Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry

Abstract

We produce two classes of solutions to the vacuum Einstein equations in U (1) symmetry: in the first we construct solutions whose total incoming H 1 energy is bounded below, but whose initial data can be supported on an arbitrarily small set. The time of existence is bounded below independently of the degree of concentration. Smallness of the support is measured in a 2 + 1 picture, which arises after quotienting out by the U (1) symmetry. Our second class of solutions is a special case of the first one: here the energy of our solutions remains concentrated along a U (1)-family of geodesics, i.e., we can arrange for any desired fraction of the total incoming energy to remain inside an arbitrarily small solid prism around this U (1) family. Our proof relies on three key ingredients: a reduction of the Einstein vacuum equations in U (1) symmetry to a system of a free wave equation coupled to ODEs; a certain scaling of the resulting coupled system that we introduce; and a novel way to make use of decay properties of waves in the large, applying them to the analysis of this scaled-up system.