There is no notion of local gravitational energy density. Gravity manifests itself as curvature of spacetime, and its strength can be measured by using variations of the elementary geometric quantities (area, volume, radius) of small balls with respect to their counterparts in flat spacetime. These variations are directly related, via the Einstein field equations, to the energy density of matter at the ball’s centre. In this talk I want to consider the vacuum case (no matter inside the ball). The geometric quantities still feel the effect of pure gravity, and the resulting variations could still be related to the gravitational strength or, in simple words, to the pure gravitational energy density. This leads to a novel prescription for an approximate quasi-local energy of the pure gravitational field.
https://indico.math.cnrs.fr/event/7248/