psi = \mathcal{S} ~, \]

is in the equation for \(\Pi\):

\[ \partial_t \Pi + \text{\{spatial derivative terms\}} = \alpha \mathcal{S} ~, \]

where \(\mathcal{S}\) is the source term (e. g. in the Klein-Gordon equation, the source term is the derivative of the scalar potential \(\mathcal{S} \equiv \partial V / \partial \Psi \).)

This function adds that contribution to the existing value of dt_pi_scalar. The wave equation terms in the scalar equation should be computed before passing the dt_pi_scalar to this function for updating.

Parameters

dt_pi_scalar	Time derivative terms of \(\Pi\). The sourceless part should be computed before with CurvedScalarWave::TimeDerivative.
scalar_source	Source term \(\mathcal{S}\) for the scalar equation.
lapse	Lapse \(\alpha\).

See also

CurvedScalarWave::TimeDerivative for details about the source-less part of the time derivative calculation.

add_stress_energy_term_to_dt_pi()

void ScalarTensor::add_stress_energy_term_to_dt_pi	(	gsl::not_null< tnsr::aa< DataVector, 3_st > * >	dt_pi,
		const tnsr::aa< DataVector, 3_st > &	trace_reversed_stress_energy,
		const Scalar< DataVector > &	lapse
	)

Add in the trace-reversed stress-energy source term to the \(\Pi\) evolved variable of the gh system.

Details

The only stress energy source term in the Generalized Harmonic evolution equations is in the equation for \(\Pi_{a b}\):

\[ \partial_t \Pi_{ab} + \text{\{spatial derivative terms\}} = \text{\{GH source terms\}} - 16 \pi \alpha (T^{(\Psi)}_{ab} - \frac{1}{2} g_{a b} g^{cd}T^{(\Psi)}_{cd}) ~. \]

(note that this function takes as argument the trace-reversed stress-energy tensor)

This function adds that contribution to the existing value of dt_pi. The spacetime terms in the GH equation should be computed before passing the dt_pi to this function for updating.

Parameters

dt_pi	Time derivative of the \(\Pi_{ab}\) variable in the gh system. The vacuum part should be computed before with gh::TimeDerivative
trace_reversed_stress_energy	Trace-reversed stress energy tensor of the scalar \(T^{(\Psi), \text{TR}}_{a b} \equiv T^{(\Psi)}_{ab} - \frac{1}{2} g_{a b} g^{cd}T^{(\Psi)}_{cd} = \partial_a \Psi \partial_b \Psi \).
lapse	Lapse \(\alpha\).

See also

gh::TimeDerivative for details about the spacetime part of the time derivative calculation.

mass_source()

void ScalarTensor::mass_source	(	gsl::not_null< Scalar< DataVector > * >	scalar_source,
		const Scalar< DataVector > &	psi,
		const double	mass_psi
	)

Computes the source term given by the mass of the scalar.

Details

For a scalar field with mass parameter \( m_\Psi \), the wave equation takes the form

\[ \Box \Psi = \mathcal{S} ~, \]

where the source is given by

\[ \mathcal{S} \equiv m^2_\Psi \Psi~. \]

Here the mass parameter value is an option that needs to be specified in the input file.

Parameters

scalar_source	Source term \(\mathcal{S}\) for the scalar equation.
psi	Scalar field \(\Psi\).
mass_psi	Mass of the scalar field \(m_\Psi\).

See also

ScalarTensor::Tags::ScalarMass for details about the mass.

scalar_charge_integrand()

void ScalarTensor::scalar_charge_integrand	(	const gsl::not_null< Scalar< DataVector > * >	result,
		const tnsr::i< DataVector, 3 > &	phi,
		const tnsr::I< DataVector, 3 > &	unit_normal_vector
	)

The scalar charge per unit solid angle.

Details

This function calculates the integrand of:

\begin{align*} q = - \dfrac{1}{4 \pi} \oint dA \Phi_i n^{i}, \end{align*}

where \( n^{i} \) is the unit (outward) normal of the surface.

For a spherically symmetric scalar, this value will coincide with the value as extracted from the \(r^{-1}\) decay in

\[ \Psi \sim \phi_\infty + q / r + \cdots~. \]

trace_reversed_stress_energy()

void ScalarTensor::trace_reversed_stress_energy	(	gsl::not_null< tnsr::aa< DataVector, 3_st > * >	stress_energy,
		const Scalar< DataVector > &	pi_scalar,
		const tnsr::i< DataVector, 3_st > &	phi_scalar,
		const Scalar< DataVector > &	lapse
	)

Compute the trace-reversed stress-energy tensor of the scalar field.

Details

The trace-reversed stress energy tensor is needed to compute the backreaction of the scalar to the spacetime evolution and is given by

\begin{align*} T^{(\Psi), \text{TR}}_{a b} &\equiv T^{(\Psi)}_{ab} - \frac{1}{2} g_{a b} g^{cd}T^{(\Psi)}_{cd} \\ &= \partial_a \Psi \partial_b \Psi ~, \end{align*}

where \(T^{(\Psi)}_{ab}\) is the standard stress-energy tensor of the scalar.

In terms of the evolved variables of the scalar,

\begin{align*} T^{(\Psi), \text{TR}}_{00} &= \alpha^2 \Pi^2 ~, \\ T^{(\Psi), \text{TR}}_{j 0} &= T^{(\Psi), \text{TR}}_{0j} = - \alpha \Pi \Phi_j ~, \\ T^{(\Psi), \text{TR}}_{ij} &= \Phi_i \Phi_j ~, \end{align*}

where \(\alpha\) is the lapse.

Parameters

stress_energy	Trace-reversed stress energy tensor of the scalar \(T^{(\Psi), \text{TR}}_{a b} \equiv T^{(\Psi)}_{ab} - \frac{1}{2} g_{a b} g^{cd}T^{(\Psi)}_{cd} = \partial_a \Psi \partial_b \Psi \).
pi_scalar	Scalar evolution variable \(\Pi\).
phi_scalar	Scalar evolution variable \(\Phi_i\).
lapse	Lapse \(\alpha\).