Relativity when you divide a trip into small time intervals

1. Jun 9, 2012

1MileCrash

Disclaimer: I am still learning physics. I think about weird things. This results in thoughts like the one that follows.

My understanding of planck length is that it is the smallest increment of length that can be measured, and to me this implies that an object can only be a whole number of planck lengths away from another.

Planck time, as I understand it, is the smallest possible increment of time in which any "event" can actually occur. Since it is derived from the time it takes moving at the fastest possible speed across the smallest possible distance, this makes perfect sense.

This also means that 1 planck length per 1 planck time = motion at the speed of light.

Does this imply that when any object moves at a speed less than C, it is really spending a fraction of this motion completely stationary?

If I watch a ball moving at 1/2 the speed of light, noting how many planck lengths it moves per planck time, the only result that makes sense with the above given is that during half of the planck time increments, it moves one planck length, and for the other half of time intervals, it doesn't move at all.

Since I couldn't observe it moving any fraction of a planck-length, and I couldn't observe anything happen in a fraction of a planck time, my only conclusion is that in this *thought experiement* is that I would observe it half the time as moving at the speed of light, and half the time not moving at all. In other words, I couldn't observe it moving one-half of a planck length per planck time.

Thus, to make my title clear, this means to me, that if something is actually moving, it is moving at C. Any speed witnessed less than C is an average speed based on how much time is spent stationary or spent moving at C - the only two possibilities.

But here's where I get a bit confused.

If an object is traveling at the 50% lightspeed relative to me, and I am Chuck Norris, I should be able to divide the journey into planck time intervals and see that half the time, it is stationary, and half the time, it is moving at light speed. However, this would also suggest that half the time, it experiences no elapsed time relative to me, and half the time, it experiences the same amount of elapsed time relative to me, suggesting that in total, it experiences half of the elapsed time relative to me. This is of course not the case, as time only elapses 1.154 times quicker at that speed for the traveler relative to me.

So why is it when I divide the trip this way, the relativistic effects don't make sense?

2. Jun 9, 2012

Muphrid

Relativity really isn't meant to "work" with discretization of the underlying space. The problems with doing so are why a theory of quantum gravity elude us so.

Nevertheless, you're correct to say that any object's velocity is $c$ at all times. This exactly gets at the idea of the four-velocity, which is usually denoted $u$. An object traveling with ordinary (or "three-") velocity $v = \beta c$ in the $e_x$ direction has four-velocity

$$u = \gamma c (e_t + \beta e_x)$$

When you take the magnitude of this vector, you always get $c$, regardless of $\beta$ (note that $\gamma = (1-\beta^2)^{-1/2}$, which is why the normalization stays fixed).

3. Jun 9, 2012