Holographic Argument for the Penrose Inequality in Ads Spacetimes

PHYSICAL REVIEW D 99, 126009 (2019)

Holographic argument for the Penrose inequality in AdS spacetimes

† Netta Engelhardt1,2,* and Gary T. Horowitz3, 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 2Gravity Initiative, Princeton University, Princeton New Jersey 08544, USA 3Department of Physics, University of California Santa Barbara, California 93106, USA

(Received 18 April 2019; published 14 June 2019)

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.

DOI: 10.1103/PhysRevD.99.126009

½σ 1=2 I. INTRODUCTION AND SUMMARY ≥ A ð Þ GM 16π : 1:1 Cosmic censorship, which states that regions of arbi- trarily large spacetime curvature are invisible to asymptotic observers, is one of the oldest conjectures about general This is known as the Penrose inequality. It is a stronger relativity. It is also one of the most important: if it is true, form of the positive mass theorem, M ≥ 0, and is con- general relativity is sufficient to predict everything that jectured to hold in the presence of an apparent horizon. happens outside black holes, while its failure raises the A violation of this inequality would indicate a failure of possibility of directly observing astronomical effects of cosmic censorship.1 quantum gravity. Despite its clear significance, however, it As stated, the inequality appears very difficult to prove. remains unproven. In the absence of a proof, theoretical Mathematicians have primarily focused on a Riemannian tests of cosmic censorship are of significant value. version of this inequality, which refers to an asymptoti- In the early 1970s, Penrose [1] proposed the following cally flat Riemannian three-dimensional manifold with test of cosmic censorship: suppose one is given asymp- nonnegative scalar curvature and a minimal surface σ. totically flat initial data for general relativity with Arnowitt- This can be taken as initial data for a solution to Einstein’s Deser-Misner mass M and an apparent horizon σ with area equation with zero extrinsic curvature and positive energy A½σ. Assuming cosmic censorship, σ lies inside a black density. In the resulting spacetime, the minimal surface is hole which is expected to settle down to a stationary Kerr an apparent horizon. After much effort, a complete proof solution. Under evolution, the area of the event horizon of this Riemannian inequality was finally given in 2001, cannot decrease and the total mass cannot increase. (Energy first for a single connected minimal surface [2] and then might be radiated away to null infinity, so the total (Bondi) for several minimal surfaces [3]. Since not all asymptoti- ’ mass may decrease.) If the final black holep isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi described cally flat Lorentzian solutions to Einstein sequation ¼ 16π by the Schwarzschild solution, then GMBH ABH= . necessarily admit such initial data, the general inequality remains open [4]. Since angular momentum decreases thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi horizon area, a final Kerr black hole satisfies GM ≥ A =16π. Since A similar inequality has been conjectured for asymp- BH BH totically anti-de Sitter (AdS) initial data with an apparent the initial quantities satisfy M ≥ M and A½σ ≤ A , BH BH horizon σ [5]. The same arguments involving cosmic this gives an immediate prediction that all initial data censorship and black holes settling down to a stationary must satisfy solution lead to the conclusion that the mass M and area A½σ of an asymptotically AdS initial data with apparent *[email protected] σ 2 † horizon in 4D should satisfy [email protected]

Published by the American Physical Society under the terms of 1The converse, however is false: a proof of the Penrose the Creative Commons Attribution 4.0 International license. inequality is not tantamount to a proof of cosmic censorship. Further distribution of this work must maintain attribution to This is clear in more than four spacetime dimensions where the the author(s) and the published article’s title, journal citation, Penrose inequality might be true, but cosmic censorship fails. and DOI. Funded by SCOAP3. 2We are setting the AdS radius to one throughout this paper.

