CurvedScalarWave::Worldtube Namespace Reference

The set of utilities for performing CurvedScalarWave evolution with a worldtube excision scheme. More...

Namespaces
namespace	Actions
	Actions for the worldtube-curved scalar wave system.
namespace	Initialization
	Initialization mutators and actions for the worldtube-curved scalar wave system.
namespace	OptionTags
	Option tags for the worldtube.
namespace	Tags
	Tags related to the worldtube.

Classes
struct	IterateAccelerationTerms
	Computes the next iteration of the acceleration due to scalar self force from the current iteration of the regular field, as well as the quantities required to compute the acceleration terms of the puncture field. More...
struct	Registration
struct	UpdateAcceleration
	Computes the final acceleration of the particle at this time step. More...
struct	WorldtubeSingleton
	The singleton component that represents the worldtube. More...

Functions
tnsr::iAA< double, 3 >	spatial_derivative_inverse_ks_metric (const tnsr::I< double, 3 > &pos)
	The spatial derivative of the zero spin inverse Kerr Schild metric, \(\partial_i g^{\mu \nu}\), assuming a black hole at the coordinate center with mass M = 1.
tnsr::iaa< double, 3 >	spatial_derivative_ks_metric (const tnsr::aa< double, 3 > &metric, const tnsr::iAA< double, 3 > &di_inverse_metric)
	The spatial derivative of the spacetime metric, \(\partial_i g_{\mu \nu}\).
tnsr::iiAA< double, 3 >	second_spatial_derivative_inverse_ks_metric (const tnsr::I< double, 3 > &pos)
	The second spatial derivative of the zero spin inverse Kerr Schild metric, \(\partial_i \partial_j g^{\mu \nu}\), assuming a black hole at the coordinate center with mass M = 1.
tnsr::iiaa< double, 3 >	second_spatial_derivative_metric (const tnsr::aa< double, 3 > &metric, const tnsr::iaa< double, 3 > &di_metric, const tnsr::iAA< double, 3 > &di_inverse_metric, const tnsr::iiAA< double, 3 > &dij_inverse_metric)
	The spatial derivative of the spacetime metric, \(\partial_i \partial_j g_{\mu \nu}\).
tnsr::iAbb< double, 3 >	spatial_derivative_christoffel (const tnsr::iaa< double, 3 > &di_metric, const tnsr::iiaa< double, 3 > &dij_metric, const tnsr::AA< double, 3 > &inverse_metric, const tnsr::iAA< double, 3 > &di_inverse_metric)
	The spatial derivative of the Christoffel symbols, \(\partial_i \Gamma^\rho_{\mu \nu}\).
tnsr::iA< double, 3 >	spatial_derivative_ks_contracted_christoffel (const tnsr::I< double, 3 > &pos)
	The spatial derivative of the zero spin Kerr Schild contracted Christoffel symbols, \(\partial_i g^{\mu \nu} \Gamma^\rho_{\mu \nu}\), assuming a black hole at the coordinate center with mass M = 1.
void	puncture_field (gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * > result, const tnsr::I< DataVector, 3, Frame::Inertial > &centered_coords, const tnsr::I< double, 3 > &particle_position, const tnsr::I< double, 3 > &particle_velocity, const tnsr::I< double, 3 > &particle_acceleration, double bh_mass, size_t order)
	Computes the puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime. described in [56]. More...
void	puncture_field_0 (gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * > result, const tnsr::I< DataVector, 3, Frame::Inertial > &centered_coords, const tnsr::I< double, 3 > &particle_position, const tnsr::I< double, 3 > &particle_velocity, const tnsr::I< double, 3 > &particle_acceleration, double bh_mass)
	Computes the puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime. described in [56]. More...
void	puncture_field_1 (gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * > result, const tnsr::I< DataVector, 3, Frame::Inertial > &centered_coords, const tnsr::I< double, 3 > &particle_position, const tnsr::I< double, 3 > &particle_velocity, const tnsr::I< double, 3 > &particle_acceleration, double bh_mass)
	Computes the puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime. described in [56]. More...
void	acceleration_terms_0 (gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * > result, const tnsr::I< DataVector, 3, Frame::Inertial > &centered_coords, const tnsr::I< double, 3 > &particle_position, const tnsr::I< double, 3 > &particle_velocity, const tnsr::I< double, 3 > &particle_acceleration, double ft, double fx, double fy, double dt_ft, double dt_fx, double dt_fy, double bh_mass)
	Computes the acceleration terms of a puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime up to zeroth order in coordinate distance. More...
