Gravity and Unification: A review

TL;DR

This review explores classical unified theories combining gravity with other interactions, focusing on internal space enlargements within four-dimensional spacetime and comparing various formalisms and their achievements.

Contribution

It provides a comprehensive comparison of different internal space unification schemes in gravity, highlighting their formalism and relation to Kaluza-Klein theories.

Findings

Enlarging internal space often yields gravity and Yang-Mills fields

Different formalisms like Einstein-Cartan and BF are reviewed and contrasted

Unified theories are compared to Kaluza-Klein approach

Abstract

We review various classical unified theories of gravity and other interactions that have appeared in the literature, paying special attention to scenarios in which spacetime remains four-dimensional, while an "internal" space is enlarged. The starting point for each such unification scenario is a particular formalism for General Relativity. We thus start by reviewing, besides the usual Einstein-Hilbert and Palatini formulations, the Einstein-Cartan, MacDowell-Mansouri and BF (both non-chiral and chiral) formulations. Each of these introduces some version of "internal" bundle and a dynamical variable that ties the internal and tangent bundles. In each of these formulations there is also an independent connection in the "internal" bundle. One can then study the effects of "enlarging the internal space", which typically leads to a theory of gravity and Yang-Mills fields. We review what has been done in the literature on each of these unification schemes, and compare and contrast their achievements to those of the better developed Kaluza-Klein scenario.