This is a follow up on Potential Energy in General Relativity. In Einstein's theory of gravity, contributions due to gravitational field on dynamics of system come from the geometry (metric $g_{\mu\nu}$) of space-time alone. The standard argument for this non-local behavior of gravitational energy is due to the principle of equivalence.
We can regard Newtonian potential energy as a fraction of conserved relativistic kinetic energy along a geodesic. Example: The conserved energy of test particle along timelike geodesics in a Schwarzschild background is given by $$E=m\left(1-\frac{2M}{r}\right)\frac{dt}{d\tau}$$ In Newtonian regime we can recover the familiar non-relativistic energy $\frac{1}{2}mv^2-\frac{Mm}{r} $ by neglecting all $O(1/c^2)$ terms in $E$. Similarly, can we recover a local description of gravitational self energy density ($-\frac{1}{8\pi G}\vec{g}.\vec{g}$) by approximating Einstein's equations in non-relativistic limit ? Are there well defined geometrical entities, which can give a measure of gravitational self energy?
Initially I wanted to start from a relativistic description of gravitational work, since in the Newtonian limit the total self energy is essentially the total work done by gravitational field in assembling a given distribution of mass (density $\rho$) quasi-statically : $$W=\frac{1}{2}\int \rho(r)\Phi(r)d^3r=-\frac{1}{8\pi G}\int\vec{g}.\vec{g}d^3r$$ where $\vec{g}=-\vec{\nabla}\Phi$. Since the gravitational force is fictitious in Einstein's theory, I am not sure if we can formulate a well defined notion of gravitational work. It seems that a relativistic description of gravitational work (if exists) can help us understand the idea of self energy.
