Properties of space

[lesliedellow]

Another way of putting it, is: how is it that the Universe has (spatially) the mathematical structure of a (locally Euclidean) manifold with three dimensions?

If you answer: because it has the spacetime structure of relativity, then how does that work?

Forgetting how it is represented mathematically, the puzzling question is, "What is space - physically?" It has a structure, so it must "be" something.

For example, if it is asked what mass is, pointing to the equation m=F/a, and saying it is an algebraic variable, doesn't tell you what it physically is. That much harder question has only recently had an answer.

 

[Radagast]

Forgetting how it is represented mathematically, the puzzling question is, "What is space - physically?" It has a structure, so it must "be" something.

You misunderstand me -- what I meant was: how is it that space in reality matches this particular mathematical structure?

Which I think is your question too, just expressed in slightly different words.

 

[Resha Caner]

You might want to point that out to Resha Caner, and justatruthseeker. Still, the really interesting question is, what is this thing, which isn't just nothing, "made of?" To put it that way is problematic, but it illustrates the problem.

I saw Radagast's post, but thought what he said was adequately addressed by others.

Yes, but why as the square, and not linear or cubic? Because space is three-dimensional. As the picture you linked to illustrates, in fact.

I guess I'm not sure how 3 dimensions relates to this at all. I thought Justatruthseeker's explanation was pretty straightfoward, and I think the same logic would produce a square law regardless of the number of dimensions.

 

[Resha Caner]

They are X, Y and Z coordinates, You could add more, but why add complexity to a coordinate system that can locate any point with 3 numbers?

Indeed. Assuming an origin is not arbitrary (which it actually is in a Cartesian system), 3 coordinates is the most parsimonious way to locate a point. I never said anything otherwise. But we must establish a few things:

1) How is our desire to locate a point translate into a property of "real" space? What if the objective is not to locate a point? What if it is something else, and there is another set of coordinates that is more parsimonious for that objective? What then?

2) Parsimony is the best approach if one's objective is engineering - the most efficient manipulation of the objects in space. But that was not the question. How does parsimony lay any requirements on the nature of anything?

 

[Radagast]

I guess I'm not sure how 3 dimensions relates to this at all. I thought Justatruthseeker's explanation was pretty straightfoward, and I think the same logic would produce a square law regardless of the number of dimensions.

Well, no. With N-dimensional space, you get an (N-1)-power law.

With N=3, you get an area on an expanding sphere (inverse square law).

With N=2, you get an arc on an expanding circle (inverse law).

With N=4, you get a volume on an expanding hypersphere (inverse cubic law).

 

[Resha Caner]

Well, no. With N-dimensional space, you get an (N-1)-power law.

And so when a wave propagates in 2 dimensions - like ripples across the surface of a pond? Isn't it the distribution of the medium that is the factor, not the properties of space?

2) Parsimony is the best approach if one's objective is engineering - the most efficient manipulation of the objects in space. But that was not the question. How does parsimony lay any requirements on the nature of anything?

But I'll give you this. With respect to my comment above, I need to rethink this. What you said may actually be a physical example of parsimony which would justify use of that principle and require me to retract that statement.

 

[Radagast]

And so when a wave propagates in 2 dimensions - like ripples across the surface of a pond? Isn't it the distribution of the medium that is the factor, not the properties of space?

No, it's a property of dimensionality.

The fact that we see an inverse-square law for light tells us that either there's no 4th spatial dimension, or else that the universe is very, very tiny along the 4th (and possibly higher) dimension.

 

[Justatruthseeker]

No, it's a property of dimensionality.

The fact that we see an inverse-square law for light tells us that either there's no 4th spatial dimension, or else that the universe is very, very tiny along the 4th (and possibly higher) dimension.

No the fact we see an inverse square law for light tells us the distance between the photons increases with distance from the source, causing less light to strike the same surface area at varying distances.

