Bekenstein Bounds, Penrose Inequalities, and Black Hole Formation

PHYSICAL REVIEW D 97, 124026 (2018)

Bekenstein bounds, Penrose inequalities, and black hole formation

Jaroslaw S. Jaracz* and Marcus A. Khuri† Department of Mathematics, Stony Brook University, Stony Brook, New York 11794, USA

(Received 14 February 2018; published 12 June 2018)

A universal geometric inequality for bodies relating energy, size, angular momentum, and charge is naturally implied by Bekenstein’s entropy bounds. We establish versions of this inequality for axisymmetric bodies satisfying appropriate energy conditions, thus lending credence to the most general form of Bekenstein’s bound. Similar techniques are then used to prove a Penrose-like inequality in which the ADM energy is bounded from below in terms of horizon area, angular momentum, and charge. Lastly, new criteria for the formation of black holes is presented involving concentration of angular momentum, charge, and nonelectromagnetic matter energy.

DOI: 10.1103/PhysRevD.97.124026

I. INTRODUCTION setting of asymptotically flat, maximal initial data. The inequality produced in [10] is not quite in the form of In [1], Bekenstein utilized heuristic arguments involving (1.2), and it is not clear if one implies the other. Both results black holes to derive an upper bound for the entropy of [9,10] are based on monotonicity of the Hawking mass along macroscopic bodies, in terms of the total energy and radius inverse mean curvature flow (IMCF), which is valid in the of the smallest sphere that encloses the object. This inequal- maximal case assuming the dominant energy condition holds. ity was later generalized [2–5] to include contributions from The purpose of the present article is threefold. The first the angular momentum J and charge Q of the body, goal is to establish Bekenstein-like inequalities closely related to (1.2) without the hypothesis of maximality for the Q2 ℏc ER 2 − 2J 2 − ≥ S initial data, and thereby generalize the works [9,10].Our ð Þ c 2 2 ; ð1:1Þ πkb approach relies on a coupling of the IMCF with an qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi embellished version of the Jang equation [11,12], which ’ where Boltzmann s constant is denoted by kb, S is entropy, E is inspired by the proof of the positive mass theorem [13]. is total energy, R is the radius described above, aℏ and c are Secondly, our techniques naturally lend themselves to the reduced Planck’s constant and speed of light. Although establish a version of the Penrose inequality [14] with the original inequality [1] without angular momentum and angular momentum and charge, for general axisymmetric charge has received much attention [6–8], the enhanced initial data without the maximal assumption. A similar relation (1.1) has not been properly investigated. An result in the maximal case was recently given in [15]. important initial step in that direction was taken by Dain Recall that Penrose [16] proposed a sharp inequality [9] who studied the inequality bounding the total energy of a black hole spacetime from below in terms of the horizon area. It serves a necessary Q4 c2J 2 condition for the cosmic censorship conjecture. Thus, a E2 ≥ þ ; ð1:2Þ 4R2 R2 counterexample would disprove cosmic censorship while verification of the Penrose inequality only lends credence which is implied by (1.1) since entropy is always non- to the conjecture’s validity. In [17,18], the Penrose inequal- negative. He was able to establish (1.2) within the context of ity has been proven in the maximal case. Moreover, electromagnetism, and also in general relativity for bodies generalizations including angular momentum and charge with zero angular momentum contained in asymptotically have been proposed [14] motivated by Penrose’s original flat, maximal initial data which are void of black holes. In this heuristic arguments. The full Penrose inequality may then result, E is given by the ADM energy. The idea is that a proof be stated as follows, of (1.2) lends indirect evidence for the full Bekenstein bound 2 (1.1). Later on, Dain’s result was extended to include a c4 A π 4πc2J 2 E2 ≥ þ Q2 þ ; ð1:3Þ contribution from angular momentum [10], again in the G 16π A A rffiffiffiffiffiffiffiffi rffiffiffiffi *[email protected] where A is the minimum area required to enclose the †[email protected] outermost apparent horizon in an axisymmetric initial data

