Einstein–Hilbert action
Concept in general relativity / From Wikipedia, the free encyclopedia
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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]
where is the determinant of the metric tensor matrix, is the Ricci scalar, and is the Einstein gravitational constant ( is the gravitational constant and is the speed of light in vacuum). If it converges, the integral is taken over the whole spacetime. If it does not converge, is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler–Lagrange equation of the Einstein–Hilbert action. The action was proposed[2] by David Hilbert in 1915 as part of his application of the variational principle to a combination of gravity and electromagnetism.[3]: 119