Howdy, folks.

As you know, I’m in the midst of writing my book on symmetry, and every now and again, I like to try to run various drafts of different snippets of the manuscript by you, the peanut gallery, for comments and suggestions. In particular, I’m working on Chapter 6, tentatively entitled, “Could we Build an Intergalactic Ansible to Communicate Instantly Across Space?” You may recall that I did an io9 column with a similar title, but a very different topic. The io9 column was about quantum entanglement. This chapter is all about Special Relativity, and in particular, how symmetry gives rise to time dilation and the like.

The text below is an early-ish draft of a few sections on how the Pythagorean Theorem leads to the Minkowski description of spacetime, and how *that* leads to time dilation. It is the most equation-dense part of the book to date. I put in 3 simple equations, all very similar to one another, but given the tone I’m going for, I still worry that that’s too much.

The images are just placeholders. Don’t worry, the final book will have a professional illustrator.

Comments, either below or via email are greatly appreciated.

**-Dave**

**The Pythagorean Theorem**

At some point in elementary school, you no doubt came across the Pythagorean Theorem:

It’s a deceptively simple equation. A and B are the lengths of the two short sides of a right triangle, and C is the length of the long side, the hypotenuse.

The Pythagorean Theorem does far more than simply tell you about triangles for their own sake. It tells you how to compute the distances between points. You may remember problems like this from school: Walk 3 miles East and then 4 miles North. Plug through the calculation and you’ll find that you’re 5 miles from where you started. To connect it to the real world, consider a small piece of a Washington D.C. transit map:

Many cities are conveniently designed so that streets run approximately along the cardinal directions of a compass. Washington D.C. is a perfect example: Numbered streets run North-South and lettered streets run East-West. So – to pick an example found by an intensive search through Google Maps – if you wanted to walk from Judiciary Square Station on the corner of 4th and E Street NW to the Chinatown Metro Station on 7th and G, you would start by walking approximately 600 meters due West (along E), and then 250 meters North (along 7th).

Of course, you could also simply take the Red Line subway, and if you plug through the numbers, you’d find that the subway trip is approximately 650 m. It’s just a practical application of the Pythagorean Theorem, albeit with the slightest relabeling of the numbers:

We owe this convention to the work of the 17th century mathematician and philosopher, Rene Descartes. The Cartesian system imagines describing all of the events and objects in the universe on a sort of map. For instance, in the East-West direction, the convention is to label positions as “x.” In the North-South direction, we normally label positions as “y.” I’m going to ignore the possibility of moving vertically in an elevator, but if you were so inclined, you might label that motion as “z.”

I’d be negligent if I didn’t point out that Cartesian system breaks down on the surface of the earth. The earth, after all, is approximately shaped like a sphere, which means that you can’t make a perfect, undistorted flat map that covers the entire thing. That’s fine. That’s why we’re talking about something much smaller, like a few city blocks.

But back to Washington D.C. Suppose you stepped down into the Judiciary Square subway station and some particularly malevolent and efficient urban planner decided to come along and rotate all of the city streets while you were underground. Instead of the numbered streets running North-South, they are turned a few degrees to the right. The streets still make a grid, just a different grid.

A pedestrian in this new version of Washington still wants to walk from Judiciary Square to the Chinatown Metro stop. He still walks down a lettered street and up a numbered one. Each leg of the trip is different than it was before, and yet the subway trip that you take underground is exactly the same as it was before!

We’ve seen rotational symmetry a number of times already from the apparent isotropy of the large-scale universe down to the experimental fact that microscopic interactions really can’t seem to distinguish one direction from another. We’ve even seen that rotational symmetry gives rise immediately to conservation of angular momentum. As a practical matter, this means that the earth will orbit around the sun at a constant rate. Long story short: you may have learned the Pythagorean Theorem as a kid, but it’s anything but child’s play.

