## A Mathematical Intro to Special Relativity

Albert Einstein was a smart guy.

I mean, he was so good at coming up with good ideas that they gave him the Nobel prize for inventing a theoretical basis for the photoelectric effect, not for any of his, perhaps more famous, work on general and special relativity. I mean, he came up with a formula more famous than the Pythagorean theorem, $E = mc^2$!

In this post, we want to talk about one of Einstein’s first famous discoveries: special relativity.1

Leading up to this, it was noticed that light was weird. Unlike most waves, it didn’t seem to propagate through anything. Sound waves propagate through air, sea waves through water, but light seems to be doing its own thing, without any sort of background.

Special relativity is Einstein’s attempt to explain this weirdness. And, to do this, he used only two assumptions.

The first was the principle of relativity. This says that you can’t tell how fast you’re traveling in absolute terms, but only how fast you are traveling relative to something else. For example, in a plane, are you traveling fast? Or sitting still? It depends on whether you’re comparing your position to a location on Earth’s surface…

…or to the position of the screaming child sitting next to you.

But neither choice is right, just different.

The second assumption was the universality of the speed of light. This says that no matter who is looking, or how, photons always travel at the same speed.2

This assumption is a bit weird. Experimentally, the Michelson-Morley experiment suggests this is true, though it’s not clear Einstein was directly influenced by this.

One explanation is that, if light propagated through something (like sound in air), it should go faster or slower depending on which direction that something was traveling. Sound, for instance, travels more slowly into the wind than with the wind.

What does the assumption of the universality of the speed of light mean?

Normally, if you’re in a car going 20 km/hr (relative to me, standing by the side of the road), and throw an orange at me at 30 km/hr (relative to yourself), the orange will be heading towards me at 50 km/hr.3

But with light, it’s different.

For that same car, if you turn on your headlights, you could measure the light leaving the car at, well, the speed of light, which is absurdly close to $3\times 10^8$ m/s. But if I were to measure the same light, I would still only see it traveling at $3\times 10^8$ m/s, despite the fact that your car is going 20 km/hr.

However, despite only assuming these two things, special relativity is… pretty weird.

To wrap our heads around all this, it will be incredibly useful to draw some “spacetime diagrams.” We start with a set of axes.

Notice that the two axes are labeled $x$ and $t$, i.e., space and time. We could draw these with more space dimensions (after all, we live in three space dimensions), but everything is clearer if we just draw one space dimension, $x$.

A good way to think about these diagrams is that you are sitting at $x=0$ not moving. You’re there at $x=0$ at the beginning, $t=0$, and you’re there later, at say $t=100$. Everything else is happening around you.

What does it look like if your friend walks by you?

Let’s say he crosses you right at $t=0$. Then, before that, he’s to the left of you. After $t=0$, he’s to the right of you. So, his path in spacetime looks like this:

At each time, he’s at a particular space location.

What happens if a beam of light shoots past you?

Unsurprisingly, it looks very similar. The only difference is the speed. The speed of light is about $3\times 10^8$ m/s, which is so much faster than most speeds you are familiar with that, on the same diagram as before, the path of the light would look almost horizontal, since its location is changing so rapidly.

In order to make these diagrams readable, we need to do something about the scale…

The standard thing to do is to measure all speeds as a ratio of the speed of light. So, the speed of light is “1,” while the speed of your friend walking might be 0.00000001.4

What’s nice about these units is that light will always follow a line at 45 degrees. This is because each time $t$ increases by 1, $x$ increases by 1 as well. So, now our diagram with your friend and the ray of light looks like this:

So far, not so weird.

Anyway, the question is this: What does it mean for two events to be simultaneous?

An event happens at a particular time and place, so, when we say an event, what we mean is a point $(x,t)$.

How can we, sitting at $x=0$, figure out when an event occurred?

One way would be to use a laser. To figure out when an event occurs, we can send out a beam of light. The light could then bounce off of a mirror that just happens to be sitting at the event, and then travel back to us. On a diagram, it would look something like this:

So since light takes the same amount of time to travel each way, we can tell exactly when the event occurs. The $t$ coordinate is exactly halfway between when we sent the light and when we received it back.

We call two different events simultaneous if, calculating this way, we find they both happened at the same time. For instance, it’s not too hard to see that all the events along the $x$-axis (i.e., events with $t=0$) are all simultaneous.

