Initial Data for General Relativity with Toroidal Conformal Symmetry

TL;DR

This paper introduces a new class of initial data for vacuum General Relativity that are globally regular, asymptotically flat, and possess a toroidal conformal symmetry, expanding the set of known solutions.

Contribution

It presents a novel construction of initial data with toroidal conformal symmetry, characterized by a decomposition into uncoupled ODEs on the orbit space.

Findings

Solutions are globally regular and asymptotically flat.

Data have no isometries but possess a $U(1)\times U(1)$ conformal symmetry.

The solutions are characterized by a countable family of uncoupled ODEs.

Abstract

A new class of time-symmetric solutions to the initial value constraints of vacuum General Relativity is introduced. These data are globally regular, asymptotically flat (with possibly several asymptotic ends) and in general have no isometries, but a $U(1)\times U(1)$ group of conformal isometries. After decomposing the Lichnerowicz conformal factor in a double Fourier series on the group orbits, the solutions are given in terms of a countable family of uncoupled ODEs on the orbit space.