An alternative approach to solving the Hamiltonian constraint

Solving Einstein's constraint equations for the construction of black hole initial data requires handling the black hole singularity. Typically, this is done either with the excision method, in which the black hole interior is excised from the numerical grid, or with the puncture method, in which the singular part of the conformal factor is expressed in terms of an analytical background solution, and the Hamiltonian constraint is then solved for a correction to the background solution that, usually, is assumed to be regular everywhere. We discuss an alternative approach in which the Hamiltonian constraint is solved for an inverse power of the conformal factor. This new function remains finite everywhere, so that this approach requires neither excision nor a split into background and correction. In particular, this method can be used without modification even when the correction to the conformal factor is singular itself. We demonstrate this feature for rotating black holes in the trumpet topology.Comment: 5 pages, 4 figures, matches version published in PR

Black Holes: from Speculations to Observations

This paper provides a brief review of the history of our understanding and knowledge of black holes. Starting with early speculations on ``dark stars'' I discuss the Schwarzschild "black hole" solution to Einstein's field equations and the development of its interpretation from "physically meaningless" to describing the perhaps most exotic and yet "most perfect" macroscopic object in the universe. I describe different astrophysical black hole populations and discuss some of their observational evidence. Finally I close by speculating about future observations of black holes with the new generation of gravitational wave detectors.Comment: 15 pages, 6 Figures; to appear in the Proceedings of the Albert Einstein Century International Conference, Paris, France, 200

The Innermost Stable Circular Orbit in Compact Binaries

Newtonian point mass binaries can be brought into arbitrarily close circular orbits. Neutron stars and black holes, however, are extended, relativistic objects. Both finite size and relativistic effects make very close orbits unstable, so that there exists an innermost stable circular orbit (ISCO). We illustrate the physics of the ISCO in a simple model problem, and review different techniques which have been employed to locate the ISCO in black hole and neutron star binaries. We discuss different assumptions and approximations, and speculate on how differences in the results may be explained and resolved.Comment: 13 pages, 2 figures, to appear in "Astrophysical Sources of Gravitational Radiation", edited by J. M. Centrella (AIP Press

Stabilization by modification of the Lagrangian

In order to reduce the error growth during a numerical integration, a method of stabilization of the differential equations of the Keplerian motion is offered. It is characterized by the use of the eccentric anomaly as an independent variable in such a way that the time transformation is given by a generalized Lagrange formalism. The control terms in the equations of motion obtained by this modified Lagrangian give immediately a completely Liapunov-stable set of differential equations. In contrast to other publications, here the equation of time integration is modified by a control term which leads to an integral which defined the time element for the perturbed Keplerian motion

Critical collapse of rotating radiation fluids

We present results from the first fully relativistic simulations of the critical collapse of rotating radiation fluids. We observe critical scaling both in subcritical evolutions, in which case the fluid disperses to infinity and leaves behind flat space, and in supercritical evolutions that lead to the formation of black holes. We measure the mass and angular momentum of these black holes, and find that both show critical scaling with critical exponents that are consistent with perturbative results. The critical exponents are universal; they are not affected by angular momentum, and are independent of the direction in which the critical curve, which separates subcritical from supercritical evolutions in our two-dimensional parameter space, is crossed. In particular, these findings suggest that the angular momentum decreases more rapidly than the square of the mass, so that, as criticality is approached, the collapse leads to the formation of a non-spinning black hole. We also demonstrate excellent agreement of our numerical data with new closed-form extensions of power-law scalings that describe the mass and angular momentum of rotating black holes formed close to criticality.Comment: 5 pages, 4 figures; version accepted for publication in PR

Luminosity versus Rotation in a Supermassive Star

We determine the effect of rotation on the luminosity of supermassive stars. We apply the Roche model to calculate analytically the emitted radiation from a uniformly rotating, radiation-dominated supermassive configuration. We find that the luminosity at maximum rotation, when mass at the equator orbits at the Kepler period, is reduced by ~36% below the usual Eddington luminosity from the corresponding nonrotating star. A supermassive star is believed to evolve in a quasistationary manner along such a maximally rotating ``mass-shedding'' sequence before reaching the point of dynamical instability; hence this reduced luminosity determines the evolutionary timescale. Our result therefore implies that the lifetime of a supermassive star prior to dynamical collapse is ~56% longer than the value typically estimated by employing the usual Eddington luminosity.Comment: 5 pages, 2 figures, uses emulateapj.sty; to appear in Ap

A formalism for the construction of binary neutron stars with arbitrary circulation

Most numerical models of binary stars - in particular neutron stars in compact binaries - assume the companions to be either corotational or irrotational. Either one of these assumptions leads to a significant simplification in the hydrodynamic equations of stationary equilibrium. In this paper we develop a new formalism for the construction of binary stars with circulation intermediate between corotational and irrotational. Generalizing the equations for irrotational flow we cast the Euler equation, which is an algebraic equation in the case of corotational or irrotational fluid flow, as an elliptic equation for a new auxiliary quantity. We also suggest a parameterized decomposition of the fluid flow that allows for a variation of the stellar circulation.Comment: 8 pages, no figures; published version with erratu