xx QuasiLocalMeasures

QuasiLocalMeasures

Erik Schnetter <schnetter@cct.lsu.edu> Roland Haas <rhaas@ncsa.illinois.edu>

2017-03-21

Abstract

Calculate quasi-local measures such as masses, momenta, or angular momenta and related quantities on closed two-dimentional surfaces, including on horizons.

1 Introduction

The module QuasiLocalMeasures implements the calculation of mass and spin multipoles from the isolated and dynamical horizon formalism [1,x2], as well as a number of other proposed formulæ for quasilocal mass, linear momentum and angular momentum that have been advanced over the years [3]. Even though there are only a few rigorous proofs that establish the properties of these latter quantities, they have been demonstrated to be surprisingly helpful in numerical simulations (see, e.g., [4]), and are therefore an indispensable tool in numerical relativity. QuasiLocalMeasures takes as input a horizon surface, or any other surface that the user specifies (like a large coordinate sphere) and can calculate useful quantities such as the Weyl or Ricci scalars or the three-volume element of the horizon world tube in addition to physical observables such as mass and momenta.

2 Physical System

The Weyl tensor is defined as the traceless part of the Riemann tensor

where , and are the Christoffel symbols, Ricci tensor and Riemann scalar respectivelyx[6]. Its self-dual is

The curvature invariants (qlmxi) and (qlmxj) are defined in terms of the self-dual of the Weyl tensorx[5]

3 A note on evaluating 3D integrals on the horizon world tube

[NOTE: Ignore the stuff below. You can do that much easier.]

3.1 Integral transformation

The papers about dynamical horizons contain integrals over the 3D horizon world tube, expressed e.g.xas

where is some quantity that lives on the horizon. These integrals have to be transformed into a form so that they can be conveniently evaluated, e.g.xas

where is the area element on the horizon cross section contained in , and is the coordinate time differential. The factor should contain the extra terms due to this coordinate transformation.

Starting from the -volume element , let us first decompose it into the -volume element and a “time” coordinate on the horizon, which we call . Note that will generally be a spacelike coordinate for dynamical horizons. Let be the induced -metric on the horizon, and be the induced -metric on the cross section. Then it is

because . The coordinate time differential and the differential will in general not be aligned because the horizon world tube will in general not have a static coordinate shape. It is where is a radial coordinate perpendicular to the horizon and also perpendicular to , and is perpendicular to and lies in the plan spanned by and . and are depend on and via a Lorentz boost. Thus we have

Putting everything together we arrive at

3.2 The “lapse” function Starting from we find, since the radius changes only in the direction, If we assume and write , and use the relations between and from above, we get [NOTE: but .] and therefore Additionally we have where is the unit vector in the direction, i.e.x

3.3 Special Behaviour In order to use the QuasiLocalMeasures thorn on existing data (postprocessing), the following procedure is necessary.

Computing time-independent quantities.

The 3-metric and the extrinsic curvature must be available in HDF5 files.

Set up a parameter file for a grid structure that contains the region around the horizon. The refinement level structure and grid spacing etc. needs to be the same as in the HDF5 files, but the grids can be much smaller. You can also leave out some finer grids, i.e., reduce the number of levels. However, the coarse grid spacing must remain the same. The symmetries must also be the same.

Use the file reader and thorn AEIThorns/IDFileADM to read in the ADM variables from the files. The parameter file does not need to activate BSSNxMoL or any time evolution mechanism. IDFileADM acts as provider for initial data, so you don’t need any other initial data either.

Set up your parameter file so that the AH finder runs, stores the horizon shape in SphericalSurface, and QuasiLocalMeasures accesses these data.

This gives you the time-independent variables on the horizon, i.e., mostly the spin. It also allows you to look for apparent horizons if you don’t know where they are.

Computing time-independent quantities, e.g.x3-determinant of the horizon

Performing some time steps is necessary. Either read in lapse and shift from files, or set them arbitrarily (e.g.xlapse one, shift zero).

Activate a time evolution thorn, i.e., BSSNxMoL, MoL, Time, etc. In order to fill the past time levels, just choose MoL::initialxdataxisxcrap. If you have hydrodynamics, read in the hydro variables as well.

Only two time steps are required. Remember that the output of QuasiLocalMeasures for iteration 0 and 1 are incorrect or very inaccurate, since the past time levels are not correct, and hence the time derivatives that QuasiLocalMeasures calculates are wrong. However, iteration 2 should be good. (One could also perform 5 iterations and cross-check.)

