Transformation of Electromagnetic Amplitude

Section 7 of Einstein’s 1905 paper on the electrodynamics of moving bodies considers an electromagnetic plane wave propagating in vacuum. Since the amplitudes of the electric and magnetic field vectors of a plane wave are equal, it follows that in terms of any standard system K of inertial coordinate x,y,z,t the components of those vectors can be written in the form

where A is the amplitude of each vector, and the phase angle Φ is given by

The symbols e_x, e_y, e_z are the direction cosines of the electric field vector, i.e., they are the cosines of the angles between the electric vector E and the x, y, and z axes respectively.  Likewise the symbols b_x, b_y, b_z denote the direction cosines of the magnetic field vector B, and the symbols n_x, n_y, n_z denote the direction cosines of the wave normal vector n (i.e., the unit vector pointing in the direction of the wave propagation).

We know the three unit vectors n, e, and b are mutually orthogonal, as are unit vectors in the x, y, and z, directions, so we have the relations

We also note that n = e x b, so the components of n can be expressed as

Obviously the squared amplitude of the electric field oscillation is the sum of squares of the components E_x, E_y, E_z at their maximum values, so we have

and similarly for the magnetic field vector.

Now, in terms of another standard system K′ of coordinates x′,y′,z′,t′ moving relative to K in the positive x direction with speed v, the components of the electric and magnetic field vectors are

where

where ω′, n_x′, n_y′, and n_z′ are the frequency and direction cosines of the wave normal in terms of the K′ system of coordinates. These are given by the Doppler formula

and the aberration formulas

Einstein concluded Section 7 of his paper by saying “We still have to find the amplitude of the waves as it appears in the moving system”. He then says “we obtain” the formula for the ratio of squared amplitudes, but doesn’t gives the actual calculation. As in the unprimed system K, the squared amplitude in the primed system K′ is equal to the sum of squares of the components E_x′, E_y′, E_z′ at their maximum values, so we have

Dividing through by A² and factoring out γ², we get

Expanding the right hand side, this gives

Making use of the identities

the equation reduces to

Since n_x = cos(ϕ) where ϕ is the angle between the wave normal and the x axis, this gives the result as it appears in Einstein’s paper

Comparing this with the Doppler formula for the frequency, we see that A′/A = ω′/ω, meaning the amplitude of an electromagnetic plane wave is strictly proportional to the frequency. Now, from Poynting’s Theorem we know the energy density is proportional to the square of the amplitude, so we might think the total energy of a given portion of a plane wave is proportional to the square of the frequency. However, as Einstein showed in Section 8 of his paper, the volume of a given “light complex” is inversely proportional to the frequency (consistent with the relation λω = c), and hence the total energy is proportional to frequency.

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