Show that Geodesic is space-like everywhere

[wam_mi]

Homework Statement

If the geodesic is space-like somewhere, show that the geodesic is space-like everywhere.

Homework Equations

Geodesic equation: $$\ddot{X}^{\mu}+\Gamma^{\mu}_{\alpha \beta}\dot{X}^{\alpha}\dot{X}^{\beta} = 0$$

The Attempt at a Solution

I looked at the metric

$$ds^{2} = g_{\alpha \beta} \dot{X}^{\alpha} \dot{X}^{\beta} = + 1$$,

where $$g_{\alpha \beta}$$ is the general curved metric in 4 dimensions of space-time. I try to write it in the form

$$g_{\alpha \beta} \dot{X}^{\alpha} \dot{X}^{\beta} = g_{\alpha \beta} \dot{X'}^{\alpha} \dot{X'}^{\beta}$$

where X is in one frame while X' is in another.

What exactly do I need to do now? I'm confused...

Thanks

 

[wam_mi]

I think I got it. Is it correct to say if one puts the inner product into the parallel transport expression, one finds that the expression vanishes as parallel transport preserves the inner product such that the character of the geodesic never changes.

Thanks