Quasilocal mass from a mathematical perspective

Quasilocal mass in general relativity is a notion defined for a closed spacelike 2-surface in spacetime. In this note, we explain the definition in [23] and [24] from a mathematical viewpoint, emphasizing the connection to differential geometry and nonlinear partial differential equations. We also discuss a minimax interpretation of the definition and compare with other notions of quasilocal mass. 1. Surface Hamiltonian Our subject of study is a two-dimensional closed embedded spacelike surface Σ in spacetime N , and thus the induced metric is Riemannian. We also assume Σ bounds a spacelike region Ω in spacetime. These are surfaces on which the notion of quasilocal energy/mass is defined in relativity. 1 We denote the Lorentz metric on N by 〈·, ·〉N and the connection by ∇ . Let e4 be the future unit timelike normal vector field of Ω and P (·, ·) be the second fundamental form of Ω with respect to e4. Let e3 denote the outward unit spacelike normal of Σ with respect to Ω. We also choose orthonormal frames {e1, e2} tangent to Σ. The mean curvature vector H is the unique normal vector field that is the normal part of ∑2 a=1 ∇eaea. Denote by k = ∑2 a=1〈∇eae3, ea〉 the mean curvature of Σ with respect e3 and, p = trΣP = ∑2 a=1〈∇eae4, ea〉, then H = −ke3 + p e4. We can reflect H along the light cone of the normal bundle to get