Penrose Inequality Research Articles

Quantum Inequalities for Quantum Black Holes

We formulate spacetime inequalities applicable to quantum-corrected black holes to all orders of backreaction in semiclassical gravity. Namely, we propose refined versions of the quantum Penrose and reverse isoperimetric inequalities, valid for all known three-dimensional asymptotically anti–de Sitter quantum black holes. Previous proposals of the quantum Penrose inequality apply in higher dimensions but fail when applied in three dimensions beyond the perturbative regime. Our quantum Penrose inequality, valid in three dimensions, holds at all orders of backreaction. This suggests cosmic censorship must exist in nonperturbative semiclassical gravity. Our quantum reverse isoperimetric inequality implies a maximum entropy state for quantum black holes at fixed volume. Published by the American Physical Society 2024

Fill-ins with scalar curvature lower bounds and applications to positive mass theorems

Given a constant C and a smooth closed (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional Riemannian manifold (Σ,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Sigma , g)$$\\end{document} equipped with a positive function H, a natural question to ask is whether this manifold can be realised as the boundary of a smooth n-dimensional Riemannian manifold with scalar curvature bounded below by C and boundary mean curvature H. That is, does there exist a fill-in of (Σ,g,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Sigma ,g,H)$$\\end{document} with scalar curvature bounded below by C? We use variations of an argument due to Miao and the author (Int Math Res Not 7:2019, 2019) to explicitly construct fill-ins with different scalar curvature lower bounds, where we permit the fill-in to contain another boundary component provided it is a minimal surface. Our main focus is to illustrate the applications of such fill-ins to geometric inequalities in the context of general relativity. By filling in a manifold beyond a boundary, one is able to obtain lower bounds on the mass in terms of the boundary geometry through positive mass theorems and Penrose inequalities. We consider fill-ins with both positive and negative scalar curvature lower bounds, which from the perspective of general relativity corresponds to the sign of the cosmological constant, as well as a fill-in suitable for the inclusion of electric charge.

A Green’s Function Proof of the Positive Mass Theorem

In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green’s function of an asymptotically flat 3-manifolds. In the same context and for 1<p<3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<3$$\\end{document}, a Geroch-type calculation is performed along the level sets of p-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality under favourable assumptions. A new characterisation of scalar curvature lower bounds in terms of the monotonicity formulas is also given.

The Riemannian Penrose inequality with matter density

The Riemannian Penrose inequality with matter density

Null Penrose inequality in a perturbed Schwarzschild spacetime

Null Penrose inequality in a perturbed Schwarzschild spacetime

A new proof of the Riemannian Penrose inequality

We give an overview of our recent new proof of the Riemannian Penrose inequality in the case of a single black hole. The proof is based on a new monotonicity formula, holding along the level sets of the p -capacitary potential of the connected boundary of an asymptotically flat 3 -manifold, with nonnegative scalar curvature.

Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature

We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1 &lt; p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's isoperimetric mass, as $p \to 1^+$. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime.

A localized spacetime Penrose inequality and horizon detection with quasi-local mass

A localized spacetime Penrose inequality and horizon detection with quasi-local mass

Nonexistence of solutions to the coupled generalized Jang equation/zero divergence system

In Bray (2011 Asian J. Math. 15 559–612), Bray and Khuri proposed coupling the generalized Jang equation to several different auxiliary equations. The solutions to these coupled systems would then imply the Penrose inequality. One of these involves coupling the generalized Jang equation to , as this would guarantee the non-negativity of the scalar curvature in the Jang surface. This coupled system of equations has not received much attention, and we investigate its solvability. We prove that there exists a spherically symmetric initial data set for the Einstein equations for which there do not exist smooth radial solutions to the system having the appropriate asymptotics for application to the Penrose inequality.

The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary

The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary

Attractive gravity probe surfaces in higher dimensions

Abstract A generalization of the Riemannian Penrose inequality in n-dimensional space (3 ≤ n &amp;lt; 8) is done. We introduce a parameter α ($-\frac{1}{n-1}\lt \alpha \lt \infty$) indicating the strength of the gravitational field, and define a refined attractive gravity probe surface (refined AGPS) with α. Then, we show the area inequality for a refined AGPS, $A \le \omega _{n-1} \left[ (n+2(n-1)\alpha )Gm /(1+(n-1)\alpha ) \right]^{\frac{n-1}{n-2}}$, where A is the area of the refined AGPS, ωn − 1 is the area of the standard unit (n − 1)-sphere, G is Newton’s gravitational constant, and m is the Arnowitt–Deser–Misner mass. The obtained inequality is applicable not only to surfaces in strong gravity regions such as a minimal surface (corresponding to the limit α → ∞), but also to those in weak gravity existing near infinity (corresponding to the limit $\alpha \rightarrow -\frac{1}{n-1}$).

Penrose Inequality as a Constraint on the Low Energy Limit of Quantum Gravity

We construct initial data violating the anti-de Sitter Penrose inequality using scalars with various potentials. Since a version of the Penrose inequality can be derived from AdS/CFT, we argue that it is a new swampland condition, ruling out holographic UV completion for theories that violate it. We produce exclusion plots on scalar couplings violating the inequality, and we find no violations for potentials from string theory. In the special case where the dominant energy condition holds, we use general relativity techniques to prove the anti-de Sitter (AdS) Penrose inequality in all dimensions, assuming spherical, planar, or hyperbolic symmetry. However, our violations show that this result cannot be generically true with only the null energy condition, and we give an analytic sufficient condition for violation of the Penrose inequality, constraining couplings of scalar potentials. Like the Breitenlohner-Freedman bound, this gives a necessary condition for the stability of asymptotically AdS (AAdS) spacetimes.

