Chapter07.pdfCan Killing Vectors Derive the Schwarzschild Metric?

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The discussion centers on the possibility of deriving the Schwarzschild metric using Killing vectors, suggesting this could simplify the process compared to using the Ricci tensor. However, it is noted that Killing vectors may not uniquely determine the metric, as demonstrated in a static case with three Killing vectors on S^2, where additional conditions are necessary to accurately recover the metric. The weak field limit is highlighted as a crucial requirement in this context. Overall, while the approach using Killing vectors is intriguing, it has limitations that must be addressed to ensure a precise derivation of the Schwarzschild metric. The conversation emphasizes the complexity of metric determination in general relativity.
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"Is it possible to derive the Schwarzschild metric from Killing vectors, thus saving all that work with the Ricci tensor etc."
 
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Given Killing vectors may not determine the metric uniquely. For example, in a specific static case with three Killing vectors on S^2 one still needs the weak field limit imposed on it to recover the metric precisely.

cf Chapt. 7 of
http://pancake.uchicago.edu/~carroll/notes/
 
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