5 dimensions

[michel123456]

From the Kaluza-Klein theory on wikipedia


From the intro

In physics, Kaluza–Klein theory (KK theory) is a unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the usual 4 of space and time

From the chapter Group Theory Interpretation

In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and the phenomenon of having a space-time with compact dimensions is referred to as compactification.

 

 

There is my question(s):

 

If we are understanding our world as four dimensional (3D of space + 1D of time) and we observe nowhere the 5th dimension, why do we suppose that the extra 5th dimension is compacted?

 

In my understanding, the 4 dimensions should interpreted as a section into a 5 dimensional world.

IOW that the 5th dimension would be of higher degree than what we actually observe. The 4D would then be embedded into a 5D reality.

That is what we do when we describe a 2D world from a 3D: we make a section, or a projection. Something like a map.

 

So that instead of "compacting" the 5th dimension, one should "unmap".

 

What would be the mathematical concept of such an inverse projection?

Edited January 6, 2015 by michel123456

[imatfaal]

From the Kaluza-Klein theory on wikipedia

 

 

From the intro

From the chapter Group Theory Interpretation

 

There is my question(s):

 

If we are understanding our world as four dimensional (3D of space + 1D of time) and we observe nowhere the 5th dimension, why do we suppose that the extra 5th dimension is compacted?

 

In my understanding, the 4 dimensions should interpreted as a section into a 5 dimensional world.

IOW that the 5th dimension would be of higher degree than what we actually observe. The 4D would then be embedded into a 5D reality.

That is what we do when we describe a 2D world from a 3D: we make a section, or a projection. Something like a map.

 

So that instead of "compacting" the 5th dimension, one should "unmap".

 

What would be the mathematical concept of such an inverse projection?

 

 

I think - but I am very far from sure - that if we have 4 spatial dimensions which are all uncompacted that we have too many degrees of freedom in many of the models that we already know work. Adding a full fourth spatial dimension would mean that most of our basic theories are incorrect - they only work if space is 3d; not if space is a 3d mapping of a 4dimensional manifold.

 

However if we add an extra tightly curled one then we get to answer some questions without raising too many more new ones.

 

I think this is AJB's field/speciality - or at least the maths of many-dimensional manifolds is; so hopefully he will see this question.

[ajb]

If we are understanding our world as four dimensional (3D of space + 1D of time) and we observe nowhere the 5th dimension, why do we suppose that the extra 5th dimension is compacted?

The ideas is that the 5th dimension should be 'small' and so we don't notice it directly. Thus, it should be 'curled up' tightly.

 

Amazingly, if you consider the Einstein field equation in 5 dimensions with a compactified 5th dimension you get the standard Einstein field equations, Maxwell's equations and the equations for an additional scalar field.

 

 

It is important that the 5th dimension is topologically the circle. This gives rise to the U(1) group of electromagnetism. You can do similar things and to include Yang-Mills theories here also.

 

The difficulty is in including chiral fermions and the tower of KK modes.

 

What would be the mathematical concept of such an inverse projection?

A local section. We can always locally split our 5-d space-time into 4-d standard space time and the compactified 5th dimension. This will give us the structure of a fibred manifold, in fact I think a fibre bundle, but the distinction is not so important at this point. You always have local sections but not necessarily global ones. If the decomposition is a global one then we for sure have global sections.

[michel123456]

The ideas is that the 5th dimension should be 'small' and so we don't notice it directly. Thus, it should be 'curled up' tightly.

 

Amazingly, if you consider the Einstein field equation in 5 dimensions with a compactified 5th dimension you get the standard Einstein field equations, Maxwell's equations and the equations for an additional scalar field.

 

 

It is important that the 5th dimension is topologically the circle. This gives rise to the U(1) group of electromagnetism. You can do similar things and to include Yang-Mills theories here also.

Very informative. Do you mean "really" a circle or any other closed curve will do?

 

 

A local section. We can always locally split our 5-d space-time into 4-d standard space time and the compactified 5th dimension. This will give us the structure of a fibred manifold, in fact I think a fibre bundle, but the distinction is not so important at this point. You always have local sections but not necessarily global ones. If the decomposition is a global one then we for sure have global sections.

That I don't understand.

If you want to "reverse engineer" a projection, you don't split things.

For example, if I have a Mercator projection of the globe, without knowing where the projection came from, what are the ways to reverse the projection and find the globe back? I suspect there are many ways to reverse the projection but only one way to find the correct answer. None of these ways will be a splitting.

 

--------------

edit

Even if you begin with the globe, the projection is not a split.

Edited January 6, 2015 by michel123456

[MigL]

For the case of a single extra compacted dimension, a simple looping line segment ( one dimensional ) at each point in space-time will do. The only guideline is that the loops be close to Planck length sized ( or else we'd 'see' them ).

In the case of multiple extra compacted dimensions, such as in SString theory, the extra dimensions are subject to rules or constraints.

 

I am nowhere near competent or confident enough to explain the fibre bundle concept, so I'll leave that to AJB.

However, the idea that our 3d+t ( four dimensional ) reality is a projection of a higher ( 5d or more ) manifold is flawed. The idea is that you would take a 3D object like a pencil, and 'project' it into a 2D manifold, like a sheet of paper. The 2D 'Flatland' inhabitants of the paper would then 'see' a circle ( the circumference of the pencil ).

