Asymptotic Structure with vanishing cosmological constant

Abstract

This is the first of two papers [1] devoted to the asymptotic structure of space-time in the presence of a non-negative cosmological constant Λ. This first paper is concerned with the case of Λ = 0. Our approach is fully based on the tidal nature of the gravitational field and therefore on the 'tidal energies' built with the Weyl curvature. In particular, we use the (radiant) asymptotic supermomenta computed from the rescaled Weyl tensor at infinity to provide a novel characterisation of radiation escaping from, or entering into, the space-time. Our new criterion is easy to implement and shown to be fully equivalent to the classical one based on the news tensor. One of its virtues is that its formulation can be easily adapted to the case with Λ > 0 covered in the second paper. We derive the general energy-momentum-loss formulae including the matter terms and all factors associated to the choices of arbitrary foliation and of supertranslation. We also revisit and present a full reformulation of the traditional peeling behaviour with a neat geometrical construction that leads, in particular, to an asymptotic alignment of the supermomenta in accordance with the radiation criterion.

The peeling theorem with arbitrary cosmological constant

“…This does not excludes that, in the Λ = 0 case, for particular γ some components may be more degenerate in the Petrov-classification sense -in concordance with the discussion in [11], see also [15]. Moreover, when studying the radiative components of the gravitational field, it is essential to take into account not only d αβγδ -this is further explained next, see also [13].…”

“…In [13] a derivation of the peeling theorem for Λ = 0 was presented based on a robust geometrical construction that indeed can be straightforwardly generalised to space-times with Λ = 0. Consider a future-directed curve γ (λ)…”

“…where ℓ α is the vector field tangent to γ(λ). Following the same scheme as in [13], which was inspired by [12], let t β α (λ i , λ j ) be the parallel propagator w.r.t. Γ a bc and t β α (λ i , λ j ) , the parallel propagator w.r.t.…”

“…Observe that such a γ(λ) is fully determined by p 0 ∈ J and an initial null vector ℓ α of T p 0 M , therefore all one has to do for the following construction is to pick up any null vector ℓ α | p 0 at the chosen point of p 0 ∈ J . Then the characterisation and properties of γ λ L β α can be found in [13], where it was shown to obey the linear system of ordinary differential equations…”

“…As explained in [13], the asymptotic behaviour of any physical field -here represented by the fully covariant version T µ 1 ...µr of its defining tensor field -can be obtained by applying the next steps…”