A gravitational collapse singularity theorem that improves Penrose's

The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in absence of predictability and hence compatibly with quatum field theory and black hole evaporation.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Characterization of some causality conditions through the continuity of the Lorentzian distance

A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proven here that a non-total imprisoning spacetime is globally hyperbolic if and only if for every metric choice in the conformal class the Lorentzian distance is continuous. Moreover, it is proven that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it vanishes. Finally, a strongly causal spacetime is causally continuous if and only if there is at least one metric in the conformal class such that the Lorentzian distance is continuous wherever it vanishes.Comment: 14 pages, 2 figure. v2: Added material on global hyperbolicity. The title has changed. Previous title: Characterization of causal simplicity and causal continuity through the continuity of the Lorentzian distance. v3: Some misprints fixed. Final versio

Orthogonal polynomial method and odd vertices in matrix models

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.Comment: 13 pages, 3 Postscript figure

Gravity theory through affine spheres

In this work it is argued that in order to improve our understanding of gravity and spacetime our most successful theory, general relativity, must be destructured. That is, some geometrical assumptions must be dropped and recovered just under suitable limits. Along this line of thought we pursue the idea that the roundness of the light cone, and hence the isotropy of the speed of light, must be relaxed and that, in fact, the shape of light cones must be regarded as a dynamical variable. Mathematically, we apply some important results from affine differential geometry to this problem, the idea being that in the transition we should preserve the identification of the spacetime continuum with a manifold endowed with a cone structure and a spacetime volume form. To that end it is suggested that the cotangent indicatrix (dispersion relation) must be described by an equation of Monge-Ampere type determining a hyperbolic affine sphere, at least whenever the matter content is negligible. Non-relativistic spacetimes fall into this description as they are recovered whenever the center of the affine sphere is at infinity. In the more general context of Lorentz-Finsler theories it is shown that the lightlike unparametrized geodesic flow is completely determined by the distribution of light cones. Moreover, the transport of lightlike momenta is well defined though there could be no notion of affine parameter. Finally, we show how the perturbed indicatrix can be obtained from the perturbed light cone

On differentiability of volume time functions

We show differentiability of a class of Geroch's volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally anti-Lipschitz can be uniformly approximated by smooth time functions with timelike gradient. Finally, we prove that in stably causal spacetimes Hawking's time function can be uniformly approximated by smooth time functions with timelike gradient

Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem

We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the $\epsilon$-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar-Wylie into a single framework.Comment: 29 page