The Schwarzschild Solution in the 4-Dimensional Kaluza-Klein Description of The Einstein's Equations

The Kaluza-Klein formalism of the Einstein's theory, based on the (2,2)-fibration of a generic 4-dimensional spacetime, describes general relativity as a Yang-Mills gauge theory on the 2-dimensional base manifold, where the local gauge symmetry is the group of the diffeomorphisms of the 2-dimensional fibre manifold. As a way of illustrating how to use this formalism in finding exact solutions, we apply this formalism to the spherically symmetric case, and obtain the Schwarzschild solution by solving the field equations.