Canonical general relativity on a null surface with coordinate and gauge fixing

Abstract

We use the canonical formalism developed together with David Robinson to study the Einstein equations on a null surface. Coordinate and gauge conditions are introduced to fix the triad and the coordinates on the null surface. Together with the previously found constraints, these form a sufficient number of second class constraints so that the phase space is reduced to one pair of canonically conjugate variables: A 3 2 and Σ 3 2 . The formalism is related to both the Bondi-Sachs and the Newman-Penrose methods of studying the gravitational field at null infinity. Asymptotic solutions in the vicinity of null infinity which exclude logarithmic behavior require the connection to fall off like 1/r 3 after the Minkowski limit. This, of course, gives the previous results of Bondi-Sachs and Newman-Penrose. Introducing terms which fall off more slowly leads to logarithmic behavior which leaves null infinity intact, allows for meaningful gravitational radiation, but the peeling theorem does not extend to Ψ 1 in the terminology of Newman-Penrose. The conclusions are in agreement with those of Chrusciel, MacCallum, and Singleton. This work was begun as a preliminary study of a reduced phase space for quantization of general relativity.

First order gravity on the light front

“…The light front formulation in Ashtekar variables was constructed in [15,16] and further investigated using the 2+2 formalism in [17,18,19]. These formulations expose additional features of the light-front theory, including the nice property that the first class part of the constraint algebra forms a Lie algebra, with proper structure constants, given by the semi-direct product of the hypersurface diffeomorphisms and the internal symmetry group.…”

The symplectic 2-form for gravity in terms of free null initial data

“…In fact canonical GR using constrained data on double null sheets has been developed by several researchers [Tor85,GRS92,GS95,d'ILV06]. Also, partial results have been obtained on the Poisson brackets of free data 5 The causal domain of dependence D[S] of a set S in a Lorentzian signature spacetime is the set of all points p such that every inextendible causal curve through p intersects S. If S is a closed achronal hypersurface one expects in physical theories that initial data on S fixes the solution in D [S].…”

“…[GR78,GS95]. In [GR78] Gambini and Restuccia give perturbation series in Newton's constant for the brackets of free data living on the bulk of N , but no brackets for other (necessary) data that live on the intersection surface S 0 .…”

Double null hamiltonian dynamics and the gravitational degrees of freedom

“…Goroff and Schwartz [35]). Some progress has already been made in this direction by Goldberg (together with Soteriou and Robinson) [16,22] for the case of a single null foliation. Furthermore based on calculations in the linearised case by Soteriou [23] there is strong support for the view that in the completely reduced phase space the gravitational degrees of freedom (of the real theory) can be identified with the real and imaginary parts of the connection component A 2 3 .…”

“…The key result that emerges from these studies is that in the double null description of metric dynamics the gravitational degrees of freedom have a simple description in terms of the conformal 2-structure (or equivalently the trace free shears in the two null directions) [10,11]. The application of a double null decomposition to connection dynamics is more recent [12][13][14] and builds very strongly on earlier work of Goldberg et al [15,16] who looked at a foliation by null hypersurfaces.…”