This way of measuring simultaneity is certainly cumbersome, but it’s hard to argue this isn’t precise. I have to admit, though, that this seems like a lot of work for something that feels obvious. But there’s one important fact that changes everything.

The universality of the speed of light.

Let’s consider your friend traveling past you. Now, walking isn’t very fast, so let’s have him on a rocket ship traveling at half the speed of light. His path will look something like this:

Which events do your friend measure as simultaneous with when he passes you?

Your friend has to measure simultaneity in the same way; he shoots out a laser beam, which is reflected. Halfway between when your friend sends out the beam and when he receives it is the time the event occurred.

The trick is that his light also travels at 45 degrees.

In your friend’s view, the moment when he passed you and the event where his light was reflected are simultaneous!

Using the same idea, it’s not hard to figure out all the events that your friend thinks are simultaneous to when he passed you.

That’s weird.

Let’s be clear about this. The fact that you and your friend disagree on what events happened at the same time is not due to a lack of cleverness in how you measured things. The complicated way we defined “simultaneous” was exactly so that you can’t argue that. It is an inescapable consequence of the universality of the speed of light that simultaneity cannot also be universal.

Let’s ask a related question: how can we measure the time between events?

Of course, since we can’t agree on simultaneity, it isn’t surprising that measuring time between events is also dependent on who’s doing the measuring.

So, what can we agree on?

Again, we all agree that light travels at 45 degrees. In terms of coordinates, that means that the change in $x$ value and the change in $t$ value have to be the same for any beam of light. If we denote “change” by $\Delta$, we could write this as $(\Delta x)^2 - (\Delta t)^2 = 0$.

Now, this is a bit unmotivated,5 but let’s define the interval $(\Delta s)^2$ between any two events to be $(\Delta s)^2 = (\Delta x)^2 - (\Delta t)^2$.6 It is one of the fundamental results in special relativity that everyone can agree on this quantity! (But let’s not prove it here…)

What does this quantity, the interval, represent?

If we measure the interval between you at time 0 and you at time 10, the $x$ coordinate didn’t change, and so $(\Delta s)^2 = -10^2 = -100$. This is a bit awkward, but if we take the negative of this, then take the square root, we get back 10.

The interval, or rather $\sqrt{-(\Delta s)^2}$, represents the amount of time you experience going between those two events. This quantity is called proper time. More generally, $\sqrt{-(\Delta s)^2}$ represents how much time someone experiences going between two events, as long as they are going on a straight path (i.e., at a constant speed.)

Let’s think about your friend, now. If he goes along his path, when you measure his time as $t=10$ seconds, his $x$ coordinate is $x = 5$. Using the formula for proper time, we can see that he thinks only $\sqrt{10^2-5^2}\approx 8.66$ seconds have passed!7

This time dilation is an important and very observable consequence of special relativity, and is one of the most precisely measured phenomena in science. In fact, I’m going to spend a lot of the next post just on that subject!

But to finish; the interval was useful for measuring the time experienced on any constant speed path. But what about nonconstant speed (i.e., curved) paths?

This is where we finally get back to metrics, as we talked about in the last post. For special relativity, instead of using the Pythagorean theorem for measuring vectors, we use the interval. So, in special relativity, the metric is $ds^2 = dx^2 - dt^2$, where $dx$ means the rate of change of your path in the $x$ direction, i.e., the derivative of $x$.8

If we want to figure out how much time passes for someone following the path, we use the tangent vector to calculate $\sqrt{-ds^2}$, similarly to how we measured the length of the tangent vector in the last post. This length is the equivalent of speed for a normal metric. However, the interpretation of this quantity is how fast time is passing for the traveler. The speed of time, if you will!

To find how much time the traveler experiences, as before, we integrate this speed of time over the entire path.

We’ll stop here for today. Next time, we’ll talk a lot more about time dilation, and, in the process, talk about the infamous twin paradox!

