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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.

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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.

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  • $\begingroup$ Their 4-velocity should be a null vector, but not identically zero; what would be meant by intrinsic 4-acceleration of zero? $\endgroup$
    – user142523
    Commented Aug 7, 2019 at 16:15

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