Special Relativity 2: The Speed of Light and its Wonders
The speed of light, a universal speed limit with a lot of interesting thought experiments.
In the last article (here), we spoke about “frames of reference” and laid down the groundwork for our dive into the theory of Special Relativity. In this article, we will go into the speed of light and all its wonders. To fully understand and appreciate Special Relativity, we first need to discuss the speed of light (after all, the theory is built for things moving close to the speed of light!). I will go into detail about what the speed of light actually is, why it is the universal speed limit and some interesting “paradoxes” that can crop up as a result. This is where the fun begins (trust me, no more trains)!
The Speed of Light
Everybody has heard the phrase before “faster than the speed of light” or “as fast as light”, but did you ever stop and think “why is it always comparing fast things with light?”. Of course, you cannot see light moving and when you switch on a lamp, the light seems to hit your eye almost instantly, so one can see why we would compare fast things with light. But light is something a bit different, it is the ultimate speed limit in the universe which nothing can exceed. That is because the “ultimate speed limit” is not actually the speed of light in general, but rather the speed of light in a vacuum. And this speed is actually the speed limit for information itself, not just light (they happen to be the same thing, as a result of the physics!). So when someone says “nothing can go faster than light”, you can smugly reply “actually, I think you mean light in a vacuum, which is the speed limit for information itself” and then lose a friend or make a stranger on the bus think you are a smart Alec.
So, what is this “speed limit” that I have been going on about so much? Well, in physics, we describe everything using equations. We cannot truly understand reality, nor appreciate it fully without them, so I will show a very important equation and break it down into pieces so we can digest what it does. Light (for our purposes here) is a wave, and so the propagation (spread or movement) of this wave is what we need to describe using our equations. For this, we use the aptly named, wave equation. The wave equation is broken down below.
The wave equation above relates how the wave moves in time (red box) with how the wave moves in space (orange box) using the speed of the wave itself (green box) as the proportional factor. Now we are familiar with this equation’s existence, let’s bring this discussion back to light. When using something called Maxwell’s equations (I won’t go through these here, but see Ref for a brief but detailed explanation), light, which is an electromagnetic wave, can be cast into the same form as the wave equation above! What does this mean? It means we can get the speed (the green box) but for a light wave! This turns out to be given by two constants of nature, the “permittivity of free space” and the “permeability of free space”. We won’t go into these two numbers, but it means that the speed of light is given by two constants, i.e. the speed of light is constant too! This speed is 299,792,458 metres per second, which can get you to the moon in around 1.3 seconds!
Speed of light and reference frames
So now we’ve discussed what the speed of light is and also what frames of reference are. Now, let’s combine these two ideas and come up with some interesting “paradoxes”! Let’s go over some quick facts:
- The speed of light (in a vacuum) is constant, this means it’s the same all the time, everywhere.
- Frames of reference are the “point of view” of someone or something in space and time. Depending on the frame of reference, there will be different observed speeds (remember the ball on the train!).
Paradox #1 — The mirror with no face.
So, armed with just these two facts, we can already see a rule-breaking scenario. Let’s imagine that you can travel at the speed of light (of course, you can’t, don’t bother trying). Now, let’s also imagine that you brought a mirror with you to check on your looks while moving at nearly 300,000 km per second. What do you see in the mirror? Let’s break down how we see objects in mirrors, first.
When looking into a mirror, how do we see what we see? Well, to see anything, the light from that object must travel to our eyes. In a mirror, the light must travel from the object, to the mirror, reflect on the surface of the mirror and then travel to our eye. So, if we are looking at ourself, the light must travel from our face, to the mirror and then back to our eyes.
But, we are moving at the speed of light, so how can the light travel from our face, to the mirror in the first place if we are moving at the same speed as the light coming from our face going to the mirror? Let’s now remind ourselves of our reference frame. The speed of light must be constant, in all frames, that includes ours, so in our frame of reference, the speed of light should be the same with respect to us and so we would see ourselves normally in the mirror. But, from someone else’s frame of reference, who is still, they will see the light from our face travelling at twice the speed of light; this breaks physics! Let’s think of the other scenario, you don’t see anything. Well, that means in your frame of reference, the light from your face has zero speed with respect to you; this also breaks physics!
So, how do we fix this?? We’ll see in future articles, but for now, you can smugly tell this one to your friends.
Paradox #2 — The car with broken headlights.
In a similar fashion, let’s imagine we are driving in a car at the speed of light. Ignoring the paradox above and the obvious question of “if we travel at the speed of light, what do we even see?”, let’s now turn on the headlights. Do we see the headlights? Once again, depending on the frame of reference, the light from the headlights will either be seen travelling at twice the speed of light, or at zero speed, both of which cannot happen!
In this article, we went into a bit of a deep dive into what light is and what exactly the speed of light is. We then, armed with the wave equation, found what the speed of light really is. Coupling this with our knowledge of reference frames, we then went into some thought experiments that seem to break physics! In the next article, we will see what really happens when you travel close to the speed of light.
