I'm watching this video (the link starts at the relevant timestamp) and am confused about one thing. Let's say I'm going at a constant speed of $0.99c$ in $+x$ direction (w.r.t Earth), and a pulse of light is also moving in the same direction. Of course, I'd see the pulse moving away from me at $c$, since I'm in an inertial frame.
But let's say I'm accelerating with some constant acceleration $a$ - my own speed will approach $c$ but never reach it, but in this scenario, how would I perceive the pulse of light? Will I still see it as moving away from me at speed $c$?
For me, can I say that for an infinitesimal duration $dt$ (duration measured in my frame), I'm in one inertial reference frame (one that's going at $0.99c$ w.r.t. Earth), then the next infinitesimal duration, I'm in the "next" inertial frame (one that's going at $0.99c+a\ dt'$ w.r.t Earth - where $dt'$ is the Lorentz transformed version of $dt$, i.e. $dt'$ is the same duration measured w.r.t. Earth), and so on? In each of those infinitesimal durations, since I'm in an inertial frame, the light still recedes a $c\ dt$ distance away from me. So that will make me think that despite accelerating, the pulse is moving away from me at speed $c$.
What are the flaws in the above argument? And how would I actually perceive the light pulse while I'm accelerating in the same direction?
