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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 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.
Inspired by Einstein's work on general relativity, the renowned mathematician David Hilbert applied the principle of least action to derive the field equations of general relativity. [25]: 186 His action, now known as the Einstein–Hilbert action, =,
The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric. To take the metric and affine connection as independent variables in the action principle was first considered by Palatini. [1]
The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on the wall of the Rijksmuseum Boerhaave in Leiden, Netherlands. where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.
A discrete version of the Einstein–Hilbert action is obtained by considering so-called deficit angles of these blocks, a zero deficit angle corresponding to no curvature. This novel idea finds application in approximation methods in numerical relativity and quantum gravity , the latter using a generalisation of Regge calculus.
In general, the Lagrangian is that function which when integrated over produces the Action functional. David Hilbert gave an early and classic formulation of the equations in Einstein's general relativity. [2] This used the functional now called the Einstein-Hilbert action.
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.