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, 2], 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 \(C_{\alpha \beta \gamma \delta }\) is defined as the traceless part of the Riemann tensor \(R_{\alpha \beta \gamma \delta }\)

\begin {eqnarray} {\Gamma ^{\alpha }}_{\beta \gamma } &=& \frac 12 g^{\alpha \mu } \left ( g_{\mu \beta ,\gamma }+g_{\mu \gamma ,\beta } - g_{\beta \gamma ,\mu } \right ), \\ {R^\alpha }_{\beta \gamma \delta } &=& {\Gamma ^\alpha }_{\beta \delta ,\gamma } - {\Gamma ^\alpha }_{\beta \gamma ,\delta } + {\Gamma ^\alpha }_{\mu \gamma } {\Gamma ^\mu }_{\beta \delta } - {\Gamma ^\alpha }_{\mu \delta } {\Gamma ^\mu }_{\beta \gamma }, \\ {R^\beta }_\delta &=& {R^{\mu \beta }}_{\mu \delta }, \\ R &=& {R^\mu }_\mu , \\ C_{\alpha \beta \gamma \delta } &=& R_{\alpha \beta \gamma \delta }+ {\frac {1}{2}}\left (R_{\alpha \delta }g_{\beta \gamma }-R_{\alpha \gamma }g_{\beta \delta }+R_{\beta \gamma }g_{\alpha \delta }-R_{\beta \delta }g_{\alpha \gamma }\right )+ {\frac {1}{6}}R\left (g_{\alpha \gamma }g_{\beta \delta }-g_{\alpha \delta }g_{\beta \gamma }\right ), \end {eqnarray}

where \({\Gamma ^{\alpha }}_{\beta \gamma }\), \({R^\beta }_\delta \) and \(R\) are the Christoffel symbols, Ricci tensor and Riemann scalar respectively [6]. Its self-dual is \begin {equation} \tilde C_{\alpha \beta \gamma \delta } = C_{\alpha \beta \gamma \delta } + (i/2)\epsilon _{\alpha \beta \mu \nu } {C^{\mu \nu }}_{\gamma \delta }. \end {equation}

The curvature invariants \(\mathcal {I}\) (qlm_i) and \(\mathcal {J}\) (qlm_j) are defined in terms of the self-dual of the Weyl tensor [5] \begin {equation} \mathcal {I} = \tilde C_{\alpha \beta \gamma \delta } \tilde C^{\alpha \beta \gamma \delta } \qquad \mbox {and}\qquad \mathcal {J} = \tilde C_{\alpha \beta \gamma \delta } {\tilde C^{\gamma \delta }}_{\mu \nu } \tilde C^{\mu \nu \alpha \beta }. \end {equation}

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. as

\begin {eqnarray} \int X\; d^3V \end {eqnarray}

where \(X\) is some quantity that lives on the horizon. These integrals have to be transformed into a \(2+1\) form so that they can be conveniently evaluated, e.g. as

\begin {eqnarray} \int X\; A\, d^2S\, dt \end {eqnarray}

where \(d^2S\) is the area element on the horizon cross section contained in \(\Sigma \), and \(dt\) is the coordinate time differential. The factor \(A\) should contain the extra terms due to this coordinate transformation.

Starting from the \(3\)-volume element \(d^3V\), let us first decompose it into the \(2\)-volume element \(d^2S\) and a “time” coordinate on the horizon, which we call \(\sigma \). Note that \(\sigma \) will generally be a spacelike coordinate for dynamical horizons. Let \(\mathbf {Q}\) be the induced \(3\)-metric on the horizon, and \(\mathbf {q}\) be the induced \(2\)-metric on the cross section. Then it is

\begin {eqnarray} d^3V & = & \sqrt {\det Q}\; d\theta \, d\phi \, d\sigma \\ & = & \frac {\sqrt {\det Q}}{\sqrt {\det q}}\; d^2S\, d\sigma \end {eqnarray}

because \(d^2S = \sqrt {\det q}\, d\theta \, d\phi \).