void	acceleration_terms_1 (gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * > result, const tnsr::I< DataVector, 3, Frame::Inertial > &centered_coords, const tnsr::I< double, 3 > &particle_position, const tnsr::I< double, 3 > &particle_velocity, const tnsr::I< double, 3 > &particle_acceleration, double ft, double fx, double fy, double dt_ft, double dt_fx, double dt_fy, double Du_ft, double Du_fx, double Du_fy, double dt_Du_ft, double dt_Du_fx, double dt_Du_fy, double bh_mass)
	Computes the acceleration terms of a puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime up to first order in coordinate distance (i.e. zeroth and first order). More...
double	smooth_broken_power_law (double orbit_radius, double alpha, double radius_at_inf, double rb, double delta)
	A smoothly broken power law that falls off to a constant value for larger radii. More...
double	smooth_broken_power_law_derivative (double orbit_radius, double alpha, double radius_at_inf, double rb, double delta)
	Returns the analytical derivative of smooth_broken_power_law.
template<size_t Dim>
tnsr::A< double, Dim >	self_force_per_mass (const tnsr::a< double, Dim > &d_psi, const tnsr::A< double, Dim > &four_velocity, double particle_charge, double particle_mass, const tnsr::AA< double, Dim > &inverse_metric)
	Computes the scalar self-force per unit mass. More...
template<size_t Dim>
tnsr::A< double, Dim >	dt_self_force_per_mass (const tnsr::a< double, Dim > &d_psi, const tnsr::a< double, Dim > &dt_d_psi, const tnsr::A< double, Dim > &four_velocity, const tnsr::A< double, Dim > &dt_four_velocity, double particle_charge, double particle_mass, const tnsr::AA< double, Dim > &inverse_metric, const tnsr::AA< double, Dim > &dt_inverse_metric)
	Computes the first time derivative of scalar self-force per unit mass, see self_force_per_mass, by applying the chain rule.
template<size_t Dim>
tnsr::A< double, Dim >	dt2_self_force_per_mass (const tnsr::a< double, Dim > &d_psi, const tnsr::a< double, Dim > &dt_d_psi, const tnsr::a< double, Dim > &dt2_d_psi, const tnsr::A< double, Dim > &four_velocity, const tnsr::A< double, Dim > &dt_four_velocity, const tnsr::A< double, Dim > &dt2_four_velocity, double particle_charge, double particle_mass, const tnsr::AA< double, Dim > &inverse_metric, const tnsr::AA< double, Dim > &dt_inverse_metric, const tnsr::AA< double, Dim > &dt2_inverse_metric)
	Computes the second time derivative of scalar self-force per unit mass, see self_force_per_mass, by applying the chain rule.
template<size_t Dim>
tnsr::A< double, Dim >	Du_self_force_per_mass (const tnsr::A< double, Dim > &self_force, const tnsr::A< double, Dim > &dt_self_force, const tnsr::A< double, Dim > &four_velocity, const tnsr::Abb< double, Dim > &christoffel)
	Computes the covariant derivative of the scalar self-force per unit mass \(f^\alpha\), see self_force_per_mass, along the four velocity \(u^\beta\), i.e. \(u^\beta \nabla_\beta f^\alpha\).
template<size_t Dim>
tnsr::A< double, Dim >	dt_Du_self_force_per_mass (const tnsr::A< double, Dim > &self_force, const tnsr::A< double, Dim > &dt_self_force, const tnsr::A< double, Dim > &dt2_self_force, const tnsr::A< double, Dim > &four_velocity, const tnsr::A< double, Dim > &dt_four_velocity, const tnsr::Abb< double, Dim > &christoffel, const tnsr::Abb< double, Dim > &dt_christoffel)
	Computes the time derivative of the covariant derivative of the scalar self-force per unit mass \(f^\alpha\), see Du_self_force_per_mass, along the four velocity \(u^\beta\), i.e. \(\frac{d}{dt}u^\beta \nabla_\beta f^\alpha\).
double	turn_on_function (double t_minus_turn_on, double turn_on_timescale)
	A function used to roll-on the self-force continuously from 0 to 1. More...
double	dt_turn_on_function (double t_minus_turn_on, double turn_on_timescale)
	The first derivative of turn_on_function
double	dt2_turn_on_function (double t_minus_turn_on, double turn_on_timescale)
	The second derivative of turn_on_function
template<size_t Dim>
void	self_force_acceleration (gsl::not_null< tnsr::I< double, Dim > * > self_force_acc, const Scalar< double > &dt_psi_monopole, const tnsr::i< double, Dim > &psi_dipole, const tnsr::I< double, Dim > &particle_velocity, double particle_charge, double particle_mass, const tnsr::AA< double, Dim > &inverse_metric, const Scalar< double > &dilation_factor)
	Computes the coordinate acceleration due to the scalar self-force onto the charge. More...
template<size_t Dim>
tnsr::I< double, Dim >	self_force_acceleration (const Scalar< double > &dt_psi_monopole, const tnsr::i< double, Dim > &psi_dipole, const tnsr::I< double, Dim > &particle_velocity, double particle_charge, double particle_mass, const tnsr::AA< double, Dim > &inverse_metric, const Scalar< double > &dilation_factor)
	Computes the coordinate acceleration due to the scalar self-force onto the charge. More...

