Forumulation of field equations

[captain]

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.

 

[haushofer]

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".

 

[Entropia]

The formulation of field equations is a fundamental aspect of physics, particularly in the study of general relativity. The field equations, also known as Einstein's equations, describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. These equations are crucial in understanding the behavior of gravity and the dynamics of the universe.

To answer your question, Einstein did indeed derive his field equations by varying the Einstein-Hilbert action. This is known as the Einstein field equations and is the most famous formulation of general relativity. In this formulation, the field equations are obtained by varying the action with respect to the metric tensor, which represents the curvature of spacetime.

However, it is worth noting that the field equations were not found first and then shown to be equivalent to the varying of the action. In fact, Einstein and Hilbert both independently arrived at the same equations by different methods. Einstein used his theory of general relativity to derive the field equations, while Hilbert used a variational approach similar to the one used in the Einstein-Hilbert action.

In summary, the Einstein field equations were originally derived by varying the Einstein-Hilbert action, but they were also found independently by Hilbert using a different approach. Both methods ultimately lead to the same equations, solidifying the importance of the field equations in understanding the behavior of gravity.