The coordinate time differential \(dt\) and the differential \(d\sigma \) will in general not be aligned because the horizon world tube will in general not have a static coordinate shape. It is

\begin {eqnarray} d\tau & = & (\cosh \alpha )\, dt + (\sinh \alpha )\, ds \\ d\sigma & = & (\cosh \alpha )\, ds + (\sinh \alpha )\, dt \end {eqnarray}

where \(s\) is a radial coordinate perpendicular to the horizon and also perpendicular to \(t\), and \(\tau \) is perpendicular to \(\sigma \) and lies in the plan spanned by \(t\) and \(s\). \(\tau \) and \(\sigma \) are depend on \(t\) and \(s\) via a Lorentz boost. Thus we have

\begin {eqnarray} \frac {d\sigma }{dt} & = & (\cosh \alpha )\, \frac {ds}{dt} + (\sinh \alpha )\, \frac {dt}{dt} \\ & = & \sinh \alpha \quad \text {.} \end {eqnarray}

Putting everything together we arrive at

\begin {eqnarray} \int X\; \frac {\sqrt {\det Q}}{\sqrt {\det q}}\; (\sinh \alpha )\, d^2S\, dt \quad \text {.} \end {eqnarray}

3.2 The “lapse” function \(N_R\)

Starting from

\begin {eqnarray} N_R & = & | \partial R | \end {eqnarray}

we find, since the radius \(R\) changes only in the \(\sigma \) direction,

\begin {eqnarray} N_R^2 & = & g^{\sigma \sigma }\, (\partial _\sigma R)\, (\partial _\sigma R) \quad \text {.} \end {eqnarray}

If we assume \(\partial _\tau R = 0\) and write \(\partial _t R = \dot R\), and use the relations between \(\sigma \) and \(t\) from above, we get

\begin {eqnarray} \dot R & = & \partial _t R \\ & = & \frac {\partial \tau }{\partial t} \partial _\tau R + \frac {\partial \sigma }{\partial t} \partial _\sigma R \\ & = & \sinh \alpha \, \partial _\sigma R \end {eqnarray}

[NOTE: but \(\partial _t \alpha \ne 0\).] and therefore

\begin {eqnarray} \partial _\sigma R & = & \frac {1}{\sinh \alpha }\; \dot R \quad \text {.} \end {eqnarray}

Additionally we have \(g^{\sigma \sigma } = g^{ab} \sigma _a \sigma _b = g_{ab} \sigma ^a \sigma ^b\) where \(\sigma ^a\) is the unit vector in the \(\sigma \) direction, i.e.

\begin {eqnarray} \tau ^a & = & (\cosh \alpha )\, t^a + (\sinh \alpha )\, s^a \\ \sigma ^a & = & (\cosh \alpha )\, s^a + (\sinh \alpha )\, t^a \end {eqnarray}

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 BSSN_MoL 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. 3-determinant of the horizon
- Performing some time steps is necessary. Either read in lapse and shift from files, or set them arbitrarily (e.g. lapse one, shift zero).
- Activate a time evolution thorn, i.e., BSSN_MoL, MoL, Time, etc. In order to fill the past time levels, just choose MoL::initial_data_is_crap. 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 \(\theta \) and \(\phi \) directions. Cactus output has only grid point indices, but does not contain the coordinates \(\theta \) and \(\phi \) themselves.

In gnuplot, one can define functions to convert indices to coordinates:

\begin {eqnarray} \theta (i) & = & (i - g\theta + 0.5) * \pi / n\theta \\ \phi (j) & = & (j - g\phi ) * 2*\pi / n\phi \end {eqnarray}

where \(g\theta \) and \(g\phi \) is the number of ghost points in the corresponding direction, and \(n\theta \) and \(n\phi \) the number of interior points. Here are the same equations in gnuplot syntax:

theta(i) = (i - nghosts + 0.5) * pi / ntheta
phi(j) = (j - nghosts) * 2*pi / nphi

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 \(\pi /2\) in the \(\theta \) and \(\pi \) in the \(\phi \) direction. Also, plotting something axisymmetric with bitant symmetry vs. \(\theta \) and vs. \(\pi -\theta \), and vs. \(\phi \) and \(2\pi -\phi \), 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) = (i + origin_theta) * delta_theta
phi(j) = (j + origin_phi) * delta_phi

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

References

[1] O. Dreyer, B. Krishnan, D. Shoemaker and E. Schnetter, Phys. Rev. D 67, 024018 (2003) doi:10.1103/PhysRevD.67.024018 [gr-qc/0206008].

