Two physical characteristics of numerical apparent horizons

This article translates some recent results on quasilocal horizons into the language of $(3+1)$ general relativity so as to make them more useful to numerical relativists. In particular quantities are described which characterize how quickly an apparent horizon is evolving and how close it is to either equilibrium or extremality.Comment: 6 pages, 2 figures, conference proceedings loosely based on talk given at Theory Canada III (Edmonton, Alberta, 2007). V2: Minor changes in response to referees comments to improve clarity and fix typos. One reference adde

Comparative Monte Carlo Efficiency by Monte Carlo Analysis

We propose a modified power method for computing the subdominant eigenvalue $\lambda_2$ of a matrix or continuous operator. Here we focus on defining simple Monte Carlo methods for its application. The methods presented use random walkers of mixed signs to represent the subdominant eigenfuction. Accordingly, the methods must cancel these signs properly in order to sample this eigenfunction faithfully. We present a simple procedure to solve this sign problem and then test our Monte Carlo methods by computing the $\lambda_2$ of various Markov chain transition matrices. We first computed ${\lambda_2}$ for several one and two dimensional Ising models, which have a discrete phase space, and compared the relative efficiencies of the Metropolis and heat-bath algorithms as a function of temperature and applied magnetic field. Next, we computed $\lambda_2$ for a model of an interacting gas trapped by a harmonic potential, which has a mutidimensional continuous phase space, and studied the efficiency of the Metropolis algorithm as a function of temperature and the maximum allowable step size $\Delta$. Based on the $\lambda_2$ criterion, we found for the Ising models that small lattices appear to give an adequate picture of comparative efficiency and that the heat-bath algorithm is more efficient than the Metropolis algorithm only at low temperatures where both algorithms are inefficient. For the harmonic trap problem, we found that the traditional rule-of-thumb of adjusting $\Delta$ so the Metropolis acceptance rate is around 50% range is often sub-optimal. In general, as a function of temperature or $\Delta$, $\lambda_2$ for this model displayed trends defining optimal efficiency that the acceptance ratio does not. The cases studied also suggested that Monte Carlo simulations for a continuum model are likely more efficient than those for a discretized version of the model.Comment: 23 pages, 8 figure

Fundamental properties and applications of quasi-local black hole horizons

The traditional description of black holes in terms of event horizons is inadequate for many physical applications, especially when studying black holes in non-stationary spacetimes. In these cases, it is often more useful to use the quasi-local notions of trapped and marginally trapped surfaces, which lead naturally to the framework of trapping, isolated, and dynamical horizons. This framework allows us to analyze diverse facets of black holes in a unified manner and to significantly generalize several results in black hole physics. It also leads to a number of applications in mathematical general relativity, numerical relativity, astrophysics, and quantum gravity. In this review, I will discuss the basic ideas and recent developments in this framework, and summarize some of its applications with an emphasis on numerical relativity.Comment: 14 pages, 2 figures. Based on a talk presented at the 18th International Conference on General Relativity and Gravitation, 8-13 July 2007, Sydney, Australi

Phase Diagrams For The Blue Phases Of Highly Chiral Liquid Crystals

Polarizing microscopy and optical-activity measurements are used to determine the phase diagram for the blue phases of chiral-racemic mixtures of terephthaloyloxy-bis-4-(2\u27-methylbutyl) benzoate. Contrary to an earlier report, it is the second blue phase (BP II) rather than the first blue phase (BP 1) that is not stable relative to the other blue phases at high chirality. With this development, all phase diagrams for the blue phases reported to date have the same topology. Using similar data for two other highly chiral systems, it is found that a simple scaling of the temperature and chiral-fraction axes produces phase diagrams in quantitative agreement with the present results. Thus, in spite of differences in molecular structure, the number of chiral centers, and phase-transition temperatures, these three systems possess remarkably similar phase diagrams and lend evidence for a universal phase diagram for the blue phases

Interpreting the extended emission around three nearby debris disc host stars

Cool debris discs are a relic of the planetesimal formation process around their host star, analogous to the solar system's Edgeworth-Kuiper belt. As such, they can be used as a proxy to probe the origin and formation of planetary systems like our own. The Herschel Open Time Key Programmes "DUst around NEarby Stars" (DUNES) and "Disc Emission via a Bias-free Reconnaissance in the Infrared/Submillimetre" (DEBRIS) observed many nearby, sun-like stars at far-infrared wavelengths seeking to detect and characterize the emission from their circumstellar dust. Excess emission attributable to the presence of dust was identified from around $\sim$ 20% of stars. Herschel's high angular resolution ($\sim$ 7" FWHM at 100 $\mu$m) provided the capacity for resolving debris belts around nearby stars with radial extents comparable to the solar system (50 to 100 au). As part of the DUNES and DEBRIS surveys, we obtained observations of three debris disc stars, HIP 22263 (HD 30495), HIP 62207 (HD 110897), and HIP 72848 (HD 131511), at far-infrared wavelengths with the Herschel PACS instrument. Combining these new images and photometry with ancilliary data from the literature, we undertook simultaneous multi-wavelength modelling of the discs' radial profiles and spectral energy distributions using three different methodologies: single annulus, modified black body, and a radiative transfer code. We present the first far-infrared spatially resolved images of these discs and new single-component debris disc models. We characterize the capacity of the models to reproduce the disc parameters based on marginally resolved emission through analysis of two sets of simulated systems (based on the HIP 22263 and HIP 62207 data) with the noise levels typical of the Herschel images. We find that the input parameter values are recovered well at noise levels attained in the observations presented here.Comment: 13 pages, 5 figures, 5 tables, accepted for publication in A&

Hamiltonian, Energy and Entropy in General Relativity with Non-Orthogonal Boundaries

A general recipe to define, via Noether theorem, the Hamiltonian in any natural field theory is suggested. It is based on a Regge-Teitelboim-like approach applied to the variation of Noether conserved quantities. The Hamiltonian for General Relativity in presence of non-orthogonal boundaries is analysed and the energy is defined as the on-shell value of the Hamiltonian. The role played by boundary conditions in the formalism is outlined and the quasilocal internal energy is defined by imposing metric Dirichlet boundary conditions. A (conditioned) agreement with previous definitions is proved. A correspondence with Brown-York original formulation of the first principle of black hole thermodynamics is finally established.Comment: 29 pages with 1 figur

Stationary untrapped boundary conditions in general relativity

A class of boundary conditions for canonical general relativity are proposed and studied at the quasi-local level. It is shown that for untrapped or marginal surfaces, fixing the area element on the 2-surface (rather than the induced 2-metric) and the angular momentum surface density is enough to have a functionally differentiable Hamiltonian, thus providing definition of conserved quantities for the quasi-local regions. If on the boundary the evolution vector normal to the 2-surface is chosen to be proportional to the dual expansion vector, we obtain a generalization of the Hawking energy associated with a generalized Kodama vector. This vector plays the role for the stationary untrapped boundary conditions which the stationary Killing vector plays for stationary black holes. When the dual expansion vector is null, the boundary conditions reduce to the ones given by the non-expanding horizons and the null trapping horizons.Comment: 11 pages, improved discussion section, a reference added, accepted for publication in Classical and Quantum Gravit