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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.
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 ...
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]
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, =,
In general relativity, Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity, [1] is a modification of the Einstein–Hilbert action to include the Gauss–Bonnet term [2] (named after Carl Friedrich Gauss and Pierre Ossian Bonnet)
The integral of is known as the Einstein–Hilbert action. The Riemann tensor is the tidal force tensor, and is constructed out of Christoffel symbols and derivatives of Christoffel symbols, which define the metric connection on spacetime. The gravitational field itself was historically ascribed to the metric tensor; the modern view is that the ...
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 ...