[2] E. Schnetter, B. Krishnan and F. Beyer, Phys. Rev. D 74, 024028 (2006) doi:10.1103/PhysRevD.74.024028 [gr-qc/0604015].

[3] L. B. Szabados, Living Rev. Rel. 7, 4 (2004).

[4] G. Lovelace et al., Phys. Rev. D 82, 064031 (2010) doi:10.1103/PhysRevD.82.064031 [arXiv:0907.0869 [gr-qc]].

[5] J. G. Baker, M. Campanelli and C. O. Lousto, Phys. Rev. D 65, 044001 (2002) doi:10.1103/PhysRevD.65.044001 [gr-qc/0104063].

[6] C. W. Misner, K. S. Thorne and J. A. Wheeler, San Francisco 1973, 1279p

5 Parameters


begin_qlm_calculations_after	Scope: private	REAL

Description: when should we start calculations?

Range		Default: 0.0
:	at/after this time (inclusively)


coordsystem	Scope: private	STRING

Description: The coordinate system to use

Range		Default: cart3d
	must be a registered coordinate system


interpolator	Scope: private	STRING

Description: The interpolator to use

Range		Default: Lagrange polynomial interpolation
	must be a registered interpolator


interpolator_options	Scope: private	STRING

Description: Options for the interpolator

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


killing_vector_method	Scope: private	KEYWORD

Description: Method for finding the Killing vector field

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


killing_vector_normalisation	Scope: private	KEYWORD

Description: Method for normalising the Killing vector field

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


num_surfaces	Scope: private	INT

Description: Number of surfaces

Range		Default: 1
0:100


output_vtk_every	Scope: private	INT

Description: Output a VTK file with the main 2D

Range		Default: (none)
	don’t output VTK file
1:*	output every so many iterations


spatial_order	Scope: private	INT

Description: Order of spatial differencing

Range		Default: 2
2	second order
4	fourth order


surface_index	Scope: private	INT

Description: Spherical surface that contains the surface shape

Range		Default: -1
-1	do not calculate
0:*	surface index


surface_name	Scope: private	STRING

Description: Spherical surface that contains the surface shape

Range		Default: (none)
	use surface_index
.*	surface name


verbose	Scope: private	BOOLEAN

Description: Produce log output while running

		Default: no


veryverbose	Scope: private	BOOLEAN

Description: Produce much log output while running

		Default: no


auto_res	Scope: shared from SPHERICALSURFACE	BOOLEAN


maxnphi	Scope: shared from SPHERICALSURFACE	INT


maxntheta	Scope: shared from SPHERICALSURFACE	INT


nghostsphi	Scope: shared from SPHERICALSURFACE	INT


nghoststheta	Scope: shared from SPHERICALSURFACE	INT


nphi	Scope: shared from SPHERICALSURFACE	INT


nsurfaces	Scope: shared from SPHERICALSURFACE	INT


ntheta	Scope: shared from SPHERICALSURFACE	INT


symmetric_x	Scope: shared from SPHERICALSURFACE	BOOLEAN


symmetric_y	Scope: shared from SPHERICALSURFACE	BOOLEAN


symmetric_z	Scope: shared from SPHERICALSURFACE	BOOLEAN

6 Interfaces

General

Implements:

quasilocalmeasures

Inherits:

admbase

sphericalsurface

tmunubase

Grid Variables

6.0.1 PRIVATE GROUPS


Group Names	Variable Names	Details

qlm_state		compact	0
	qlm_calc_error	description	Status information
	qlm_have_valid_data	dimensions	0
	qlm_have_killing_vector	distribution	CONSTANT
	qlm_timederiv_order	group type	SCALAR
	qlm_iteration	timelevels	1
		vararray_size	num_surfaces
		variable type	INT

qlm_state_p		compact	0
	qlm_have_valid_data_p	description	Previous status information
	qlm_have_valid_data_p_p	dimensions	0
	qlm_have_killing_vector_p	distribution	CONSTANT
	qlm_have_killing_vector_p_p	group type	SCALAR
		timelevels	1
		vararray_size	num_surfaces
		variable type	INT