2470-0010=2019=99(12)=126009(7) 126009-1 Published by the American Physical Society NETTA ENGELHARDT and GARY T. HOROWITZ PHYS. REV. D 99, 126009 (2019) ½σ 1=2 ½σ 3=2 ≥ A þ A ð Þ recover (1.2). As we discuss in the next section, we will GM 16π 16π : 1:2 require an extra condition on the apparent horizon which is generically satisfied. Since this constitutes what we believe Since the horizon radius of a Schwarzschild AdS black is the first general argument for a Lorentzian Penrose 3 hole, rþ, is related to its mass by M ¼ðrþ þ rþÞ=2, this is inequality from first principles, it is possible that the correct just the statement that A½σ is bounded from above by the general form of the inequality (1.3) also requires this extra 3 horizon area of a static black hole with the same mass. condition. Despite some partial results (see, e.g., [6–9]) this conjecture The fact that the Penrose inequality follows so simply is largely open. from holography raises the possibility that holography So far we have been assuming four-dimensional space- might imply a relaxed version of cosmic censorship as times, but there is no obstruction to considering the Penrose described above. Another piece of evidence in favor of this inequality in higher dimensions. The general form of the possibility is the following. Since our construction involves Penrose inequality in AdS in higher dimensions is given a spacetime with a wormhole, the bulk theory must satisfy in [5]; a proof of the Riemannian Penrose inequality for the weak gravity conjecture by the arguments in [17]. asymptotically flat Riemannian manifolds of dimension While the relevance of the weak gravity conjecture to less than eight is given in [10]. In higher dimensions, Penrose’s inequality may not a priori be clear, an intriguing however, the Penrose inequality loses its connection with connection has been discovered between cosmic censorship cosmic censorship, since there are unstable black holes and the weak gravity conjecture: it was found in [18,19] in higher dimensions that develop singularities on their that cosmic censorship can be violated in AdS in a theory horizon when they pinch off, violating cosmic censorship. involving only a Maxwell field coupled to gravity. However, this type of naked singularity is rather mild, in However, under inclusion of a charged scalar field with that its resolution in quantum gravity is almost certainly to mass and charge satisfying the weak gravity conjecture, let the horizon bifurcate (or change its topology) with only the Einstein-Maxwell counterexamples to cosmic censor- of order a Planck energy emitted in the process. It is still ship require fine tuning and are not generic [20,21]. This possible that some relaxed version of cosmic censorship, triumvirate connection between cosmic censorship, the which permits such mild singularities but forbids large- weak gravity conjecture, and the Penrose inequality is scale violations, remains valid. Such a reformulation of therefore suggestive that some principle that rules out large cosmic censorship could still imply the Penrose inequality. violations of cosmic censorship may be generically sat- Another possibility is that the Penrose inequality may be isfied in holography. false as a broad conjecture about general relativity but could It is possible to construct a quantum generalization of be valid for theories of gravity coupled to low energy matter the construction in [14,15], where A½σ is replaced by the 4 ℏ fields that admit a UV completion within quantum gravity. generalized entropy G Sgen [22]. A natural question is This is a statement that can be tested within the framework of then whether holography implies a quantum generalization holographic quantum gravity [11–13]. The classical limit of of the Penrose inequality. Indeed, in order to prove holography relates classical properties of gravity to proper- Penrose’s inequality in the semiclassical regime, we need a b ties of a dual quantum field theory (QFT) and one can use to replace the null energy condition (Tabk k ≥ 0 for all this dual description to try to derive new inequalities. null vectors ka) with the so-called quantum focusing We will show that a precise formulation of the conjecture [23]. The final statement of the AdS Penrose 4 ℏ (Lorentzian) AdS Penrose inequality follows from standard inequality, however, is not particularly interesting: G Sgen ideas in holography without assuming cosmic censorship. differs from A½σ by a perturbative correction involving the The basic idea is very simple. Given the initial data above, entropy of quantum fields on the background classical it is possible to construct a spacetime with two asymptotic spacetime. Since it is a perturbative correction, it can only boundaries and the same mass M on each boundary such make a difference in the case when the classical Penrose that the dual two-boundary QFT state has the property that inequality is saturated. But in that case, it reduces to the the reduced density matrix of one boundary, ρ0, has von well-known statement that the entropy of a quantum field Neumann entropy S½ρ0¼A½σ=4Gℏ [14,15]. This entropy on a static black hole background is maximized by the is clearly less than the maximum entropy of any density Hartle-Hawking state. matrix with the same energy M. But the bulk dual to a Saturation of the Penrose inequality is interesting in maximum entropy state in a microcanonical ensemble is the its own right, as the existing proofs of the Riemannian static AdS black hole [16].So Penrose inequality for asymptotically flat initial data show that it is saturated only for the Schwarzschild solution. ½σ¼4 ℏ ½ρ ≤ 4 ℏ ½ρ¼ ð Þ ð Þ A G S 0 G max S ABH M ; 1:3 So the Penrose inequality provides a rigidity result for fix M Schwarzschild black holes. However, in the context of ð Þ where ABH M is the area of a static AdS black hole with mass M. After solving the right-hand side for M we 3We thank T. Jacobson for discussions on this.

