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Antonio N. Bernal

Publications and source records attributed to Antonio N. Bernal.

10 recordsLinked to original sources
arXiv2017

On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism

We study the most general solution for affine connections that are compatible with the variational principle in the Palatini formalism for the Einstein-Hilbert action (with possible minimally coupled matter terms). We find that there is a family of solutions generalising the Levi-Civita connection, characterised by an arbitrary, non-dynamical vector field ${\cal A}_μ$. We discuss the mathematical properties and the physical implications of this family and argue that, although there is a clear mathematical difference between these new Palatini connections and the Levi-Civita one, both unparametrised geodesics and the Einstein equation are shared by all of them. Moreover, the Palatini connections are characterised precisely by these two properties, as well as by other properties of its parallel transport. Based on this, we conclude that physical effects associated to the choice of one or the other will not be distinguishable, at least not at the level of solutions or test particle dynamics. We propose a geometrical interpretation for the existence and unobservability of the new solutions.

gr-qc↗
arXiv2008

Physics from scratch. Letter on M. Tegmark's ``The Mathematical Universe''

In a recent article, M. Tegmark poses the hypothesis that our known universe is a ``baggage free'' mathematical structure among many other possible ones, which also correspond to other physical universes --Mathematical Universe Hypothesis, MUH. Naturally, questions arise, such as how to obtain the physical properties of our world from the mathematical structure, or how many possibilities exist for a Universe minimally similar to ours. In this letter we present some results which can be regarded as a strengthening of MUH, as they give some hints on the derivation of spacetime in current physics from a baggage free mathematical structure. Concretely, we argue that the set of mathematical structures which can be interpreted as a description of a spacetime is drastically reduced, if one admits some natural postulates on minimal symmetry. Furthermore, the apparently very particular form of classical Galilei-Newton and relativistic spacetimes, is not arbitrary and cannot be regarded as ``two possibilities among arbitrarily many others''. In fact, such theories are determined by a single mathematical structure which only permits four possible types of spacetimes. Finally, we show how the minimal postulates on symmetry can be endowed with a simple physical interpretation, i.e., they acquire ``baggage'' in a natural way.

gr-qc↗
arXiv2006

Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"

The classical definition of {\em global hyperbolicity} for a spacetime $(M,g)$ comprises two conditions: (A) compactness of the diamonds $J^+(p)\cap J^-(q)$, and (B) strong causality. Here we show that condition (B) can be replaced just by causality. In fact, we show first that the classical definition of causal simplicity (which impose to be distinguishing, apart from the closedness of $J^+(p)$, $J^-(q)$) can be weakened in causal instead of distinguishing. So, the full consistency of the causal ladder (recently proved by the authors in a definitive way) yields directly the result.

gr-qc↗
arXiv2005

Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R \times S$ in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).

gr-qc↗
arXiv2005

Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes

The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $τ$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting $M= \R \times {\cal S}$, $g= - β(τ,x) dτ^2 + \bar g_τ$, (b) if a spacetime $M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $τ$ with timelike gradient $\nabla τ$ on all $M$.

gr-qc↗
arXiv2004

Smooth globally hyperbolic splittings and temporal functions

Geroch's theorem about the splitting of globally hyperbolic spacetimes is a central result in global Lorentzian Geometry. Nevertheless, this result was obtained at a topological level, and the possibility to obtain a metric (or, at least, smooth) version has been controversial since its publication in 1970. In fact, this problem has remained open until a definitive proof, recently provided by the authors. Our purpose is to summarize the history of the problem, explain the smooth and metric splitting results (including smoothability of time functions in stably causal spacetimes), and sketch the ideas of the solution.

gr-qc↗
arXiv2002

Leibnizian, Galilean and Newtonian structures of spacetime

The following three geometrical structures on a manifold are studied in detail: (1) Leibnizian: a non-vanishing 1-form $Ω$ plus a Riemannian metric $\h$ on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized. (2) Galilean: Leibnizian structure endowed with an affine connection $\nabla$ (gauge field) which parallelizes $Ω$ and $\h$. Fixed any vector field of observers Z ($Ω(Z) = 1$), an explicit Koszul--type formula which reconstruct bijectively all the possible $\nabla$'s from the gravitational ${\cal G} = \nabla_Z Z$ and vorticity $ω= rot Z/2$ fields (plus eventually the torsion) is provided. (3) Newtonian: Galilean structure with $\h$ flat and a field of observers Z which is inertial (its flow preserves the Leibnizian structure and $ω= 0$). Classical concepts in Newtonian theory are revisited and discussed.

gr-qc↗