Relate remnant DHS multipoles to progenitors

Derive explicit relations connecting the multipole moments and other invariant properties of the remnant dynamical horizon segment to those of the two progenitor dynamical horizon segments, establishing continuous links through the merger.

Background

Numerical evolutions show that progenitor DHSs evolve continuously and merge into the common DHS of the remnant, with inner horizon segments exhibiting intricate bifurcations and self-intersections. These complexities obscure direct comparisons of invariant quantities such as multipoles.

A relation between progenitor and remnant multipoles would bridge inspiral and ringdown regimes quasi-locally and reduce reliance on waveform matching, offering deeper insight into strong-field dynamics.