**What does “distance” mean in space and time?**

Space and time are very similar to one another, but not identical. If Einstein’s postulates of Special Relativity are correct – and to date, they have passed every experimental test thrown at them – then we’re going to have to figure out a way of jamming space and time into a single “spacetime.” Einstein himself warned of the danger of trying too hard to think in four-dimensional spacetime:

No man can visualize four dimensions, except mathematically … I think in four dimensions, but only abstractly. The human mind can picture these dimensions no more than it can envisage electricity. Nevertheless, they are no less real than electro-magnetism, the force which controls our universe, within, and by which we have our being.

Let me put this in familiar terms –- or at least familiar if you’ve memorized the entire Star Trek canon. The Vulcan homeworld is approximately 16 light-years from earth and right now (Footnote: In short order, we’ll find out how poorly defined the concept of “right now” really is, but for now, humor me.) Solkar, the great-grandfather of our own Mr. Spock, is a young spaceship pilot. Because we are separated from Solkar in space, but not in time, calculating the distance is easy: 16 light-years.

Let’s add in time. You are now reading these words and ten seconds ago you were reading “Let me put this in familiar terms.” Assuming you are sitting perfectly still, these two events are separated in time by ten seconds, and not separated in space at all.

But what if events are separated in both space and time? If we were to point a ridiculously powerful telescope at Vulcan right now, we wouldn’t see Solkar flying around his spaceship. Instead, we’d see the events on Vulcan unfolding from 16 years ago. This is, of course, because we’re seeing the signals traveling at the speed of light. The events we see are separated from us by 16 light-years of space and 16 years of time. Light signals will always have this one-to-one separation of space and time.

On earth, of course, events are also separated by both space and time. Go to a baseball game and watch and listen to the batter hit a ball. You probably know from experience that you’ll see the hit before you hear it. That’s because the speed of sound is far slower –- by roughly a factor of a million –- than the speed of light. The delay between a batter hitting the ball and you hearing it is perhaps half a second. The distance that the signal travels, on the other hand, is far less -– only about 500 billionths of a light-second.

On earth, generally, separations tend to be much larger in space than time – at least on the human scale. The circumference of the earth is only about an eighth of a light-second, but the time that it takes us to travel that distance is many hours. We may as well be standing still.

See? We can compare totally space and time, but unlike with different dimensions of space, there doesn’t immediately seem to be any equivalent of the Pythagorean Theorem that tells us how to add them.

I’m going to describe things a bit ahistorically, in large part, because some of the most relevant results to our purpose – symmetry – weren’t discovered all at once. Einstein came up with his theory of Special Relativity in 1905, based in large part on the work of James Clerk Maxwell and a number of mathematicians and physicists who laid the groundwork over the decade prior. It wasn’t until 1907 that the German mathematician Hermann Minkowski finally showed how space and time really fit together in a way that would have made Pythagoras proud.

Minkowski realized that in some sense, space and time work in opposite directions. There’s some suggestion, for instance, that Betelgeuse, the bright red star in the constellation Orion, might go supernova any day now (In astronomical terms, “any day now,” means that it might go off in 100,000 years or more.). Betelgeuse is about 600 light-years from earth, which means that even if we saw it blow itself up tomorrow, the actual explosion took place 600 years ago. How far away is the explosion really? We could describe the distance in space (600 light-years) or time (600 years), but combining the two is a bit trickier.

We have a hint, though. If the light from a supernova explosion is just now reaching us then there is an immediacy to it. It is, in a real sense, happening in the here and now, since the speed limit of light prevented us from learning about it earlier. Minkowski created a variant of the Pythagorean theorem where time behaves almost exactly the same as distance, except for a minus sign:

This “interval” is just a fancy way of combining distances in both space and time (Footnote: The mathematically savvy among you may have noticed that for events that are separated more in time than in space, the interval squared turns out to be a negative number, making the interval an imaginary number. Don’t worry about it. This is just a mathematical device to let us know that the two events are “time-like separated,” which simply means that one event could affect the other. If the interval-squared is positive, they’re space-like separated, which means that causality simply can’t come into play. If math makes you want to rock yourself in a corner, please move on. There’s nothing to see here.). It’s also pretty clever. By definition anything that we’re just seeing now –- regardless of how far away it is in space –- has an interval of zero. More importantly for our purposes, the interval between two events is completely independent on your perspective.

