On (special) relativity: the Lorentz transformations

Pre-scriptum (dated 26 June 2020): These posts on elementary math and physics have not suffered much the attack by the dark force—which is good because I still like them. While my views on the true nature of light, matter and the force or forces that act on them have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think most (if not all) of the analysis in this post remains valid and fun to read. In fact, I find the simplest stuff is often the best. 🙂

Original post:

I just skyped to my kids (unfortunately, we’re separated by circumstances) and they did not quite get the two previous posts (on energy and (special) relativity). The main obstacle is that they don’t know much – nothing at all actually – about integrals. So I should avoid integrals. That’s hard but I’ll try to do so in this post, in which I want to introduce special relativity as it’s usually done, and so that’s not by talking about Einstein’s mass-energy equivalence relation first.

Galilean/Newtonian relativity

A lot of people think they understand relativity theory but they often confuse it with Galilean (aka Newtonian) relativity and, hence, they actually do not understand it at all. Indeed, Galilean or Newtonian relativity is as old as Galileo and Newton (so that’s like 400 years old), who stated the principle of relativity as a corollary to the laws of motion: “The motions of bodies included in a given space are the same amongst themselves, whether that space is at rest or moves uniformly forward in a straight line.”

The Galilean or Newtonian principle of relativity is about adding and subtracting speeds: if I am driving at 120 km/h on some highway, but you overtake me at 140 km/h, then I will see you go past me at the rather modest speed of 20 km/h. That’s all what there is to it.

Now, that’s not what Einstein‘s relativity theory is about. Indeed, the relationship between your and my reference frame (yours is moving with respect to mine, and mine is moving with respect to yours but with opposite velocity) is very simple in this example. It involves a so-called Galilean transformation only: if my coordinate system is (x, y, z, t), and yours is (x‘, y‘, z‘, t‘), then we can write:

(1) x’ = x – ut (or x = x’ + ut),  (2) y’ = y, (3) z’ = z and (4) t’ = t

To continue the example above: if we start counting at t = t’ = 0 when you are overtaking me, and if we both consider ourselves to be at the center of our reference frame (i.e. x = 0 where I am and x’ = 0 where you are), then you will be at x = 10 km after 30 minutes from my point of view, and I will be at x’ = –10 km (so that’s 10 km behind) from your point of view. So x’ = x – ut indeed, with u = 20 km/h.

Again, that’s not what Einstein’s principle of relativity is about. They knew that very well in the 17th century already. In fact, they actually knew that much earlier but Descartes formalized his Cartesian coordinate system only in the first half of the 17th century and, hence, it’s only from that time onwards that scientists such as Newton and Huygens started using it to transform the laws of physics from one frame of reference to another. What they found is that those laws remained invariant.

For example, the conservation law for momentum remains valid even if, as illustrated below, an inertial observer will see an elastic collision, such as the one illustrated, differently than a observer who’s moving along: for the observer who’s moving along, the (horizontal) speed of the blue ball will be zero, and the (horizontal) speed of the red ball will be twice the speed as observed by the inertial observer. That being said, both observers will find that momentum (i.e. the product of mass and velocity: p = mv) is being conserved in such collisions.

But, again, that’s Galilean relativity only: the laws of Newton are of the same form in a moving system as in a stationary system and, therefore, it is impossible to tell, by making experiments, whether our system is moving or not. In other words: there is no such thing as ‘absolute speed’. But, so – let me repeat it again – that is not what Einstein’s relativity theory is about.

Let me give a more interesting example of Galilean relativity, and then we can see what’s wrong with it. The speed of a sound wave is not dependent on the motion of the source: the sound of a siren of an ambulance or a noisy car engine will always travel at a speed of 343 meter per second, regardless of the motion of the ambulance. So, while we’ll experience a so-called Doppler effect when the ambulance is moving – i.e. a higher pitch when it’s approaching than when it’s receding – this Doppler effect does not have any impact on the speed of the sound wave. It only affects the frequency as we hear it. The speed of the wave depends on the medium only, i.e. air in this case.

Indeed, the speed of sound will be different in another gas, or in a fluid, or in a solid, and there’s a surprisingly simple function for that – the so-called Newton-Laplace equation: v_sound = (k/ρ)². In this equation, k is a coefficient of ‘stiffness’ of the medium (even if ‘stiffness’ sounds somewhat strange as a concept to apply to gases), and ρ is the density of the medium (so lower or higher air density will increase/decrease the speed of sound).

This has nothing to do with speed being absolute. No. The Galilean relativity principle does come into play, as one would expect: it is actually possible to catch up with a sound wave (or with any wave traveling through some medium). In fact, that’s what supersonic planes do: they catch up with their own sound waves. However, in essence, planes are not any different from cars in terms of their relationship with the sound that they produce. It’s just that they are faster: the sound wave they produce also travels at a speed of 1,235 km/h, and so cars can’t match that, but supersonic planes can!

[As for the shock wave that is being produced as these planes accelerate and actually ‘break’ the ‘sound barrier’, that has to do with the pressure waves the plane creates in front of itself (just like a traveling compresses the air in front of it). These pressure waves also travel at the speed of sound. Now, as the speed of the object increases, the waves are forced together, or compressed, because they cannot get out of the way of each other. Eventually they merge into one single shock wave, and so that’s what happens and creates the ‘sonic boom’, which also travels at the speed of sound. However, that should not concern us here. For more information on this, I’d refer to Wikipedia, as I got these illustrations from that source, and I quite like the way they present the topic.] 

The Doppler effect looks somewhat different (it’s illustrated above) but so, once again, this phenomenon has nothing to do with Einstein’s relativity theory. Why not? Because we are still talking Galilean relativity here. Indeed, let’s suppose our plane travels at twice the speed of sound (i.e. Mach 2 or almost 2,500 km/h). For us, as inertial observers, the speed of the sound wave originating at point 0 in the illustration above (i.e. the reference frame of the inertial observer) will be equal to dx/dt = 1235 km/h. However, for the pilot, the speed of that wave will be equal to

dx’/dt = d(x – ut)/dt = dx/dt – d(ut)/dt = dx/dt – d(ut)/dt = 1235 km/h – u

= 1235 km/h – u = 1235 km/h – 2470 km/h = – 1235 km/h

In short, from the point of view of the pilot, he sees the wave front of the wave created at point 0 traveling away from him (cf. the negative value) at 1235 km/h, i.e. the speed of sound. That makes sense obviously, because he travels twice as fast. However – I cannot repeat it enough – this phenomenon has nothing to do with Einstein’s theory of relativity: if they could have imagined supersonic travel, Galileo, Newton and Huygens would have predicted that too.

So what’s Einstein’s theory of (special) relativity about?

Einstein’s principle of relativity

In 1865, the Scottish mathematical physicist James Clerk Maxwell –  I guess it’s important to note he’s Scottish with that referendum coming 🙂 – finally discovered that light was nothing but electromagnetic radiation – so radio waves, (visible) light, X-rays, gamma rays,… It’s all the same: electromagnetic radiation, also known as light tout court.

Now, the equations that describe how electromagnetic radiation (i.e. light) travels through space are beautiful but involve operators which you may not recognize and, hence, I will not write them down. The point to note is that Maxwell’s equations were very elegant but… There were two major difficulties with them:

1. They did not respect Galilean relativity: if we transform them using the above-mentioned Galilean transformation (x’ = x – ut, y’ = y, z’ = z and t’ = t) then we do not get some relative speed of light. On the contrary, according to Maxwell’s equations, from whatever reference frame you look at light, it should always travel at the same (absolute) speed of light c = 299,792 km/h. So c is a constant, and the same constant, ALWAYS.
2. Scientists did not have any clue about the medium in which light was supposed to travel. The second half of the 19th century saw lots of experiments trying to discover evidence of a hypothetical ‘luminiferous aether’ in which light was supposed to travel, and which should also have some ‘stiffness’ and ‘density’, but so they could not find any trace of it. No one ever did, and so now we’ve finally accepted that light can actually travel in a vacuum, i.e. in plain nothing.

