MVLSC Home · Issue Contents · Forthcoming Papers
Remarks on Antichains in the Causality Order of Space-time
Stephan Foldes
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.
Keywords: space-time, causality, subluminal causality, partial order, world line, chain, antichain, cutset, graded poset, rank, level, space-like vector, light-like vector, time-like vector, separation line, optical line, optical hyperplane, inertia plane, Lorentz group, Lorentz transformation, Poincar´e group