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Miguel Sánchez

Publications and source records attributed to Miguel Sánchez.

At least 37 records · Page 2Linked to original sources
arXiv2022

Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

The Cauchy slicings for globally hyperbolic spacetimes and their relation with the causal boundary are surveyed and revisited, starting at the seminal conformal boundary constructions by R. Penrose. Our study covers: (1) adaptive possibilities and techniques for their Cauchy slicings, (2) global hyperbolicity of sliced spacetimes, (3) critical review on the conformal and causal boundaries for a globally hyperbolic spacetime, and (4) procedures to compute the causal boundary of a Cauchy temporal splitting by using isocausal comparison with a static product. New simple counterexamples on $\mathbb{R}^2$ illustrate a variety of possibilities related to these splittings, such as the logical independence (for normalized sliced spacetimes) between the completeness of the slices and global hyperbolicity, the necessity of uniform bounds on the slicings in order to ensure global hyperbolicity, or the insufficience of these bounds for the computation of the causal boundary. A refinement of one of these examples shows that the space of all the (normalized, conformal classes of) globally hyperbolic metrics on a smooth product manifold $\mathbb{R}\times S$ is not convex, even though it is path connected by means of piecewise convex combinations.

gr-qc↗
arXiv2022

On the significance of the stress-energy tensor in Finsler spacetimes

We revisit the physical arguments which lead to the definition of the stress-energy tensor $T$ in the Lorentz-Finsler setting $(M,L)$ starting at classical Relativity. Both the standard heuristic approach using fluids and the Lagrangian one are taken into account. In particular, we argue that the Finslerian breaking of Lorentz symmetry makes $T$ an anisotropic 2-tensor (i. e., a tensor for each $L$-timelike direction), in contrast with the energy-momentum vectors defined on $M$. Such a tensor is compared with different ones obtained by using a Lagrangian approach. The notion of divergence is revised from a geometric viewpoint and, then, the conservation laws of $T$ for each observer field are revisited. We introduce a natural {\em anisotropic Lie bracket derivation}, which leads to a divergence obtained from the volume element and the non-linear connection associated with $L$ alone. The computation of this divergence selects the Chern anisotropic connection, thus giving a geometric interpretation to previous choices in the literature.

gr-qc↗
arXiv2021

Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes

The notion of wind Finslerian structure is developed; this is a generalization of Finsler metrics where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which provides: (a) a model for classical Zermelo's navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The wind Finslerian structure is described in terms of two (conic, pseudo) Finsler metrics, one with a convex indicatrix and the other with a concave one. However, the spacetime viewpoint for the wind Riemannian case gives a useful unified viewpoint. A thorough study of the causal properties of such a spacetime is carried out in Finslerian terms. Randers-Kropina metrics appear as the Finslerian counterpart to the case of an SSTK when the Killing vector field is either timelike or lightlike. Among the applications, we obtain the solution of Zermelo's navigation with arbitrary stationary wind, metric-type properties (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat's principle of independent interest for arbitrary spacetimes, as well as its applications to SSTK spacetimes and Zermelo's navigation, are also provided.

math.DG↗
arXiv2021

Applications of cone structures to the anisotropic rheonomic Huygens' principle

A general framework for the description of classic wave propagation is introduced. This relies on a cone structure $C$ determined by an intrinsic space $Σ$ of velocities of propagation (point, direction and time-dependent) and an observers' vector field $\partial_t$ whose integral curves provide both a Zermelo problem for the wave and an auxiliary Lorentz-Finsler metric $G$ compatible with $C$. The PDE for the wavefront is reduced to the ODE for the $t$-parametrized cone geodesics of $C$. Particular cases include time-independence ($\partial_t$ is Killing for $G$), infinitesimally ellipsoidal propagation ($G$ can be replaced by a Lorentz metric) or the case of a medium which moves with respect to $\partial_t$ faster than the wave (the strong wind case of a sound wave), where a conic time-dependent Finsler metric emerges. The specific case of wildfire propagation is revisited.

math.DG↗
arXiv2020

Foundations of Finsler spacetimes from the Observers' Viewpoint

Physical foundations for relativistic spacetimes are revisited, in order to check at what extent Finsler spacetimes lie in their framework. Arguments based on inertial observers (as in the foundations of Special Relativity and Classical Mechanics) are shown to correspond with a double linear approximation in the measurement of space and time. While General Relativity appears by dropping the first linearization, Finsler spacetimes appear by dropping the second one. The classical Ehlers-Pirani-Schild approach is carefully discussed and shown to be compatible with the Lorentz-Finsler case. The precise mathematical definition of Finsler spacetime is discussed by using the {\em space of observers}. Special care is taken in some issues such as: the fact that a Lorentz-Finsler metric would be physically measurable only on the causal directions for a cone structure, the implications for models of spacetimes of some apparently innocuous hypotheses on differentiability, or the possibilities of measurement of a varying speed of light.

gr-qc↗
arXiv2019

Ehlers-Kundt Conjecture about Gravitational Waves and Dynamical Systems

Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${\mathbb R}^2$ with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potential $V(z,u)$, $(z,u)\in {\mathbb R}^2\times {\mathbb R}$, harmonic in $z$ (i.e. source-free), the trajectories of its associated dynamical system $\ddot{z}(s)=-\nabla_z V(z(s),s)$ are complete (they live eternally) if and only if $V(z,u)$ is a polynomial in $z$ of degree at most $2$ (so that $V$ is a standard mathematical idealization of vacuum). Here, the conjecture is solved in the significative case that $V$ is bounded polynomially in $z$ for finite values of $u\in {\mathbb R}$. The mathematical and physical implications of this {\em polynomial EK conjecture}, as well as the non-polynomial one, are discussed beyond their original scope.

