Is our universe a 4D object?

[Yeezi]

I haven't put much thought into this question (and don't have much evidence for it, if any at all), so please poke holes in it, it's fun to see what you think .

 

 

 

I've been thinking recently that our universe is a 4D object/shape.

 

Whenever in my physics class, or on the internet, someone asks why everywhere is the centre of expansion for the universe, the analogy I hear a lot is that of a balloon inflating (I'm not going to explain it, so feel free to watch this video https://youtu.be/i1UC6HpxY28?t=57).

 

 

Basically could our universe be the same concept as the balloon inflating; a 4D object expanding and our 3D world is on the surface.

I think this would also would explain the fact that there is no centre to the universe.

 

 

It's just a thought, but I would like to hear your opinion on it.

[Strange]

Close. The balloon analogy is intended to show that a 2D surface does not have an edge or a center, but can still expand.

 

Unfortunately, we see the balloon expanding in 3D space. There isn't any equivalent for the way the universe expands. However, someone has suggested that you can view the radius of the balloon (the fourth dimension) as time and so you see the surface expanding as it moves through time. That is close the the way the universe is described in general relativity.

[Phi for All]

Do you mean a fourth spatial dimension? You're right, there is no evidence of this.

 

For a fourth spatial dimension, you need to visualize a 3D cube, then move 90 degrees away from every point in that cube to establish a 4D coordinate system. Right now, we can find the location of anything we want if we know its x, y, and z coordinates, and we can establish when with a temporal dimension (x and y tell us longitude and latitude, z tells us altitude, t tells us the time, so we can say meet at the Empire State Building, 27th floor, at 08:30 tomorrow).

 

What else would a fourth spatial dimension tell us about coordinates in our universe?

[Moontanman]

Do you mean a fourth spatial dimension? You're right, there is no evidence of this.

 

For a fourth spatial dimension, you need to visualize a 3D cube, then move 90 degrees away from every point in that cube to establish a 4D coordinate system. Right now, we can find the location of anything we want if we know its x, y, and z coordinates, and we can establish when with a temporal dimension (x and y tell us longitude and latitude, z tells us altitude, t tells us the time, so we can say meet at the Empire State Building, 27th floor, at 08:30 tomorrow).

 

What else would a fourth spatial dimension tell us about coordinates in our universe?

 

 

Wouldn't time qualify as a forth spatial dimension in your analogy?

[Mordred]

No spatial dimensions are used to measure geometric objects.

 

There are 3 spatial dimensions one time dimension.

[Rumfrd]

If space-time is itself four dimensional, and in addition it is curved, this means there must be an additional spatial dimension for the curvature to extend into. You can not curve a one dimensional line without a dimension for that curvature to extend into. You can not curve a surface without a dimension for that curvature to extend into. Would not four dimensional space-time, which is curved by the presence of mass, require a fifth dimension to accommodate that curvature's extension?

[ajb]

If space-time is itself four dimensional, and in addition it is curved, this means there must be an additional spatial dimension for the curvature to extend into.

This is not true. We have notions of intrinsic curvature of spaces which do not require any kind of embedding into higher dimensional spaces.

 

 

 

You can not curve a one dimensional line without a dimension for that curvature to extend into.

Not true, I can consider the circle quite independently of any embedding into the plane.

 

You can not curve a surface without a dimension for that curvature to extend into.

Simply not true.

 

What you are talking about is extrinsic curvature due to a choice of embedding.

 

 

Would not four dimensional space-time, which is curved by the presence of mass, require a fifth dimension to accommodate that curvature's extension?

No...

 

 

It is possible to consider space-time as being embedded into a higher dimensional space, you can even make this space flat (if you pick high enough dimensions). But you do not need to do this to understand the intrinsic properties of space-time.

 

Bottom line here is learn some differential geometry before making statements like this.

[Rumfrd]

It was not a statement, it was a question. Thank you for your responses. But please Chikis, show me how a circle can exist without two dimensions of space. Its line has one dimension surely, but its curvature defines a plane, no? Again, a question.

So, if I think what you are saying is that my suggested requirement for a higher space dimension does not apply because the time dimension of the continuum differs from the space dimension. How can this be if continua are homogenous and isotropic? Could not the obvious difference between time and space dimensions be a phenomenon of perception. Again, all questions. I don't know.

Edited April 23, 2016 by Rumfrd

[Strange]

So, if I think what you are saying is that my suggested requirement for a higher space dimension does not apply because the time dimension of the continuum differs from the space dimension.

