Geometric inequalities for black holes

Transcription

1 Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 26 July, 2013

2 Geometric inequalities Geometric inequalities have an ancient history in Mathematics. A classical example is the isoperimetric inequality for closed plane curves given by L 2 4πA (= circle) where A is the area enclosed by a curve C of length L, and where equality holds if and only if C is a circle. L A L A L 2 > 4πA L 2 = 4πA

3 The inequality L 2 4πA applies to complicated geometric objects (i.e. arbitrary closed planar curves). The equality L 2 = 4πA is achieved only for an object of optimal shape (i.e. the circle). This object has a variational characterization: the circle is uniquely characterized by the property that among all simple closed plane curves of given length L, the circle of circumference L encloses the maximum area.

4 Geometrical inequalities in General Relativity General Relativity is a geometric theory, hence it is not surprising that geometric inequalities appear naturally in it. Many of these inequalities are similar in spirit as the isoperimetric inequality. However, General Relativity as a physical theory provides an important extra ingredient. It is often the case that the quantities involved have a clear physical interpretation and the expected behavior of the gravitational and matter fields often suggests geometric inequalities which can be highly non-trivial from the mathematical point of view. The interplay between physics and geometry gives to geometric inequalities in General Relativity their distinguished character.

5 Plan of the talk Part I: Physical picture. Part II: Theorems. Part III: Open problems and recent results on bodies.

6 Part I Physical picture

7 Positive mass theorem Let m be the total ADM mass on an asymptotically flat complete initial data such that the dominant energy condition is satisfied, then we have: 0 m (= Minkowski). The mass of the spacetime measures the total amount of energy and hence it should be positive from the physical point of view. The mass m in General Relativity is represented by a geometrical quantity on a Riemannian manifold. From the geometrical mass definition, without the physical picture, it would be very hard to conjecture that this quantity should be positive. In fact the proof turn out to be very subtle (Schoen-Yau 79, Witten 81).

8 The positive mass theorem is proved on initial conditions: Lorentzian proofs (particular cases): Penrose-Sorkin-Woolgar 99, Chrusciel-Galloway 04. There is up to now no general Lorentzian proof of the positive mass theorem.

9 A key assumption in the positive mass theorem is that the matter fields should satisfy an energy condition. This condition is expected to hold for all physically realistic matter. This kind of general properties which do not depend very much on the details of the model are not easy to find for a macroscopic object. And hence it is difficult to obtain simple and general geometric inequalities among the parameters that characterize ordinary macroscopic objects. Black holes represent a unique class of very simple macroscopic objects. They are natural candidates for geometrical inequalities.

10 Stationary black holes The black hole uniqueness theorem ensures that stationary black holes in vacuum (for simplicity we will not consider the electromagnetic field) are characterized by the Kerr exact solution of Einstein equations (important aspects of black hole uniqueness remain open see Chruściel Lopes Costa Heusler, Living Review 12). The Kerr metric depends on two parameters: the mass m and the angular momentum J. The Kerr metric is well defined for any choice of the parameters. However, it represents a black hole if and only if the following inequality holds J m.

11 Kerr solution m Extreme Black Hole Extreme Naked Schwarzschild Naked Singularity Singularity J

12 Kerr black hole: geometrical inequalities The area of the horizon is given by the important formula A = 8π (m 2 + ) m 4 J 2. This formula implies the following three geometric inequalities between the three relevant parameters (A, m, J): A 16π m (= Schwarzschild) J m (= Extreme Kerr) 8π J A (= Extreme Kerr)

13 Physical interpretation A 16π m A The difference m 16π is the rotational energy of the Kerr black hole. This is the maximum amount of energy that can be extracted by the Penrose process (Christodoulou). J m From Newtonian considerations, we can interpret this inequality as follows (Wald): in a collapse the gravitational attraction ( m 2 /r 2 ) at the horizon (r m) dominates over the centrifugal repulsive forces ( J 2 /mr 3 ). 8π J A The black hole temperature κ = 1 ) (1 (8πJ)2 4m A 2, is positive κ 0 and it is zero if and only if the black hole is extreme.

