Spacelike geodesics parameter

[Henrique Mello]

Guys, does anyone know which affine parameter should I use to study spacelike geodesics? I mean, for timelike curves we use the proper time. I've read somewhere that I should use the proper distance. Is it correct? And why (or why not)?

 

Thanks to all.

[elfmotat]

Ah, but that's the beauty of it: the length of the curve is invariant with respect to any choice of affine parameter. This should make sense - why would proper time (a measurable quantity) depend on how a physicist chooses to parametrize the clock's curve? So you can choose your parameters based on how well they simplify your calculations.

 

So, for your example of proper time, yes it is completely valid to use proper time as the parameter:

 

[math]\tau=\int \sqrt{-g_{\mu \nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}}~d\tau[/math]

 

But if you can write [math]\tau[/math] so that it is a function of some other parameter (for example you can use coordinate time [math]t[/math], so [math]\tau = \tau (t)[/math]), then via chain rule we have:

 

[math]\tau=\int \sqrt{-g_{\mu \nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}}~d\tau=\int \sqrt{-g_{\mu \nu} \frac{dx^\mu}{dt} \frac{dt}{d\tau} \frac{dx^\nu}{dt} \frac{dt}{d\tau}} ~d\tau=\int \sqrt{-g_{\mu \nu} \frac{dx^\mu}{dt} \frac{dx^\nu}{dt}} ~\frac{dt}{d\tau} d\tau=\int \sqrt{-g_{\mu \nu} \frac{dx^\mu}{dt} \frac{dx^\nu}{dt}} ~dt[/math]

 

If you have a diagonalized metric, then the above form (parametrized W.R.T. coordinate time) can greatly simplify some problems. So, to answer you question more explicitly, yes you can use proper distance [math]ds[/math]. But you can also use any affine variable which might make your calculations simpler. I.e. if I was working in Schwarzschild coordinates, I would likely use the radial coordinate [math]r[/math] to parametrize radial distances.

[Henrique Mello]

Thanks, elfmotat. I'll work on this problem here and if I can't do it I call for help here

[ajb]

I've read somewhere that I should use the proper distance. Is it correct? And why (or why not)?

Already answered really, but Yes, you can use proper distance.