Geometry of black hole thermodynamics:Energy VS Entropy representation

Narit Pidokrajt

Department of Physics, Stockholm Universitynarit@fysik.su.se

GRSweden 2011

Linkoping20 May 2011

Brian Edgar and Narit at the same conferences

1. GRSweden day, Fysikum Stockholm University, February, 2003

2. GRSweden meeting, Matematiska institutionen, Linkopings universitet, June 3,2003

3. GRSweden meeting, Matematiska institutionen, Linkopings universitet, April 23,2004

4. Geometry and Relativity Meeting, Matematiska institutionen, Linkopingsuniversitet, Feb 18, 2005

5. GRSweden meeting, Matematiska institutionen, Linkopings universitet, June14-15, 2006

6. Eleventh Marcel Grossmann Meeting on General Relativity Berlin, Germany, July23 - 29, 2006

7. XXIX Spanish Relativity Meeting (E.R.E 2006), Palma de Mallorca, Spain,September 4-8, 2006

8. BritGrav7, University of Cambridge, Cambridge, UK. 3-4 April 2007

9. GRSweden meeting, Matematiska institutionen, Linkopings universitet, 26-27April 2007

10. Spanish Relativity Meeting (ERE2007), 10-14 September 2007, Tenerife, Spain

11. MG12 in Paris, France, 12-18 July 2009

12. GRSweden meeting in Linkopings, 4 August 2009

13. ERE2009 in Bilbao, Spain, 7-11 September 2009

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SPIRES topcites for this research field

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Special thanks to

I Kvant- och Faltteori, Fysikum, Stockholms universitet

I KVA: The Royal Swedish Academy of Sciences (travel grant, FOA10V-116)

I Helge Ax:son Johnson Stiftelse (research grant 2010-2011)

I Physics Department, Szegedi Tudomanyegyetem (University of Szeged)

I IGAIA3 (The third conference on Information Geometry and its Applications2010), Leipzig, Germany

A book to read this summer about Roy Kerr and his achievements:

Cracking the Einstein Code: RELATIVITY AND THE BIRTH OF BLACK HOLE PHYSICS

University of Chicago Press

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Who is who?

Collaborators:

1. Jan E. Aman (Stockholm U.)

2. James Bedford (Imperial College London)

3. Ingemar Bengtsson (Stockholm U.)

4. Laszlo A. Gergely (Szeged U.)

5. Daniel Grumiller (TU Wien)

6. John Ward (U. of Victoria)

7. Sergei Winitzki (NEXT Munich, IT business)

Consultants:

1. Roberto Emparan (Barcelona U.)

2. Gary Gibbons (Cambridge U.)

3. Des Johnston (Hariott-Watt U.)

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Plan of this talk:

1. Information geometry and applications

2. Black hole thermodynamics & problems that come with it

3. Some results (selected)

I Kerr-Newman family and AdS black holesI Myers-Perry black holes and black objectsI Energy, Entropy or Free energy? ⇐

4. Ongoing projects

5. Outlook

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Leipzig, August 2 - 6, 2010Max Planck Institute for Mathematics in the Sciences

in den NaturwissenschaftenMathematik Max-Planck-Institut für

Scientifi c Organizers

Nihat AyMax Planck Institute for Mathematics in the Sciences

Information Theory of Cognitive Systems GroupGermany

Paolo GibiliscoUniversità degli Studi di Roma ˝Tor Vergata“

Facoltà di EconomiaItaly

František MatúšAcademy of Sciences of the Czech Republic

Institute of Information Theory and AutomationCzech Republic

Scientifi c Committee

Shun-ichi AmariRIKEN Brain Science Institute,

Mathematical Neuroscience LaboratoryJapan

Imre CsiszárHungarian Academy of Sciences

Alfréd Rényi Institute of MathematicsHungary

Dénes PetzBudapest University of

Technology and EconomicsDepartment for Mathematical Analysis

Hungary

Giovanni PistoneCollegio Carlo Alberto

Italy

Administrative Contact

Antje VandenbergMax Planck Institute for Mathematics in the Sciences

Germany

Information Geometry and its Applications

Registration and further information: www.mis.mpg.de/calendar/conferences/2010/infgeo.html