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<– Previous Post: The Birth of Metrics
First post on manifolds and metrics: Asteroids on a Donut
–> Next Post: The Speed of Time: The Twin Paradox

1. Why do I say one of his first, instead of his first? Because, in one year, he published four groundbreaking papers. Because, you know, Einstein
2.  In a vacuum.
3.  Yes, I’m an American. Yes, I refuse to use Imperial units in public.
4.  These speeds are measured as ratios, so they are unitless. Often, how this is explained is that we choose to measure time in terms of meters. This is the opposite of measuring distances in terms of time, which you do when you use lightyears. So, 1 meter of time is the time it takes for light to travel 1 meter. Thus, the speed of light is 1 meter per meter, i.e., 1 without units. These are called geometrized units.
5. So, I’m going to do something, and it’s not immediately obvious why I’m doing this thing. But, just trust me, it’ll magically work out. (This is what “unmotivated” means in this context.)
6. If we had three space dimensions, this would be $(\Delta s)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - (\Delta t)^2$ instead.
7. This is similar to the Pythagorean theorem, but, because of the minus sign on the $(\Delta t)^2$ term, the hypotenuse is always shorter than the vertical height.
8. Technically, this metric and those of the previous post are different. Normal metrics, like those in the last post, always give positive $ds^2$, and are called Riemannian metrics. This new metric gives negative $ds^2$ for some vectors, and we call it a pseudo-Riemannian metric.

## The Birth of Metrics

In 1851, Bernhard Riemann got his PhD in mathematics. In the German university system at the time, in order to become a professor1, you had write an extra thesis, which Riemann did on what we would now call Fourier series. In addition, you had to give a probationary first lecture.

To make it worse, you didn’t even get to choose the topic of your talk. You were to propose three different subjects, and then the faculty, which at that time was headed by Friedrich Gauss, chose between them. Two of the subjects Riemann proposed were ones he had already worked extensively in. The third was “On the foundations of geometry.”

Unfortunately for Riemann, Gauss was a troll. Against all tradition, he picked the third topic.

Fortunately for mathematicians, Riemann was a mathematical boss.

Of course, Riemann was also researching in mathematical physics and working as what we would probably call a TA (teaching assistant). The strain of another major research project on top of all that led to a temporary mental breakdown. Fortunately, he quickly recovered, and a few months later, gave his talk.

In his probationary lecture, Riemann invented the idea of a manifold, as we talked about in previous posts. He also introduced the important idea of removing manifolds from some space like $\mathbb{R}^n$. And, finally, he came up with a way of measuring lengths on these manifolds, without having to refer to $\mathbb{R}^n$.

That way of measuring distances is the topic of this post.

We’re going to slowly build up ideas in order understand Riemann’s idea.

How do we measure lengths?

Let’s start in the plane, $\mathbb{R}^2$. Suppose we want to measure the distance between two points, like these:

In this case, we can just use the Pythagorean Theorem, $a^2 + b^2 = c^2$. Here, $a$ is the horizontal distance and $b$ is the vertical distance between the points.

Now, that’s fine and dandy for $\mathbb{R}^2$, but it doesn’t really work on other shapes. After all, what’s a “straight line” on a torus?2

We need a way to measure the length of a path between the two points, not just the straight one.

How can we do that?

The key idea actually comes from a car. When we drive, our speedometer tells us our speed. If we traveled at the same speed for an entire trip, the length of our travel would just be our speed times the time we traveled. Now, of course you don’t travel at a constant speed. Sometimes you go slower, sometimes faster. But our odometer figures out our total distance traveled just the same.

One way to think about what the odometer is doing is it is adding up the distance traveled for each second. If you’re going faster for a second, the odometer ticks up a bit more; if slower, a bit less. At any given second, you’re almost going a constant speed, and so this is a good approximation of your distance.

Now, of course, your speed isn’t constant during that second, but the odometer is essentially adding up the distances you travel over each infinitesimal interval of time.

Not to scare you, but it will be convenient to introduce some calculus here.

Don’t worry, we won’t actually do any calculations. We just need the basic ideas of the derivative and the integral of a function. We’ll illustrate with a basic example.

A function takes some input and gives you some output. For our example, let’s look at the function $s(t)$. The input, $t$, is the time since leaving your house, while the output, $s(t)$ measures the distance you are away from your house.

The derivative of a function measures the  rate of change of the output. So, the derivative of $s(t)$, which we’ll write as $s'(t)$ (read “s prime of t”), is the rate of change of position, i.e., the speed.

The integral is essentially the opposite of the derivative. The integral of speed (i.e., of $s'(t)$) from $t=0$ to $t=10$ adds up the rates of change in order to calculate the total change in position from time 0 to time 10. The notation here is $\int_0^{10} s'(t) dt$, but we shouldn’t need to use that here.

