- #1

alex3

- 44

- 0

[tex]ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2)[/tex]

using the Euler-Lagrange equations reveal the following equations of motion

[tex]\ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0[/tex]

[tex]\ddot{\theta} - \sin{\theta}\cos{\theta}\dot{\phi}^2 = 0[/tex]

Using the general geodesic equation form [itex]\ddot{x}^{\alpha} + \Gamma^{\alpha}_{\beta\gamma}\dot{x}^\beta\dot{x}^{\gamma}=0[/itex], we infer that the equations derived describe geodesics. This shows that the only non-zero terms of the metric connection [itex]\Gamma^{\alpha}_{\beta\gamma}[/itex] are

[tex]\Gamma^{\theta}_{\phi\phi} = -\sin{\theta}\cos{\theta},\quad \Gamma^{\phi}_{\theta\phi} = \Gamma^{\phi}_{\phi\theta} = \frac{\cos{\theta}}{\sin{\theta}}[/tex]

My problem is comprehending where the factor of two has gone for the [itex]\Gamma^{\phi}_{\phi\theta}[/itex] term. Is it due to fact that it's a coefficient of a mixed derivative, why is that?