Gravity: Kaluza-Kline Theory

March 25, 2009, 8:24 pm
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Unification

There is also the question of just what is meant by unification. Is it  sufficient for all fields to appear in the same equation, or should fields appear as various internal flavors of a single super-field? For example, mathematical representations of physical  phenomena in general relativity include the metric, gmn which  follows Einstein's interpretation  that gravity is automatically being included in the  description of all other fields in Nature, even though these fields retain their distinguishability. Maxwell's equations for the electromagnetic  field can be written in the beautiful tensorial mathematics of general relativity, and shows how the electromagnetic field, Am and  the gravitational field,  gmn are seemingly tied together. But there is a difference in the way that gmn and Am relate to the  actual geometry of spacetime itself.   gmn does so directly since, by its definition, it is the embodiment of this  geometry. Am  on the other hand, only relates to spacetime indirectly through  the inclusion of  gmn  in the machinery of the equation. Can this duality be  resolved? Do there exist characteristics of gmn  that can be directly  interpreted as  Am so that only one superfield needs to appear in the equations?

In 1914, just before general relativity was announced by Einstein, Gunnar Nordstrom devised a theory in which gravity and electromagnetism  were seemlessly combined into a single field,  but to do so, Nordstrom proposed that spacetime was 5-D not 4-D.  This meant that instead of Maxwell's  equations describing a four-element, electromagnetic field, they described  a five-element field. The new fifth element of this field behaved just like  the Newtonian gravitational potential, which for centuries was all that was needed to define the gravitational field.

 In 1919, once  general relativity was published,  another solution to unifying gravity and electromagnetism was announced by Theodor Kaluza. Kaluza like Nordstrom before him, discovered that by extending  general relativity in ordinary 4-D  spacetime to a 5-D theory, this kind of unification could also be acheived. Geometrically, the introduction of this new coordinate means  that an additional degree of freedom has been introduced in the specification  of all geometric quantities. The general relativity of 5-D spacetime requires not only the 10 quantities defining ,  gmn but an additional 5 new quantities. These quantities played the same role, and had  the same characteristics, as the electromagnetic field Am.

Oscar Klein re-discovered Kaluza's 5-dimensional version of general relativity in 1926, and by 1938, Einstein and Bergmann together with Einstein, Bargmann and  Bergmann also had a go at exploring this idea, but with the caviat that the extention of spacetime  through the 5th dimension was limited. The additional spatial coordinate was  finite in size and comparable in circumference to that of sub-atomic  particles.

Kaluza's work was sufficiently bizarre that few took it very seriously. It languished for years as a mathematical curiosity. It is interesting to see where Kaluza  had gotten this idea. By 1921, not even quantum mechanics was yet in place. The famous eclipse of 1919 which had vindicated general relativity had only  just been observed, and although Nordstrom had long since published a 5- dimensional, special relativity  theory of his own, there was no acknowledgment of it by Kaluza. Even  Einstein had ignored Nordstrom's ground-breaking forray into this  subject, presumably because Einstein didn't like any of Nordstrom's  other work in relativity either.

How ironic that it would be Nordstrom's ideas, not Einsteins, that would eventually come closer to the core doctrine of unification theory by the end of the 20th century. By 1963, Bryce DeWitt went on to suggest that Yang-Mills theory  might be unified with gravity by using Kaluza-Klein's approach. The  first complete accounting of just how such a program might be designed  was given by Cho and Freund in 1975. The problem with this  approach turned out to be that there were no good reasons why the new  dimensions which had to be tacked-on to spacetime should be tiny. This problem was eased in the late 1970's with the exploration of "dimensionally-extended supergravity". It didn't resolve all of the problems of the particle mismatches, but it did show that some  interesting simplifications were possible in the cumbersome SO(8) theory.

In the spirit of the Nordstrom-Kaluza-Klein theory which involved adding a 5th dimension to spacetime, (TBD) discovered that SO(8) in 4-D could be recast as the far-simpler SO(1) theory, but in 11-D. By adding additional dimensions the mathematics, and the compliment of fundamental fields, became much simpler in 11-dimensions, but when it was reduced back to a 4-dimensional theory, you could recover the full compliment of fields predicted by the more complex SO(8) theory. Still, there were problems that seemed insurmountable no matter what trick you tried to work.

During the late 70's an increasing fraction of theoretical attention became focussed on the search for a self-consistent, infinity-free GUT. Supergravity and Supersymmetric GUTS (SUSY GUTS) were the object of a veritable feeding frenzy of activity as this new motherlode was mined. A bewildering number of ideas were spawned that impacted the theoretical structure of spacetime and the vacuum. New phase transitions were identified at energies between 100 and 1015 GeV implying new patinas of vacuum states litering spacetime. Dimensional-extension was applied to Supersymmetry theory, but required spacetime to have far more than 4-D; perhaps as many as 11. These additional dimensions would be rolled-up into sub-spaces resembling tiny spheres attached to each point in spacetime, but other more exotic geometries for these sub-spaces were also investigated. These ideas resulted in an explosion of popular articles as physicists boldy went into a landscape previously reserved for science fiction authors.  A Scientific American article "The Hidden Dimensions of Spacetime" written by Freeman and  van Nieuwenhuizen tried to set the record straight on what all of this meant, but any intuitive understanding of 11-D spacetime seemed to elude the grasp of even the most ardent science popularizers.

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Citation

Odenwald, Sten, Ph.D. (Contributing Author); Bernard Haisch (Topic Editor). 2008. "Gravity: Kaluza-Kline Theory." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published February 12, 2008].
<http://www.cosmosportal.org/articles/view/135647/>

 

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(2009). Gravity: Kaluza-Kline Theory. Retrieved from http://www.cosmosportal.org/view/article/135647

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