Kaluza Klein Theory
Sol - Dec 18, 2003 6:39 am
Edited Feb 23, 2004 4:39 pm
Kaluza and Klein showed in the 1920's that the Maxwell's equations can be derived by extending general relativity into five dimensions. This strategy of using higher dimensions to unify different forces is an active area of research in particle physics.
Kaluza-Klein theory
From Wikipedia, the free encyclopedia.
Kaluza-Klein theory is a model which unifies classical gravity and electromagnetism. It was discovered by the mathematician Theodor Kaluza that if general relativity is extended to a five-dimensional spacetime, the equations can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton". Oskar Klein proposed that the fourth spatial dimension is curled up with a very small radius, i.e. 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, in fact, also gives rise to quantization of charge, as waves directed along a finite axis can only occupy discrete frequencies.
Kaluza-Klein theory can be extended to cover the other fundamental forces - namely, the weak and strong nuclear forces - but a straightforward approach, if done using an odd dimensional manifold runs into difficulties involving chirality. The problem is that all neutrinos appear to be left-handed, meaning that they are spinning in the direction of the fingers of the left hand when they are moving in the direction of the thumb. All anti-neutrinos appear to be right-handed. Somehow particle reactions are asymmetric when it comes to spin and it is not straightforward to build this into a Kaluza-Klein theory since the extra dimensions of physical space are symmetric with respect to left-hand spinning and r-hand spinning particles.
But see Orbifold for one possible solution.
Some physicists have speculated that in the early universe, cosmic inflation causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.
Moreover the theory as it stands does not deal with quantum effects. However, the same approach to unification of forces is taken by some more modern theories, notably string theory and the related M-theory.
Differential geometry and topology
(Redirected from Differential geometry)
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using calculus. These fields are adjacent, and have many applications in physics, notably in the theory of relativity. Together they make up the geometric theory of differentiable manifolds - which can also be studied directly from the point of view of dynamical system...
Sol
(15 previous messages)Sol
- Jan 7, 2004 11:28 pm (#16 of 19)
In 1921, the mathematician Kaluza posited that if we could formulate a fifth dimension, we might discover the shape of the universe. In 1926, Kaluza and the physicist Klein proposed that the fifth dimension is round and not large. In 1927, Lemaître presented his Big Bang theory. In 1979, Alan Guth developed the theory of inflation. These concepts are in line with wave theory, which, like all natural theories (for example, Darwin’s), pulls together many observations and experiments. Kaluza and Klein’s fifth dimension matches the properties of a magnetic loop. Guth’s inflation is the primary force and behaviour of energetic matter.
Sol
- Apr 1, 2004 4:30 am (#17 of 19)
The fashionable fifth dimension at the turn
The are several different ways in which one can "roll-up" a non compact manifold, like the one on the left, into a compact space, like the other surfaces
http://www.tech.port.ac.uk/staffweb/seahras/neat_physics/extra_dimensions/fifth.htm
Sol
- May 8, 2004 6:29 am (#18 of 19)
sol2 <sol2@physicsforums.com> wrote in message news:<sol2.15wlbg-100000@physicsforums.com>...
I have been trying to distill a framework that is consistent geometrically from different perspectives, and in looking at stringtheory/M Theory.The Monte Carlo effect and the idea of quantum gravity from this geometrical consideration.
Is there geometrizational aspect in stringtheory right from the string as a point to brane world? Is such a method availiable?
[Moderator's note: I am afraid that it will be difficult for most of us to understand this question. LM]
The Klein bottle and the mobius strip are images that are caught in mind when thinking of chirality and parity in symmetry. What way is there to remain geometrical consistent in a action that reveals planck length as something other then what we had been concerned with in matter considerations in reductionism, has now found its way to energy determinations?
Klein`s Ordering of the Geometries
"A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry."
http://www.ensc.sfu.ca/people/grad/brassard/personal/THESIS/node21.html
"Brane World models only carry this unique topology solution a bit further returning in part, though often modified in present format, to Klien's solution to the unification of gravity to electromagnetism through an additional dimensional set. The beauty of this path as we should rightly call it is that it is a natural progression of Einstein's dream of a pure geometric explanation for everything we see around us in nature."
