arXiv:2410.07231v1 Announce Type: new
Abstract: We point out that the geometry of connected totally geodesic compact null hypersurfaces in Lorentzian manifolds is only slightly more specialized than that of Riemannian flows over compact manifolds, the latter mathematical theory having been much studied in the context of foliation theory since the work by Reinhart (1959). We are then able to import results on Riemannian flows to the horizon case, so obtaining theorems on the dynamical structure of compact horizons that do not rely on (non-)degeneracy assumptions. Furthermore, we clarify the relation between isometric/geodesible Riemannian flows and non-degeneracy conditions. This work also contains some positive results on the possibility of finding, in the degenerate case, lightlike fields tangent to the horizon that have zero surface gravity.
Future Roadmap for Readers: Challenges and Opportunities on the Horizon
To better understand the conclusions of the mentioned text and to highlight potential challenges and opportunities in the future, we provide a roadmap for readers:
1. Background: Riemannian Flows over Compact Manifolds
Start by revisiting the mathematical theory of Riemannian flows over compact manifolds. Understand the key concepts and results, with a focus on the work by Reinhart in 1959. Explore the relationship between Riemannian flows and foliation theory.
2. Connected Totally Geodesic Compact Null Hypersurfaces in Lorentzian Manifolds
Examine the geometry of connected totally geodesic compact null hypersurfaces in Lorentzian manifolds. Compare this specialization to that of Riemannian flows over compact manifolds. Understand how this connection opens up new possibilities and applications.
3. Importing Results on Riemannian Flows to the Horizon Case
Investigate how the results on Riemannian flows can be imported and applied to the horizon case. Explore the theorems on the dynamical structure of compact horizons that do not rely on (non-)degeneracy assumptions. Recognize the significance of this importation for understanding compact horizons.
4. Relationship between Isometric/Geodesible Riemannian Flows and Non-Degeneracy Conditions
Gain insights into the relation between isometric/geodesible Riemannian flows and non-degeneracy conditions. Explore the implications of this relationship for understanding the behavior of flows and the possibility of finding lightlike fields tangent to the horizon with zero surface gravity.
5. Positive Results on the Degenerate Case
Analyze the positive results presented in the work regarding the possibility of finding lightlike fields tangent to the horizon in the degenerate case with zero surface gravity. Understand the implications and potential applications of these findings.
Overall, this roadmap provides an overview of the important concepts and conclusions discussed in the mentioned text. It offers readers the opportunity to delve into the specialized field of connected totally geodesic compact null hypersurfaces in Lorentzian manifolds and the application of Riemannian flow theory. While the importation of results and the exploration of the relationship between different mathematical concepts present exciting opportunities, challenges may arise in understanding the complex geometry and applying the theorems in practical scenarios. The future endeavors in this field hold the potential for further advancements in the dynamical structure of compact horizons and finding zero surface gravity lightlike fields.