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The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the (− + + +) metric signature , the gravitational part of the action is given as [ 1 ]
The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be defined. However, the use of the Einstein–Hilbert action is appropriate only when the underlying spacetime manifold M {\displaystyle {\mathcal {M}}} is closed , i.e., a manifold which is both ...
The second-order formalism action is then acquired by substituting this expression for the spin connection back into the action, yielding additional quartic gravitino vertices, with the Einstein–Hilbert and Rarita–Schwinger actions now being written with a torsionless spin connection that explicitly depends on the vielbeins.
Another important action is the Plebanski action (see the entry on the Barrett–Crane model), and proving that it gives general relativity under certain conditions involves showing it reduces to the Palatini action under these conditions. Here we present definitions and calculate Einstein's equations from the Palatini action in detail.
Hilbert claimed priority for the introduction of the Riemann scalar into the action principle and the derivation of the field equations from it," [B 6] (Sauer mentions a letter and a draft letter where Hilbert defends his priority for the action functional) "and Einstein admitted publicly that Hilbert (and Lorentz) had succeeded in giving the ...
In general relativity and gravitation the Palatini variation is nowadays thought of as a variation of a Lagrangian with respect to the connection.. In fact, as is well known, the Einstein–Hilbert action for general relativity was first formulated purely in terms of the spacetime metric.
The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field. Maxwell's equations can be derived as conditions of stationary action. The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.
Action principles start with an energy function called a Lagrangian describing the physical system. The accumulated value of this energy function between two states of the system is called the action. Action principles apply the calculus of variation to the action. The action depends on the energy function, and the energy function depends on ...