Curvature Invariants and the Geometric Horizon Conjecture in a Binary Black Hole Merger

TL;DR

This paper investigates curvature invariants in binary black hole mergers to test the geometric horizon conjecture, finding that certain level sets of a complex scalar invariant closely track the true geometric horizon during the merger.

Contribution

It provides numerical evidence that level--$ ext{0}$ sets of a complex scalar polynomial invariant can approximate the geometric horizon in binary black hole mergers.

Findings

Level--$ ext{0}$ sets of the invariant closely match the geometric horizon.

The best approximation occurs at $ ext{epsilon} = 10^{-3}$.

Level--$ ext{epsilon}$ sets track the horizon during merger.

Abstract

We study curvature invariants in a binary black hole merger. It has been conjectured that one could define a quasi-local and foliation independent black hole horizon by finding the level--$0$ set of a suitable curvature invariant of the Riemann tensor. The conjecture is the geometric horizon conjecture and the associated horizon is the geometric horizon. We study this conjecture by tracing the level--$0$ set of the complex scalar polynomial invariant, $\mathcal{D}$, through a quasi-circular binary black hole merger. We approximate these level--$0$ sets of $\mathcal{D}$ with level--$\varepsilon$ sets of $|\mathcal{D}|$ for small $\varepsilon$. We locate the local minima of $|\mathcal{D}|$ and find that the positions of these local minima correspond closely to the level--$\varepsilon$ sets of $|\mathcal{D}|$ and we also compare with the level--$0$ sets of $\text{Re}(\mathcal{D})$. The analysis provides evidence that the level--$\varepsilon$ sets track a unique geometric horizon. By studying the behaviour of the zero sets of $\text{Re}(\mathcal{D})$ and $\text{Im}(\mathcal{D})$ and also by studying the MOTSs and apparent horizons of the initial black holes, we observe that the level--$\varepsilon$ set that best approximates the geometric horizon is given by $\varepsilon = 10^{-3}$.