Conformal Infinity

Abstract

The notion of conformal infinity has a long history within the research in Einstein’s theory of gravity. Today, “conformal infinity” is related to almost all other branches of research in general relativity, from quantisation procedures to abstract mathematical issues to numerical applications. This review article attempts to show how this concept gradually and inevitably evolved from physical issues, namely the need to understand gravitational radiation and isolated systems within the theory of gravitation, and how it lends itself very naturally to the solution of radiation problems in numerical relativity. The fundamental concept of null-infinity is introduced. Friedrich’s regular conformal field equations are presented and various initial value problems for them are discussed. Finally, it is shown that the conformal field equations provide a very powerful method within numerical relativity to study global problems such as gravitational wave propagation and detection.

Asymptotic directional structure of radiative fields in spacetimes with a cosmological constant

“…The above mentioned analysis lead Penrose to the elegant idea of (weakly) asymptotically simple spacetimes -those having a smoothg and Ω on I (see, e.g., [27,29,[33][34][35][60][61][62] and references therein for the precise definition). Because it entails a certain fall-off behaviour of the physical metric near I, this is a fruitful rigorous concept for studying asymptotic radiation properties of isolated systems in general relativity.…”

“…We will thus only assume the falloff typical for zero-rest-mass fields, without engaging in a study of the field equations. Motivated by discussion of behaviour of fields with a consistent field equation (s ≤ 2) in asymptotically flat spacetimes ( [29,33] or, e.g., [51,62] for gravitational field) we will assume…”

Constraint-preserving Sommerfeld conditions for the harmonic Einstein equations

“…In a realistic problem, such as the binary black hole problem, a global scheme is necessary to provide physically correct outer boundary data, either by using hyperboloidal time slices to extend the Cauchy evolution to infinity (see [22,23] for reviews) or by matching to an exterior characteristic or perturbative solution (see [24] for a review). Harmonic evolution offers important advantages for Cauchy-characteristic matching (CCM) which have led to successful matching in the linearized regime [10].…”

An elementary notion of gauge equivalence

“…11 Let d : Φ → d · Φ denote the obvious action of Diff(V ) on K (according to which a d acts on each component field of Φ in the usual way). We say that d ∈ Diff(V ) is a spatiotemporal symmetry of our theory it maps solutions to solutions (spacetime diffeomorphisms act locally, so a spacetime symmetry is a symmetry in the sense discussed above).…”