Linear algebra nugget of the day. You can classify a 2d quadratic form without finding eigenvalues. Det tells you if the eigenvalues are the same sign and if det is positive, trace tells you which sign. Exercise: what if det is 0?
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No excuse: in GR you don't have "one" metric, you have many: one for each solution. So you gotta pay attention to which maps depend on the metric and which don't. I know you folks know this... but the kids reading this need to be told, early and often!
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I completely agree that it is wise to remain at least dimly aware of how the many geometrical structures present in each solution of the Einstein equation depend upon the spacetime metric (just as it is wise to understand Euclidean geometry as a thing of many interrelated parts.)
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