Calculation of percolation thresholds in high dimensions for fcc, bcc, and diamond lattices

In a recent article, Galam and Mauger proposed an invariant for site and bond percolation thresholds, based on known values for twenty lattices (Eur. Phys. J. B 1 (1998) 255-258). Here we give a larger list of values for more than forty lattices in two to six dimensions. In this list are new results for fcc, bcc, and diamond lattices in 4, 5, and 6 dimensions. The list contains examples of lattices with equal site percolation thresholds, but different bond percolation thresholds. These and other examples show that there are deviations from the proposed invariant of up to 12% in two dimensions, increasing to 69% in higher dimensions.Comment: 12 pages, 3 figures (EPS), LaTe

Spectral methods in general relativistic astrophysics

We present spectral methods developed in our group to solve three-dimensional partial differential equations. The emphasis is put on equations arising from astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures, submitted to Journal of Computational & Applied Mathematic

Darwin-Riemann problems in general relativity

A review is given of recent results about the computation of irrotational Darwin-Riemann configurations in general relativity. Such configurations are expected to represent fairly well the late stages of inspiralling binary neutron stars.Comment: 20 pages, 11 PostScript figures, uses PTPTeX, to appear in the Proceedings of Yukawa International Seminar 99 "Black Holes and Gravitational Waves", edited by T. Nakamura & H. Kodama, Prog. Theor. Phys. Supp

Site percolation and random walks on d-dimensional Kagome lattices

The site percolation problem is studied on d-dimensional generalisations of the Kagome' lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d. The site percolation thresholds are calculated numerically for d= 3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually assumed to hold for all lattices in high dimensions. A series expansion is calculated, in order to understand the different behaviour of the Kagome' lattice. The return probability of a random walker on these lattices is also shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices these results imply p_c ~ 1/(q-1).Comment: 11 pages, LaTeX, 8 figures (EPS format), submitted to J. Phys.

A Dynamical Systems Approach to Schwarzschild Null Geodesics

The null geodesics of a Schwarzschild black hole are studied from a dynamical systems perspective. Written in terms of Kerr-Schild coordinates, the null geodesic equation takes on the simple form of a particle moving under the influence of a Newtonian central force with an inverse-cubic potential. We apply a McGehee transformation to these equations, which clearly elucidates the full phase space of solutions. All the null geodesics belong to one of four families of invariant manifolds and their limiting cases, further characterized by the angular momentum L of the orbit: for |L|>|L_c|, (1) the set that flow outward from the white hole, turn around, then fall into the black hole, (2) the set that fall inward from past null infinity, turn around outside the black hole to continue to future null infinity, and for |L|<|L_c|, (3) the set that flow outward from the white hole and continue to future null infinity, (4) the set that flow inward from past null infinity and into the black hole. The critical angular momentum Lc corresponds to the unstable circular orbit at r=3M, and the homoclinic orbits associated with it. There are two additional critical points of the flow at the singularity at r=0. Though the solutions of geodesic motion and Hamiltonian flow we describe here are well known, what we believe is a novel aspect of this work is the mapping between the two equivalent descriptions, and the different insights each approach can give to the problem. For example, the McGehee picture points to a particularly interesting limiting case of the class (1) that move from the white to black hole: in the limit as L goes to infinity, as described in Schwarzschild coordinates, these geodesics begin at r=0, flow along t=constant lines, turn around at r=2M, then continue to r=0. During this motion they circle in azimuth exactly once, and complete the journey in zero affine time.Comment: 14 pages, 3 Figure

Black hole tidal problem in the Fermi normal coordinates

We derive a tidal potential for a self-gravitating fluid star orbiting Kerr black hole along a timelike geodesic extending previous works by Fishbone and Marck. In this paper, the tidal potential is calculated up to the third and fourth-order terms in $R/r$, where $R$ is the stellar radius and $r$ the orbital separation, in the Fermi-normal coordinate system following the framework developed by Manasse and Misner. The new formulation is applied for determining the tidal disruption limit (Roche limit) of corotating Newtonian stars in circular orbits moving on the equatorial plane of Kerr black holes. It is demonstrated that the third and fourth-order terms quantitatively play an important role in the Roche limit for close orbits with R/r \agt 0.1. It is also indicated that the Roche limit of neutron stars orbiting a stellar-mass black hole near the innermost stable circular orbit may depend sensitively on the equation of state of the neutron star.Comment: Correct typo