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[Clarification: I'm not claiming GR is wrong or anything. I'm asking if there are issues in interpreting GR as an action principle, and since I assume not, why not?]

We can get the equations of GR from the Einstein-Hilbert action by finding the extremal action in variations of the metric. However, the action does not appear to be bounded below as the curvature could just go arbitrarily negative for a finite period of time. Similarly it doesn't seem to be bounded from above. So are the equations we find when solving for the extremal action actually some kind of "saddle point"?

Doesn't this prevent us from considering GR in terms of a least action principle?

Is this issue of being unbounded from below in anyway related to the problems of trying to quantize gravity?

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# Einstein-Hilbert Action -> GR is a saddle point?

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