How do I interpret this statement?

Question

This is about a derivation of the geodesic equation (Carrol, 2003), in which the author gave 2 definitions of the geodesic curve; the first one is that the tangent vector is parallel transported on the curve, and the second is that the curve is the extremal of the path length functional. I have no problem understanding everything but this paragraph:

What was hidden in our derivation of (3.47) was that the demand that the tangent vector be parallel transported actually constrains the parameterization of the curve, specifically to one related to the proper time by (3.58).

Where (3.47) is the equation derived from the first definition:

$$ \frac{\mathrm d^2x^\mu}{\mathrm d\lambda^2}+\Gamma^\mu_{\rho\sigma}\frac{\mathrm dx^\rho}{\mathrm d\lambda}\frac{\mathrm dx^\sigma}{\mathrm d\lambda}=0 $$

and $\lambda$ is an arbitrary parameter when doing the calculation. (3.58) is the definition of affine parameters. The second definition leads to almost the same equation, but with the parameter restricted to $\tau$.

I am a little stuck. How exactly does the parameterization be constrained? Any help is appreciated.

Answer 1

So, when I was writing about the question I understood. I'll post my question anyway, and there might be better solutions than mine.

So, when using an arbitrary parameter, the tangent vector is different from that of an affine parameter in magnitude only. So, when applying the equation (3.47) as defined above, the magnitude of the tangent vector is demanded to be constant. Therefore, the parameter of the curve must be 'moving at a constant velocity', that is, affinely connected to the proper time of the curve.

Answer 2

What was hidden in our derivation of (3.47) was that the demand that the tangent vector be parallel transported actually constrains the parameterization of the curve, specifically to one related to the proper time by (3.58).

For a given initial condition and initial tangent vector, for the Christoffel connection, (3.47) gives you the geodesic. But it doesn't give you all the ways you can parameterise this geodesic. The parameterisations given by (3.47) are all necessarily related to proper time linearly. If you wanted an arbitrary parameterisation, you could transform the curve you got from (3.47), $x^\mu(\lambda)$ to $x^\mu(\alpha)$, where $\alpha$ non-linearly related to $\lambda$. But this parameterisation of the curve, you couldn't have got from the definition given in block quote.

That the tangent vector is parallel transported can only be seen in some kinds of parameterisations manifestly. This is what I understand.