Discussion Overview
The discussion revolves around finding an example of a two-dimensional metric that lacks any Killing vectors. Participants explore the challenges of proving that a metric is unsymmetrical and consider various approaches to tackle the problem.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant requests examples of two-dimensional metrics without Killing vectors and notes the difficulty in proving unsymmetry due to potential coordinate issues.
- Another participant references a paper that discusses a spacetime with no Killing vectors, although it is not specifically two-dimensional, suggesting it may provide useful insights.
- A different participant proposes a potentially absurd metric as a candidate, indicating a willingness to explore unconventional solutions.
- Another participant suggests that a metric induced from three-dimensional flat Euclidean space on a two-dimensional ellipsoid with unequal sides may not possess any Killing vectors, questioning the preservation of the ellipsoid by the Killing vectors of Euclidean space.
- One participant expresses frustration and curiosity about the problem without providing a specific solution.
Areas of Agreement / Disagreement
Participants do not reach a consensus on a specific example of a two-dimensional metric without Killing vectors, and multiple competing ideas and approaches are presented.
Contextual Notes
The discussion highlights the complexity of proving the absence of Killing vectors and the potential influence of coordinate choices on the characterization of metrics.
Who May Find This Useful
Researchers and students interested in differential geometry, general relativity, and the properties of metrics in theoretical physics may find this discussion relevant.