Paul Karl Hoiland "The Importance of Tipping Light Cones" 2/23/04 7:24pm
http://physicsweb.org/objects/world/13/11/9/pw1311091.gif
"a) Compactifying a 3-D universe with two space dimensions and one time dimension. This is a simplification of the 5-D spacetime considered by Theodor Kaluza and Oskar Klein. (b) The Lorentz symmetry of the large dimension is broken by the compactification and all that remains is 2-D space plus the U(1) symmetry represented by the arrow. (c) On large scales we see only a 2-D universe (one space plus one time dimension) with the "internal" U(1) symmetry of electromagnetism."
http://physicsweb.org/box/world/13/11/9/pw1311091
http://cerncourier.com/objects/2000/cernnews5_3-00.gif
"Of course, if this third dimension were infinite in size, as it is in our world, then the flatlanders would see a 1/r2 force law between the charges rather than the 1/r law that they would predict for electromagnetism confined to a plane. If, on the other hand, the extra third spatial dimension is of finite size, say a circle of radius R, then for distances greater than R the flux lines are unable to spread out any more in the third dimension and the force law tends asymptotically to what a flatlander physicist would expect: 1/r. However, the initial spreading of the flux lines into the third dimension does have a significant effect: the force appears weaker to a flatlander than is fundamentally the case, just as gravity appears weak to us. Turning back to gravity, the extra-dimensions model stems from theoretical research into (mem)brane theories, the multidimensional successors to string theories (April 1999 p13). One remarkable property of these models is that they show that it is quite natural and consistent for electromagnetism, the weak force and the inter-quark force to be confined to a brane while gravity acts in a larger number of spatial dimensions."
http://cerncourier.com/main/article/40/2/6/1
The ole view of the quark to quark measure revealled a metric field for consideration? Had dimensional significance?
What geometrial consideration would arise from U(1)*SU(2)*SU(3) and it seemed to me that this rise would have been revealing in the standard model? Not only the quark to quark measure of that energy in distance, but revealling a spherical consideration, when viewing Greg Egan's animations. Rotations?
Sorry if it seems I am all over the map, I have never known one subject that could be so diverse and it is only a theory? Nima's illucidation of a photon of spin 2 quality, as a undertsanding if this extra dimension seemed a bit strange to me, as it signalled "a law," whether he wanted to show this kind of thinking or not, as a consequence of what the future might reveal in string/M theory.
HOPE THIS HELPS CLARIFY THE QUESTION?:)
Sol
- Jan 15, 2005 2:05 am (#19 of 19)
What is it?
KK Tower 
Now part of the problem of visualization here is what and how the cosmic string could have developed. Now, determination of the various sizing of these strings would have had to incorporate the value of the energy involved, in terms of 1r and using the KK tower, such classifcations help in this direction.
Kaluza-Klein theory is a model which unifies classical gravity and electromagnetism. It was discovered by the mathematician Theodor Kaluza that if general relativity is extended to a five-dimensional spacetime, the equations can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton". Oskar Klein proposed that the fourth spatial dimension is curled up with a very small radius, i.e. 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, in fact, also gives rise to quantization of charge, as waves directed along a finite axis can only occupy discrete frequencies.
Kaluza-Klein theory can be extended to cover the other fundamental forces - namely, the weak and strong nuclear forces - but a straightforward approach, if done using an odd dimensional manifold runs into difficulties involving chirality. The problem is that all neutrinos appear to be left-handed, meaning that they are spinning in the direction of the fingers of the left hand when they are moving in the direction of the thumb. All anti-neutrinos appear to be right-handed. Somehow particle reactions are asymmetric when it comes to spin and it is not straightforward to build this into a Kaluza-Klein theory since the extra dimensions of physical space are symmetric with respect to left-hand spinning and r-hand spinning particles.
So in order to get to the summation, views of hidden dimenisons had to be mathematically described for us, so a generalization here would suffice in the following diagram.
Now, not having the room to explain, and having linked previous information on extension of KK theory, I wondered about the following. If we understood well, the leading perspective that lead us through to the dynamical realizations, then the road Gauss and Reimann lead us to would help us to understand the visualization materializing by the calorimeter disciptions of each energy placement harmonically describing each particle's value?
Electromagnetic Calorimeter of the Phenix
If one understood well enough about the direction of discernation of early universe consideration and microstates, then such questions would have been of value in the ideas of topological considerations?
Here again I would point to the Glast determinations and of how we percieve these comological interactions, that continue to be built mathematically? Cosmic string developement would have shown energy valuation that would have continued to expand if we see this as a continous fucntion of particle identification? The matter states would have become distinctive inregards to the weak field manifestations represented in the comsological functions of our universe now?
We would have had to learn to map topological considerations, and the only way is how we see the calorimeter is used?
Kaluza Klein Theory