 

It is not because of the nature of the space or time dimensions. It is always possible to describe curvature as intrinsic, i.e. not requiring a higher dimension.

[Rumfrd]

Moontanman, I believe it would tell us the direction in which spacetime is curved. But I take your point. Am I correct that you are saying, that if it is a dimension of space it should be visible to us, and it should be capable of having a metric assigned to it, meters for instance? Could it not be visible in cross section, as the distortion produced by mass induced curvature?

Strange, Are you saying the time dimension can be intrinsically curved, or that the continuum can be intrinsically curved?

Is the spacetime continuum intrinsically curved by mass? Is that what you are saying Strange? Anyone else know?

Edited April 23, 2016 by Rumfrd

[Strange]

Is the spacetime continuum intrinsically curved by mass? Is that what you are saying Strange? Anyone else know?

 

Yes, it is curved by the presence of mass (or energy) and that curvature can be (and is) described as intrinsic curvature.

 

My understanding is that curvature is curvature. You can choose to describe it extrinsically by embedding the surface in a higher dimension, or intrinsically where you don't need any higher dimensions.

[ajb]

It was not a statement, it was a question.

In the post I replied to you made statements rather than asked questions.

 

But please Chikis, show me how a circle can exist without two dimensions of space.

You can think about the one point compactifiation of the real line.

 

Its line has one dimension surely, but its curvature defines a plane, no? Again, a question.

This is a question... gramatically it is okay and we have a question mark.

 

I do not understand the question. I do not see how the curvature of a line, which is intrinsically flat, to define a plane.

[Rumfrd]

The intrinsic designation, applied to curvature, means that the curvature is testable. For instance, a space mapped to a sphere would show a different value of pi than a flat space, which makes the curvature of the spherical space intrinsic. A space mapped to a cylinder would not show a difference in pi, and it thus has extrinsic curvature, but both the sphere and the cylinder require an additional dimension. No question mark this time.

Edited April 24, 2016 by Rumfrd

[ajb]

The intrinsic designation, applied to curvature, means that the curvature is testable.

I am not sure what you mean here.

 

 

For instance, a space mapped to a sphere would show a different value of pi than a flat space, which makes the curvature of the spherical space intrinsic. A space mapped to a cylinder would not show a difference in pi, and it thus has extrinsic curvature,

I am not quite sure what you are saying here. I think you are telling us thet the cylinder is flat while a sphere is not. This is true.

 

 

 

 

but both the sphere and the cylinder require an additional dimension.

This is not true.

 

You can define both of these spaces as topological (or smooth) manifolds using the local chart definition. In this way we avoid having to look at embeddings.

 

 

No question mark this time.

Okay, but your statements are muddled and some of them are wrong.

[Rumfrd]

A circle is a line mapped to a cylinder. A cylinder is not one dimensional. But I don't know much topology, I may be missing something. What do you mean by "local chart definition"?

[ajb]

A circle is a line mapped to a cylinder.

 

I can imagine (and even define!) curves on a cylinder that are not (topologically) the circle. For instance the curve could spiral around and not close.

 

 

A cylinder is not one dimensional.

This is true.

 

What do you mean by "local chart definition"?

This is the common definition on a manifold. We have a topological space that is locally isomorphic to [math]\mathbb{R}^{n}[/math] for some n known as the dimension of the manifold. We can define a manifold (topological) using small patches of [math]\mathbb{R}^{n}[/math] together with a rule on how we glue these patches together. We can define the circle and cylinder in this way without any reference to an embedding.

[Rumfrd]

Does this mean that everything describable in topology can exist in this Universe? Mathematical describability is not equivalent to reality. It is perhaps a necessary condition, but in itself, is it not insufficient to guarantee existence? In other words, OK, you can imagine and define such a spiral, but can it exist in the real world?

 

Let me rephrase my question: In the case we are discussing (the curvature of spacetime by the presence of mass) is the consensus of scientific opinion that this curvature does not require mapping to an additional dimension? If so, then why is it intrinsic? Would not the value of pi be reduced if a black hole were approached?

Edited April 28, 2016 by Rumfrd

[ajb]

Does this mean that everything describable in topology can exist in this Universe?

Probabily not, but I have no idea how you would test this.

 

 

Mathematical describability is not equivalent to reality.

Okay, in general we have mathematical constructions that seem not to have anything to do with physics (yet!).

