general relativity - On motivation for the definition of ADM mass

The ADM mass is expressed in terms of the initial data as a surface integral over a surface $S$ at spatial infinity:

$$M:=-\frac{1}{8\pi}\lim_{r\to \infty}\int_S(k-k_0)\sqrt{\sigma}dS$$ where $\sigma_{ij}$ is the induced metric on $S$, $k=\sigma^{ij}k_{ij}$ is the trace of the extrinsic curvature of $S$ embedded in $\Sigma$ ($\Sigma$ is a hypersurface in spacetime containing $S$). and $k_0$ is the trace of extrinsic curvature of $S$ embedded in flat space.

Can someone explain to me why ADM mass is defined so. Why is integral of difference of traces of extrinsic curvatures important?