StaticConformal

Tom Goodale et al

Date

Abstract

Base thorn to provide the variables for the static conformal factor

1 Purpose

This thorn provides the variables defining a static conformal factor which is used to transform the physical metric. If this thorn is active and the ADMBase::metric_type parameter is set to static conformal, then the ADMBase::g... variables are the conformal values as opposed to the physical values.

The transformation is

g_{i j}^{physical} = ψ^{4} g_{i j}^{conformal}

The extrinsic curvature is not transformed.

Memory is provided for the conformal factor psi, its first derivatives psix, psiy, psiz, and its second derivatives psixx, psixy, psixz, psiyy, psiyz, and psizz depending on the setting of the conformal_storage parameter.

Note that the first and second “derivative” grid functions have an additional factor of $1 ∕ ψ$ normalisation since this is the most common use of the derivative. I.e., the grid functions are

\begin{array}{rcl} p s i & = & ψ, \\ p s i x & = & ψ_{x} ∕ ψ, etc \\ p s i x x & = & ψ_{i j} ∕ ψ etc \end{array}

Thorns need to check the value of the grid scalar conformal_state to determine how many levels of these variables have actually been calculated before using the conformal factor:

conformal_state=0: No conformal factor has been calculated — thorns may assume the conformal factor is 1 at all points. (I.e., the metric is physical.)
conformal_state=1: The conformal factor has been calulated, but no derivatives.
conformal_state=2: The conformal factor and its first derivatives have been calculated.
conformal_state=3: The conformal factor and its first and second derivatives have been calculated.

Note that this means that if you only want to know whether psi contains the values for the conformal factor you can check for conformal_state > 0.

2 Utilities

StaticConformal provides aliased functions to convert between physical and conformal 3-metric values. It is very important to understand that these functions apply the conversion in place. That is, if gxx contains the conformal metric value, when the routine is exited it will now contain the physical metric value. These functions do not change the value of conformal_state and should be used with due care. (These functions are for example used by some analysis thorns who work only with the physical metric, they apply the transformation on entry to the analysis routine and switch it back on exit).

Convert from conformal to physical:: subroutine ConfToPhysInPlace (nx, ny, nz,
                              psi,
                              gxx, gxy, gxz, gyy, gyz, gzz)
   implicit none
   CCTK_INT,                         intent(in)    :: nx, ny, nz
   CCTK_REAL, dimension(nx, ny, nz), intent(in)    :: psi
   CCTK_REAL, dimension(nx, ny, nz), intent(inout) :: gxx, gxy, gxz, gyy, gyz, gzz
end subroutine ConfToPhysInPlace
Convert from physical to conformal:: subroutine PhysToConfInPlace (nx, ny, nz,
                              psi,
                              gxx, gxy, gxz, gyy, gyz, gzz)
   implicit none
   CCTK_INT,                         intent(in)    :: nx, ny, nz
   CCTK_REAL, dimension(nx, ny, nz), intent(in)    :: psi
   CCTK_REAL, dimension(nx, ny, nz), intent(inout) :: gxx, gxy, gxz, gyy, gyz, gzz
end subroutine ConfToPhysInPlace

3 Comments

The StaticConformal thorn itself does not calculate any conformal factor, but does initialise the conformal_state variable to 0.

4 Parameters


conformal_storage	Scope: restricted	KEYWORD

Description: How much conformal storage do we have ?

Range		Default: factor+derivs+2nd derivs
factor	Just the conformal factor
factor+derivs	Conformal factor plus first derivatives
see [1] below	Conformal factor plus first and second derivatives

[1]

factor+derivs+2nd derivs


metric_type	Scope: shared from ADMBASE	KEYWORD

Extends ranges:

static conformal	Metric is conformal with static conformal factor, extrinsic curvature is physical

5 Interfaces

General

Implements:

staticconformal

Inherits:

grid

Grid Variables

5.0.1 PUBLIC GROUPS


Group Names	Variable Names	Details

conformal_state	conformal_state	compact	0
		dimensions	0
		distribution	CONSTANT
		group type	SCALAR
		timelevels	1
		variable type	INT

confac		compact	0
	psi	description	Conformal factor
		dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar” Prolongation=”None” InterpNumTimelevels=1
		timelevels	1
		variable type	REAL

confac_1derivs		compact	0
	psix	description	First spatial derivatives of conformal factor divided by psi
	psiy	dimensions	3
	psiz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”U” Prolongation=”None” InterpNumTimelevels=1
		timelevels	1
		variable type	REAL

confac_2derivs		compact	0
	psixx	description	Second spatial derivatives of conformal factor divided by psi
	psixy	dimensions	3
	psixz	distribution	DEFAULT
	psiyy	group type	GF
	psiyz	tags	tensortypealias=”dd_sym” Prolongation=”None” InterpNumTimelevels=1
	psizz	timelevels	1
		variable type	REAL

Provides:

ConfToPhysInPlace to

PhysToConfInPlace to

6 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinBase/StaticConformal. 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.

Storage

Always:	Conditional:
conformal_state	confac[1]
	confac[1] confac_1derivs[1]
	confac[1] confac_1derivs[1] confac_2derivs[1]

Scheduled Functions

CCTK_INITIAL

staticconformal_initialisestate

set the conformal_state variable to 0

	Before:	admbase_initialdata
	Language:	c
	Type:	function
	Writes:	staticconformal::conformal_state(everywhere)