Geodesics: a (for me) "mysterious" property related to affine parameters.

Question

Consider a Riemannian manifold $(M,g)$ with the Levi-Civita connection $\nabla$. If $D_t$ is the covariant derivative along curves descending from $\nabla$, a geodesic is a curve $\gamma: I\subseteq\mathbb R\longrightarrow M$ such that $D_t\gamma'=0$ where $\gamma'$ is the velocity vector field along $\gamma$. In coordinates (respect the coordinare frame $\frac{\partial}{\partial x^1},\ldots\frac{\partial}{\partial x^n}$) a geodesic is a curve that satisfies the following equation(s):

$$\ddot \gamma^k(t)+ \dot\gamma^j(t)\dot\gamma^i(t)\Gamma^k_{ij}(\gamma(t))=0$$

It is evident that there aren't constraints for the parameter $t$, but on the book "Carroll - Spacetime and relativity" I read the following mysteriuous phrases:

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Notation: Here by the parametrization with the "proper time" $\tau$, he means the parametrization with the arclength parameter. The equation $3.44$ is the same that I've just written above and moreover the others cited equations deal with the variational approach to geodesics.

So I don't understand why, according to Carroll, if a curve satisfies the above equation(s), then its parametrization should be affine. The author doesn't explain this point.

Answer 1

To elaborate on the comment by @PavelC, the geodesic equation $\ddot \gamma^k(t)+ \dot\gamma^j(t)\dot\gamma^i(t)\Gamma^k_{ij}(\gamma(t))=0$ actually implies that the curve is constant speed. This is what the author seems to be saying: the parameter is necessarily affine with respect to the unit speed parametrisation. The proof of this can be found in my course notes here, Lemma 8.3.5 on page 67.

Answer 2

If $\gamma(t)$ is a geodesic then $$\frac{d}{dt}\left<\gamma'(t),\gamma'(t)\right>=2\left<D_t\gamma'(t),\gamma'(t)\right>=0,$$ thus the magnitude of the velocity field (i.e., the speed) is necessarily constant. The "proper time" parameter for a geodesic is then the parameter $\tau$ such that the constant magnitude of its velocity field is equal to 1.