So what? Well… Let’s first look at the first point. Just like a sound wave, the motion of the source does not have any impact on the speed of light: it goes out in all directions at the same speed c, whether it is emitted from a fast-moving car or from some beacon near the sea. However, unlike sound waves, Maxwell’s equations imply that we cannot catch up with them. That’s troublesome, very troublesome, because, according to the above-mentioned Galilean transformation rules,

i.e. v’ = dx’/dt = dx/dt – u = v – u,   

some light beam that is traveling at speed v = c past a spaceship that itself is traveling at speed u – let’s say u = 0.2c for example – should have a speed of c‘ = c – 0.2c = 0.8c = = 239,834 km/h only with respect to the spaceship. However, that’s not what Maxwell’s equations say when you substitute x, y, z and t for x‘, y‘, z‘ and t‘ using those four simple equations x’ = x – ut, y’ = y, z’ = z and t’ = t. After you do the substitution, the transformed Maxwell equations will once again yield that c’ = c = 299,792 km/h, and not c’ = 0.8×299,792 km/h = 239,834 km/h.

That’s weird ! Why? Well… If you don’t think that this is weird, then you’re actually not thinking at all ! Just compare it with the example of our sound wave. There is just no logic to it !

The discovery startled all scientists because there could only be possible solutions to the paradox:

1. Either Maxwell’s equations were wrong (because they did not observe the principle of (Galilean relativity) or, else,
2. Newton’s equations (and the Galilean transformation rules – i.e. the Galilean relativity principle) are wrong.

Obviously, scientists and experimenters first tried to prove that Maxwell had it all wrong – if only because no experiment had ever shown Newton’s Laws to be wrong, and so it was probably hard – if not impossible – to try to come up with one that would ! So, instead, experimenters invented all kinds of wonderful apparatuses trying to show that the speed of the light was actually not absolute.

Basically, these experiments assumed that the speed of the Earth, as it rotates around the Sun at a speed of 108,000 km per hour, would result in measurable differences of c that would depend on the direction of the apparatus. More specifically, the speed of the light beam, as measured, would be different if the light beam would be traveling parallel to the motion of the Earth, as opposed to the light beam traveling at right angle to the motion of the Earth. Why? Well… It’s the same idea as the car chasing its own light beams, but I’ll refer to you to other descriptions of the experiment, because explaining these set-ups would take too much time and space. 🙂 I’ll just say that, because 108,000 km/h (on average) is only about 30 km per second (i.e. 0.0001 times c), these experiments relied on (expected) interference effects. The technical aspect of these experiments is really quite interesting. However, as mentioned above, I’ll refer you to Wikipedia or other sources if you’d want more detail.

Just note the most famous of those experiments: the 1887 Michelson-Morley experiment, also known as ‘the most famous failed experiment in history’ because, indeed, it failed to find any interference effects: the speed of light always was the speed of light, regardless of the direction of the beam with respect to the direction of motion of the Earth.

The Lorentz transformations

Once the scientists had recovered from this startling news (Michelson himself suffered from a nervous breakdown for a while, because he really wanted to find that interference effect in order to disprove Maxwell’s Laws), they suggested solutions.

The math was solved first. Indeed, just before the turn of the century, the Dutch physicist Hendrik Antoon Lorentz suggested that, if material bodies would contract in the direction of their motion with a factor (1 – u²/c²)^1/2 and, in addition, if time would also be dilated with a factor (1 – u²/c²)^–1/2, then the Michelson-Morley results could be explained. Of course, scientists objected to this ‘explanation’ as being very much ‘ad hoc’.

So then came Einstein. He just took the math for granted, so Einstein basically accepted the so-called Lorentz transformations that resulted from it, and corrected Newton’s Law in order to set physics right again.

And so that was it. As it turned out, all that was needed in fact, was to do away with the assumption that the inertia (or mass) of an object is a constant and, hence, that it does not vary with its velocity. For us, today, it seems obvious: mass also varies, and the factor involved is the very same Lorentz factor that we mentioned above: γ = (1 – u²/c²)^–1/2. Hence, the m in Newton’s Second Law (F = d(mv)/dt) is not a constant but equal to m = γm₀. For all speeds that we, human beings, can imagine (including the astronomical speed of the Earth in orbit around the Sun), the ‘correction’ is too small to be noticeable, or negligible, but so it’s there, as evidenced by the Michelson-Morley experiment, and, some hundred years later, we can actually verify it in particle accelerators.

As said, for us, today, it’s obvious (in my previous post, I mention a few examples: I explain how the mass of electrons in an electron beam is impacted by their speed, and how the lifetime of muon increases because of their speed) but one hundred years ago, it was not. Not at all – and so that’s why Einstein was a genius: he dared to explore and accept the non-obvious.

Now, what then are the correct transformations from one reference frame to another? They are referred to as the Lorentz transformations, and they can be written down (in a simplified form, assuming relative motion in the x direction only) as follows:

Now, I could point out many interesting implications, or come up with examples, but I will resist the temptation. I will only note two things about them:

1. These Lorentz transformations actually re-establish the principle of relativity: the Laws of Nature – including the Laws of Newton as corrected by Einstein’s relativistic mass formula – are of the same form in a moving system as in a stationary system, and therefore it is impossible to tell, by making experiments, whether the system is moving or not.

2. The second thing I should note is that the equations above imply that the idea of absolute time is no longer valid: there is no such thing as ‘absolute’ or ‘universal’ time. Indeed, Lorentz’ concept of ‘local time’ is a most profound departure from Newtonian mechanics that is implicit in these equations.

Indeed, space and time are entangled in these equations as you can see from the –ut and –ux/c² terms in the equation for x’ and t’ respectively and, hence, the idea of simultaneity has to be abandoned: what happens simultaneously in two separated places according to one observer, does not happen at the same time as viewed by an observer moving with respect to the first. Let me quickly show how.

Suppose that in my world I see two events happening at the same time t₀ but so they happen at two different places x₁ and x₂. Now, if you are movingaway from me at a (uniform) speed u, then equation (4) tells us that you will see these two events happen at two different times t₁‘ and t₂‘, with the time difference t₁‘ – t₂‘ equal to t₁‘ – t₂‘ = γ[u(x₁ – x₂)/c²], with γ the above-mentioned Lorentz factor. [Just do the calculation for yourself using equation 4.]

Of course, the effect is negligible for most speeds that we, as human beings, can imagine, but it’s there. So we do not have three separate space coordinates and one time coordinates, but four space-time coordinates that transform together, fully entangled, when applying those four equations above. 

That observation led the German mathematician Hermann Minkowski, who helped Einstein to develop his theory of four-dimensional space-time, to famously state that “Space of itself, and time of itself, will sink into mere shadows, and only a kind of union between them shall survive.”

Post scriptum: I did not elaborate on the second difficulty when I mentioned Maxwell’s equations: the lack of a need for a medium for light to travel through. I will let that rest for the moment (or, else, you can just Google some stuff on it). Just note that (1) it is kinda convenient that electromagnetic radiation does not need any medium (I can’t see how one would incorporate that in relativity theory) and (2) that light does seem to slow down in a medium. However, the explanation for that (i.e. for light to have an apparently lower speed in a medium) is to be found in quantum mechanics and so we won’t touch upon that complex matter here (for now that is). The point to note is that this slowing down is caused by light interacting with the matter it encounters as it travels through the medium. It does not actually go slower. However, I need to stop here as this is, yet again, a post which has become way too long. On the other hand, I am hopeful my kids will actually understand this one, because it does not involve integrals. 🙂