A (Very) Informal Introduction To Einstein’s Theory Of Special Relativity

¹In the following pages, I will explain Einstein’s special theory of relativity.²

Let’s start with some background you’ll need to know. They are the things that seem obvious, which is why it’s important to clear them up before we get started.³

In high school algebra, you learned about the Cartesian plane. That’s a fancy term for a big plus-sign-looking thing you usually end up drawing on grid paper. You might enjoy referring to the lines of the + as “axes” and labeling them “x” and “y.” Go right ahead, but it’s not mandatory.

What is great about the Cartesian plane is that not only does it take care of horizontal and vertical, but all the diagonals in between.

This, the +, is the simplest most boring thing I could write about. It merits an irritated response like “of course, I already know that.” But before you roll your eyes too hard, think about this. With the purchase of just two lines, you get every line you could ever want, tossed in for free. You can describe any location on a map, or draw⁴ any route from here to there, or go swimming for a long, long time.⁵ What else could you ever want or need?⁶

At some point in middle school you read Flatland, and you don’t remember it.⁷ You were supposed to get an intuition for how interesting three-dimensional space can be. In my own (inaccurate⁸) memory of the book, the Flatlanders experience all sorts of strange phenomena. These mysteries are resolved when it turns out there is a third dimension, and sentient three-dimensional forms have been drifting through the formerly peaceful and straighforward two-dimensional universe.⁹

Educators of all philosophies agree that if you haven’t read this book by the tenth grade, you are doomed to miss the excitement of the third dimension. This is because you spend your entire existence in three spatial dimensions. Familiarity makes it hard to describe, in the same way that it’s hard to describe what the word “the” means.¹⁰ You don’t need to think about it, so you can’t find it too exciting, and it’s very hard to explain what it is, or to imagine what other kind of space there could possibly be.¹¹

My advice is to think of it like this: you have a lot more options when you are given an extra dimension. For every point on every plane (and there are an uncountably¹² infinite number of them), there are an infinite¹³ number of points above and below it. That’s a lot of bonus points.

That takes care of three things¹⁴ that are obvious. The one remaining obvious thing we need to discuss is the idea that time is completely unlike space.¹⁵ You can’t explain how you know that time is different from space, because the whole idea of asking a question like that makes no sense. It’s like explaining how the number two is different from a piece of broccoli. They’re so different you can’t begin to explain it, and anyone who wants it explained is just being annoying. Except that, as it happens, time is in fact just a fourth dimension. Space and time are inextricably linked, and we need to think about the universe in terms of space-time, not space-and-time.

This is the first interesting thing Einstein’s special theory wants us to accept, and probably the easiest. Fine, time’s a dimension just like space. I can draw it on a graph if I consider the Flatlanders, stuck on their dreary two-dimensional plane. I can draw an axis for their mysterious third dimension and label it “time” without a problem.¹⁶ There’s a little matter of right and left, up and down, plus and minus, though. Dimensions go in all directions. Time doesn’t. Einstein didn’t figure out the answer to this flaw in his formulation, so we’ll just let it go.¹⁷

The really counterintuitive thing about the theory of relativity is that it turns out to be a theory of sameness, not of difference. The soul¹⁸ of the theory is that light travels at the same speed, regardless of the reference frame. Don’t try to understand that sentence yet.¹⁹ Just notice the words “same” and “regardless,” which are very different from the word “relative.” Anything “relative” about this theory is just a consequence of the speed of light being constant.²⁰

You could be forgiven for being fairly disappointed right about now. Not only doesn’t this seem to be a theory of relativity, it doesn’t seem particularly special either. So, the speed of light is constant. OK. That doesn’t seem hard to accept. When are we going to get to the part about E=mc²? Space travelers that never age? The speed of light being constant just doesn’t feel weird enough to live up to the promise of life-altering revelations about the physical universe.²¹

Don’t worry. This is where we get to cash in on all those free diagonals from the Cartesian plane. Let’s pretend there is a sequel to Flatland you forgot to read called Lineland, that imagines the plight of sentient dots living in a one-dimensional world. If we draw their single dimension of space as the horizontal axis of the Cartesian plane, and time as the vertical axis, then we can imagine how moving objects in their universe will trace diagonals through that space-time plane. A quick refresher on the mathematics of lines and their slopes will remind us that objects which are moving quickly will have shallower slopes than those which are moving more slowly.²² And, more to the point, objects moving at the same speed will trace out identical diagonals.

Keep that in mind and think back to the reference frames that I asked you to ignore earlier.²³ The idea of reference frames is that objects²⁴ sometimes move at constant speed relative²⁵ to one another. Like, for example, a train moving at constant speed past a station platform.²⁶ The critical point Einstein made is that every reference frame is equal. The laws of physics must behave the same way, regardless of which reference frame you are in. Including the law that no matter where you are when you measure the speed of light, and no matter how many times you measure it, you will always get the same answer.

