Geometric inequalities for black holes - IME-USPgelosp2013/files/dain.pdf · Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de C ordoba, CONICET, Argentina.

Geometric inequalities for black holes

Sergio Dain

FaMAF-Universidad Nacional de Cordoba, CONICET, Argentina.

26 July, 2013

Geometric inequalities

Geometric inequalities have an ancient history in Mathematics. Aclassical example is the isoperimetric inequality for closed planecurves given by

L2 ≥ 4πA (= circle)

where A is the area enclosed by a curve C of length L, and whereequality holds if and only if C is a circle.

AL

L2 > 4πA

AL

L2 = 4πA

I The inequality L2 ≥ 4πA applies to complicated geometricobjects (i.e. arbitrary closed planar curves).

I The equality L2 = 4πA is achieved only for an object of“optimal shape” (i.e. the circle). This object has a variationalcharacterization: the circle is uniquely characterized by theproperty that among all simple closed plane curves of givenlength L, the circle of circumference L encloses the maximumarea.

Geometrical inequalities in General Relativity

I General Relativity is a geometric theory, hence it is notsurprising that geometric inequalities appear naturally in it.Many of these inequalities are similar in spirit as theisoperimetric inequality.

I However, General Relativity as a physical theory provides animportant extra ingredient. It is often the case that thequantities involved have a clear physical interpretation and theexpected behavior of the gravitational and matter fields oftensuggests geometric inequalities which can be highly non-trivialfrom the mathematical point of view.

I The interplay between physics and geometry gives togeometric inequalities in General Relativity their distinguishedcharacter.

Plan of the talk

I Part I: Physical picture.

I Part II: Theorems.

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

Part IPhysical picture

Positive mass theorem

Let m be the total ADM mass on an asymptotically flat completeinitial 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 ofenergy and hence it should be positive from the physical pointof view.

I The mass m in General Relativity is represented by ageometrical 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 bepositive. In fact the proof turn out to be very subtle (Schoen-Yau79, 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 positivemass theorem.

I A key assumption in the positive mass theorem is that thematter fields should satisfy an energy condition. Thiscondition is expected to hold for all physically realistic matter.

I This kind of general properties which do not depend verymuch on the details of the model are not easy to find for amacroscopic object. And hence it is difficult to obtainsimple and general geometric inequalities among theparameters that characterize ordinary macroscopic objects.

I Black holes represent a unique class of very simplemacroscopic objects. They are natural candidates forgeometrical inequalities.

Stationary black holes

I The black hole uniqueness theorem ensures that stationaryblack holes in vacuum (for simplicity we will not consider theelectromagnetic field) are characterized by the Kerr exactsolution of Einstein equations (important aspects of black holeuniqueness remain open see Chrusciel – Lopes Costa –Heusler, Living Review 12).

I The Kerr metric depends on two parameters: the mass m andthe angular momentum J.

I The Kerr metric is well defined for any choice of theparameters. However, it represents a black hole if and only ifthe following inequality holds√

|J| ≤ m.

Kerr solutionm

JExtr

emeExtrem

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 +√

m4 − J2).

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

A

16π≤ m (= Schwarzschild)√

|J| ≤ m (= Extreme Kerr)

8π|J| ≤ A (= Extreme Kerr)

Physical interpretation

I

√A

16π ≤ m

The difference m −√

A16π is the rotational energy of the Kerr

black hole. This is the maximum amount of energy that canbe extracted by the Penrose process (Christodoulou).

I√|J| ≤ m

From Newtonian considerations, we can interpret thisinequality as follows (Wald): in a collapse the gravitationalattraction (≈ m2/r 2) at the horizon (r ≈ m) dominates overthe centrifugal repulsive forces (≈ J2/mr 3).

I 8π|J| ≤ AThe black hole temperature

κ =1

4m

(1− (8πJ)2

A2

),

is positive κ ≥ 0 and it is zero if and only if the black hole isextreme.

Geometric inequalities for dynamical black holes

I Black holes are not stationary in general:

I Astrophysical phenomena like the formation of a black hole bygravitational collapse or a binary black hole collision are highlydynamical.

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

I Remarkably, these inequalities extend to the fully dynamicalregime.

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

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

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

I The total ADM mass m of the spacetime: the sum of theblack hole mass and the mass of the gravitational wavessurrounding it. In the stationary case, the mass of the blackhole is equal to the total mass of the spacetime. The mass mis a global quantity: it carries information on the wholespacetime.

I The area A of the horizon. The area A is a quasi-localquantity: it carries information on a bounded region of thespacetime.

What are the quasi-local mass and quasi-local angularmomentum of a dynamical black hole? In general, it is difficultto find physically relevant quasi-local quantities like mass andangular momentum (see Szabados, Living Review, 09).