While we won’t be worrying about formulas or calculating these things, it is important to note that, given a function, it’s a relatively simple thing to find derivatives and integrals of it.3

In terms of our car description, the speedometer measures the derivative of your position, which is your speed, while the odometer integrates your speed in order to calculate your total distance traveled.

Riemann’s idea is to measure lengths the same way.

How can we calculate speed along a path?

If we have a path in, for instance, $\mathbb{R}^2$, we can think of a person driving along that path. Her current position is given by a point on the path, which we can label as $(x(t), y(t))$. So, $x(t)$ gives her position in the $x$ direction $t$ seconds after beginning, and similarly for $y(t)$.

With a derivative, it’s easy to say how much her position is changing. The function $x'(t)$, by definition, is the rate of change of her position in the $x$ direction, and similarly for $y'(t)$. If we put these together as $(x'(t),y'(t))$, this tangent vector tells us what direction the driver is going, and its length tells us how quickly her position is changing.

To calculate speed, we need to calculate the length of a vector in $\mathbb{R}^2$. Fortunately, that’s as simple as using the Pythagorean theorem. Since the tangent vector is $x'$ wide and $y'$ tall (i.e., represented by the vector $(x',y')$), the Pythagorean theorem says that the length $s'$ of the vector (and thus the speed) is given by $s'^2 = x'^2 + y'^2$, and so $s' = \sqrt{x'^2 + y'^2}$.

Let’s write that rule a bit differently. Let’s write that as $ds^2 = dx^2 + dy^2$. You can think of $dx$ as the derivative of the $x$ position, while $ds$ is the derivative of the position, or, in other words, the speed.

This way of measuring vectors, $ds^2 = dx^2 + dy^2$, is called a metric.4

So, to review, to find the length of a path, we first find its tangent vector (via a derivative). We calculate the length of these vectors using the metric, which tells us the speed. Then we add up the speeds (via an integral) to get the length of the path.

If we had a path in space (i.e., in $\mathbb{R}^3$) instead, the only difference is that we have three directions (x, y and z) instead of two (x and y). We can find the length of the tangent vector in the same way, only using $ds^2 = dx^2 + dy^2 + dz^2$. The term $dz$ is the same as the others, just for the extra $z$ direction.

How does this help us with manifolds?

Let’s look at a specific case first. Think of the sphere $\mathbb{S}^2$ sitting in space $\mathbb{R}^3$. The first thing to notice is that a path on the sphere is also a path in space!

This means that we can use the metric in space to find the lengths of the tangent vectors, and thus the speed. The integration works the same for any path, so that gives us what we need.

But we want to be able to define distances without thinking of the manifold in space.

To get around that, recall that the definition of a manifold tells us that, for a 2-dimensional manifold, we always have two coordinates. For the sphere, we can choose the two coordinates to be latitude and longitude. Longitude is measured as an angle $\theta$ from some fixed north-south line.5 Latitude is measured as an angle $\phi$ from the north pole.6

A path on the surface of the sphere in $\mathbb{R}^3$ could be written like $(x(t), y(t), z(t))$, but we could also write it in terms of latitude and longitude, as $(\theta(t), \phi(t))$. It’s the same path, just written in terms of the coordinates for the sphere.

Similarly, instead of writing the tangent vector as $(x'(t), y'(t), z'(t))$, we can write it in terms of the new coordinates, $(\theta'(t), \phi'(t))$. Again, these are the same vectors, just with different names, if you will.

And if they’re the same vectors, they should have the same lengths. So, we want to figure out what the metric should look like in terms of the new coordinates. After a calculation I’ll explain in a footnote7, we find that the metric on a sphere of radius 1 is $ds^2 = \sin^2(\phi) d\theta^2 + d\phi^2$. In this, $d\phi$ represents the change in the $\phi$ coordinate (i.e., $\phi'(t)$), while $\phi$ represents the value of the $\phi$ coordinate (i.e., $\phi(t)$).

This is the same metric as before, but it looks very different because we’re using different coordinates. But the same vector will have the same length.

The important thing to note here is that this new form of the metric makes no reference to $\mathbb{R}^3$. It only refers to the coordinates on the manifold!

This immediately gives us how to measure lengths on any manifold.

On any manifold, we can choose some coordinates, say $(x,y,z)$. Then we can write down a metric using those coordinates. It might be ugly, like $ds^2 = dx^2 +\cos^4(z) dydx + dy^2 +e^{xy}dz^2$, but that’s okay. There are a few things we generally want to worry about (like “smoothness”), but we won’t here. We can then use that metric to measure the lengths of tangent vectors, then integrate those lengths (i.e., speeds) to find the total length of the path.