a a b set satisfying the relevant energy condition, and G is the and n is the unit timelike normal to M, then μ ¼ Tabn n −1 −1 a gravitational constant. This comes with a rigidity statement and c Ji ¼ c Tain are the matter energy and momen- asserting that equality holds if and only if the initial data tum density of the slice. These must satisfy the constraint arise from an embedding into the Kerr spacetime. We also equations note that the Bekenstein bound (1.1), when applied to black holes, implies the Penrose inequality (1.3). To see this, 16πG μ ¼ R þðTr kÞ2 − jkj2; simply recall that for a black hole with event horizon area c4 g Ae the radius and entropy are given by 8πG J ¼ divðk − ðTr kÞgÞ; ð2:1Þ c4 g A k A R e S b e ¼ 4 ; ¼ 2 ; ð1:4Þ where R is the scalar curvature of g. We will also be π 4lp rffiffiffiffiffiffi interested in the electromagnetic field, and let E and B 3 denote the electric and magnetic fields induced on M, where l ¼ Gℏ=c is the Planck length. Inequality (1.3) p respectively. Assuming that all measured energy densities has been established in the maximal case without the pffiffiffiffiffiffiffiffiffiffiffiffiffi are non-negative implies μ ≥ jJj, which is referred to angular momentum term in a series of papers [19–22]. as the dominant energy condition. It is also useful to single However, there has been very little to no progress on out the nonelectromagnetic matter fields for which the including angular momentum. The only result known to the author in this direction is [15]. Here we will establish a energy and momentum densities are obtained from μEM ¼ − 1 2 2 1 version of (1.3) valid in the general nonmaximal setting, μ 8π ðjEj þjBj Þ and JEM ¼ J þ 4π E × B. The charged assuming the existence of solutions to a canonical coupling dominant energy condition is then μEM ≥ jJEMj. of the Jang equation to IMCF; such solutions are known to A body Ω will be described as a connected open subset exist in spherical symmetry. of M having compact closure and smooth boundary ∂Ω. Lastly, the methods used to study the Penrose inequality The total charge within the body is then given by above lead to new criteria for black hole formation, as well 1 2 1 2 as inequalities for bodies involving size, angular momen- 2 Q ¼ divEdxg þ divBdxg ; ð2:2Þ tum, and charge. Recall that Thorne’s hoop conjecture [23] 4π Ω 4π Ω Z Z roughly states that if enough matter/energy is condensed in an appropriately small region, then gravitational collapse and it will always be presumed that there is no charged will ensue. Mathematically this assertion may be translated matter outside Ω. In order to characterize the angular into a heuristic inequality, momentum of the body, the initial data will be assumed to be axisymmetric. That is, there is a Uð1Þ subgroup MassðΩÞ > C · SizeðΩÞ; ð1:5Þ within the group of isometries of the Riemannian manifold ðM; gÞ, and all relevant quantities are invariant under the which if satisfied for a body Ω, then implies that Ω must be Uð1Þ action. Without axisymmetry it is problematic to contained within an apparent horizon; here C is a universal define quasi-local angular momentum [29]. Moreover, with constant. One of the primary difficulties in establishing this hypothesis all angular momentum is contained within such a result is finding a proper notion of quasi-local mass the matter fields, as gravitational waves carry no angular to use in the left-hand side of (1.5). As it turns out, mass/ momentum. Let η be the generator of the Uð1Þ symmetry, energy is not the only quantity that is appropriate to place then the angular momentum of the body is on the left-hand side of the inequality. We will show below 1 that angular momentum and charge also naturally arise on i J ¼ Jiη dxg: ð2:3Þ the left-hand side, and thus provide extra means to satisfy c Ω Z (1.5). This will be rigorously proven in spherical symmetry, and motivation will be given to indicate why the result The basic strategy to obtain Bekenstein type bounds should hold in generality. Related results concerning black (1.2) is to use monotonicity of the Hawking mass along hole existence due to concentration of angular momentum inverse mean curvature flow. This worked well in [9,10,15] 0 or charge have been given in [24–26], using different because of the maximal assumption Trgk ¼ . More methods. See also [27,28]. precisely, monotonicity of the Hawking mass relies on non-negativity of the scalar curvature, and this is achieved with the dominant energy condition if the data are maximal. II. BEKENSTEIN BOUNDS Here we do not assume that the data are maximal, and thus Consider a spacelike slice M of an asymptotically flat this method breaks down. However, we may follow an four-dimensional spacetime. The induced positive definite approach similar to that in the proof of the positive mass metric g and extrinsic curvature k together yield an initial theorem [13], where the initial data are deformed by 2 data set ðM; g; kÞ.IfTab denotes the stress energy tensor ðM; g; kÞ → ðM; g¯Þ with g¯ij ¼ gij þ u fifj for some