 

It is perhaps a necessary condition, but in itself, is it not insufficient to guarantee existence?

 

Physical realisation you mean? Well who knows.

 

But what you have to be careful with is mixing a physical theory with nature. A physical theory is a mathematical description of what we see. It is not the same as nature itself.

 

 

In other words, OK, you can imagine and define such a spiral, but can it exist in the real world?

Well, if I take a finite cylinder, say a the cardboard tube inside a roll of toilet paper then I can draw on it such a spiral. I can mathematically model this as a map from an interval to the cylinder. In this sense I have a physical realisation...taking into account that I cannot actually draw a line of zero width or thickness!

 

 

 

Let me rephrase my question: In the case we are discussing (the curvature of spacetime by the presence of mass) is the consensus of scientific opinion that this curvature does not require mapping to an additional dimension?

Yes. We have the mathematics to understand space-time without having extra dimensions that we cannot detect. But again, it is possible to think of space-time as sitting inside something larger, and this is done in string theory for example, but today there is no compelling evidence that extra dimensions exists. Thus, when dealing with general relativity we do not need to think of higher dimensions.

 

 

If so, then why is it intrinsic? Would not the value of pi be reduced if a black hole were approached?

No as pi is a natural constant related to the geometry of spheres.

 

If you are talking about the angles of a 'triangle' in different geometries then 'pi' (not really pi) will change. Look up sphereical and hyperbolic geometries.

Edited April 28, 2016 by ajb

[Rumfrd]

I meant spaces mapped to spherical surfaces, or even circles and diameters inscribed on spheres. But never mind.

 

I have read that some physicists propose that scale is a dimension, representing another degree of freedom for geometric objects. Perhaps dimensions are not so easy to recognise.

 

Thanks, I will look up the geometries you suggest.

[petrushka.googol]

We could have as many as 11 dimensions as per string theory, 4 are apparent, 7 are tightly curled up. (too small to isolate).

[ajb]

I have read that some physicists propose that scale is a dimension, representing another degree of freedom for geometric objects.

I am not quite sure what you mean by scale, but it is common to treat variables in a theory as being 'dimensions'.

 

For example, we have phase space where position and momentum are the 'geometric directions' on a manifold. So we can and do think of spaces other than space-time.

[Rumfrd]

There is quite a bit on the web about a Scale dimension, including its mathematical statement.

 

Another suggested dimension is the Information Dimension, which is the freedom of matter to move on the Information/Entropy axis (not a geometric degree of freedom). When dead matter is consumed by a living thing (say water) which takes it into itself and organizes it, it moves that matter in the Information Direction when it becomes part of a life. When the life that does the organizing ends, the matter moves to a lower organization state i.e. the organization of the life decays, its matter moves in the entropic direction, as it acquires many more possible states. The equations for entropy and information can be put in the same form, except for the sign. Are you interested in Information Theory or Thermodynamics?

Edited April 29, 2016 by Rumfrd

[ajb]

There is quite a bit on the web about a Scale dimension, including its mathematical statement.

It is not a concept I am an aware of.

 

 

Another suggested dimension is the Information Dimension, which is the freedom of matter to move on the Information/Entropy axis (not a geometric degree of freedom).

Information dimension is realted to Shannon entropy and so on. There are other things called 'dimension' that are not the topological dimension of a space, which is what we usually mean in the context of geometric methods of physics. So be a little careful with the term 'dimension'.

 

When dead matter is consumed by a living thing (say water) which takes it into itself and organizes it, it moves that matter in the Information Direction when it becomes part of a life. When the life that does the organizing ends, the matter moves to a lower organization state i.e. the organization of the life decays, its matter moves in the entropic direction, as it acquires many more possible states. The equations for entropy and information can be put in the same form, except for the sign.

This sounds a bit strange... I assume this is something to do with Shannon entropy?

 

 

Are you interested in Information Theory or Thermodynamics?

I am aware of the basic notions of information theory, statistical mechanics and thermodynamics. But I am not professionally intersted in these topics, at the moment anyway. I know what you would expect anyone with a degree in physics to know anout these topics.

[Rumfrd]

Yes, Shannon's formula. Thank you, enjoyed your input!

[Rumfrd]

There is a very lucid discussion of this issues here: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_variable/index.html

 

It states what I have been maintaining. That both intrinsic and extrinsic curvature require a higher dimension to accommodate that curvature.