Inverse-square law - Wikipedia, the free encyclopedia

Again, no magic or dimensions need be invoked, just physics.

 

[Justatruthseeker]

I saw Radagast's post, but thought what he said was adequately addressed by others.



I guess I'm not sure how 3 dimensions relates to this at all. I thought Justatruthseeker's explanation was pretty straightfoward, and I think the same logic would produce a square law regardless of the number of dimensions.

It would produce the same result in one dimension. Draw lines radiating out from a source on a sheet of paper. The further one goes, the less light rays will strike any line segment drawn perpendicular or at an arc to that source.

http://wlym.com/antidummies/part58_files/afigure02%20riemann58.gif

Again, we need apply no dimensions to the problem to solve it. It can be solved in 1, 2, 3, 4 or 20, by the same simple law of physics. In a radial emission photons must of necessity increase in distance between them as they radiate outwards. The number of dimensions is irrelevant to the problem.

The problem is that if the surface is moved perpendicular to one photon, and only one photon passes thru that surface, light will never be an inverse square to any surface placed along that line as it will always receive 1 photon. Unless of course one decreases the distance, then light will eventually increase as an inverse square. This is of course assuming a perfect light source which emits photons along the same path each time.

 

[Justatruthseeker]

Forgetting how it is represented mathematically, the puzzling question is, "What is space - physically?" It has a structure, so it must "be" something.

For example, if it is asked what mass is, pointing to the equation m=F/a, and saying it is an algebraic variable, doesn't tell you what it physically is. That much harder question has only recently had an answer.

Yah, energy, in a "neutral" universe.

 

[Resha Caner]

It would produce the same result in one dimension. Draw lines radiating out from a source on a sheet of paper. The further one goes, the less light rays will strike any line segment drawn perpendicular or at an arc to that source.

No. If you do that math for your drawing, you'll see that the arc lengths are changing linearly with distance, not the square.

No, it's a property of dimensionality.

The fact that we see an inverse-square law for light tells us that either there's no 4th spatial dimension, or else that the universe is very, very tiny along the 4th (and possibly higher) dimension.

This doesn't address what I said. What you are speaking of is a point source of light radiating into a uniform quantum foam. So, it's the same spherical problem as before.

My example was of a wave radiating in 2 dimensions. When a ripple spreads across a pond, according to what you say the 3 dimensions of "space" should make it obey a square law. But because it's only radiating in 2 of those 3 dimensions (the media - the surface of the pond - is only 2 dimensions) - it no longer obeys a square law.

To use your example of light, the same is true of lasers. The media has been changed to created a (nearly) 1-dimensional radiation of light, and so the dissipation of that light doesn't (theoretically) obey a square law. Since no laser is perfect, there is some dispersion of the light that doesn't follow the theory ... but again that's a consequence of the media used to propagate the light.

AFAIK, physics is no longer claiming real things will move through a vacuum. We've never tested anything in a real vacuum. The foam is always present - everywhere - uniformly.

Yet whenever any of these waves we're talking about encounters some change in the media, it changes how the wave propagates. It seems it's not dependent on the dimensions of space, but on the characteristics of the media through which the wave propagates. I'm in new territory, so I haven't yet been able to think of a situation where the media is not what I would call "uniform", but I imagine one could mathematically construct a media that would cause a wave to propagate according to a cubic law. For example, I know of some materials that approximate a cubic stiffness, so it might work there ... though the math would get very hairy.

 

[Radagast]

My example was of a wave radiating in 2 dimensions. When a ripple spreads across a pond, according to what you say the 3 dimensions of "space" should make it obey a square law.

Well, no, because the surface of a pond is 2-dimensional.

It would produce the same result in one dimension. Draw lines radiating out from a source on a sheet of paper. The further one goes, the less light rays will strike any line segment drawn perpendicular or at an arc to that source.