We saw with the Pythagorean theorem that it doesn’t matter how you rotate the orientation of your streets, the distance between any two subway stations will always remain the same. The interval is exactly the same. Einstein said that all inertial observers should measure the same speed of light, which means that the interval between any two events should be the same no matter how fast you’re traveling through space.

We’re in deep now. Spend a moment to think about how abstract thinking of symmetries ends up revealing surprising connections. We just found that, from a mathematical perspective, rotating a coordinate axis is the exact same sort of symmetry in space as moving at different speeds does on spacetime. Both of these two transformations leave something invariant. For rotations, the distance between two points stays the same; for different speeds – “boosts” as relativists call them – the interval stays the same. Who could have seen that coming?

If you have trouble picturing this, imagine a Vulcan astronaut flying the earth to Betelgeuse route at a sizeable fraction of the speed of light. As we’ll see, he’ll measure the total distance of the route to be less than 600 light-years. However, he’ll also measure the delay in between Betelgeuse going kablooey and us measuring it as less than 600 years. All combined, he’ll measure the same zero interval as we do no matter how fast he’s going.

**How Time Gets Stretched**

It’s not enough to simply say that the speed of light is the same for all observers. Clearly time gets messed up, but so far, we haven’t gotten any idea as to how. Suppose Solkar decided to buzz past the earth at half the speed of light. Provided he’s moving in a constant speed and direction, he feels as though he is sitting still. That is, of course, one way of thinking about Einstein’s first postulate of Special Relativity. As Solkar goes about his business – reading the paper, taking a nap, browsing the intergalactic net -– he is, as far as he’s concerned, moving through time and not through space.

On earth, on the other hand, we see him moving through both space and time. If Solkar settles down for what seems to us to be an eight-hour nap, by the time he wakes up, the ship will have traveled four light-hours.

The beauty of Minkowski’s interval is that it’s the same for all observers. From the earth’s perspective, the space and time between the beginning and end of Solkar’s nap partially cancel each other out. On the other hand, as far as Solkar’s concerned, he’s slept for less than 8 hours. As it turns out when you crunch the numbers, he’s actually only gotten about 7 hours of sleep. Relativity can be even more disruptive to your circadian rhythm than daylight’s savings.

Moving clocks run slow. This is not some trick of measurement; it’s a real effect, albeit a very small one, at least under normal circumstances. To put things in perspective, even on the highest speed Japanese bullet trains, time only appears to run slower by less than one part in a trillion.

At only half the speed of light, things only appear a little bit off. The second hands on clocks would appear to tick only about 52 times for every one of our minute, for instance. If, instead, he sped past at 99% the speed of light, things become crazy – clocks appear to be slowed down by a factor of seven! And understand, this isn’t some weird optical illusion or mechanical effect because of the strain of the speed. Everything appears to be slowed down by the same factor. Solkar’s heart – and all of his metabolic processes – would beat slower than normal; his computers would appear sluggish by normal standards; every single device capable of measuring time would appear pear to have slowed down to a crawl. And yet, from Solkar’s perspective, everything appears to be running perfectly normal within the ship.

While we can’t actually build spaceships capable of moving at relativistic speeds, we can measure time dilation here on earth using particles called muons. A muon is almost identical to an electron – but 200 times heavier. As we’ve seen before, heavy particles, whenever possible will decay into lighter ones, and the muon is no exception. After about 2 millionths of a second on average, a muon will decay into an electron and a couple of neutrinos.

Since muons decay so quickly, it’s a wonder that they’re around to be observed at all. Fortunately, the universe is nothing if not dedicated to the task of producing massive particles. When extremely high-energy particles from space – cosmic rays – strike the upper atmosphere, a cascade of particles get created, culminating in the production of muons. This means that the bulk of muons get produced more than ten kilometers above the surface of the earth. This would be no big deal except for their incredibly short lifetime. Even traveling at the speed of light, a muon could only cover about 600 meters in that time. We’d reasonably suppose that virtually no muons should ever reach detectors on the surface of the earth.

