Thread: Is the Universe all connected?

1. If the Universe is connected, than why are atoms that make up air not connected? For example in water, the atoms are all spaced apart and able to move freely.

Even if they were connected, then it would make the matter solid.

2.

3. Originally Posted by noSkillz
If the Universe is connected, than why are atoms that make up air not connected? For example in water, the atoms are all spaced apart and able to move freely.

Even if they were connected, then it would make the matter solid.
What do you mean by connected ?

Generally if someone says that the universe is connected they mean that the spacetime manifold of general relativity is toplogically connected. In lay terms that means "one piece".

That in no way implies that atoms are touching one another. In fact, it is not all clear what "touching" means in the atomic quantum mechanical realm.

4. Space contains a minuscule amount of atoms, space is not made of atoms. Virtually all the atoms are clustered in space, in galaxies. As space-time expands the galaxies are pushed apart.

5. Originally Posted by Geo
Space contains a minuscule amount of atoms, space is not made of atoms. Virtually all the atoms are clustered in space, in galaxies. As space-time expands the galaxies are pushed apart.
And the relationship of this observation to connectedness is what ?

6. I'm not sure I understand what you are referring to when you say "connectedness." However, it is indeed true that space is discontinuous--what this implies is that it possesses a metric that can't be infinitesimally subdivided. Mathematicians and physicists that work in continuity mechanics measure a distance relationship in a metric in order to validate a space's continuity. It's based around this rather rudimentary idea:

Assume that there are two points in space, a and b, with an arbitrary third-point, c. Where d(x,y) designates the distance between two point x and y, than where the following occurs the space is continous:

According to Einstein, gravity is what results in the curvature of the four-dimensional space-time continuum, however, and therefore while space is essentially discontinous (as we know by viewing it at the quantum "Planck" scale), it can be a continuum when fused with other mediums.

An example of mathematically abstract space that is continous is the real-number line (the underlying space that the functions being discussed take place).

7. Originally Posted by Ellatha
I'm not sure I understand what you are referring to when you say "connectedness." However, it is indeed true that space is discontinuous--what this implies is that it possesses a metric that can't be infinitesimally subdivided.
I don't know who told you this but it is wrong.

The model used for both quantum mechanics and for general relativity is that of a continuum. Distance has no known minimal value. Discontinuous is not a valid adjective in this context. Space is modeled as a connected manifold, a continuum. It is generally taken to be simply connected as well. In layman's terms it is in one piece and has no holes or gaps.

Originally Posted by Ellatha
Mathematicians and physicists that work in continuity mechanics measure a distance relationship in a metric in order to validate a space's continuity. It's based around this rather rudimentary idea:

Assume that there are two points in space, a and b, with an arbitrary third-point, c. Where d(x,y) designates the distance between two point x and y, than where the following occurs the space is continous:

This is part of the definition of a topological metric. The rest is that and if then

But it has nothing whatever to do with the space being "continuous". Continuity is a property of a function between two topological spaces and is not a property of a topological space itself.

You also ought to be told that the term "metric" is used in differential geometry to mean an non-degenerate quadratic form on the tangent bundle, and that is the sense in which it is used in general relativity. The relationship between this metric and the topological metric is this: If the metric happens to be positive definite then it can also be used to define a topological metric in which the distance between two points is the minima of the arc lengths of paths joining the two points, with path length determined by an integral involving the quadratic form. That metric can also be shown to induce the topology of the manifold.

Originally Posted by Ellatha
According to Einstein, gravity is what results in the curvature of the four-dimensional space-time continuum, however, and therefore while space is essentially discontinous (as we know by viewing it at the quantum "Planck" scale), it can be a continuum when fused with other mediums.
No such thing is known. No one knows what happens at the Planck scale. No one. The current physical models assume that the manifold model holds at all scales, and while that might change any statement to the contrary is just speculation.

Originally Posted by Ellatha
An example of mathematically abstract space that is continous is the real-number line (the underlying space that the functions being discussed take place).
You have the wrong adjective here. The real numbers are a connected continuum, but the word "continuous" does not apply. It is also complete as a metric space, and that is an important property.

You might want to look up the definition of "connected" in a topology book. A topological space is connected if it is not the union of two disjoint non-empty open sets. In lay terms that means that it is just one piece. [/tex]

8. Originally Posted by DrRocket

And the relationship of this observation to connectedness is what ?
I'm sorry, I didn't know what noskillz meant by "connected".

I try to answer according to my interpretation of the questionnaire's level of knowledge, I must have got this wrong.

9. Originally Posted by Geo
Originally Posted by DrRocket

And the relationship of this observation to connectedness is what ?
I'm sorry, I didn't know what noskillz meant by "connected".

I try to answer according to my interpretation of the questionnaire's level of knowledge, I must have got this wrong.
I'm not sure noskillz knows what he meant by "connected' either. You may be right, but if so his definition is unconventional to say the least.

It is pretty hard to tell since he has not been back.

