Optics and Relativity

The fundamental fact of light is that it moves by propagation; that is, it moves as a wave does. A change in the amplitude of the wave at some point causes an identical change at a neighboring point, thereby making the wave reproduce itself point by point and thus propagate from point to point.

In a beam of light a sinusoidal electric field oscillates in a way that makes it appear to move sideways (in a direction perpendicular to the direction in which the electric vector points). That oscillation, in accordance with Ampere’s law, generates a sinusoidal magnetic field that also oscillates and appears to move sideways with the electric field, though oriented perpendicular to the electric field vector. In accordance with Faraday’s law, the oscillation of the magnetic field generates an electric field oriented perpendicular to the magnetic field. Maxwell’s equations tell us further that if the fields appear to move at a certain speed, they will support each other so perfectly that the beam won’t fade with distance. That fact must be true to Reality for all observers measuring the speed of the light in the beam. Those observers must thus infer that the electric and magnetic fields are each other’s medium of propagation, so there is no luminiferous aether to provide a reference for an absolute state of rest. Therefore, the beam propagates as a wave past all observers at the same speed of light, regardless of how those observers move relative to one another. That’s the fundamental fact of optics.

Imagine that we can see the crests of the waves and count them. The distance between two neighboring crests represents one cycle of the field oscillation, so if we count N crests passing a given stationary point in a time interval T (measured by a clock on that stationary point), then we can calculate the ratio as the frequency of the light, f=N/T, in cycles per second. But we prefer to use angular measure and we know that there are 2π radians in one cycle, so we multiply the frequency to get the angular frequency, ω=2πf, in radians per second.

Imagine a beam of light frozen in place at one instant. Along a line drawn perpendicular to the wavefronts, a line parallel to the direction of propagation, count N wave crests over a distance X. Taking the ratio of those numbers gives us the wave number, W=N/X, in cycles per meter, the reciprocal of the wavelength. Again we convert to angular measure to turn the wave number into the propagation number (or propagation vector), k=2πW, in radians per meter. We call it a propagation vector because, in addition to having a numerical magnitude, it points in the direction of propagation.

The angular frequency and the propagation vector are the fundamental kinematic properties of light. If we know the numerical values of those functions, we can plot the motion of a ray of light by following the zeroes in the wave function,

(Eq’n 1)

As an aside I’ll point out that if we multiply the propagation vector and the angular frequency be Dirac’s constant (aitch-bar), we get a quantum-mechanical description of the linear momentum and kinetic energy carried by any photon in the ray.

Imagine creating a pulse of light in the form of a plane
wave. We do that by exploiting Christian Huygens’ wavelet model of the
propagation of light We have a flat plate of length x_{p} floating
parallel to the x-y plane of a Cartesian coordinate grid. We cover one side of
the plate with a large number of tiny emitters with miniature clocks that are
all synchronized with each other. When the emitters all flash at the same
instant, the spherical wavelets emerging from them interfere with each other and
produce a plane wave of pulsed light propagating in the z-direction. We also
have a thin pole running parallel to the z-axis and some of the emitters next to
the pole emit only red light, so we see a red patch moving down the pole at the
speed of light, illuminating a series of clocks mounted on the pole.

An outside observer enters the scene from the right, moving in the negative x-direction at a speed v. That observer carries a pole oriented parallel to the inside observer’s pole so that when the two poles pass each other their ends come within less than a millimeter of touching. Both observers must see the close approach of the poles’ ends, so they must infer that length and distances oriented perpendicular to a relative motion are not altered by that motion; algebraically, y=Y and z=Z.

In the outside observer’s frame the inside observer and their apparatus move in the positive X-direction at a speed V. That fact necessitates that the plane wave have a component of motion in the X-direction. The wave’s total speed remains unchanged by the shift between reference frames, so the velocity vector of the plane wave, in conformity with the Pythagorean theorem, has components

(Eq’ns 2)

The second of those equations tells us that the outside observer will see the red patch moving down the inside observer’s pole more slowly than the inside observer sees it moving. But the clocks on the pole display the same readings that the inside observer sees when the red patch illuminates them. Thus the outside observer must infer that the clocks on the pole count time more slowly than their own clocks do, in accordance with

(Eq’n 3)

Whereas the inside observer sees the light moving straight along the z-axis,
the outside observer must see the light moving at some angle away from the
Z-axis. Light cannot propagate sideways: it always propagates in the direction
perpendicular to its wavefronts, so the light in this case must be tilted at an
angle θ
away from the Z-axis in accordance with sinθ=V/c.
That fact necessarily implies that in the outside observer’s frame the emitters
on the left side of the plate flashed before the emitters on the right side did.
The time difference between the flashing of those emitters is reflected in the
distance that the wave emitted by the left emitters has traveled before the
right emitters flash, a distance determined by the length X_{p} of the
plate and the sine of the angle of tilt. Dividing that distance by the light’s
speed of propagation yields the time difference that the outside observer would
measure,

(Eq’n 4)

The propagation vector, as seen by the outside observer, must also tilt away from the Z-axis. In the inside observer’s frame the propagation vector is

(Eq’n 5)

in which the lower-case ee with a hat represents the unit vector pointing in the direction indicated by the subscript (the positive z-direction in this case). For the same light the outside observer uses the propagation vector

(Eq’n 6)

As indicated in the comment prefacing Equation 2, we track the motion of the wave through the equation

(Eq’n 7)

The zeroes of the wave are the same for both observers, so we can write

(Eq’n 8)

Substituting from Equation 3 transforms that equation into

(Eq’n 9)

If we imagine freezing our plane wave in place at some instant in time and look at how it illuminates the inside observer’s pole, we will see between any two given wavecrests a certain number of radians of angular vibrations (some integer multiple of 2 pi). Both observers will see the same number of radians, so we know that

(Eq’n 10)

in which z and Z represent the distance between the waves on the pole. But we
have already determined that z=Z, so we must infer that k_{z}=K_{Z}.
We can use that fact to remove the z-components from Equation 9, which then
becomes, with a little algebraic rearrangement,

(Eq’n 11)

Because we can calculate the angle at which the plane wave
tilts away from the Z-axis, we can calculate a value for K_{X}. We have

(Eq’n 12)

With that result Equation 11 becomes

(Eq’n 13)

which looks like an analogue of the Lorentz-Fitzgerald contraction, especially given the way we derived it. It tells us that Ω is bigger than ω, which we expect due to the Doppler shift.

Making the appropriate substitution from the Lorentz Transformation, we have the first part of Equation 8 in the form

(Eq’n 14)

Comparing the temporal terms on both sides of the equality sign leads us to write

(Eq’n 15)

Comparing that equation to Equation 11 tells us that |v|=|V|, so comparing the spatial terms on both sides of the equality sign in Equation 14 gives us

(Eq’n 16)

Equations 11 and 16, along with

(Eq’ns 17)

comprise a relativistic transformation of the fundamental kinematic properties of light. They resemble in form the equations of the Lorentz Transformation. But the Lorentz Transformation applies to spatio-temporal intervals, so we need a different name for this one. Let’s call it the Einstein Transformation.

eabf