4-Dimensional Kaluza-Klein Approach to General Relativity in the (2,2)-Splitting of Spacetimes

Details 4-Dimensional Kaluza-Klein Approach to General Relativity in the (2,2)-Splitting of Spacetimes 4-Dimensional Kaluza-Klein Approach to General Relativity in the (2,2)-Splitting of Spacetimes

1996-11-21

arxiv.org/abs/gr-qc/9611050v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

37 pages, RevTex, contains 1 ps file for one figure. Downloading the PostScript file is recommended for Web users. No change w

Scientific paper

We propose a Kaluza-Klein approach to general relativity of 4-dimensional spacetimes. This approach is based on the (2,2)-splitting of a generic 4-dimensional spacetime, which is viewed as a local product of a (1+1)-dimensional base manifold and a 2-dimensional fibre space. In this Kaluza-Klein formalism we find that general relativity of 4-dimensional spacetimes can be interpreted in a natural way as a (1+1)-dimensional gauge theory. In a gauge where the (1+1)-dimensional metric can be written as the Polyakov metric, the action is describable as a (1+1)-dimensional Yang-Mills type gauge theory action coupled to a (1+1)-dimensional non-linear sigma field and a scalar field, subject to the constraint equations associated with the diffeomorphism invariance of the (1+1)-dimensional base manifold. Diffeomorphisms along the fibre directions show up as the Yang-Mills type gauge symmetries, giving rise to the Gauss-law constraints. We also present the Einstein's equations in the Polyakov gauge and discuss them from the (1+1)-dimensional gauge theory point of view. Finally we show that this Kaluza-Klein formalism is closely related to the null hypersurface formalism of general relativity.

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