If data for the extrinsic curvature is not available, but those for the 3-metric, lapse, and shift for consecutive time steps are (that is, if you have data suitable for finding event horizons), then one needs to reconstruct the extrinsic curvature first. There is a thorn AEIThorns/CalcK that helps with that. It reads the data for the 4-metric timestep after timestep, calculates the time derivative of the 3-metric through finite differencing in time, and then determines the extrinsic curvature from that, and writes it to a file. Once you have it, you can go on as above. CalcK has a small shell script that tells you what to do.

In general, things become more interesting if a static conformal factor is involved (since more variables are present), especially if it is output only once (since it is static), which means that one has to mix variables from different time steps.

The thorns involved in this procedure have some examples. In general, this is NOT a “just do it” action; you have to know what you are doing, since you have to put the pieces together in your parameter file and make sure that everything is consistent. We may have a vision that you just call a script in a directory that contains output files and the script figures out everything else, but we’re not there yet. All the ingredients are there, but you’ll have to put them together in the right way. Think Lego.

4 Interpreting 2D output

2D output is given on a rectangular grid. This grid has coordinates which are regular and have a constant spacing in the and directions. Cactus output has only grid point indices, but does not contain the coordinates and themselves.

In gnuplot, one can define functions to convert indices to coordinates: where and is the number of ghost points in the corresponding direction, and and the number of interior points. Here are the same equations in gnuplot syntax:

theta(i)x=x(ix-xnghostsx+x0.5)x*xpix/xntheta
phi(j)x=x(jx-xnghosts)x*x2*pix/xnphi

Usually, nghosts=2, ntheta=35, and nphi=72. i and j are is the integer grid point indices. Note that ntheta and nphi in the parameter file include ghost zones, while their definitions here do not include them. In general, nphi is even and ntheta is odd, because the points are staggered about the poles.

A test plot shows whether the plot is symmetric about in the and in the direction. Also, plotting something axisymmetric with bitant symmetry vs.x and vs.x, and vs.x and , should lie exactly on top of each other.

There are also scalars origin/delta_theta/phi which one can use in the above equations. Then the equations read

theta(i)x=x(ix+xorigin_theta)x*xdelta_theta
phi(j)x=x(jx+xorigin_phi)x*xdelta_phi

but, of course, these four quantities are all irrational and don’t look nice.

References

[1]xxxO.xDreyer, B.xKrishnan, D.xShoemaker and E.xSchnetter, Phys.xRev.xD 67, 024018 (2003) doi:10.1103/PhysRevD.67.024018 [gr-qc/0206008].

[2]xxxE.xSchnetter, B.xKrishnan and F.xBeyer, Phys.xRev.xD 74, 024028 (2006) doi:10.1103/PhysRevD.74.024028 [gr-qc/0604015].

[3]xxxL.xB.xSzabados, Living Rev.xRel.x7, 4 (2004).

[4]xxxG.xLovelace et al., Phys.xRev.xD 82, 064031 (2010) doi:10.1103/PhysRevD.82.064031 [arXiv:0907.0869 [gr-qc]].

[5]xxxJ.xG.xBaker, M.xCampanelli and C.xO.xLousto, Phys.xRev.xD 65, 044001 (2002) doi:10.1103/PhysRevD.65.044001 [gr-qc/0104063].

[6]xxxC.xW.xMisner, K.xS.xThorne and J.xA.xWheeler, San Francisco 1973, 1279p 5 Parameters

11beginxqlmxcalculationsxafter 10Scope: privatex01  REAL

11Description: when should we start calculations?

11 Range 10 x01 Default: 0.0
11*:*  10at/after this time (inclusively)

11

11coordsystem  10Scope: privatex01  STRING

11Description: The coordinate system to use

11 Range 10 x01 Default: cart3d
11                      10must be a registered coordinate system

11

11interpolator  10Scope: privatex01  STRING

11Description: The interpolator to use

11 Range 10 x01 Default: Lagrange polynomial interpolation
11                      10must be a registered interpolator

11

11interpolatorxoptions  10Scope: privatex01  STRING

11Description: Options for the interpolator

11 Range 10 x01 Default: order=2
11                      10must be a valid options specification

11

11killingxvectorxmethod 10Scope: privatex01  KEYWORD

11Description: Method for finding the Killing vector field

11 Range 10 x01 Default: eigenvector
11axial  10Assume that d/dphi is a Killing vector
11eigenvector  10Solve the Killing vector equation as eigenvector equation
11gradient  10Calculate the normal to the gradient of a scalar

