Extremal black hole initial data deformations

TL;DR

This paper investigates how small deformations of axially symmetric initial data in Einstein-Maxwell theory affect the horizon structure, providing insights into the stability and uniqueness of black hole initial data.

Contribution

It demonstrates that $t$-$$ symmetry ensures a family of deformed data with unchanged horizon structure, aiding in understanding solution stability.

Findings

Deformations preserve the horizon structure under $t$-$$ symmetry.

Quantitative measures of solution proximity are established.

Results apply to initial data with specific asymptotic behaviors.

Abstract

We study deformations of axially symmetric initial data for Einstein-Maxwell equations satisfying the time-rotation ($t$-$\phi$) symmetry and containing one asymptotically cylindrical end and one asymptotically flat end. We find that the $t$-$\phi$ symmetry implies the existence of a family of deformed data having the same horizon structure. This result allows us to measure how close solutions to Lichnerowicz equation are when arising from nearby free data.