Calculus will tell you it has to be an (N-1)-power law for N dimensions. Conversely, when energy radiates in all directions, the nature of the power law tells you the value of N.

But I give up.

 

[Resha Caner]

Well, no, because the surface of a pond is 2-dimensional.

Yes, I know. But it happens in what you claim is some intrinsic 3-space structure. So, it's an example of something that exists in your 3-space that doesn't have to obey a square law, because ... well, of the medium of propagation.

Anyway, I did pick up a few nuances to the discussion. So, thanks. I'm sorry I couldn't do the same for anyone else.

 

[Michael]

Yes, I know. But it happens in what you claim is some intrinsic 3-space structure. So, it's an example of something that exists in your 3-space that doesn't have to obey a square law, because ... well, of the medium of propagation.

Anyway, I did pick up a few nuances to the discussion. So, thanks. I'm sorry I couldn't do the same for anyone else.

I also gained some insights from this conversation. Thank you very much!

 

[Justatruthseeker]

Well, no, because the surface of a pond is 2-dimensional.



Calculus will tell you it has to be an (N-1)-power law for N dimensions. Conversely, when energy radiates in all directions, the nature of the power law tells you the value of N.

But I give up.

It does produce the same results, the surface area of a line placed on any distance r from the wave center, would receive less of the wave, and less of the force from the wave, the further from the center.

Simple mathematics which you all claim to follow. As the radius increases, that same surface area receives less of the wavefront.

Besides a pond is NOT a 2D structure. The wave also goes below the surface, and does not as you all are trying to imply, simply skim across the surface. Also if a rock is dropped into that pond - cause followed by effect - air waves will also propagate outward - wavefronts will emerge from the center in all 3 dimensions. And in each of those 3 dimensions it will decrease as the square of the distance. As it does with all electric, magnetic, light, gravitational, sound and radiation phenomenon.

http://www.acoustics.salford.ac.uk/feschools/waves/propagation.php

This is because in reality - there is no such thing as one or two dimensions - not even 3. There are no strings, no zero volume nothings of infinite mass. There is "EMPTY space" - composed of energy, through which particles of matter move. We "map" them in 3 dimensions - a coordinate system, nothing more. We have never measured space, just the distance between two points in emptiness. There is no up or down, left or right, forwards or backwards, merely what we choose to define it an any given point in time.

 

[Resha Caner]

Besides a pond is NOT a 2D structure. The wave also goes below the surface, and does not as you all are trying to imply, simply skim across the surface. Also if a rock is dropped into that pond - cause followed by effect - air waves will also propagate outward - wavefronts will emerge from the center in all 3 dimensions. And in each of those 3 dimensions it will decrease as the square of the distance. As it does with all electric, magnetic, light, gravitational, sound and radiation phenomenon.

Given we're both saying that space is not intrinsically made of 3 dimensions, I don't understand why you're disagreeing with examples that show that.

I am aware the amplitude of a wave in a pond rises above and drops below a theoretically flat surface, but that is different from the direction of propagation of the wave. My work as an engineer is in the area of mechanics, and wave mechanics was a big part of that. If you want a reference that derives the linear inverse relationship for waves that propagate in 2D (cylindrical waves), look here: https://www.cis.rit.edu/class/simg303/Notes/Ch7-PropagationofWaves.pdf

Then, for examples of this occurring in the real world for various media, look here:

Sound: The inverse distance law 1/r and the sound pressure effect decay no square scale sound field quantity no sound intensity as sound energy quantity acoustic audio sound reduction free field particle amplitude volume loudness level distance laws dB deci

Earth mechanics (earthquakes): Wilcox : Examples of Cylindrical Shock Wave Conversion by Focusing

Light: Diffraction

Music (drumheads): Vibrations of a circular membrane - Wikipedia, the free encyclopedia

Ocean sciences: (p. 244) Zoological Physics: Quantitative Models of Body Design, Actions, and ... - Boye Ahlborn - Google Books