Something to ponder
Here is a motion paradox of my own invention. Here we deal in the movement of information. Einstein says that information cannot be communicated faster than the speed of light. Imagine a small massless particle hurtling through space at the speed of light. This is a very small particle (like an electron) and thus cannot be said to exist at a definite position at any time. Instead, it’s position is represented as a probability field within a bounded region (remember electron clouds in high school chemistry?) Ok. So starting at the position x = 0, this particle moves a distance at the speed of light for an amount of time. Let’s say it’s a nanosecond, which is the amount of time it takes light to travel a foot. After this amount of time, the particle has traveled a foot. But since the particle exists as a probability field, there is a non-zero probability of the particle existing at position x + (w/2), where w is the width of the of bounded region where we expect to find the particle. Since there is a possibility of finding the particle at x + (w/2), we have sent information, in a non-deterministic way, faster than the speed of light. If the possibility of finding a single particle at x + (w/2) is small, we can still send information with a high degree of confidence by sending lots of particles in parallel.
Have we sent a signal faster than light?
I’ve been thinking about whether or not this is a paradox at all. It might be an argument in favor of the existence of special relativity. If we allow space to compress as the particle increases in speed, w will have a limit of 0 as v approaches c. On the other hand, if we allow w to go to zero, then it is possible at any point in time to describe the exact position of the particle, which I am pretty sure is forbidden in quantum mechanics. Things that make you go hmmm….
The Diagram: I drew the diagram above in paint shop to help visualize the problem. We have a yellow particle traveling at the speed of light. The red circle around the particle has diameter w and marks the bounds of the probability field in which the particle might exist at any particular point in time. The white lines are meant the recall the compression of space-time as any particle accelerates, though that this compression has nothing to do with the paradox as posed. The punchline is illustrated by the dotted red line to the far right, which marks the furthest distance from the starting position of the particle at which we might observe the particle. It is further away from x = 0 than should be possible for light to travel in a nanosecond.
Another possible solution is that only deterministic signalling must obey the speed of light limit. I don’t know if this is actually the case, but I wouldn’t be that surprised. This is the kind of weirdness that quantum mechanics allows.
Anyways, I’ve been thinking about paradoxes all week and I might have a couple more to post after I’ve thought about them a bit more.
