On the constraint algebra of quantum gravity in the loop representation

Although an important issue in canonical quantization, the problem of representing the constraint algebra in the loop representation of quantum gravity has received little attention. The only explicit computation was performed by Gambini, Garat and Pullin for a formal point-splitting regularization of the diffeomorphism and Hamiltonian constraints. It is shown that the calculation of the algebra simplifies considerably when the constraints are expressed not in terms of generic area derivatives but rather as the specific shift operators that reflect the geometric meaning of the constraints.Comment: 17 pages, LaTeX, 2 figures included with eps

Monte Carlo simulations of 4d simplicial quantum gravity

Dynamical triangulations of four-dimensional Euclidean quantum gravity give rise to an interesting, numerically accessible model of quantum gravity. We give a simple introduction to the model and discuss two particularly important issues. One is that contrary to recent claims there is strong analytical and numerical evidence for the existence of an exponential bound that makes the partition function well-defined. The other is that there may be an ambiguity in the choice of the measure of the discrete model which could even lead to the existence of different universality classes.Comment: 16 pages, LaTeX, epsf, 4 uuencoded figures; contribution to the JMP special issue on "Quantum Geometry and Diffeomorphism-Invariant Quantum Field Theory

Spinning black hole in the puncture method: Numerical experiments

The strong-field region inside a black hole needs special attention during numerical simulation. One approach for handling the problem is the moving puncture method, which has become an important tool in numerical relativity since it allows long term simulations of binary black holes. An essential component of this method is the choice of the '1+log'-slicing condition. We present an investigation of this slicing condition in rotating black hole spacetimes. We discuss how the results of the stationary Schwarzschild '1+log'-trumpet change when spin is added. This modification enables a simple and cheap algorithm for determining the spin of a non-moving black hole for this particular slicing condition. Applicability of the algorithm is verified in simulations of single black hole, binary neutron star and mixed binary simulations

Comparison between numerical relativity and a new class of post-Newtonian gravitational-wave phase evolutions: the non-spinning equal-mass case

We compare the phase evolution of equal-mass nonspinning black-hole binaries from numerical relativity (NR) simulations with post-Newtonian (PN) results obtained from three PN approximants: the TaylorT1 and T4 approximants, for which NR-PN comparisons have already been performed in the literature, and the recently proposed approximant TaylorEt. The accumulated phase disagreement between NR and PN results over the frequency range $M\omega = 0.0455$ to $M\omega = 0.1$ is greater for TaylorEt than either T1 or T4, but has the attractive property of decreasing monotonically as the PN order is increased.Comment: 6 pages, 4 figure

More on the exponential bound of four dimensional simplicial quantum gravity

A crucial requirement for the standard interpretation of Monte Carlo simulations of simplicial quantum gravity is the existence of an exponential bound that makes the partition function well-defined. We present numerical data favoring the existence of an exponential bound, and we argue that the more limited data sets on which recently opposing claims were based are also consistent with the existence of an exponential bound.Comment: 10 pages, latex, 2 figure

High-spin binary black hole mergers

We study identical mass black hole binaries with spins perpendicular to the binary's orbital plane. These binaries have individual spins ranging from $s/m^2=-0.90$ to 0.90, ($s_1 = s_2$ in all cases) which is near the limit possible with standard Bowen-York puncture initial data. The extreme cases correspond to the largest initial spin simulations to date. Our results expand the parameter space covered by Rezzolla {\it et al.} and, when combining both data sets, we obtain estimations for the minimum and maximum values for the intrinsic angular momenta of the remnant of binary black hole mergers of $J/M^2=0.341 \pm 0.004$ and $0.951 \pm 0.004$ respectively. Note, however, that these values are reached through extrapolation to the singular cases $|s_1| = |s_2| = 1$ and thus remain as {\it estimates} until full-fledged numerical simulations provide confirmation.Comment: 8 pages, 7 figures. Changed to match the version accepted for publication in PR

How the Jones Polynomial Gives Rise to Physical States of Quantum General Relativity

Solutions to both the diffeomorphism and the hamiltonian constraint of quantum gravity have been found in the loop representation, which is based on Ashtekar's new variables. While the diffeomorphism constraint is easily solved by considering loop functionals which are knot invariants, there remains the puzzle why several of the known knot invariants are also solutions to the hamiltonian constraint. We show how the Jones polynomial gives rise to an infinite set of solutions to all the constraints of quantum gravity thereby illuminating the structure of the space of solutions and suggesting the existance of a deep connection between quantum gravity and knot theory at a dynamical level.Comment: 7p

Binary Black Hole Mergers in 3d Numerical Relativity

The standard approach to the numerical evolution of black hole data using the ADM formulation with maximal slicing and vanishing shift is extended to non-symmetric black hole data containing black holes with linear momentum and spin by using a time-independent conformal rescaling based on the puncture representation of the black holes. We give an example for a concrete three dimensional numerical implementation. The main result of the simulations is that this approach allows for the first time to evolve through a brief period of the merger phase of the black hole inspiral.Comment: 8 pages, 9 figures, REVTeX; expanded discussion, results unchange