Thread by @MarkusDeserno

May I invite you to a fun thread about a delightful quirk of relativity theory? Starting with a simple fact about rotations, I’ll hope to give you some intuition about something that’s considered wildly counterintuitive: velocity addition. Intrigued? Buckle up!

Great, welcome aboard! Let’s go! We shall begin by looking at a 2-dimensional rotation around the origin in the (x,y)-plane. We can quantify it by an angle 𝜙 and visualize it by a line through the origin that is rotated by that angle 𝜙.

Let’s now consider 𝘵𝘸𝘰 rotations, 𝜙1 and 𝜙2, as well as the two corresponding lines through the origin at angles 𝜙1 and 𝜙2. We are interested in an overall rotation 𝜙12 by both angles together.

Evidently, the total angle is 𝜙12=𝜙1+𝜙2. One way to see this in the image is to realize that this also implies that 𝜙2=𝜙12–𝜙1, and so rotating back from the sum angle by one of the two angles gives us the other one.

OK, none of this is remotely surprising. But things get interesting if we—for whatever reason—decide to describe these rotated lines in a different way. Specifically, how about we describe these lines by their 𝘴𝘭𝘰𝘱𝘦 m, “rise over run”.

This is of course a very common way to describe tilted lines, and it has the redeeming quality that the equation for the line is dead easy: y=m·x.

Just as before, we can now picture two lines, characterized by slopes m1 and m2, and ask what is the slope m12 of the line that rotates by the sum angle? This turns out to be a much trickier question, because 𝘢𝘯𝘨𝘭𝘦𝘴 𝘢𝘥𝘥, 𝘣𝘶𝘵 𝘴𝘭𝘰𝘱𝘦𝘴 𝘥𝘰𝘯’𝘵.

Fortunately, angle and slope are related by a fairly simple relation: the slope is the tangent of the angle: m=tan(𝜙 ). Since we know that angles add, and by exploiting some trigonometric identities for the tangent, we can find the new slope. Let’s do this!

Evidently, if 𝜙12=𝜙1+𝜙2, we also have

arctan(m12) = arctan(m1) + arctan(m2)

Moreover, exploiting the trigonometric identity

tan(x+y) = [tan(x) + tan(y)] / [1 – tan(x)·tan(y)]

this leads to

m12 = [m1 + m2] / [1 – m1·m2]

This is the formula for how rotations combine if we for whatever reason decide to describe a rotated line not by its angle but by its slope. Clearly, this looks more complicated, but we see what’s going on. Everything’s fine. We’re good!

But imagine now that we restrict ourselves to really small angles, and correspondingly small slopes. Indeed, think of m1 and m2 being very much smaller than 1, such that “m1·m2” in the denominator can be safely ignored compared to “1”.

In this small-angle-limit the slope-addition-formula simplifies tremendously: up to a tiny correction, m12=m1+m2. Slopes add! How nice! Of course, it’s just an approximation, but it surely makes life easier if we happen to be in a small angle regime.

So far so good. Now let’s go to the next step. Imagine a group of “practical geometers” who (for whatever reason) have never really dealt with large angles. (Weird, I know—but bear with me!) For practical purposes, they always work in the small angle regime.

For them, m12=m1+m2 always holds with excellent approximation. In fact, they might not be able to tell the difference, because it’s too small to measure. They might even start to think of this formula as being how rotations 𝘢𝘤𝘵𝘶𝘢𝘭𝘭𝘺 combine. You add slopes!

It is easy to see how they might develop some “intuition” for why this should be so. And how, as time passes, they would start to think of this formula not as an approximation but as the 𝘛𝘳𝘶𝘵𝘩, with a capital “T”. Habit is a powerful drug!

Until, one day, a particularly deep-thinking geometer, Al Unapietra, starts to think hard and deep about the true geometry of rotations, and he “re-discovers” the actual truth, and the more complicated formula. Everyone’s surprised. Most people are confused.

Precision measurements show that Al is right: the more complicated formula is really correct. But damnit, it is so unintuitive! Al’s discovery is simultaneously hailed as a breakthrough and as mathematical challenge too difficult for everyday people to comprehend.

Is it, though? It’s only unintuitive if you insist on describing the rotation of lines by their slope m. But this is just not a very smart way of doing it if you want to add rotations! If you re-calibrate your thinking and return to the angle 𝜙, things greatly simplify!

So far, the moral of our story is this: whether things look simple or not—intuitive or not—often depends on how you describe them. If you insist on the wrong mental framework, a simple fact might look needlessly opaque.

At this point you might be asking, “Hello? Relativity? Didn’t you promise us a lesson in relativity?” Yes, I did. But I needed to 𝘱𝘳𝘦𝘱𝘢𝘳𝘦 you for it. That’s done now, and we’re ready for the harvest!

Let’s say we have a spaceship that moves away from us at some sizable speed v1. And let’s say that inside the spaceship an astronaut fires a railgun, shooting a bullet forward with velocity v2 relative to the rocket. What is the bullet’s speed v12 relative to us?

