ADMAnalysis

The trace of the extrinsic curvature at each point on the grid is placed in the grid function trK. The algorithm for calculating the trace uses the physical metric, that is it includes any conformal factor.

\begin {equation} {\tt trK} \equiv tr K = \frac {1}{\psi ^4} g^{ij} K_{ij} \end {equation}

3 Determinant of 3-Metric

The determinant of the 3-metric at each point on the grid is placed in the grid function detg. This is always the determinant of the conformal metric, that is it does not include any conformal factor.

\begin {equation} {\tt detg} \equiv det g = -g_{13}^2*g_{22}+2*g_{12}*g_{13}*g_{23}-g_{11}*g_{23}^2- g_{12}^2*g_{33}+g_{11}*g_{22}*g_{33} \end {equation}

4 Transformation to Spherical Cooordinates

The values of the metric and/or extrinsic curvature in a spherical polar coordinate system \((r,\theta ,\phi )\) evaluated at each point on the computational grid are placed in the grid functions (grr, grt, grp, gtt, gtp, gpp) and (krr, krt, krp, ktt, ktp, kpp). In the spherical transformation, the \(\theta \) coordinate is referred to as q and the \(\phi \) as p.

\begin {eqnarray*} A_{rr}&=& \sin ^2\theta \cos ^2\phi A_{xx} +\sin ^2\theta \sin ^2\phi A_{yy} +\cos ^2\theta A_{zz} +2\sin ^2\theta \cos \phi \sin \phi A_{xy} \\ && +2\sin \theta \cos \theta \cos \phi A_{xz} +2\sin \theta \cos \theta \sin \phi A_{yz} \\ A_{r\theta }&=& r(\sin \theta \cos \theta \cos ^2\phi A_{xx} +2*\sin \theta \cos \theta \sin \phi \cos \phi A_{xy} +(\cos ^2\theta -\sin ^2\theta )\cos \phi A_{xz} \\ && +\sin \theta \cos \theta \sin ^2\phi A_{yy} +(\cos ^2\theta -\sin ^2\theta )\sin \phi A_{yz} -\sin \theta \cos \theta A_{zz}) \\ A_{r\phi }&=& r\sin \theta (-\sin \theta \sin \phi \cos \phi A_{xx} -\sin \theta (\sin ^2\phi -\cos ^2\phi )A_{xy} -\cos \theta \sin \phi A_{xz} \\ && +\sin \theta \sin \phi \cos \phi A_{yy} +\cos \theta \cos \phi A_{yz}) \\ A_{\theta \theta }&=& r^2(\cos ^2\theta \cos ^2\phi A_{xx} +2\cos ^2\theta \sin \phi \cos \phi A_{xy} -2\sin \theta \cos \theta \cos \phi A_{xz} +\cos ^2\theta \sin ^2\phi A_{yy} \\ && -2\sin \theta \cos \theta \sin \phi A_{yz} +\sin ^2\theta A_{zz}) \\ A_{\theta \phi }&=& r^2\sin \theta (-\cos \theta \sin \phi \cos \phi A_{xx} -\cos \theta (\sin ^2\phi -\cos ^2\phi )A_{xy} +\sin \theta \sin \phi A_{xz} \\ && +\cos \theta \sin \phi \cos \phi A_{yy} -\sin \theta \cos \phi A_{yz}) \\ A_{\phi \phi }&=& r^2\sin ^2\theta (\sin ^2\phi A_{xx} -2\sin \phi \cos \phi A_{xy} +\cos ^2\phi A_{yy}) \end {eqnarray*}

If the parameter normalize_dtheta_dphi is set to yes, the angular components are projected onto the vectors \((r d\theta , r \sin \theta d \phi )\) instead of the default vector \((d \theta , d\phi )\). That is,

\begin {eqnarray*} A_{\theta \theta } & \rightarrow & A_{\theta \theta }/r^2 \\ A_{\phi \phi }& \rightarrow & A_{\phi \phi }/(r^2\sin ^2\theta ) \\ A_{r\theta } & \rightarrow & A_{r\theta }/r \\ A_{r\phi } & \rightarrow & A_{r\phi }/(r\sin \theta ) \\ A_{\theta \phi } & \rightarrow & A_{\theta \phi }/r^2\sin \theta ) \end {eqnarray*}

5 Computing the Ricci tensor and scalar

The computation of the Ricci tensor uses the ADMMacros thorn. The calculation of the Ricci scalar uses the generic trace routine in this thorn.

6 Parameters

7 Interfaces

General

Grid Variables

7.0.1 PUBLIC GROUPS

8 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinAnalysis/ADMAnalysis. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.


normalize_dtheta_dphi	Scope: private	BOOLEAN

Description: Project angular components onto rdtheta and rsin(theta)*dphi?

		Default: no


ricci_persist	Scope: private	BOOLEAN

Description: Keep storage of the Ricci tensor and scalar around?

		Default: no


ricci_prolongation_type	Scope: private	KEYWORD

Description: The kind of boundary prolongation for the Ricci tensor and scalar

Range		Default: none
Lagrange	standard prolongation (requires several time levels)
copy	use data from the current time level (requires only one time level)
none	no prolongation (use this if you do not have enough time levels active)


ricci_timelevels	Scope: private	INT

Description: Number of time levels for the Ricci tensor and scalar

Range		Default: 1
1:3


Group Names	Variable Names	Details

trace_of_k		compact	0
	trK	description	trace of extrinsic curvature
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”none”
		timelevels	1
		variable type	REAL

detofg		compact	0
	detg	description	determinant of the conformal metric
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” Prolongation=”none”
		timelevels	1
		variable type	REAL

spherical_metric		compact	0
	grr	description	Metric in spherical coordinates
	gqq	dimensions	3
	gpp	distribution	DEFAULT
	grq	group type	GF
	grp	tags	Prolongation=”none”
	gqp	timelevels	1
		variable type	REAL

spherical_curv		compact	0
	krr	description	extrinsic curvature in spherical coordinates
	kqq	dimensions	3
	kpp	distribution	DEFAULT
	krq	group type	GF
	krp	tags	Prolongation=”none”
	kqp	timelevels	1
		variable type	REAL

ricci_tensor		compact	0
	Ricci11	description	Components of the Ricci tensor
	Ricci12	dimensions	3
	Ricci13	distribution	DEFAULT
	Ricci22	group type	GF
	Ricci23	tags	tensortypealias=”dd_sym” ProlongationParameter=”ADMAnalysis::ricci_prolongation_type”
	Ricci33	timelevels	3
		variable type	REAL

ricci_scalar		compact	0
	Ricci	description	The Ricci scalar
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”scalar” ProlongationParameter=”ADMAnalysis::ricci_prolongation_type”
		timelevels	3
		variable type	REAL

ADMAnalysis

Abstract

1 Purpose

2 Trace of Extrinsic Curvature