Arclength on a PseudoRiemann Manifold

[Tac-Tics]

Wikipedia gives a confusing definition of a path's length and I would like some clarity.

Let M be a pseudo-Riemann manifold with metric g and let a and b be points in M.If y is a smooth function from R->M where y(0) = a and y(1) = b, then it's length is the integral

\int_0^1\sqrt{\pm g(y'(t), y'(t))} dt

Now, why they decided to put a plus-minus in there, I don't know, but it seems like they MEANT

\int_0^1\sqrt{|g(y'(t), y'(t))|} dt

Where |.| is the absolute value.

Now on the other hand, this definition of length doesn't distinguish between time-distance and space-distance. It seems like perhaps if we left off the sign-changing nonsense

\int_0^1\sqrt{g(y'(t), y'(t))} dt

We would get real-valued lengths for space-like paths and imaginary-valued lengths for time-like paths. That's just what my intuition suggests, though. How close would such a guess be to the reality of it?

 

[Fredrik]

Why would you want it to be imaginary when you can define it to be real just by changing the sign under the square root? I mean, everything else in the theory is real, so why introduce a complex number unless you have to?

 

[Tac-Tics]

Why would you want it to be imaginary

I just want to know the correct interpretation. Is the plus-minus "fixed" at the start of the evaluation of the integral? Or is it chosen at evaluation of every infinitesimal segment (which would simply mean the absolute value)?

 

[Fredrik]

The integral is only defined along curves such that the quantity under the square root has the same sign everywhere on the curve. If it's negative, you flip the sign to make sure that you take the square root of a positive number. (So it's the first of the two options you suggested: It's fixed at the start of the evaluation). In relativity this means that the integral is defined for timelike curves and for spacelike curves, but not for any other curves. In the case of timelike curves, we call the result of the integration "proper time" and in the case of spacelike curves, we call it "proper length".

There may be generalizations of this that I'm not aware of. I learned this stuff in the context of general relativity.

 

[Tac-Tics]

If it's negative, you flip the sign to make sure that you take the square root of a positive number.

That sounds like you take the absolute value then (which is what I thought). I don't understand why they don't just word it as such on Wikipedia >____>

Thanks =-)