Geometric inequalities for black holes

Geometric inequalities for black holes

Sergio Dain

FaMAF-Universidad Nacional de C´ordoba, CONICET, Argentina.

26 July, 2013

Geometric inequalities

Geometric inequalities have an ancient history in Mathematics. A classical example is theisoperimetric inequality for closed plane curves given by

L² ≥4πA (= circle)

whereAis the areaenclosed by a curve C of lengthL, and where equalityholds if and only ifC is a circle.

A L

L²>4πA

A L

L² = 4πA

I The inequalityL² ≥4πAapplies to complicated geometric objects (i.e. arbitrary closed planar curves).

I The equality L²= 4πAis 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 circumferenceL encloses the maximum area.

Geometrical inequalities in General Relativity

I General Relativity is ageometric 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.

I 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.

I The interplay between physics and geometrygives to geometric inequalities in General Relativity their distinguished character.

Plan of the talk

I Part I: Physical picture.

I Part II: Theorems.

I Part III: Open problems and recent results on bodies.

Part I

Physical picture

Positive mass theorem

Letm 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).

I The mass of the spacetime measures the total amount of energy and hence it should be positive from the physical point of view.

I The massm 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).

I The positive mass theorem is proved on initial conditions:

I Lorentzian proofs (particular cases):

Penrose-Sorkin-Woolgar 99,Chrusciel-Galloway 04.

I There is up to now no general Lorentzian proof of the positive mass theorem.

I 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.

I 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.

I Black holes represent a unique class of very simple macroscopic objects. They are natural candidates for geometrical inequalities.

Stationary black holes

I 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 Chru´sciel – Lopes Costa – Heusler, Living Review 12).

I The Kerr metric depends on two parameters: themassm and theangular momentum J.

I 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

p|J| ≤m.

Kerr solution

m

J Extreme

Extrem e

Schwarzschild

Naked Singularity

Naked Singularity Black Hole

Kerr black hole: geometrical inequalities

The area of the horizon is given by the important formula A= 8π

m²+p

m⁴−J²

.

This formula implies the following three geometric inequalities between the three relevant parameters (A,m,J):

r A

16π ≤m (= Schwarzschild)

p|J| ≤m (= Extreme Kerr) 8π|J| ≤A (= Extreme Kerr)

Physical interpretation

I

q A 16π ≤m

The difference m−q

A

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).

I p

|J| ≤m

From Newtonian considerations, we can interpret this inequality as follows (Wald): in a collapse the gravitational attraction (≈m²/r²) at the horizon (r ≈m) dominates over the centrifugal repulsive forces (≈J²/mr³).

I 8π|J| ≤A

The black hole temperature κ= 1

4m

1−(8πJ)² A²

,

is positiveκ≥0 and it is zero if and only if the black hole is extreme.

Geometric inequalities for dynamical black holes

I Black holes are not stationary in general:

I Astrophysical phenomena like the formation of a black hole by gravitational collapse or a binary black hole collision are highly dynamical.

I Dynamical black holes can not be characterized by few parameters as in the stationary case.

I Remarkably, these inequalities extend to the fully dynamical regime.

I The inequalities are deeply connected with global properties of the gravitational collapse: cosmic censorship conjecture.

The inequality 8π | J | ≤ A for dynamical black holes

Consider a dynamical black hole. Physical quantities that are well defined for this spacetime are:

I 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 massm is a global quantity: it carries information on the whole spacetime.

I The area A of the horizon. The areaA 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 (seeSzabados, Living Review, 09).

Axial symmetry

I 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.

I 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.

I Using AandJ we can define the quasi-local mass of the black hole by the Kerr formula, that is:

mbh= r A

16π + 4πJ² A .

This quasi-local mass is well defined for axially symmetric dynamical black holes.

Evolution of the quasi-local mass m

I By the area theorem, we know that the horizon areaA increase with time.

I 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).

I Then, both the area A and the quasi-local mass m_bh should monotonically increase with time.

Variations ofm_bh (first law of black hole thermodynamics):

δm_bh = κ

8πδA+ Ω_HδJ, where

κ= 1 4m_bh

1−(8πJ)² A²

, Ω_H = 4πJ A m_bh. In axial symmetryδJ = 0 and hence we obtain

δm_bh = κ 8πδA.

