zonde said:
If you view it as a physical question then sure there are no such a thing as physical singularity. So the question is mathematical.
Well, it is the physical aspects that provide the necessary insight, while the mathematics describes them. If we define a singularity as a place with infinite acceleration, then that would lie at a point mass singularity with Newtonian gravity, for example, the place where the acceleration of gravity would become infinite. In GR, it would lie at the event horizon at r = 2m with that definition, although some might still consider the point at r = 0 to be the singularity, depending upon whether one defines a singularity as infinite acceleration or the central gravitational point that all mass must fall to internally, I would say the former, although the coordinate system can also be changed from that of Schwarzschild so that the event horizon can be shrunken to a point and the interior coordinates are no longer accounted for, leaving only the exterior coordinate system that external observers can physically measure with a singular point mass at the center, which I personally prefer.
So with Rindler, going on the definition of infinite acceleration, if a photon beyond a certain distance cannot ever catch up to an observer with a finite constant proper acceleration, then a massive particle at that distance with a speed of less than c will certainly never catch up either. At x > 0, a photon could catch up within finite time. At x = 0, the photon would not catch up or would just catch up after infinite time. But since the speed right at x = 0 must be c to catch up just at infinite time, and since a massive particle must infinitely accelerate to get to a speed of c, then infinite proper acceleration must be applied to the massive particle at x = 0 to catch up to the accelerating observer, so that point where infinite acceleration must be applied could be considered the "singularity" in a sense.
But while x < 0 can always be plotted from the perspective of an inertial frame, it cannot in the accelerating frame. Let's say we have a ship where the accelerating observer is at the front, with the front accelerating with some finite constant proper acceleration and the back of the ship lies at the Rindler horizon in relation to that proper acceleration at the front. In order for the ship to maintain Born rigidity, so that if all parts of it were to stop accelerating at some point, different parts at different times depending on the inertial frame of observation due to relativity of simultaneity and the different instantaneous speeds at different times along the ship, such that it will have been contracted correctly according to SR upon reaching the final frame and becoming inertial again, then we will have different constant proper accelerations applied all along the length of the ship. But in order for the back of the ship to continue to remain caught up with the front, since the back lies at the Rindler horizon, it must infinitely accelerate.
In other words, the ship must remain intact in order for the ship's coordinate system to also remain intact and applicable. But if the back of the ship lies beyond the Rindler horizon, it will not remain intact, but will break up, since it would require greater than infinite acceleration at points x < 0 to remain caught up with the front. So those points in the ship's coordinate system are no longer applicable, as they will no longer be attached to the rest of the ship, but rather will have detached from the ship and its overall coordinate system. I suppose the difference between a physical singularity and a coordinate singularity, then, would be that with a physical singularity, infinite acceleration acts upon a particle naturally as with gravity, but with a coordinate singularity, infinite acceleration must be applied to the particle in order for the coordinate system to remain intact and include the particle, but does not naturally occur.