Approaching relativity - The doppler radar

This is yet another blog post that is seemingly unrelated to the theme of the blog. My interest in the subject of Special Relativity began relatively recently, partly from a curiosity to learn more about it and partly from perhaps being able to bring some clarity over the controversies surrounding it, controversies that I had not been aware of when I first studied the subject as an undergraduate. For the most part, the controversy concerns the question of who was the true originator of the theory of Special Relativity, was it Einstein or perhaps Poincaré or Lorentz? For the rest, the controversy is about the theory itself, whether it is false or true or whether it is even meaningful in a physical sense. Sometimes these discussions get mixed up in a very confusing way. Here we will not touch the priority dispute at all but will only be concerned with the postulate of Special Relativity stating that the speed of light c is constant in every inertial reference frame, together with its consequences in terms of time dilation, length contraction and so on. This demarcation is suitable since "Relativity Theory" is often associated with other postulates such as the formula E = mc², the bending of light in gravity fields, that are only weakly related to, or even unrelated to, the postulate of the constancy of the speed of light. Before we move on to the main arguments we have to warm up with some concepts from classical physics.


Galilean relativity

The theory of relativity formulated by Galileo Galilei is a cornerstone of classical mechanics. It could be stated saying that there does not exist any measurable "absolute" motion in space but only relative motion between interacting bodies. For example, when you sit in an airplane you can still play with your jo-jo as if you were both at rest in your comfy chair at home. This despite of the fact that, according to an observer on the ground, both you and the jo-jo travel at a considerable speed. Another way of putting it is to say that the laws of physics stay the same in all reference frames in uniform motion relative to each other. Newtonian mechanics is a perfect example of a physical theory that obeys Galilean relativity. As we will see further on, it is when Maxwell's equations of classical electrodynamics are confronted with Galilean relativity that the seed of Special relativity is born. If you speak of the "speed of light" being 300 000 km/s the question becomes: in which reference frame?


The classical doppler effect

If we now move our attention instead to sound waves waves the question of Galilean relativity becomes a much simpler matter. Why so? because now we have a medium where the waves travel as our natural reference frame. Sound waves need some medium, such as air or water, in which to travel. Hence, when we speak of the "speed of sound" we mean, naturally, the speed of the sound waves relative to the medium in which they travel. However, for a vehicle traveling in the medium it is still quite possible to approach the crests of the sound wave with a relative speed exceeding the speed of sound relative to the medium. This variability in relative speed gives rise to the doppler effect. When a vehicle approaches the sound wave with a speed v relative to the medium it will reflect the crests of the sound waves at shorter intervals giving rise to an increase in frequency of the reflected wave. We will not go into the detailed theoretical derivation of this phenomena here, but merely point out some main bullet points to take away from this section:

1. In the case of sound waves, the question of Galilean relativity is unproblematic since, in this case, the speed of sound refers to the motion of the waves relative to the medium where they travel.

2. It is possible for a vehicle traveling in the medium to approach (or move away from) a sound wave with a relative speed that is higher than (or lower than) the speed of the sound wave relative to the medium. This gives rise to the doppler effect.

3. In the case of sound waves, the magnitude of the doppler effect depends on whether it is the source or the receiver that moves in the medium.


Maxwell's equations and Galilean relativity

We now come to the important question: Do Maxwell's equations obey Galilean relativity? In other words, do they predict the same measurable phenomena under a Galilean transformation, that is, switching to a reference frame in uniform motion relative to the former. One thought experiment, which is also  mentioned in the introduction to Einstein's 1905 paper, concerns the moving magnet and conductor problem. We will not go into the details of that problem here but only point out that, in this case, the measurable outcome remains the same even after a Galilean transformation, hence, the moving magnet and conductor problem seems to be an example where Maxwell's equations elegantly cope with a change of reference frame. An apparently more difficult problem arises when we consider instead the solution to Maxwell's equations in vacuum, that is, when we are left without any electromagnetic source terms in the form of charged particles or currents. In this case, it can be shown that Maxwell's equations boil down to a wave-equation whose solution is a wave propagating in space with velocity c, and now the Galilean relativity problem arises: The speed c in which reference frame?


Lorentz, the ether theory and the Michelson-Morley experiment

Whether it was considerations of Galilean relativity that led Lorentz to formulate his ether theory will be left unspoken. However, the idea of a motionless light medium, called the "ether", could provide a natural reference frame in which light always travel at the speed c, just like sound waves travel at a constant speed relative to their medium. In order to test this hypothesis, Michelson and Morley conducted an experiment set out to detect the motion of the earth relative to the motionless ether. The result was negative, no ether wind could be detected. Lorentz, nevertheless, continued to pursue his ether theory and attempted to explain the negative result of the Michelson-Morley experiment with notions such as length-contraction and time-dilation, formulas that still remain with us today although with a somewhat different interpretation. The stage was now set for Einstein to enter.


19th century dogma turned into 20th century dogma?

In short, Einsteins solution to the problems discussed in the previous section has been formulated as follows:

The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

It could be disputed whether this exact formulation ever occurs explicitly in Einstein's 1905-paper, but that is another story. As mentioned in the introduction, we are not interested here in the contribution of various individuals to the theory of relativity but merely to examine the theory as it stands today.


The doppler radar

I would like the reader to have a look at the following article from Wikipedia explaining the working of the doppler radar, a hand-held equipment used primarily by military och law-enforcement officers to measure the speed of moving vehicles.

https://en.wikipedia.org/wiki/Doppler_radar

 It turns out that, the classical formula for the doppler effect remains valid with the slight modification that we do not make any distinction between whether it is the source or receiver that is moving, only the relative motion comes into play. Curiously enough, the article states explicitly:

There is no need to invoke Einstein's theory of special relativity, because all observations are made in the same frame of reference.

Fair enough, but now we come to the question: How are we to explain this "classical" doppler effect of light if we insist that "The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source." Clearly, when we derive the doppler shift for sound waves we assume that it is possible for the vehicle to approach the sound wave at variable speeds. To add to confusion there does also exist a hypothetical relativistic doppler effect, but the formula for this theoretical effect is different in nature and smaller in magnitude than that which we observe. I will leave to the reader to try to make sense out of all of this, if it is even possible, but until the doppler radar is explained in accordance with relativity theory I will remain doubtful as to the validity of the main postulate of Special Relativity.


The roots of the difficulties

Here we will make some very tentative remarks about the source of all confusion, namely the solution to Maxwell's equations in vacuum. First I would like to point out that light essentially is a force, or at least becomes a force when it comes in contact with charged particles. So, in essence, the solution to Maxwell's equations in vacuum is a solution that contains only forces and no particles. What is wrong with that picture? I would say that such a system is necessarily incomplete, because forces never exist on their own, they are always mediators between particles. If a particle is accelerated by an electric field then, according to Newton's third law, some other particle in the universe needs to take the recoil. However, saying that a reference frame is incomplete is not the same thing as saying that it is useless. For example, often in mechanics the earth is replaced by a constant gravitational force pointing downwards, thus we forget that every time we jump upwards the earth needs to take a recoil downwards. In summary, posing an extremely fundamental question such as that of Galilean relativity so a system that is by necessity incomplete might lead you astray. Very, very astray.