It is a little hard to think about the speed of light concretely, what with light being both a wave and a particle and all.²⁷ So, let’s pretend for a moment that we are talking about the speed of billiard balls.²⁸ Imagine a billiard ball rolling down the train station platform you’re standing on, at a speed of one meter per second. Now imagine another billiard ball that is also rolling at a speed of one meter per second, but on a train that is moving past you at a speed of two meters per second. You would measure that train-riding billiard ball as having a speed of 3 meters per second.²⁹ If there was a law of physics asserting that billiard balls will always be measured as moving at the same speed, this would be vexing.

At last we are equipped to answer³⁰ the question of why it is surprising that the speed of light doesn’t change, even if measured in one reference frame that is traveling at some speed relative to another reference frame.

To make this more concrete again, let’s go back to the Linelanders. Imagine that there are three reference frames moving at different constant speeds relative to one another. Imagine that the space-time of each reference frame is the two-dimensional spacetime we talked about above, when we got into the whole thing with diagonals and slopes. If the Linelander on reference frame A draws the path of light as it moves through space-time in reference frames B and C, the diagonals should have different slopes, because the reference frames are moving at different speeds. But the theory of special relativity tells us the slopes traced by light moving through space-time must be the same, regardless of who is measuring it and what reference frame they are in. Hence: surprising.

To be the same, the shallower line drawn for the faster reference frame would have to get sort of stretched back up to match the steeper line drawn for the slower one, effectively saying that the Linelanders on the fast reference frame³¹ are moving more slowly in time.³²

And this, finally, is where all the sci-fi tropes³³ come from, about people never aging when they are in a spaceship moving near the speed of light. Moving more slowly through time implies aging more slowly.³⁴ Sadly, the accompanying sci-fi trope of the two differently-aging space-traveling sweethearts meeting up later and being horrified³⁵ by their radical age difference doesn’t work. All this talk of reference frames requires them to move at constant velocity. In order to meet back up, one of the space travelers would have to turn their ship around. That acceleration will undo all this nice math, and they would never get to stare in horror³⁶ at their unaged beloved.

Well, we’ve reached the part where life-altering truths³⁷ have been revealed and corny sci-fi plots have been explained or debunked. My work here is done.³⁸

[Inspired by David Foster Wallace³⁹]

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Shani Offen got her Ph.D. in Neuroscience from NYU at exactly the moment⁴⁰ that she stopped being a graduate student. She managed to get home despite the Google maps fiasco described in footnote ⁴, so she now has a job. She would specify that job here, except that she doesn’t know when you will be reading this, so she suggests that you Google her⁴¹ and see what the internet thinks she’s doing right now. It will almost certainly involve division, which has always been her favorite algorithm.⁴²