Axial symmetry

I However, in axial symmetry, there is a well defined notion ofquasi-local angular momentum: the Komar integral of theKilling axial Killing vector. Moreover, the angular momentumis conserved in vacuum. That is, axially symmetricgravitational waves do not carry angular momentum.

I For axially symmetric dynamical black holes we have twowell defined quasi-local quantities: the area of the horizon Aand the angular momentum J.

I Using A and J we can define the quasi-local mass of the blackhole by the Kerr formula, that is:

mbh =

√A

16π+

4πJ2

A.

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

Evolution of the quasi-local mass mbh

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

I In axial symmetry there is not transfer of angularmomentum by gravitational waves. Then, the quasi-localmass of the black hole should increase with the area, sincethere is no mechanism at the classical level to extract energyfrom the black hole (no Penrose process in axialsymmetry).

I Then, both the area A and the quasi-local mass mbh

should monotonically increase with time.

Variations of mbh (first law of black hole thermodynamics):

δmbh =κ

8πδA + ΩHδJ,

where

κ =1

4mbh

(1− (8πJ)2

A2

), ΩH =

4πJ

A mbh.

In axial symmetry δJ = 0 and hence we obtain

δmbh =κ

8πδA.

By the area theorem we have

δA ≥ 0.

Then δmbh ≥ 0 if and only if κ ≥ 0, namely

A ≥ 8π|J|.It is natural to conjecture that this inequality should besatisfied for any axially symmetric black hole.It implies the existence of a non trivial monotonic quantity forvacuum axially symmetric spacetimes (in addition to the black holearea) mbh ≥ 0.

Penrose inequality in axial symmetry

I Penrose proposed a remarkable physical argument thatconnects global properties of the gravitational collapsewith geometric inequalities on the initial conditions.

I This argument implies the Penrose inequality√

A16π ≤ m for

dynamical black holes.

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

Assume that the following statements hold in a gravitationalcollapse:

(i) Gravitational collapse results in a black hole (weak cosmiccensorship)

(ii) The spacetime settles down to a stationary final state. Wewill further assume that at some finite time all the matterhave fallen into the black hole and hence the exterior region isvacuum (we ignore electromagnetic field for simplicity).

Conjectures (i) and (ii) constitute the standard picture of thegravitational collapse. Let us denote by m0, A0 and J0 the mass,area and angular momentum of the remainder Kerr blackhole.

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

I Gravitational waves carry positive energy

m ≥ m0.

I A is the area of the intersection of the event horizon with theCauchy surface. By the black hole area theorem we have

A0 ≥ A.

I Since we have assumed axial symmetry, angular momemtum isconserved

J0 = J

I Since we have assumed that 8π|J| ≤ A then mbh also increasewith 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 Penroseinequality with angular momentum:√

A

16π+

4πJ2

A≤ m.

The inequality above implies√|J| ≤ m.

In fact, this inequality can be deduced directly by the sameargument without using the area theorem, it depends only on thefollowing 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 afinal Kerr black hole.

Summary

For axially symmetric, dynamical black holes, the followinggeometrical inequalities are expected:

Quasi-local

8π|J| ≤ A (= Extreme Kerr horizon)

Global √A

16π+

4πJ2

A≤ m (= Kerr black hole)

The global inequality implies:

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

16π≤ m (= Schwarzschild).

I Inequality between angular momentum and mass:√|J| ≤ m. (= extreme Kerr black hole).

It is important that in this inequality the area of the horizondo not appears.

The three geometrical inequalities valid for the Kerr blackholes 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 and J) can be calculated on the initial surface.

I A counter example of the global inequality will imply thatcosmic censorship is not true. Conversely a proof of it givesindirect evidence of the validity of censorship, since it is veryhard to understand why this highly nontrivial inequality shouldhold unless censorship can be thought of as providing theunderlying physical reason behind it.

I There are two group of geometrical inequalities:

1.√

A16π ≤ m: the area appears as lower bound.

2.√|J| ≤ m and 8π|J| ≤ A: the angular momentum appears

as lower bound.

The mathematical methods used to study these two groupsare, up to now, very different. This talk is mainly concernedwith the second group.

Part IITheorems

Penrose inequality

The Penrose inequality√A

16π≤ m (= Schwarzschild)

is expected to be valid also without axial symmetry. It has beenproved in the Riemannian case (i.e. initial data with zero extrinsiccurvature) by Huisken-Ilmanen 01 and Bray 01.For a recent review on this subject see Mars, CQG, TopicalReview, 09.There is up to now no result including angular momentum in thePenrose inequality.

