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

TL;DR

This review surveys how total energy-momentum and mass are defined in general relativity when the cosmological constant is strictly positive. It contrasts Λ>0 with the familiar Λ=0 and Λ<0 cases, detailing ADM-type charges anchored to spatial infinity, and TBS-type charges defined on cuts of the conformal boundary, with spinorial and twistor techniques unifying the approaches. Key insights include the role of the conformal boundary ${\cal I}^+$, the absence of canonical translations near ${\cal I}^+$ in de Sitter backgrounds, and the various positivity and rigidity results under specific boundary conditions. The work also highlights Penrose’s proposals, the mass-loss behavior in asymptotically de Sitter spacetimes, and the construction of mass for closed universes, providing a comprehensive roadmap for defining and understanding gravitational energy in expanding cosmologies. Collectively, these results illuminate how energy-momentum notions adapt to a universe with accelerated expansion and guide future developments in quasi-local and global mass concepts in de Sitter-like spacetimes.

Abstract

A review is given of the various approaches to and expressions for total energy-momentum 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.

Figures (4)

• Figure 1: Foliations of the Minkowski spacetime: i. The hypersurfaces $\Sigma_t$ form a global foliation. Each of the leaves is a global Cauchy surface with $\mathbb{R}^3$ topology, and all these extend to spatial infinity $i^0$. These hypersurfaces are both intrinsically and extrinsically asymptotically flat. The $t={\rm const}$ hyperplanes in the Cartesian coordinates are like this. ii. The hypersurfaces $\Sigma_\tau$, which are only partial Cauchy surfaces, extend to the future null infinity $\mathscr{I}^+$ and foliate both the spacetime and $\mathscr{I}^+$. These hypersurfaces are intrinsically asymptotically hyperboloidal and the extrinsic curvature is asymptotically proportional to the intrinsic metric. For fixed $T>0$, in the Cartesian coordinates, the hypersurfaces $\tau:=t-\sqrt{T^2+r^2}={\rm const}$ have this character. iii. However, for fixed $\tau$ and variable $T$, the hypersurfaces $\Sigma _T$, given by $T^2:=(t-\tau)^2-r^2={\rm const}$, foliate only the chronological future of the point $(\tau,0,0,0)$ but do not provide a foliation of (even an open subset of) $\mathscr{I}^+$.
• Figure 2: Foliations of the anti-de Sitter spacetime: i. The hypersurfaces $\Sigma_t$ form a global foliation. The leaves $\Sigma_t$ are partial Cauchy surfaces with $\mathbb{R}^3$ topology, and these foliate the (timelike) conformal boundary $\mathscr{I}$, too. The $t={\rm const}$ hypersurfaces in the global coordinates of the anti-de Sitter spacetime are like this. These hypersurfaces are intrinsically hyperboloidal and extrinsically flat. ii. The hypersurfaces $\Sigma_T$ foliate neither the whole spacetime nor its conformal boundary $\mathscr{I}$. They foliate only a globally hyperbolic open subset. These hypersurfaces are intrinsically asymptotically hyperboloidal, and their extrinsic curvature is asymptotically proportional to the intrinsic metric. Such a $T$, whose level sets are the hypersurfaces $\Sigma_T$, is the time coordinate in the FRW form of the anti-de Sitter metric. (See also Figure 20 of HE.)
• Figure 3: Foliations of the de Sitter spacetime: i. The hypersurfaces $\Sigma_t$ form a global foliation. The leaves $\Sigma_t$ are global Cauchy surfaces with $S^3$ topology. The $t={\rm const}$ hypersurfaces in the global coordinates of the de Sitter spacetime are like this, which are metric spheres. ii. The hypersurfaces $\Sigma_{\hat{t}}$ foliate the 'steady state' part of (or 'Poincaré patch' in) the (half) de Sitter spacetime. Their topology is $\mathbb{R}^3$, they are intrinsically asymptotically flat and their extrinsic curvature is asymptotic to a nonzero value. Their one-point compactification yields a 'spatial infinity' $i^0$, which is a point of the future conformal boundary $\mathscr{I}^+$. These hypersurfaces do not foliate $\mathscr{I}^+$. (See also Figure 17 of HE.)
• Figure 4: Foliations of the de Sitter spacetime: iii. The hypersurfaces $\Sigma _{\hat{\tau}}$ foliate both the 'steady state' part of the de Sitter spacetime and the (spacelike) future conformal boundary $\mathscr{I}^+$ (minus $i^0$). The leaves $\Sigma_{\hat{\tau}}$ are intrinsically asymptotically hyperboloidal, and their extrinsic curvature is asymptotically proportional to the intrinsic metric. iv. The hypersurfaces $\Sigma_T$ foliate the disjoint union of the past domain of dependence of the $r\in[0,\pi/2)$ and the $r\in(\pi/2,\pi]$ hemispheres of the conformal boundary $\mathscr{I}^+\approx S^3$. They do not provide a foliation of the conformal boundary itself.