124026-2 BEKENSTEIN BOUNDS, PENROSE INEQUALITIES, AND … PHYS. REV. D 97, 124026 (2018) functions u>0 and f.In[13], the function u ¼ 1 and f is 4 t2 ¯ ¯ ¯ c jStj ¯ chosen to solve the so called Jang equation, which is EHðSt2 Þ − EHðSt1 Þ ≥ R: ð2:8Þ 16 16 ¯ πG t1 sffiffiffiffiffiffiffiffiπ S designed to impart positivity properties to the scalar Z Z t curvature R¯ of g¯. In the present setting, it is more appropriate to utilize an embellished version of the Jang The first two terms on the right-hand side in the expression equation (2.5) will provide lower bounds for (2.8) involving the charge and angular momentum, while the divergence expression will contribute to the Hawking energies. u2fifj u∇ f þ u f þ u f gij − ij i j j i − k ¼ 0; Consider now the case when the flow starts from a point 1 þ u2j∇fj2 1 2 ∇ 2 ij Ω þ u j fj x0 within the body on the axis of rotation, so that the starting time is t0 ¼ −∞. Observe that in (2.8) with t1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:4Þ −∞ and t2 ¼ ∞ several simplifications occur. Namely, ∇ since the Hawking energy of a point is zero and the limit of where ij are second covariant derivatives with respect to g → ∞ i ij Hawking energies as t is no larger than the ADM and f ¼ g fj. This equation also yields desirable features (total) energy, the left-hand side of (2.8) may be replaced for the scalar curvature which now takes the form with the ADM energy E. Note that this total energy is a priori with respect to the deformed metric g¯.However,by 16 ¯ πG 2 2 −1 placing the natural boundary conditions at infinity for R ¼ ðμ −JðwÞÞþjh − kj¯ þ 2jqj¯ −2u div¯ ðuqÞ; c4 g g g solutions of the Jang equation, namely f → 0 in the ð2:5Þ asymptotic end, the total energy of g and g¯ are equivalent [13]. Furthermore if the charged dominant energy condition holds then μ − J w ≥ 1 E 2 B 2 , as it may be where ð Þ 8π ðj j þj j Þ assumed without loss of generality in axisymmetry that the electric and magnetic fields have no component in the u∇ f þ u f þ u f uf ij i j j i i Killing direction so that E × BðwÞ¼0.In[19], a deforma- hij ¼ 2 2 ;wi ¼ 2 2 ; 1 þ u j∇fj 1 þ u j∇fj tion of the electromagnetic field ðE; BÞ → ðE;¯ B¯ Þ, tailored to ufj the Jang metric g¯, was given which preserves total charge as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qi ¼ 2 2 ðhij kijÞ: ð2:6Þ well as zero charge density and has less energy density than 1 þ u j∇fj the original field jEj ≥ jE¯ j, jBj ≥ jB¯ j. From this, a lower pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bound for the right-hand side of (2.8) is obtained in terms of These formulas along with their geometric interpretations the energy density of ðE;¯ B¯ Þ, and since the surface integrals are explained in [11,12]. Observe that the first term on the are computed with respect to g¯ a relation with total charge is right-hand side of (2.5) is non-negative if the dominant produced as in [19]. In particular, energy condition is satisfied, since jwj ≤ 1. Furthermore, all other terms are manifestly non-negative except possibly the divergence term. The deformed scalar curvature may ∞ jS¯ j t ðμ − JðwÞÞ then be described as “weakly” non-negative, since inte- 16 ¯ −∞ sffiffiffiffiffiffiffiffiπ St grating it against u produces a non-negative quantity Z Z modulo boundary terms. 1 ∞ jS¯ j Q2 ≥ t ¯ 2 ¯ 2 ≥ In order to optimize the positivity of R¯ with regards to ðjEj þjBj Þ ¯ ; ð2:9Þ 8π t sffiffiffiffiffiffiffiffi16π S¯ 2R ¯ ∞ Z Ã Z t tÃ IMCF, we choose u as follows. Let fStgt¼t0 be an IMCF in the deformed data ðM; g¯Þ, where t0 ¼ 0 or −∞ depending where R¯ ¼ jS¯ j=4π is the area radius of S¯ . The time t on whether the flow starts at a surface or a point. A weak tÃ tÃ tÃ Ã version of the flow always exists [18] in the asymptotically may be chosenqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arbitrarily; however, in order to obtain the flat setting, although for the purposes of exposition we may optimal inequality for the body, tÃ will denote the first ¯ ¯ (smallest) time such that the flow surface S¯ completely assume that the flow is smooth. Then set u ¼ jStj=16πH tÃ to be the product of the square root of area and mean encloses Ω. Moreover, since the flow will change depending pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi curvature for the flow surfaces. Consider now the Hawking on the choice of its starting point x0, optimization requires energy of the flow surfaces within the deformed data: that we choose the x0 for which the area radius at tÃ is smallest. Such a starting point exists within the body since Ω is compact. The radius R¯ of Ω will then be defined as in [9] to c4 jS¯ j 1 ¯ ¯ t ¯ 2 be this optimal area radius, and in (2.9) the radius Rt may be EHðStÞ¼ 1 − H : ð2:7Þ Ã G sffiffiffiffiffiffiffiffi16π 16π S¯ R¯ Z t replaced with . Within the scalar curvature formula (2.5) the second term A well-known computation [18] asserts that if t2 >t1, then on the right-hand side encodes a contribution from angular

