HYPERBOLOIDAL FOLIATIONS WITH ℐ-FIXING IN SPHERICAL SYMMETRY

Abstract

We study in spherical symmetry the conformal compactification for hyperboloidal foliations with nonvanishing constant mean curvature. The conformal factor and the coordinates are chosen such that null infinity is at a fixed radial coordinate location. Hyperboloidal surfaces in asymptotically flat spacetimes have first been used for an initial value formulation of the Einstein equations by Friedrich. 4 Instead of approaching spatial infinity as Cauchy surfaces do, they reach null infinity, I, which makes them suitable for radiation extraction. Contrary to characteristic surfaces, these spacelike surfaces are as flexible as Cauchy surfaces and they can be used in numerical calculations with the 3+1 approach based on a hyperboloidal initial value problem. 3,5–7 We want to study the conformal compactification 11 of hyperboloidal foliations in spherical symmetry. It has been suggested 1,2,8 that conformal compactifications in which I is kept at a fixed spatial coordinate location might be useful for testing new ideas in numerical calculations. Here we explicitly discuss the simplest cases, namely the Minkowski and Schwarzschild spacetimes. The physical line element in spherical symmetry can be written as