[Paper Review] Penrose inequalities and a positive mass theorem for charged black holes in higher dimension

Levi Lopes de Lima, Frederico Girão+2 more|arXiv (Cornell University)|Jan 5, 2014

Geometric Analysis and Curvature Flows30 references3 citations

TL;DR

This paper establishes Penrose-type inequalities and proves a positive mass theorem for time-symmetric Einstein-Maxwell initial data sets representing charged black holes in higher dimensions (n ≥ 3), using the inverse mean curvature flow on hypersurfaces embedded in ℝⁿ⁺¹. The key result shows that the total mass is bounded from below by a function of the horizon area and charge, with equality only for the Reissner-Nordström-Tangherlini solution, confirming the physical consistency of charged black holes under geometric constraints.

ABSTRACT

We use the inverse mean curvature flow to establish Penrose-type inequalities for time-symmetric Einstein-Maxwell initial data sets which can be suitably embedded as a hypersurface in Euclidean space $\mathbb R^{n+1}$, $n\geq 3$. In particular, we prove a positive mass theorem for this class of charged black holes. As an application we show that the conjectured upper bound for the area in terms of the mass and the charge, which in dimension $n=3$ is relevant in connection with the Cosmic Censorship Conjecture, always holds under the natural assumption that the horizon is stable as a minimal hypersurface.

Motivation & Objective

• To establish Penrose-type inequalities for time-symmetric Einstein-Maxwell initial data sets in higher dimensions (n ≥ 3) that can be embedded in ℝⁿ⁺¹.
• To prove a positive mass theorem for this class of charged black holes, showing mass is bounded below by a function of charge and horizon geometry.
• To demonstrate that the charge contributes explicitly to the total mass through integral-geometric invariants of the horizon, independent of the global initial data set.
• To validate the conjectured upper bound for horizon area in terms of mass and charge under the assumption of horizon stability as a minimal hypersurface.
• To extend the applicability of the inverse mean curvature flow method to charged black hole systems beyond vacuum solutions.

Proposed method

• The inverse mean curvature flow (IMCF) is applied to time-symmetric Einstein-Maxwell initial data sets embedded in ℝⁿ⁺¹ to analyze the evolution of hypersurfaces and derive monotonicity properties.
• The method relies on the weak formulation of IMCF to handle non-smooth or outer-minimizing horizons, ensuring robustness in geometric analysis.
• Key geometric inequalities—specifically, a sharp Alexandrov-Fenchel-type inequality for hypersurfaces in warped product spaces—are used to control curvature and area evolution.
• The analysis incorporates the Gauss equation and stability inequality for minimal hypersurfaces, linking intrinsic curvature to extrinsic geometry and electric field strength.
• The proof uses Newton-Maclaurin and Cauchy-Schwarz inequalities to bound the total charge and relate it to the horizon's mean curvature and scalar curvature.
• Equality cases are analyzed via the constancy of scalar curvature and spherical symmetry, suggesting that equality holds only for Reissner-Nordström-Tangherlini-type solutions.

Experimental results

Research questions

• RQ1 Does the Penrose inequality for charged black holes in higher dimensions hold under the assumption that the initial data set can be embedded in ℝⁿ⁺¹?
• RQ2 Can a positive mass theorem be established for charged black holes in dimensions n ≥ 4 using geometric flows, independent of other methods?
• RQ3 How does the electric charge influence the total mass of a black hole system in terms of geometric invariants of the horizon?
• RQ4 Is the conjectured upper bound for the horizon area in terms of mass and charge valid when the horizon is stable as a minimal hypersurface?
• RQ5 Under what geometric conditions does equality in the Penrose inequality occur, and does it correspond to a known black hole solution?

Key findings

• The Penrose inequality (3.18) holds for Einstein-Maxwell initial data sets embedded in ℝⁿ⁺¹ when the horizon is stable as a minimal hypersurface, confirming a geometric constraint on black hole mass and charge.
• A positive mass theorem is proven for this class of charged black holes, showing that the total mass is bounded from below by a function of the horizon area and charge, with equality only for the Reissner-Nordström-Tangherlini solution.
• The charge contributes to the total mass through the integral-geometric invariant ∫Σ₀ HdΣ₀, with the bound Q² ≤ 𝒴̃(Σ₀,k)ℛΣ₀² implying the validity of the Penrose inequality.
• Equality in the main inequality (3.23) occurs only when each level surface Σₜ is a round sphere, and the scalar curvature is constant, indicating that the spacetime is isometric to a Reissner-Nordström-Tangherlini solution.
• The result extends to outer-minimizing horizons via the weak formulation of IMCF, broadening the applicability beyond star-shaped boundaries.
• The Alexandrov-Fenchel inequality (4.39) provides a lower bound on the total scalar curvature of the horizon, which, when combined with curvature estimates, ensures the validity of the Penrose inequality under 2-convexity or star-shapedness.