Understanding Einstein-Hilbert Action: Detailed Explanation from an Expert

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Can someone explain to me why this is the appropriate action?It makes some sens that that would be used, but I'd like a detailed explanation from someone familiar with the topic.

Why is it the one that yields the proper equations?
 
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Well, for a start, you can derive the field equations by setting variations of this action with respect to the metric equal to zero. This is done in many GR texts.
 
Yes but what is the intuition behind it? What lead hilbert to choose that particular action?
 
Can you think of a simpler scalar that encodes the curvature of spacetime? The Ricci scalar is a fairly obvious choice.
 
So essentially it used because its the simplest, most general(encodes curvature) involves the inverse metric, the ricci tensor, etc.. and such. So essentially its simple and encodes all the information that we may want tin the field equations...


This is the first time I've emcountered this topic(variational mechanics) so it seems quite exotic. Its very interesting though.

It would however be nice, to gain whatever insight I can into this principle and this topic in general.
 
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