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 $P_{z}=(0M,2M,5M)$ and vanishing spin. Also for $P_{z}=2M$ and a spin of $5M$ in the x direction the horizon converges. Figure D7.1 shows the case with $P_{z}=2M$ and vanishing spin.

Figure D7.1: Convergence of the horizon area for $P_{z}=2M$

[angle=0,width=8cm]/home/cactus_web/CheckOut/CactusStable/arrangements/CactusEinstein/AHFinder/doc/p2_areacomp

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 D7.2. 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).

Figure D7.2: Horizon positions for BL data

[angle=0,width=10cm]/home/cactus_web/CheckOut/CactusStable/arrangements/CactusEinstein/AHFinder/doc/hori_bl

The dashed lines show inner trapped surfaces in the left figure and the surface where the algorithm stopped without finding a horizon in the right figure.

Also the Misner case was checked. Here for $\mu$ = 1.35 a common horizon is found. For $\mu$ = 1.37 separated horizons are found. From the literature we know that (e.g. gr-qc/9809004) the critical value of $\mu$ is 1.36. This is confirmed by the horizon finder.

The information of when a horizon was found can be seen in the cactus-logfile. There will be output from the thorn even if no horizon was found.