Detailed Description

The set of utilities for performing CurvedScalarWave evolution with a worldtube excision scheme.

Details

The worldtube excision scheme is a method that aims to enable NR evolutions of intermediate mass ratio binary black hole simulations. In standard BBH simulations two excision spheres are cut out from the domain within the apparent horizons of the respective black holes. For larger mass ratios, this introduces a scale disparity in the evolution system because the small grid spacing in the elements near the smaller black hole constrain the time step to be orders of magnitude smaller than near the larger black hole due to the CFL condition. The worldtube excision scheme avoids this by excising a much larger region (the worldtube) around the smaller black hole. Since the excision boundary no longer lies within the apparent horizon, boundary conditions are required. These are derived by approximating the solution inside the worldtube using a perturbative solution - a black hole perturbed by another black hole. The solution is calibrated by the evolved metric on the worldtube boundary and in turn provides boundary conditions to the evolution system.

Here, we test this scheme using a toy problem of a scalar charge in circular orbit around a Schwarzschild black hole.

Function Documentation

acceleration_terms_0()

void CurvedScalarWave::Worldtube::acceleration_terms_0	(	gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * >	result,
		const tnsr::I< DataVector, 3, Frame::Inertial > &	centered_coords,
		const tnsr::I< double, 3 > &	particle_position,
		const tnsr::I< double, 3 > &	particle_velocity,
		const tnsr::I< double, 3 > &	particle_acceleration,
		double	ft,
		double	fx,
		double	fy,
		double	dt_ft,
		double	dt_fx,
		double	dt_fy,
		double	bh_mass
	)

Computes the acceleration terms of a puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime up to zeroth order in coordinate distance.

Details

The appropriate expression can be found in Eq. (37) of [207]. The values ft, fx, fy are the time, x and y component of the self force per unit mass evaluated at the position of the particle; dt_ft, dt_fx, dt_fy are the respective total time derivatives. The code in this function was auto-generated by generating the full expressions with Mathematica and employing common subexpression elimination with sympy. The mathematica file and generating script can be found at https://github.com/nikwit/puncture-field.

acceleration_terms_1()

void CurvedScalarWave::Worldtube::acceleration_terms_1	(	gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * >	result,
		const tnsr::I< DataVector, 3, Frame::Inertial > &	centered_coords,
		const tnsr::I< double, 3 > &	particle_position,
		const tnsr::I< double, 3 > &	particle_velocity,
		const tnsr::I< double, 3 > &	particle_acceleration,
		double	ft,
		double	fx,
		double	fy,
		double	dt_ft,
		double	dt_fx,
		double	dt_fy,
		double	Du_ft,
		double	Du_fx,
		double	Du_fy,
		double	dt_Du_ft,
		double	dt_Du_fx,
		double	dt_Du_fy,
		double	bh_mass
	)

Computes the acceleration terms of a puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime up to first order in coordinate distance (i.e. zeroth and first order).