Inequality between mass and angular momentum

TheoremConsider an axially symmetric, vacuum, asymptotically flat andmaximal initial data set with two asymptotics ends. Let m andJ denote the total mass and angular momentum at one of theends. Then, the following inequality holds√

|J| ≤ m (= Extreme Kerr).

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

I Chrusciel 08, Chrusciel-Li-Weinstein 08: generalizationand simplification of the proof. Important partial results formultiple black holes.

I Chrusciel-Lopes Costa 09, Lopes Costa 10: electriccharge.

I Schoen-Zhou 12: relevant improvements on the rigidity, firstrigidity result with charge, measure of the distance to theextreme 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 and J are properties of the whole spacetime.

Kerr black hole initial data

Non extreme Extreme

i0i0S

Cauchy horizon

Event horizon

Singularity

i0S

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

TheoremGiven an axisymmetric closed marginally trapped and stablesurface S, in a spacetime with non-negative cosmological constantand 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 assumethe existence of a maximal slice.

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

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

I Acena-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: electriccharge.

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

I Gabach-Reiris 13: shape.

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

I Null vectors `a and ka spanning the normal plane to S andnormalized as `aka = −1.

I Expansion: θ(`) = qab∇a`b.

I Marginal trapped surface: θ(`) = 0.

I Stable (Andersson-Mars-Simon 05): given a closedmarginally trapped surface S we will refer to it as spacetimestably outermost if there exists an outgoing (−ka-oriented)vector X a = γ`a − ψka, with γ ≥ 0 and ψ > 0, such that thevariation of θ(`) with respect to X a fulfills the condition

δX θ(`) ≥ 0.

I Riemannian proof: minimal surfaces on maximal slices.

I Lorentzian proof: trapped surfaces.

Extreme Kerr throat geometry

i0S

Event horizon

S

Singularity

Intrinsic metric of S:

|J|(

(1 + cos2 θ)dθ2 +4 sin2 θ

(1 + cos2 θ)dφ2

)

Rigidity

Generic trapped surface Extreme Kerr throat

Part IIIOpen problems and recent results on bodies

Open problems

√|J| ≤ m

I Remove the maximal condition (small trace Zhou 12).

I Prove it for initial data for multiple black holes. This problemis related with the uniqueness of the Kerr black hole withdegenerate and disconnected horizons. It is probably a hardproblem. (Chrusciel-Li-Weinstein 08 partial results,Dain-Ortiz 09 numerical evidences)

8|J| ≤ A

I Does some version of this inequality holds without axialsymmetry? This requires a notion of quasi-local angularmomentum for black holes without symmetries.√

A16π + 4πJ2

A ≤ 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 allbodies of the form

R2(Ω) ?G

c3|J(Ω)|,

where G is the gravitational constant and c the speed of light.The symbol ? is intended as an order of magnitude.

Physical argumentsThe arguments in support of this inequality are based in thefollowing three physical principles:

(i) The speed of light c is the maximum speed.

(ii) For bodies which are not contained in a black hole thefollowing inequality holds (hoop conjecture)

R(Ω) ?G

c2m(Ω),

where m(Ω) is the mass of the body and R(Ω) is somemeasure of size of Ω, which can in principle be different fromR(Ω).

(iii) The conjectured inequality for bodies holds for black holes.The inequality

8π|J|Gc3≤ A

can be interpreted as a version of this inequality for blackholes.

Constant density bodies

TheoremLet (S , hij ,Kij , µ, j

i ) be an initial data set that satisfy the energycondition. We assume that the data are maximal and axiallysymmetric. Let Ω be an open set in S. Assume that the energydensity µ is constant on Ω. Then the following inequality holds

R2(Ω) ≥ 24

π3

G

c3|J(Ω)|.

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

I The radius R is defined in terms of the Schoen-Yau, OMurchadha radius.

Schoen-Yau radius

RSY

Ω

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

I Let p be largest constant such that the set of pointswithin a distance p of Γ is contained within Ω andforms a proper torus.

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

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

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

Consider a region Ω with a axial Killing vector with square norm η.The radius R is defined by

R(Ω) =2

π

(∫Ω

√η dv

)1/2

RSY (Ω).

The most natural normalization for R is to require that R = b foran sphere in flat space of radius b. This is the reason for the factor2/π.

For more details see the review article:

Geometric inequalities for axially symmetric black holesS. Dain

Class.Quant.Grav. 29 (2012) 073001 “Topical Review”http://arxiv.org/abs/1111.3615

and the forthcoming:

Geometric inequalities bounding angular momentum andcharges in General Relativity

C. Cederbaum, S. Dain, M. E. Gabach Clement.

Living Review in Relativity