You might say, “Easy! It’s obviously v12=v1+v2!” And that’s indeed what you would learn in any introductory course in classical mechanics. But the answer is wrong. If you make very careful measurements, you get a slightly different answer!

To write down what the true answer is, let me introduce one more piece of notation that is very common in relativity: we measure speeds in fractions of the speed of light, c, and we call that fraction β. So β=v/c, or equivalently, v = βc.

In this notation, you might expect to find

β12 = β1 + β2

However, the 𝘢𝘤𝘵𝘶𝘢𝘭 answer is this:

β12 = [β1 + β2] / [1 + β1·β2]

Does this remind you of something? Up to a + vs. – difference in the denominator (I’ll get back to that later!), this is basically the same as our fancy slope-addition formula! And the beautiful thing is: this is not a coincidence! Let me explain.

It turns out that describing the motion of a “frame of reference” (e.g. a spaceship) by its speed is equivalent to describing the rotation of a line by its slope. It works, but it can get you in trouble, especially for large angles—or here: large speeds.

Successive changes of reference frames, which in a “Galilean mindset” you want to think of as “adding their speeds”, really are more akin to rotations, and it’s these rotations that add, not the speeds!

“But wait,” you say, “what’s rotating?” If the spaceship moves to the right, and the railgun inside it is 𝘢𝘭𝘴𝘰 fired to the right, everything happens along the same direction! Where is the rotation? Great question! And this is where relativity is 𝘳𝘦𝘢𝘭𝘭𝘺 weird!

There are 𝘵𝘸𝘰 qualitatively different things at play now. Let me address them one at a time.

First, the rotation is indeed not a rotation in 𝘴𝘱𝘢𝘤𝘦. It is a rotation in 𝘴𝘱𝘢𝘤𝘦𝘵𝘪𝘮𝘦! Relativity insists that changes between moving coordinate systems mix up space- and time-coordinates. That, indeed, is 𝘷𝘦𝘳𝘺 unexpected for our Galilean minds!

And second, I haven’t yet addressed that pesky minus sign difference between our “addition formulas”. It turns out that this is where it now matters. Our transformation is indeed not 𝘦𝘹𝘢𝘤𝘵𝘭𝘺 a rotation. Instead, it’s a so-called “hyperbolic rotation”.

Before I tell you how to write this down in mathematical notation, let me show you what it looks like in two simple animations.

First, normal rotation. You are surely familiar with how this “works”. Here’s an animation that rotates a coordinate system by some angle 𝜙. Both axes tilt by the same amount in the same direction, and the orbits are circles.

Now, hyperbolic rotation. This animation shows that, again, both axes tilt by the same amount, but in 𝘰𝘱𝘱𝘰𝘴𝘪𝘵𝘦 directions. Furthermore, the orbits are now hyperbolas, not circles.

OK, this doesn’t really look like a rotation 𝘢𝘵 𝘢𝘭𝘭—so why do I call it “hyperbolic rotation”?

The reason is that it’s mathematically 𝘷𝘦𝘳𝘺 similar. First, it’s a linear transformation. Second, its matrix has determinant 1. And third, that matrix even 𝘭𝘰𝘰𝘬𝘴 almost like a rotation matrix! Except all the trigonometric functions are replaced by hyperbolic ones!

For direct comparison: here’s what an (active) space rotation by an angle 𝜙 looks like—when applied to the (x,y)-coordinates from our above animation, and when written succinctly as a matrix equation:

And here’s an (active) hyperbolic rotation by an “angle” 𝜙, applied to the spacetime coordinates (ct,x)—i.e. speed of light c times time t, paired up with an x-coordinate:

I think this is similar enough to warrant a terminology that at least “reminds” us of rotations!

It gets better: recall that in the "normal" rotation case we had a connection between angle and slope: m=tan(𝜙). We also have a corresponding relation between the hyperbolic rotation angle 𝜙 and the relativistic equivalent of the slope, the scaled speed β. It is: β=tanh(𝜙)!

Again, up to a “trig goes hyperbolic” replacement, everything is identical. And since the “sum of angles” identity for tanh versus tan has a + vs. – difference in the denominator, that also explains the difference in our addition formulas!

Incidentally, since angles add, both for rotations as well as for relativistic changes of reference frames, the angle 𝜙 also has a special name in relativity. It’s called “𝘳𝘢𝘱𝘪𝘥𝘪𝘵𝘺”. And as far as velocity addition is concerned, we can now see: rapidities add!

You will of course not be surprised to learn that these hyperbolic rotations probably play a big role in relativity. And indeed, they do. An 𝘦𝘯𝘰𝘳𝘮𝘰𝘶𝘴𝘭𝘺 big role. In fact, all of physics nowadays must play nicely with these rotations.

Except, they are usually not called “hyperbolic rotations” by physicists. They are called “𝙇𝙤𝙧𝙚𝙣𝙩𝙯 𝙩𝙧𝙖𝙣𝙨𝙛𝙤𝙧𝙢𝙖𝙩𝙞𝙤𝙣𝙨”.

[END]

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