The text I'm reading claims that in an infinitesimally small region of the manifold, we can always find observers that locally see the metric as either Minkowski, $\eta_{\mu\nu}$, or Euclidean, $\delta_{\mu\nu}$. The observer seeing a Minkowski metric makes sense to me, if we look at a small enough region the curvature is small enough that the metric reduces locally to that of flat space, at least that's what I think. But finding the metric to be Euclidean implies $diag(1,1,1,1)$, unless I am wrong about that. My question is in what physical situation could an observer see a Euclidean metric?
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He can't see a Euclidean metric, except by restricting to space.
I think the writer did not mean what he appears to say, as quoted in the comment. I think he only intended to give two examples of diagonal metrics, not to give two examples of metrics which an observer can actually see.
Charles Francis
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