Multidimensional Unified Theories

Abstract

An extended spacetime, M4+N, is a Riemannian (4 + N)‐dimensional manifold which admits an N‐parameter group G of (spacelike) isometries and is such that ordinary spacetime M4 is the space M4+N/G of the equivalence classes under G‐transformations of M4+N. A multidimensional unified theory (MUT) is a dynamical theory of the metric tensor on M4+N, the metric being determined from the Einstein‐Hilbert action principle: in absence of matter, the Lagrangian is (essentially) the total curvature scalar of M4+N. A MUT is an extension of the Cho‐Freund generalization of Jordan's five‐dimensional theory. A MUT can be faithfully translated in four‐dimensional language: as a theory on M4, a MUT is a gauge field theory with gauge group G. A unifying aspect of MUT's is that all fields occur as elements of the metric tensor on M4+N. When the isometry generators are subjected to strongest constraints, a MUT becomes the De Witt‐Trautman generalization of Kaluza's five‐dimensional theory; in four‐dimensional language, this is the theory of Yang‐Mills gauge fields coupled to gravity. With weaker constraints, a MUT appears to be more natural than a Yang‐Mills theory as a physical realization of the gauge principle for an exact symmetry of gauged confined color. Such weakly‐constrained MUT leads to bag‐type models without the need for ad hoc surgery on the basic. Lagrangian. The present paper provides a detailed introduction to the formalism of multidimensional unified gauge field theory.

Kaluza-Klein gravity

“…In general, however, spacetime can still be coaxed into compactifying in the desired manner -at the cost of altering the higher-dimensional vacuum Einstein equations, either by incorporating torsion [65][66][67][68], adding higher-derivative terms (eg., R 2 ) onto the Einstein action [69], or -last but not leastadding an explicit higher-dimensional energy-momentum tensor to the theory. If chosen judiciously, this last will induce "spontaneous compactification" of the extra dimensions, as first demonstrated by Cremmer & Scherk [70,71].…”

“…The point of this exercise is that all four of the metrics (68), (83), (85) and (90) are flat in five dimensions, although they would be perceived very differently by four-dimensional observers (as evinced by their expansion factors and equations of state). The reason for these differences is the ψ-dependence of the coordinate transformations, and the fact that the theory is covariant with respect to five-, not four-dimensional coordinates.…”

“…The cosmological metric (68), and the others mentioned in § 6.5, are all fivedimensional generalizations of spatially flat four-dimensional FRW spacetimes. One could also consider a curved version of the homogeneous and isotropic line element (65):…”

“…In the previous section we worked entirely in coordinates defined by the cosmological metric (68). We are of course free to transform to coordinates in which the spatial components of the five-velocity are not comoving.…”

Construction of a Kaluza-Klein type Theory from One Dimension

“…The unifying framework for gravitation and gauge theories reviewed here is constructed in the mathematical setting of a principle fibre bundle. Keeping within the spirit of Einstein's original 4-dimensional spacetime theory of gravitation and the incorporation of electromagnetism in the extension to a 5-dimensional arena by Kaluza and Klein [1,2], the generalisation for geometric unification with non-Abelian gauge theory is founded upon a metric tensor ǧ, now defined upon the manifold of the principle bundle P = (M 4 , G) itself ( [12], see also [13], [14] sections I-V and [15]).…”