Naked and Thunderbolt Singularities in Black Hole Evaporation

If an evaporating black hole does not settle down to a non radiating remnant, a description by a semi classical Lorentz metric must contain either a naked singularity or what we call a thunderbolt, a singularity that spreads out to infinity on a spacelike or null path. We investigate this question in the context of various two dimensional models that have been proposed. We find that if the semi classical equations have an extra symmetry that make them solvable in closed form, they seem to predict naked singularities but numerical calculations indicate that more general semi classical equations, such as the original CGHS ones give rise to thunderbolts. We therefore expect that the semi classical approximation in four dimensions will lead to thunderbolts. We interpret the prediction of thunderbolts as indicating that the semi classical approximation breaks down at the end point of black hole evaporation, and we would expect that a full quantum treatment would replace the thunderbolt with a burst of high energy particles. The energy in such a burst would be too small to account for the observed gamma ray bursts.Comment: 21 pages (10 diagrams available on request

Collaborative ERP Curriculum Developing Using Industry Process Models

This paper presents and discusses the design of a problem based learning approach that seeks to embed industrial knowledge in the curriculum. It describes a project currently underway that is developing a business reference model using Process Engineering techniques. This reference model is being implemented in the leading Enterprise-wide System (also known as Enterprise Resource Planning System) SAP R/3. Teaching cases are being developed through collaboration between universities and industry. These teaching cases are to be available for use in the IS curriculum, irrespective of which faculty in which this curriculum is found. The teaching cases will also be available for wide distribution. This paper argues that this approach is in alignment with the recommendations of key curriculum documents and educational approaches. It also argues that the resultant teaching cases will be attractive to students, meet the current requirements of industry while maintaining the focus on education and the fundamentals of the IS Curriculum. This paper is the result of collaborative activity of two Australian Universities and one American University seeking to develop appropriate curriculum material and working collaboratively with other universities

Black hole pair creation and the stability of flat space

We extend the Gross-Perry-Yaffe approach of hot flat space instability to Minkowski space. This is done by a saddle point approximation of the partition function in a Schwarzschild wormhole background which is coincident with an eternal black hole. The appearance of an instability in the whole manifold is here interpreted as a black hole pair creation.Comment: 11 pages,RevTeX4, 2 figures. Accepted for publication in Int. J. Mod. Phys.

On Further Generalization of the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

A rigidity theorem that applies to smooth electrovac spacetimes which represent either (A) an asymptotically flat stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics was given in a recent work \cite{frw}. Here we enlarge the framework of the corresponding investigations by allowing the presence of other type of matter fields. In the first part the matter fields are involved merely implicitly via the assumption that the dominant energy condition is satisfied. In the second part Einstein-Klein-Gordon (EKG), Einstein-[non-Abelian] Higgs (E[nA]H), Einstein-[Maxwell]-Yang-Mills-dilaton (E[M]YMd) and Einstein-Yang-Mills-Higgs (EYMH) systems are studied. The black hole event horizon or, respectively, the compact Cauchy horizon of the considered spacetimes is assumed to be a smooth non-degenerate null hypersurface. It is proven that there exists a Killing vector field in a one-sided neighborhood of the horizon in EKG, E[nA]H, E[M]YMd and EYMH spacetimes. This Killing vector field is normal to the horizon, moreover, the associated matter fields are also shown to be invariant with respect to it. The presented results provide generalizations of the rigidity theorems of Hawking (for case A) and of Moncrief and Isenberg (for case B) and, in turn, they strengthen the validity of both the black hole rigidity scenario and the strong cosmic censor conjecture of classical general relativity.Comment: 25 pages, LaTex, a shortened version, including a new proof for lemma 5.1, the additional case of Einstein-Yang-Mills-Higgs systems is also covered, to appear in Class. Quant. Gra