14 Geometric inequalities for dynamical black holes Black holes are not stationary in general: Astrophysical phenomena like the formation of a black hole by gravitational collapse or a binary black hole collision are highly dynamical. Dynamical black holes can not be characterized by few parameters as in the stationary case. Remarkably, these inequalities extend to the fully dynamical regime. The inequalities are deeply connected with global properties of the gravitational collapse: cosmic censorship conjecture.

15 The inequality 8π J A for dynamical black holes Consider a dynamical black hole. Physical quantities that are well defined for this spacetime are: The total ADM mass m of the spacetime: the sum of the black hole mass and the mass of the gravitational waves surrounding it. In the stationary case, the mass of the black hole is equal to the total mass of the spacetime. The mass m is a global quantity: it carries information on the whole spacetime. The area A of the horizon. The area A is a quasi-local quantity: it carries information on a bounded region of the spacetime. What are the quasi-local mass and quasi-local angular momentum of a dynamical black hole? In general, it is difficult to find physically relevant quasi-local quantities like mass and angular momentum (see Szabados, Living Review, 09).

16 Axial symmetry However, in axial symmetry, there is a well defined notion of quasi-local angular momentum: the Komar integral of the Killing axial Killing vector. Moreover, the angular momentum is conserved in vacuum. That is, axially symmetric gravitational waves do not carry angular momentum. For axially symmetric dynamical black holes we have two well defined quasi-local quantities: the area of the horizon A and the angular momentum J. Using A and J we can define the quasi-local mass of the black hole by the Kerr formula, that is: A m bh = 16π + 4πJ2 A. This quasi-local mass is well defined for axially symmetric dynamical black holes.

17 Evolution of the quasi-local mass m bh By the area theorem, we know that the horizon area A increase with time. In axial symmetry there is not transfer of angular momentum by gravitational waves. Then, the quasi-local mass of the black hole should increase with the area, since there is no mechanism at the classical level to extract energy from the black hole (no Penrose process in axial symmetry). Then, both the area A and the quasi-local mass m bh should monotonically increase with time.

18 Variations of m bh (first law of black hole thermodynamics): δm bh = κ 8π δa + Ω HδJ, where κ = 1 ) (1 (8πJ)2 4m bh A 2, Ω H = 4πJ. A m bh In axial symmetry δj = 0 and hence we obtain δm bh = κ 8π δa. By the area theorem we have δa 0. Then δm bh 0 if and only if κ 0, namely A 8π J. It is natural to conjecture that this inequality should be satisfied for any axially symmetric black hole. It implies the existence of a non trivial monotonic quantity for vacuum axially symmetric spacetimes (in addition to the black hole area) ṁ bh 0.

19 Penrose inequality in axial symmetry Penrose proposed a remarkable physical argument that connects global properties of the gravitational collapse with geometric inequalities on the initial conditions. This argument implies the Penrose inequality dynamical black holes. A 16π m for In the following we present this argument in axial symmetry, where angular momentum is conserved. We include a relevant new ingredient: the inequality 8π J A.

20 Assume that the following statements hold in a gravitational collapse: (i) Gravitational collapse results in a black hole (weak cosmic censorship) (ii) The spacetime settles down to a stationary final state. We will further assume that at some finite time all the matter have fallen into the black hole and hence the exterior region is vacuum (we ignore electromagnetic field for simplicity). Conjectures (i) and (ii) constitute the standard picture of the gravitational collapse. Let us denote by m 0, A 0 and J 0 the mass, area and angular momentum of the remainder Kerr black hole.

21 Take a Cauchy surface in the spacetime such that the collapse has already occurred. Gravitational waves carry positive energy m m 0. A is the area of the intersection of the event horizon with the Cauchy surface. By the black hole area theorem we have A 0 A. Since we have assumed axial symmetry, angular momemtum is conserved J 0 = J Since we have assumed that 8π J A then m bh also increase with time, and hence we get m bh m 0 m

22 Singularity (J 0, m 0, A 0 ) Radiation Horizon A A A 0 m 0 m J 0 = J (J, m)

23 That is, we arrive to the following generalization of Penrose inequality with angular momentum: A 16π + 4πJ2 A m. The inequality above implies J m. In fact, this inequality can be deduced directly by the same argument without using the area theorem, it depends only on the following assumptions Gravitational waves carry positive energy. Angular momentum is conserved in axial symmetry. In a gravitational collapse the spacetime settles down to a final Kerr black hole.

