The Action for General Relativity in ADM Decomposition
The Einstein-Hilbert action reads: $$ S=\int{\sqrt{-g}d^4x\:R} $$ Suppose $\Sigma$ is a 3-dimensional manifold, and suppose our universe is a 4-dimensional oriented manifold $M$ with a smooth function $t$ defined on it, such that each level set $\Sigma_{t}$ of $t$ is a hypersurface that is diffeomorphic to $\Sigma$. We define a smooth vector field $t^{a}$ on $M$ such that $t^{a} \nabla_{a} t = 1$. So far, we have not specified a metric on $M$. Now, suppose we equip $M$ with a metric $g_{ab}$, such that each level set $\Sigma_{t}$ is a spacelike hypersurface, and $t^{a}$ is a future-directed timelike vector field. In this case, $\left\{ \Sigma_{t} \right\}$ and $t^{a}$ define a ADM decomposition of the spacetime $(M, g_{ab})$. The vector field $t^{a}$ can then be expressed as ...