(2,2)-formalism of general relativity: an exact solution

We discuss the (2,2)-formalism of general relativity based on the (2,2)-fibration of a generic four-dimensional spacetime of the Lorentzian signature. In this formalism general relativity is describable as a Yang-Mills gauge theory defined on the (1 + 1)-dimensional base manifold, whose local gauge symmetry is the group of the diffeomorphisms of the two-dimensional fibre manifold. After presenting the Einstein's field equations in this formalism, we solve them for the spherically symmetric case to obtain the Schwarzschild solution. Then we discuss possible applications of this formalism.