xx ADMAnalysis

ADMAnalysis

Tom Goodale et al

Date

Abstract

Basic analysis of the metric and extrinsic curvature tensors

1 Purpose This thorn provides analysis routines to calculate the following quantities:

The trace of the extrinsic curvature ().

The determinant of the 3-metric ().

The components of the 3-metric in spherical coordinates ().

The components of the extrinsic curvature in spherical coordinates ().

The components of the 3-Ricci tensor in cartesian coordinates () for .

The Ricci scalar ().

2 Trace of Extrinsic Curvature

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.

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.

4 Transformation to Spherical Cooordinates

The values of the metric and/or extrinsic curvature in a spherical polar coordinate system 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 coordinate is referred to as q and the as p.

The general transformation from Cartesian to Spherical for such tensors is

If the parameter normalizexdthetaxdphi is set to yes, the angular components are projected onto the vectors instead of the default vector . That is,

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

11normalizexdthetaxdphi 10Scope: privatex01  BOOLEAN

11Description: Project angular components onto r*dtheta and r*sin(theta)*dphi?

11                  10  x01  Default: no

11

11riccixpersist  10Scope: privatex01  BOOLEAN

11Description: Keep storage of the Ricci tensor and scalar around?

11                  10  x01  Default: no

11riccixprolongationxtype 10Scope: privatex01  KEYWORD

11Description: The kind of boundary prolongation for the Ricci tensor and scalar

11 Range 10 x01 Default: none
11Lagrange  10standard prolongation (requires several time levels)
11copy  10use data from the current time level (requires only one time level)
11none  10no prolongation (use this if you do not have enough time levels active)

11

11riccixtimelevels  10Scope: privatex01  INT

11Description: Number of time levels for the Ricci tensor and scalar

11  Range  10  x01  Default: 1

111:3  10

11

11metricxtype  10Scope: shared from ADMBASEx01KEYWORD

11

7 Interfaces

General Implements: admanalysis

Inherits: admbase staticconformal grid admmacros

Grid Variables

7.0.1 PUBLIC GROUPS

11x Group Names x10x Variable Names xx01 Details x11x
11  tracexofxk  10  x01  compact110
11                 10  trK  x01 description11trace of extrinsic curvature
11                 10  x01 dimensions113
11                 10  x01 distribution11DEFAULT
11                 10  x01 group type11GF
11 10 x01 tags11tensortypealias=”scalar” Prolongation=”none”
11                 10  x01 timelevels111
11                 10  x01variable type11REAL
11  detofg  10  x01  compact110
11                 10  detg  x01 description11determinant of the conformal metric
11                 10  x01 dimensions113
11                 10  x01 distribution11DEFAULT
11                 10  x01 group type11GF
11 10 x01 tags11tensortypealias=”scalar” Prolongation=”none”
11                 10  x01 timelevels111
11                 10  x01variable type11REAL
11 sphericalxmetric 10  x01  compact110
11                 10  grr  x01 description11Metric in spherical coordinates
11                 10  gqq  x01 dimensions113
11                 10  gpp  x01 distribution11DEFAULT
11                 10  grq  x01 group type11GF
11                 10  grp  x01  tags11Prolongation=”none”
11                 10  gqp  x01 timelevels111
11                 10  x01variable type11REAL
11 sphericalxcurv 10  x01  compact110
11                 10  krr  x01 description11extrinsic curvature in spherical coordinates
11                 10  kqq  x01 dimensions113
11                 10  kpp  x01 distribution11DEFAULT
11                 10  krq  x01 group type11GF
11                 10  krp  x01  tags11Prolongation=”none”
11                 10  kqp  x01 timelevels111
11                 10  x01variable type11REAL
11  riccixtensor  10  x01  compact110
11                 10  Ricci11  x01 description11Components of the Ricci tensor
11                 10  Ricci12  x01 dimensions113
11                 10  Ricci13  x01 distribution11DEFAULT
11                 10  Ricci22  x01 group type11GF
11                 10  Ricci23  x01  tags11tensortypealias=”ddxsym” ProlongationParameter=”ADMAnalysis::riccixprolongationxtype”
11                 10  Ricci33  x01 timelevels113
11                 10  x01variable type11REAL
11  riccixscalar  10  x01  compact110
11                 10  Ricci  x01 description11The Ricci scalar
11                 10  x01 dimensions113
11                 10  x01 distribution11DEFAULT
11                 10  x01 group type11GF
11 10 x01 tags11tensortypealias=”scalar” ProlongationParameter=”ADMAnalysis::riccixprolongationxtype”
11                 10  x01 timelevels113
11                 10  x01variable type11REAL

11

Uses header: Symmetry.h

Provides: CartToSphere to Trace to

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.