11

11killingxvectorxnormalisation 10Scope: privatex01  KEYWORD

11Description: Method for normalising the Killing vector field

11 Range 10 x01 Default: average
11average  10Average several integral lines
11median  10Use the median integral line

11

11numxsurfaces  10Scope: privatex01  INT

11Description: Number of surfaces

11 Range 10 x01 Default: 1
110:100  10

11

11outputxvtkxevery  10Scope: privatex01  INT

11Description: Output a VTK file with the main 2D

11 Range 10 x01 Default: (none)
11                      10don’t output VTK file
111:*  10output every so many iterations

11spatialxorder  10Scope: privatex01  INT

11Description: Order of spatial differencing

11 Range 10 x01 Default: 2
112  10second order
114  10fourth order

11

11surfacexindex  10Scope: privatex01  INT

11Description: Spherical surface that contains the surface shape

11 Range 10 x01 Default: -1
11-1  10do not calculate
110:*  10surface index

11

11surfacexname  10Scope: privatex01  STRING

11Description: Spherical surface that contains the surface shape

11 Range 10 x01 Default: (none)
11                      10use surfacexindex
11.*  10surface name

11

11verbose  10Scope: privatex01  BOOLEAN

11Description: Produce log output while running

11                      10  x01  Default: no

11

11veryverbose  10Scope: privatex01  BOOLEAN

11Description: Produce much log output while running

11                      10  x01  Default: no

11

11autoxres  10Scope: shared from SPHERICALSURFACEx01BOOLEAN

11

11maxnphi  10Scope: shared from SPHERICALSURFACEx01INT

11

11maxntheta  10Scope: shared from SPHERICALSURFACEx01INT

11

11nghostsphi  10Scope: shared from SPHERICALSURFACEx01INT

11

11nghoststheta  10Scope: shared from SPHERICALSURFACEx01INT

11

11nphi  10Scope: shared from SPHERICALSURFACEx01INT

11

11nsurfaces  10Scope: shared from SPHERICALSURFACEx01INT

11

11ntheta  10Scope: shared from SPHERICALSURFACEx01INT

11

11symmetricxx  10Scope: shared from SPHERICALSURFACEx01BOOLEAN

11symmetricxy  10Scope: shared from SPHERICALSURFACEx01BOOLEAN

11

11symmetricxz  10Scope: shared from SPHERICALSURFACEx01BOOLEAN

11

6 Interfaces

General Implements: quasilocalmeasures

Inherits: admbase sphericalsurface tmunubase

Grid Variables

6.0.1 PRIVATE GROUPS

11x Group Names x10 x Variable Names x x01 Details x11x
11  qlmxstate  10  x01  compact110
11                 10  qlmxcalcxerror  x01 description11Status information
11                 10 qlmxhavexvalidxdata x01 dimensions110
11                 10 qlmxhavexkillingxvector x01 distribution11CONSTANT
11                 10 qlmxtimederivxorder x01 group type11SCALAR
11 10 qlmxiteration x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11INT
11 qlmxstatexp 10 x01 compact110
11                 10 qlmxhavexvalidxdataxp x01 description11Previous status information
11                 10 qlmxhavexvalidxdataxpxp x01 dimensions110
11                 10 qlmxhavexkillingxvectorxp x01 distribution11CONSTANT
11                 10qlmxhavexkillingxvectorxpxpx01 group type11SCALAR
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11INT
11 qlmxgridxint 10  x01  compact110
11                 10  qlmxnghoststheta  x01 description11Grid description
11                 10  qlmxnghostsphi  x01 dimensions110
11                 10  qlmxntheta  x01 distribution11CONSTANT
11                 10  qlmxnphi  x01 group type11SCALAR
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11INT
11 qlmxgridxreal 10  x01  compact110
11                 10  qlmxoriginxx  x01 description11Grid description
11                 10  qlmxoriginxy  x01 dimensions110
11                 10  qlmxoriginxz  x01 distribution11CONSTANT
11                 10  qlmxoriginxtheta  x01 group type11SCALAR
11                 10  qlmxoriginxphi  x01 timelevels111
11                 10  qlmxdeltaxtheta  x01vararrayxsize11numxsurfaces
11                 10  qlmxdeltaxphi  x01variable type11REAL
11 qlmxgridxrealxp 10  x01  compact110
11                 10  qlmxoriginxxxp  x01 description11Previous grid description
11                 10  qlmxoriginxyxp  x01 dimensions110
11                 10  qlmxoriginxzxp  x01 distribution11CONSTANT
11                 10  qlmxoriginxxxpxp  x01 group type11SCALAR
11                 10  qlmxoriginxyxpxp  x01 timelevels111
11                 10  qlmxoriginxzxpxp  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11REAL
11  qlmxshapes  10  x01  compact110
11                 10  qlmxshape  x01 description11Shape of the surface
11                 10  x01 dimensions112
11                 10  x01 distribution11CONSTANT
11                 10  x01 group type11ARRAY
11                 10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                 10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                 10  x01  tags11convergencexpower=1
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11REAL

