- #1

Tac-Tics

- 816

- 7

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

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

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

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

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

[tex]\int_0^1\sqrt{g(y'(t), y'(t))} dt[/tex]

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?