Foundations of Finsler Spacetimes from the Observers’ Viewpoint

Abstract

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 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.MSC: 53C60, 83D05 83A05, 83C05.

Predictions of Ultra-High Energy Cosmic Ray Propagation in the Context of Homogeneously Modified Special Relativity

“…On the contrary, a Finsler geometry is constructed starting from a positive defined metric. Some issues can emerge in defining a pseudo-Finsler structure [23][24][25][26][27], but, in the case of HMSR, the function f is supposed to be of a tiny magnitude compared to the other MDR quantities, so it represents a perturbation; therefore, as a result, it is possible to deal with these issues. In the actual HMSR formulation, MDR are supposed CPT even, since they do not present an explicit dependence on particle helicity or spin.…”

Cosmological Finsler Spacetimes

“…This is a refined version of the definition of Finsler spacetimes in Reference [11] and basically covers, if one chooses A = T , the improper Finsler spacetimes defined in Reference [24].…”

“…If N = 0 and M = 0, then the first equation in (24) implies immediately that Ω(t, s) = 0 and thus the Finsler Lagrangian is L = 0.…”

Causal hierarchy in modified gravity

“…This extension is a particular case of the wider notion of convex cones which, aside from being applicable to multi-metric frameworks, would also allow us to describe other situations in which the cones degenerate either totally (obtaining hyperplanes, as in the case of Newtonian gravity) or partially (as in Casimir type effects, where superluminal propagation is allowed only in a specific spacelike direction). A detailed discussion of this formalism is out of the scope of this paper (we direct the reader to the recent works [77,78]), although in the discussion below we include brief remarks in specific places where this formalism would be most useful. For this aim, we just need to define C as the convex hull (i.e.…”