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.
Highlights
A plethora of alternatives to classical general relativity has been developed since its very beginning.Many of them were motivated by the search of a unified theory which solved disturbing issues of compatibility with quantum mechanics (Kaluza–Klein, M-theory, quantum field gravity, etc.) while, since the 90s, unexpected cosmological measurements led to further alternatives
A deduction of the existence of a projective structure starting at a general version of the law of inertia. This would exclude the time-like pregeodesics for a Lorentz–Finsler metric, but again, the proof crucially relies on an argument of C2 -differentiability, which is related to nontrivial issues on Finslerian metrics
A posteriori, if we are able to measure by using some sort of symmetry, the spacetime itself will be endowed with the geometric structure which codifies such symmetries
Summary
A plethora of alternatives to classical general relativity has been developed since its very beginning. The generality of Finsler geometry in comparison with the Riemannian setup (namely, analogous to the generality of the convex open subsets of an affine space in comparison with the ellipsoids) is a big drawback, as the number of new variables and parameters would seem immeasurable. This is similar to the generality of general relativity in comparison with the special one (see Remark 13). There is no reason to assume that the physical reality will satisfy such requirements in an exact way—even though, certainly, the existence of such approximated symmetries are meaningful and useful for modeling.
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