  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 will be 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.
