Generating Functions in the (2+2)-Formalism of General Relativity

In the (2+2)-formalism, the vacuum Einstein equations can be decomposed into four divergence-type equations and the Hamilton's equations of motion that follow from a non-vanishing Hamiltonian. In this paper, we report that the quasi-local flux integrals associated with divergence-type equations are the generating functions of translations in the (1+1)-dimensional manifold and of the diffeomorphisms of the compact two-surface, which enables us to interpret them as quasi-local flux integrals of energy, linear momentum and angular momentum, respectively. At the null infinity, these fluxes reduce to the Bondi flux of energy, linear momentum and angular momentum. This formalism may be regarded as a 4-dimensional Kaluza-Klein theory without isometries in the (2,2)-splitting of spacetimes.