Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
1996-06-15
Phys.Rev. D54 (1996) 7311-7321
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
12 pages, REVTeX 3.0, also available (with additional pictures and numerical data) from http://doppler.thp.univie.ac.at/~shusa
Scientific paper
10.1103/PhysRevD.54.7311
Asymptotically flat, time-symmetric, axially symmetric and conformally flat initial data for vacuum general relativity are studied numerically on $R^3$ with the interior of a standard torus cut out. By the choice of boundary condition the torus is marginally outer trapped, and thus a surface of minimal area. Apart from pure scaling the standard tori are parameterized by a radius $a\in [0,1]$, where $a=0$ corresponds to the limit where the boundary torus degenerates to a circle and $a=1$ to a torus that touches the axis of symmetry. Noting that these tori are the orbits of a $U(1)\times U(1)$ conformal isometry allows for a simple scheme to solve the constraint, involving numerical solution of only ordinary differential equations.The tori are unstable minimal surfaces (i.e. only saddle points of the area functional) and thus can not be apparent horizons, but are always surrounded by an apparent horizon of spherical topology, which is analyzed in the context of the hoop conjecture and isoperimetric inequality for black holes.
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