I’m finally back from my tour of the Iberian peninsula — a conference in Porto, a summer school in the Azores, and then a week’s vacation in Spain. (I know, I know, it’s so hard being a graduate student in cosmology!) But I’ve got a grant application due in three weeks, a talk to prepare for two weeks after that, and a paper to finish up in the same timeframe. And then the real crunch begins, with dozens of job applications and a thesis to finish.
Enough complaining. It’s good to be back, and today I’ve got the perfect excuse to talk about one of my favorite things ever. This link has been making the rounds today. Apparently, a team of scientists working at CERN have found evidence suggesting that neutrinos can travel faster than the speed of light. While this has turned a few heads, the wide consensus is that this will probably turn out to be some kind of systematic error and not a real effect. Extraordinary claims require extraordinary evidence, and this evidence has to pass a hell of a lot more tests — and be reproduced by a lot more people — before anyone, including the team who produced the results, will feel comfortable saying that they’ve found a way around the ultimate speed limit.
But why would this be such an extraordinary claim? The speed of light is a hard limit in Einstein’s relativity, but why? There are several reasons to claim that nothing can go faster than the speed of light: some of the most common things you’ll hear are that it takes an infinite amount of energy to accelerate to the speed of light, or that you get infinitely massive as you approach the speed of light. Both of those things are true, but that still doesn’t tell you why nothing can go faster than the speed of light. After all, if the goal were to get something to go at the speed of light, that’s easy — you and I can’t do it, but light does so quite easily, thank you very much. But relativity tells us that very strange things would happen if you can send any thing or signal faster than the speed of light.
Relativity talks a lot about what happens when we switch frames of reference. When we move to a new frame of reference1 — when we go cycling, or get in a train, or go jogging — our intuition and our senses tell us that distances and times in the world around us don’t change. After all, when you’re driving down the highway, the distance from your house to the airport isn’t any different than it was when you were sitting in your chair at home, nor does your flight leave any sooner.
Relativity throws all of this out in the trash. Our intuitions about the things which remain the same when we move to a new frame of reference are totally incorrect: distances and times can both change, sometimes dramatically. You were wrong all along! Your flight really is earlier, and your house actually is closer. The name “relativity” might suggest that these changes in size and time are illusory, a matter of mere perspective — but the real illusion is that space and time don’t change when we change our frame of reference. They do change, and they always have. The only reason we ever thought that distances and times remain the same from frame to frame was that the changes are minuscule unless we’re moving at a significant fraction of the speed of light. (An example: if you’re moving at the cruising speed of a Boeing 747, the change in distance between New York and San Francisco in your frame is about 1/100th the width of a human hair.)
But relativity also teaches us that there are things that don’t change when we move from one frame to another — they just aren’t the ones that we thought they were. For example, while the distance and time between a pair of events depends on your frame of reference, there’s a way of combining that distance and time into a quantity called the “invariant spacetime interval,” and that will never change from frame to frame.2 This quantity (which we usually call S) is just the square of the distance between the events subtracted from the square of the distance light could travel in the time between the two events3 The sign of S tells us whether light can travel between a pair of events: if S is positive, then light can make it from one event to the other, but if S is negative, then light can’t make it.
For example, say that Event A is me stubbing my toe in my room this morning, and Event B is you sitting down to use your computer today. You and I, here on Earth, may measure the distance between us as several hundred kilometers, and the time between my toe-stubbing and your computer usage as a couple of hours; Elton John, on a rocket flying past the Solar System at nearly the speed of light, will find both the distance and the time between the two events to be much smaller. But all three of us — you, me, and the Rocket Man — will agree about the value of S for our two events. And since S doesn’t change from frame to frame, neither does the question of whether light can make it from my toe-stubbing to your computer usage. If light can make it between two events from the perspective of one observer, then light can make it between those events from the perspective of any potential observer.
