tl;dr (and mathlessly) Colloquially, you'd hold something other than the photon stationary and consider the light's path in that stationary observer's frame, an

tl;dr (and mathlessly) Colloquially, you'd hold something other than the photon stationary and consider the light's path in that stationary observer's frame, and then it doesn't matter whether proper time is rigorously defined. For a handful of others in this thread: While it's strictly true that proper time is hard to define usefully for a classical massless particle, you can still parameterize a null curve any way you want, depending on what it is you're trying to do. In flat spacetime you're rarely trying to eke out a transformation much more complex than a Lorentz transform (the matrix for which involves \gamma, which is undefined for v=c), and so you're stuck with ds = 0. On the other hand, your objects on lightlike curves are probably light itself (classical or otherwise), so you'd generally look to something like an affine parameterization like k^b\Delta_{b}k^a=0 and k_{a}k^a=0 where k^a(x) are the components of the photon's momentum, or (aiming for E=pc) parameterize on the wave vector k (where v=c implies k^2={\omega}^2, where \omega is the wave angular frequency). Other sensible choices of affine parameter are available, and typically give you freedom to pick out a momentum vector k^\mu = \dot A^\mu where A is the affine time; usually you'd do this so as to match the momentum-energy at the emitter or absorber (or the redshift along the null curve).