On the exact evaluation of spin networks

We introduce a fully coherent spin network amplitude whose expansion generates all SU(2) spin networks associated with a given graph. We then give an explicit evaluation of this amplitude for an arbitrary graph. We show how this coherent amplitude can be obtained from the specialization of a generating functional obtained by the contraction of parametrized intertwiners a la Schwinger. We finally give the explicit evaluation of this generating functional for arbitrary graphs

Spacetime structure of static solutions in Gauss-Bonnet gravity: charged case

We have studied spacetime structures of static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-Maxwell-$\Lambda$ system. Especially we focus on effects of the Maxwell charge. We assume that the Gauss-Bonnet coefficient $\alpha$ is non-negative and $4{\tilde \alpha}/\ell^2\leq 1$ in order to define the relevant vacuum state. Solutions have the $(n-2)$-dimensional Euclidean sub-manifold whose curvature is $k=1,~0$, or -1. In Gauss-Bonnet gravity, solutions are classified into plus and minus branches. In the plus branch all solutions have the same asymptotic structure as those in general relativity with a negative cosmological constant. The charge affects a central region of the spacetime. A branch singularity appears at the finite radius $r=r_b>0$ for any mass parameter. There the Kretschmann invariant behaves as $O((r-r_b)^{-3})$, which is much milder than divergent behavior of the central singularity in general relativity $O(r^{-4(n-2)})$. Some charged black hole solutions have no inner horizon in Gauss-Bonnet gravity. Although there is a maximum mass for black hole solutions in the plus branch for $k=-1$ in the neutral case, no such maximum exists in the charged case. The solutions in the plus branch with $k=-1$ and $n\geq6$ have an "inner" black hole, and inner and the "outer" black hole horizons. Considering the evolution of black holes, we briefly discuss a classical discontinuous transition from one black hole spacetime to another.Comment: 20 pages, 10 figure

Spin-Raising Operators and Spin-3/2 Potentials in Quantum Cosmology

Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This paper derives the conditions under which spin-raising operators preserve these local boundary conditions on a 3-sphere for fields of spin 0,1/2,1,3/2 and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3/2 is applied to the case of a 3-sphere boundary. It is shown that such boundary conditions can only be imposed in a flat Euclidean background, for which the gauge freedom in the choice of the potentials remains.Comment: 13 pages, plain-tex, recently appearing in Classical and Quantum Gravity, volume 11, April 1994, pages 897-903. Apologies for the delay in circulating the file, due to technical problems now fixe

The Seeds of Cosmic structure as a door to New Physics

There is something missing in our understanding of the origin of the seeds of Cosmic Structuture. The fact that the fluctuation spectrum can be extracted from the inflationary scenario through an analysis that involves quantum field theory in curved space-time, and that it coincides with the observational data has lead to a certain complacency in the community, which prevents the critical analysis of the obscure spots in the derivation. The point is that the inhomogeneity and anisotropy of our universe seem to emerge from an exactly homogeneous and isotropic initial state through processes that do not break those symmetries. This article gives a brief recount of the problems faced by the arguments based on established physics, which comprise the point of view held by a large majority of researchers in the field. The conclusion is that we need some new physics to be able to fully address the problem. The article then exposes one avenue that has been used to address the central issue and elaborates on the degree to which, the new approach makes different predictions from the standard analyses. The approach is inspired on Penrose's proposals that Quantum Gravity might lead to a real, dynamical collapse of the wave function, a process that we argue has the properties needed to extract us from the theoretical impasse described above.Comment: Prepared for the proceedings of the conference NEBXII " Recent Developments in Gravity", Napfio Grece June 2006. LateX, 15 page

Comment on "Absence of trapped surfaces and singularities in cylindrical collapse"

Recently, the gravitational collapse of an infinite cylindrical thin shell of matter in an otherwise empty spacetime with two hypersurface orthogonal Killing vectors was studied by Gon\c{c}alves [Phys. Rev. {\bf D65}, 084045 (2002).]. By using three "alternative" criteria for trapped surfaces, the author claimed to have shown that {\em they can never form either outside or on the shell, regardingless of the matter content for the shell, except at asymptotical future null infinite}. Following Penrose's original idea, we first define trapped surfaces in cylindrical spacetimes in terms of the expansions of null directions orthogonal to the surfaces, and then show that the first criterion used by Gon\c{c}alves is incorrect. We also show that his analysis of non-existence of trapped surfaces in vacuum is incomplete. To confirm our claim, we present an example that is a solution to the vacuum Einstein field equations and satisfies all the regular conditions imposed by Gon\c{c}alves. After extending the solution to the whole spacetime, we show explicitly that trapped surfaces exist in the extended region.Comment: latex, 2 figures, the last version to appear in Phys. Rev.