124026-3 JAROSLAW S. JARACZ and MARCUS A. KHURI PHYS. REV. D 97, 124026 (2018) momentum. In order to extract this contribution, we first Theorem 1: Let ðM; g; k; E; BÞ be a complete, axisym- make some observations. The metric g¯ arises as the induced metric, asymptotically flat initial data set for the Einstein- metric on the graph of the function f [12], and the surfaces Maxwell equations, satisfying the charged dominant ¯ ≥ St may be interpreted as a flow within the graph. There is energy condition μEM jJEMj and without apparent hori- ¯ Ω ⊂ then a natural projection of St into ðM;gÞ which will be zons. Suppose that M is a body outside of which there denoted St. Since the flow starts from a point on the is no charge density or momentum density in the direction ¯ of axisymmetry. If the Jang/IMCF system of equations symmetry axis, each of the surfaces St, St is axisymmetric. As is shown in the Appendix under mild hypotheses, it then admits a solution, then 0 follows that hðη; νÞ¼ on St, where ν is the unit normal to Q2 G J 2 S . Therefore, assuming that angular momentum density E ≥ þ : ð2:13Þ t 2R¯ 2c2 R¯ R¯ 2 vanishes outside the body and using Hölder’s inequality c produces This theorem generalizes the results of [9,10] to the nonmaximal setting. Although it is in the spirit of the 2 2 2 8πG 2 Bekenstein bound (1.2), these two inequalities are distinct J ¼ kðη;νÞ ¼ ½kðη;νÞ − hðη;νÞ c3 in that one does not directly imply the other. Nevertheless, ZSt ZSt 2 as will be shown in the next section inequality (2.13) does 2 2 2 ≤ jk − hj jηj ≤ jk −hj¯ jηj ; indirectly imply a lower bound for E which has the same g ¯ g ¯ ZSt ZSt ZSt structure as (1.2). ð2:10Þ III. PENROSE INEQUALITIES where we have also used the fact that g¯ measures areas to be at least as large as does g. This estimate is suited to give a In this section, we will adapt the techniques discussed lower bound for the ADM energy which may be expressed above to establish a version of the Penrose inequality with properly with the ‘circumference’ radius angular momentum and charge (1.3). This will then yield an alternate version of the Bekenstein bound (2.13). Recall ⊂ ∞ ¯ that an apparent horizon is a surface S M which has zero ¯ −2 ¯ jStj Rc ¼ jSt j 2 : ð2:11Þ null expansion, that is, a shell of light emitted from the Ã ¯ η tÃ S j j surface is (infinitesimally) neither growing nor shrinking in qffiffiffiffiffiffiffiffi Z pt ffiffiffiffiffiffiffi area as it leaves the surface. These surfaces indicate the ¯ R The radius Rc was used and studied in [10,15], where presence of a strong gravitational field, and may be inter- it was shown that if the flow has reasonably nice preted as quasi-local versions of black hole event horizons properties then this radius may be related to more tradi- from the initial data point of view. Mathematically they are 0 tional measures of size for the body. In particular, if the expressed by one of the two equations θÆ ≔ H Æ TrSk ¼ , flow remains convex outside of Ω, as it is known to be for where the signs þ=− indicate a future/past horizon. An ¯ large times jtj ≫ 0 or in spherical symmetry, then Rc ≤ apparent horizon is called outermost within an initial data set if it is not enclosed by any other apparent horizon. 5=2maxS jηj which is proportional to the circumference tÃ In contrast to the previous section, here we will work of the largest orbit within S . Because it provides and upper pffiffiffiffiffiffiffiffi tÃ with an IMCF starting at a closed axisymmetric surface S R¯ bound for c, when the flow is convex the circumference so that t0 ¼ 0 is the starting time of the flow, and S will may be used in place of the this radius in either be an outermost apparent horizon or the boundary of a body ∂Ω. First consider the case in which S ¼ ∂Ω, and 4 ∞ ¯ 2 assume that the boundary of the body is completely c jStj 2 G J jh − kjg¯ ≥ 2 ¯ ¯ 2 : ð2:12Þ untrapped H>jTr kj. This allows for the prescription 16πG 0 sffiffiffiffiffiffiffiffi16π ¯ 2c RR S Z ZSt c of a Neumann type boundary condition for solutions of the It is now possible to combine (2.8), (2.9), and (2.12) to Jang equation (2.4) ∂ obtain a Bekenstein-type bound. Note that the proof above u νf −1 relies on the existence of a solution to the Jang equation 2 2 ¼ H TrSk: ð3:1Þ 1 þ u j∇fj coupled to IMCF through the choice of the function u.Due to the fact that solutions to the Jang equation tend to blow- It was shownp inffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi[11,31], in the context of spherical up at apparent horizons [30], it will be assumed that the symmetry, that solutions of the Jang/IMCF system exist initial data are devoid of these surfaces. Under this satisfying this boundary condition. Moreover, it was also hypothesis, the desired solutions to the Jang/IMCF system shown that with (3.1) the boundary integrals arising from have been shown to always exist in spherical symmetry the divergence expression associated with R¯ in (2.8), [11], and it is reasonable to expect that existence continues combine with the Hawking energy on the left-hand side to hold at least in a weak sense in axisymmetry. of (2.8), to yield