Invited Speakers

Shun-ichi Amari (RIKEN, Japan)Pierre-Yves Bourguignon (Max Planck Institute for Mathematics in the Sciences, Germany)Dorje Brody (Imperial College London, United Kingdom)Frank Critchley (The Open University, United Kingdom)Imre Csiszár (Hungarian Academy of Sciences, Hungary)Philip Dawid (University of Cambridge, United Kingdom)Christopher Dodson (University of Manchester, United Kingdom)Shinto Eguchi (Institute of Statistical Mathematics, Japan)Christopher A. Fuchs (Perimeter Institute for Theoretical Physics, Canada)Akio Fujiwara (Osaka University, Japan)Kenji Fukaya (Kyoto University, Japan)Kenji Fukumizu (The Institute of Statistical Mathematics, Japan)Shigeru Furuichi (Nihon University, Japan)Frank Hansen (University of Copenhagen, Denmark)Peter Harremoës (Copenhagen Business College, Denmark)Fumio Hiai (Tohoku University, Japan)Shiro Ikeda (Institute of Statistical Mathematics, Japan)Oliver Johnson (University of Bristol, United Kingdom)Jürgen Jost (Max Planck Institute for Mathematics in the Sciences, Germany)Peter Jupp (University of St. Andrews, United Kingdom)Christian Léonard (Université Paris Ouest, France)Shunlong Luo (Chinese Academy of Sciences, China)Paul Marriott (University of Waterloo, Canada)Hiroshi Matsuzoe (Nagoya Institute of Technology, Japan)Noburu Murata (Waseda University, Japan)Hiroshi Nagaoka (The University of Electro-Communica-tions, Japan)Hiroyuki Nakahara (RIKEN, Japan)Jan Naudts (Universiteit Antwerpen, Belgium)Richard Nock (Université des Antilles et de la Guyane, France)Atsumi Ohara (Osaka University, Japan)Jan Peters (MPI for Biological Cybernetics, Germany)Dénes Petz (Budapest University of Technology and Economics, Hungary)Narit Pidokrajt (Stockholm University, Sweden)Giovanni Pistone (Collegio Carlo Alberto, Italy)Johannes Rauh (Max Planck Institute for Mathematics in the Sciences, Germany)Soo-Jong Rey (Seoul National University, South Korea)Jim Smith (The University of Warwick, United Kingdom)Raymond Streater (King's College London, United Kingdom)Arleta Szkoła (Max Planck Institute for Mathematics in the Sciences, Germany)Flemming Topsøe (University of Copenhagen, Denmark)Constantino Tsallis (Centro Brasileiro de Pesquisas Fisicas, Brazil, and Santa Fe Institute, USA)Armin Uhlmann (Universität Leipzig, Germany)Keiko Uohashi (Tohoku Gakuin University, Japan)Igor Vajda † (Academy of Sciences, Czech Republic)Christophe Vignat (Université d'Orsay, France)Stephan Weis (Universität Erlangen, Germany)Jun Zhang (University of Michigan/AFOSR, USA)C

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Information Geometry (IG)

I IG is the study of probability and information by way of differentialgeometry.

I Treat structures in probability theory, information theory and statistics asstructures in differential geometry by regarding a space of probabilitydistributions as a differentiable manifold endowed with a Riemannianmetric.

I Some key people: S.-I. Amari, H. Nagaoka, Sir R. A. Fisher, O. E.Barndorff-Nielsen, C. R. Rao, I. Csiszar, R. F. Streater, P. Gibilisco,Janyszek, Mruga la, Ruppeiner, Weinhold.

I We focus on the Fisher information (to come)

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Applications of IG

1. Neural networks and cognitive systems: Brain science

2. Computational and systems biology: Bioinformatics

3. Mathematical finance

4. Statistical mechanics

5. Quantum information and quantum statistics

6. Gravitational physics ⇐7. Statistical modelling

8. Unique applications: US and Greek Naval research teams work on IGmethods for wave predictions for the North Atlantic ocean.

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IG – Fisher information metric

For our purpose we study metrics that are defined (in some preferred affinecoordinate system) by

gab = ∂a∂bψ (1)

The potential ψ can be any reasonable function. One example is

ψ =

N∑i=1

xi lnxi , xi > 0 (2)

This potential is a negative of Shannon entropy. The Hessian of ψ is the Fisherinformation matrix. With some affined coordinate system it is regarded as ametric. If ψ is free energy then this metric is called Fisher-Rao metric. Thederived geodesic distance, known as Rao distance

ds2 =∂2F

∂Xa∂XbdXadXb (3)

And ds provides a measure of difference between two probability distributions.

For more detailed information, see Geometry of Quantum States, printed byCambridge University Press. Authors: Ingemar Bengtsson and K. Zyczkowski.

