|Mar29-04, 01:41 AM||#1|
Kaluza-Klein Theory and spatial curvature
In Kaluza-Klein theory, the gauge symmetries for all the fundamental
forces are mapped onto the higher spatial dimensions.
So the internal symmetries are now externalised.
Does this imply that you can extend the analogy with gravity further:
so for example, if the 5th dimension contains the guage symmetry of
EM, do electromagnetic charges produce distortions in the 5th
dimension, in the same way that mass produces distortions in 4D
If so then presumably you can rewrite Maxwell's equations in a 1D 5th
dimensional sub-space, as the sum of a curvature scalar and a metric
scalar field equals the charge density scalar field, just as the
gravitational field equation is written as a sum of a curvature tensor
and a metric tensor field equals the energy-momentum tensor field?
The relative field strength of electromagnetism, compared with the
much weaker gravitational field, could then explain why the 5th
dimension is compactified; whereas space-time is not.
I don't have enough mathematical tools to understand high powered
string theory; but I'd just be interested to know whether what I've
written is reflected in current theory.
It just seems like common sense to me that the relative
compactification of the spatial dimensions, could be related to the
strength of the fundamental forces with which they are associated by
Btw, since the compactification of all the spatial dimensions becomes
identical, at energies above the symmetry breaking point between
gravity and the other fundamental forces; I wondered does anything
happen to the time dimension?
If not then you're still left with a rather ugly asymmetry between
space and time.
Would be much nicer if the spatial dimensions became more temporal,
and the time dimension more spatial, until they meet in the middle.
I don't know if such hybrid dimensions are mathematically possible?
|Apr2-04, 02:58 PM||#2|
4D gravitation was described in a geometric fashion, and Kaluza-Klein wanted to unify gravitation and electromagnetism such that the unified theory was geometrical as well. Hence, start with 5D general relativity (a theory with local 5D coordinate invariance). Compactify on a circle (using one of the spatial dimensions). The general coordinate transformations in 5D becomes general coordinate transformations in 4D plus seperate local U(1) transformations. The metric tranforms as it should under the local 4D coordinate transformations (you have 4D gravity). In addition, from the 4D perspective certain modes of the 5D metric look like a vector field and a scalar field. The vector field transforms as a gauge field under the local U(1) transformations (you get electromagnetism). You are left with a scalar field which can have a non-zero vacuum expectation value. This value in part determines the strength of Newton's constant as well as the couplings in electromagnetism. It was shown that this model alone does not agree with observations. However, string theory (and the low energy limit supergravity) revived this idea since now the situation is more complex than pure gravity in 5D...there are more symmetries, more fields, more dimensions, and more interesting compactifications than on a circle).
|Apr5-04, 12:44 PM||#3|
Thanks, I think I follow that; but it still doesn't really explain why we compactify the 5th dimension?
That's why I thought it might be compactified by charge, just as space-time is distorted by mass.
I figured that since space-time curvature is non-uniform, but depends on the motion of massive bodies in the Universe; similarly the compactification of the higher spatial dimensions must also vary over space-time.
I pictured a 5th dimension which contracts whenever a charged virtual particle pair pops out of the vacuum, and then starts to unfurl when the particles annihilate. But then another particle pair appears, and causes it to contract again, thus maintaining the compaction.
Similarly virtual quark pairs could maintain the compactification of 3 of the other spatial dimensions with their colour charge, etc, etc...
Perhaps occasionally there is enough time for the 5th dimension to unfurl completely and become momentarily open, which would be even more intriguing!
I think what you've written however, implies a uniform and fixed compactification of the 5th dimension across all of space-time, with the EM coupling strength being determined by the amount of compaction? But how could such a fixed compaction arise?
|Apr7-04, 11:21 AM||#4|
Kaluza-Klein Theory and spatial curvature
|Apr8-04, 06:26 AM||#5|
What I'm struggling to understand is why the curvature of the space-time hypersurface evolves; but the curvature of the higher spatial dimensions supposedly remains constant?
Also since gravity produces spatial curvature; why is it that the other forces do not?
They must've done originally, because all 4 forces were unified before symmetry breaking occurred.
According to general relativity, gravity effectively "is" space-time curvature; so shouldn't the other forces also be analagous to dimensional curvature?
Perhaps they are equivalent to higher dimensional curvature? I'm still not clear on this point.
|Apr8-04, 08:43 AM||#6|
What I was suggesting was that the field equation for this scalar field within a single loop of the 5th dimension, should look identical to the gravitational field equation in 4D space-time (except that it would be in 1D instead of 4D, so the 2nd rank tensor fields would be replaced by scalar fields).
I'm assuming here that the unbroken unified field equation in 10D space-time has the same form as the field equation of general relativity, but extrapolated to 10 dimensions. Possibly this is a bad assumption?
What is it that fixes the expectation value of the scalar field to be a constant? I don't understand why it should be?
|Apr8-04, 03:09 PM||#7|
For every wrap of the cylinder, the dynamics of its length changes? It becomes a energy consideration?
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