I've recently been studying about the General relativity and Einstein field equation. When I reading of the derivation of the field equation, I encounterd a method called Einstein-Hilbert action. This action is given as \begin{align} S_{EH}[g_{ij}]\,=\,\intop_M \sqrt{-g}R\,dx_3dx_2dx_1dt \end{align} where $S_{EH}[g_{ij}]$ is the functional of $g_{ij}$, $g=\det(g_{ij})$ and $\sqrt{-g}R\,dx_3dx_2dx_1dt$ is volume element of $M$.
I don't have a deep understanding in term of physics for this E-H action and how Hilbert - Einstein find them in over 100 years ago. However, in term of mathematics, here is what I have learnt from school :
A spacetime $M$ is a smooth, connected, orientable manifold with a Lorentz metric tensor $g$, a Levi-Civita connection $\nabla$ and be time oriented with a time orientation $T$ (a smooth timelike vectorfield on $M$).
$g$ is a metric tensor on $M$, which means $g$ is a (0,2)_tensor field that assign each point $p\in M$ with a scalar product $g_p:\ T_pM\times TM_p\longrightarrow \mathbb R$. The coefficients $g_{ij}(p)$ is given by \begin{align} g_{ij}(p)\,=\,g_p\left(\frac{\partial}{\partial x_i}\bigg|_p,\frac{\partial}{\partial x_j}\bigg|_p \right), 0\leq i,j\leq 3. \end{align}
The definition of variational operator (reference from Haim Brezis - functional analysis and pde)
Let $H$ be a Hilbert space with a subset $A\subset H$ and a subspace $M\leq H $. Consider the functional $J:\ A\longrightarrow\mathbb R,\ u\longmapsto J[u]$. Then the vatiational operator is defined as the Gateaux derivative of $J$ at $u\in A $ in the direction $\varphi\in M$ \begin{align} \frac{\delta J}{\delta\varphi}[u]\,=\,\lim_{\varepsilon\to0}\frac{J[u+\varepsilon\varphi]-J[u] }{\varepsilon}. \end{align} If $J[u_0]=\displaystyle\min_{u\in A}J[u]$ then $\displaystyle \frac{\delta J}{\delta\varphi}[u_0]=0$.
So, with these all background knowledge, is it enough for me to get understand the E-H action ? I feel quite ambigous about the symbols is used in the action, I don't even know what is the domaind that the action acts to, so I really hope someone with knowledge would help me to clarify this ambigousness. Thanks.