QuasoLocalMeasures

Erik Schnetter <schnetter@cct.lsu.edu>

2010-04-02

Abstract

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

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

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

1.1 Integral transformation

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

Xd3V (1)

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

XAd2Sdt (2)

where d2S is the area element on the horizon cross section contained in Σ, 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 d3V , let us first decompose it into the 2-volume element d2S and a “time” coordinate on the horizon, which we call σ. Note that σ will generally be a spacelike coordinate for dynamical horizons. Let Q be the induced 3-metric on the horizon, and q be the induced 2-metric on the cross section. Then it is

d3V = det Qd𝜃dϕdσ (3) = detQ det q d2Sdσ (4)

because d2S = det qd𝜃dϕ.

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

dτ = (coshα)dt + (sinhα)ds (5) dσ = (coshα)ds + (sinhα)dt (6)

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

dσ dt = (coshα)ds dt + (sinhα)dt dt (7) = sinhα. (8)

Putting everything together we arrive at

Xdet Q det q (sinhα)d2Sdt. (9)

1.2 The “lapse” function NR

Starting from

NR = |R| (10)

we find, since the radius R changes only in the σ direction,

NR2 = gσσ( σR)(σR). (11)

If we assume τR = 0 and write tR = , and use the relations between σ and t from above, we get

= tR (12) = τ t τR + σ t σR (13) = sinhασR (14)

[NOTE: but tα0.] and therefore

σR = 1 sinhα. (15)

Additionally we have gσσ = gabσaσb = gabσaσb where σa is the unit vector in the σ direction, i.e. 

τa = (coshα)ta + (sinhα)sa (16) σa = (coshα)sa + (sinhα)ta (17)

1.3 Special Behaviour

In order to use the IsolatedHorizon thorn on existing data (postprocessing), the following procedure is necessary.

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.

2 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 𝜃 and ϕ directions. Cactus output has only grid point indices, but does not contain the coordinates 𝜃 and ϕ themselves.

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

𝜃(i) = (i g𝜃 + 0.5) πn𝜃 (18) ϕ(j) = (j gϕ) 2 πnϕ (19)

where g𝜃 and gϕ is the number of ghost points in the corresponding direction, and n𝜃 and nϕ 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 π2 in the 𝜃 and π in the ϕ direction. Also, plotting something axisymmetric with bitant symmetry vs. 𝜃 and vs. π 𝜃, and vs. ϕ and 2π ϕ, 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