Spacetime Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Initial Data for the Einstein Equations

Abstract

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth 'spacetime harmonic functions' in addition to the energymomentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved. D. Kazaras acknowledges the support of NSF Grant DMS-1547145. M. Khuri acknowledges the support of NSF Grant DMS-1708798.

The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets With Toroidal Infinity and Related Rigidity Results

“…Spacetime harmonic functions satisfy a Bochner-type identity, which when integrated produces a natural relation between the dominant energy condition and the boundary geometry of initial data sets. This observation leads to a proof of the spacetime version of the positive mass theorem in the asymptotically flat and hyperboloidal settings [3,14]. Here we will present a version of the resulting integral identity suitable for the purposes of this paper.…”

“…where K is the Gauss curvature of regular level sets Σ t . Observe that by Sard's theorem, the set of values in [u, u] which are critical for u on Ω or ∂Ω is of measure zero; see [14,Remark 3.3] for the applicability of Sard's theorem under the current regularity hypotheses. Thus, on the right-hand side of (3.2) we may restrict attention to regular level sets Σ t for which t is also a regular value of u| ∂Ω .…”

“…The unit normal which takes this set of orientations at the various components will be denoted by υ. We then propose the following boundary value problem which is closely related to that used in [14]:…”

“…(1) The manifold These two theorems are established using the level set technique associated with spacetime harmonic functions. This approach has recently been used to prove the positive mass theorem in the asymptotically flat and asymptotically hyperboloidal (spherical infinity) settings [3,4,14], and was inspired by the work of Stern [25] where the level sets of harmonic maps were used to study scalar curvature on compact 3-manifolds. We refer the reader to the survey [2] for these and other developments concerning the level set method.…”

On a spacetime positive mass theorem with corners

“…This has given rise to new proofs of the positive mass theorem for both the Riemannian case ( [12], cf. [7]) and the spacetime case ( [30]). We refer readers to the comprehensive survey [11] by Bray, Hirsch, Kazaras, Khuri and Zhang for applications of the level set method to the study of ADM mass.…”