Hamiltonian structure and constraint algebra in the (2+2) formalism

Abstract

The canonical formalism of the (2+2) formulation of general relativity of four spacetime dimensions is studied under no symmetry assumptions, where the spacetime is viewed as a local product of a two dimensional base manifold of Lorentzian signature with the vertical space as its complement. The affine null parameter is chosen as the time coordinate whose level surfaces are three dimensional spacelike hypersurfaces. From the first-order action principle, Hamilton's equations of motion and the constraints are obtained, which are found to be equivalent to the Einstein's equations. The constraint algebra is also presented, which has interesting subalgebras such as the infinite dimensional Lie algebra of the diffeomorphisms of the two dimensional vertical space, infinite dimensional Virasoro algebra associated with the two dimensional base manifold, and an analogue of supertranslation. The symmetry algebra may be viewed as a generalization of the BMS or Spi group to a finite distance.