Yay! We now understand Riemann’s idea for measuring lengths on a manifold!

How did Gauss react to Riemann’s lecture? Recall that Gauss was the foremost mathematician of his time. In 1827, he had proven a theorem about geometry so awesome that, to this day, it is officially called “The Awesome Theorem.”8 Gauss’s influence would decide whether Riemann got the job.

Riemann blew Gauss away.

Dedekind, who later replaced Gauss as head of the faculty, said that Gauss sat at the lecture “which surpassed all his expectations, in the greatest astonishment, and on the way back from the faculty meeting he spoke to Wilhelm Weber, with the greatest appreciation, and with an excitement rare for him, about the depth of the ideas presented by Riemann.”9

Needless to say, Riemann got the job.

There is a lot more we could talk about metrics and manifolds. And we will. After all, I do want to tell you about “The Awesome Theorem.” But, before that, I want to talk about the mathematics of something important that Riemann’s ideas allowed to be invented: Einstein’s theory of special relativity.

<– Previous Post: Manifold Menagerie
First post in this series: Asteroids on a Donut
–> Next Post: A Mathematical Intro to Special Relativity

1. The position (Privatdocent) is a bit different than how we think of a professor. You did not have a regular salary, but were simply forwarded fees paid by any students who chose to attend your lectures. Riemann did eventually get a regular salary, but not for a few years.
2. There definitely is an idea of a straight line, called a geodesic, and they’re super important. We’ll talk about them at some point, but not in this post.
3. Well, at least, mathematically, it’s simple to say that the derivative or integral exists, though writing out an explicit formula is sometimes hard or impossible. Fortunately, that doesn’t stop us from using the concepts.
4. There is another concept called a metric space. A manifold with a metric is a metric space, though a metric space is more general. The “metric” on a metric space defines the distances between points, rather than giving a way of measuring speed. Clearly related, but not quite the same.
5. On Earth, this line is the north-south line through Greenwich, England.
6. Latitudes on Earth usually measure from the equator, rather than the north pole, but mathematicians will do as they want. Also, following a master plan to confuse everyone, physicists reverse the names of these coordinates, switching $\phi$ and $\theta$. We’ll use the mathematical names.
7. This calculation is actually fairly simple. The coordinates are related by, for instance, $z = \cos(\phi)$. By taking the (exterior) derivative of both sides, we see that $dz = -\sin(\phi) d\phi$. You do this for each coordinate, then substitute what you get in $ds^2 = dx^2 + dy^2 + dz^2$ and simplify what you get.
8. Okay, technically, it’s called “Theorema Egregium.” But that’s Latin for “The Totally Awesome Theorem,” so I stand by my claim. We’ll definitely be talking about this theorem when we eventually talk about curvature.
9. My story about Riemann comes from Spivak’s Introduction to Differential Geometry, Volume 2, which includes the translated lecture in chapter 4. Though probably only a few in the audience were able to appreciate the lecture at the time, it was the pivot point in the history of geometry.

## Manifold Menagerie

In the last post, we introduced the idea of manifolds. A two-dimensional manifold is essentially a shape on which we can go in two directions, say up-down and left-right, but not forward-backward as well. We talked about some simple examples, like a sphere or a torus, but there are so many more wonderful manifolds than that!

Let me give you a brief tour of the manifold menagerie.

The first manifold I want to show you is an interesting variation of the torus. Last time, we showed how you can make a torus by considering a rectangle, and identifying opposite sides.

Now, notice how the arrows are pointing in the same direction. What would happen if we changed the direction of one of the arrows? You’d get what we call a Klein bottle.1

What this means is that the left edge is still identified with the right edge, but with, for instance, points near the bottom left being identified with points near the top right edge.

Another way to visualize this is that if you played Asteroids on a Klein bottle, and you flew through the left side facing up, you would come out the right side, facing down!

What does this Klein bottle thing even look like, anyway?

Well, first we can roll up the rectangle, like we did with the torus. But we can’t bend the two ends together quite like before because the orientation (i.e., the direction of the arrows) doesn’t match up. Instead, we have to do something funny.

Take the end of the tube through the wall near the other end, then bend it up through the end. Then, flange over that end to match up the two ends. Okay… that description needs a picture.

Weird looking, right?

