%0 Journal Article
%T Where Is Phase Velocity in Minkowski Space?
%A Antony J. Bourdillon
%J Journal of Modern Physics
%P 1555-1566
%@ 2153-120X
%D 2024
%I Scientific Research Publishing
%R 10.4236/jmp.2024.1510065
%X In the special theory of relativity, massive particles can travel at neither the speed of light <i>c</i> nor faster. Meanwhile, since the photon was quantized, many have thought of it as a point particle. How pointed? The idea could be a mathematical device or physical simplification. By contrast, the preceding notion of wave-group duality has two velocities: a group velocity <i>v</i><i><sub>g</sub></i> and a phase velocity <i>v</i><i><sub>p</sub></i>. In light <i>v</i><i><sub>p</sub></i> = <i>v</i><i><sub>g</sub></i> = c; but it follows from special relativity that, in massive particles, <i>v</i><i><sub>p</sub></i> &gt; <i>c</i>. The phase velocity is the product of the two best measured variables, and so their product constitutes internal motion that travels, verifiably, faster than light. How does <i>v</i><i><sub>p</sub></i> then appear in Minkowski space? For <i>light</i>, the spatio-temporal Lorentz invariant metric is <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
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, the same in whatever frame it is viewed. The space is divided into 3 parts: firstly a cone, symmetric about the vertical axis <i>ct</i> &gt; 0 that represents the world line of a stationary particle while the conical surface at <i>s</i> = 0 represents the locus for light rays that travel at the speed of light <i>c</i>. Since no real thing travels faster than the speed of light c, the surface is also a horizon for what can be seen by an observer starting from the origin at time <i>t</i> = 0. Secondly, an inverted cone represents, equivalently, time past. Thirdly, outside the cones, inaccessible space. The phase velocity <i>v</i><i><sub>p</sub></i>, group velocity <i>v</i><i><sub>g</sub></i> and speed of light are all equal in free space, <i>v</i><i><sub>p</sub></i> = <i>v</i><i><sub>g</sub></i> = <i>c,</i> constant. By contrast, for <i>particles,</i> where causality is due to<i> particle</i> interactions having rest mass <i>m</i><i><sub>o</sub></i> &gt; 0, we have to employ the Klein-Gordon equation with <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
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%K Event Horizon
%K Scattering Range
%K Wave Packet
%K Phase Velocity
%K Group Velocity
%K Dispersion Dynamics
%K Quantum Physics
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=136257