Conformal structures of static vacuum data

Details Conformal structures of static vacuum data Conformal structures of static vacuum data

2012-03-28

arxiv.org/abs/1203.6125v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

65 pages

Scientific paper

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is {\it necessary} that the Cotton tensor of $h$ satisfies a certain conformally invariant condition at space-like infinity, it is {\it sufficient} that the initial three-metric $h$ be asymptotically static at space-like infinity. The purpose of this article is to characterize the possible gap between these conditions. We show that with the class of metrics which satisfy the first condition and a certain non-degeneracy requirement is associated a one-form $\kappa$ with conformally invariant differential $d\kappa$ which allows us to formulate a criterion under which $h$ is asymptotically conformal to static data at space-like infinity. With the understanding presently available, showing that smoothness at the critical sets implies this criterion amounts essentially to showing that necessary and sufficient for the solution-jets to be smooth at the critical sets is that the initial metric $h$ behaves at space-like infinity asymptotically like static data.

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