By the area theorem we have

δA≥0.

Thenδmbh ≥0 if and only ifκ≥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)m˙bh ≥0.

Penrose inequality in axial symmetry

I Penrose proposed a remarkable physical argument that connects global properties of the gravitational collapse with geometric inequalities on the initial conditions.

I This argument implies the Penrose inequality q A

16π ≤m for dynamical black holes.

I In the following we present this argument in axial symmetry, whereangular momentum is conserved. We include a relevant new ingredient: the inequality 8π|J| ≤A.

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 bym₀,A₀ andJ₀ the mass, area and angular momentum of the remainder Kerr black hole.

I Take a Cauchy surface in the spacetime such that the collapse has already occurred.

I Gravitational waves carry positive energy m≥m₀.

I A is the area of the intersection of the event horizon with the Cauchy surface. By the black hole area theoremwe have

A₀≥A.

I Since we have assumed axial symmetry, angular momemtum is conserved

J0 =J

I Since we have assumed that 8π|J| ≤A thenmbh also increase with time, and hence we get

mbh ≤m0 ≤m

Radiation Singularity

Horizon A

(J0, m0, A0)

(J, m) A≤A0

m0≤m J0=J

That is, we arrive to the following generalization of Penrose inequality with angular momentum:

r A

16π +4πJ² A ≤m.

The inequality above implies

p|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

I Gravitational waves carry positive energy.

I Angular momentum is conserved in axial symmetry.

I In a gravitational collapse the spacetime settles down to a final Kerr black hole.

Summary

Foraxially symmetric, dynamical black holes, the following geometrical inequalities are expected:

Quasi-local

8π|J| ≤A (= Extreme Kerr horizon) Global

r A

16π +4πJ²

A ≤m (= Kerr black hole)

The global inequality implies:

I Penrose inequality (valid also without axial symmetry) r A

16π ≤m (= Schwarzschild).

I Inequality between angular momentum and mass:

p|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.

Comments on the physical arguments

I All the quantities involved in the geometrical inequalities (A, m andJ) can be calculated on the initial surface.

I A counter example of the global inequality will imply that cosmic censorship is not true. Conversely a proof of itgives 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.

I There are two group of geometrical inequalities:

1.

q A

16π ≤m: thearea appears as lower bound.

2. p

|J| ≤m and8π|J| ≤A: theangular 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.

Part II

Theorems

Penrose inequality

The Penrose inequality r 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 01andBray 01.

For a recent review on this subject seeMars, CQG, Topical Review, 09.

There is up to now no result including angular momentum in the Penrose inequality.

Inequality between mass and angular momentum

Theorem

Consider an axially symmetric,vacuum, asymptotically flat and maximalinitial 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

p|J| ≤m (=Extreme Kerr).

I Dain 06, 08: formulation of the variational problem, first proof of the global inequality.

I Chrusciel 08, Chrusciel-Li-Weinstein 08: generalization and simplification of the proof. Important partial results for multiple black holes.

I Chrusciel-Lopes Costa 09, Lopes Costa 10: electric charge.

I Schoen-Zhou 12: relevant improvements on the rigidity, first rigidity result with charge, measure of the distance to the extreme Kerr initial data

I Zhou 12: small trace non-maximal data.

I Gabach 12: results on the force between black holes.

I The theorem is proved on initial conditions:

I However, m andJ are properties of the whole spacetime.

Kerr black hole initial data

Non extreme Extreme

i0

i0 S

Cauchy horizon Event horizon

Singularity

i0

S

Event horizon ic

Singularity

Rigidity

Two asymptotic flat ends (non-extreme Kerr,

Schwarzschild, Bowen-York, etc.)

Extreme Kerr initial data

Area-angular momentum quasi-local inequality

Theorem

Given an axisymmetric closed marginally trappedand stable surfaceS, 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 ofS.

This theorem does not assume vacuumanddoes not assume the existence of a maximal slice.

I Ansorg–Pfister 08,Hennig–Ansorg–Cederbaum 08, Hennig– Cederbaum–Ansorg 11: stationary case with surrounding matter.