24 Summary For axially symmetric, dynamical black holes, the following geometrical inequalities are expected: Quasi-local 8π J A (= Extreme Kerr horizon) Global A 16π + 4πJ2 A m (= Kerr black hole)

25 The global inequality implies: Penrose inequality (valid also without axial symmetry) A 16π m (= Schwarzschild). Inequality between angular momentum and mass: J m. (= extreme Kerr black hole). It is important that in this inequality the area of the horizon do not appears. The three geometrical inequalities valid for the Kerr black holes are expected to hold also for axially symmetric, dynamical black holes.

26 Comments on the physical arguments All the quantities involved in the geometrical inequalities (A, m and J) can be calculated on the initial surface. A counter example of the global inequality will imply that cosmic censorship is not true. Conversely a proof of it gives indirect evidence of the validity of censorship, since it is very hard to understand why this highly nontrivial inequality should hold unless censorship can be thought of as providing the underlying physical reason behind it. There are two group of geometrical inequalities: 1. m: the area appears as lower bound. A 16π 2. J m and 8π J A: the angular momentum appears as lower bound. The mathematical methods used to study these two groups are, up to now, very different. This talk is mainly concerned with the second group.

27 Part II Theorems

28 Penrose inequality The Penrose inequality A 16π m (= Schwarzschild) is expected to be valid also without axial symmetry. It has been proved in the Riemannian case (i.e. initial data with zero extrinsic curvature) by Huisken-Ilmanen 01 and Bray 01. For a recent review on this subject see Mars, CQG, Topical Review, 09. There is up to now no result including angular momentum in the Penrose inequality.

29 Inequality between mass and angular momentum Theorem Consider an axially symmetric, vacuum, asymptotically flat and maximal initial data set with two asymptotics ends. Let m and J denote the total mass and angular momentum at one of the ends. Then, the following inequality holds J m (= Extreme Kerr).

30 Dain 06, 08: formulation of the variational problem, first proof of the global inequality. Chrusciel 08, Chrusciel-Li-Weinstein 08: generalization and simplification of the proof. Important partial results for multiple black holes. Chrusciel-Lopes Costa 09, Lopes Costa 10: electric charge. Schoen-Zhou 12: relevant improvements on the rigidity, first rigidity result with charge, measure of the distance to the extreme Kerr initial data Zhou 12: small trace non-maximal data. Gabach 12: results on the force between black holes.

31 The theorem is proved on initial conditions: However, m and J are properties of the whole spacetime.

32 Kerr black hole initial data Non extreme Extreme Singularity Singularity Cauchy horizon Event horizon i 0 S Event horizon i 0 i c S i 0

33 Rigidity Two asymptotic flat ends (non-extreme Kerr, Schwarzschild, Bowen-York, etc.) Extreme Kerr initial data

34 Area-angular momentum quasi-local inequality Theorem Given an axisymmetric closed marginally trapped and stable surface S, in a spacetime with non-negative cosmological constant and fulfilling the dominant energy condition, it holds the inequality 8π J A (= Extreme Kerr), where A and J are the area and angular momentum of S. This theorem does not assume vacuum and does not assume the existence of a maximal slice.

35 Ansorg Pfister 08, Hennig Ansorg Cederbaum 08, Hennig Cederbaum Ansorg 11: stationary case with surrounding matter. Dain 10: heuristic arguments for the full dynamical regime, formulation of the variational problem and proof locally near extreme. Aceña-Dain-Gabach 11: proof for particular cases. Dain-Reiris 11: first general proof for minimal surfaces, introduction of the stability condition. Jaramillo-Reiris-Dain 11: stable trapped surfaces, non-vacuum, non-maximal data. Gabach-Jaramillo 11, Gabach-Jaramillo-Reiris 12: electric charge. Holland 11, Paetz Simon 13: higher dimensions. Gabach-Reiris 13: shape. Yazadjiev 12, 13: Einstein-Maxwell dilaton gravity.