Details

The appropriate expression can be found in Eq. (37) of [207]. The values ft, fx, fy are the time, x and y component of the self force per unit mass evaluated at the position of the particle; dt_ft, dt_fx, dt_fy are the respective total time derivatives. The code in this function was auto-generated by generating the full expressions with Mathematica and employing common subexpression elimination with sympy. The mathematica file and generating script can be found at https://github.com/nikwit/puncture-field.

puncture_field()

void CurvedScalarWave::Worldtube::puncture_field	(	gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * >	result,
		const tnsr::I< DataVector, 3, Frame::Inertial > &	centered_coords,
		const tnsr::I< double, 3 > &	particle_position,
		const tnsr::I< double, 3 > &	particle_velocity,
		const tnsr::I< double, 3 > &	particle_acceleration,
		double	bh_mass,
		size_t	order
	)

Computes the puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime. described in [56].

Details

The field is computed using a Detweiler-Whiting singular Green's function and perturbatively expanded in the geodesic distance from the particle. It solves the inhomogeneous wave equation

\begin{align*} \Box \Psi^\mathcal{P} = -4 \pi q \int \sqrt{-g} \delta^4(x^i, z(\tau)) d \tau \end{align*}

where \(q\) is the scalar charge and \(z(\tau)\) is the worldline of the particle. The expression is expanded up to a certain order in geodesic distance and transformed to Kerr-Schild coordinates.

The function given here assumes that the particle has scalar charge \(q=1\) and is on a fixed geodesic orbit. It returns the singular field at the requested coordinates as well as its time and spatial derivative. For non-geodesic orbits, corresponding acceleration terms have to be added to the puncture field.

Note

The expressions were computed with Mathematica and optimized by applying common subexpression elimination with sympy. The memory allocations of temporaries were optimized manually.

puncture_field_0()

void CurvedScalarWave::Worldtube::puncture_field_0	(	gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * >	result,
		const tnsr::I< DataVector, 3, Frame::Inertial > &	centered_coords,
		const tnsr::I< double, 3 > &	particle_position,
		const tnsr::I< double, 3 > &	particle_velocity,
		const tnsr::I< double, 3 > &	particle_acceleration,
		double	bh_mass
	)

Computes the puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime. described in [56].

Details

The appropriate expression can be found in Eq. (36) of [207]. For non-geodesic orbits, there are additional contributions, see acceleration_terms_0.

puncture_field_1()

void CurvedScalarWave::Worldtube::puncture_field_1	(	gsl::not_null< Variables< tmpl::list< CurvedScalarWave::Tags::Psi, ::Tags::dt< CurvedScalarWave::Tags::Psi >, ::Tags::deriv< CurvedScalarWave::Tags::Psi, tmpl::size_t< 3 >, Frame::Inertial > > > * >	result,
		const tnsr::I< DataVector, 3, Frame::Inertial > &	centered_coords,
		const tnsr::I< double, 3 > &	particle_position,
		const tnsr::I< double, 3 > &	particle_velocity,
		const tnsr::I< double, 3 > &	particle_acceleration,
		double	bh_mass
	)

Computes the puncture/singular field \(\Psi^\mathcal{P}\) of a scalar charge on a generic orbit in Schwarzschild spacetime. described in [56].

Details

For non-geodesic orbits, there are additional contributions, see acceleration_terms_0.

self_force_acceleration() [1/2]

template<size_t Dim>

tnsr::I< double, Dim > CurvedScalarWave::Worldtube::self_force_acceleration	(	const Scalar< double > &	dt_psi_monopole,
		const tnsr::i< double, Dim > &	psi_dipole,
		const tnsr::I< double, Dim > &	particle_velocity,
		double	particle_charge,
		double	particle_mass,
		const tnsr::AA< double, Dim > &	inverse_metric,
		const Scalar< double > &	dilation_factor
	)

Computes the coordinate acceleration due to the scalar self-force onto the charge.

