We develop the general theory of relativity in a formalism with extended causality that describes physical interaction through discrete, transversal and localized pointlike fields. The essence of this approach is of working with fields defined with support on straight lines and not on hypersurfaces as usual. The general relativity homogeneous field equations are then solved for a finite, singularity-free, point-like field that we associate with a `classical graviton'. The standard Einstein continuous formalism is retrieved by means of an averaging process, and its continuous solutions are determined by the chosen imposed symmetry. The Schwarzschild metric is obtained by imposing spherical symmetry on the averaged field.