10. Originally Posted by DrRocket
But it has nothing whatever to do with the space being "continuous". Continuity is a property of a function between two topological spaces and is not a property of a topological space itself.
The continuity of a function and that of a space are not the same thing. When taking a course in calculus of one variable, we are introduced to the rigorous definition of continuity relatively early. It's based around the idea that the limit as x approaches a particular value of a function is identical to the evaluation of that function at that input. Mathematically, this relationship is shown as follows:

Loosely speaking, the continuity of a space implies that there are an infinite number of points between any two points. The aformentioned model that you spoke of refers to the space-time continuum (four dimensions), and not uniquely space (three dimensions).

But it has nothing whatever to do with the space being "continuous".
It does. Continuity is the property of a space's metric to potentially be infinitesimally subdvided. Where the previously mentioned distance relationship holds true no matter how small the values continuity is mantained. The least upper bound property, the theorem that seperates the continuity of the real numbers as compared with the rational numbers is one such example, and demonstrates where calculus can or cannot be applied.

The current physical models assume that the manifold model holds at all scales, and while that might change any statement to the contrary is just speculation.
The interpretation of the manifold was not made rigorous in mathematics until the work of Riemann in the 19th century, whereas Einstein published his papers early in the 20th century. What a manifold does is merely take some geometric space, whether it be hyperbolic or double-elliptic, and view it at a near enough level such that it holds Euclidean properties. There is no need to use a manifold here other than conventional applications to the theories of special and general relativity; the continuity of a space breaks down at the Planck [10^(-33)] scale.

No such thing is known. No one knows what happens at the Planck scale. No one. The current physical models assume that the manifold model holds at all scales, and while that might change any statement to the contrary is just speculation.
It is a general consenses among physicists that space is discontinous at the Planck (and certainly sub-Planck) scales. I could easily bring up more sources than I care to name confirming this. If (and it is certainly not unlikely that it is) it is true that space is discontinous at such a scale, than it certainly is not counter-intuitive that space is discontinous, period.

You also ought to be told that the term "metric" is used in differential geometry to mean an non-degenerate quadratic form on the tangent bundle, and that is the sense in which it is used in general relativity. The relationship between this metric and the topological metric is this: If the metric happens to be positive definite then it can also be used to define a topological metric in which the distance between two points is the minima of the arc lengths of paths joining the two points, with path length determined by an integral involving the quadratic form. That metric can also be shown to induce the topology of the manifold
Following your previous advice, I held off on learning advanced topology or differential geometry. I still taught myself some basics, but I'm afraid that your paragraph is difficult to understand with much fluency. Still, I understand the basic idea.

You might want to look up the definition of "connected" in a topology book. A topological space is connected if it is not the union of two disjoint non-empty open sets. In lay terms that means that it is just one piece
Thanks.

You have the wrong adjective here.
I apologize about that; I tried to write my sentence out such that the syntax did not make the sentence more complicated than it had to open (otherwise I would have generated some oxymorons).

The real numbers are a connected continuum, but the word "continuous" does not apply. It is also complete as a metric space, and that is an important property.
I was under the impression that a continuum referred to a continuous space (this means that it can be infinitesimally subdivided).

11. Connectedness means that things join together, touch, or relate in some way. Like the plants are all connected to the earth, birdhouses connect to trees, your water bottle connects to the table in your dining room and your table is connected to the floor which is in your house, which is connected on the ground. You and your parents are connected because you were born from them, and you carry your moms and dads genes. Supposedly, earth and the sun and all the other planets are held together because of gravity that holds them in place. But I don't know what to believe, because after all, gravity is just a theory and has not been proven as a fact.

12. There are many things that haven't been proven and may never be. In these situations, you must simply trust what you believe. The predicate
can't be proven without being based around axioms to begin with, but many individuals choose to believe it anyway. Gravity is something I (and certainly any Ph.D. physicist) believe exists. So many theories are based around the concept of gravity that if it's existence collapsed that extant physics (and in particular classical mechanics) would require almost entire revision (the concepts of motion in one and two dimensions for example).

13. Originally Posted by Ellatha

Loosely speaking, the continuity of a space implies that there are an infinite number of points between any two points. The aformentioned model that you spoke of refers to the space-time continuum (four dimensions), and not uniquely space (three dimensions).
With this definition the rational numbers would be continuous. They are totally disconnected.

[quot="DrRocket"]]But it has nothing whatever to do with the space being "continuous".[/quote]

I
Originally Posted by Ellatha"
t does. Continuity is the property of a space's metric to potentially be infinitesimally subdvided. Where the previously mentioned distance relationship holds true no matter how small the values continuity is mantained. The least upper bound property, the theorem that seperates the continuity of the real numbers as compared with the rational numbers is one such example, and demonstrates where calculus can or cannot be applied.
The real numbers are connected while the rational numbers are not. That is the result of the least upper bound property. It has absolutely nothing to do with the distance relationship. The statement tnat "the previously mentioned distance relationship holds true no matter how small the values continuity is mantained" makes no sense whatever -- it is just word salad.

Originally Posted by Ellatha
The interpretation of the manifold was not made rigorous in mathematics until the work of Riemann in the 19th century, whereas Einstein published his papers early in the 20th century. What a manifold does is merely take some geometric space, whether it be hyperbolic or double-elliptic, and view it at a near enough level such that it holds Euclidean properties. There is no need to use a manifold here other than conventional applications to the theories of special and general relativity; the continuity of a space breaks down at the Planck [10^(-33)] scale.
Yep, Riemann preceeded Einstein and Einstein used Riemann's work in the development of general relativity. What is your point?

