Positivity of Quasi-local Mass Ii

A spacetime is a four-manifold with a pseudo-metric of signature (+,+,+,−). A hypersurface or a 2-surface in a spacetime is spacelike if the induced metric is positive definite. A quasi-local energy-momentum vector is a vector in R associated to a spacelike 2-surface which depends on its first and second fundamental forms and the connection to its normal bundle in the spacetime. The time component of the four-vector is called quasi-local energy (mass). Similar to [10, 8], we require the quasi-local energy-momentum vector to satisfy the following properties. (1) It should be zero for the flat spacetime. (2) The quasi-local mass should be equivalent to the standard definition if the spacetime is spherically symmetric and the quasi-local mass is evaluated on the spheres [7]. (We say that two masses m1 and m2 are equivalent if there is a universal constant c > 0 such that cm1 ≤ m2 ≤ cm1.) In particular, for the centered spheres in the Schwarzschild spacetime, the quasi-local mass should be equivalent to the standard mass. (3) For an asymptotically flat slice, the quasi-local mass of the coordinate sphere should be asymptotic to the ADM energy-momentum vector. (4) For an asymptotically null slice, the quasi-local mass of the coordinate sphere should be asymptotic to the Bondi energy-momentum vector. (5) For an apparent horizon Σ, the quasi-local mass should be no less than a (universal) constant multiple of the irreducible mass, which is √ Area(Σ)/16π. (6) The quasi-local energy-momentum vector should be nonspacelike and the quasi-local mass should be nonnegative. Our definition of quasi-local energy [22] arises naturally from calculations in the second author’s work [36] on black holes and is strongly motivated by our ability to prove its positivity. After the second author proposed our definition, we were informed of the existence of much earlier works by Brown-York [3, 4] and others [21, 20, 11]. The main goal of this paper is to provide a complete proof of a stronger version of the positivity stated in [22]. The rest of the paper is organized as follows. In Section 2, we recall our definition of quasi-local energy, discuss its properties, and state the main result (positivity of