Construction of Hyperboloidal Initial Data

Let (M,g,Ω) be the conformal rescaling of a 3+1 dimensional vacuum spacetime (M̃, g̃), with g = Ωg̃, where Ω is a smooth function on M vanishing on the boundary ∂M. If M has smooth null boundary I , with connected components of topology S2×R, it follows that the Weyl tensor of (M,g) vanishes on I . From this follows peeling properties for (M̃, g̃). This picture of isolated systems in general relativity, developed by Penrose, see [?, ?], is useful for studying the mass and angular momentum of spacetimes, as well as gravitational radiation. Friedrich has found a first order symmetric–hyperbolic version of the Einstein evolution equations, called the conformally regular field equations, which may be extended through I [?]. The Cauchy data for this system on a future Cauchy surface M , asymptotic to I so that M̄ ∩ ∂M = ∂M , include gab,Kab,Ω as well as components of the rescaled Weyl tensor ΩCβγδ. In order to get a regular evolution at I , these data must be regular up to ∂M . It was shown by the author, Chrusciel and Friedrich [?, ?, ?, ?] that under certain conditions on the boundary geometry of M , the Cauchy data for the conformally regular field equations, are smooth up to ∂M . Using the conformally regular field equations, Friedrich has shown that the maximal vacuum development of data (called hyperboloidal data) on a future Cauchy surface intersecting I , regular up to ∂M is a spacetime which has a “smooth piece of I ”. Further, for small data the maximal vacuum development has a null boundary with future complete null generators and a regular timelike infinity. A programme has been initiated to numerically evolve the Einstein equations using the conformally regular field equations [?, ?, ?, ?]. This approach may have advantages over the “traditional” approach which treats the asymptotic flatness condition by introducing boundary conditions far away from the isolated system under study, and attempts to observe for example gravitational wave signatures on this boundary. In this note we will discuss the conformal procedure for constructing solutions (gab,Kab,Ω) to the constraint equations and the geometric conditions for regularity at I . I will also briefly discuss the Cauchy problem for the Einstein evolution equations at I , directly from the point of view of the Einstein equations in (M,g).