I was having a little thought experiment about Lorentz contraction, and I couldn't really figure out what would actually happen. Note that I'm not looking for a answer 'this effect is barely noticeable'; consider I have a cat that steps on my keyboard.
Say, we are a stationary observer, and we observe a rod continuously accelerating along its axis. As far as the rod is concerned, it is accelerating uniformly.
Being a stationary observer, my guess is that we don't see it accelerating uniformly. It is exhibiting Lorentz contraction, so the rod seems to be getting shorter as it accelerates. In order for the tip and the end of the rod to end up closer to each other after some time accelerating, the end of the rod must appear to be moving faster than the tip of the rod.
Now, stuff gets weird. The end appears to be moving faster for us, stationary observers, so it must exhibit more Lorentz contraction than the tip, repeat ad nausea. The opposite holds for the tip of the rod. So, am I right in presuming that the rod will end up as such (moving left-to-right)
->) [=======-======-======-======-======] < Stationary rod (equal spacing of '-') ->) [=-==-===-====-=====] < Accelerating rod (unequal spacing of '-')
And then there is also the relativity of simultaneity... is that another factor affecting the apparent shape of the rod, or is that another way of arriving at the same result?
Alternatively, do I have all of this wrong, and am I applying all the wrong principles to the poor rod?