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 \(\varepsilon = 10^{-3}\).
Data Availability
Complete data for this work is presented in [49], which is available on https://dalspace.library.dal.ca/handle/10222/79721, and is also available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported financially by NSERC (AAC and ES). JMP would like to thank AAC for supervising his masters thesis and ES for numerical assistance and useful discussions, and the Perimeter Institute for Theoretical Physics for hospitality during this work.
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Peters, J.M., Coley, A. & Schnetter, E. Curvature invariants in a binary black hole merger. Gen Relativ Gravit 54, 65 (2022). https://doi.org/10.1007/s10714-022-02944-1
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DOI: https://doi.org/10.1007/s10714-022-02944-1
Keywords
- Geometric horizon
- Geometric horizon conjecture
- Marginally outer trapped surface
- Apparent horizon
- Event horizon
- Black hole horizon
- Binary black hole merger
- Petrov classification
- Boost weight classification
- Scalar polynomial invariant
- Differential scalar polynomial invariant
- Extended cartan invariant
- Weyl tensor
- Level sets
- Numerical relativity
- Differential geometry