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0 16 ∞ ¯ 4 horizon θþθ− ¼ so that ESHðSÞ¼ jSj= π. With this a jStj c 2 E − ESHðSÞ ≥ ðμ − JðwÞÞ þ jh− kjg¯ version of the Penrose inequality follows. 0 sffiffiffiffiffiffiffiffi16π S¯ 16πG pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z t Theorem 2: Let ðM; g; k; E; BÞ be an axisymmetric, ð3:2Þ asymptotically flat initial data set for the Einstein-Maxwell equations, satisfying the charged dominant energy con- where the spacetime Hawking energy is given by dition μEM ≥ jJEMj and with outermost apparent horizon boundary having one component. Suppose further that c4 jSj 1 there is no charge density or momentum density in the E ðSÞ¼ 1 − θ θ− : ð3:3Þ direction of axisymmetry. If the Jang/IMCF system of SH G 16π 16π þ rffiffiffiffiffiffiffiffi ZS equations admits a proper solution then

It should be pointed out that (3.2) depends on appropriate 4 2 2 2 c j∂Mj π c J behavior of the IMCF. For instance in the weak formulation E2 ≥ þ Q2 þ : ð3:5Þ G 16π j∂Mj 4R¯ 2 of Huisken/Ilmanen [18], the flow may instantaneously rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi c jump from the desired starting surface S to another surface This result is similar to the conjectured Penrose inequal- S˜ enclosing it with less area. If this occurs, then in ˜ ity (1.3) with the primary difference arising in the angular inequality (3.2) the role of S should be replaced by S. momentum term. Instead of area, this term involves the Such ‘jumping’ behavior can be prevented by requiring that squared radius defined in the previous section. The proof of S be outer area minimizing in ðM; g¯Þ, in that any surface this theorem requires one more observation in order that it which encloses S should have greater area. In order to follow from (3.4). Namely, upon multiplying (3.4) by the achieve this property with respect to the deformed data first two terms on the right-hand side, we find metric g¯, further geometric hypotheses on S with respect to the original initial data may be required. For the purposes of Q2 Q2 2 2 ≥ ≥ the present article, which does not seek to fully examine the E ESHðSÞþ ¯ E ESHðSÞþ ¯ 2R0 2R0 analytical problem of solving the Jang/IMCF system in G J 2 generality, we will simply refer to solutions with these E S : : þ SHð Þ 2 2 ¯ ¯ 2 ð3 6Þ suitable properties as proper solutions. As pointed out, it is c R0Rc known that proper solutions always exist under the hypothesis of spherical symmetry and small perturbations thereof. From this the desired inequality in Theorem 2 arises from the As in the previous section, the two terms on the right- arguments above. Furthermore, (3.6) may be used to yield a ∂Ω hand side of (3.2) yield contributions of angular momentum Bekenstein bound. Suppose that S ¼ is the boundary of and charge. More precisely, applying (2.9) and (2.12) a body immersed in a strong gravitational field. By this we ≔ 1 − 16 0 produces mean that λ ðjSj= πÞsupSθþθ− > , or rather that θþθ− has sufficiently small positive part. In particular, Q2 G J 2 surfaces S which are close to being an apparent horizon E ≥ E ðSÞþ þ ; ð3:4Þ satisfy this property, as do trapped surfaces. For surfaces S SH 2R¯ 2 2 R¯ R¯ 2 0 c 0 c which satisfy this property, the spacetime Hawking energy is ¯ bounded below by the product of λ and the area radius up to a where R0 is the area radius of S0 ¼ S, and J , Q denote the universal constant. Let λ0 > 0 be fixed and consider the angular momentum and charge contained within S. This class of bodies with boundaries experiencing a appropriately inequality will lead to a Bekenstein bound for bodies in the strong gravitational field so that λ ≥ λ0. Then for bodies of presence of a sufficiently strong gravitation field, as well as this type a Bekenstein bound follows immediately from (3.6) a version of the Penrose inequality. The arguments above seem to rely on the assumption that 4 2 2 2 Q c J S is untrapped, as otherwise the boundary condition (3.1) E ≥ ¯ 2 þ λ0 ¯ 2 : ð3:7Þ 4R0 4Rc would imply that u∂νf ¼Æ∞. However, for the Jang equation, blow-up solutions are natural as first observed in This inequality has the same structure as the Bekenstein the proof of the positive mass theorem [13]. Blow-up inequality (1.2), although the radius associated with the occurs at apparent horizons, and can be prescribed at angular momentum term is more complicated than the area outermost apparent horizons as well [30]. Therefore, in radius. place of the boundary condition (3.1), at an outermost apparent horizon S we will prescribe blow-up as the boundary condition. In this situation, the graph of the IV. BLACK HOLE FORMATION solution to Jang’s equation asymptotes to a cylinder over S, It is a basic folklore belief that if enough matter/energy is and the area of this surface in the deformed metric and concentrated in a sufficiently small region, then gravita- the original coincide jS¯j¼jSj. Moreover, at an apparent tional collapse must ensue. This is typically referred to as

