One principle in general relativity is that the wordlines of massless particles are null geodesics. It also seem to be a commonly stated fact (for instance see eq. (3.62) in Section 3.4 of Carroll's GR text) that one can parameterize the worldline of a massless particle as $x(\lambda)$ such that $$\frac{dx^{\mu}}{d\lambda} = p^{\mu}, \tag{1}$$ where $p^{\mu}$ is the four-momentum of the massless particle. To clarify what is meant by four-momentum, the energy of a particle with four-momentum $p$ as measured by an observer moving with four-velocity $U$ is $-p_\mu U^\mu$.
Now I understand why one could pick a parameterization that satisfies (1) at a particular point. But why must such a parameterization satisfy (1) globally?
The only argument I have is that such a parameterization always exists for the world-line of a particle with mass, and taking the limit to light speed suggests that such a parameterization should also hold for the world-lines of massless particles.
My question is: what are some other arguments for why the parameterization should exist?
