Unique continuation and extensions of Killing vectors at boundaries for stationary vacuum space-times

TL;DR

This paper proves unique continuation and extension properties of Killing vectors in stationary vacuum space-times, showing that local boundary data determines the space-time geometry and symmetries near boundaries, generalizing previous Riemannian results.

Contribution

It extends Riemannian theorems to Lorentzian stationary vacuum space-times, establishing boundary uniqueness and symmetry extension results in both asymptotically AdS and general settings.

Findings

Two stationary vacuum space-times coinciding up to first order along a timelike hypersurface are locally isometric.

Killing initial data sets (KIDS) on the boundary extend to Killing vectors nearby.

Unique continuation holds near conformal infinity if metrics share the same conformal boundary and undetermined terms.

Abstract

Generalizing Riemannian theorems of Anderson-Herzlich and Biquard, we show that two $(n+1)$-dimensional stationary vacuum space-times (possibly with cosmological constant $\Lambda \in \R$) that coincide up to order one along a timelike hypersurface $\mycal T$ are isometric in a neighbourhood of $\mycal T$. We further prove that KIDS of $\partial M$ extend to Killing vectors near $\partial M$. In the AdS type setting, we show unique continuation near conformal infinity if the metrics have the same conformal infinity and the same undetermined term. Extension near $\partial M$ of conformal Killing vectors of conformal infinity which leave the undetermined Fefferman-Graham term invariant is also established.