Related papers: Timelike Hilbert and Funk geometries
We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its…
Semiclassical gravity and the holographic description of the static patch of de Sitter space appear to disagree about properties of correlation functions. Certain holographic correlation functions are necessarily real whereas their…
Let $X$ be a (real or complex) rearrangement-in\-va\-riant function space on $\Om$ (where $\Om = [0,1]$ or $\Om \subseteq \bbN$) whose norm is not proportional to the $L_2$-norm. Let $H$ be a separable Hilbert space. We characterize…
Analogous to Weil-Petersson quasicircles, we investigate infinite circle patterns in the Euclidean plane parameterized by discrete harmonic functions of finite Dirichlet energy. The space of such circle patterns forms an…
We explore connections furnished by the Funk metric, a relative of the Hilbert metric, between projective geometry, billiards, convex geometry and affine inequalities. We first show that many metric invariants of the Funk metric are…
For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The…
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…
Hilbert space combines the properties of two fundamentally different types of mathematical spaces: vector space and metric space. While the vector-space aspects of Hilbert space, such as formation of linear combinations of state vectors,…
It is shown that the problem of a possible violation of the Lorentz transformations at Lorentz factors $\gamma >5\times 10^{10} ,$ indicated by the situation which has developed in the physics of ultra-high energy cosmic rays (the absence…
The space time that is used in relativistic Quantum Mechanics and Quantum Field Theory is the Minkowski space time. Yet, as pointed out by several scholars this classical space time is incompatible with the Heisenberg Uncertainity…
The notion of null distance was introduced by Sormani and Vega as part of a broader program to develop a theory of metric convergence adapted to Lorentzian geometry. Given a time function $\tau$ on a spacetime $(M,g)$, the associated null…
In 1914, F. Hausdorff defined a metric on the set of closed subsets of a metric space $X$. This metric induces a topology on the set $H$ of compact subsets of $X$, called the Hausdorff topology. We show that the topological space $H$…
Time-asymmetric spacetime structures, in particular those representing black holes and the expansion of the universe, are intimately related to other arrows of time, such as the second law and the retardation of radiation. The nature of the…
For non-empty sets X we define notions of distance and pseudo metric with values in a partially ordered set that has a smallest element $\theta $. If $h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the pair $(X,h_X)$…
This paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time…
Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and S\"amann (Ann. Glob. Anal. Geom. 54(3):399--447, 2018) we introduce a notion of a hyperbolic angle, an angle between timelike curves and…
A directional time-frequency localization measure for functions defined on the $d$-dimensional Euclidean space is introduced. A connection between this measure and its periodic counterpart is established. For a class of functions, an…
A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of…
In this paper, we {\it find} the infinitesimal structure of Funk-Finsler metric in spaces of constant curvature. We investigate the geometry of this Funk-Finsler metric by explicitly computing its $S$-curvature, Riemann curvature, Ricci…
The geometry of the Sagnac effect in a stationary region of a spacetime is reviewed with the aim of emphasizing the role of asymmetry of a Finsler metric defined on a spacelike hypersurface associated to a stationary splitting and related…