Your statement regarding manifolds is garbled. It makes no sense.

There is no proof whatever that the model breaks down at the Planck scale. Current models are assumed to hold at all scales, although there is speculation and concern that the manifold model may cease to work at some sufficiently small scale. Repeat -- the breakdown is speculative, not established.

I have no idea what your statement that "There is no need to use a manifold here other than conventional applications to the theories of special and general relativity" is supposed to mean and I doubt that you do either. Word salad.

Originally Posted by Ellatha"}It is a general consenses among physicists that space is discontinous at the Planck (and certainly sub-Planck) scales. I could easily bring up more sources than I care to name confirming this. If (and it is certainly not unlikely that it is) it is true that space is discontinous at such a scale, than it certainly is not counter-intuitive that space is discontinous, period.[/quote

Bullshit.

Nobody knows what happens at the Planck scale and a consensus is utterly meaningless. I think you mean that ther is a consensus among those who write speculative books that masquerade as science for the general public.

[/quote="Ellatha]I was under the impression that a continuum referred to a continuous space (this means that it can be infinitesimally subdivided).
There is no such thing as a "continuous space" and "infiiniteseimally subdivided" is also meaningless. There are connected spaces and that is what I think you are calling "continuos", but that is not good terminology.

I think I may have mispoken earlier. The topological definition of a continuun is a compact connected Hausdorff space. The unit interval is a continuum.

14. The concept of a continuum is a very simple one--in fact, I have had a professor of mathematics at MIT (David Jerison [http://www-math.mit.edu/people/profile.php?pid=112]) confirm the previously provided definition, including the continuity of the real number line. A simple google search could easily provide all of the information I have made in my last few posts: according to you these sources would all also be playing "word salad" and be "garbled." I assure you that I am not randomly typing information as I go along.

Repeat -- the breakdown is speculative, not established.
What is your point?--you went out of your way to make sure that the poster I directed my post to understood that my statement was speculative?

There is no such thing as a "continuous space"
In fact, there is more than one:

"A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material."

http://en.wikipedia.org/wiki/Continuum_mechanics

Your statement regarding manifolds is garbled. It makes no sense
It is not garbled in the least. All I said is that if a space, at a close enough level, is similar to a Euclidean space than it is a manifold. The surface of the Earth, being similar to the surface of a sphere and therefore in spherical space, can be viewed at a close enough level such that it is "flat," and therefore Euclidean (of course, more rigorously, possessing the properties of Euclidean space based around the postulates of Euclid [only one line goes through any two points, etc.]). This is an example of a manifold.

I do not understand what you mean by"There is no need to use a manifold here other than conventional applications to the theories of special and general relativity"...
All I mean is that using a manifold makes things "simple" since we can work in Euclidean space than (this is a very simple geometry--the kind tenth grade students in America learn).

Bullshit.
What exactly did I state there that was incorrect?

With this definition the rational numbers would be continuous. They are totally disconnected.
Would they? I was unaware that there were an infinite number of rational numbers between any two rational numbers. The infinite number of points between any two points definition was something that I thought of and believed was correct based off of the rigorous definitions I provided earlier (which are not my own), but I apparently didn't know how many rational numbers there were between any two rational numbers.

The topological definition of a continuun is a compact connected Hausdorff space. The unit interval is a continuum
I don't know what a Hausdorff space is, but the definition you provided of "connectedness" earlier sounds similar to a continuum as provided in continuum mechanics.

I apologise for the bad-sounding terminology.

15. Originally Posted by Ellatha

I don't know what a Hausdorff space is, but the definition you provided of "connectedness" earlier sounds similar to a continuum as provided in continuum mechanics.
The notion of a continuum in continjuum mechanics has little to do with the definition of a continuum in topology.

Topology -- a continuum is a connected Hausdorff space

Continuum mechanics -- a continuum is a material that is describable as a manifold and for which the material properties are describable as tensors.

16. Originally Posted by Ellatha
The concept of a continuum is a very simple one--in fact, I have had a professor of mathematics at MIT (David Jerison [http://www-math.mit.edu/people/profile.php?pid=112]) confirm the previously provided definition, including the continuity of the real number line. A simple google search could easily provide all of the information I have made in my last few posts: according to you these sources would all also be playing "word salad" and be "garbled." I assure you that I am not randomly typing information as I go along.
I don't care what you think Jerison said or what Google says, the definition that you provided is word salad.

A continuum in topology is a connected Hausdorff Space. "Infinitely divisible" without an attendent definition is meaningless.

Topological spaces are not continuous, they are perhaps connected. Functions can be continuous.

If you do a search on "continuous space" you find it a synonym for "topological space" and I assure you that is not common terminology. Neither is it helpful as there are all sorts of topological spaces, including discrete ones. http://mathworld.wolfram.com/ContinuousSpace.html

The concept that you want for the real line is connectedness, not continuity.

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