And yet, we do detect atmospheric muons all the time. We can even tell that they originate from cosmic rays because we can see a big empty spot – a shadow – where the moon is. In 1941, Bruno Rossi and David Hall, both from the University of Chicago, measured the number of muons coming from sky as measured at the top of a two kilometer high mountain and at ground level. If Galileo were right, and time flowed the same for everybody, then all of the muons should have decayed from top to bottom. Instead, Rossi and Hall found very clearly that the muons’ “internal clock” seemed to be slowed by roughly a factor of five, meaning that they were hauling ass at roughly 98% the speed of light.

But relativity says far crazier things than “moving clocks run slow.” Einstein’s first postulate of special relativity was that you can’t ever tell if you’re the one moving or standing still. It’s easy to imagine things from Rossi and Hall’s perspective. They’re people, after all, and just like in sci-fi movies, we tend to have an anthropocentric view of things.

But if you can manage a little empathy, put yourself in the muons’ position. The muons don’t feel like they are moving at all. Here they are, newly born, and all of a sudden they see the ground – and Rossi and Hall – hurtling toward them at 98% the speed of light. The muons, provided they have the presence of mind to do the experiments, find that Rossi and Hall seem to be living in slow motion, by the same factor of five that we saw before.

Despite having studied relativity for a long time, this still seems crazy to me. If we pass one another in spaceships traveling close to the speed of light, we each measure the other to be aging slowly and we each measure the other’s ship to be shortened in the direction of motion. This simply seems to be logically inconsistent. And yet, it isn’t.

Wasn’t put off by math!

But since the + turned into a – , I did wonder why and how the equations could be said to be the same or alike or related. is it just the obvious ‘everything else is the same’?

Also, it might be helpful to go over why the Pythagorean theorem uses rectangles connected to the edges forming triangle because they’re used in drawing (though you could have a diagram without the rectangles) but never commented on.

Great read.

Rock on.

Hi Jin. The diagram was just something I grabbed to remind people what right triangles looked like, and which sides were A, B, and C. I certainly don’t want to prove the Pythagorean Theorem in the text.

But as for your other point, I’m trying to motivate the minus sign with the whole discussion of the constancy of the speed of light. Yes, the comparison with the Pythagorean Theorem is an analogy, but I’d have thought it was a sound one.

Hmm… what do you think would better motivate the argument that:

1. Coordinates add using a square

2. Time is just another coordinate

3. It’s different, though, so it gets a minus sign

?

Thoughts on where I lost you would be appreciated.

The equations didn’t throw me off at all. I’d never seen the Minkowski equation before but it was close enough to the Pythagorean Theorum that it wasn’t difficult to relate the two. Had you thrown it in without the Pythagorean Theorum I’d have had a bit of trouble trying to recall middle school & high school math.

I didn’t just zip through the post- it took a fair amount of time for me to read it, but everything was understandable (or well explained- some things will just always seem crazy to me, even though they’re true) in the end.

Thanks for the sneak peek.

I really appreciate your writing. It’s good practice in a few different ways of thinking. I can understand it, but not without drawing pictures and animations in my brain, so it’s good exercise. The math is helpful. The Pythagorean Theorem was one I understood while I never paid attention in math class, so it was easy to build on.

For some reason, my mind skipped over the fact that only the numbered streets are rotated by the malevolent and efficient city planner, and I was thinking the grid rotated as it was. It was making no sense until I finally read the words that were right there on the page. I was going to say that it’s an important part of the theory, but no more so than almost every other sentence. There aren’t many unnecessary words in here.

I didn’t have issues with the math, but I did have issues connecting the dots because, I think, it wasn’t broken up and labeled such that the key points stood out as connected. Also, the interval is the same for all observers? A diagram might help. And invariant? Not exactly clear to the casual reader.

The muon was the clearest part. I wish that part came. First.

Interesting. I wasn’t sure whether to draw spacetime diagrams because even explaining them gets a bit tough. More to the point, the curve of constant intervals are represented by hyperbolas, which are not well known to the casual reader. Let me see if I can figure out a way to include it.

As for the muons, it’s not clear that it’s connected pedagogically if it comes first. Why would anyone not already familiar with the possibility suspect that time _should_ be dilated for observers moving close to the speed of light?

At any rate, good food for thought.