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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]
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 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 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 ...
For comparison with alternatives, the formulas of General Relativity [4] [5] are: = = = which can also be written = (). The Einstein–Hilbert action for general relativity is:
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, =,
For the second step see the article about the Einstein–Hilbert action. Since δ Γ μ ν λ {\displaystyle \delta \Gamma _{\mu \nu }^{\lambda }} is the difference of two connections, it should transform as a tensor.