gr-qc↗
arXiv2019

On the definition and examples of cones and Finsler spacetimes

A systematic study of (smooth, strong) cone structures $\C$ and Lorentz-Finsler metrics $L$ is carried out. As a link between both notions, cone triples $(Ω,T, F)$, where $Ω$ (resp. $T$) is a 1-form (resp. vector field) with $Ω(T)\equiv 1$ and $F$, a Finsler metric on $\ker (Ω)$, are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures $\C$ are bijectively associated with classes of anisotropically conformal metrics $L$, and the notion of {\em cone geodesic} is introduced consistently with both structures. As a non-relativistic application, the {\em time-dependent} Zermelo navigation problem is posed rigorously, and its general solution is provided.

math.DG↗
arXiv2019

Structure of globally hyperbolic spacetimes with timelike boundary

Globally hyperbolic spacetimes with timelike boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\overline{M}$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of $\partial M$, the splitting of any globally hyperbolic $(\overline{M},g)$ as an orthogonal product ${\mathbb R}\times \barΣ$ with Cauchy slices with boundary $\{t\}\times \barΣ$ is proved. This is obtained by constructing a Cauchy temporal function $τ$ with gradient $\nabla τ$ tangent to $\partial M$ on the boundary. To construct such a $τ$, results on stability of both, global hyperbolicity and Cauchy temporal functions are obtained. Apart from having their own interest, these results allow us to circumvent technical difficulties introduced by $\partial M$. As a consequence, the interior $M$ both, splits orthogonally and can be embedded isometrically in ${\mathbb L}^N$, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.

gr-qc↗
arXiv2018

A note on stability and Cauchy time functions

Since the solution of the so-called folk problems of smoothability, there has been a special interest in the properties of classical time and volume functions of spacetimes. Here we supply some information that complements the one provided in arXiv:1108.5120v3 and arXiv:1301.2909v1, and discuss some results. As an illustration, we show that any Cauchy temporal function for a globally hyperbolic spacetime remains Cauchy temporal for close metrics ---which, in particular, implies stability for global hyperbolicity.

gr-qc↗
arXiv2017

Some criteria for Wind Riemannian completeness and existence of Cauchy hypersurfaces

Recently, a link between Lorentzian and Finslerian Geometries has been carried out, leading to the notion of wind Riemannian structure (WRS), a generalization of Finslerian Randers metrics. Here, we further develop this notion and its applications to spacetimes, by introducing some characterizations and criteria for the completeness of WRS's. As an application, we consider a general class of spacetimes admitting a time function $t$ generated by the flow of a complete Killing vector field (generalized standard stationary spacetimes or, more precisely, SSTK ones) and derive simple criteria ensuring that its slices $t=$ constant are Cauchy. Moreover, a brief summary on the Finsler/Lorentz link for readers with some acquaintance in Lorentzian Geometry, plus some simple examples in Mathematical Relativity, are provided.

math.DG↗
arXiv2017

Wind Riemannian spaceforms and Randers metrics of constant flag curvature

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

math.DG↗
arXiv2014

Finsler metrics and relativistic spacetimes

Recent links between Finsler Geometry and the geometry of spacetimes are briefly revisited, and prospective ideas and results are explained. Special attention is paid to geometric problems with a direct motivation in Relativity and other parts of Physics.

math.DG↗
arXiv2014

An Invitation to Lorentzian Geometry

The intention of this article is to give a flavour of some global problems in General Relativity. We cover a variety of topics, some of them related to the fundamental concept of 'Cauchy hypersurfaces': (1) structure of globally hyperbolic spacetimes, (2) the relativistic initial value problem, (3) constant mean curvature surfaces, (4) singularity theorems, (5) cosmic censorship and Penrose inequality, (6) spinors and holonomy.

math.DG↗
arXiv2013

On the completeness of trajectories for some mechanical systems

The classical tools which ensure the completeness of vector fields and second order differential equations for mechanical systems are revisited. Possible extensions in three directions are discussed: infinite dimensional Banach and Hilbert manifolds, Finsler metrics and pseudo-Riemannian spaces, including links with some relativistic spacetimes. Special emphasis is taken in the cleaning up of known techniques, the statement of open questions and the exploration of prospective frameworks.

math.DG↗
arXiv2013

Remarks on the completeness of trajectories of accelerated particles in Riemannian manifolds and plane waves

Recently, classical results on completeness of trajectories of Hamiltonian systems obtained at the beginning of the seventies, have been revisited, improved and applied to Lorentzian Geometry. Our aim here is threefold: to give explicit proofs of some technicalities in the background of the specialists, to show that the introduced tools allow to obtain more results for the completeness of the trajectories, and to apply these results to the completeness of spacetimes that generalize classical plane and pp-waves.

math.DG↗
arXiv2012

Completeness of trajectories of relativistic particles under stationary magnetic fields

The second order differential equation $\frac{D\dotγ}{dt}(t) = F_{γ(t)}(\dotγ(t)) - \nabla V(γ(t))$ on a Lorentzian manifold describes, in particular, the dynamics of particles under the action of a electromagnetic field $F$ and a conservative force $-\nabla V$. We provide a first study on the extendability of its solutions, by imposing some natural assumptions.

math.DG↗