126009-2 HOLOGRAPHIC ARGUMENT FOR THE PENROSE … PHYS. REV. D 99, 126009 (2019) holography this is not the case: maximum entropy static but we will only need to apply it to complete components black holes need not be unique. For example, at low energy of the asymptotic boundary. It is often the case that the 5 in AdS5 × S , localized ten dimensional black holes have surface XHRT wraps the internal dimensions, so that 5 ¼ ðdÞ Y more entropy than Schwarzschild AdS5 × S , while the XHRT XHRT × D−d, where the full spacetime is given situation is reversed at high energy. Clearly, there is a by Md × YD−d, and Md is asymptotically AdS. In this case, particular energy at which these two different static black we obtain holes have the same entropy. Even without including the S5 ðdÞ (or other compact extra dimensions) surprisingly little is A½X S ½ρ¼ HRT ; ð2:2Þ known about the uniqueness of static AdS black holes. vN 4GðdÞℏ Even Schwarzschild AdS has not been shown to be unique. ðdÞ ’ The best one has is a proof that there are no nearby static where G is the d-dimensional Newton s constant. black holes [24]. We will also need a more recent addition to the holo- There are applications of the AdS Penrose inequality graphic dictionary, which relates the area of a close variant to the area law for apparent horizons [25–29] and a of apparent horizons to a coarse-graining of the von proposed quasilocal mass formula [30]. We will discuss Neumann entropy. Recall that an apparent horizon is a these applications in Sec. III, after deriving the inequality in type of marginally trapped surface: that is, a compact, σ the next section. There is also a generalization to charged codimension-two surface whose area is stationary under black holes, which we describe in Sec. III. deformations in an outgoing null direction. Here outgoing is defined with respect to the anti–de Sitter (AdS) boundary (in the situation that there are multiple connected compo- II. CONSTRUCTION nents to the asymptotic boundary we define outgoing with In this section, we will first review the requisite concepts respect to a particular connected component). More explic- for our argument. This includes a review of holographic itly, if ka is the outgoing, future-directed orthogonal null entanglement entropy, our assumptions about apparent vector to σ, and hab is the induced metric on σ, then σ is horizons, and of the dual to the area of apparent horizons. marginally trapped if the expansion We will then present our argument for the Penrose inequality. ab θ ≡ h ∇akb ð2:3Þ A. Background review vanishes everywhere on σ. The usual definition of an Assumptions and conventions: we work in the large-N, apparent horizon is the outermost, marginally trapped strong coupling limit of gauge/gravity duality. We will surface on a Cauchy slice Σ.In[14,15], a closely related make all of the same assumptions as [14,15], including the type of surface called a “minimar” surface was defined. null energy condition. Since our construction relies in a A compact, marginally trapped surface σ is said to be large part on [14,15], we will give a rough sketch of that minimar if: construction and refer the reader to the original papers for (1) σ is homologous to a (complete connected) compo- technical details. Finally, we assume reflecting boundary nent of the asymptotic boundary. That is, there exists conditions at the asymptotic boundary. In this section we a hypersurface H such that ∂H ¼ σ ∪ B, where B will restrict to time-independent boundary sources for our is a Cauchy slice of (a connected component of) original QFT state (we follow a construction that results in the asymptotic boundary. The outer wedge of σ— a new QFT state, to which this assumption may not apply). the region spacelike to σ and between it and the This restriction will be lifted in Sec. III. asymptotic boundary—is the domain of dependence ½σ We make use of the Hubeny-Rangamani-Takayangi of H, and is denoted OW (see Fig. 1). ½σ σ (HRT) prescription for holographic entanglement entropy (2) There exists a Cauchy slice H of OW such that is [31,32] (see [33] for a review): the minimal area surface on H which is homologous to the boundary. ½ σ A XHRT (3) is stable: consider the null geodesic congruence S ½ρ¼ ; ð2:1Þ a λ λ ¼ 0 vN 4GðDÞℏ generated by k with affine parameter (with on σ) and let la be an ingoing future-directed where ρ is the reduced density matrix on a single connected null vector orthogonal to surfaces of constant λ. ðDÞ la component B1 of the asymptotic boundary, G is the bulk Then there exists a parametrization of such that a∇ θ ≤ 0 θ Newton’s constant in D ¼ 10 or 11-dimensions (depending k a ðlÞ , where the expansion, ðlÞ, is defined l on whether we are in string theory or M-theory), and XHRT as in (2.3) with kb replaced by b. Equality can hold is the minimal area spacelike codimension-two surface in only if θðlÞ ¼ 0 everywhere on σ. the full (10 or 11-dimensional) bulk which is (i) a stationary Since apparent horizons are outermost on a Cauchy point of the area functional and (ii) is homologous to B1. slice, they always satisfy requirement 1. One can show The original prescription works for arbitrary subregions, that generic apparent horizons satisfy the other two

