AHFinder

Miguel Alcubierre/Latex version by Peter Diener

Date

Abstract

Finding Apparent Horizons in a numerical spacetime

1 Purpose

Thorn AHFinder finds Apparent Horizons (AHs) in numerical spacetimes. It calulates various quantities like horizon area and its corresponding mass.

2 Using AHFinder

Thorn AHFinder can be used either with a minimization or a flow algorithm.

2.1 Parameters

General parameters

Parameters used in evolutions

Parameters specifying the expansion of the surface in sperical harmonics.

Looking for three horizons

The finder can also be used to look for three horizons each time. This is done by just running the algorithm three consecutive times with different initial guesses, and is useful for simulations of black hole collisions.

Further parameters for the initial guess

The initial guess can be furthermore controlled by some parameters which are set to ”no” by default.

Parameters for surface intergrals

Parameters indicating symmetries

Parameters for minimization algorithm

Parameters for the flow algorithm

The character of the different flows and the α and β parameters are described in Carsten Gundlach’s paper on his pseudo-spectral apparent horizon finder (gr-qc/9707050).

Parameters for output

Parameters for mask

2.2 Minimal parameter settings

Usually only a few of the parameters described above are needed in the parameter file. The simplest parameter settings for using the flow algorithm for a full 3D horizon with a large sphere as initial guess is

interpolation_order = 2     # Second order interpolation  
ahf_active          = "yes"  
ahf_flow            = "yes"  
ahf_phi             = "yes"

This looks for a non-axisymmetric horizon around the origin with lmax = 8 and using the flow algorithm. It starts with the largest sphere that fits in the 3D grid and outputs 2D grid functions. The other parameters can be used if needed.

2.3 Hints for parameter settings

In full 3D the flow algorithm is faster than the minimization algorithm. However, in cases when there are very few terms in the expansion in spherical harmonics the minimization can be faster. In axisymmetry this typically happens for lmax 10.

While the default settings usually work fine, they can be changed to meet special purposes:

2.4 Output to Files

The output of the thorn consists of two gridfunctions and several one dimensional output files.

2.5 Some results with the finder

The finder has been examined with puncture initial data for single and binary-black hole scenarios.

Calculations with different grid spacings but constant grid size show convergence of the horizon area.

This has been checked with different linear momenta in the z direction Pz = (0M,2M,5M) and vanishing spin. Also for Pz = 2M and a spin of 5M in the x direction the horizon converges. Figure 1 shows the case with Pz = 2M and vanishing spin.


PIC


Figure 1: Convergence of the horizon area for Pz = 2M

Further, not only the area converges but also the shape of the horizon. For both the minimization and the flow algorithm the horizon converges to the same shape, as can be seen from the coefficients fo the expansion. The order of convergence for the coefficients is between 1.4 and 1.7.

By using the parameters ahf_xc, ahf_yc, ahf_zc it can also be shown that the finder also locates horizons which are not centered. This works in general as long as the surface can be expanded in spherical harmonics around this point, but the error increases with the off-centering.

The parameter ahf_r0 can be used e.g. when dealing with two black holes. If one searches for separate horizons one can center the finder on one of the locations of the holes and use an initial radius ahf_r0 smaller than the coordinate distance of the holes. With this parameter settings the single horizon can be found faster. But also a setup with an initial sphere of maximum radius should work at least for the flow algorithm. This has been checked with puncture data for two holes with vanishing linear and angular momentum for each hole (equivalent to Brill-Lindquist data) and is shown in Figure 2. Here for a coordinate distance of the holes of 1.6M the separated horizons for the holes are found but no common horizon. For a coordinate distance of 1.5M a common horizon is found and also single ones, which are inner surfaces in this case. This coincides with other work where the critical coordinate distance for a single horizon is between 1.53M and 1.56M (gr-qc/9809004).


PIC


Figure 2: Horizon positions for BL data