11x Group Names x10x Variable Names xx01 Details x11x
11 qlmxshapesxp 10 x01 compact110
11                 10 qlmxshapexp x01 description11Previous shapes of the surface
11                 10 qlmxshapexpxp x01 dimensions112
11                 10  x01 distribution11CONSTANT
11                 10  x01 group type11ARRAY
11                 10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                 10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                 10  x01  tags11convergencexpower=1
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11REAL
11 qlmxcoordinates 10  x01  compact110
11                 10  qlmxx  x01 description11Cartesian coordinates of the grid points on the surface
11                 10  qlmxy  x01 dimensions112
11                 10  qlmxz  x01 distribution11CONSTANT
11                 10  x01 group type11ARRAY
11                 10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                 10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                 10  x01  tags11convergencexpower=1
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11REAL
11 qlmxcoordinatesxp 10  x01  compact110
11                 10  qlmxxxp  x01 description11Past Cartesian coordinates of the grid points on the surface
11                 10  qlmxyxp  x01 dimensions112
11                 10  qlmxzxp  x01 distribution11CONSTANT
11                 10  qlmxxxpxp  x01 group type11ARRAY
11                 10  qlmxyxpxp  x01  size11SPHERICALSURFACE::MAXNTHETA
11                 10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                 10  qlmxzxpxp  x01  tags11convergencexpower=1
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11REAL
11 qlmxtetradxl 10  x01  compact110
11                 10  qlmxl0  x01 description11Tetrad vector lm̂ u
11                 10  qlmxl1  x01 dimensions112
11                 10  qlmxl2  x01 distribution11CONSTANT
11                 10  qlmxl3  x01 group type11ARRAY
11                 10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                 10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                 10  x01  tags11convergencexpower=1
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11REAL
11 qlmxtetradxn 10  x01  compact110
11                 10  qlmxn0  x01 description11Tetrad vector nm̂ u
11                 10  qlmxn1  x01 dimensions112
11                 10  qlmxn2  x01 distribution11CONSTANT
11                 10  qlmxn3  x01 group type11ARRAY
11                 10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                 10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                 10  x01  tags11convergencexpower=1
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11REAL
11 qlmxtetradxm 10  x01  compact110
11                 10  qlmxm0  x01 description11Tetrad vector mm̂ u
11                 10  qlmxm1  x01 dimensions112
11                 10  qlmxm2  x01 distribution11CONSTANT
11                 10  qlmxm3  x01 group type11ARRAY
11                 10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                 10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                 10  x01  tags11convergencexpower=1
11                 10  x01 timelevels111
11                 10  x01vararrayxsize11numxsurfaces
11                 10  x01variable type11COMPLEX