I Dain 10: heuristic arguments for the full dynamical regime, formulation of the variational problem and proof locally near extreme.

I Ace˜na-Dain-Gabach 11: proof for particular cases.

I Dain-Reiris 11: first general proof for minimal surfaces, introduction of the stability condition.

I Jaramillo-Reiris-Dain 11: stable trapped surfaces, non-vacuum, non-maximal data.

I Gabach-Jaramillo 11, Gabach-Jaramillo-Reiris 12: electric charge.

I Holland 11, Paetz–Simon 13: higher dimensions.

I Gabach-Reiris 13: shape.

I Yazadjiev 12, 13: Einstein-Maxwell dilaton gravity.

I Null vectors `<sup>a</sup> andk<sup>a</sup> spanning the normal plane toS and normalized as `^ak_a =−1.

I Expansion: θ<sup>(`)</sup>=q<sup>ab</sup>∇a`b.

I Marginal trapped surface: θ^(`)= 0.

I 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 =γ`<sup>a</sup>−ψk<sup>a</sup>, with γ≥0 and ψ >0, such that the variation of θ<sup>(`)</sup> with respect toX^a fulfills the condition

δ_Xθ^(`)≥0.

I Riemannian proof: minimal surfaces on maximal slices.

I Lorentzian proof: trapped surfaces.

Extreme Kerr throat geometry

i0

S

Event horizon S

Singularity

Intrinsic metric ofS:

|J|

(1 + cos²θ)dθ²+ 4 sin²θ (1 + cos²θ)dφ²

Rigidity

Generic trapped surface Extreme Kerr throat

Part III

Open problems and recent results on bodies

Open problems

p|J| ≤m

I Remove the maximal condition (small traceZhou 12).

I 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 08partial results, Dain-Ortiz 09numerical evidences)

8|J| ≤A

I Does some version of this inequality holds without axial symmetry? This requires a notion of quasi-local angular momentum for black holes without symmetries.

q A

16π +^4πJ_A² ≤m

I Include the angular momentum in the Penrose inequality.

Bodies

I Rotating body Ω.

I J(Ω): angular momentum of the body.

I R(Ω): a measure of the size of the body.

We conjecture that there exists an universal inequality for all bodies of the form

R²(Ω)? G

c³|J(Ω)|,

whereG is the gravitational constant andc the speed of light.

The symbol?is intended as an order of magnitude.

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)

Re(Ω)? G c²m(Ω),

wherem(Ω) is the mass of the body andRe(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 c³ ≤A

can be interpreted as a version of this inequality for black holes.

Constant density bodies

Theorem

Let(S,h_ij,K_ij, µ,jⁱ) be an initial data set that satisfy the energy condition. We assume that the data are maximal and axially symmetric. Let Ωbe an open set in S . Assume that the energy densityµ is constantonΩ. Then the following inequality holds

R²(Ω)≥ 24 π³

G

c³|J(Ω)|.

I This theorem is a direct consequence of the Schoen-Yau 83 bound for the minimum of the scalar curvature.

I The radius Ris defined in terms of the Schoen-Yau, ´O Murchadha radius.

Schoen-Yau radius

RSY

Ω

I Let Γ be a simple closed curve in Ω which bounds a disk in Ω.

I 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.

I Then p is a measure of the size of Ω with respect to the curve Γ.

I The radius RSY(Ω) is defined as the largest value of p we can find by considering all curves Γ.

That is,RSY(Ω) is expressed in terms of the largest torus that can be embedded in Ω.

Consider a region Ω with aaxial Killing vector with square normη.

The radiusRis defined by R(Ω) = 2

π R

Ω

√ηdv1/2

RSY(Ω) .

The most natural normalization forR is to require thatR=b for an sphere in flat space of radiusb. This is the reason for the factor 2/π.

For more details see the review article:

Geometric inequalities for axially symmetric black holes S. Dain

Class.Quant.Grav. 29 (2012) 073001 “Topical Review”

http://arxiv.org/abs/1111.3615 and the forthcoming:

Geometric inequalities bounding angular momentum and charges in General Relativity

C. Cederbaum, S. Dain, M. E. Gabach Cl´ement.

Living Review in Relativity