Angular Momentum Penrose Inequality (2512.06918v1)

Abstract: We prove the Angular Momentum Penrose Inequality for axisymmetric vacuum initial data satisfying the dominant energy condition. This inequality establishes a sharp lower bound on the ADM mass in terms of both the horizon area and the Komar angular momentum of a black hole, with equality achieved precisely by the Kerr solution. The proof combines four main ingredients: solving an axisymmetric Jang equation where twist enters as a lower-order perturbation, establishing conformal factor bounds via a divergence identity, proving angular momentum conservation along level sets using de Rham cohomology, and applying the proven Dain-Reiris area-angular momentum inequality for sub-extremality. The monotonicity of a combined area-angular momentum functional along the Agostiniani-Mazzieri-Oronzio flow yields the result. This provides the first geometric inequality incorporating both horizon area and angular momentum, with implications for cosmic censorship, black hole thermodynamics, and gravitational wave observations of spinning black holes.

• The paper establishes a rigorous lower bound for ADM mass based on black hole horizon area and angular momentum, with equality only for Kerr slices.
• It employs advanced techniques including the Jang–conformal method and p-harmonic flows to extend classical Penrose proofs to rotating configurations.
• Numerical validations across 199 configurations confirm that no physically admissible data violates the inequality, reinforcing cosmic censorship.

The Angular Momentum Penrose Inequality: An Authoritative Synthesis

Introduction

The Angular Momentum Penrose Inequality establishes a sharp lower bound for the ADM mass of asymptotically flat, axisymmetric, vacuum initial data admitting a marginally outer trapped surface (MOTS) of specified area and angular momentum. Unlike the classical Penrose inequality, which neglects the role of spin, this result provides a geometric inequality that incorporates both the horizon area ($A$) and angular momentum ($J$), with the extremal case realized precisely for slices of Kerr spacetime. This addresses a long-standing open question in the study of general relativistic black holes and their connection to cosmic censorship.

Statement and Significance of the Main Result

Let $(M^3, g, K)$ be an asymptotically flat, axisymmetric, vacuum initial data set satisfying the dominant energy condition, with an outermost stable MOTS $\Sigma$ of area $A$ and Komar angular momentum $J$. The main theorem proven is

$$M_{ADM} \geq \sqrt{\frac{A}{16\pi} + \frac{4\pi J^2}{A}}$$

with equality if and only if the data arises from a Kerr slice. Equivalent formulations in terms of $M_{ADM}$, $A$, and the irreducible mass $M_{irr}$ are established. For $J=0$ this reduces to the Riemannian Penrose inequality, and for nonzero $J$, it encodes sub-extremality, providing $A \geq 8\pi |J|$ as a necessary condition.

This inequality is significant in several respects:

• It formalizes, with mathematical rigor, the minimal ADM mass compatible with specified horizon area and angular momentum, thus making precise the intuition that Kerr realizes the absolute lower bound for such physical parameters.
• It is fully geometric, depending entirely on initial data and not on spacetime evolution.
• It confirms and extends the Dain–Reiris area-angular momentum inequality into the context of mass bounds.
• The proof technique synthesizes advanced geometric PDE methods, conformal deformation, and monotonicity via p-harmonic flows, demonstrating the feasibility of extending known Penrose-type proofs to incorporate rotational degrees of freedom.

Methodological Framework

The approach generalizes techniques used for the Riemannian and spacetime Penrose inequalities, adapting them to axisymmetric, rotating configurations using the "Jang–conformal–AMO" method:

1. Axisymmetric Jang Equation: The Jang deformation is constructed in the presence of rotation, with twist contributions entering as subleading perturbations. Detailed analysis confirms the twist term is negligible in the singular limit near the MOTS, thus preserving the essential structure needed for barrier arguments and asymptotic cylindrical geometry.
2. Angular-Momentum-Modified Lichnerowicz Equation: The conformal transformation incorporates a TT-tensor contribution $\Lambda_J$ encoding angular momentum. Existence and comparison principles yield a conformal factor $\phi\leq 1$.
3. p-Harmonic/AMO Flow: The area-angular momentum functional

$$\mathcal{M}_{1,J}(t) = \sqrt{\frac{A(t)}{16\pi} + \frac{4\pi J^2}{A(t)}}$$

is shown to be non-decreasing along the AMO flow; crucially, $J$ is shown to be exactly conserved across all level sets via a topological (de Rham cohomological) argument, under the vacuum constraint.

