Hawking Mass Monotonicity for Initial Data Sets

TL;DR

The paper introduces a two-function PDE framework on initial data sets $(M,g,k)$ to model double-null foliations and derive a divergence identity that generalizes Hawking mass monotonicity under inverse mean curvature flow to data satisfying the dominant energy condition. By coupling equations for $(u,v)$ and proving a robust existence theory (notably for $a=0$ in full generality and in spherical symmetry for $a=1$), the authors unify IMCF, spacetime harmonic analyses, and Penrose-type results within a single analytic scheme. They demonstrate the approach via Minkowski and Schwarzschild models, derive spacetime charged harmonic variants, and establish geometric applications including Penrose-type inequalities in spherical symmetry. Overall, the work provides a new analytic pathway toward spacetime Penrose-type results by blending double-null foliations with a two-function PDE system and associated divergence identities, under DEC.

Abstract

We introduce new systems of PDE on initial data sets $(M,g,k)$ whose solutions model double-null foliations. This allows us to generalize Geroch's monotonicity formula for the Hawking mass under inverse mean curvature flow to initial data sets satisfying the dominant energy condition. We study the existence theory of these systems and give geometric applications.

Figures (3)

• Figure 1: The double null foliation ($\Sigma_u$,$\Sigma_v$) for the initial data set $(M,g,k)\subset \mathbb{R}^{3,1}$ is obtained by intersecting past and future directed lightcones in $\mathbb{R}^{3,1}$ with $(M,g,k)$.
• Figure 2: There are no monotone quantities associated with the level sets $\Sigma_u$ and $\Sigma_v$ individually. Hence, a divergence identity as in Theorem \ref{['T:main']} is a more general concept than a monotonicity formula.
• Figure 3: Assuming the final state conjecture holds, an initial data set $(M,g,k)$ evolves to a slice of Kerr. In Kerr $\mathfrak m\ge\frac{|\Sigma|}{16\pi}$ where $\Sigma$ is the intersection of the event horizon with $(M,g,k)$. Since matter is radiating away to infinity, and due to Hawking's area theorem HawkingWald, we also have $\mathfrak m\ge\frac{|\Sigma|}{16\pi}$ on $(M,g,k)$. Combining the Cosmic Censorship with Penrose's singularity theorem Penrose2Wald allows us to replace the event horizon with the minimal area enclosure of the apparent horizon.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of Theorem \ref{['T:Minkowski']}
• proof : Proof of Theorem \ref{['T:main']}
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Theorem 3.4