Low Frequency Gravitational Waves from Black Hole MACHO Binaries

Nakamura, Sasaki, Tanaka, and Thorne have recently estimated the initial distribution of binary MACHOs in the galactic halo assuming that the MACHOs are primordial half solar mass black holes, and considered their coalescence as a possible source for ground-based interferometer gravitational wave detectors such as LIGO. Evolving their binary distribution forward in time to the present, the low-frequency (10^{-5} < f < 10^{-1} Hz) spectrum of gravitational waves associated with such a population of compact binaries is calculated. The resulting gravitational waves would form a strong stochastic background in proposed space interferometers such as LISA and OMEGA. Low frequency gravitational waves are likely to become a key tool for determining the properties of binaries within the dark MACHO population.Comment: 8 pages + 2 ps figures; AASTe

Testing a Simplified Version of Einstein's Equations for Numerical Relativity

Solving dynamical problems in general relativity requires the full machinery of numerical relativity. Wilson has proposed a simpler but approximate scheme for systems near equilibrium, like binary neutron stars. We test the scheme on isolated, rapidly rotating, relativistic stars. Since these objects are in equilibrium, it is crucial that the approximation work well if we are to believe its predictions for more complicated systems like binaries. Our results are very encouraging.Comment: 9 pages (RevTeX 3.0 with 6 uuencoded figures), CRSR-107

Negative-weight percolation

We describe a percolation problem on lattices (graphs, networks), with edge weights drawn from disorder distributions that allow for weights (or distances) of either sign, i.e. including negative weights. We are interested whether there are spanning paths or loops of total negative weight. This kind of percolation problem is fundamentally different from conventional percolation problems, e.g. it does not exhibit transitivity, hence no simple definition of clusters, and several spanning paths/loops might coexist in the percolation regime at the same time. Furthermore, to study this percolation problem numerically, one has to perform a non-trivial transformation of the original graph and apply sophisticated matching algorithms. Using this approach, we study the corresponding percolation transitions on large square, hexagonal and cubic lattices for two types of disorder distributions and determine the critical exponents. The results show that negative-weight percolation is in a different universality class compared to conventional bond/site percolation. On the other hand, negative-weight percolation seems to be related to the ferromagnet/spin-glass transition of random-bond Ising systems, at least in two dimensions.Comment: v1: 4 pages, 4 figures; v2: 10 pages, 7 figures, added results, text and reference

Computing the Similarity Between Moving Curves

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality

Implementing an apparent-horizon finder in three dimensions

Locating apparent horizons is not only important for a complete understanding of numerically generated spacetimes, but it may also be a crucial component of the technique for evolving black-hole spacetimes accurately. A scheme proposed by Libson et al., based on expanding the location of the apparent horizon in terms of symmetric trace-free tensors, seems very promising for use with three-dimensional numerical data sets. In this paper, we generalize this scheme and perform a number of code tests to fully calibrate its behavior in black-hole spacetimes similar to those we expect to encounter in solving the binary black-hole coalescence problem. An important aspect of the generalization is that we can compute the symmetric trace-free tensor expansion to any order. This enables us to determine how far we must carry the expansion to achieve results of a desired accuracy. To accomplish this generalization, we describe a new and very convenient set of recurrence relations which apply to symmetric trace-free tensors.Comment: 14 pages (RevTeX 3.0 with 3 figures