Penrose inequality for integral energy conditions

TL;DR

This work extends the Penrose inequality beyond pointwise energy conditions by proving a local Penrose inequality under a spherical, averaged dominant energy condition, and a modified Penrose inequality for evaporating black holes tied to Quantum Energy Inequalities. The approach relies on averaging the energy condition via a $d\nu$-ADEI and leveraging spherical symmetry; it recovers the classical inequality when the average bound vanishes, while allowing systematic departures otherwise. For dynamical spacetimes, the paper connects horizon evolution to averaged energy bounds, demonstrating a bound on the ADM mass in terms of horizon area that remains valid under evaporation when a QEIs-inspired bound is imposed. The results provide a semiclassical framework where Penrose-type relations can still test weak cosmic censorship, and they include concrete models (effective quantum potentials, vacuum polarization, IMCF, and $SU(d+1)$-invariant data) illustrating both compliance and violations of pointwise energy conditions. Overall, the findings offer a path to Penrose inequalities compatible with semiclassical gravity and QEIs, with practical tests and explicit model-based bounds.

Abstract

The classical Penrose inequality, a relation between the ADM mass and the area of any cross section of the black hole event horizon, was introduced as a test of the weak cosmic censorship conjecture: if it fails, the trapped surface is not necessarily behind the event horizon and a naked singularity could form. Since that original derivation, a variety of proofs have developed, mainly focused on the initial data formulation on maximal spacelike slices of spacetime. Most of these proofs are applicable only for classical fields, as the energy conditions required are violated in the context of quantum field theory. In this work we provide two generalizations of the Penrose inequality for spherically symmetric spacetimes: a proof of a classical Penrose inequality using initial data and an average energy condition, and a proof of a modified Penrose inequality for evaporating black holes with a connection to the weak cosmic censorship conjecture. The latter case could also be applicable to quantum fields as it uses a condition inspired by quantum energy inequalities. Finally, we provide physically motivated examples for both.

Figures (5)

• Figure 2: The integrand of S-DEC for the third-order modified Schwarzschild solution with $B = M = 1$ and varying $A$ demonstrates different behaviors.
• Figure 3: The figures show how much the inequality of Eq. \ref{['eqn:snecpi']} differs from the original Penrose one in terms of the affine distance of the two initial data sets ($s$). We see that with increasing temperature and $B$ (both meaning more negative energy allowed) the factor goes to zero faster. As reference temperature ($T_{\text{sun}}$) we are using the Hawking temperature of a solar mass black hole. Most astrophysical black holes would have a lower temperature.
• Figure : (a)
• Figure : (a)
• Figure : (b)

Theorems & Definitions (18)

• Definition 3.1: Averaged Dominant Energy Inequality
• Theorem 3.1
• Corollary 3.2
• Corollary 3.3
• Corollary 3.4
• proof : Proof of \ref{['PI-SADEC']}
• Theorem 4.1
• proof
• Theorem 4.2
• Theorem 4.3