# Is there a smallest distance?

## Main Question or Discussion Point

My friend and I stayed up one night when we were in junior high school, and we were discussing science, and he asked me if I knew if there was a smallest amount of distance that an object could move.

I was really into computer graphics (1980s-90s), so I said space might be continuous, or there might be increments (like pixels or quanta) that objects use to move in little jumps, if you will.

Later I pointed out that most small constituents are vibrating anway, and only slow to a stop at a temperature of Absolute 0. However, this is sort of an aside answer, not a real answer to the stated question.

Anyone know or care to offer their thoughts? I hope this is not another overly used thread topic...

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Chronos
Gold Member
Try googling on the Heisenberg uncertainty principle.

f-h
Here is a little homework, using the uncertainty principle calculate how strongly you can localize an object before it turns into a black hole.

In response to the uncertainty principle response: what if the moving object is macroscopic and not a single particle? Would there still be uncertainty in its position and momentum when trying to determine the continuity of its motion?

f-h: I did some quick Internet research and came across a paper with the sentence: It is commonly believed that quantum gravity implies the existence of a minimum length [7]. This is in the third paragraph after the abstract on http://www.iop.org/EJ/article/0264-9381/20/15/101/q315l1.html" [Broken].

Am I getting warmer? From the article I gleaned that the minimum length delta x sub min = 2 (alpha')(Planck's length).

However, this alone does not clear up for me enough whether or not motion is continuous and whether we live on a grid, like graph paper, of space.
Planck's length 1
-------------
..------------- (dots are simply placeholders)
Planck's length 2
Above, we can see that there is still locations in between two points a Planck's length apart.

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f-h
Just do the calculation yourself it's straightforward highschool algebra!

As for the second part of your question I think the question is insufficiently precise to have a well defined answer.

For example: Basic quantummechanics says a particle can have spin up or down, but it can "move" between these states continuously in some sense (the expectation value can). Now in a theory of QG spacetime intervals required for operationally defining velocities should be QM measurements. How to correctly interpret these is a subtle and open question in general.

Chronos
Gold Member
But it does make for a great ATM argument. Shifting states instantaneously looks a lot like spooky action at a distance - assuming you make a few bold assumptions.

Although I don't know the value of alpha', I can see that it is on the scale of a Planck's length. Isn't that the length of a string? Therefore, how does motion occur, one string length at a time?

I should have entitled the thread: "Is there a shortest distance for motion?"

I talked to my friend again over the phone just now. He said his question had to do with something he read in Discover magazine back in the 1980s or early 1990s about the smallest amount of volume that space can be. He read something about loops and some sort of loop theory of space, and he wasn't sure what was outside the individual loop (inbetween loops, I suppose) -- as it would not be space, one would infer.

Any thoughts on this?

o| Hiram

f-h
This is Loop Quantum Gravity, please use the search function on the forum and you will find a wealth of information on this.

josh1
My friend and I stayed up one night...
Oops, I thought you we`re going in a different direction.:surprised

mjsd
Homework Helper
If you start talking about strings and branes, it is perhaps difficult to visualise what is going on in 10 dimensions, let alone "distance" they propagate. I don't really know how to properly answer your original question. But I can make these remarks: Physics as we know it is only good up to a certain point; in terms of length/energy scale, Planck length is the limit, and that's why we need a new theory... string theory is one of them. Strings and branes propagate and move on the Worldsheet and how would that correspond to physical movement in 3+1 dimensions.... can be a bit tricky to visualise. I am sure a string expert can answer that much better than I can.

msjd et al.: Thanks for a valiant effort. I had read 10 chapters from The Elegant Universe and I didn't know how the Planck's length applied to the smallest increment of motion of a meter-or-so sized body. I guess it would be difficult to verify anyway.

Chronos
Gold Member
Keep in mind the HUP limits our ability to measure, it does not forbid the universe from acting on scales beyond the reach of our instruments. From a scientific viewpoint, actions beyond observational limits complicate our efforts to do 'science'. That does not mean they are impossible, or necessarily irrelevant. We have taken some leaps of faith in cosmology that incorporate unobserved aspects of the universe into theory - e.g., dark matter. If DM resides in hidden dimensions, as suggested by ST, we may never be able to detect it by means other than gravitational effects.

-The question is, if there is a minimum amount of space then why can't we have Quantum Gravity??, since the Feynmann Path-integrals would become just ordinary multi-dimensional integrals that could be evaluated by Numerical methods to calculate propagators without recalling to (divergent) perturbation theory.

-Another Question is how do you define 'Geommetry' in a discrete space-time?, since Riemann only worked with continous surfaces so $$g_{ab}dx^{a}dx^{b}$$ were just the line element, although they have mentioned uncertainty principle to show that there's a minimum distance (could anyone provide the demonstration please??) i think this fact should come up from the theory when 'Quantizying' the surface.

-It really surprises me (not in bad sense) how this question arose to 'eehiram' and his friend using computers ,since there's hardly a connection between informatics (in classical sense) and Gravity, ain't this amazing??

Chronos
Gold Member
Your argument is fascinating. I'm curious why you think Riemann failed to consider, or mention this potential flaw in his argument.

Gib Z
Homework Helper
Doesn't HUP and QM intrigue you >.< Apparently stars that we can't see are in a superposition of states where it is in nuclear fusion and where it isnt >.< When Observed HUP applies and thats why the Sun can do it...

Chronos