TITLE:
Physical Constraints Simplify 5 Dimensions in Relativistic Waves
AUTHORS:
Antony J. Bourdillon
KEYWORDS:
5-Fold Dimensionality, Relativity, Wave Packet, Phase Velocity, Group Velocity, Dispersion Dynamics, Quanta
JOURNAL NAME:
Journal of Modern Physics,
Vol.17 No.2,
February
13,
2026
ABSTRACT: The classical mechanics of Newton—where momentum energy is less than rest mass energy, pc moc2 = Eo—transitions, with increasing momentum, to the relativistic mechanics of Einstein and Klein-Gordon—where pc > moc2. The transition replaces the classical rest mass energy constant Eo by Einstein’s varying relativistic energy
E=
m
′
c
2
=
p
2
c
2
+
m
o
2
c
4
. We proceed to apply this 5-dimensional relativistic equation to dispersion dynamics in electron microscope probes1. The two scalers, E and mo are initially presented as concentric, spherically-symmetric, spheres that are constrained by the Pythagorean triangle that is implicit in the relativistic expression
E
2
=
p
2
c
2
+
m
o
2
c
4
, where the speed of light c is constant. Then p, E and mo are co-planar, and all act like vectors in 5 independent dimensions. The fact is significant because E is conjugate to time t in the free particle wave equation. Band structures and band gaps are derived. Remarkable physical effects are explained with mathematics hardly more complex than Pythagoras’s theorem in a realistic framework of conventional and verified hypotheses. Consistent with Occham’s razor, a proper economy in dimensionality is retained. On this firm footing, other dimensions extend to 2-dimensional displays of physical properties including Minkowski space and phase velocity in internal motion.