KIDs are non-generic

TL;DR

This paper proves that generic solutions to the vacuum Einstein equations lack both global and local symmetries, including Killing vectors, on various types of initial data surfaces, highlighting the rarity of symmetries in such solutions.

Contribution

It establishes that, generically, vacuum Einstein solutions do not admit any local or global Killing vectors or conformal Killing vectors, extending previous symmetry non-existence results.

Findings

Generic vacuum solutions lack global Killing vectors.

Generic vacuum solutions lack local Killing vectors.

Generic metrics do not have conformal Killing vectors.

Abstract

We prove that generic solutions of the vacuum constraint Einstein equations do not possess any global or local space-time Killing vectors, on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that non-existence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors.