(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

How do I show the following metric have time-like geodesics, if [tex]\theta[/tex] and [tex]R[/tex] are constants

[tex]ds^{2} = R^{2} (-dt^{2} + (cosh(t))^{2} d\theta^{2}) [/tex]

2. Relevant equations

[tex]v^{a}v_{a} = -1[/tex] for time-like geodesic, where [tex]v^{a}[/tex] is the tangent vector along the curve

3. The attempt at a solution

First, I write it as the Lagrangian

[tex] L = -R^{2}\dot{t}^{2} + (cosh(t))^{2} \dot{\theta}^{2} = -R^{2}\dot{t}^{2} [/tex]

as [tex]\theta[/tex] is a constant.

How do I proceed to show that this indeed gives us a time-like geodesic.

Could someone also tell me if I have computed the Christoffel symbol components correctly? My result is

[tex] \Gamma^{t}_{\theta \theta} = 0-sinh(t) \times cosh(t) [/tex]

[tex] \Gamma^{\theta}_{t \theta} = tanh(t) [/tex]

and all other components vanish.

Cheers!

P.S. How do I type minus sign? It doesn't seem to work if I have left the 0 out at above.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Timelike Geodesic and Christoffel Symbols

**Physics Forums | Science Articles, Homework Help, Discussion**