# On the local extension of Killing vector fields in electrovacuum spacetimes

## TL;DR

This paper extends the understanding of Killing vector fields in electrovacuum spacetimes, demonstrating conditions for their extension and providing examples where such extensions are impossible, thus advancing the geometric analysis of Einstein-Maxwell solutions.

## Contribution

It generalizes the extension of Killing vector fields to strong null convex domains in electrovacuum spacetimes and constructs examples of non-extendible stationary solutions.

## Key findings

- Extension of Killing vector fields to null convex domains in electrovacuum spacetimes.
- Existence of local stationary electrovacuum solutions without extension of the Hawking vector field.
- Generalization of Ionescu-Klainerman's results to the Einstein-Maxwell setting.

## Abstract

We revisit the problem of extension of a Killing vector field in a spacetime which is solution to the Einstein-Maxwell equation. This extension has been proved to be unique in the case of a Killing vector field which is normal to a bifurcate horizon by Yu. Here we generalize the extension of the vector field to a strong null convex domain in an electrovacuum spacetime, inspired by the same technique used by Ionescu-Klainerman in the setting of Ricci flat manifolds. We also prove a result concerning non-extendibility: we show that one can find local, stationary electrovacuum extension of a Kerr-Newman solution in a full neighborhood of a point of the horizon (that is not on the bifurcation sphere) which admits no extension of the Hawking vector field. This generalizes the construction by Ionescu-Klainerman to the electrovacuum case.