What is the velocity of a photon through space-time? What is the 4-velocity of a photon? What is the velocity of a photon through space-time?  What is the 4-velocity of a photon?
 A: The problem with the four velocity for anything travelling at the speed of light (i.e. any massless particle) is that we define the four velocity using the proper time $\tau$. If we choose some coordinates $(t,x,y,z)$ then the four velocity in our coordinates is:
$$ \mathbf U = \left(\frac{dt}{d\tau}, \frac{dx}{d\tau}, \frac{dy}{d\tau}, \frac{dz}{d\tau}\right) $$
The problem is that the proper time $\tau$ is the elapsed time in the rest frame of the moving object, and photons don't have a rest frame so the proper time is not defined. That means the four velocity is not defined either.
There are some workarounds. The four velocity is the tangent vector to a world line, and null or indeed spacelike world lines still have tangent vectors. But we'd have to write the four velocity using some affine parameter not the proper time and this isn't terribly useful.
Alternatively you could consider the behaviour as the speed tends towards $c$. You find that the norm of the four velocity stays equal to $c$ but the components of the four vector tend towards infinity.
