Maxwell’s EM Wave Equation — Does It Hold Good for Obliquely Moving Observers?
Maxwell’s electromagnetic wave equation is derived from his set of four equations for electromagnetic fields which together describe how fluctuations in electromagnetic fields propagate at a constant speed c.
Light is considered an electromagnetic wave and therefore, must follow Maxwell’s electromagnetic wave equation.
Since light propagating with a constant speed c is the vehicle for derivation of Lorentz Transformation (also known as Special Relativity relations), its conformity to the wave equation, or otherwise, from both the frames — stationary as well as moving — becomes very important.
While Voigt, Lorentz and Poincare followed the Maxwell’s wave equation in their separate contribution to arrive at the present form of Lorentz Transformation, Einstein applied kinematics, though with mistakes, to work it out.
A study of history of Lorentz transformation reveals that certain terms had to be added to the Galilean Transformation relations to arrive at Lorentz Transformation (for parallel observations, of course) so as to strike compatibility with the Maxwell’s wave equation from both the frames — stationary as well as moving. For an electromagnetic wave propagating in x-direction under observation from a frame moving in x-direction itself, Voigt first suggested in 1897 a correction of vx/c2 to be applied to the time of stationary frame t to get the corresponding time of the moving frame t’ for maintaining compatibility with the wave equation. This was adopted by Lorentz in his 1904 paper and Poincare suggested in 1906 multiplication by a constant factor, which is known as Lorentz Factor 𝛾 today.
Attention is drawn to the fact that all the above steps were taken in the context of an observer moving in the same direction as that of the light wave. Consequently, the resultant Lorentz Transformation relations, which continue even today without any modification, are valid for parallel observations i.e. when the observer is moving parallel to the light.
A question looms large, which has not attracted anybody's attention so far, is — Does the Maxwell’s wave equation hold good for observers moving obliquely (at an Inclination) to the direction of light? If it does, what are the consequences?
Before we go further, let me tell you that I have already derived the Lorentz Transformation relations for such observers in my book titled “Refining Relativity Part 1 (The Special Theory)” available on Amazon. The broad details may also be seen on my blog at https://refiningrelativity.blogspot.com.
As the invariance of c has been maintained for oblique observations too in chapter 5 of the book, the results obtained there should also conform to the wave equation. I am proud to tell you that it does with the transformed expressions of distance and time. The same is shown below.
Let me first inform you that I have used the letter U instead of c for reasons described in my book. Further, the constant used here in place of the Lorentz Factor 𝛾 is a and it had to be taken to maintain constancy of light speed in both the frames, with no wave equation in picture at all. Similarly, the correction to time t has been derived there to get t’, rather than making any guess or ansatz.
As the exercise is for obliquely moving observers, I have taken the most general case of a 3D distance denoted by r in place of x. Further, introduction of angles becomes inescapable. The 3D angle between the directions of the observer and the light is 𝜃 and that between the directions of the light pulse observed from origins of the stationary and the moving frame is 𝜑. In other words, the angle between v and r is 𝜃 and angle between r and r’ is 𝜑. As usual, the primed parameters are for the moving frame. The following fig. show the angles.
