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

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 $varepsilon = 10^{-3}$.