Freely falling 2-surfaces and the effective gravitational mass

We derive an expression for the effective gravitational mass for any closed spacelike 2-surface. This effective gravitational energy is defined direct ly through the geometrical quantity of the freely falling a-surface and thu s is well adapted to intuitive expectation that the gravitational mass shou ld be determined by the motion of a test body moving freely in the gravitat ional field. We find that this effective gravitational mass has a reasonabl e positive value for a small sphere in the non-vacuum space-times and can b e negative for the vacuum case. Further, this effective gravitational energ y is compared with the quasi-local energy based on the (2 + 2) formalism of General Relativity. Although some gauge freedoms exist, analytic expressio ns of the quasi-local energy for vacuum cases are the same as the effective gravitational mass. Especially, we see that the contribution from the cosm ological constant is the same in general cases.