A review of total energy–momenta in GR with a positive cosmological constant

Abstract

A review is given of the various approaches to and expressions for total energymomentum and mass in the presence of a positive cosmological constant in Einstein's field equations, together with a discussion of the key conceptual questions, main ideas and techniques behind them. * Dedicated to our friend, Jörg Frauendiener, on the occasion of his 60th birthday.Ach, wie schön muss sich's ergehen, Dort in Newmans Himmelsraum. Und Lichtkugeln auf den Höhen -O wie lab'n sie meinen Traum ! from Ein theoretisch-physikalischer Schiller-Traum by N. V. Mitskievitch future of the point with Cartesian coordinates (τ, 0, 0, 0). All the leaves of this latter foliation intersect the future null infinity at one and the same cut u = τ , rather than providing a foliation of the null infinity (see Fig. 1.iii.).3 This can be proved, given certain causality assumptions; see [16].

Conserved charges in asymptotically de Sitter spacetimes

“…Our work explores notions of asymptotically de Sitter spacetimes and associated conserved charges. Over the last two decades there have been many discussions of asymptotically de Sitter spacetimes and de Sitter charges, see, e.g., [1,2,3,4,5,6,7,8,9] and reviews [10,11,12] for further references. A well studied notion of asymptotically de Sitter spacetimes in non-linear general relativity is "Dirichlet" or "reflective" boundary conditions at future infinity I + .…”

(A)dS 4 in Bondi gauge

“…To accommodate radiating spacetimes one needs to consider asymptotically locally (A)dS spacetimes with a time dependent boundary metric. While we now know the general asymptotic solution for such spacetimes, we do not know yet what is the correct identification of the appropriate notion of mass that accounts for the radiation (but see [8,[12][13][14][15][16][17]). Such mass should be monotonic and it is not yet clear whether the Bondi mass defined using (5.38) has such monotonicity properties.…”

On the canonical energy of weak gravitational fields with a cosmological constant $$\varLambda \in \mathbb {R}$$

“…Most of the results derived here have been summarised in [12]. Further results on, or related to, this problem can be found in [1][2][3]7,24,25,35,42,43,49,51,52].…”