First and foremost, let me go on the record, once again, and say that I think that the faster than light neutrino result is almost certainly flawed. I’m not going to debate the point, since I’ve already made a number of objections, here and elsewhere, and at the end of the day, the results will either be experimentally confirmed or they won’t. Probably on the “won’t” side.
That said, a number of you emailed me asking about the implications of what would happen if we could send a signal back in time. I got an email from a reader named Jason (who previously asked me about Hawking radiation), who wanted to know:
Even if the communication took place instantaneously, I don’t see how one would use that to communicate with one’s own past. Can you explainit in plain English?
Specifically, Alice and Bob want to chat, Bob sends a message and N seconds later, Alice sends a reply. Even if the transit time for the message were zero, N seconds have still elapsed for both individuals. How does a superluminal transit speed equate to negative elapsed time? Wouldn’t, at best, a message have a zero delay?
Well, I’m not going to answer Jason’s question entirely intuitively (that’s why there’s a “technical” flag), but hopefully I can shed a little light (no pun intended) on the situation.
Also, in order to make some really kickass spacetime diagrams, I wrote a nice little package for pylab. Download my code, as well as a sample program showing how to use it if you’d like to make your own.
So first things first: Jason is absolutely correct in one very important respect. Say we’re standing about 1 light-second away from one another, and I send him a beam which (for convenience) travels at . I realize that this is much faster than the purported neutrino effect, but the difference is only one of degree. In that case, it’ll take 1/4 of a second to send a signal. If Jason then responds immediately, I get his response 1/4 of a second after that.
The entire series of events can be summed up in a spacetime diagram:
Jason and I are both represented by vertical lines in this diagram. We’re not moving through space — just through time.
The situation is exactly as Jason said it would be. I receive a response to my signal exactly half a second after I send my original message. Or, to put it another way event “3” (receiving the response) occurs after event “1” (sending my original signal) exactly as I expected it to.
But things get kind of strange once we consider frames moving compared to one another. For instance, if Jason is flying to the right at speed (I’ll be using so-called “natural units” throughout, so that c=1, light-seconds and seconds can be used interchangeably, unit-wise, and all speeds are in fractions of the speed of light), then the time and space interval as seen by each of us will be different.
For instance, if the time between the two events (as seen by me) is and the space interval is , then Jason will see:
If Lorentz transforms, and the idea that two different observers can measure the amount of distance or time between two different events bother you, there’s little I can do — short of doing a complete course on special relativity.
If you want a quick explanation, the basic idea is that those transformations are basically unique in that they produce the same speed of light for all inertial observers.
What this means is that suppose I measure two events occurring to Jason and me simultaneously (), Jason won’t necessarily measure it that way. Supposing the event in question was Jason and me both popping a balloon, and that at that instant, I measure Jason’s ship to be about 1 light-second away. In that case, . If Jason is traveling at 0.6 (60% the speed of light) then , and thus, using the relation above, we’ll find that:
In other words, he’ll claim that my balloon popped first.
Even if I sent him a message at the speed of light (warning him about the balloon poppage, presumably), my message won’t reach him until after his own balloon has popped. In other words, because of finite signal speed, this difference in frame normally doesn’t matter.
But what happens if you can send a signal faster than light? Then things get very interesting.
We’ll call our signal speed, and rather than go through all of the math, I’ll show you the result of a communication graphically. The only thing you need to verify is that in my diagram, photons always make 45 degree angles with the horizontal (because they cover 1 ls/s), and that from the perspective of the person sending it, neutrinos always have a slope corresponding to 4ls/s (much closer to the horizontal).
What I’m plotting here is exactly what we described before. I send a signal to Jason (event “1”), and Jason receives it (event “2”). I also shot off a photon at the same time, just for reference (the squiggly line). Hold off on worrying about event “3” for the moment.
Let’s look at things from Jason’s perspective. That’s the whole point of relativity. According to Einstein, there’s nothing preventing Jason from claiming that he’s at rest, and that I’m the one who’s moving, and hence he’s represented by a vertical line in his own space-time diagram:
The first thing you’ll notice is that, according to Jason, event “2” occurs before event “1.” In other words, he receives my signal before I sent it. In fact, there’s nothing terrible about this, since these two events occur in the same place. The craziness occurs when Jason sends his signal back. He sends his signal to me at very high speed, and what looks to be his future. However, seen from my perspective (look at the “unprimed” figure above again), event “3” (me receiving his message) occurs before the signal was sent out.
In other words, if you can send a signal faster than light, then it really is possible to communicate with the past, provided the observers are moving away from each other fast enough. How fast? If two observers are moving away from one another at speed, , then the signal needs to propagate at at least:
which for our example where Jason is traveling at v=0.6, corresponds to .
In other words, if you want a succinct answer to “Why would it matter if we could send particles faster than light?” You have one: You’d be able to send messages into the freakin’ past!