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Bekenstein Bounds, Penrose Inequalities, and Black Hole Formation

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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 2470-0010=2018=97(12)=124026(8) 124026-1 © 2018 American Physical Society JAROSLAW S. JARACZ and MARCUS A. KHURI PHYS. REV. D 97, 124026 (2018) 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.
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