Gravitational Waves in Open de Sitter Space

We compute the spectrum of primordial gravitational wave perturbations in open de Sitter spacetime. The background spacetime is taken to be the continuation of an O(5) symmetric instanton saddle point of the Euclidean no boundary path integral. The two-point tensor fluctuations are computed directly from the Euclidean path integral. The Euclidean correlator is then analytically continued into the Lorentzian region where it describes the quantum mechanical vacuum fluctuations of the graviton field. Unlike the results of earlier work, the correlator is shown to be unique and well behaved in the infrared. We show that the infrared divergence found in previous calculations is due to the contribution of a discrete gauge mode inadvertently included in the spectrum.Comment: 17 pages, compressed and RevTex file, including one postscript figure fil

Comments on gauge-invariance in cosmology

We revisit the gauge issue in cosmological perturbation theory, and highlight its relation to the notion of covariance in general relativity. We also discuss the similarities and differences of the covariant approach in perturbation theory to the Bardeen or metric approach in a non-technical fashion.Comment: 7 pages, 1 figure, revtex4; v3: minor changes, typos corrected, discussion extended; v4: typos corrected, corresponding to published versio

The Singularity Problem for Space-Times with Torsion

The problem of a rigorous theory of singularities in space-times with torsion is addressed. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors have done, because their definition of geodesics only involves the Christoffel connection, though studying theories with torsion. We propose a preliminary definition of singularities which is based on timelike or null geodesic incompleteness, even though for theories with torsion the paths of particles are not geodesics. The study of the geodesic equation for cosmological models with torsion shows that the definition has a physical relevance. It can also be motivated, as done in the literature, remarking that the causal structure of a space-time with torsion does not get changed with respect to general relativity. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Hawking's theorem can be generalized, provided the torsion tensor obeys some conditions. Thus our result can also be interpreted as a no-singularity theorem if these additional conditions are not satisfied. In other words, it turns out that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work is to be compared with previous papers in the literature. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy momentum tensor,Comment: 17 pages, plain-tex, published in Nuovo Cimento B, volume 105, pages 75-90, year 199

Higher order corrections to primordial spectra from cosmological inflation

We calculate power spectra of cosmological perturbations at high accuracy for two classes of inflation models. We classify the models according to the behaviour of the Hubble distance during inflation. Our approximation works if the Hubble distance can be approximated either to be a constant or to grow linearly with cosmic time. Many popular inflationary models can be described in this way, e.g., chaotic inflation with a monomial potential, power-law inflation and inflation at a maximum. Our scheme of approximation does not rely on a slow-roll expansion. Thus we can make accurate predictions for some of the models with large slow-roll parameters.Comment: 13 pages, 1 figure; section on consistency relations of inflation added; accepted by Physics Letters

Geometrization of metric boundary data for Einstein's equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein's equations.Comment: Article to appear in Gen. Rel. Grav. volume in memory of Juergen Ehler

Spacetime geometry of static fluid spheres

We exhibit a simple and explicit formula for the metric of an arbitrary static spherically symmetric perfect fluid spacetime. This class of metrics depends on one freely specifiable monotone non-increasing generating function. We also investigate various regularity conditions, and the constraints they impose. Because we never make any assumptions as to the nature (or even the existence) of an equation of state, this technique is useful in situations where the equation of state is for whatever reason uncertain or unknown. To illustrate the power of the method we exhibit a new form of the ``Goldman--I'' exact solution and calculate its total mass. This is a three-parameter closed-form exact solution given in terms of algebraic combinations of quadratics. It interpolates between (and thereby unifies) at least six other reasonably well-known exact solutions.Comment: Plain LaTeX 2e -- V2: now 22 pages; minor presentation changes in the first part of the paper -- no physics modifications; major additions to the examples section: the Gold-I solution is shown to be identical to the G-G solution. The interior Schwarzschild, Stewart, Buch5 XIII, de Sitter, anti-de Sitter, and Einstein solutions are all special cases. V3: Reference, footnotes, and acknowledgments added, typos fixed -- no physics modifications. V4: Technical problems with mass formula fixed -- affects discussion of our examples but not the core of the paper. Version to appear in Classical and Quantum Gravit