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IG – Thermodynamic geometryIn late 1970s, thermodynamic geometry program was started. If the potentialψ is a negative entropy, namely

ψ = −S(X,N i) (4)

the corresponding metric gab = −∂a∂bS is known as the Ruppeiner metric. For

ψ = M(S,N i) (5)

it is called Weinhold metric. The parameters N i are mechanically conserved(extensive thermodynamic variables).

The two metrics are related by a conformal relation:

ds2W = Tds2R , T ≡(∂M

∂S

)(6)

The Ruppeiner curvature scalar is physically meaningful. For ideal gas RR = 0.For a system with interacting statistics such as van der Waals gas, the Isingmodel, RR diverges at the critical point. Metric signature is related to sign ofspecific heat (stability...?).

An intriguing question is: What does RR mean for black holes? . The answeris: It is a very interesting question, indeed.

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Pre-Hawking black holes

Schwarzschild BH is described solely by M ;

ds2 = −(

1−2M

r

)dt2 +

(1−

2M

r

)−1

dr2 + r2dS2 (7)

It is a member of the Kerr-Newman black hole family characterized by (M,J,Q).

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BH thermodynamics I

I One of the most puzzling aspects of the BH is that it is a thermodynamical systemwith entropy (Bekenstein): S ∼ A.

I Hawking: BHs ain’t black! There is a thermal radiation with a black body spectrumemitted by black holes due to quantum effects. BH temperature: T ∝ κ and Hawkingfixed the proportionality constants for T and S

TBH =~κ

2πckBSBH =

kBA

4`2P(8)

where `P is the Planck length, `P =√G~/c3.

Hawking will be in Uppsala on 2 July 2011 to give a public lecture.

NB: Extremal BH has T = 0. Due to degenerate horizons. In some sense: mass isbalanced with charge. For BHs to exist physically mass should be larger than charge(in normalized units).

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Today’s black holes

A black hole which forms, and then completely evaporates away and we areback to the Minkowski space (but not without any problem!). Information

falling into the black hole will hit the singularity.

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BH thermodynamics II

The four laws of black hole mechanics analogous to thermodynamic laws:

I Zeroth law: Surface gravity κ is constant on the BH event horizon ↔ TBHis constant throughout a body in thermal equilibrium and TBH = κ

2π

I First law:dM =

κ

8πdA+ Ω dJ + Φ dQ (9)

Because S = A4

thus

dM = TBH dS + Ω dJ + Φ dQ (10)

with TH = κ2π

.By definition: T = ∂SM,Ω = ∂JM and Φ = ∂QM .

I Second law: dA > 0 ↔ dS > 0

I Third law: T = 0 cannot be reached (weak version) ↔ the BH extremalstates are inaccessible. Exception: Schwarzschild BH with M →∞.It is OK that T ⇒ 0 as S ⇒ 0 (strong version), then we have extremal BHwith T = 0 but S 6= 0, some violation there. So the third law of BHthermodynamics is unsettling.

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BH thermodynamics III

I Gravity is long-range interaction ⇒ gravitating bodies including BHs arenonextensive systems.

S(λU, λV ) 6= λS(U, V ) (11)

with λ a scaling variable

I Most BHs have negative specific heat (the more it radiates the hotter itbecomes)

I Astrophysical observations show that most BHs in nature are Kerr BHs (in factthey spin ultrafast!), i.e. most BHs are near-extremal BHs.

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BH thermodynamics IV

BH thermodynamics shows that there is a very deep and fundamental relationshipbetween gravitation, thermodynamics, and quantum theory but there are some bigproblems e.g.

I Information loss problem

Hawking: Information gets lost! Gravity and Quantum Mechanics are just plainincompatible. Susskind: Hawking was wrong! Black holes are not information-erasersbut information-scramblers!

I Origin of BH entropy ⇔ BH statistical mechanics

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BH thermodynamics V

I Thermodynamics is a well-defined macroscopic theory and its microscopicdescription is the quantum statistical mechanics.

S = kB ln Ω Ω = a number of microstates

I As of today we do not have a fully consistent microscopic description of blackholes. Since the microscopic description seems to require a quantum theory ofgravity ⇒ study of BH entropy is the study of some aspects of quantum gravity.

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Who will solve the big problem of physics?

Edward Witten—probably the most brilliant theoretical physicist of our time, and

Narit at KVA in Stockholm. Witten has not yet been able to erect the full-fledged

quantum theory of gravity .