Metrics with $$\lambda _1(-\Delta + k R) \ge 0$$ and Flexibility in the Riemannian Penrose Inequality

Metrics with $$\lambda _1(-\Delta + k R) \ge 0$$ and Flexibility in the Riemannian Penrose Inequality

Doubling of Asymptotically Flat Half-spaces and the Riemannian Penrose Inequality

Building on previous works of Bray, of Miao, and of Almaraz, Barbosa, and de Lima, we develop a doubling procedure for asymptotically flat half-spaces (M, g) with horizon boundary Sigma subset M and mass min {mathbb {R}}. If 3le dim (M)le 7, (M, g) has non-negative scalar curvature, and the boundary partial M is mean-convex, we obtain the Riemannian Penrose-type inequality m≥12nn-1|Σ|ωn-1n-2n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} m\\ge \\left( \\frac{1}{2}\\right) ^{\\frac{n}{n-1}}\\,\\left( \\frac{|\\Sigma |}{\\omega _{n-1}}\\right) ^{\\frac{n-2}{n-1}} \\end{aligned}$$\\end{document}as a corollary. Moreover, in the case where partial M is not totally geodesic, we show how to construct local perturbations of (M, g) that increase the scalar curvature. As a consequence, we show that equality holds in the above inequality if and only if the exterior region of (M, g) is isometric to a Schwarzschild half-space. Previously, these results were only known in the case where dim (M)=3 and Sigma is a connected free boundary hypersurface.

Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS *

We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.

Penrose inequality in holography

The recent holographic deduction of the Penrose inequality only assumes the null energy condition, while a weak or dominant energy condition is required in the usual geometric proof. We take a step toward filling the gap between these two approaches. For planar or spherically symmetric asymptotically Schwarzschild anti-de Sitter (AdS) black holes, we give a purely geometric proof for the Penrose inequality by assuming the null energy condition. We also point out that two naive generalizations of the charged Penrose inequality are generally not true, and we propose two new candidates. When the spacetime is asymptotically AdS but not Schwarzschild-AdS, the total mass is defined according to holographic renormalization and depends on the scheme of quantization. In this case, the holographic argument implies that the Penrose inequality should still be valid, but we use a concrete example to show that whether the Penrose inequality holds or not will depend on what kind of quantization scheme we employ.

Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass

In [52], Mantoulidis and Schoen constructed 3-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be “far away” from being round. The resulting manifolds, called extensions, are geometrically not “close” to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality as discussed in [53,19]. Their construction was later adapted to n+1 dimensions by Cabrera Pacheco and Miao [22]. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum [1] and by Cabrera Pacheco, Cederbaum, and McCormick [21], a similar construction was performed for 3-dimensional asymptotically Euclidean, electrically charged Riemannian manifolds and for asymptotically hyperbolic Riemannian manifolds, respectively.This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in n+1 dimensions for n≥2. We study in detail the sub-extremality of these manifolds, consider the so far unstudied case of extremality in extensions with electric charge and allow more general conditions for the metric of our extensions.Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.

Spherically symmetric counter examples to the Penrose inequality and the positive mass theorem under the assumption of the weak energy condition

Of the various energy conditions which can be assumed when studying mathematical general relativity, intuitively the simplest is the weak energy condition which simply states that the observed mass-energy density must be non-negative. This energy condition has not received as much attention as the so-called dominant energy condition. When the natural question of the Penrose inequality in the context of the weak energy condition arose, we could not find any results in the literature, and it was not immediately clear whether the inequality would hold, even in spherical symmetry. This led us to constructing a spherically symmetric asymptotically flat initial data set satisfying the weak energy condition which violates the ‘usual’ formulations of the Penrose conjecture. This does not contradict (Malec and Murchadha 1994 Phys. Rev. D 49 6931–4), as there the data is assumed to be maximal. Our result is unexpected and interesting because the heuristic argument behind the Penrose inequality relies on the black hole area law which holds when the weak energy condition is satisfied. Our methods also lead to a counter example to the positive mass theorem assuming the weak energy condition.

Tightening the Penrose inequality

The Penrose inequality estimates the lower bound of the mass of a black hole in terms of the area of its horizon. This bound is relatively loose for extremal or near extremal black holes. We propose a new Penrose-like inequality for static black holes involving the mass, area of the black hole event horizon and temperature. Our inequality includes the Penrose inequality as its corollary, and it is saturated by both the Schwarzschild and Reissner-Nordström black holes. In the spherically symmetric case, we prove this new inequality by assuming both the null and trace energy conditions.

Geometry of massive particle surfaces

We propose a generalization of the photon surfaces of Claudel, Virbhadra and Ellis to the case of massive charged particles, considering a timelike hypersurface such that any worldline of a particle with mass $m$, electric charge $q$ and a fixed total energy $\mathcal{E}$, initially touching it, will forever remain in this hypersurface. Such surfaces can be defined not only with particle motion equations, but instead using the partially umbilic nature of the surface geometry. Such an approach should be especially useful in the case of nonintegrable equations of motion. It can be applied to the theory of nonthin accretion disks, and can also serve as a new tool for some general problems such as uniqueness theorems, Penrose inequalities, and hidden symmetries. A condition for the stability of worldlines is derived, which reduces to differentiation along the flow of surfaces of a certain energy. A number of examples of electrovacuum and dilaton solutions are considered; conditions are found for marginally stable surfaces of massive particles, regions of stable or unstable surfaces of massive particles and photons, as well as solutions that satisfy the no-force condition.