Measurement of distance in space

[Uncertainty01]

First of all I wanted that I am not a physicist. I have an introductory knowledge gained from college, tv, news, magazines etc. So that's why came here with this question. I thought maybe you could help me with.

So the question is simple. As technology has advanced we are able to measure distances at a smaller and smaller scale (e.g. nanometer). My question is when does this ever smaller and smaller scale stop? Or does it stop? For example, if you had one object that was measured to be 14.99 inches away from another object. How many more decimal places (e.g. 9's) can we add to this distance before one must stop? For some reason I believe that it could be infinite which makes me question what space really is. I guess that is good start so anyway I hope this interests someone.

 

[CompuChip]

Welcome to PF Uncertainty.

First some practical info... to measure something on scale x, you need something which is of that scale. For example, when you look through a microscope, you are sending light at an object and looking at the reflection (optically magnified). Light has a wave-length of several nanometers (visible light is approximately 300 - 800 nm). Light can reflect of things which are larger than this wavelength, but when the object is smaller the light will not reflect (an intuitive, but probably wrong, picture I have is that the wave is too stretched to actually hit the object and will just pass it). So with microscopes we can only see objects to about 300 nm. If we use, for example, X-rays, we can go a bit further. Quantummechanically, however, all particles have a wavelength (because they are simultaneously waves) called the de Broglie wavelength. The wavelength of electrons is still much smaller than that of electromagnetic radiation such as light or X-rays. This is being applied in electron microscopes, where instead of light (photons), electrons are shot at a target (and processed by a computer to produce an image), allowing us to see much smaller scales and indeed resolve separate atoms.

So currently, the scale to which we can zoom is limited by technological factors. If we want to see smaller scales, we will need smaller objects to probe them. Compare it to this: if you shoot tennis balls at a wall and look how it comes back to you, you only get rough information about the surface of the wall. But if you shoot ping pong balls, they will also recoil from smaller cracks in the surface and you will also get information about a finer structure.

Now, as for the theoretical part.
As far as we know now, we can measure positions to arbitrary precision. However, there is a quantummechanical "uncertainty" principle, named after Heisenberg, which tells us that we have to pay a price for such precision. For example: the better we know the position of a particle, the less we know about its momentum (velocity). This has to do - again - with particles actually being waves in quantummechanics.

We think that all our current theories break down at some point. At very small length scales, very high energies and very large momenta, the theory we have will stop being valid, in much the same way as classical mechanics makes way to special relativity for large momenta and for quantum mechanics at small length scales (and quantum field theory at large momenta and small length scales, at the same time).
Currently theoretical physicists are speculating what would lie beyond this point. One general idea is that below a certain scale (called Planck scale) space(time) will start looking granular. Then arbitrary measurements are no longer possible, we can only measure distances in terms of some fundamental distance for example. This is comparable to electric charge, which can only be measured in terms of some fundamental charge e (electron charge). However, the Planck scale lies far beyond anything we can measure today, or even in the near future. Maybe we will never be able to measure it at all. Therefore the greatest challenge for such theories is to postulate some microscopical structure (say, on the Planck scale) of space and try to average out these effects to predict something on a scale which can be reached now or in the near future.

 

[Uncertainty01]

Hi,

Thank you for the response. I am sorry it has taken so long to respond but I had to think some more about what you said. So assuming there is a fundamental distance and no measurement can be made on a smaller scale, that would still leave some interval of space which theoretically could be divided up into smaller intervals as you do anytime you measure in space. What would stop you from dividing that interval of space into smaller intervals? I guess I don't accept that there necessarily must be some fundamental distance. There isn't anything about space that points to a fundamental distance.

And I am interested in the theoretical aspects of the possibility that space can scale down infinitely when measured. One hypothesis that might explain this apparent inability to measure space to any finite precision is that there really is no true "empty space" and that the universe is interconnected by some substance. Perhaps it is the interaction of multiple substances. Maybe dark matter or some other substance we have yet to discover. But if the universe were an interconnected medium, then it would be impossible to break that medium into finite intervals (as one attempts to do when measuring) because it is of a continuous nature. There are other potential implications for this theory in terms of gravity. In this view an object with mass would have its effect on the the hypothetical substance which would in turn act upon other objects through changes to the properties of this substance. No mysterious action at a distance.

So I would like to hear what you think. I know there are a lot of gaps and unknowns but it has been an interesting intellectual exercise for me.

 

[pallidin]

Finite intervals.

This appears to be your basic question.

As a bullet from a gun approaches your head, one can "mathematically" divide the approaching distance such that the bullet never impacts your head.
But what happens in reality?

So, either "that" math is wrong or the reality is wrong. OK?

 

[Qui]

Pallidin, that was a extreme example. I prefare the race between the turtle and bunny :)

And I would like to thank CompuChip for his explanation, good reading!

 

[Uncertainty01]

Finite intervals

Thanks for your response I guess. No what I am saying is that when we "attempt" to "measure" distance in space we end up with infinite intervals. I argued this may be because the universe is continuous and cannot be divided into some some finite fundamental interval. It is not that there is some infinite distance between objects as your example seems to dispute, its that there can be no absolutely precise measurement of fundamental distance because space is by definition extent in space and any extent by defintion involves some distance which can be further broken down. This lack of precision causes the uncertainty of our measurements.

 

[f95toli]

As a bullet from a gun approaches your head, one can "mathematically" divide the approaching distance such that the bullet never impacts your head.

No, this is not correct. Yes, you can divide the distance into an infinite number of intervals but that does NOT mean that the bullet will never hit you.
The key here is to realize that you can sum an infinite number of terms and get a finite answer.

The Greeks did not know this which is why they considered the Tortoise-Hare story to be a demonstration of a paradox.

 

[pallidin]

I did not say it was correct, as it obviously violates the reality experience.
That was my point.

 

[f95toli]

True, but my point was that the "solution" to this paradox has nothing to do with reality; i.e. there is no need to assume anything about "properties of space" or anything along those lines;it is a purely mathematical results.
If you want you can actually use the data from the original story and simply calculate how long it will take for the hare to overtake the tortoise. The only reason why people thought this was a paradox is because they did not know enough about math.

The fact that the solution relies on mathematics and not physics is a subtle -but very important- point.

 

[sonofahb]

I actually read an article about this very subject. If space is finite, then we are all living in a hologram lol!