Now, you may want to raise an objection. Dr. Dilts, you may say, isn’t that not a manifold? After all, where the tube intersects itself, it doesn’t look like $\mathbb{R}^2$!

You’d be right to mention that. In fact, that’s one of the interesting things about this example.

If you look back at the rectangle representation of the Klein bottle, there’s nothing that stops it from being a manifold. There’s no self-intersection there. The problem is not in the Klein bottle itself, but in how we chose to visualize it in three-dimensional space, $\mathbb{R}^3$!

As an anology, consider a circle.2 We can think of it as a loop of string. In that standard configuration, it looks like a manifold, since if you zoom in, it looks like a line, i.e., a one-dimensional space. But we could pick up the end and pull it over itself like such:

It no longer looks like a manifold, but, again if we think of it as a string, the places where the loop overlaps itself are not the same points; they just look like it because they overlap. The crossing lines don’t “see” each other.

Again, the problem is not with our loop of string, which certainly is a manifold. The problem is with how we are visualizing it in the plane, $\mathbb{R}^2$.

So, the Klein bottle is a manifold.

This raises a question. Can we visualize the Klein bottle so that it doesn’t intersect itself?3

The answer is yes, but, surprisingly, you can’t do it in $\mathbb{R}^3$. You need a fourth dimension to do it!

The idea is something like this.

If we go back to the loop of string example, where the string overlaps itself, the visualization intersects itself. However, it’s easy to resolve this problem. Simply pick up the string near where the intersection is; in other words, pull it from two dimensions into the third.

In order to visualize the Klein bottle without an intersection, we have to do the same thing, only with a fourth dimension. You grab the Klein bottle near the intersection, and pull it from three dimensions into the fourth.

Weird, huh?

Now, this worked for the Klein bottle, this trick of pulling it into a higher dimension. But you can imagine there are other manifolds that have this same problem of self-intersection. (In fact, we’ll even get to another one in this post!)

If you pick any manifold, is there always a visualization that doesn’t have self-intersections?

The answer is yes, though it’s hardly obvious how to prove such a statement. The Whitney embedding theorem says that any manifold can be visualized without self-intersection, but it may require a high dimension to do so. Precisely, it may take as many as twice as many dimensions as the manifold has.

If the manifold has $n$ dimensions, this visualization may require as high as $2n$ dimensions. Our Klein bottle (with $n=2$) is an example of this, as it requires $4$ dimensions. A six-dimensional manifold might take twelve dimensions!

Okay, back to the funky Klein bottle thing.

Guess what? It doesn’t have an inside or an outside! Weird, yes? This fun fact about the Klein bottle is exactly like how a Möbius strip has no front or back.4

With a sphere, if you think of yourself walking on the outside surface, you can’t “walk” to the inside of the surface; you’d have to punch a hole through. But look closely at the Klein bottle, if you started walking on the “outside,” and followed along just like you know you should, you can end up underneath where you began.

Trippy.

Okay, the Klein bottle is weird.5

Let’s talk about some more pedestrian6 examples.

In the last post, we talked about the sphere, by which I mean a shape like the surface of the Earth. We call this the two-dimensional sphere, $\mathbb{S}^2$ (read “S 2”), since there are two directions you can go.

Along with that, you have the one-dimensional sphere, $\mathbb{S}^1$. This has one direction you can go… in other words, it’s a circle!

We can go higher in dimensions as well. What is $\mathbb{S}^3$, the three-dimensional sphere?

Well, it’s not the inside of the sphere, which is what your mind might go to first. Instead, it will help to use analogy.

If you look at $\mathbb{S}^2$, horizontal slices of it are just circles, $\mathbb{S}^1$.

So, as we take slices of $\mathbb{S}^2$, we have small circles that get larger till we pass the equator, then get smaller again till they disappear.

We can visualize $\mathbb{S}^3$ the same way. The problem is, it’s naturally in $\mathbb{R}^4$ instead of $\mathbb{R}^3$, so it’s harder to see. To help us, let’s think of $\mathbb{R}^4$ as the three normal dimensions, plus time.

So, if we take $\mathbb{S}^3$, and we look at what we can see at any instant in time (which is equivalent to looking at horizontal slices in the previous example), we can see a sphere, $\mathbb{S}^2$, which starts small, grows till it reaches the “equator,” then shrinks back down till it disappears.

Yeah, four dimensions is hard.

MORE manifolds!

