Forumulation of field equations

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this is a question more about the history of the field equations, but did Einstein get his field equations originally by varying the Einstein-Hilbert action or was it that the field equations was were found first then the they can also be shown by the varying the action. my feeling is that its the latter but i just want be cleared about any doubts.
 

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haushofer
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this is a question more about the history of the field equations, but did Einstein get his field equations originally by varying the Einstein-Hilbert action or was it that the field equations was were found first then the they can also be shown by the varying the action. my feeling is that its the latter but i just want be cleared about any doubts.
This has a nice piece of history :)

David Hilbert first found the field equations, by varying his action. Einstein found the same equations, but his motivation was a covariant extension of Poisson's equation for the gravitational field. I believe that Hilbert found the field equations 5 days earlier than Einstein did, but he was influenced by Einstein's equivalence principe which led to the idea of gravity as space-time geometry. So the equations are rightly named after Einstein, whereas the action is called the "Hilbert action".
 
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