qlm_grid_int		compact	0
	qlm_nghoststheta	description	Grid description
	qlm_nghostsphi	dimensions	0
	qlm_ntheta	distribution	CONSTANT
	qlm_nphi	group type	SCALAR
		timelevels	1
		vararray_size	num_surfaces
		variable type	INT

qlm_grid_real		compact	0
	qlm_origin_x	description	Grid description
	qlm_origin_y	dimensions	0
	qlm_origin_z	distribution	CONSTANT
	qlm_origin_theta	group type	SCALAR
	qlm_origin_phi	timelevels	1
	qlm_delta_theta	vararray_size	num_surfaces
	qlm_delta_phi	variable type	REAL

qlm_grid_real_p		compact	0
	qlm_origin_x_p	description	Previous grid description
	qlm_origin_y_p	dimensions	0
	qlm_origin_z_p	distribution	CONSTANT
	qlm_origin_x_p_p	group type	SCALAR
	qlm_origin_y_p_p	timelevels	1
	qlm_origin_z_p_p	vararray_size	num_surfaces
		variable type	REAL

qlm_shapes		compact	0
	qlm_shape	description	Shape of the surface
		dimensions	2
		distribution	CONSTANT
		group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL


Group Names	Variable Names	Details

qlm_shapes_p		compact	0
	qlm_shape_p	description	Previous shapes of the surface
	qlm_shape_p_p	dimensions	2
		distribution	CONSTANT
		group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_coordinates		compact	0
	qlm_x	description	Cartesian coordinates of the grid points on the surface
	qlm_y	dimensions	2
	qlm_z	distribution	CONSTANT
		group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_coordinates_p		compact	0
	qlm_x_p	description	Past Cartesian coordinates of the grid points on the surface
	qlm_y_p	dimensions	2
	qlm_z_p	distribution	CONSTANT
	qlm_x_p_p	group type	ARRAY
	qlm_y_p_p	size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
	qlm_z_p_p	tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_tetrad_l		compact	0
	qlm_l0	description	Tetrad vector lu
	qlm_l1	dimensions	2
	qlm_l2	distribution	CONSTANT
	qlm_l3	group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_tetrad_n		compact	0
	qlm_n0	description	Tetrad vector nu
	qlm_n1	dimensions	2
	qlm_n2	distribution	CONSTANT
	qlm_n3	group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_tetrad_m		compact	0
	qlm_m0	description	Tetrad vector mu
	qlm_m1	dimensions	2
	qlm_m2	distribution	CONSTANT
	qlm_m3	group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	COMPLEX


Group Names	Variable Names	Details

qlm_newman_penrose		compact	0
	qlm_npkappa	description	Newman-Penrose quantities
	qlm_nptau	dimensions	2
	qlm_npsigma	distribution	CONSTANT
	qlm_nprho	group type	ARRAY
	qlm_npepsilon	size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
	qlm_npgamma	tags	Checkpoint=”no”
		tags	convergence_power=1
	qlm_npbeta	timelevels	1
	qlm_npalpha	vararray_size	num_surfaces
	qlm_nppi	variable type	COMPLEX

qlm_weyl_scalars		compact	0
	qlm_psi0	description	Weyl scalars (aka Newman-Penrose spin coefficients)
	qlm_psi1	dimensions	2
	qlm_psi2	distribution	CONSTANT
	qlm_psi3	group type	ARRAY
	qlm_psi4	size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
	qlm_i	tags	Checkpoint=”no”
		tags	convergence_power=1
	qlm_j	timelevels	1
	qlm_s	vararray_size	num_surfaces
	qlm_sdiff	variable type	COMPLEX

qlm_ricci_scalars		compact	0
	qlm_phi00	description	Ricci scalars
	qlm_phi11	dimensions	2
	qlm_phi01	distribution	CONSTANT
	qlm_phi12	group type	ARRAY
	qlm_phi10	size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
	qlm_phi21	tags	Checkpoint=”no”
		tags	convergence_power=1
	qlm_phi02	timelevels	1
	qlm_phi22	vararray_size	num_surfaces
	qlm_phi20	variable type	REAL

qlm_twometric		compact	0
	qlm_qtt	description	2-metric
	qlm_qtp	dimensions	2
	qlm_qpp	distribution	CONSTANT
	qlm_rsc	group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	Checkpoint=”no”
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_killing_vector		compact	0
	qlm_xi_t	description	Killing vector field
	qlm_xi_p	dimensions	2
	qlm_chi	distribution	CONSTANT
		group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_killed_twometric		compact	0
	qlm_lqtt	description	Lie derivative of the 2-metric along the Killing vector field
	qlm_lqtp	dimensions	2
	qlm_lqpp	distribution	CONSTANT
		group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	Checkpoint=”no”
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL


