Complex Numbers

No Space for Time

The Big Think article that came up for me today involves math. Fair warning so you don't end up defenestrating your device.

    Time: Yes, it’s a dimension, but no, it’s not like space  
The fabric of spacetime is four-dimensional, with three for space and only one for time. But wow, time sure is different from space!

Time is different from space? I never would have guessed, what with them having different names and all.

When did you first realize that the shortest distance connecting any two points in space is going to be a straight line?

I'm not sure that's a fair question. It's something I'd consider intuitive. What's hard to grasp, sometimes, are the cases where the shortest distance between two points isn't a straight line, because that runs counter to our everyday experience.

In fact, that realization, as far as human knowledge is concerned, comes from a place we might not realize: the Pythagorean theorem.

"In fact," I think they've got this backwards. The Pythagorean Theorem may quantify the "shortest distance" intuition, but I'm pretty sure humans knew about the straight-line thing before they had numbers or geometry. (Also, the idea predated Pythagoras by hundreds or thousands of years; the Greeks didn't invent everything.)

Taking all three of these dimensions into account — so long as we assume that space is still flat and universal — how would we then figure out what the distance is between any two points in space? Perhaps surprisingly, we’d use the exact same method as we used in two dimensions, except with one extra dimension thrown in.

I feel like the only "surprising" thing here is that the math is basically the same.

Thinking about distances like we just did provides a great description of what we’ll wind up with if we consider flat, uncurved space on its own. What will happen when we fold time, as a dimension, into the equation as well? You might think, “Well, if time is just a dimension, too, then the distance between any two points in spacetime will work the same way.”

At which point, unsurprisingly, the math isn't basically the same.

There are two fundamental ways that time, as a dimension, is different from your typical spatial dimension. The first way is a small but straightforward difference: you can’t put space (which is a measurement of distance, with units like feet or meters) and time (which is a measurement of, well, time, with units like seconds or years) on the same footing as each other right from the start.

Feet? Footing? Get it? Haha.

Fortunately, one of the great revelations of Einstein’s theory of relativity was that there is an important, fundamental connection between distance and time: the speed of light.

Yes, and the invariance of that speed is still very, very hard to wrap your head around, because it, unlike the "straight line" thing, runs counter to everyday experience.

However, there’s also a second way that time is fundamentally different from space, as a dimension, and this second difference requires an enormous leap to understand. In fact, it’s a difference that eluded many of the greatest minds of the late 19th and early 20th centuries. The key idea is that all observers, objects, quanta, and other entities that populate our Universe all actually move through the fabric of the Universe — through both space and time — simultaneously.

Turns out that everything in the universe is moving at a constant speed... through spacetime. Once that was pointed out to me, a whole lot of other stuff started to make more sense.

It turns out that the faster (and the greater the amount) you move through space, the slower (and the lesser the amount) you move through time.

Like that, for instance.

There’s an even deeper insight to be gleaned from these thoughts, which initially eluded even Einstein himself. If you treat time as a dimension, multiply it by the speed of light, and — here’s the big leap — treat it as though it were an imaginary mathematical quantity, rather than a real one, then we can indeed define a “spacetime interval” in much the same fashion that we defined a distance interval earlier...

Great. Wonderful. Now we'll get "time is imaginary" on top of "time is an illusion" nonsense. I'd forestall this by pointing out that imaginary numbers aren't actually imaginary (or at least they're no more abstract than the "real" numbers), but that's not going to stop the airy pseudophilosophy.

There is, of course, quite a bit more at the link. I'm not sure if it's really that useful; I feel like there's either too much or not enough math to keep people interested enough to follow the arguments. But if all we can get out of it is "spacetime is weird, and time is different from space," maybe that's good enough.