Remarks on Antichains in the Causality Order of Space-time

: Foldes, Stephan

Publisher: Old City Publishing

: 2024

: Journal of Multiple-Valued Logic and Soft Computing

: 42

: 4

: 375

: 383

: 1542-3999

: https://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-42-number-4-2024/

We study partial orders defined on the set of points of space-time that are invariant under Lorentz transformations. Kirszbraun’s Theorem allows to show that world lines of particles evolving in space-time are precisely the maximal chains in the causality order. We show that the causality order is well behaved in the sense that it is gradable and level sets under various gradings are precisely the anti-chain cutsets. We also show that the causality orders corresponding to different light speed parameters c are essentially the only partial orders invariant under Lorentz transformations and under some other, more obvious affine transformations of space-time. We characterize optical lines and hyperplanes, inertial lines and planes, and separation lines as well, in terms of the causality order and use these characterizations to provide a variant proof of the Alexandrov-Zeeman Theorem.

:
This work has been co-funded by Marie Curie Actions (European Union) and supported by the National Development Agency (NDA) of Hungary and the Hungarian Scientific Research Fund (OTKA), within a project hosted by the University of Miskolc, Department of Analysis. The work was also completed as part of the TAMOP-4.2.1.B.- 10/2/KONV-2010-0001 project at the University of Miskolc, with support from the European Union, co-financed by the European Social Fund. The author wishes to thank S\u00E1ndor Radeleczki and Mikl\u00F3s Ront\u00F3 for useful comments and discussions. The anonymous referee brought to the author\u2019s attention the fundamental connection with Kirszbraun\u2019s Theorem that explains and proves Proposition 2.3.