1. Sub-Extremality via Dain–Reiris: The established area-angular momentum lower bound $A \geq 8\pi |J|$ holds globally by stability and applies to all level sets, precluding violations associated with super-extremal data.

The synthesis of these steps yields monotonicity from the horizon to infinity, with the horizon value providing the lower bound in the mass inequality.

Rigidity and Sharpness

A detailed rigidity analysis confirms that the only case of saturation is precisely for slices of the Kerr solution. The monotonicity argument implies all terms contributing to strictness must vanish, which forces the solution to be (up to diffeomorphism) a Kerr initial slice. This is consistent with the uniqueness theorems of Carter–Robinson and the positive mass theorem rigidity.

Numerical Validation and No Genuine Counterexamples

An extensive numerical investigation is reported, spanning 199 configurations across various data families (Kerr, Bowen-York, Brill waves, black holes with matter, multi-black-hole data, etc.). All physically meaningful configurations satisfy the inequality strictly, with Kerr saturating the bound as expected. Apparent violations are traced to incorrect parametrization, unphysical matter/initial data, or configurations violating cosmic censorship (i.e., $|J| > M_{ADM}^2$). No counterexamples exist within the physically admissible set.

Theoretical and Practical Implications

Theoretical Implications: The result resolves a major open question about the geometric structure of black holes with spin:

• It provides evidence supporting cosmic censorship in axisymmetric vacuum setups and links the Kerr geometry directly to optimal configurations in the mass-area-angular momentum phase space.
• It sets a new standard for geometric inequalities in general relativity, indicating the methodology can be extended, at least in principle, to treat more general conserved charges and possibly to non-axisymmetric or dynamical situations with further work.

Practical Implications: This inequality underpins a variety of applications:

• It sets physical mass-spin limits relevant for numerical relativity, black hole merger simulations, and the analysis of gravitational wave signals.
• It applies to discussions of horizon formation and observability in high-angular-momentum collapse scenarios, providing a sharp minimum mass for black hole formation for given spin.

Generalizations and Open Problems

The paper conjectures natural extensions:

• Inclusion of Charge: An analogous inequality with electric charge $Q$ is postulated, which would generalize to the Kerr–Newman case.
• Multiple Horizons: The extension to multiple horizon components is conjectured.
• Non-Axisymmetric Data: Fully non-axisymmetric initial data remains open, with challenges associated with defining and conserving total angular momentum quasi-locally.
• Dynamical Horizons: Extension to time-dependent or non-stationary horizons, with appropriate definitions of angular momentum, remains for future investigation.

Conclusion

The Angular Momentum Penrose Inequality,

$$M_{ADM} \geq \sqrt{\frac{A}{16\pi} + \frac{4\pi J^2}{A}},$$

has been rigorously established for vacuum, asymptotically flat, axisymmetric initial data with stable outermost MOTS. The result is sharp and achieved precisely for Kerr initial data. Its proof combines advanced geometric analysis, PDE, and differential topology, and sets the foundation for further extension of geometric inequalities in general relativity. The implications are significant for both mathematical relativity and astrophysical modeling of black hole systems.

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References:

• Dain, S. and Reiris, M.E. "Area-angular momentum inequality for axisymmetric black holes" (Dain et al., 2011).
• Han, Q. and Khuri, M. "Existence and blow-up behavior for solutions of the generalized Jang equation" (Kirilova, 2011).
• Agostiniani, V., Mazzieri, L., Oronzio, F. "A Green's function proof of the positive mass theorem" (Agostiniani et al., 2021).
• Bray, H. and Khuri, M. "A Jang equation approach to the Penrose inequality" (0905.2622).