arXiv:2504.01207v1 Announce Type: new
Abstract: We introduce the concept of $k-$future convex spacelike/null hypersurface $Sigma$ in an $n+1$ dimensional spacetime $M$ and prove that no $k-$dimensional closed trapped submanifold (k-CTM) can be tangent to $Sigma$ from its future side. As a consequence, k-CTMs cannot be found in open spacetime regions foliated by such hypersurfaces. In gravitational collapse scenarios, specific hypersurfaces of this kind act as past barriers for trapped submanifolds. A number of examples are worked out in detail, two of them showing 3+1 spacetime regions containing trapped loops ($k=1$) but no closed trapped surfaces ($k=2$). The use of trapped loops as an early indicator of black hole formation is briefly discussed.
Future Convex Spacelike/Null Hypersurfaces and Trapped Submanifolds: A Roadmap
Introduction
In this article, we will explore the concept of $k-$future convex spacelike/null hypersurfaces in an $n+1$ dimensional spacetime. We will discuss the implications of these hypersurfaces on the existence of $k-$dimensional closed trapped submanifolds (k-CTMs) and their relationship to open spacetime regions. Additionally, we will delve into the role of specific hypersurfaces as past barriers in gravitational collapse scenarios and their potential use in identifying black hole formation.
Understanding Future Convex Spacelike/Null Hypersurfaces
First, we will define and explain the concept of $k-$future convex spacelike/null hypersurfaces. These hypersurfaces exist in $n+1$ dimensional spacetimes and play a crucial role in understanding the behavior of trapped submanifolds.
Implications for Trapped Submanifolds
Next, we will examine the relationship between future convex spacelike/null hypersurfaces and $k-$dimensional closed trapped submanifolds (k-CTMs). We will present a proof demonstrating that no k-CTM can be tangent to the hypersurface from its future side. This result leads to the conclusion that k-CTMs cannot exist in open spacetime regions that are foliated by these hypersurfaces.
Gravitational Collapse Scenarios and Past Barriers
One of the significant findings of our study is the identification of specific hypersurfaces as past barriers for trapped submanifolds in gravitational collapse scenarios. We will explore the implications of these past barriers and their role in defining the behavior of trapped submanifolds during collapse.
Worked Examples
In this section, we will provide detailed worked examples to illustrate the concepts discussed. We will present two examples that depict 3+1 spacetime regions containing trapped loops ($k=1$) but no closed trapped surfaces ($k=2$). These examples will help solidify the understanding of the relationships between hypersurfaces, trapped submanifolds, and collapse scenarios.
Trapped Loops as Indicators of Black Hole Formation
Finally, we will briefly discuss the potential use of trapped loops as early indicators of black hole formation. This section will explore the significance of identifying trapped loops and their implications for understanding the formation and behavior of black holes.
Roadmap Challenges and Opportunities
- Challenge: The concept of future convex spacelike/null hypersurfaces may be complex for readers unfamiliar with advanced spacetime theories. Clear and concise explanations will be required to ensure understanding.
- Challenge: The proof that no k-CTM can be tangent to the hypersurface from its future side may require mathematical comprehension beyond the scope of some readers. Efforts should be made to present the proof in a simplified manner.
- Opportunity: The worked examples will provide concrete illustrations of the concepts discussed, aiding readers in visualizing the relationships between hypersurfaces, trapped submanifolds, and collapse scenarios.
- Opportunity: The potential use of trapped loops as indicators of black hole formation offers an exciting avenue for further research and exploration. This section may inspire readers to delve deeper into this area of study.
Conclusion
This roadmap has outlined the key elements of the article on future convex spacelike/null hypersurfaces, trapped submanifolds, and their implications in gravitational collapse scenarios. By addressing potential challenges and highlighting opportunities for further exploration, readers will gain a comprehensive understanding of the subject matter and its significance in advanced spacetime theories.