Is there a hyperplane hyperbolic orthogonal to a light-like curve? In Special Relativity, a hyperplane that is hyperbolic orthogonal to a worldline at a point p can be viewed as the simultaneity plane with respect to the worldline at p. What if, however, the worldline coincides with a light-like path? What is a hyperbolic orthogonal hyperplane of a light-like curve? Should we say that there is no hyperplane hyperbolic orthogonal to a light-like curve? Or, rather, should we say that the light-light curve is hyperbolic orthogonal to itself?
 A: In 1+1 dimensions, it's true that the direction orthogonal to the worldline (the particle's space direction) merges with its time direction once the speed becomes $c$ (a.k.a. the time direction becomes "null").  So you might be justified in saying that the (one-dimensional!) orthogonal hyperplane has "disappeared".
But if we have more than one spatial dimension, we can still find other directions that are orthogonal to that null direction.  Let's say we're in a frame where the worldline is along the $+x$ direction, so the 4-velocity, with $c$ normalized to 1, is $v^{\mu} = (1,1,0,0)$.  Let an orthogonal direction be $u^{\mu} = (a,b,c,d)$.   Then all we need for orthogonality, using the Minkowski metric $\eta_{\mu\nu}$ with signature (-+++) is
$\eta_{\mu\nu}v^{\mu}u^{\nu} = -1\cdot a + 1\cdot b + 0\cdot c + 0\cdot d = 0 \\ \rightarrow -a + b = 0 \\ \rightarrow a = b$
So our set of orthogonal directions becomes all 4-vectors of the form $(a, a, c, d)$.  In other words, it is still 3-dimensional, still a hyperplane.  The only thing is, it now includes the 4-velocity itself.
But perhaps more importantly, the description in terms of simultaneity kind of gets lost.  That's because, in general, the simultaneity of all events in the orthogonal hyperplane is as perceived by the particle, but the null worldline has a proper time of zero (which is why it's called null).  So if the speed-of-light particle experiences no proper time anyway, how can we say it would perceive any set of events to "occur at the same time?"  If anything, all events in spacetime occur at a single instant!  But that's more a matter of interpretation.