¹ Before we even get started, it’s only fair to mention that sometimes I think of things I want to say but don’t really need to say, and I try to stick those in footnotes. There’s no point in getting annoyed this early, so if footnotes annoy you, just ignore them completely from here on out.
² I don’t mean that I think it’s special. I mean that it’s not his general theory. Usually special is more complicated than general, but not in physics. The general theory is way too hard to explain.
³ What I’m saying is that obvious things are the trickiest ones. I wasn’t sure if that was obvious or not.
⁴ But I have to suggest that rather than trying to draw the route yourself, you use Google maps, which knows more than you do. Except when it shows you the route to the local train station so you can go home after a very tiring day of interviews, and you get to the end of the route, and Google maps congratulates you for having arrived, but you are standing in front of an apartment building and there are no tracks in sight.
⁵ Or biking or running or rolling or cartwheeling or whatever you like. The point is, the plane is infinite in any direction.
⁶ I don’t actually mean this rhetorically. There’s an answer to this question: more dimensions. Things change dramatically when you bump up your dimensionality. The same way that adding a vertical axis to the horizontal one suddenly allows you to move in infinitely more directions, adding a third (or a fourth or fifth) dimension similarly opens up infinitely more possibilities.
⁷ That’s ok, no one does.
⁸ As it turns out, the book is more of a social commentary than a mystery novel. I had to check Wikipedia.
⁹ The plot of the actual book is that a sentient sphere visits a different two-dimensional Flatlander every year, in the hope of convincing the inhabitants of Flatland of the existence of a third dimension. The sphere manifests as a circle that appears and disappears and grows and shrinks mysteriously, because it is moving in and out of the plane along the heretical third orthogonal dimension. I prefer the (nonexistent) version that I remember, which was more of a Sherlock-Holmes-style locked-room mystery novel.
¹⁰ I said describe, not define. I know that it’s easy to google the grammatical term for the role “the” plays in the language. But can you describe it? (Easily?)
¹¹ If I haven’t just blown your mind, you didn’t understand what I just said. Go back and read it again, then come back to this footnote. Welcome back. Now you got it?
¹² This means really infinite, with all the holes plugged in, like tar instead of charcoal.
¹³ Yes, still the tar type, I just didn’t feel like saying ‘uncountably’ again.
¹⁴ The x,y,z of space. (Sorry, this one’s a traditional footnote. Or it was, until I added this first-person parenthetical.)
¹⁵ Aside from the inherent pleasure of revisiting middle school experiences, this is why I brought up Flatland. We, too, experience all sorts of strange phenomena. And as it turns out, being forced to think about time as an additional dimension is critical to understanding those phenomena.
¹⁶ Unless we think too hard about what it means to move around in time as if it were space. Then it becomes a problem. So let’s avoid that for now.
¹⁷ I find it satisfying to keep a record of all the things Einstein didn’t know or just got plain wrong, like quantum mechanics. The fallibility of genius can be very reassuring when your own fallibility is getting you down.
¹⁸ Referring to it as the soul rather than the heart is a pointlessly subtle nod to Einstein’s famous rejection of quantum theory for reasons related to his beliefs about God’s gambling habits.
¹⁹ I wouldn’t be able to follow an instruction like that either. So, let me explain reference frames. They’re just sections of space-time moving with respect to one another. Now let’s go back to not worrying about it, shall we?
²⁰ I leave it as an exercise to the reader to decide why they didn’t name it the special theory of immutability
²¹ If anything, the weirdest thing about it is the idea that light has a speed at all, and that’s not even what this theory is about.
²² If you are lucky enough to know a third grader, you can ask them to explain this to you. Otherwise, go ahead and sketch out for yourself the trajectory something would trace if it moved two space units for every time unit, vs one space unit for every two time units. Voila.
²³ Assuming you are a footnote reader (which seems a fair assumption given the fact that you are reading this footnote), feel free to skip this part, since we already covered it in an earlier footnote.
²⁴ “Object” is really not the right term. But what else could I use? I could be more technical and say “units of mass” or “distinct space-time coordinate systems,” but that would be pedantic and potentially confusing. I could go in the other direction and just say “stuff,” but that would be too informal, even for me. So, we are stuck with the worst of all worlds, a word that is neither vague enough nor specific enough to be correct. There is a lesson in there somewhere that I choose to ignore.
²⁵ Yes, fine. This is why it is called the theory of relativity. And the reason it is called special is that reference frames moving at constant relative speeds are just a special case of a larger class of reference frames that can move at non-constant (accelerating) relative speeds. Now what does that have to do with gravity, you ask? That is a topic for another article. And since I can’t put footnotes on footnotes, you’re really just going to have to deal with waiting this time.
²⁶ Trains and platforms are always used as the example when talking about special relativity. I suppose that’s because they are familiar and therefore provide some intuition. Unfortunately, most people’s train experience doesn’t include trains that travel near the speed of light, so the intuition they provide can be more confusing than helpful. Consider yourself warned.
²⁷ That’s what quantum mechanics is about. And we’ve already discussed Einstein’s feelings on quantum mechanics.
²⁸ Everything in physics that isn’t a train is a billiard ball. No one knows why.
²⁹ The problem with the train analogy is that it’s obvious to us that the train is moving with respect to everything else, and not vice versa. So we automatically say “oh, but it’s the train that’s moving faster, the billiard ball is still moving at the same speed.” For any human thinking about trains and platforms, one of the reference frames is more authoritative than the others. Einstein’s insight depends on the assertion that no reference frame is created more equal than any other. There is no way to pick one reference point and say that everything else is moving relative to it. You can’t just reject the fact that the billiard ball is moving at 3 meters per second relative to the person on the platform, by claiming that it’s the train’s motion that accounts for the difference.
³⁰ Or, at the very least, to ask it more clearly.
³¹ More formally, this means that to translate between the space-times of different reference frames, you can no longer rely on your friends in third grade. You need much fancier math, called Lorentz transforms.
³² Here’s where the headache begins. How can you move more slowly in time? It’s like saying you moved less space in space. Actually, that’s not a bad way to think of it. Let’s go with that.
³³ Understanding special relativity is mostly useful for thinking about sci-fi plots. It doesn’t come up much in daily life otherwise, what with us Earthlings moving at a speed that is effectively 0 relative to the speed of light.
³⁴ This has actually been measured and confirmed in muons, which move with a speed much closer to the speed of light than we do.
³⁵ Or delighted. Depends on what kind of sci fi you like to read.
³⁶ Or delight.
³⁷ I’m referring to the way that space and time stretch and compress to accommodate the constant speed of light. Though the part about diagonal lines was pretty great too.
³⁸ Oh wait! I didn’t explain the part about E=mc² yet, the part where energy and matter are interchangeable. That’s also a consequence of the weirdness of space-time in the presence of near-light speeds, but I don’t find it as fun as the part about time slowing down or space compressing. Plus, it always makes me think of nuclear bombs, and that’s a downer. So this footnote is all you’re going to get on that topic.
³⁹ Which should be pretty obvious to you if you’ve read through to the 39^th footnote.
⁴⁰ Another consequence of asserting that no reference frame is more definitive than any other, is that there can be no absolute concept of simultaneity. However, the kind of semantic simultaneity employed here remains intact.
⁴¹ She finds it amusing to write about herself in the third person for this bio, when she just spent quite a bit of time writing in first person for the article itself.
⁴² As you might have already gathered from her passion for diagonal lines and their slopes. She wasn’t kidding about how great she thinks those are.