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

TL;DR

This paper develops the Hamiltonian formalism for the (2+2) decomposition of general relativity without symmetry assumptions, revealing a rich algebraic structure including diffeomorphisms, Virasoro, and supertranslation-like symmetries.

Contribution

It derives the Hamiltonian and constraint algebra in the (2+2) formalism, uncovering new infinite-dimensional symmetry subalgebras generalizing BMS and Spi groups.

Findings

Constraint algebra includes diffeomorphisms, Virasoro, and supertranslation-like symmetries.

Hamilton's equations are equivalent to Einstein's equations in this formalism.

The algebraic structure suggests a generalization of asymptotic symmetry groups.

Abstract

The canonical formalism of the (2+2) formulation of general relativity of 4 spacetime dimensions is studied under no symmetry assumptions, where the spacetime is viewed as a local product of a 2 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 3 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 2 dimensional vertical space, infinite dimensional Virasoro algebra associated with the 2 dimensional base manifold, and an analog of supertranslation. The symmetry algebra may be viewed as a generalization of the BMS or Spi group to a finite distance.