Group Names	Variable Names	Details

qlm_invariant_coordinates		compact	0
	qlm_inv_z	description	Invariant coordinates on the surface
		description	assuming axisymmetry
		dimensions	2
		distribution	CONSTANT
		group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	Checkpoint=”no”
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_multipole_moments		compact	0
	qlm_mp_m0	description	Mass and spin multipole moments
	qlm_mp_m1	dimensions	0
	qlm_mp_m2	distribution	CONSTANT
	qlm_mp_m3	group type	SCALAR
	qlm_mp_m4	tags	Checkpoint=”no”
	qlm_mp_m5	timelevels	1
	qlm_mp_m6	vararray_size	num_surfaces
	qlm_mp_m7	variable type	REAL

qlm_3determinant		compact	0
	qlm_3det	description	3-Determinant of H for a special choice of the triad
		dimensions	2
		distribution	CONSTANT
		group type	ARRAY
		size	SPHERICALSURFACE::MAXNTHETA
		size	SPHERICALSURFACE::MAXNPHI
		tags	Checkpoint=”no”
		tags	convergence_power=1
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

qlm_scalars		compact	0
	qlm_time	description	Scalar quantities on the surface
	qlm_equatorial_circumference	dimensions	0
	qlm_polar_circumference_0	distribution	CONSTANT
	qlm_polar_circumference_pi_2	group type	SCALAR
	qlm_area	timelevels	1
	qlm_irreducible_mass	vararray_size	num_surfaces
	qlm_radius	variable type	REAL

qlm_scalars_p		compact	0
	qlm_time_p	description	Some scalar quantities on the surface at previous times
	qlm_time_p_p	dimensions	0
	qlm_radius_p	distribution	CONSTANT
	qlm_radius_p_p	group type	SCALAR
		timelevels	1
		vararray_size	num_surfaces
		variable type	REAL

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

Always:
qlm_state qlm_scalars
qlm_state_p qlm_scalars_p
qlm_grid_int qlm_grid_real qlm_grid_real_p
qlm_shapes qlm_tetrad_l qlm_tetrad_n qlm_tetrad_m
qlm_shapes_p
qlm_killing_vector
qlm_coordinates qlm_coordinates_p
qlm_newman_penrose qlm_weyl_scalars
qlm_ricci_scalars qlm_twometric qlm_killed_twometric
qlm_invariant_coordinates qlm_multipole_moments qlm_3determinant

Scheduled Functions

CCTK_PARAMCHECK

qlm_paramcheck

check quasi-local parameter settings

	Language:	fortran
	Options:	global
	Type:	function

CCTK_INITIAL

qlm_init

initialise quasi-local calculations

	Language:	fortran
	Options:	global
	Type:	function

CCTK_ANALYSIS

qlm_calculate

calculate quasi-local quantities

	After:	sphericalsurface_hasbeenset
		settmunu
	Language:	fortran
	Options:	global
	Storage:	qlm_coordinates
		qlm_coordinates_p
		qlm_newman_penrose
		qlm_weyl_scalars
		qlm_ricci_scalars
		qlm_twometric
		qlm_killed_twometric
		qlm_invariant_coordinates
		qlm_multipole_moments
		qlm_3determinant
	Triggers:	qlm_state
		qlm_grid_int
		qlm_grid_real
		qlm_shapes
		qlm_tetrad_l
		qlm_tetrad_n
		qlm_tetrad_m
		qlm_coordinates
		qlm_newman_penrose
		qlm_weyl_scalars
		qlm_ricci_scalars
		qlm_twometric
		qlm_killing_vector
		qlm_killed_twometric
		qlm_invariant_coordinates
		qlm_multipole_moments
		qlm_3determinant
		qlm_scalars
	Type:	function