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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]
Albert Einstein's discovery of the gravitational field equations of general relativity and David Hilbert's almost simultaneous derivation of the theory using an elegant variational principle, [B 1]: 170 during a period when the two corresponded frequently, has led to numerous historical analyses of their interaction.
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.
Albert Einstein presented the theories of special relativity and general relativity in publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental results on which he based his theories, most notably to the work of Henri Poincaré and Hendrik Lorentz for special relativity, and to the work of David ...
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 ...
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 action consists of the Einstein–Hilbert action and the Rarita–Schwinger action. The theory was first formulated by Daniel Z. Freedman, Peter van Nieuwenhuizen, and Sergio Ferrara, and independently by Stanley Deser and Bruno Zumino in 1976.
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.