The spacetime Penrose inequality for cohomogeneity one initial data

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.

Our current picture of black hole gravitational collapse relies on two assumptions: i) the resulting singularity is hidden behind an event horizon -- weak cosmic censorship conjecture -- and ii) spacetime eventually settles down to a stationarity state. In this setting, it follows that the minimal area containing an apparent horizon is bound by the square of the total ADM mass (Penrose inequality conjecture). Following Dain et al. 2002, we construct numerically a family of axisymmetric initial data with one or several marginally trapped surfaces. Penrose and related geometric inequalities are discused for these data. As a by-product, it is shown how Penrose inequality can be used as a diagnosis for an apparent horizon finder numerical routine.

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.

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.

A spherically symmetric spacetime is presented with an initial data set that is asymptotically flat, satisfies the dominant energy condition, and such that on this initial data $M<\sqrt{A/16\pi}$, where M is the total (ADM) mass and A is the area of the apparent horizon. This provides a counterexample to a commonly stated version of the Penrose inequality, though it does not contradict the ``true'' Penrose inequality.

The Penrose inequality gives a lower bound for the total mass of a spacetime in terms of the area of suitable surfaces that represent black holes. Its validity is supported by the cosmic censorship conjecture, and therefore its proof (or disproof) is an important problem in relation with gravitational collapse. The Penrose inequality is a very challenging problem in mathematical relativity and it has received continuous attention since its formulation by Penrose in the early seventies. Important breakthroughs have been made in the last decade or so, with the complete resolution of the so-called Riemannian Penrose inequality and a very interesting proposal to address the general case by Bray and Khuri. In this review, the most important results on this field will be discussed and the main ideas behind their proofs will be summarized, with the aim of presenting what is the status of our present knowledge in this topic.

The Penrose inequality has so far been proven in cases of spherical symmetry and in cases of zero extrinsic curvature. The next simplest case worth exploring would be nonspherical, nonrotating black holes with nonzero extrinsic curvature. Following Karkowski et al.'s construction of prolate black holes, we define initial data on an asymptotically flat spacelike 3-surface with nonzero extrinsic curvature that may be chosen freely. This gives us the freedom to define the location of the apparent horizon such that the Penrose inequality is violated. We show that the dominant energy condition is violated at the poles for all cases considered.

We give a holographic argument in favor of the AdS Penrose inequality, which conjectures a lower bound on the total mass in terms of the area of apparent horizons. This inequality is often viewed as a test of cosmic censorship. We further find a connection between the area law for apparent horizons and the Penrose inequality. Finally, we show that the argument also applies to solutions with charge, resulting in a charged Penrose inequality in AdS.

The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with nonnegative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian Penrose inequality has also been established for the Einstein–Maxwell equations where the lower bound on mass depends also on electric charge, a charged Riemannian Penrose inequality. Here, we establish some quasi-local charged Penrose inequalities for surfaces isometric to closed surfaces in a suitable Reissner–Nordström manifold, which serves as a reference manifold for the quasi-local mass. In the case where the reference manifold has zero mass and nonzero electric charge, the lower bound on quasi-local mass is exactly the lower bound on the ADM mass given by the charged Penrose inequality.

affected by initial degrees of freedom in early Universe cosmology. This leads to an open question as to the applicability of the Riemannian Penrose inequality, in early universe conditions , if the mass m, is a sum of prior universe gravitons, and if the area A is due to either a quantum bounce, or due to Non Linear Electrodynamics scale factor a not being zero. Note that the Riemannian Penrose inequality is for Black hole physics. Its application to our problem is solely due to a nonzero, but extremely small initial scale factor. If the initial scale factor goes to zero, then of course, this inequality no longer holds.

Riemannian Penrose Inequalities are precise geometric statements that imply that the total mass of a zero second fundamental form slice of a spacetime is at least the mass contributed by the black holes, assuming that the spacetime has nonnegative matter density everywhere. In this paper, we remove this last assumption, and prove stronger statements that the total mass is at least the mass contributed by the black holes, plus a contribution coming from the matter density along the slice. We use the first author's conformal flow to achieve this, combined with Stern's harmonic level set techniques in the first case, and spinors in the second case. We then compare these new results to results previously known from Huisken-Ilmanen's inverse mean curvature flow techniques.

The Riemannian Penrose inequality Get access G. Huisken, G. Huisken Search for other works by this author on: Oxford Academic Google Scholar T. Ilmanen T. Ilmanen Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1997, Issue 20, 1997, Pages 1045–1058, https://doi.org/10.1155/S1073792897000664 Published: 01 January 1997 Article history Published: 01 January 1997 Received: 27 August 1997 Revision received: 10 October 1997

We prove that the Riemannian Penrose inequality holds for asymptotically flat 3‐manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the mass being a well‐defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential‐theoretic version of it, recently introduced by Agostiniani, Oronzio, and the third named author. As a consequence, we establish the equality between mass and Huisken's isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose inequality in terms of the isoperimetric mass on any 3‐manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well‐posed notion of weak inverse mean curvature flow (IMCF). In particular, such isoperimetric Riemannian Penrose inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's isoperimetric mass and the Hawking mass.

We study the Sobolev stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of manifolds $M^3_i$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled for time $t \in [0,T]$. In particular, we consider a sequence of regions of manifolds $U_T^i\subset M_i^3$, foliated by a IMCF, $\Sigma_t$, such that if $\partial U_T^i = \Sigma_0^i \cup \Sigma_T^i$ and $m_H(\Sigma_T^i) \rightarrow 0$ then $U_T^i$ converges in $W^{1,2}$ to a flat annulus or in the hyperbolic setting it converges to a annulus portion of hyperbolic space. If instead $m_H(\Sigma_T^i)-m_H(\Sigma_0^i) \rightarrow 0$ and $m_H(\Sigma_T^i) \rightarrow m >0$ then we show that $U_T^i$ converges in $W^{1,2}$ to a topological annulus portion of the Schwarzschild metric or in the Hyperbolic case to a topological annulus portion of the Anti-de~Sitter Schwarzschild metric.

We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically flat manifold $M^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically flat manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i = \Sigma_0^i \cup \Sigma_T^i$ and $m_H(\Sigma_T^i) \rightarrow 0$ then $U_T^i$ converges to a flat annulus with respect to $L^2$ metric convergence. If instead $m_H(\Sigma_T^i)-m_H(\Sigma_0^i) \rightarrow 0$ and $m_H(\Sigma_T^i) \rightarrow m >0$ then we show that $U_T^i$ converges to a topological annulus portion of the Schwarzschild metric with respect to $L^2$ metric convergence.