## Einstein’s light postulate saves Galileo

Good evening (local time)! I’d like to share with you the answer I gave to one of my students earlier today. He asked how Einstein’s postulate of a constant light speed (independent from an observer’s own state of motion) can save Galilean relativity.

In Galilean relativity, the laws of physics are the same for all observers that have a constant speed relative to each other (read this post to be reminded of how that works). The key here is constant motion; as soon as the observers start accelerating, inertial forces start playing a role – and Galilean relativity no longer holds.

## All’s well that Maxwell*?

However, in the the nineteenth century a new type of force was discovered: the electromagnetic, or Lorentz, -force. This force differs from Newton’s gravity in that it depends on the speed of a particle (and not only acceleration): the strength of the Lorentz-force on a charged particle moving in a magnetic field depends on the speed of the particle with respect to the field (see picture above).

In Maxwell’s theory of electromagnetism the magnetic field is at rest with respect to absolute space. So we need absolute space to determine how strong the electromagnetic force is, but Einstein’s theory is famous for rejecting the notion of absolute space. As a result, there is no notion of absolute motion, and therefore no way to determine the strength of the Lorentz force. Einstein solved this by introducing his light postulate. Does the light postulate solve the problem? Why is it that a constant light speed – a speed that is the same for all observers regardless of their own state of motion – saves galilean relativity?

## Lorentz Relativity

To answer this question, we must first understand what it means to ‘save’ galilean relativity. What is Galilean relativity? Okay, the laws of physics must be the same for all observers, but what does that mean? It means that all observers can agree on the amount of force working on some object if they know the speed that they themselves have relative to each other and the object. But therein lies the catch! Because, without a notion of absolute space, there is no notion of absolute motion, and therefore it is meaningless to say of two moving objects that one is moving faster than the other (the only thing we measure is a relative velocity so we can only say that they move relative to each other). And if we do not know the speed of some charged particle, we cannot say how strong the Lorentz force is that works on it.

## Lightspeed as a lifebuoy

How does a constant light speed solve this? Let us look at a simple example: suppose we want to measure the speed of a charged particle by detecting photons that it reflects at two points in time (t_1 and t_2; see image below). The speed of the particle is the length of the path it has travelled divided by the time interval in between the two photon-detections. But how do we know the length of this path? We can calculate it by comparing the particle’s position at t_1 and its position at t_2. But only if we know the speed of the photons (which, in classical physics, we wouldn’t know because in an absolute space we can’t be sure that we ourselves are at rest).

Below you’ll find the detailed calculation. We see that we can determine delta x if we know the value of c, since the t-values – in the equation at the bottom – can be measured. 