Storage x

11x11Conditional:
11x 11riccixtensor[1] riccixscalar[1]
11x 11riccixtensor[2] riccixscalar[2]
11x 11riccixtensor[3] riccixscalar[3]

11x 11x

11

Scheduled Functions

CCTKxPARAMCHECK x admanalysisxparamcheck x check that the metricxtype is recognised x

11x11Language:11c
11 x 11Options: 11global
11 x 11Type: 11function

11

CCTKxWRAGH (conditional) x admanalysisxregistersymmetry x register symmetry of ricci tensor and scalar x

11x11Language:11c
11 x 11Options: 11global
11 x 11Type: 11function

11

CCTKxANALYSIS (conditional) x riccigroup x calculate ricci tensor, with boundary conditions x

11x11Storage: 11riccixtensor[1]
11 x 11x  11riccixscalar[1]
11 x 11x  11detofg
11 x 11Triggers:11riccixtensor
11 x 11x  11riccixscalar
11 x 11Type: 11group

RicciGroup x admanalysisxricci x calculate ricci tensor, with boundary conditions x

11x11Language:11c
11 x 11Reads: 11admbase::metric
11 x 11Type: 11function
11 x 11Writes: 11detg(everywhere)
11 x 11x  11riccixtensor(everywhere)
11 x 11x  11riccixscalar(everywhere)

11

RicciGroup x ricciboundariesgroup x set ricci tensor on the boundary x

11x11After:11admanalysisxricci
11 x 11Type:11group

11

RicciBoundariesGroup x admanalysisxriccixboundaries x select boundary conditions for the ricci tensor x

11x11Language:11c
11 x 11Options: 11level
11 x 11Sync:  11riccixtensor
11 x 11x  11riccixscalar
11 x 11Type: 11function

11

RicciBoundariesGroup x applybcs x apply boundary conditions to the ricci tensor x

11x11After:11admanalysisxricci
11 x 11Type:11group

11

CCTKxANALYSIS x admanalysisxevaltrk x compute the trace of the extrinsic curvature and the determinant of the metric

11x11Language:11c
11 x 11Reads: 11admbase::metric
11 x 11x  11admbase::curv
11 x 11x  11staticconformal::psi
11 x 11x  11staticconformal::conformalxstate
11 x 11Storage: 11tracexofxk
11 x 11x  11detofg
11 x 11Sync:  11tracexofxk
11 x 11x  11detofg
11 x 11Triggers: 11tracexofxk
11 x 11x  11detofg
11 x 11Type: 11function
11 x 11Writes: 11trk(everywhere)
11 x 11x  11detg(everywhere)

11

CCTKxANALYSIS x admanalysisxmetriccarttosphere x calculate the spherical metric in r,theta(q), phi(p) x

11x11Language:11c
11 x 11Reads: 11grid::coordinates
11 x 11x  11admbase::metric
11 x 11Storage: 11sphericalxmetric
11 x 11Sync:  11sphericalxmetric
11 x 11Triggers: 11sphericalxmetric
11 x 11Type: 11function
11 x 11Writes: 11sphericalxmetric(everywhere)

11

CCTKxANALYSIS x admanalysisxcurvcarttosphere x calculate the spherical ex. curvature in r, theta(q), phi(p) x

11x11Language:11c
11 x 11Reads: 11grid::coordinates
11 x 11x  11admbase::curv
11 x 11Storage: 11sphericalxcurv
11 x 11Sync:  11sphericalxcurv
11 x 11Triggers: 11sphericalxcurv
11 x 11Type: 11function
11 x 11Writes: 11sphericalxcurv(everywhere)

11

CCTKxPOSTINITIAL (conditional) x riccigroup x calculate ricci tensor, with boundary conditions x

11x11After: 11molxpoststep
11 x 11x  11molxpostinitial
11 x 11Storage:11detofg
11 x 11Type: 11group

11

CCTKxPOSTREGRID (conditional) x ricciboundariesgroup x set ricci tensor on the boundary

11x11Storage:11detofg
11 x 11Type: 11group

11

CCTKxPOSTRESTRICT (conditional) x ricciboundariesgroup x set ricci tensor on the boundary x

11x11Type:11group

11

CCTKxPOSTxRECOVERxVARIABLES (conditional) x riccigroup x calculate ricci tensor, with boundary conditions x

11x11After: 11molxpoststep
11 x 11x  11molxpostinitial
11 x 11Storage:11detofg
11 x 11Type: 11group

11

CCTKxEVOL (conditional) x riccigroup x calculate ricci tensor, with boundary conditions x

11x11After: 11molxevolution
11 x 11Storage:11detofg
11 x 11Type: 11group

11

Aliased Functions x

11Alias Name: xxxxxxx11Function Name:
11ApplyBCs  11ADMAnalysisxApplyBCs