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Kerr-Newman family I - Kerr

S = 2M2 −Q2 + 2M2

√1−

Q2

M2−

J2

M4(12)

The Ruppeiner metric of the Kerr black hole (Q = 0) is

ds2R =2(

1− J2

M4

) −2

[(1−

J2

M4

)3/2

+ 1−3J2

M4

]dM2 −

4J

M3dMdJ +

dJ2

M2

(13)

The diagonalized metric reads (using v = JM2 ; −1 6 v 6 1.)

ds2R = −2

(1 +

2√

1− v2

)dM2 +

2M2

(1− v2)3/2dv2 (14)

The Ruppeiner curvature scalar is

R =1

4M2

√1− J2

M4 − 2√1− J2

M4

. (15)

This curvature scalar diverges in the extremal limit (J = M2).

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Kerr-Newman family II - Reissner-Nordstrom

S(M,Q) = M2 −Q2 +M2

√1−

Q2

M2(16)

ds2W =1

8S32

[−(

1−3Q2

S

)dS2 − 8QdSdQ+ 8SdQ2

](17)

with u = Q√S

; −1 6 u 6 1

ds2R = −dS2

2S+ 4S

du2

1− u2(18)

This is a flat metric and can be embedded in Minkowskian-like cone

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Reissner-Nordstrom AdS

M =

√S

2

(1 +

S

l2+Q2

S

);

Q2

S= 1 +

3S

l2, (19)

which is consistent with the extremal limit of the ordinary RN black hole when thecosmological constant is switched off (Λ = 0 i.e. l→∞).

ds2W =1

8S3/2

[−(

1−3S

l2−

3Q2

S

)dS2 − 8QdSdQ+ 8SdQ2

](20)

ds2R =1

1 + 3τ2

2l2− u2

[−(

1−3τ2

2l2− u2

)dτ2 + 2τ2du2

](21)

We can observe that the metric above changes its properties as the signature of themetric changes. It is related to the Hawking-Page phase transition.

R =9

l2

(3Sl2

+ Q2

S

)(1− S

l2− Q2

S

)(

1− 3Sl2− Q2

S

)2 (1 + 3S

l2− Q2

S

) (22)

Curvature scalar diverges both in the extremal limit and along the curve where themetric changes signature.

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Einstein black holes in hyperspaceEvent horizon is obtained through:

r2+ + a2 −µ

rd−5+

= 0. (23)

but not analytic. Jan Aman was first to obtain Kerr black hole mass (as a function ofthermodynamic parameters) in D dimension

M =D − 2

4S

D−3D−2

(1 +

4J2

S2

)1/(D−2)

(24)

RR = −1

S

1− 12d− 5

d− 3

J2

S2(1− 4

d− 5

d− 3

J2

S2

)(1 + 4

d− 5

d− 3

J2

S2

) . (25)

For d > 6 we have a curvature blow-up but not in the limit of extremal black hole,rather at

4J2 =d− 3

d− 5S2. (26)

Emparan and Myers (2003) suggest that the Kerr black hole becomes unstable andchanges its behavior to be like a black membrane approximately at this point. So RRpredicts the onset of ultraspinning instability of Myers-Perry black holes.

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Black hole phase transitions

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Ruppeiner, Weinhold or Fisher-Rao?

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What else am I doing?

Gravitational Wave emissions and information geometry.How would these fit together?We only worry about the plunge/merger casei.e. two equal-mass BHs merge to become a single BH.This is the case of strongest GW emission. BH merger isby far the most energetic event in the post big-bang era.

The upper limit for Gravitational Wave emission is 29 %based on Hawking’s thermodynamic argument[Phys. Rev. Lett. 26, 1344 (1971)].Based on no-hair theorem.

η = 1−mf

m1 +m2(27)

But the upper limit from numericalGR gives betweeen 4-11 % [e.g. Pretorius 0710.1338]

We want sophisticaled thermodynamic argumentgiving numbers much close to thatcomes out from numerical works.

Information geometry = theway to improve this ? L. Gergely and I are trying anyway.

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Summary and outlook

I Ruppeiner geometry is physically suggestive whereas its counterpart (Weinhold) ishelpful in most calculations (when Ruppeiner coordinates are complicated).

I Some of our results have been supported by physical results in the literature in thatultraspinning instability of Myers-Perry Kerr black holes obtained from the curvaturesingularity of the Ruppeiner metric matches the results found earlier by Emparan andMyers.

I State space diagrams can be very useful when studying extremal limits of BH solutions.

Some issues

1. Profusion of black hole solutions lead to incongruities of results (based on differentversions of geometry)

2. BH phase transition. Who is least wrong?

3. (Quantum) statistical version of BH information geometry?

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