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Freely falling 2-surfaces and the quasi-local energy

We derive an expression for effective gravitational mass for any closed spacelike 2-surface. This effective gravitational energy is defined directly through the geometrical quantity of the freely falling 2-surface and thus is well adapted to intuitive expectation that the gravitational mass should be determined by the motion of test body moving freely in gravitational field. We find that this effective gravitational mass has reasonable positive value for a small sphere in the non-vacuum space-times and can be negative for vacuum case. Further, this effective gravitational energy is compared with the quasi-local energy based on the $(2+2)$ formalism of the General Relativity. Although some gauge freedoms exist, analytic expressions of the quasi-local energy for vacuum cases are same as the effective gravitational mass. Especially, we see that the contribution from the cosmological constant is the same in general cases.