In V. Faraoni's textbook "Special relativity", on p. 173 we find the statement
"A parameter $\lambda$ such that $$\frac{d^2x^\mu}{d\lambda^2}=0$$ is called an affine parameter."
(Here $x^\mu=x^\mu(\lambda)$ are the parametric equations of the null curve)
This would mean that null geodesics have zero acceleration. Is this always true; or is it true only in some specific coordinates? Or is it true in flat spacetime only.
Geodesics (by definition) have an intrinsic 4-acceleration zero. However, when expressed in terms of coordinates, the coordinate acceleration $\mathrm{d}^2 x^{i}/\mathrm{d} t^{2}$ can very easily be non-zero, and the coordinate velocity $ \mathrm{d} x^i/\mathrm{d}t$ can behave unexpectedly.