Discussion Overview
The discussion revolves around the representation of black holes and their event horizons from a manifold perspective within the framework of General Relativity (GR). Participants explore the nature of singularities and their embeddings in spacetime manifolds, considering both theoretical implications and mathematical representations.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant questions whether black holes can be viewed as submanifolds within a smooth 4-manifold endowed with a metric, specifically regarding the event horizon and singularity.
- Another participant agrees that black holes can be considered submanifolds but notes that the singularity's status is more complex, suggesting it is not part of the manifold.
- A different viewpoint suggests that if the singularity were to be defined as part of the manifold, it would not be a point but rather a spacelike surface.
- Concerns are raised about the nature of the singularity, with one participant describing it as approaching a collapsed 3-cylinder structure, complicating its representation.
- Some participants argue that the singularity could be conceptualized as a line in spacetime, but this perspective is challenged by others who emphasize the topological issues involved.
- There is a discussion about the dimensionality of the singularity, with one participant noting that the dimensionality can vary based on the choice of coordinates, particularly in the context of Schwarzschild spacetime.
- Another participant highlights the difficulty in defining the singularity's dimensionality due to the absence of a metric at that point, suggesting that indirect methods may be necessary for such definitions.
Areas of Agreement / Disagreement
Participants express differing views on the nature of singularities and their representation within manifolds. There is no consensus on whether singularities can be treated as points, lines, or surfaces, and the discussion remains unresolved regarding the implications of these representations.
Contextual Notes
The discussion touches on complex mathematical and topological issues related to the representation of singularities and event horizons, with limitations stemming from the absence of a well-defined metric at singularities in certain coordinate systems.