Particle disappearing from view if it exceeds c

[NanakiXIII]

I was working on a relativity problem in the context of electrodynamics and the problem was to show that a particle in hyperbolic motion (i.e. under the influence of a constant force) in one spatial dimension would never disappear from view for an observer at a position x once it was in view, providing its velocity did not exceed the speed of light. The idea was to draw some light cones on different points of the trajectory and indeed you can see that once the particle was visible at a point x at time t, that point x would lie within every light cone for later points in time.

What I don't see, however, is how the particle would ever disappear from view even if its velocity did exceed c. If you draw a trajectory for a particle with velocity 2c or 20c, you can still draw a line with slope -1 from each point on the time axis to some point on the trajectory and the light from the particle emitted at that point could still reach the observer at x=0. The light would be badly Dopplershifted and from some points you would see the particle moving backwards, but you would still see it.

My book, however, explicitly states that the particle would disappear from view if its velocity exceeds the speed of light, so apparently I'm making some error in my thinking here. In my reasoning there also seems to be an asymmetry in the situation (namely that when the particle is moving towards an observer, it seems it would be moving backwards while it would just seem Dopplershifted when moving away from an observer - this I conclude from looking at subsequent light cones I drew), which I'm not sure should be there. If anyone could point out what my error is, I'd appreciate it.

 

[Mentz114]

Just looking at the space-time diagram, I agree with you. Any signal emitted by the particle intercepts my world-line sometime in the finite-future. But there will come a time when the signal is red-shifted to zero, which is the same thing as disappearing from view (in my interpretation).

 

[NanakiXIII]

Ah, that is true. If its velocity exceeds c at some point due to the acceleration, it'll be shifted to zero as it crosses that barrier. I wonder what the meaning of imaginary red-shift would be. Thanks for your reply.