On the Nature of Discrete Space-Time: Part 1: The distance formula, relativistic time dilation and length contraction in discrete space-time

Abstract

In this work, the relativistic phenomena of Lorentz-Fitzgerald contraction and time dilation are derived using a modified distance formula that is appropriate for discrete space. This new distance formula is different than the Pythagorean theorem but converges to it for distances large relative to the Planck length. First, four candidate formulas developed by different people over the last 70 years is discussed. Three of the formulas are shown to be identical for conditions that best describe discrete space. It is shown that this new distance formula is valid for all size-scales -from the Planck length upwards -and solves two major problems historically associated with the discrete space-time (DST) model. One problem it solves is the widely believed anisotropic nature of most discrete space models. Just as commonly believed is the second problem -the incompatibility of DST's concept of an immutable atom of space and the length contraction of this atom required by special relativity. The new formula for distance in DST solves this problem. It is shown that length contraction of the atom of space does not occur for any relative velocity of two reference frames. It is also shown that time dilation of the atom of time does not occur. Also discussed is the possibility of any object being able to travel at the speed of light for specific temporal durations given by an equation derived in this work. Also discussed is a method to empirical verify the discreteness of space by studying any observed anomalies in the motion of astronomical bodies, such as differences in the bodies' inertial masses and gravitational masses. The importance of the new distance formula for causal set theory and other theories of quantum gravity is also discussed.

Quantum Cosmology: The Integration of Special Relativity and Quantum Mechanics - Part 1

“…Arguments in favor of the discretization of space have been proposed based upon mathematics, electrodynamics, quantum electrodynamics, loop quantum gravity, loop quantum cosmology, string theory, discrete lattice, asymptotic safe gravity, causal sets, spin foams, deformed special relativity, causal dynamical triangulation, quantum graphity, and black hole theory among others. (Crouse, 2016); (Crouse & Skufca, 2018); (Hagar, 2014); (Hossenfelder, 2013(Hossenfelder, , 2014. 28 The precise shape and size of Planck Spheres is unknown, and the description of " Planck Spheres" is illustrative only.…”

Quantum Cosmology: The Integration of Special Relativity and Quantum Mechanics Part I

“…Arguments in favor of the discretization of space have been proposed based upon mathematics, electrodynamics, quantum electrodynamics, loop quantum gravity, loop quantum cosmology, string theory, discrete lattice, asymptotic safe gravity, causal sets, spin foams, deformed special relativity, causal dynamical triangulation, quantum graphity, and black hole theory among others. (Crouse, 2016); (Crouse & Skufca, 2018); (Hagar, 2014); (Hossenfelder, 2013(Hossenfelder, , 2014. 28 The precise shape and size of Planck Spheres is unknown, and the description of " Planck Spheres" is illustrative only.…”

“…83 As used in this analysis, the terms local and non-local are causal concepts connected with the 4D spacetime concepts of time, space, and the maximum speed of light. 84 Despite the importance of the local vs. non-local debate in 4D spacetime, the concepts of time, space, and volume do not apply to Planck Space. In Planck Space, a quantum state's Bell Quantum Hyper-Point is neither space-like separated nor separable.…”

“…53, Note 1). 84 (Ney, 2021, p. 96). See regarding different uses of the term "non-local" in Einstein/de Broglie Box experiments.…”