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Σ FIG. 1. Adapted from [15]. An apparent horizon σ (purple) and FIG. 2. Adapted from [15]. The initial data slice (green) used its outer wedge O ½σ (shaded gray). By assumption there exists to prepare the doubled spacetime with an HRT surface X whose W σ ½σ a Cauchy slice H of O ½σ on which σ is the minimal area surface area is identical to the area of . Note that OW is fixed by W Σ ½σ homologous to the boundary. taking the component of in OW to be identical in the doubled spacetime and in the original spacetime. The entanglement wedge bounded by X and the right boundary is dual to ρ0. requirements also [15]. From here on, we will assume that our apparent horizons are minimar. In [14,15], it was argued that the area of apparent A½σ¼A½X; ð2:5Þ horizons is computed by a coarse-grained entropy called the outer entropy, obtained by maximizing the von where X is the HRT surface of ðM0;g0Þ. Assuming the Neumann entropy over all possible spacetimes that can HRT prescription, we have be glued into the interior of σ: ðDÞ A½σ¼A½X¼4G ℏS ½ρ0; ð2:6Þ Area½σ vN ¼ max S ½ρ ≡ Souter½σ; ð2:4Þ 4 ℏ ρ∈H vN G for some QFT state ρ0 (and all quantities in the above H equation are strictly finite in the large-N limit, with the where is the set of all QFT states with a semiclassical ðDÞ bulk dual which is identical to our original bulk in the understanding that G should be replaced by the appropriate power of 1=N according to the holographic region O ½σ. The proof identifies the state ρ ¼ ρ0 that W dictionary).5 We note here two technical points. First, we maximizes SvN as above by explicitly constructing the dual ð 0 0Þ bulk spacetime ðM0;g0Þ: this spacetime has two asymptotic are assuming that M ;g has a CFT dual. We have not regions, agrees with the original spacetime ðM; gÞ on proven that it always will, but we think that this is very likely. Second, since the proof of the HRT proposal in OW½σ, and has an HRT surface X whose area is the same as the area of σ. The spacetime is constructed by preparing [37,38] applies to states that can be constructed by path ð 0 0Þ an initial data slice Σ and time evolving it to generate the integrals, and it is not obvious that M ;g can be maximal evolution M. It can then be shown that the HRT constructed in this way, it is in principle possible that ð 0 0Þ surface X is spacelike separated from both asymptotic M ;g has a QFT dual but that its von Neumann entropy is boundaries, is null related to σ, and has the same area as σ not computed by the area of the HRT surface. Thus our (see Fig. 2).4 result could be framed as an exclusive alternative: either Note that we are not assuming cosmic censorship, so the Penrose inequality holds, or the HRT prescription is it is possible that M will have a Cauchy horizon and be incomplete even in the regime of classical general relativity extendible. As noted in [15], the same result still holds in in the bulk. ½σ this case. The outer wedge OW is by construction identical in ðM; gÞ and in ðM0;g0Þ. This immediately implies that asymptotic charges are identical in both spacetimes. In B. An AdS Penrose inequality particular, the total mass within OW½σ is identical in both The immediate conclusion that follows from the con- spacetimes. This in turn implies that the QFT stress struction reviewed above is that there exists a spacetime tensor integrated on any slice of ∂M ∩ OW½σ—the QFT ðM0;g0Þ in which the following equality holds: 5We are taking the N → ∞ limit in a way that keeps the ratio 4The regularity of the characteristic initial data specified on Σ of the black hole radius to AdS length scale nonzero. If it does is expected to result in a locally unique Cauchy evolution [34]; go to zero, one ends up with a black hole in an symptotically see e.g., [35,36]. The data satisfies the constraint equations, and is flat spacetime and radiation can dominate the microcanonical consistent with minimally coupled scalar and Maxwell fields. ensemble.