124026-5 JAROSLAW S. JARACZ and MARCUS A. KHURI PHYS. REV. D 97, 124026 (2018) the hoop conjecture or trapped surface conjecture [23,32], ¯ t2 jS j and is quite difficult to formulate precisely, see the t ðμ − JðwÞÞ −∞ sffiffiffiffiffiffiffiffi16π S¯ references in [26]. One of the most general results in this Z Z t direction is due to Schoen and Yau [33], who exploited the t1 ¯ jStj techniques developed in their proof of the positive mass ≥ ðμEM − JEMðwÞÞ −∞ sffiffiffiffiffiffiffiffi16π ¯ theorem [13] to prove the existence of apparent horizons Z ZSt whenever matter density is highly concentrated. Their ¯ 1 t2 jS j strategy is to show that the concentration hypothesis forces þ t ðjE¯ j2 þjB¯ j2Þ 8π sffiffiffiffiffiffiffiffi16π ¯ solutions of the Jang equation to blow-up, and since blow- Zt1 ZSt up can only occur at an apparent horizon the existence of 4 2 ¯ π ¯ 3 Q R1 such a surface in the initial data is established. Here we will ≥ R1minðμEM − jJEMjÞ þ 1 − ; ð4:2Þ 3 Ω˜ 2R¯ R¯ combine this strategy with the techniques used in the 1 1 sffiffiffiffiffiffi2 previous two sections to obtain a black hole existence Ω˜ ¯ ¯ ¯ result due to concentration of nonelectromagnetic matter where 1 is the domain enclosed by St1 and R1, R2 are the ¯ ¯ energy, charge, or angular momentum. In addition, the area radii of St1 , St2 . Notice that if the charged dominant measure of size used in our result will differ considerably energy condition is valid then the first term on the right in from the complicated measure in [13]. (4.2) is non-negative, and the second term also has this Consider two concentric bodies Ω1 ⊂ Ω2, each having the property since areas are nondecreasing in an IMCF. topology of a three-dimensional ball, inside an axisymmet- Similarly, applying the arguments of (2.12) to the current ric asymptotically flat initial data set ðM; g; k; E; BÞ. The setting produces model astronomical body in this context is a typical star, where there is a highly dense core and interior (represented 4 ¯ 2 t2 J Ω c jStj 2 G by 1) compared to the outermost layer or corona (repre- jh − kjg¯ ≥ 2 ¯ ¯ 2 ; ð4:3Þ 16πG −∞ s16ffiffiffiffiffiffiffiffiπ S¯ 2c R1R sented by Ω2nΩ1) with very little matter density. For Z Z t ac simplicity of the model we will assume that the charge where the circumference radius is with respect to the density and momentum density in the Killing direction annular domain vanish in the annular region Ω2nΩ1, so that 0 ¯ divE ¼ divB ¼ JðηÞ¼ . If there are no apparent