We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown and York (Contemporary mathematics, vol 132, American Mathematical Society, Providence, pp 129-142, 1992; Phys Rev D (3) 47(4):1407-1419, 1993) and Liu and Yau (Phys Rev Lett 90(23):231102, 2003; J Am Math Soc 19(1):181-204, 2006). Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed two dimensional surfaces evolving in an ambient three dimensional manifold. As a by-product, we are able to write the ADM mass (Arnowitt et al. in Phys. Rev. (2), 122:997-1006, 1961) of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere Sr and an integral of the scalar curvature plus a geometrically constructed function Φ(x) in the asymptotic region outside Sr. In the third part, we prove that for any closed, spacelike, 2-surface Σ in the Minkowski space R3 for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in R3,1 then the Liu-Yau mass of Σ is strictly positive unless Σ lies on a hyperplane. We also show that the examples given by Ó Murchadha et al. (Phys Rev Lett 92:259001, 2004) are special cases of this result.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics