Vacuum Stability in Kaluza-Klein Theories

Abstract

When it was discovered that string theories are inconsistent except when they are formulated in spacetimes of either 26 or 10 dimensions, a new reason for investigating dynamical geometry was born. Up to this point the general theory of relativity had stood in isolation. The gravitational force, based as it was in the geometry of spacetime, seemed different from the other “fundamental” forces. It is true that, among the proposals for unification, there was an early suggestion by Kaluza² that spacetime should be viewed as a 5-dimensional cylinder in order to put electromagnetism on the same geometrical footing as gravitation. Interesting as it was, this idea was somewhat premature in that many fundamental forces had yet to be discovered. This may yet be the case, if electromagnetism is to be explained by such a mechanism then the distance scales (e. g. the cylinder’s radius) must be smaller by many orders than what is experimentally resolvable. This means that the most novel and distinctive qualities of the 5-dimensional geometry, the most characteristic predictions of the theory, cannot be tested. For this reason, the Kaluza- Klein theories have remained for many years a curiosity, and Einstein’s theory of gravitation maintains in isolation.