Why c Speed Limit Doesn't Prove Space is Continuous?

[Andrea Panza]

Hi everyone,

First of all thank you for all the amazing amounts of information on this forum!

I have a very stupid question, which is probably due to a deep misunderstanding about space quantization.

I was wandering why the fact that no mass could move at the speed of light is not per se a proof that space is not quantized.

I’ll try to clarify my idea through this thought experiment:-Imagine a source that at t0 emits both a photon (that obviously moves at c) and a particle that moves at 0.99c

-at t=100 the photon would have covered 100 units of space so x=100, while the particle would have covered only 99 units of space so x =99. This is true both in the case of quantized space or continuous space.

-at t = 1, if the space is continuous the photon will always be ahead of the particle (the photon will be at x =1, while the particle will be at x=0.99). If the space is quantized (with x being the minimal unit of space) at t = 1 both the photon and the particle should be at x =1, which is impossible.This leaves me to 2 possibilities:

a) I’m smarter than people that dedicated their life and professional career to the study of physics and with this simple trick I demonstrated that space is not quantized.

b) I’m wrong, somewhere along my way of reasoningDespite my ego would choose “a” I suspect the true answer is “b” so please help me because this is driving me nuts!

Thanks in advance for your help!

 

[Mentz114]

If you quantize time as well as space then at t=0 they are both at 0, but at t=1 the matter will still be at 0.

 

[Isaac0427]

If I understand this correctly, I have an analogy to what the last comment said:

Say you have a clock and I have a clock (both are analogue). Your clock (the electron in this case) is slow and only shows 59.9 seconds have passed during a time span of 60 seconds. The minute hand only moves when it hits 60 seconds, not at 59.9 seconds. So, if we both start our clocks at the same time as s=0, when it is s=60 my minute hand will have moved and yours will not have.

This is not a stupid question (you should see some of mine).

 

[newjerseyrunner]

I you observe your photon traveling 100 units and your other object moving 99 units, did it really move 99 units? The object thinks it only traveled 14 units. Remember, the faster you are moving, the shorter space gets: http://en.wikipedia.org/wiki/Length_contraction

 

[newjerseyrunner]

I you observe your photon traveling 100 units and your other object moving 99 units, did it really move 99 units? The object thinks it only traveled 14 units. Remember, the faster you are moving, the shorter space gets: http://en.wikipedia.org/wiki/Length_contraction

EDIT: In response to your followup, the object and observer will also disagree about how much time has passed. If an observer sees an object traveling for 60 seconds at .99c, the object has only experienced 8 seconds.

 

[Andrea Panza]

Thank you for your answers, but... ehm... I'm not sure I got them right.
I was considering an inertial system where all the acceleration to bring the particle at .99 c ends at t0 when the photon is emitted.
If my detector is at x=1, at t=1 (in my frame of reference which is stationary with the emitter) I will detect the photon, but I will not detect the particle.
Since in my frame of reference x has the minimal possible length there should be no other possible 'space' states between x= 0 and x = 1, so if I don't detect the particle at x=1 it means that the particle should still be at x=0 (and in my understanding this can be explained with the particle time dilation because in the frame of reference of the particle 'it takes more time -with respect of my frame of reference- to complete a unit of time and change its position).
However if I know that the particle at t=0 was at x = 0 and at t = 1 is still at x=0 (this is what I observe in my frame of reference) and I know that no other forces were applied to the particle after t=0 (the particle acceleration is stopped at t=0) my conclusion is that the speed of the particle must be 0 and not .99 c

Again I got something wrong...

 

[Mentz114]

Thank you for your answers, but... ehm... I'm not sure I got them right.
I was considering an inertial system where all the acceleration to bring the particle at .99 c ends at t0 when the photon is emitted.
If my detector is at x=1, at t=1 (in my frame of reference which is stationary with the emitter) I will detect the photon, but I will not detect the particle.
Since in my frame of reference x has the minimal possible length there should be no other possible 'space' states between x= 0 and x = 1, so if I don't detect the particle at x=1 it means that the particle should still be at x=0 (and in my understanding this can be explained with the particle time dilation because in the frame of reference of the particle 'it takes more time -with respect of my frame of reference- to complete a unit of time and change its position).
However if I know that the particle at t=0 was at x = 0 and at t = 1 is still at x=0 (this is what I observe in my frame of reference) and I know that no other forces were applied to the particle after t=0 (the particle acceleration is stopped at t=0) my conclusion is that the speed of the particle must be 0 and not .99 c

Again I got something wrong...

The situation you are trying to describe is absurd, so expect trouble. You can't build a particle detector on those scales, even in principle. So as a thought experiment it falls down.

 

[wabbit]

Maybe I am misunderstanding this but it seems to me in your setup you are using classical notions of position and velocity instead of operators, so it isn't clear what conclusions can be reached. For instance the velocity of the particle will be wildly indefinite if its position is determined to within the minimum length, and since the time interval here is also the minimum interval, location in time to such precision involves a very high energy interaction.

 

[andresB]

Will there even be "particles" at those scales?

 

[PeterDonis]

if I know that the particle at t=0 was at x = 0 and at t = 1 is still at x=0 (this is what I observe in my frame of reference) and I know that no other forces were applied to the particle after t=0 (the particle acceleration is stopped at t=0) my conclusion is that the speed of the particle must be 0 and not .99 c

No, your conclusion should be that you need to look at more than one quantized unit of time to measure the speed of an object that goes slower than light. After 100 units of time, the particle will be at x = 99, not x = 0. In other words, for every 100 units of time, the particle moves by 99 units of space, hence its speed is 0.99 (vs. a speed of 1 for the photon).

To put it another way, if space and time are quantized, then the concept of "speed" as you're used to using it can only apply over scales much larger than the minimum unit of space and time. Over small enough scales, it simply is not possible to distinguish a speed of 1 from a speed of 0.99 if space and time are quantized.

 

[Ibix]

That reminds me of movement in some tabletop wargames based on square or hexagonal grids. If you were moving at speed 32 you got to move every turn; if you were moving at speed 31 you moved every time except the thirty-second turn; speed 16 moved every other turn; speed 1 moved every thirty second turn (and was only for use if you enjoyed being used for target practice).

This required that each unit remember its speed relative to the grid and to keep track of when its (notional) movement had accumulated to the point that it could move one step on the grid. In this context, that means particles that can do arithmetic, or else some kind of stochastic process. That's rather a different concept from the GR concept of velocity, I think.

 

[PeterDonis]

That's rather a different concept from the GR concept of velocity, I think.

Yes, certainly. GR models spacetime as a continuum.