Discussion Overview
The discussion revolves around the embedding of the Schwarzschild metric within higher-dimensional spaces, specifically exploring whether pseudoremannian manifolds can be isometrically embedded in pseudoeuclidean spaces. The scope includes theoretical considerations of general relativity and mathematical implications of embedding theorems.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- Some participants reference Nash's theorem, questioning its applicability to pseudoremannian manifolds and the possibility of finding a submanifold in pseudoeuclidean space that induces the Schwarzschild metric.
- One participant cites Chris Clarke's work, stating that every 4-dimensional spacetime can be isometrically embedded in a higher-dimensional flat space, with 90 dimensions being sufficient for all spacetimes.
- Another participant claims that 6 dimensions are needed to embed a Schwarzschild solution, suggesting that all general relativity solutions can be locally embedded in 10 dimensions.
- A follow-up question challenges the assertion about needing 6 dimensions for the Schwarzschild solution, asking for the basis of that claim.
Areas of Agreement / Disagreement
Participants express differing views on the dimensionality required for embedding the Schwarzschild solution, with some asserting 6 dimensions and others referencing the broader context of 90 dimensions for general spacetimes. The discussion remains unresolved regarding the exact dimensional requirements.
Contextual Notes
The discussion includes assumptions about the nature of embeddings and the specific characteristics of the Schwarzschild metric, which may not be fully articulated or agreed upon by all participants.