horizons t2 jS j R¯ −2 ¯ t in the initial data then, as discussed in Sec. II, we may take a ac ¼ jSt1 j 2 : ð4:4Þ t1 ¯ jηj solution of the Jang/IMCF system of equations with the flow qffiffiffiffiffiffiffiffi Z pSt ffiffiffiffiffiffiffi emanating from a point x0 ∈ Ω1 on the axis of rotation. Let R Furthermore assuming that the outer surface St2 is t1 and t2 be the first times for which the flow completely untrapped, so that H>jTrSt kj, implies that the Hawking encloses the boundaries ∂Ω1 and ∂Ω2, respectively. From 2 energy may be estimated above by the area radius the arguments used to obtain (3.2), together with the fact that 4 E ≤ c R¯ – the Hawking energy of a point is zero, we find that SHðSt2 Þ 2G 2. Therefore, combining (4.1) (4.3), yields 4 4 ¯ 4 c ¯ π ¯ 3 t2 jS j c R2 ≥ R minðμ − jJ jÞ t 2 2 3 1 ˜ EM EM ESHðSt2 Þ ≥ ðμ − JðwÞÞ þ jh −kj¯ G Ω1 16 ¯ 16 g −∞ sffiffiffiffiffiffiffiffiπ St πG Z Z 2 ¯ 2 Q R1 G J ð4:1Þ 1 − : 4:5 þ 2 ¯ ¯ þ 2 2 ¯ ¯ 2 ð Þ R1 sRffiffiffiffiffiffi2 c R1Rac if the Jang solution f is prescribed to be zero (or more generally constant) on St2 . Note that this boundary condition The geometric inequality (4.5) relates the size of the differs from (3.1) which is used to obtain (3.2). This is due to body Ω2 ⊃ Ω1 to its core nonelectromagnetic matter con- the fact that the boundary integrals that arise from the tent, total charge, and total angular momentum. It may be divergence term in R¯ have different signs on the inner and interpreted as stating that a material body of fixed size can outer boundaries [31]. In fact, the boundary terms at the only contain a certain fixed amount of matter energy, outer boundary have an advantageous sign, and it is likely charge, and angular momentum. The primary hypotheses that this Dirichlet boundary condition used for (4.1) is not which were used to derive this inequality consist of the needed. assumption that the outer region is untrapped, the annular Proceeding as in Sec. II, lower bounds for the right-hand region Ω2nΩ1 has no charge and momentum density in the side of (4.1) may be extracted in terms of the total charge Killing direction, and most importantly that the initial data and angular momentum of Ω1. In addition, a contribution are void of apparent horizons. This latter assumption is from the nonelectromagnetic matter fields will also occur. used to obtain regular solutions of the Jang equation, and To see this, observe that as in (2.9) following [33] we may turn this around to obtain a black