126009-4 HOLOGRAPHIC ARGUMENT FOR THE PENROSE … PHYS. REV. D 99, 126009 (2019) energy E—is identical in ρ and ρ0 to leading order in 1=N. Since we have assumed that any sources are time independent in OW½σ, this energy is independent of time. Since ρ0 is a state (on one connected component of the asymptotic boundary) with energy E, its entropy must be smaller than the entropy in the microcanonical ensemble:

ðDÞℏ ½ρ ≤ ðDÞℏ ¼ ðDÞℏ ½ρ ð Þ G SvN 0 G max SvN G SvN micro 2:7 EδE where the right-hand side is a maximization of SvN over all QFT states with energy in the range E δE at a fixed boundary Cauchy slice, where δE is much larger than the difference between energy eigenvalues but much smaller than E. It has recently been argued in FIG. 3. The area increases in a spacelike direction along a [16] using the Euclidean path integral that the bulk dual future holographic screen, which corresponds to time increase of the microcanonical ensemble is a static black hole of on the boundary. Allowing time-dependent sources on the mass E, whose Bekenstein-Hawking entropy is precisely boundary results in an increase in E. We maximize SvN subject ½σ S ½ρ . to E at the past boundary of OW , so the mass in the Penrose vN micro inequality increases correspondingly with the apparent horizon We thus find: area increase. ½σ ≤ ½ ð Þ A ABH E : 2:8 Recall now that hypersurfaces foliated by marginally Since we have not required the dominating static black hole trapped surfaces—so-called future holographic screens or σ to be a product with the internal space, this formula [39]—satisfy an area monotonicity theorem [25–29].In applies to the full ten or eleven dimensional spacetime. particular, the spacelike component of a future holographic 7 When the spacetime is asymptotically AdS4 × Y and both screen is foliated by minimar surfaces. If the (minimar) σ and the dominating saddle of the microcanonical ensem- apparent horizons in the foliations are labeled σðrÞ, with r ble are products with Y7, we recover precisely Eq. (1.2), the foliation parameter, then evolving forwards to increas- the Penrose inequality in four-dimensional asymptotically ing r along the holographic screen corresponds to evolving AdS spacetimes. along the boundary towards the future: the past boundary of OW½σð1Þ ∩ ∂M is to the past of the past boundary of OW½σð1 þ ϵÞ ∩ ∂M. See Fig. 3. The energy increase due to III. GENERALIZATIONS AND APPLICATIONS boundary sources corresponds to an area increase, and the A. Time dependent sources two are related via the Penrose inequality. In the previous section, we restricted to time-independent In [14,15], the area increase was interpreted as a sources in the QFT to simplify the construction. We will thermodynamic second law: a coarse-grained QFT entropy now relax that condition and allow arbitrary time-dependent increase. Here we see that this thermodynamic entropy sources on the boundary. Since time-dependent sources will increase is also related to an energy increase via the Penrose by definition result in changes to the total energy, we will inequality. need to be more precise about the asymptotic energy that goes into Eq. (2.8). Operating under our prior assumption of B. Connection to quasilocal mass reflecting boundary conditions, turning on boundary sources There is an interesting connection between our deriva- typically increases the energy, so if we choose to evaluate tion of the Penrose inequality and a recent definition of a ¼ the energy at a time slice t t1 on the boundary, it will quasilocal gravitational mass associated to a “normal” generically be smaller than the energy at a boundary time (non-trapped, θ > 0) surface. Bousso et al. [30] proposed ¼ k slice t t2 >t1. that the outer entropy of a normal surface ν should be To obtain the tightest bound, we consider smooth thought of as defining a quasilocal mass Mν associated to ν. spacelike cross sections of the boundary that are contained They defined Mν using the relation between mass and area ½σ in OW and compute the energy on each. We then take the of a Schwarzschild black hole. However, since we are ½σ minimum of these energies, Emin . We may execute the considering asymptotically AdS spacetimes, it seems more ½σ full construction above while keeping Emin fixed, which appropriate to use the Schwarzschild AdS solution. In four yields the general inequality: dimensions, this AdS version of the proposal in [30] is

