Asymptotics with a positive cosmological constant. IV. The no-incoming radiation condition

Abstract

Consider compact objects -such as neutron star or black hole binaries-in full, non-linear general relativity. In the case with zero cosmological constant Λ, the gravitational radiation emitted by such systems is described by the well established, 50+ year old framework due to Bondi, Sachs, Penrose and others. However, so far we do not have a satisfactory extension of this framework to include a positive cosmological constant -or, more generally, the dark energy responsible for the accelerated expansion of the universe. In particular, we do not yet have an adequate gauge invariant characterization of gravitational waves in this context. As the next step in extending the Bondi et al framework to the Λ > 0 case, in this paper we address the following questions: How do we impose the 'no incoming radiation' condition for such isolated systems in a gauge invariant manner? What is the relevant past boundary where these conditions should be imposed, i.e., what is the physically relevant analog of past null infinity I − o used in the Λ = 0 case? What is the symmetry group at this boundary? How is it related to the Bondi-Metzner-Sachs (BMS) group? What are the associated conserved charges? What happens in the Λ → 0 limit? Do we systematically recover the Bondi-Sachs-Penrose structure at I − o of the Λ = 0 theory, or do some differences persist even in the limit? We will find that while there are many close similarities, there are also some subtle but important differences from the asymptotically flat case. Interestingly, to analyze these issues one has to combine conceptual structures and mathematical techniques introduced by Bondi et al with those associated with quasi-local horizons. The framework introduced in this paper will serve as the point of departure in the construction of the analog(s) of future null infinity, I + o where the radiation emitted by isolated systems can be analyzed systematically.

Non-expanding horizons: multipoles and the symmetry group

“…This section is divided into three parts. In the first we introduce the symmetry Lie algebra g and the symmetry group G; in the second we discuss salient properties first of g and then of G; and in the third we compare G with the BMS group B and also with similar symmetry groups associated with internal boundaries that have been introduced in the literature [16,17,[31][32][33].…”

“…Interestingly, G is isomorphic to a one-dimensional extension of the (BMS) group B. In particular, in contrast to other works [16,17], the semi-direct product between the supertranslation subgroup and the Lorentz group is exactly the same as that of B. As already remarked, in the companion paper [5] we will discuss charges and fluxes associated with symmetries in G.…”

“…The first extension makes use of a simple observation [16] that has received relatively little attention so far: each NEH ∆ comes with a three parameter family of unit, round 2sphere metrics qab that are conformally related to the physical (degenerate) metric q ab on ∆. Using spherical harmonics of these round metrics, we will define multipole moments that provide an invariant characterization of the NEH geometry.…”

Charge algebra in Al(A)dSn spacetimes

“…While the holographic correspondence is less clear in asymptotically de-Sitter (dS) spacetimes [36][37][38], the latter are of major interest in cosmology since they offer simple mathematical models for the observed accelerating expansion of the universe. Imposing boundary conditions at the future spacelike boundary I + dS [39][40][41][42][43] delicate issue since it can drastically restrict the Cauchy problem by eliminating late-time radiation. One is naturally led to allow some non-vanishing flux going through the spacetime boundary and leaky boundary conditions become an essential ingredient.…”

Asymptotic Symmetries in the Gauge fixing Approach and the BMS group

“…The geometric approach was essentially used in gravity theory and led to much progress in the study of symmetries and symplectic structures for asymptotically flat spacetimes at null infinity [21][22][23][24] and spatial infinity [25,26]. It was also considered to study asymptotically (A)dS spacetimes [27][28][29][30]. Moreover, this framework was recently applied to study boundary conditions and associated phase spaces on null hypersurfaces [31].…”