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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.
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.
Action principles are "integral" approaches rather than the "differential" approach of Newtonian mechanics.[2]: 162 The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.
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.
Comparison of the Ricci tensor with the Ricci tensor calculated from the principle of least action, Theoretical motivation for general relativity#Principle of least action in general relativity identifying the stress–energy tensor with the Hilbert stress-energy, and remembering that A+B=1 removes the ambiguity in A, B, and C.
The second-order formalism action is then acquired by substituting this expression for the spin connection back into the action, yielding additional quartic gravitino vertices, with the Einstein–Hilbert and Rarita–Schwinger actions now being written with a torsionless spin connection that explicitly depends on the vielbeins.
Another important action is the Plebanski action (see the entry on the Barrett–Crane model), and proving that it gives general relativity under certain conditions involves showing it reduces to the Palatini action under these conditions. Here we present definitions and calculate Einstein's equations from the Palatini action in detail.