Inertial frames without the relativity principle

Abstract

Ever since the work of von Ignatowsky circa 1910 it has been known (if not always widely appreciated) that the relativity principle, combined with the basic and fundamental physical assumptions of locality, linearity, and isotropy, leads almost uniquely to either the Lorentz transformations of special relativity or to Galileo's transformations of classical Newtonian mechanics. Thus, if one wishes to entertain the possibility of Lorentz symmetry breaking within the context of the class of local physical theories, then it seems likely that one will have to abandon (or at the very least grossly modify) the relativity principle. Working within the framework of local physics, we reassess the notion of spacetime transformations between inertial frames in the absence of the relativity principle, arguing that significant and nontrivial physics can still be extracted as long as the transformations are at least linear. An interesting technical aspect of the analysis is that the transformations now form a groupoid/pseudo-group --- it is this technical point that permits one to evade the von Ignatowsky argument. Even in the absence of a relativity principle we can nevertheless deduce clear and compelling rules for the transformation of space and time, rules for the composition of 3-velocities, and rules for the transformation of energy and momentum. As part of the analysis we identify two particularly elegant and physically compelling models implementing "minimalist" violations of Lorentz invariance --- in the first of these minimalist models all Lorentz violations are confined to carefully delineated particle physics sub-sectors, while the second minimalist Lorentz-violating model depends on one free function of absolute velocity, but otherwise preserves as much as possible of standard Lorentz invariant physics.Comment: V1: 42 pages; V2: now 43 pages; added 8 references, added brief discussion of Carroll kinematics, added brief discussion of Robertson-Mansouri-Sexl framework, added various minor clarifications. V3: now 51 pages; added another 34 references; more discussion of DSR and relative locality; various clarifications and extensions; this version accepted for publication in JHE

Free Particle in Very Special Relativity, Gauge Symmetry and Two-Time Physics

“…This line element also was found in [11]. Some works about Lorentz violation can be seen in [12,13,14,15,16,44,18,19,20,21,22,23,24,25,26,27,28] and works about VSR can be found in [29,30,31,32,33,34,35,36].…”

Einstein’s physical chronogeometry

“…Berzi and Gorini (1969) consider that taking the transformation functions to be real and continuous is a mathematical assumption. Baccetti, Tate, and Visser (2012) consider the description of space and time using real numbers as an assumption. Levy-Leblond (1976) also calls the attention to a causality assumption related to the notion of flow of time, differentiating clearly time from space.…”

Symmetry and Special Relativity

“…The first was by Ignatowsky [1] in 1910. Many other derivations followed; see [2] for a full list of references. Nevertheless, the approach here, based on symmetry, is new.…”