Abstract
The causal set program and the Wolfram physics project leave open the problem of how a graph that is a (3+1)-dimensional Minkowski spacetime according to its simple geodesic distances could be generated solely from simple deterministic rules. This paper provides a solution by describing simple rules that characterize discrete Lorentz boosts between 4D lattice graphs, which combine further to form Wigner rotations that produce isotropy and lead to the emergence of the continuous Lorentz group and the (3+1)-dimensional Minkowski spacetime. On such graphs, the speed of light, the proper time interval, as well as the proper length are all shown to be highly accurate.
Highlights
We present novel discrete structures generated solely by local deterministic rules of remarkable simplicity that fulfill all of the above requirements
We describe the construction of the GRIDS example M∞ and go on to calculate the accuracy of both the speed of light and the proper time interval
We show how accurately this geodesic distance approximates the formula of the proper time interval for any v that is slower than the speed of light c = 1 by some arbitrarily small constant δ
Summary
We present novel discrete structures generated solely by local deterministic rules of remarkable simplicity that fulfill all of the above requirements This is different from previous approaches to Lorentz symmetry, such as randomly sprinkled causal sets, which date back to Bombelli [1,3,7]. Bolognesi [8,9] achieved Lorentz symmetry deterministically, without presupposing a continuous space His emerging spacetimes were restricted to 1+1 dimensions only. Due to their high regularity, such structures could in principle be reformulated and generated within the frameworks of causal sets, pure lambda calculus [15], graph rewriting systems [16], Wolfram models [6], and others.
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