Classical static final state of collapse with supertranslation memory

TL;DR

This paper explores how supertranslation symmetries influence the final static state of gravitational collapse, linking the supertranslation field to collapse details and deriving related energy conservation laws.

Contribution

It introduces a method to generate static solutions with supertranslation fields and relates the final state to collapse dynamics through energy conservation laws.

Findings

Final state characterized by mass, angular momentum, and supertranslation field.

Derived angle-dependent energy conservation law.

Predicted deviations from Schwarzschild metric based on energy flux modeling.

Abstract

The Kerr metric models the final classical black hole state after gravitational collapse of matter and radiation. Any stationary metric which is close to the Kerr metric has been proven to be diffeomorphic to it. Now, finite supertranslation diffeomorphisms are symmetries which map solutions to inequivalent solutions as such diffeomorphisms generate conserved superrotation charges. The final state of gravitational collapse is therefore parameterized by its mass, angular momentum and supertranslation field, signaled by its conserved superrotation charges. In this paper, we first derive the angle-dependent energy conservation law relating the asymptotic value of the supertranslation field of the final state to the details of the collapse and subsequent evolution of the system. We then generate the static solution with an asymptotic supertranslation field and we study some of its properties. Up to a caveat, the deviation from the Schwarzschild metric could therefore be predicted on a case-by-case basis from accurate modeling of the angular dependence of the ingoing and outgoing energy fluxes leading to the final state.