I grew up with science fiction stories that grappled with the speed of light, sometimes treating it as an inviolable barrier and sometimes as an inconvenience to be papered over in the interests of advancing the story. Viewed as a natural law, the inability to exceed the speed of light somehow comes across as a traffic ordinance that's begging to be violated. We know light can go slower, as it does through glass or water, for instance, so why not faster? Learning a little bit more about it as an amateur, I now gather it's more a question of light-speed as an inherent quality of a specific thing, like the length of a pencil, that only seems to have maximum and less-than-maximum manifestations because we're looking at it in more or less foreshortened perspectives, so to speak. So your pencil might look longer or shorter depending on your angle, but it's always as long as it is, and no longer.
My Great Courses lecturer used an analogy that leapt out at me. Drawn from an excellent 1965 popular textbook by Edwin Taylor and John Wheeler (Spacetime Physics, 1st ed., PDF link to 1st chapter here), the analogy is "the parable of the surveyors." The king asks his surveyors to figure out how far the smithy is from point "X." One uses a coordinate system based on true North, while the other uses magnetic North, so they get different recipes for "go so many feet East, then so many feet North."
But the two vectors we add to get to the straight-line distance can be nearly anything, depending on how we rotate the frame of reference. The analogy is to space and time as the elements that make up the speed of light: from some points of view, the distance will be one thing and the time elapsed another, but those two elements can change. What will never change is the speed of light. It's not so much that light isn't "allowed" to go faster; it's more like the fact that the smithy is a certain distance from "X." Would it be allowed to be farther? Sure, but that's not where it happens to be.
The way the lecturer puts it is that space and time are aspects of the same thing, and the speed of light, c, is the conversion factor needed to switch back and forth between them. Similarly, mass and energy are aspects of the same thing, and the speed of light squared is the conversion factor needed to switch back and forth between them. Why is one just c and the other c-squared? No idea.
15 comments:
I've never seen the "smithy" analogy before, but it makes wonderful sense. Not sure how VALID it might be, but as a way to feel better about the complex topic, it works very well indeed. Thanks for sharing.
Also, consider this an endorsement from a homeschooling parent for the Great Courses videos. Kids and adults alike benefit.
The problem with the analogy is in an unspoken assumption: that the space isn't curved, but flat ("Euclidean").
Even at the level of a smithy, it's possible to imagine that one of your surveyors goes over a hill and the other along a relatively flat piece of land just because of the way their initial frames of reference point. Now the distance is going to vary. But because the graph holds space flat, "c" will appear to be constant even though it isn't really.
In terms of lightspeed, relativity theory is one reason we've decided an assumption of Euclidean space is false. But to get there, we've chosen to hold "c" constant rather than the flatness of the space. That seems to produce a lot of useful results. Both General and Special Relativity have produced a lot that would suggest it's at least a very useful way of looking at the world. But it could be wrong.
The theory doesn't prevent something from going faster than light, "c", what the theory says, that something that is going slower than "c" cannot go faster than "c", and something that is going faster than "c" cannot go slower than "c". The theory implies, that nothing can cross the light speed boundary, either by going faster than "c" if slower, or slower than "c" if faster.
Not proven, but needs to be true if tachyons exist.
Grim, the analogy doesn't assume that space is flat. It only suggests a way of thinking about definite dimensions and how they can be expressed by multiple vectors, all with equal validity. You strain an analogy significantly if you try to take it too literally. For that matter, space and time aren't like North and East, either, because they're not at right angles to each other in a plane--but that's not the point!
Analogies always break, Tex: that is their character. If they didn't break, they wouldn't be analogies. This is because, to be perfectly analogous such that the comparison didn't break, you'd have to be talking about things that were actually identical -- and that's not an analogy at all!
...space and time aren't like North and East, either, because they're not at right angles to each other in a plane...
Actually, that's debatable too; we just don't have a good way of conceptualizing what it means to be at right angles in this case. But you can usefully draw spacetime diagrams in two or three dimensions by suppressing one or two spatial dimensions and substituting a time dimension (at right angles to the spatial ones). I was trained to do this, and it really does seem to work in making sense of relativistic problems.
In some contexts it's best not to attempt to use analogies at all.
Kinetic energy is 1/2 m v^2 (the units of energy are mass times distance^2/time^2). Hence the c^2.
You can always define orthogonality locally. In a non-flat spacetime you may have some issues--the axes may not look the same along an object's path through spacetime.
FWIW, it turns out to often be convenient to do calculations using units in which c=1. There are jokes about setting π=1, but that really only happens by embarrassing accident.
I didn't so much mean that I didn't know what equation the c-squared comes from, or even that a velocity-squared element is always present in converting between mass and kinetic energy. I just meant I have no intuition whatever for why these two great equivalences--space-time and mass-energy--should both involve the speed of light, but in one case it's squared. It just seems kind of funny, as if it should make a lightbulb come on, but it doesn't.
The dimensional analysis kind of demands it. dim(c) = distance/time
dim(x)=distance = dim(c)*dim(t) = (distance/time) * time
dim(E)= mass * distance^2/time^2 = dim(m) * dim(c)^2
BTW, just as you can define a kind of invariant "distance" between two events: (δx)^2 - c^2 (δx)^2 (which can be positive or negative for space-like or time-like separations), you can find an invariant mass for a particle E^2 - c^2 p^2 (where p is the momentum)--the same no matter what reference frame you view it from.
Don't feel bad for not having an intuition about it, though. The neat thing about relativity theory is that we poor humans managed to discover it even though it goes against all our ordinary intuitions. That's really the amazing part of the story.
I see from Wheeler's textbook--the one I linked to-- that you can state an invariant spacetime interval as
interval^2 = (c^2)(t^2) - x^2,
basically the Pythagorean theorem, where the interval is the hypotenuse, right? So I probably was latching on erroneously to something the lecturer said about a plain c conversion factor for spacetime. The fact is, c-squared crops up both ways, whether it's spacetime or mass-energy. There's no real sense in which c is the factor with one and c-squared with the other.
Whoops, that's not right--I've got a minus sign there, not a plus sign, so the hypotenuse-like thing would have to be one of the other elements. The (c^2)(t^2) part? Oh, well.
Grim, of course I don't feel "bad" for not having an intuition, it's just that an almost-intuition is so tantalizing, when you're trying to learn about something.
That Wheeler textbook is really good. Most of our untutored intuitions about space, time, and the laws of motion are gammon; he makes it clear how much easier it is to understand things if we get our intuitions on the right footing. Kind of like putting the Sun in the middle of the solar system and getting rid of the pesky epicycles: the epicycles work, but they're inordinately complicated. The first guy to figure these things out has to do really hard mental work, but then he can put things in such a way that we ordinary mortals can readily learn about them.
Well, and I'm being a little too pat: it wasn't "we poor humans" who worked it out, it was Einstein and a very narrow band of physicists who were working at or near his level. Einstein himself worked against them, mostly, but without them he couldn't probably have done what he did. And he did it by working backwards, finding where they couldn't be right and then examining their assumptions by tracking backwards through the history of physics and the philosophy that underlies it. That's where he found the right stones to move, in the deep history of the ideas.
I just meant I have no intuition whatever for why these two great equivalences--space-time and mass-energy--should both involve the speed of light, but in one case it's squared. It just seems kind of funny, as if it should make a lightbulb come on, but it doesn't.
I think if they unsimplified the equation, it might make more sense.
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