The Mass-Angular Momentum Inequality for Multiple Black Holes

TL;DR

The paper proves the mass-angular momentum inequality $m \ge \sqrt{|\,\mathcal{J}\,|}$ for a broad class of axisymmetric, multi-end initial data, unless there exists an ADM-minimizing counterexample to extreme black hole uniqueness. The core technique is a flow of singular harmonic maps $\Phi$ into $\mathbb{H}^2$ whose renormalized energy $\mathcal{E}(\Phi)$ decreases along a puncture-flow $dz_i/dt=-b_i$, with punctures representing black hole horizons. Collisions and scatterings of punctures are analyzed via collision/separation configuration maps, providing energy inequalities that, together with an induction on the number of punctures, yield the desired inequality and the rigidity statement that equality characterizes a slice of extreme Kerr. The work further develops the linearized theory near punctures and proves differentiability of singular harmonic maps with respect to puncture locations, underpinning the smooth dependence needed for the flow and its energy estimates.

Abstract

This is the second in a series of two papers to establish the conjectured mass-angular momentum inequality for multiple black holes, modulo the extreme black hole 'no hair theorem'. More precisely it is shown that either there is a counterexample to black hole uniqueness, in the form of a regular axisymmetric stationary vacuum spacetime with an asymptotically flat end and multiple degenerate horizons which is 'ADM minimizing', or the following statement holds. Complete, simply connected, maximal initial data sets for the Einstein equations with multiple ends that are either asymptotically flat or asymptotically cylindrical, admit an ADM mass lower bound given by the square root of total angular momentum, under the assumption of nonnegative energy density and axisymmetry. Moreover, equality is achieved in the mass lower bound only for a constant time slice of an extreme Kerr spacetime. The proof is based on a novel flow of singular harmonic maps with hyperbolic plane target, under which the renormalized harmonic map energy is monotonically nonincreasing. Relevant properties of the flow are achieved through a refined asymptotic analysis of solutions to the harmonic map equations and their linearization.

Figures (3)

• Figure 1: Domain of integration and boundary components.
• Figure 2: Case 1: collision of two punctures.
• Figure 3: Scattering along two rods with four punctures.

Theorems & Definitions (75)

• Theorem 1.1
• Remark 1.2
• Theorem 3.1
• Corollary 3.2
• Conjecture 3.3
• Proposition 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3