11 x Group Names x 10x Variable Names xx01 Details x11x
11qlmxnewmanxpenrose10 x01 compact110
11                  10 qlmxnpkappa x01 description11Newman-Penrose quantities
11 10 qlmxnptau x01 dimensions112
11                  10 qlmxnpsigma x01 distribution11CONSTANT
11 10 qlmxnprho x01 group type11ARRAY
11                  10 qlmxnpepsilon x01  size11SPHERICALSURFACE::MAXNTHETA
11 10 x x01 size11SPHERICALSURFACE::MAXNPHI
11                  10 qlmxnpgamma x01  tags11Checkpoint=”no”
11                  10  x  x01  tags11convergencexpower=1
11                  10  qlmxnpbeta  x01 timelevels111
11                  10  qlmxnpalpha  x01vararrayxsize11numxsurfaces
11                  10  qlmxnppi  x01variable type11COMPLEX
11 qlmxweylxscalars 10  x01  compact110
11                  10  qlmxpsi0  x01 description11Weyl scalars (aka Newman-Penrose spin coefficients)
11                  10  qlmxpsi1  x01 dimensions112
11                  10  qlmxpsi2  x01 distribution11CONSTANT
11                  10  qlmxpsi3  x01 group type11ARRAY
11                  10  qlmxpsi4  x01  size11SPHERICALSURFACE::MAXNTHETA
11                  10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                  10  qlmxi  x01  tags11Checkpoint=”no”
11                  10  x  x01  tags11convergencexpower=1
11                  10  qlmxj  x01 timelevels111
11                  10  qlmxs  x01vararrayxsize11numxsurfaces
11                  10  qlmxsdiff  x01variable type11COMPLEX
11 qlmxriccixscalars 10  x01  compact110
11                  10  qlmxphi00  x01 description11Ricci scalars
11                  10  qlmxphi11  x01 dimensions112
11                  10  qlmxphi01  x01 distribution11CONSTANT
11                  10  qlmxphi12  x01 group type11ARRAY
11                  10  qlmxphi10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                  10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                  10  qlmxphi21  x01  tags11Checkpoint=”no”
11                  10  x  x01  tags11convergencexpower=1
11                  10  qlmxphi02  x01 timelevels111
11                  10  qlmxphi22  x01vararrayxsize11numxsurfaces
11                  10  qlmxphi20  x01variable type11REAL
11 qlmxtwometric 10  x01  compact110
11                  10  qlmxqtt  x01 description112-metric
11                  10  qlmxqtp  x01 dimensions112
11                  10  qlmxqpp  x01 distribution11CONSTANT
11                  10  qlmxrsc  x01 group type11ARRAY
11                  10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                  10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                  10  x01  tags11Checkpoint=”no”
11                  10  x  x01  tags11convergencexpower=1
11                  10  x01 timelevels111
11                  10  x01vararrayxsize11numxsurfaces
11                  10  x01variable type11REAL
11 qlmxkillingxvector 10  x01  compact110
11                  10  qlmxxixt  x01 description11Killing vector field
11                  10  qlmxxixp  x01 dimensions112
11                  10  qlmxchi  x01 distribution11CONSTANT
11                  10  x01 group type11ARRAY
11                  10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                  10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                  10  x01  tags11convergencexpower=1
11                  10  x01 timelevels111
11                  10  x01vararrayxsize11numxsurfaces
11                  10  x01variable type11REAL
11qlmxkilledxtwometric10  x01  compact110
11                  10  qlmxlqtt  x01 description11Lie derivative of the 2-metric along the Killing vector field
11                  10  qlmxlqtp  x01 dimensions112
11                  10  qlmxlqpp  x01 distribution11CONSTANT
11                  10  x01 group type11ARRAY
11                  10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                  10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                  10  x01  tags11Checkpoint=”no”
11                  10  x  x01  tags11convergencexpower=1
11                  10  x01 timelevels111
11                  10  x01vararrayxsize11numxsurfaces
11                  10  x01variable type11REAL

