Freely Falling 2-Surfaces and the Effective Gravitational Mass

We derive an expression for the effectivegravitational mass for any closed spacelike 2-surface.This effective gravitational energy is defined directlythrough the geometrical quantity of the freely falling 2-surface and thus is well adapted to intuitiveexpectation that the gravitational mass should bedetermined by the motion of a test body moving freely inthe gravitational field. We find that this effective gravitational mass has a reasonable positivevalue for a small sphere in the non-vacuum space-timesand can be negative for the vacuum case. Further, thiseffective gravitational energy is compared with the quasi-local energy based on the (2 + 2)formalism of General Relativity. Although some gaugefreedoms exist, analytic expressions of the quasi-localenergy for vacuum cases are the same as the effective gravitational mass. Especially, we see that thecontribution from the cosmological constant is the samein general cases.