36 Null vectors l a and k a spanning the normal plane to S and normalized as l a k a = 1. Expansion: θ (l) = q ab a l b. Marginal trapped surface: θ (l) = 0. Stable (Andersson-Mars-Simon 05): given a closed marginally trapped surface S we will refer to it as spacetime stably outermost if there exists an outgoing ( k a -oriented) vector X a = γl a ψk a, with γ 0 and ψ > 0, such that the variation of θ (l) with respect to X a fulfills the condition δ X θ (l) 0.

37 Riemannian proof: minimal surfaces on maximal slices. Lorentzian proof: trapped surfaces.

38 Extreme Kerr throat geometry Singularity Event horizon S S i 0 Intrinsic metric of S: ( J (1 + cos 2 θ)dθ ) sin2 θ (1 + cos 2 θ) dφ2

39 Rigidity Generic trapped surface Extreme Kerr throat

40 Part III Open problems and recent results on bodies

41 Open problems J m Remove the maximal condition (small trace Zhou 12). Prove it for initial data for multiple black holes. This problem is related with the uniqueness of the Kerr black hole with degenerate and disconnected horizons. It is probably a hard problem. (Chrusciel-Li-Weinstein 08 partial results, Dain-Ortiz 09 numerical evidences) 8 J A Does some version of this inequality holds without axial symmetry? This requires a notion of quasi-local angular momentum for black holes without symmetries. A 16π + 4πJ2 A m Include the angular momentum in the Penrose inequality.

42 Bodies Rotating body Ω. J(Ω): angular momentum of the body. R(Ω): a measure of the size of the body. We conjecture that there exists an universal inequality for all bodies of the form R 2 (Ω) G c 3 J(Ω), where G is the gravitational constant and c the speed of light. The symbol is intended as an order of magnitude.

43 Physical arguments The arguments in support of this inequality are based in the following three physical principles: (i) The speed of light c is the maximum speed. (ii) For bodies which are not contained in a black hole the following inequality holds (hoop conjecture) R(Ω) G c 2 m(ω), where m(ω) is the mass of the body and R(Ω) is some measure of size of Ω, which can in principle be different from R(Ω). (iii) The conjectured inequality for bodies holds for black holes. The inequality 8π J G c 3 A can be interpreted as a version of this inequality for black holes.

44 Constant density bodies Theorem Let (S, h ij, K ij, µ, j i ) be an initial data set that satisfy the energy condition. We assume that the data are maximal and axially symmetric. Let Ω be an open set in S. Assume that the energy density µ is constant on Ω. Then the following inequality holds R 2 (Ω) 24 π 3 G c 3 J(Ω). This theorem is a direct consequence of the Schoen-Yau 83 bound for the minimum of the scalar curvature. The radius R is defined in terms of the Schoen-Yau, Ó Murchadha radius.

45 Schoen-Yau radius Ω R SY Let Γ be a simple closed curve in Ω which bounds a disk in Ω. Let p be largest constant such that the set of points within a distance p of Γ is contained within Ω and forms a proper torus. Then p is a measure of the size of Ω with respect to the curve Γ. The radius R SY (Ω) is defined as the largest value of p we can find by considering all curves Γ. That is, R SY (Ω) is expressed in terms of the largest torus that can be embedded in Ω.

46 Consider a region Ω with a axial Killing vector with square norm η. The radius R is defined by R(Ω) = 2 ( ) 1/2 Ω η dv. π R SY (Ω) The most natural normalization for R is to require that R = b for an sphere in flat space of radius b. This is the reason for the factor 2/π.

47 For more details see the review article: Geometric inequalities for axially symmetric black holes S. Dain Class.Quant.Grav. 29 (2012) Topical Review and the forthcoming: Geometric inequalities bounding angular momentum and charges in General Relativity C. Cederbaum, S. Dain, M. E. Gabach Clément. Living Review in Relativity