Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass

Abstract

Abstract. We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." Such flows generalize inverse mean curvature flow, which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black hole. A flow of time flat surfaces may have connections to the problem in general relativity of bounding the mass of a spacetime from below by the quasi-local mass of a spacelike 2-surface contained therein.

Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II

“…Oriented orthonormal bases { ν θ , ν ⊥ θ } of the normal bundle, with outward-spacelike ν θ , are in one-to-one correspondence with smooth functions θ : Σ −→ R according to ν θ = cosh θ ν H + sinh θ ν ⊥ H . Recall (equation (3.3) of [3]) that the connection one-form α ν θ relates to α H as α ν θ = α H − dθ. and Θ ∈ C ∞ ((−1, 1), R) is any nondecreasing function.…”

“…Theorem 1.11. [Corollary 1.4 of [3]] For a uniformly area expanding family of connected time flat surfaces Σ(s) whose orthogonal flow velocity is achronal (e.g., spacelike) in a spacetime satisfying the dominant energy condition, d ds (m H (Σ(s))) ≥ 0.…”

“…Finally, the fifth line is zero by the time flat assumption. Theorem 1.13 (Theorem 1.1 of [3]). Given a uniformly area expanding family of surfaces Σ(s),…”

Rigidity of time-flat surfaces in the Minkowski spacetime

“…implies that the isometric embedding of Σ into R 3 ⊂ R 3,1 is an optimal isometric embedding in the sense of [13,14]. Recently, Bray and Jauregui [2] discovered a very interesting monotonicity property of the Hawking mass along surfaces that satisfy the condition (1.1). Such surfaces are said to be "time-flat" in [2] and include all 2-surfaces in a time-symmetric initial data set.…”

“…

A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to the constant torsion condition for curves in a threedimensional space and has been studied in [2,5,6,13,14]. In particular, any 2-surface in a static slice of a static spacetime is time-flat.

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