Consider a space ship, undergoing constant acceleration (which for our purposes means that the same amount of energy is being used per second to increase its speed). According to special relativity the ship will accelerate but in such a way that it will approach the speed of light $c$, but never reach the speed itself or cross it.
This means however that the more time we spend accelerating the space ship, the closer and closer it will get to the speed of light.
Which basically amounts to: our knowledge, or accuracy of our velocity is going to steadily go up.
Heisenberg's Uncertainty principle states that there is limit to how accurately we can know our velocity before we need to give up some information about our position?
What does this mean in our context? Say we reach the critical threshold where we are accurate to some 35+ significant digits aka 0.999999999999999999999999999999999999999999999999999999999999999999c is our speed.
We need to lose accuracy of our position between our start and endpoint (assuming this trip has a definite endpoint)?
Whats happening? Is the ship suddenly making jumps to random locations? Is it spreading out like a wave, where we the faster we go, the less likely we are to know when we reached our endpoint?
I'm very very curious.
EXTENSION: Additional Thought
Consider an object in circular orbit around a black hole, outside of the event horizon (the black hole itself is stationary) and these are the only 2 systems present, with the object in question having very negligible mass compared to the black hole.
As the object is brought closer and closer to the black hole, the centripetal force it experiences, obviously goes up, causing to orbit at a faster speed. The closer you bring it towards the center of the black hole, the faster the object will orbit, allowing you to bring it arbitrary close to value of C. According to heisenberg as the velocity increases to an accuracy beyond the reduced planck's constant (0.9999...)c the position of the object becomes increasingly unknown. It would start to "smear" out in a circle around the orbit, transforming into a haze which can be observed.
Or So I think... Is this correct intuition?