Null geodesics and affine parameters

I've been away for a while preparing my IELTS test and I hope I'll be ok, whatever my fate is! So, concerning this entry, it came to mind this Physics SE answer I wrote some time ago, where the original poster commented he was only concerned with proper time. In turn I posted this link to another Physics SE answer which deals with what an affine parameter is.

So, the thing is pretty straightforward: the definition of proper time is just the arc length
$$\begin{equation}\tau=\int\sqrt{-ds^2}\end{equation}$$ and for null geodesics, (by definition) $ds^2=0$, so that's why "proper time assigns the same value to all points (on a null) geodesic". For spacelike and timelike geodesics, $ds^2\neq0$ and $\dot{s}^2=\pm1$ when proper time is the parameter.

Now, recall the geodesic equations for a curve $x=x(\lambda)$ on an arbitrary parameter $\lambda$ with a tangent vector $U\equiv\frac{dx}{d\lambda}$,
$$\begin{equation}\left(\frac{dU}{d\lambda}\right)^\alpha=\frac{d^2x^\alpha}{d\lambda^2}+\frac{dx^\beta}{d\lambda}\frac{dx^\mu}{d\lambda}{\Gamma^\alpha}_{\mu\beta}=0\end{equation}$$ usually written shorthand as
$$\begin{equation}(\nabla_{U}{U})^\alpha\equiv{U}^\beta\nabla_\beta{U}^\alpha=0\label{dagger}\end{equation}$$
So, let's now consider another arbitrary parameter ${\sigma=\sigma(\lambda)}$. Then,
$$\begin{align}\frac{d}{d\lambda}&=\frac{d\sigma}{d\lambda}\frac{d}{d\sigma}\nonumber\\[0.2in] \frac{d^2}{d\lambda^2}&=\left(\frac{d\sigma}{d\lambda}\frac{d}{d\sigma}\right)\left(\frac{d\sigma}{d\lambda}\frac{d}{d\sigma}\right)\nonumber\\&=\frac{d\sigma}{d\lambda}\left[\frac{d}{d\sigma}\left(\frac{d\sigma}{d\lambda}\right)\frac{d}{d\sigma}+\frac{d\sigma}{d\lambda}\frac{d^2}{d\sigma^2}\right]\nonumber\\&=\frac{d\sigma}{d\lambda}\left[\frac{\left(\frac{d^2\sigma}{d\lambda^2}\right)}{\frac{d\sigma}{d\lambda}}\frac{d}{d\sigma}+\frac{d\sigma}{d\lambda}\frac{d^2}{d\sigma^2}\right]\end{align}$$ and the geodesic equations on $\sigma$ are
$$\begin{equation}\frac{\left(\frac{d^2\sigma}{d\lambda^2}\right)}{\frac{d\sigma}{d\lambda}}\frac{dx^\alpha}{d\sigma}+\frac{d\sigma}{d\lambda}\frac{d^2x^\alpha}{d\sigma^2}+\frac{d\sigma}{d\lambda}\frac{dx^\beta}{d\sigma}\frac{dx^\mu}{d\sigma}{\Gamma^\alpha}_{\mu\beta}=0\end{equation}$$ i.e., from ($\ref{dagger}$),
$$\begin{equation}U^\beta\nabla_\beta{U}^\alpha=-\frac{\left(\frac{d^2\sigma}{d\lambda^2}\right)}{\left(\frac{d\sigma}{d\lambda}\right)^2}U^\alpha\end{equation}$$ so that from the start, in principle, one could've defined the geodesic equations as ${\nabla_UU\propto{U}}$; however, if
$$\begin{equation}\frac{d^2\sigma}{d\lambda^2}=0\,\,\Longrightarrow\,\,\sigma=a\lambda+b\label{ddagger}\end{equation}$$ ($\ref{dagger}$) is recovered. These $\sigma$'s are called affine parameters, and this is why they have this simple form.

Now, (I think) this is where trouble comes. Some authors, e.g. Carroll, define an affine parameter as any parameter related to the proper time $\tau$ as $\sigma=a\tau+b$. The thing is that also, usually it is said that one should use an affine parameter for null geodesics (as in the previous Physics SE answer I mentioned before), but still with this last definition you end up with a constant parameter. So what is generally understood as an affine parameter is simply that with which ($\ref{dagger}$) is satisfied. For null geodesics, ($\ref{dagger}$) is trivially satisfied with $\tau$ (or any linear combination whatsoever) so that this parameter is indeed useless; however one is free to use any affine parameter as defined in ($\ref{ddagger}$) with non-constant $\lambda\in\mathbb{R}$ regardless of its physical relevance.

Now, the term $\nabla_{U}{U}=\frac{dU}{d\lambda}=\frac{d^2x}{d\lambda^2}$ is indeed the acceleration of an observer along the curve $x=x(\lambda)$, so that when using an affine parameter, the observer is not accelerating, while au contraire, with a non-affine parameter, the observer will be accelerated parallel to the direction of movement. This is the main relevance of this sort of parameter; in General Relativity it's the custom to deal with affine parameters where $\tau$, of course, is our favorite one for timelike geodesics, whereas with null geodesics people usually don't care what the parameter is as long as it works, at the end it's all about a goofy parameter ;-)