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]
S={1\over2\kappa}\intR\sqrt{-g}d4x,
where
g=\det(g\mu\nu)
R
\kappa=8\piGc-4
G
c
S
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.
The Einstein equations in the presence of matter are given by adding the matter action to the Einstein–Hilbert action.
Suppose that the full action of the theory is given by the Einstein–Hilbert term plus a term
l{L}M
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
\begin{align} 0&=\deltaS\\ &=\int\left[
1 | |
2\kappa |
\delta\left(\sqrt{-g | |
R\right)}{\delta |
g\mu\nu
Since this equation should hold for any variation
\deltag\mu\nu
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,
T\mu\nu:=
-2 | |
\sqrt{-g |
To calculate the left hand side of the equation we need the variations of the Ricci scalar
R
The variation of the Ricci scalar follows from varying the Riemann curvature tensor, and then the Ricci curvature tensor.
\deltaR\sigma\nu\equiv\delta
\rho} | |
{R | |
\sigma\rho\nu |
= \nabla\rho\left(\delta
\rho | |
\Gamma | |
\nu\sigma |
\right)-\nabla\nu\left(\delta
\rho | |
\Gamma | |
\rho\sigma |
\right)
R=g\sigma\nuR\sigma\nu
\begin{align} \deltaR&=R\sigma\nu\deltag\sigma\nu+g\sigma\nu\deltaR\sigma\nu\\ &=R\sigma\nu\deltag\sigma\nu+\nabla\rho\left(g\sigma\nu
\rho | |
\delta\Gamma | |
\nu\sigma |
-g\sigma\rho\delta
\mu | |
\Gamma | |
\mu\sigma |
\right), \end{align}
\nabla\sigmag\mu\nu=0
(\rho,\nu) → (\mu,\rho)
When multiplied by
\sqrt{-g}
\nabla\rho\left(g\sigma\nu
\rho | |
\delta\Gamma | |
\nu\sigma |
-g\sigma\rho
\mu | |
\delta\Gamma | |
\mu\sigma |
\right)
Aλ
\sqrt{-g}Aλ
\sqrt{-g}
λ | |
A | |
;λ |
= \left(\sqrt{-g}
λ\right) | |
A | |
;λ |
= \left(\sqrt{-g}
λ\right) | |
A | |
,λ |
\sqrt{-g}\nabla\muA\mu= \nabla\mu\left(\sqrt{-g}A\mu\right)= \partial\mu\left(\sqrt{-g}A\mu\right)
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
\deltag\mu\nu,
\partialλ\deltag\mu\nu\equiv\delta\partialλg\mu\nu
\deltag\mu\nu
at events not in the closure of the boundary.
Jacobi's formula, the rule for differentiating a determinant, gives:
\deltag=\delta\det(g\mu\nu)=gg\mu\nu\deltag\mu\nu
or one could transform to a coordinate system where
g\mu\nu
\delta\sqrt{-g}=-
1 | |
2\sqrt{-g |
In the last equality we used the fact that
g\mu\nu\deltag\mu\nu=-g\mu\nu\deltag\mu\nu
which follows from the rule for differentiating the inverse of a matrix
\deltag\mu\nu=-g\mu\alpha\left(\deltag\alpha\beta\right)g\beta\nu
Thus we conclude that
Now that we have all the necessary variations at our disposal, we can insert and into the equation of motion for the metric field to obtain
which is the Einstein field equations, and
\kappa=
8\piG | |
c4 |
has been chosen such that the non-relativistic limit yields the usual form of Newton's gravity law, where
G
When a cosmological constant Λ is included in the Lagrangian, the action:
S=\int\left[
1 | |
2\kappa |
(R-2Λ)+l{L}M\right]\sqrt{-g}d4x
Taking variations with respect to the inverse metric:
\begin{align}\deltaS &=\int\left[
\sqrt{-g | |
Using the action principle:
0=\deltaS=
1 | |
2\kappa |
\deltaR | |
\deltag\mu |
+
R | |
2\kappa |
1 | |
\sqrt{-g |
Combining this expression with the results obtained before:
\begin{align}
\deltaR | |
\deltag\mu |
&=R\mu\\
1 | |
\sqrt{-g |
We can obtain:
\begin{align}
1 | |
2\kappa |
R\mu+
R | |
2\kappa |
-g\mu | |
2 |
-
Λ | |
\kappa |
-g\mu | |
2 |
+\left(
\deltal{L | |
M |
With , the expression becomes the field equations with a cosmological constant:
R\mu-
1 | |
2 |
g\muR+Λg\mu=
8\piG | |
c4 |
T\mu.