That statement is indeed a bit vague. Let me elaborate: In the process of deriving the said equations, an equation for the radial coordinate of the photon was

That statement is indeed a bit vague. Let me elaborate: In the process of deriving the said equations, an equation for the radial coordinate of the photon was achieved. This was identified with a classical, Newtonian system of one particle with unity mass. As the real, massless photon lives in four-dimensional spacetime and the said massive "test particle" lives in three-dimensional space, these systems just can't be dynamically the same (in spacetime, the massive particle would take a timelike curve). To reword the statement, the derived "equation of motion" will yield the same trajectory in the spacelike components (x, y, z), but possibly with a different parametrization - in the classical system, we're integrating the equations of motion with respect to the time. However, this has nothing to do with the "coordinate time" of the four-vectors nor the proper time of the particle (for the photon, proper time doesn't even make sense). Hope this helps! You could also see [0] for an alternative take on this derivation. I will try and clarify my article a bit as well. [0] http://rantonels.github.io/starless/ http://rantonels.github.io/starless/