½σ ≤ ½ ½σ ð Þ outer 1=2 outer 3=2 A ABH Emin : 3:1 2GMν ¼ðaS ½νÞ þðaS ½νÞ ð3:2Þ

126009-5 NETTA ENGELHARDT and GARY T. HOROWITZ PHYS. REV. D 99, 126009 (2019) where a ¼ Gℏ=π. Bousso et al. [30] also construct a The holographic argument is easily generalized to generalization of the doubled spacetime construction include charge. One can again construct a spacetime so reviewed in Sec. II for normal surfaces. They provide that the dual state, ρ0, has the same mass and charge as the ð 00 00Þ ðDÞ initial data for a spacetime M ;g with an HRT surface original one and satisfies A½σ¼4G ℏSvN½ρ0. This is outer Xν whose area is given by S ½ν, and where the outer because the arguments in [14,15] included a possible ½ν wedge OW is unchanged. Maxwell field. One can now maximize SvN½ρ over all Now we can apply our above argument. Let ρ1 be the states holding the charge fixed as well as the mass. The 00 00 dual to one side of ðM ;g Þ. The von Neumann entropy of argument in [16] is easily generalized to include charge, ρ1 is given by the area of Xν, and is smaller than the von with the result that the bulk dual to a maximum entropy Neumann entropy in the microcanonical ensemble with the state at fixed energy and charge is the maximum area same energy. Therefore: static black hole with the same conserved quantities.6 ½σ ð Þ outer This implies that A

This is because if the final black hole is Reissner-Nordstrom, ACKNOWLEDGMENTS ¼ ¼ theinequality is saturated with A ABH and M MBH.Ifthe final black hole is rotating, the left-hand side is reduced. It is a pleasure to thank R. Bousso, X. Dong, R. Referring back to the original quantities only reduces the Emparan, S. Fischetti, G. Gibbons, T. Jacobson, D. left-hand side further and increases the right-hand side. Marolf, and J. Santos for discussions. N. E. is supported Equation(3.4) can beviewed as a strengtheningofthe positive by the Princeton University Gravity Initiative and by NSF mass theorem in the presence of charge: GM ≥ jQj.This Grant No. PHY-1620059. G. H. is supported in part by NSF argument also extends to AdS; the statement is imply that the Grant No. PHY-1801805. initial area cannot be greater than the area of a Reissner- Nordstrom AdS black hole with the same mass and charge. 6We thank D. Marolf for a discussion about this.

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