Items related to solving the puncture equation. More...

Namespaces
namespace	Tags
	Tags related to the puncture equation.

Classes
struct	FirstOrderSystem
	The puncture equation, formulated as a set of coupled first-order partial differential equations. More...
struct	LinearizedSources
	The linearization of the sources \(S\) for the first-order formulation of the puncture equation. More...
struct	Sources
	The sources \(S\) for the first-order formulation of the puncture equation. More...

Functions
void	add_sources (gsl::not_null< Scalar< DataVector > * > puncture_equation, const Scalar< DataVector > &alpha, const Scalar< DataVector > &beta, const Scalar< DataVector > &field)
	Add the nonlinear sources for the puncture equation.
void	add_linearized_sources (gsl::not_null< Scalar< DataVector > * > linearized_puncture_equation, const Scalar< DataVector > &alpha, const Scalar< DataVector > &beta, const Scalar< DataVector > &field, const Scalar< DataVector > &field_correction)
	Add the linearized sources for the puncture equation.

void	adm_mass_integrand (const gsl::not_null< Scalar< DataVector > * > result, const Scalar< DataVector > &field, const Scalar< DataVector > &alpha, const Scalar< DataVector > &beta)
	The volume integrand for the ADM mass \(M_\mathrm{ADM}\).
Scalar< DataVector >	adm_mass_integrand (const Scalar< DataVector > &field, const Scalar< DataVector > &alpha, const Scalar< DataVector > &beta)
	The volume integrand for the ADM mass \(M_\mathrm{ADM}\).

Detailed Description

Items related to solving the puncture equation.

The puncture equation

\begin{equation}\label{eq:puncture_eqn} -\nabla^2 u = \beta \left(\alpha \left(1 + u\right) + 1\right)^{-7} \end{equation}

is a nonlinear Poisson-type elliptic PDE for the "puncture field" \(u\). See Eq. (12.52) and surrounding discussion in [14], or [31] for an introduction. To arrive at the puncture equation we assume conformal flatness and maximal slicing in vacuum so the Einstein momentum constraint becomes homogeneous:

\begin{equation}\label{eq:mom_constraint} \nabla_j \bar{A}^{ij} = 0 \end{equation}

Here, \(\nabla\) is the flat-space covariant derivate. \(\bar{A}^{ij}\) is the conformal traceless extrinsic curvature that composes the extrinsic curvature as

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

(where \(K=0\) under maximal slicing). \(\psi\) is the conformal factor that composes the spatial metric as

\begin{equation}\gamma_{ij} = \psi^4 \bar{\gamma}_{ij} \end{equation}

(where \(\bar{\gamma}_{ij} = \delta{ij}\) under conformal flatness and in Cartesian coordinates).

The momentum constraint ( \(\ref{eq:mom_constraint}\)) is solved analytically by the Bowen-York extrinsic curvature

\begin{equation}\bar{A}^{ij} = \frac{3}{2} \frac{1}{r_C^2} \left( 2 P^{(i} n^{j)} - (\delta^{ij} - n^i n^j) P^k n^k + \frac{4}{r_C} n^{(i} \epsilon^{j)kl} S^k n^l\right) \end{equation}

representing a black hole with linear momentum \(\mathbf{P}\) and angular momentum \(\mathbf{S}\) at position \(\mathbf{C}\). The quantity \(r_C=||\mathbf{x}-\mathbf{C}||\) is the Euclidean coordinate distance to the black hole, and \(\mathbf{n}=(\mathbf{x}-\mathbf{C})/r_C\) is the radial unit normal to the black hole. Since the momentum constraint is linear, any superposition of \(\bar{A}^{ij}\) is also a solution to the momentum constraint, allowing to represent multiple black holes.

Only the Einstein Hamiltonian constraint remains to be solved numerically for the conformal factor:

\begin{equation}\nabla^2 \psi = \frac{1}{8} \psi^{-7} \bar{A}_{ij} \bar{A}^{ij} \end{equation}

It reduces to the puncture equation ( \(\ref{eq:puncture_eqn}\)) when we decompose the conformal factor as:

\begin{equation}\psi = 1 + \frac{1}{\alpha} + u \end{equation}

where we define

\begin{equation}\frac{1}{\alpha} = \sum_I \frac{M_I}{2 r_I} \end{equation}

and

\begin{equation}\beta = \frac{1}{8} \alpha^7 \bar{A}_{ij} \bar{A}^{ij}. \end{equation}

Here, \(M_I\) is the "puncture mass" (or "bare mass") parameter for the \(I\)th black hole at position \(\mathbf{C}_I\), and \(\bar{A}_{ij}\) is the superposition of the Bowen-York extrinsic curvature of the black holes with the parameters defined above. Note that the definition of \(\frac{1}{\alpha}\) in Eq. (12.51) in [14] is missing factors of \(\frac{1}{2}\), but their Eq. (3.23) includes them, as does Eq. (8) in [31] (though the latter includes the unit offset in the definition of \(u\)).

Function Documentation

◆ add_linearized_sources()

void Punctures::add_linearized_sources	(	gsl::not_null< Scalar< DataVector > * >	linearized_puncture_equation,
		const Scalar< DataVector > &	alpha,
		const Scalar< DataVector > &	beta,
		const Scalar< DataVector > &	field,
		const Scalar< DataVector > &	field_correction )

Add the linearized sources for the puncture equation.

Adds \(-\frac{d}{du}(\beta \left(\alpha \left(1 + u\right) + 1\right)^{-7})\).

See also: Punctures

◆ add_sources()

void Punctures::add_sources	(	gsl::not_null< Scalar< DataVector > * >	puncture_equation,
		const Scalar< DataVector > &	alpha,
		const Scalar< DataVector > &	beta,
		const Scalar< DataVector > &	field )

Add the nonlinear sources for the puncture equation.

Adds \(-\beta \left(\alpha \left(1 + u\right) + 1\right)^{-7}\).

See also: Punctures

◆ adm_mass_integrand() [1/2]

void Punctures::adm_mass_integrand	(	const gsl::not_null< Scalar< DataVector > * >	result,
		const Scalar< DataVector > &	field,
		const Scalar< DataVector > &	alpha,
		const Scalar< DataVector > &	beta )

The volume integrand for the ADM mass \(M_\mathrm{ADM}\).

The ADM mass for Punctures is (Eq. (12.56) in [14])

\begin{equation}M_\mathrm{ADM} = \sum_I M_I + \frac{1}{2\pi} \int \beta \left(\alpha \left(1 + u\right) + 1\right)^{-7} \end{equation}

See also: Punctures

◆ adm_mass_integrand() [2/2]

Scalar< DataVector > Punctures::adm_mass_integrand	(	const Scalar< DataVector > &	field,
		const Scalar< DataVector > &	alpha,
		const Scalar< DataVector > &	beta )