11 x Group Names x 10 x Variable Names x x01 Details x11x
11qlmxinvariantxcoordinates10  x01  compact110
11                      10  qlmxinvxz  x01 description11Invariant coordinates on the surface
11                      10  x  x01 description11assuming axisymmetry
11                      10  x01 dimensions112
11                      10  x01 distribution11CONSTANT
11                      10  x01 group type11ARRAY
11                      10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                      10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                      10  x01  tags11Checkpoint=”no”
11                      10  x  x01  tags11convergencexpower=1
11                      10  x01 timelevels111
11                      10  x01vararrayxsize11numxsurfaces
11                      10  x01variable type11REAL
11 qlmxmultipolexmoments 10  x01  compact110
11                      10  qlmxmpxm0  x01 description11Mass and spin multipole moments
11                      10  qlmxmpxm1  x01 dimensions110
11                      10  qlmxmpxm2  x01 distribution11CONSTANT
11                      10  qlmxmpxm3  x01 group type11SCALAR
11                      10  qlmxmpxm4  x01  tags11Checkpoint=”no”
11                      10  qlmxmpxm5  x01 timelevels111
11                      10  qlmxmpxm6  x01vararrayxsize11numxsurfaces
11                      10  qlmxmpxm7  x01variable type11REAL
11 qlmx3determinant 10  x01  compact110
11                      10  qlmx3det  x01 description113-Determinant of H for a special choice of the triad
11                      10  x01 dimensions112
11                      10  x01 distribution11CONSTANT
11                      10  x01 group type11ARRAY
11                      10  x01  size11SPHERICALSURFACE::MAXNTHETA
11                      10  x  x01  size11SPHERICALSURFACE::MAXNPHI
11                      10  x01  tags11Checkpoint=”no”
11                      10  x  x01  tags11convergencexpower=1
11                      10  x01 timelevels111
11                      10  x01vararrayxsize11numxsurfaces
11                      10  x01variable type11REAL
11  qlmxscalars  10  x01  compact110
11                      10  qlmxtime  x01 description11Scalar quantities on the surface
11                      10 qlmxequatorialxcircumference x01 dimensions110
11                      10 qlmxpolarxcircumferencex0 x01 distribution11CONSTANT
11                      10qlmxpolarxcircumferencexpix2x01 group type11SCALAR
11 10 qlmxarea x01 timelevels111
11                      10 qlmxirreduciblexmass x01vararrayxsize11numxsurfaces
11                      10  qlmxradius  x01variable type11REAL
11  qlmxscalarsxp  10  x01  compact110
11                      10  qlmxtimexp  x01 description11Some scalar quantities on the surface at previous times
11                      10  qlmxtimexpxp  x01 dimensions110
11                      10  qlmxradiusxp  x01 distribution11CONSTANT
11                      10  qlmxradiusxpxp  x01 group type11SCALAR
11                      10  x01 timelevels111
11                      10  x01vararrayxsize11numxsurfaces
11                      10  x01variable type11REAL

11

7 Schedule This section lists all the variables which are assigned storage by thorn EinsteinAnalysis/QuasiLocalMeasures. 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

11Always:  11x
11qlmxstate qlmxscalars  11x
11qlmxstatexp qlmxscalarsxp  11x
11qlmxgridxint qlmxgridxreal qlmxgridxrealxp  11x
11qlmxshapes qlmxtetradxl qlmxtetradxn qlmxtetradxm  11x
11qlmxshapesxp  11x
11qlmxkillingxvector  11x
11qlmxcoordinates qlmxcoordinatesxp  11x
11qlmxnewmanxpenrose qlmxweylxscalars  11x
11qlmxriccixscalars qlmxtwometric qlmxkilledxtwometric  11x
11qlmxinvariantxcoordinates qlmxmultipolexmoments qlmx3determinant11x

11x  11x

11

Scheduled Functions

CCTKxPARAMCHECK x qlmxparamcheck x check quasi-local parameter settings x

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

11

CCTKxINITIAL x qlmxinit x initialise quasi-local calculations x

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

11

CCTKxANALYSIS x qlmxcalculate x calculate quasi-local quantities

11x11After: 11sphericalsurfacexhasbeenset
11 x 11x  11settmunu
11 x 11Language:11fortran
11 x 11Options: 11global
11 x 11Storage: 11qlmxcoordinates
11 x 11x  11qlmxcoordinatesxp
11 x 11x  11qlmxnewmanxpenrose
11 x 11x  11qlmxweylxscalars
11 x 11x  11qlmxriccixscalars
11 x 11x  11qlmxtwometric
11 x 11x  11qlmxkilledxtwometric
11 x 11x  11qlmxinvariantxcoordinates
11 x 11x  11qlmxmultipolexmoments
11 x 11x  11qlmx3determinant
11 x 11Triggers: 11qlmxstate
11 x 11x  11qlmxgridxint
11 x 11x  11qlmxgridxreal
11 x 11x  11qlmxshapes
11 x 11x  11qlmxtetradxl
11 x 11x  11qlmxtetradxn
11 x 11x  11qlmxtetradxm
11 x 11x  11qlmxcoordinates
11 x 11x  11qlmxnewmanxpenrose
11 x 11x  11qlmxweylxscalars
11 x 11x  11qlmxriccixscalars
11 x 11x  11qlmxtwometric
11 x 11x  11qlmxkillingxvector
11 x 11x  11qlmxkilledxtwometric
11 x 11x  11qlmxinvariantxcoordinates
11 x 11x  11qlmxmultipolexmoments
11 x 11x  11qlmx3determinant
11 x 11x  11qlmxscalars
11 x 11Type: 11function