Relativistic Time Dilation and Length Contraction in Discrete Space-Time using a Modified Distance Formula

In this work, the relativistic phenomena of Lorentz contraction and time dilation are derived using a modified distance formula appropriate for discrete space. This new distance formula is different than Pythagoras's 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 will be considered. Three of the formulas are shown to be identical for conditions that best describe discrete space; this equation is then used in the rest of the paper. It is shown that this new distance formula is applicable to all size-scales, from the Planck length upwards, and solves two major historical problems associated with a discrete space-time model. One problem it solves is maintaining isotropy in discrete space. The second problem it solves is the commonly perceived incompatibility of the model's concept of an immutable "atom" of space and the Lorentz contraction of this atom required by special relativity. With the new distance formula, it is shown that the Lorentz contraction of the atom of space does not occur regardless of the relative velocities 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 derived in this work.