The torus we talked about before is just one example of a torus. Since it’s a two-dimensional manifold, we call it $\mathbb{T}^2$ (read “T 2”). Using the same idea, it’s not hard to make higher dimensional versions.

If we take a cube (instead of a rectangle), we can identify opposite sides of the cube. So, if you are playing (3-d!) Asteroids in this, and you fly out the top, you’d fly in from the bottom. This gives us a new, higher dimensional torus, $\mathbb{T}^3$.

Pretty cool. Of course, it’s much harder to visualize as sitting inside of $\mathbb{R}^4$. Good thing we don’t have to!

There’s another variation we can do.

Instead of taking a square, take an octagon. Now, identify the sides in this pattern.

Now, it’s much harder to see what’s going on without a hint. What is this shape?

It’s a two-holed torus.

You can actually do this for any number of holes. You just need 4 sides for each hole you want your torus to have.

Let’s see, what else weird can we do?

Oh, I know!

Okay. Take the sphere ($\mathbb{S}^2$). Now, for each point, identify it with the point on the exact opposite end of the sphere.

Another way to think of this is to think just of the upper hemisphere of the sphere. Then, just on the equator, again identify the points on the opposite side.

In other words, if you fly south, and go past the equator, you end up coming north out of the equator on the other side!

At first consideration, this really seems to just be a sphere, since on a sphere, you can fly through the south pole just the same.  But it really is different. It’s called $\mathbb{RP}^2$ (read “R P 2”), where the $\mathbb{RP}$ stand for “real projective” space.

And it’s really weird.

To explain it, we’re going to need one of those funny mathematical objects; an infinitely stretchable and infinitely compressible rubber band.

Look at the sphere. If you put a rubber band on it through the north and south poles, you can always have the rubber band shrink down, like this:

However, if we took the equivalent path on $\mathbb{RP}^2$, it can’t shrink down! See, if you try to move the parts that hit the equator closer to each other, you can’t, since they, by the definition of the space, have to stay on the opposite sides of the hemisphere! Essentially, you can only rotate the rubber band, but never have it come off.

But, if we wrap the rubber band around twice, all of a sudden we can shrink it down.

Wait, what?

The trick is to run one part down each side of the hemisphere. Thus we can have two small half circles getting smaller, and finally shrink to nothing like on a sphere.

So, in $\mathbb{RP}^2$, one closed path is stuck. But if you wrap it around twice, all of a sudden you can shrink it down to nothing!7

So this new space is not a sphere!

Again, we call this space $\mathbb{RP}^2$. As you can see, it turns out to be about as interesting as the Klein bottle. In fact, we can even use a similar rectangle to define it, like this!

Like the Klein bottle, $\mathbb{RP}^2$ cannot be visualized in three dimensions without self-intersections. Here’s one way to look at it, with those opposite parts of the equator matched up.

$\mathbb{RP}^2$ also doesn’t have an inside or an outside. Which is really weird, since it seems like if you take a sphere, anything you get from it should still have an inside… But, you know… strange but true statements like this are the heart of mathematics.

There are plenty of other weird manifolds out there, but that’s a good introduction to some interesting and important ones.

Next, we want to come up with a way to measure distances on a manifold, without visualizing it in a space like $\mathbb{R}^3$.

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1. Apparently, it was originally called Kleinsche Fläche (Klein surface), but, perhaps due to its appearance, it was misinterpreted as Kleinsche Flasche (Klein bottle), which may have led to the use of this term in German as well.
2. Remember, a circle is the outer edge, not the filled-in shape.
3. A visualization of a manifold (in $\mathbb{R}^n$, for instance, and with a few other conditions) is called an immersion. If the immersion doesn’t intersect itself, it’s called an embedding. So, our current visualization of the Klein bottle is an immersion, and the question is whether we can find an embedding.
4. In fact, a Klein bottle is two Möbius strips with their edges glued together.
5. Klein bottles are so cool that you can get blown-glass Klein bottles. The man who runs that site, Cliff Stoll, looks like a mad scientist. And he’s obsessed… Check out this interview of Cliff Stoll on Numberphile. He also shows off his Klein bottle hats and Möbius scarfs!
6. Pun! Ba-dum-ching!
7. This idea of drawing paths and asking whether or not they can be shrunk to nothing is an important one in topology, called the fundamental group of the manifold. I was about to link to the Wikipedia page, but on second glance it’d be a horrible introduction…