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 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 Deriving equations of motion from an action has several advantages.
First, it allows for easy unification of general relativity with other classical field theories (such as Maxwell theory), which are also formulated in terms of an action.
In the process, the derivation identifies a natural candidate for the source term coupling the metric to matter fields.
Moreover, symmetries of the action allow for easy identification of conserved quantities through Noether's theorem.
In general relativity, the action is usually assumed to be a functional of the metric (and matter fields), and the connection is given by the Levi-Civita connection.
The Palatini formulation of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integer spin.
describing any matter fields appearing in the theory.
The stationary-action principle then tells us that to recover a physical law, we must demand that the variation of this action with respect to the inverse metric be zero, yielding Since this equation should hold for any variation
, it implies that is the equation of motion for the metric field.
The right hand side of this equation is (by definition) proportional to the stress–energy tensor,[4] To calculate the left hand side of the equation we need the variations of the Ricci scalar
These can be obtained by standard textbook calculations such as the one given below, which is strongly based on the one given in Carroll (2004).
[5] The variation of the Ricci scalar follows from varying the Riemann curvature tensor, and then the Ricci curvature tensor.
The first step is captured by the Palatini identity Using the product rule, the variation of the Ricci scalar
, we have By Stokes' theorem, this only yields a boundary term when integrated.
The boundary term is in general non-zero, because the integrand depends not only on
; see the article Gibbons–Hawking–York boundary term for details.
Thus, we can forget about this term and simply obtain at events not in the closure of the boundary.
Jacobi's formula, the rule for differentiating a determinant, gives: or one could transform to a coordinate system where
Using this we get In the last equality we used the fact that which follows from the rule for differentiating the inverse of a matrix Thus we conclude that Now that we have all the necessary variations at our disposal, we can insert (3) and (4) into the equation of motion (2) for the metric field to obtain which is the Einstein field equations, and has been chosen such that the non-relativistic limit yields the usual form of Newton's gravity law, where
When a cosmological constant Λ is included in the Lagrangian, the action: Taking variations with respect to the inverse metric: Using the action principle: Combining this expression with the results obtained before: We can obtain: With
, the expression becomes the field equations with a cosmological constant: