\( \newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}}
\newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label}
\newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}}
\newcommand{\third}{{\scriptstyle \frac{1}{3}}}
\newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm
\partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}}
\newtheorem{problem}{Problem}[chapter]
\newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)
XI. Field Equations
The Einstein–Hilbert action in general relativity is the
action that yields the Einstein field equations through the
stationary-action principle. This allows comparison of general
relativity with other classical field theories formulated in terms
of an action (such as electromagnetism). Most importantly, matter
can be coupled to gravity by adding to the Einstein–Hilbert action a
matter action describing any matter fields appearing in the theory.
By general covariance, an action for the metric ${g_{\mu \nu }}$
will have to take the form
\[S[g] := \int {{d^4}x} \sqrt
{\left| g \right|} R[{g_{\mu \nu }}] {\rm{ }}\]
Here, ${d^4}x\sqrt {\left| g \right|}$ is the invariant volume
measure and $R$ a scalar constructed from the metric. The simplest
choice is the Ricci scalar (10.18). It is also the unique
choice if one is looking for a scalar constructed only from first
and second derivatives of the metric. This simple and elegant
action, known as the Einstein-Hilbert action, was proposed
by Hilbert a few days before Einstein presented his final form of
the gravitational field equations.
To give the Einstein-Hilbert action the dimension of an action a
pre-factor is required, conventionally written as $1/2\kappa $,
where $\kappa = 8\pi G/{c^{ - 4}}$ is the Einstein gravitational
constant. One may also include a cosmological constant
$\Lambda $ by the substitution $R \to R + 2\Lambda $:
\[{S_{{\text{EH}}}}[g] =
\frac{1}{{2\kappa }}\int {{d^4}x} \sqrt {\left| g \right|} \left(
{R[{g_{\mu \nu }}] + 2\Lambda } \right)\]
Since in GRT the Levi-Civita connection can be expressed in terms of
the metric, see (1.29), the Einstein-Hilbert action is taken to be a
functional of the (inverse) metric tensor. It can be shown that the
Euler-Lagrange equations following upon variation of (11.6) indeed
lead to the correctly normalized Einstein equations. [Wikipedia:
Einstein–Hilbert action]