Physics, Secondary ed level, Tutorial

Einstein’s special relativity in under 6.999 minutes for people on the move

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In 1905, Albert Einstein published an article on moving bodies and electrodynamics. He noticed that Newton’s mechanics of moving bodies weren’t compatible with Maxwell’s equations of electromagnetism. In this article, he reconciled the two by modifying the first. These ideas and mathematical derivations became what we now know as Einstein’s special (theory of) relativity. In this post, we will describe some of its important bits. We start with two fundamental propositions. For the geeks, we will end with answering why special relativity is special.

Einstein’s postulates

His first postulate is basically that, whether or not you’re standing on a moving object, such as a ship, a train, or in your car, the same laws of physics apply. If you’re standing on a platform at the train station and you throw a ball in the air, the laws of physics ensure your ball comes down again. If you’re standing in a moving train, that ball still comes down again because the same laws of physics apply. The fact that you’re in motion doesn’t change anything to the rules of the Universe.

Einstein’s second postulate is basically that, by extension of the first one, the speed of light is the same for everyone. He recognised that Maxwell and colleagues correctly describe light as an electromagnetic disturbance propagating according to the electromagnetic laws of physics. He also recognised that the velocity of the light source plays no role in Maxwell’s equations. And so, if the first postulate is correct, then irrespective of the velocity of an observer relative to the light source, light travels at the same speed $c$ ($c$ is about 300 000 000 m/s) and nothing can go faster.

An image of the interior of an underground train. Passengers are sitting across each other.
Whether you’re on the train or standing on the platform, the laws of physics are the same. The propagation of light is a law of electrodynamics not involving the velocity of its source. Hence, its velocity is the same on the train as on the platform, irrespective of its source or the train’s velocity.

Intuition works mostly, just not really

Suppose, you’re standing on a train station’s platform. A train passes at a speed of 30 metre per second. From inside the train, our friend throws a tennis ball out the window but in the direction of where the train is headed, at 2 metre per second, right at you. You catch it. At what speed does the ball hit your hand?

Intuitively, you might say, that’s 30 + 2 = 32 metre per second. This way of calculating is very useful, most of the time. Instinctively, you would simply add the train’s speed and the throwing speed together. This is how Isaac Newton1 would want you to do it. And, mostly, he’s not wrong. Except, well, he is kinda.

Let’s wonder what would happen if our friend didn’t throw a ball, but, instead, switched on a pocket torch. The train still passes at 30 m/s. Light, however, flies out the torch at a speed of about 300 000 000 m/s. You lift up your hand. It ‘catches’ the light. At what speed does it hit your hand?

You might say, that’s 30 + 300 000 000 = 300 000 030 m/s. But no. That’s wrong. Remember Einstein’s second postulate? Irrespective of the motion of the observer, light always travels at 300 000 000 m/s and nothing goes faster, full stop. So, by our simple addition, we would have invented a way for light to go faster than light! We would be Nobel Prize winners, surely. Except, it doesn’t, and we’re not.

Something’s gotta give

So, if the speed of light is the same to our friend, on the moving train, as it is to us, standing on the platform (do read this again and realise how bonkers this is), then how the Helheim does the Universe achieve this? After he did some relatively simple mathematics – which a student in secondary education can do – Einstein realised that something was up with metres and seconds. He proved that what is a metre to us isn’t a metre to our friend and vice versa. Furthermore, what is a second to us isn’t a second to our friend either.

Basically, since we, on the platform, measure light to be going at 300 000 000 metre per second, and so does our friend on the train, well, that means that our understanding of what metres and seconds are is wrong.

The point

Here’s what’s happening. If two ‘things’, people, ‘objects’, or reference frames as physicists call them, move with respect to each other, weird things happen to space and time. Yes, the actual space and time. They are weird. We thought they were just there. Static. Always and everywhere the same. Two unchanging entities. Well, they’re not.

In the Dutch town of Leiden, the project Leiden Wall Formulas have scattered physics equations throughout the town centre. This is the Lorentz contraction, aptly placed alongside a train track: it describes how space (in this case, a one-dimensional length) is contracted, according to Einstein’s special relativity. Click here for Google Street View.

