Double null hamiltonian dynamics and the gravitational degrees of freedom

Abstract

In this paper we review the Hamiltonian description of General Relativity using a double null foliation. We start by looking at the 2+2 version of geometrodynamics and show the role of the conformal 2-structure of the 2-metric in encoding (through the shear) the 2 gravitational degrees of freedom. In the second part of the paper we consider instead a canonical analysis of a double null 2+2 Hamiltonian description of General Relativity in terms of self-dual 2-forms and the associated SO(3) connection variables. The algebra of first class constraints is obtained and forms a Lie algebra that consists of two constraints that generate diffeomorphisms in the two surface, a constraint that generates diffeomorphisms along the null generators and a constraint that generates self-dual spin and boost transformations.

A boundary term for the gravitational action with null boundaries

“…In this section, we shall detail the construction of a coordinate system adapted to a null surface and also choose a form for k a . This coordinate system has the double null coordinate system [33] as a special case. In fact, the form of the metric on the null surface will be identical to that in double null coordinates.…”

Wheeler-DeWitt equation in Null-foliated spacetimes

“…An attempt is made to do a 2+2 foliation of our spacetime which can be achieved through [5] σ µν = g µν + 2l (µ n ν) (8) where σ is the metric on the folia where l µ and n ν are null vectors normal to the foliation satisfying…”

“…However, for integrable configurations, it is desirable to have ω µ = 0. To understand this geometrically, consider the two projectors [8], [9] σ µν = g µν + l µ n ν + l ν n µ (31)…”

The extended Kerr-Schild approach to general relativity

“…We draw connections between the xKS form and another '(2+2)-split' ansatz, namely the doublenull foliation [14][15][16], and show that one can identify them in special gauges. We temporarily drop tilde for the double-null foliation formulae below.…”

“…It was shown [15,16] that gauge choices can be made so that s a A vanish, in which case it is easier to bring the 2-d metric h ab dθ a dθ b to a conformally flat form W (x, y) 2 (dx 2 + dy 2 ) in the so-called isothermal coordinates [17]. Then the metric (69) is in exactly the xKS form:…”