Dispersion and Geometry
Dispersion diagrams are typically used in the theory of waves propagating along transmission lines, or through wave guides [4]. The local distribution of events in space-time determines the shape and size of these wave guides. The dispersion diagram illustrates the relationship between frequency and wavelength.
FIG. 1
The equation plotted here in blue for a mass of
1 and a range of momentum from 
, is the total energy, which is the
hyperbola,
(48)
Graphs for conditions of zero mass photons 
, and rest
energy 
are also shown as diagonal and horizontal straight lines
respectively.
The interesting plots are the group velocity 
, and the phase
velocity 
.
The group velocity, shown in red is tangent to the hyperbola, and
is defined as the derivative,
(49)
The phase velocity shown in green goes through the origin and intersects the hyperbola at the point tangent to the group velocity. It is defined by the values of energy and momentum at that point,
(50)
For a null geodesic line of a ray of light, the group and phase velocities are equal.
(51)
Noting that the intersection of the group and phase velocities can be transposed to the origin of a Cartesian coordinate system. The result is a Minkowski space-time diagram.
FIG. 2
The group velocity is then,
(52)
Therefore this diagram shows that the Dispersion interpretation and Minkowski’s Geometrical interpretation of Relativity are equivalent.