Suppose, we are on the train this time. We are in motion relative to our friend on the platform. Our friend then observes that space along our direction of motion becomes smaller, it contracts. They will actually measure our train to be shorter as compared to when it was standing still relative to our friend. They will also observe that our time is being stretched, i.e. our clock slows down. If one second goes by on their clock, they see only 0.9999999995… seconds have past on ours2.

And to us, being on the train, it’s our friend who is travelling (backwards!) relative to us, in fact, the whole world is travelling relative to us, backwards. So, indeed, we measure the world to be shorter in the opposite direction of our motion as well as their time being slowed down.

Twin paradox

Now, you might say, hold on: if both of us see each other’s clock slow down, then surely, there is no difference between our clocks. However, when we ride back to our friend, stop at the platform, and compare our clock to our friend’s clock, we do see that our clock is behind. This is the so-called twin paradox. If both can state the same thing, how do their clocks still differ in the end, causing one half of the twin (on the train) to be younger than the half who stayed behind (on the platform)?

This is due to the switching of reference frames: we were on a moving frame (the train), switched to a moving frame in the opposite direction (returning to our friend), and, lastly, switched to the platform’s frame, standing next to our friend, to compare clocks, while our friend never switched – he stayed on the platform3. So, the situation isn’t symmetric.

Length contraction and time dilation. They’re not illusory, they’re real. Many experiments showed that space contracts and time dilates for things in motion as observed by things with a different motion.

Newton vs Einstein

To summarise in a slightly more mathematical way – Newton taught us that one platform metre equals one train metre and that the same is true for seconds:

1 platform metre = 1 train metre,
1 platform second = 1 train second.

Einstein, however, taught us that:

1 platform metre $\equiv \dfrac{1\text{ train metre}}{\gamma}$,
1 platform second $\equiv \dfrac{1\text{ train second}}{\gamma}$.

This $\gamma$ (Greek letter gamma) is crucial. It’s a factor necessary to make sure that the speed of light doesn’t get any faster than the speed of light, even if it’s on-board a moving train. This factor is called the Lorentz factor4.

We never notice these things though. Usually, our speeds are way to slow for the effects of special relativity to be noticeable. Except for things that do go fast. Without Einstein’s special relativity, GPS satellites wouldn’t work5. Research institutes such as CERN and Fermilab also need to take special relativity in account for the high-energy, fast-flying particles.

So, our intuition (Newton) is mostly just fine though not precisely right.

What’s so special about special relativity

This is a somewhat technical question, requiring a somewhat technical answer, our apologies. Contrary to what is being said in many popular science texts, special relativity isn’t really about constant speeds as opposed to general relativity dealing with acceleration. In fact, special relativity is able to deal with acceleration. It’s just that it works fine as long as we’re assuming things exist in so-called Euclidean space, adhering to Euclidean geometry6. However, after ample deliberation, Einstein concluded through his general relativity that we’re not living in Euclidean-geometric reality at all. We’re living in a Riemannian manifold7. In short, Euclidean space is a special case of the more general Riemannian manifold8. Hence, we have special relativity as opposed to general relativity.

Featured image by StockSnap from Pixabay, modified by @kjrunia (added equations).

  1. And Galileo Galilei before him as this is the so-called Galilean transformation.[↩]
  2. This number is a metaphor. The real time difference is too small for my calculator to show.[↩]
  3. Contrary to many popular and even introductory physics texts, this has less to do with acceleration, even though this does plays a role – without it, in the real world, one wouldn’t be able to switch reference frames. However, mathematically, acceleration isn’t necessary for solving the so-called twin paradox, switching frames is.[↩]
  4. Named after Hendrik Lorentz. The expression for his Lorentz factor is as follows: \[ \gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}, \] where $v$ is the speed of the other relative to us and $c$ is the speed of light.[↩]
  5. Of course, we also need Einstein’s General Relativity for that, but that’s for another time.[↩]
  6. Named after Euclid.[↩]
  7. Named after Bernhard Riemann.[↩]
  8. If we’re being precise, Hermann Minkowski developed a modified version of Euclidean space which we now call Minkowski space, while Riemannian manifold should actually be named pseudo-Riemannian manifold but many physicists simply call this Riemannian manifold anyway, perhaps because they’re not mathematicians.[↩]