In all models, the curvature has to be intrinsic, i.e. it doesn't curve in a higher dimension, there is no embedding.

( not sure about recent work in M-theory, can branes have curvature and even loop back on themselves ? maybe AJB can shed some light on that also )

[derek w]

zero dimension (1st dimension)

one dimensions (2nd dimension)

two dimensions (3rd dimension)

three dimensions(4th dimension)

time (5th dimension)

[michel123456]

The ideas is that the 5th dimension should be 'small' and so we don't notice it directly. Thus, it should be 'curled up' tightly.

 

Amazingly, if you consider the Einstein field equation in 5 dimensions with a compactified 5th dimension you get the standard Einstein field equations, Maxwell's equations and the equations for an additional scalar field.

 

 

It is important that the 5th dimension is topologically the circle. This gives rise to the U(1) group of electromagnetism. You can do similar things and to include Yang-Mills theories here also.

 

The difficulty is in including chiral fermions and the tower of KK modes.

 

 

A local section. We can always locally split our 5-d space-time into 4-d standard space time and the compactified 5th dimension. This will give us the structure of a fibred manifold, in fact I think a fibre bundle, but the distinction is not so important at this point. You always have local sections but not necessarily global ones. If the decomposition is a global one then we for sure have global sections.

 

 

For the case of a single extra compacted dimension, a simple looping line segment ( one dimensional ) at each point in space-time will do. The only guideline is that the loops be close to Planck length sized ( or else we'd 'see' them ).

In the case of multiple extra compacted dimensions, such as in SString theory, the extra dimensions are subject to rules or constraints.

 

I am nowhere near competent or confident enough to explain the fibre bundle concept, so I'll leave that to AJB.

However, the idea that our 3d+t ( four dimensional ) reality is a projection of a higher ( 5d or more ) manifold is flawed. The idea is that you would take a 3D object like a pencil, and 'project' it into a 2D manifold, like a sheet of paper. The 2D 'Flatland' inhabitants of the paper would then 'see' a circle ( the circumference of the pencil ).

In all models, the curvature has to be intrinsic, i.e. it doesn't curve in a higher dimension, there is no embedding.

( not sure about recent work in M-theory, can branes have curvature and even loop back on themselves ? maybe AJB can shed some light on that also )

 

 

I am not sure that there is no embedding.

 

To keep things simple, first of all what we call a "dimension" is not "something" that is straigth or curved. It is not an object, it is simply a mathematical or geometrical tool. So, if the 5th dimension has to exist, it should be a tool like the other dimensions. It should be a geometrical axis perpendicular to the 4 others. If we know from mathematical research that the 5th dimension is spacelike, it means that this tool is perpendicular to any of the 3 other spatial dimensions.

 

Which means that the 5th dimension should be a geometric line exactly as the 3 spatial dimensions are defined. Not to say all spacelike dimensions should be interchangeable.

 

If the facts are that we cannot observe this 5th dimension anywhere, then one solution to the enigma could be that because we are looking from inside what we are observing is a projection, or a section, of the 5 dimensions world. Because we have defined (have we?) the 5th dimension as a tool perpendicular to the others, the section or projection, being orthogonal or not, will be a point. That is the result of the geometrical definition of a line.

And a point can be geometrically described as a circle of null radius.

Edited January 7, 2015 by michel123456

[ajb]

Do you mean "really" a circle or any other closed curve will do?

The standard construction is to use a circle, all closed curves are topologically the circle so I doubt you can really use much more that that.

 

 

That I don't understand.

If you want to "reverse engineer" a projection, you don't split things.

For example, if I have a Mercator projection of the globe, without knowing where the projection came from, what are the ways to reverse the projection and find the globe back? I suspect there are many ways to reverse the projection but only one way to find the correct answer. None of these ways will be a splitting.

The kind of mappings you are thinking of here are charts. That is local maps from the manifold in question to R^{n}.

 

That is not usually what one means by a projection in differential geometry. For Kaluza-Klien theory it is standard to take the topology globally to be [math]S\times M[/math] and so we have a projection onto [math]M[/math]. We could assume that this splitting is just a local one, and then we have a fibre bundle. Anyway, even in this more general situation you always have local sections.

[michel123456]

The standard construction is to use a circle, all closed curves are topologically the circle so I doubt you can really use much more that that.

 

 

 

The kind of mappings you are thinking of here are charts. That is local maps from the manifold in question to R^{n}.

 

That is not usually what one means by a projection in differential geometry. For Kaluza-Klien theory it is standard to take the topology globally to be [math]S\times M[/math] and so we have a projection onto [math]M[/math]. We could assume that this splitting is just a local one, and then we have a fibre bundle. Anyway, even in this more general situation you always have local sections.

I understand nothing. I see things in a very Euclidian way. I hope you understood my post #7.

[ajb]

It should be a geometrical axis perpendicular to the 4 others. If we know from mathematical research that the 5th dimension is spacelike, it means that this tool is perpendicular to any of the 3 other spatial dimensions.

 

 

Which means that the 5th dimension should be a geometric line exactly as the 3 spatial dimensions are defined.

So locally this would always be true; or really you can find coordinates such that this is true.