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For time-like geodesic, the affine parameter is the proper time $\tau$ or its linear transform, and the geodesic equation is

$$\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma_{\rho\sigma}^{\mu}\frac{dx^{\rho}}{d\tau}\frac{dx^{\sigma}}{d\tau}=0. $$

But proper time $\Delta\tau=0$ for null paths, so what the physical meaning of is the affine parameter for null geodesic?

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Spacetime is locally flat, and in any flat space you have parallelism. This structure of parallels is completely independent of whether you think of the points as representing points in relativistic spacetime, points in Newtonian spacetime, or points in Euclidean space. It could even be a space such as a graph of temperature versus time. Once you've got a notion of parallelism, you are automatically able to construct a system of measurement along any given line. You can see the construction worked out here: – Ben Crowell Nov 27 '11 at 15:00

If you forget about the affine-ness for a moment: you can parametrize a null geodesic in any way you want. Actually, you can parametrize any geodesic (heck, even any curve) in any way you want; all you need is a monotonic function that maps points on the geodesic to unique values of the parameter. But for timelike geodesics, you almost always use the proper time because it's a nice, sensible physical quantity that also happens to work as a parameter.

With null geodesics, you don't have the proper time as an option because the proper time mapping assigns the same value to all points on the geodesic. So you have to pick some other parametrization. In principle, again, it can be any monotonic function that maps points on the geodesic to unique values of the parameter.

However, it's possible to pick a way to parametrize the null geodesic in a way that is "sensible" in the same way that proper time is "sensible" for a timelike geodesic. This is called an affine parameter. In particular, one way to define an affine parameter is that it satisfies the geodesic equation. (Note: the geodesic equation does not work for just any arbitrary parametrization of a geodesic. You have to use an affine parameter.) Another way is to say that iff the parametrization is affine, parallel transport preserves the tangent vector, as Wikipedia does. Another way is to say that the acceleration is perpendicular to the velocity given an affine parameter, as Ron did. All these definitions are equivalent.

It turns out, although I don't know the details of a proof, that there is a unique affine parameter for any geodesic, up to transformations of the form $t \to at+b$.

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The property you mentioned in the last paragraph is what affine means in mathematics. – timur Oct 31 '14 at 13:31

Following David Z's answer, the proof for the last paragraph is:

  • since $t$ is an affine parameter it satisfies: \begin{equation} \frac{d^2x^a}{dt^2}-\Gamma^a_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}=0 \tag1 \end{equation}

  • the parameter $t'$ must be related in some way to $t$, that is: $$t'=t'(t) \tag2$$

  • use the chain rule to get:

$$\frac{dt'}{dt}\frac{d}{dt'}\left(\frac{dt'}{dt}\frac{dx^a}{dt'}\right)-\Gamma^a_{bc}\left(\frac{dt'}{dt}\frac{dx^b}{dt'}\right)\left(\frac{dt'}{dt}\frac{dx^c}{dt'}\right)=0 \\ \Leftrightarrow \left(\frac{dt'}{dt}\right)^2\frac{d^2x^a}{dt'^2}+\frac{d^2t'}{dt^2}\frac{dx^a}{dt'}-\Gamma^a_{bc}\left(\frac{dt'}{dt}\right)^2\frac{dx^b}{dt'}\frac{dx^c}{dt'}=0 \tag3 $$

  • the affine parameter $t'$ must also satisfy: $$\frac{d^2x^a}{dt'^2}-\Gamma^a_{bc}\frac{dx^b}{dt'}\frac{dx^c}{dt'}=0 \tag4$$

  • comparing $(3)$ and $(4)$ we get the conditions that: $$ \left(\frac{dt'}{dt}\right)^2\neq0 \wedge \frac{d^2t'}{dt^2}=0 \tag5 $$

  • which yields $t'=\alpha t+\beta$ with $\alpha \neq0$

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An affine parameter is one which makes the acceleration perpendicular to the velocity. It is the limit of proper time as you take the limit that the geodesic becomes null, rescaled to make the total length stay finite.

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