Details

It is given by

\begin{equation} (u^0)^2 \ddot{x}^i_p = \frac{q}{\mu}(g^{i \alpha} - \dot{x}^i_p g^{0 \alpha} ) \partial_\alpha \Psi^R \end{equation}

where \(\dot{x}^i_p\) is the position of the scalar charge, \(\Psi^R\) is the regular field, \(q\) is the particle's charge, \(\mu\) is the particle's mass, \(u^\alpha\) is the four-velocity and \(g^{\alpha \beta}\) is the inverse spacetime metric in the inertial frame, evaluated at the position of the particle. An overdot denotes a derivative with respect to coordinate time. Greek indices are spacetime indices and Latin indices are purely spatial. Note that the coordinate geodesic acceleration is NOT included.

self_force_acceleration() [2/2]

template<size_t Dim>

void CurvedScalarWave::Worldtube::self_force_acceleration	(	gsl::not_null< tnsr::I< double, Dim > * >	self_force_acc,
		const Scalar< double > &	dt_psi_monopole,
		const tnsr::i< double, Dim > &	psi_dipole,
		const tnsr::I< double, Dim > &	particle_velocity,
		double	particle_charge,
		double	particle_mass,
		const tnsr::AA< double, Dim > &	inverse_metric,
		const Scalar< double > &	dilation_factor
	)

Computes the coordinate acceleration due to the scalar self-force onto the charge.

Details

It is given by

\begin{equation} (u^0)^2 \ddot{x}^i_p = \frac{q}{\mu}(g^{i \alpha} - \dot{x}^i_p g^{0 \alpha} ) \partial_\alpha \Psi^R \end{equation}

where \(\dot{x}^i_p\) is the position of the scalar charge, \(\Psi^R\) is the regular field, \(q\) is the particle's charge, \(\mu\) is the particle's mass, \(u^\alpha\) is the four-velocity and \(g^{\alpha \beta}\) is the inverse spacetime metric in the inertial frame, evaluated at the position of the particle. An overdot denotes a derivative with respect to coordinate time. Greek indices are spacetime indices and Latin indices are purely spatial. Note that the coordinate geodesic acceleration is NOT included.

self_force_per_mass()

template<size_t Dim>

tnsr::A< double, Dim > CurvedScalarWave::Worldtube::self_force_per_mass	(	const tnsr::a< double, Dim > &	d_psi,
		const tnsr::A< double, Dim > &	four_velocity,
		double	particle_charge,
		double	particle_mass,
		const tnsr::AA< double, Dim > &	inverse_metric
	)

Computes the scalar self-force per unit mass.

Details

It is given by

\begin{equation} f^\alpha = \frac{q}{\mu} (g^{\alpha \beta} + u^\alpha u^\beta) \partial_\beta \Psi^R \end{equation}

where \(\Psi^R\) is the regular field at the position of the particle, \(q\) is the particle's charge, \(\mu\) is the particle's mass, \(u^\alpha\) is the four-velocity and \(g^{\alpha \beta}\) is the inverse spacetime metric in the inertial frame, evaluated at the position of the particle.

smooth_broken_power_law()

double CurvedScalarWave::Worldtube::smooth_broken_power_law	(	double	orbit_radius,
		double	alpha,
		double	radius_at_inf,
		double	rb,
		double	delta
	)

A smoothly broken power law that falls off to a constant value for larger radii.

Details

The function is given by Eq. (3) of [206]

\begin{equation} f(r) = R_\infty \left(\frac{r}{r_b}\right)^{\alpha} \left( 1 + \left(\frac{r}{r_b}\right)^{1 / \Delta}\right)^{-\alpha\Delta}. \end{equation}

For radii \(r \ll r_b\), the function obeys the power law \(f(r) \propto r^{\alpha}\). For radii \(r \gg r_b\), the function asymptotes to \(R_\infty\). The parameter \(\Delta\) determines the width of the transition region with a larger value of \(\Delta\) leading to a more gradual transition.

This function is used to control the worldtube radius for more eccentric orbits so the radius does not grow too large during the apoapsis passage as this does not lead to performance gains and can cause problems with the domain.

turn_on_function()

double CurvedScalarWave::Worldtube::turn_on_function	(	double	t_minus_turn_on,
		double	turn_on_timescale
	)

A function used to roll-on the self-force continuously from 0 to 1.

Details

It is given by Eq.(60) of [207]

\begin{equation} w(t) = 1 - \exp \left( - \left( (t - t_{\mathrm{turn_on}}) / \sigma \right)^4 \right), \end{equation}

where \(t\) is the current simulation time, \(t_{\mathrm{turn_on}}\) is the time where the self-force is turned on and \(\sigma\) dictates the timescale over which it is turned on. The function is \(\mathcal{C}^3\), i.e. three times continuously differentiable, assuming \(w(t) = 0\) for \(t < t_{\mathrm{turn_on}}\).