124026-6 BEKENSTEIN BOUNDS, PENROSE INEQUALITIES, AND … PHYS. REV. D 97, 124026 (2018) hole existence result. More precisely, if a body with the the tensor associated with the solution f of Jang’s equation hypotheses above, minus any assumption on apparent and is given in (2.6). Geometrically the tensor h represents horizons, satisfies the extrinsic curvature of the graph of f in a static spacetime constructed from the metric g¯ and function u [12]. The 4 4 c ¯ π ¯ 3 R2 < R minðμ − jJ jÞ vanishing of this particular component of h is used 2 3 1 ˜ EM EM G Ω1 throughout the main body of the paper in order to allow 2 ¯ 2 for angular momentum contributions to the various inequal- Q R1 G J þ 1 − þ ; ð4:6Þ ities. Here we will confirm this property of h. 2R¯ R¯ 2c2 R¯ R¯ 2 1 sffiffiffiffiffiffi2 1 ac In [35], it was shown that if M is asymptotically flat and simply connected then a global cylindrical coordinate then an apparent horizon must be present within the initial system exists, denoted by ðρ;z;ϕÞ and referred to as Brill data. The reasoning is that if there were no apparent coordinates, such that the metric takes the following form horizons, then we may apply the arguments above to conclude that (4.5) holds, a contradiction. This relies on g ¼ e−2Uþ2αðdρ2 þ dz2Þþρ2e−2Uðdϕ þ Adρ þ BdzÞ2 the analysis of the Jang/IMCF system of equations, which has been established rigorously in the case of spherical ðA1Þ symmetry [11]. This conclusion concerning the existence of an apparent horizon implies that the spacetime arising from for some functions U, α, A, and B all depending only on the initial data contains a singularity, or more accurately is ðρ;zÞ. The Killing field is given by η ¼ ∂ϕ, and if U ¼ α ¼ null geodesically incomplete by the Hawking-Penrose A ¼ B ¼ 0 then g reduces to the typical expression of the flat singularity theorems [34], and assuming cosmic censorship metric on Euclidean 3-space in cylindrical coordinates. For it must, therefore, possess a black hole. This result may be simplicity it will be assumed that A ¼ B ¼ 0 so that η is interpreted as stating that if a body of fixed size contains perpendicular to the orbit space or ρz-half plane. Observe sufficient amounts of nonelectromagnetic matter energy, that since u and f are axisymmetric, that is ∂ϕf ¼ ∂ϕu ¼ 0, charge, or angular momentum, then it must collapse to form it follows that a black hole. ρ z u∇ f uðΓ ∂ρf þ Γ ∂zfÞ ACKNOWLEDGMENTS hðη; νÞ¼ ϕν ¼ − ϕν ϕν ; ðA2Þ 1 þ u2j∇fj2 1 þ u2j∇fj2 M. A. K. acknowledges the support of NSF Grant No. DMS-1708798. pΓlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the ij are Christoffel symbols. Since the surface is axisymmetric, ∂ is tangent to S, and thus g η; ν 0.A APPENDIX: VANISHING EXTRINSIC ϕ ð Þ¼ CURVATURE straightforward calculation then yields Consider an axisymmetric closed surface S within an 1 1 Γρ gρi∂ g 0; Γz gzi∂ g 0; A3 axisymmetric Riemannian 3-manifold ðM; gÞ.Ifη is the ϕν ¼ 2 ν ϕi ¼ ϕν ¼ 2 ν ϕi ¼ ð Þ generator for the axisymmetry and ν is the unit normal to S, then under mild hypotheses hðη; νÞ¼0 along S, where h is and the desired conclusion follows.

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