Alrighty, a quick course in curve lengths in Special Relativity coming right up. Hopefully I won’t make a complete mess out of it.

Think about geometry in the plane first. If we have two points on the plane, the distance between them is given by sqrt(([symbol]D[/symbol]x)[sup]2[/sup] + ([symbol]D[/symbol]y)[sup]2[/sup]). This is called an *invariant*, because it doesn’t depend on the coordinates we choose. As a concrete example of this, suppose we have two surveyors measuring distances on a farm. One of them, though, has a broken compass, so instead of measuring north-south lines he’s actually measuring lines that run from northwest to southeast. Despite this discrepancy, they will still both agree that the distance between the pond and the old oak tree is one-quarter of a mile, since the absolute distance between two points doesn’t depend on the coordinates you use.

Here’s where it gets interesting. In special relativity, we have to introduce the concept of an event; instead of just saying, “I was at position x”, we have to say, “I was at position x at time t.” We can then define something called the “space-time interval” between two events:

([symbol]D[/symbol]s)[sup]2[/sup] = c[sup]2[/sup] ([symbol]D[/symbol]t)[sup]2[/sup] - ([symbol]D[/symbol]x)[sup]2[/sup]

where c is the speed of light. What this says is that if you take the time separation between two events and the distance separation between two events and combine them in this odd way, you get the space-time interval between these two events.

What special relativity says is that ([symbol]D[/symbol]s)[sup]2[/sup] is *also* an invariant. If someone is whizzing past me in a rocket ship, her coordinates are just as good as mine, and for all she knows, it’s me who’s whizzing past him in the other direction. She’ll measure a different distance-separation between the two events than I will (since she’ll move between the times of the two events), but according to special relativity she’ll measure the same ([symbol]D[/symbol]s)[sup]2[/sup] that I will.

What does this have to do with curve length? Well, suppose my friend blasts off in her rocket ship and follows (according to me) some arbitary curve x(t) through space. According to her, though, she’s staying in one place while the world is moving around her. In particular, if we take a tiny chunk of her curve, the amount of time she sees elapse is just equal to the space-time interval, since according to her the event at the start of this chunk and at the end of this chunk take place exactly where she is at those times. So the amount of time that elapses according to her clock is given by

[symbol]Dt[/symbol] = sqrt(c[sup]2[/sup] ([symbol]D[/symbol]t)[sup]2[/sup] - ([symbol]D[/symbol]x)[sup]2[/sup]) = sqrt(c[sup]2[/sup] - ([symbol]D[/symbol]x/[symbol]D[/symbol]t)[sup]2[/sup]) [symbol]D[/symbol]t

and if we add up all these little chunks to find the total time elapsed on my friend’s clock, we get

[symbol]t[/symbol] = Int sqrt(c[sup]2[/sup] - (dx/dt)[sup]2[/sup]) dt

This is kind of like the curve length, except for the factor of c and the minus sign. It’s the closest thing I could think of in physics to a curve length that has physical importance. There’s a **lot** of special relativity I didn’t get to here, though. If you’re interested in learning more about it, try reading *A Traveler’s Guide to Spacetime* by Thomas A. Moore; it’s a good introduction to the subject for someone at your level.