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I consider Minkowski space $M$ in this question. My question is about the following. For lightlike worldlines we can define the geodesic equation as follows:

$$\nabla_TT=0,$$

where $$T = \gamma^\prime(s)$$ with $s$ an affine parameter and $T$ the tangent vector field to the smooth curve $\gamma$ in $M$ and $\nabla$ denoting the covariant derivative. Since we want $g(T,T)$ to be constant (with $g$ the metric tensor) for an affine parameter we choose $s=\tau$ for lightlike worldlines. Using the convention $ g= diag(-1,1,1,1)$ we then see that we indeed find that the 4-velocity is:

$$u = \gamma^\prime(\tau) = \frac{dx^\alpha}{d\tau}.$$

Such that we get $$g(u,u)=-1$$ so indeed we have that $\tau$ is an affine parameter describing the curve. But for null worldlines we do not choose $\tau$ as our affine parameter but we keep it abstract and state it as $s$ or $\lambda$ which is more common notation. If I write out the definition of the proper time:

$$d\tau^2 =-ds^2$$

we see that $d\tau = 0$ for null worldlines and therefore we get $\tau=0$ for every path in spacetime. Is this the reason why people do not use the proper time as an affine parameter since it is always zero independent of the path and that the 4-velocity is ill-defined since $d\tau=0$? But why do we need another parameter $\lambda$ mathematically, could someone make this more rigorous?

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    $\begingroup$ Yes, you're right. Or if you want a physical reason: the proper time experienced by light is zero, all along the null geodesic. But in order to uniquely label each point on a geodesic, we need a parameter that varies all along the geodesic path. See this: physics.stackexchange.com/q/17509/133418 $\endgroup$ – Avantgarde Oct 4 '18 at 19:41
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    $\begingroup$ Consider to back up the different conventions for the words null & light-like by references. $\endgroup$ – Qmechanic Oct 5 '18 at 17:18

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