Madarász, J.X., Stannett, M. orcid.org/0000000227948614 and Székely, G. (Submitted: 2020) Groups of worldview transformations implied by isotropy of space. arXiv. (Submitted)
Abstract
Given any Euclidean ordered field, Q, and any 'reasonable' group, G, of (1+3)dimensional spacetime symmetries, we show how to construct a model MG of kinematics for which the set W of worldview transformations between inertial observers satisfies W=G. This holds in particular for all relevant subgroups of Gal, cPoi, and cEucl (the groups of Galilean, Poincaré and Euclidean transformations, respectively, where c∈Q is a modelspecific parameter orresponding to the speed of light in the case of Poincaré transformations). In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set W of worldview transformations satisfies either W⊆Gal, W⊆cPoi, or W⊆cEucl for some c>0. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are Poincaré transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of Q to the field of reals.
As part of this work, we also prove the rather surprising result that, for any G containing translations and rotations fixing the timeaxis t, the requirement that G be a subgroup of one of the groups Gal, cPoi or cEucl is logically equivalent to the somewhat simpler requirement that, for all g∈G: g[t] is a line, and if g[t]=t then g is a trivial transformation (i.e. g is a linear transformation that preserves Euclidean length and fixes the timeaxis setwise).
Metadata
Authors/Creators: 



Copyright, Publisher and Additional Information:  © 2020 The Author(s). For reuse permissions, please contact the Author(s).  
Keywords:  mathph; mathph; math.LO; math.MG; math.MP  
Dates: 


Institution:  The University of Sheffield  
Academic Units:  The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield)  
Funding Information: 


Depositing User:  Symplectic Sheffield  
Date Deposited:  06 Aug 2020 08:26  
Last Modified:  07 Aug 2020 18:09  
Published Version:  https://arxiv.org/abs/2007.14261v1  
Status:  Submitted  
Related URLs: 