We’re not done trashing our intuitions about the things that remain unchanged when switching frames. Not only do clocks slow down and objects shrink, but events which are simultaneous in one frame may not be simultaneous in another. If, in your frame, you see two fireworks go off at the same time on opposite sides of the sky, and I’m passing by in Elton John’s super-fast rocket, I’ll disagree with you about both the distance between the two fireworks and whether or not the fireworks went off at the same time. Better still: say that I pass by in one direction and my friend Andy passes by in the opposite direction. While you say that the red firework went off at the same time as the blue firework, I will say that the red one went off before the blue one — and Andy will say that the blue one went off before the red one. Furthermore, we will all be totally correct. The time between events does not remain the same when going from one frame to another; it never did. Simultaneity, independent of a frame of reference, is an illusion.
But how can we disagree about the order of events? If I’m right about the red firework going off before the blue one, and Andy is also right about the blue firework going off before the red one, what’s to stop someone from saying that I swore and hopped around on one foot before I stubbed my toe this morning? If the order of events can change, then why can’t effects precede causes?4 And if effects can precede causes, what’s to stop me from sending signals back in time? Well, you saw those fireworks going off at the same time; that means the difference in time between the two events must be zero. Therefore, S must be negative, since the distance between the two events is (clearly) greater than the distance that light could travel in the time between then. And S must be the same in every frame, so while Andy and I disagree about which firework went off first, we agree with each other — and with you — that the spacetime interval between the two events must be negative. In other words, we may not agree which firework went off first, or even if one firework went off first, but we all agree that you’d have to go faster than the speed of light to make it from one firework’s explosion to the other. This is what saves us. There’s a rule in relativity, one which I won’t bother proving here:5 it’s only possible for two observers to disagree about the sequence of two events if the spacetime interval between those two events is negative. Thus, if it’s possible for light (or something moving at the speed of light) to make it from one event to another, everyone will agree about which of those two events happened first. Otherwise, the sequence of events is up for grabs. So, in order for an effect to precede a cause in someone else’s reference frame, that cause would have to create that effect faster than the speed of light in your frame. This is one of the great insights into the nature of space and time that relativity has given us: sending any sort of signal faster than the speed of light is tantamount to sending that signal back in time.6 And this, in turn, is one of the major reasons why you’ll often hear that “nothing can go faster than the speed of light.”
Of course, while this is all totally awesome, none of what I’ve said actually constitutes a proof that it’s impossible to go faster than the speed of light — all I’ve done is tell you that going faster than the speed of light would amount to getting in the DeLorean and heading back to 1955. But since going back in time has its own set of issues ranging from the physical to the philosophical to the fictional, going faster than the speed of light is certainly problematic. Even if these neutrino results are right, though, don’t expect a tachyonic antitelephone anytime soon; in that case, it’s rather likely that relativity is wrong in some important way, in which case all of this is out the window. But since relativity is one of the best-tested theories in all of science, it’ll take a lot more than a single result to convince us that it’s wrong. More on that this weekend, I think; once the actual data come out tomorrow, you can expect physicists the world over to tear it apart.
- When I talk about “your frame of reference” or “my frame of reference,” what I really mean is a frame of reference in which you are (or I am) at rest. If you’re on a train, and I’m on an embankment by the tracks, then in my frame of reference, you (and the train) go flying past me at 150 kph. But in your frame of reference, I (and the embankment) go flying past you at 150 kph. Either choice is perfectly valid — the laws of physics operate precisely the same way in both frames. If this bothers you — if you feel that the reference frame in which the embankment is at rest is a truer or better frame in some way — remember that from the perspective of, say, the Sun, the embankment most certainly isn’t at rest: it’s spinning around once a day with the rest of the Earth, and hurtling many times faster than that around the Sun. And, lest you’re tempted to take the frame of the Sun, remember that the Sun itself revolves around the center of the galaxy…. [↵]
- This is part of why we physicists talk about spacetime, rather than space and time individually — spacetime isn’t frame-dependent, but space and time are. [↵]
- Nitpickers: yes, technically this is S2, but it doesn’t matter in the slightest. [↵]
- We’re sidestepping the extraordinarily thorny philosophical question of what “causes” and “effects” are. [↵]
- This fact falls directly out of the Lorentz transformations, and takes about two lines of algebra to prove. [↵]
- Hence this comic. [↵]