Covariant double-null dynamics: 2 + 2-splitting of the Einstein equations

Abstract

The paper develops a (2 + 2)-imbedding formalism adapted to a double foliation of spacetime by a net of two intersecting families of lightlike hypersurfaces. The formalism is two-dimensionally covariant, and leads to simple, geometrically transparent and tractable expressions for the Einstein field equations and the Einstein-Hilbert action, and it should find a variety of applications. It is applied here to elucidate the structure of the characteristic initial-value problem of general relativity.

Double null hamiltonian dynamics and the gravitational degrees of freedom

“…Note that these are just formulae (45) and (48) in Brady et al [11]. By setting a = 0 and a = 1 respectively the above quantities give essentially (modulo signs and factors of 2) the same as θ,θ and σ,σ of Hayward [9].…”

“…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.…”

“…For this reason we prefer to work with the 2+2 formalism of d'Inverno and Smallwood [28] since this applies to a general co-dimension 2 foliation. See the article by Brady et al [11] for a general discussion of a double null splitting of the Einstein equations. Throughout the paper Greek indices run from 0 to 3, early Latin indices (a, b, .…”

Metric perturbation approach to gravitational waves in isotropic cosmologies

“…The coordinates y, z, u, v are such that the surfaces u = const, v = const are two families of intersecting null hypersurfaces. The general form of line-element in a coordinate system based upon two families of intersecting null hypersurfaces is given in [9]. For our purposes we write this as…”

“…Next making use of the Ricci identities 9) and recalling that γ ab is divergenceless and orthogonal to u a we find (since C abcd = 0 in the background)…”