Minimal data at a given point of space for solutions to certain geometric systems

We consider a geometrical system of equations for a three dimensional Riemannian manifold. This system of equations has been constructed as to include several physically interesting systems of equations, such as the stationary Einstein vacuum field equations or harmonic maps coupled to gravity in three dimensions. We give a characterization of its solutions in a neighbourhood of a given point through sequences of symmetric trace free tensors (referred to as `null data'). We show that the null data determine a formal expansion of the solution and we obtain necessary and sufficient growth estimates on the null data for the formal expansion to be absolutely convergent in a neighbourhood of the given point. This provides a complete characterization of all the solutions to the given system of equations around that point.Comment: 26 pages, no figure

Degeneracy measures for the algebraic classification of numerical spacetimes

We study the issue of algebraic classification of the Weyl curvature tensor, with a particular focus on numerical relativity simulations. The spacetimes of interest in this context, binary black hole mergers, and the ringdowns that follow them, present subtleties in that they are generically, strictly speaking, Type I, but in many regions approximately, in some sense, Type D. To provide meaning to any claims of "approximate" Petrov class, one must define a measure of degeneracy on the space of null rays at a point. We will investigate such a measure, used recently to argue that certain binary black hole merger simulations ring down to the Kerr geometry, after hanging up for some time in Petrov Type II. In particular, we argue that this hangup in Petrov Type II is an artefact of the particular measure being used, and that a geometrically better-motivated measure shows a black hole merger produced by our group settling directly to Petrov Type D.Comment: 14 pages, 7 figures. Version 2 adds two references

Decoherence of Macroscopic Closed Systems within Newtonian Quantum Gravity

A theory recently proposed by the author aims to explain decoherence and the thermodynamical behaviour of closed systems within a conservative, unitary, framework for quantum gravity by assuming that the operators tied to the gravitational degrees of freedom are unobservable and equating physical entropy with matter-gravity entanglement entropy. Here we obtain preliminary results on the extent of decoherence this theory predicts. We treat first a static state which, if one were to ignore quantum gravitational effects, would be a quantum superposition of two spatially displaced states of a single classically well describable ball of uniform mass density in empty space. Estimating the quantum gravitational effects on this system within a simple Newtonian approximation, we obtain formulae which predict e.g. that as long as the mass of the ball is considerably larger than the Planck mass, such a would-be-coherent static superposition will actually be decohered whenever the separation of the centres of mass of the two ball-states excedes a small fraction (which decreases as the mass of the ball increases) of the ball radius. We then obtain a formula for the quantum gravitational correction to the would-be-pure density matrix of a non-relativistic many-body Schroedinger wave function and argue that this formula predicts decoherence between configurations which differ (at least) in the "relocation" of a cluster of particles of Planck mass. We estimate the entropy of some simple model closed systems, finding a tendency for it to increase with "matter-clumping" suggestive of a link with existing phenomenological discussions of cosmological entropy increase.Comment: 11 pages, plain TeX, no figures. Accepted for publication as a "Letter to the Editor" in "Classical and Quantum Gravity

The Positivity of Energy for Asymptotically Anti-de Sitter Spacetimes

We use the formulation of asymptotically anti-de Sitter boundary conditions given by Ashtekar and Magnon to obtain a coordinate expression for the general asymptotically AdeS metric in a neighbourhood of infinity. From this, we are able to compute the time delay of null curves propagating near infinity. If the gravitational mass is negative, so will be the time delay (relative to null geodesics at infinity) for certain null geodesics in the spacetime. Following closely an argument given by Penrose, Sorkin, and Woolgar, who treated the asymptotically flat case, we are then able to argue that a negative time delay is inconsistent with non-negative matter-energies in spacetimes having good causal properties. We thereby obtain a new positive mass theorem for these spacetimes. The theorem may be applied even when the matter flux near the boundary-at-infinity falls off so slowly that the mass changes, provided the theorem is applied in a time-averaged sense. The theorem also applies in certain spacetimes having local matter-energy that is sometimes negative, as can be the case in semi-classical gravity.Comment: (Plain TeX - figures not included

Spherical gravitational collapse in N-dimensions

We investigate here spherically symmetric gravitational collapse in a spacetime with an arbitrary number of dimensions and with a general {\it type I} matter field, which is a broad class that includes most of the physically reasonable matter forms. We show that given the initial data for matter in terms of the initial density and pressure profiles at an initial surface $t=t_i$ from which the collapse evolves, there exist rest of the initial data functions and classes of solutions of Einstein equations which we construct here, such that the spacetime evolution goes to a final state which is either a black hole or a naked singularity, depending on the nature of initial data and evolutions chosen, and subject to validity of the weak energy condition. The results are discussed and analyzed in the light of the cosmic censorship hypothesis in black hole physics. The formalism here combines the earlier results on gravitational collapse in four dimensions in a unified treatment. Also the earlier work is generalized to higher dimensional spacetimes to allow a study of the effect of number of dimensions on the possible final outcome of the collapse in terms of either a black hole or naked singularity. No restriction is adopted on the number of dimensions, and other limiting assumptions such as self-similarity of spacetime are avoided, in order to keep the treatment general. Our methodology allows to consider to an extent the genericity and stability aspects related to the occurrence of naked singularities in gravitational collapse.Comment: Revtex4, The replaced version matches the published on