QUANTUM BITS FROM EVAPORATING BLACK HOLES...STRING THEORY String Theory provides quantum models of black holes in HAMILTONIAN form, with respect to asymptotic time frames Hence, it

QUANTUM BITS

FROM

EVAPORATING

BLACK HOLES

Jose L. F. BarbonIFT UAM/CSIC

Madrid, January 2005.

INTRODUCTION

Classically, black holes have little hair

� ������� ����� ��� �����������

Quantum mechanically, they are thermal radiators with in-trinsic temperature (Hawking, 1974)

��� � ���� � �

Therefore, they have effective entropy (Bekenstein 1972)

� �"! � # $ � %� � & � � � ' � ( )+*-,/.10"*32& �

In a full theory of quantum gravity, they might be eigenstatesof some Hamiltonian, with density of levels (’t Hooft 1985)

4 5�6 7 � 8:9+; <=& � � 6 '?>

Conversely, such a branch of states is a test to be met byany candidate theory of quantum gravity

January 2005 @ ACBEDGFHAJILKNMOFHPNQSRHTEIVU=WXBZY[TER\BXF]M^DJ_`KNa^BGb\cedfTGaOghP i 1 Jose L. F. Barbon

INTRODUCTION

Physics of Hawking emission:

Tidal forces can materialize virtual pairs

jlk�m1nporqts u v w x ormzy{ | } ~��at radial separation �Potential energy

� k�mznporq�s w x orm1y� u v ��� � �can overcome the virtual energy w x orm1y at distance � s u vand produce a real pair

Higher particle escapes with positive energy at infinity

Lower particle falls in negative energy orbit

January 2005 � �C�E�G�H�J�L�N�O�H�N�S�H�E�V�=�X�Z�[�E�\�X�]�^�J�`�N�^�G�\�e�f�G�O�h� � 2 Jose L. F. Barbon

INTRODUCTION

V

d

ω

λ

Vtidal

ω

eff

Tunneling amplitude goes as

� �¡ £¢ ¤ ¥+¦ § ¢©¨«ª ¬®­ ¯ ° ± ²´³rµ1¶¸·corresponding to effective temperature ¹ º § ¯ °

The entangled state of escaping » ± ¼?½f¾À¿ and infalling » ± ¼ÂÁE³rÃzÃpairs is called the Unruh state

»ÅÄ ¼ §Æ ÇÉÈ

ÆËÊÍÌÏÎ » ± ¼ ½f¾Ð¿�Ñ » ± ¼ ÁE³rÃzÃ

January 2005 Ò ÓCÔEÕGÖHÓJ×LØNÙOÖHÚNÛSÜHÝE×VÞ=ßXÔZà[ÝEÜ\ÔXÖ]Ù^ÕJá`ØNâ^ÔGã\äeåfÝGâOæhÚ ç 3 Jose L. F. Barbon

INTRODUCTION

IN

OUTp

p−

yM

U

Background matter and gravitational field field èêé ë ì treatedas CLASSICAL spectator.

èêí î ï ì ð èÅé ë ìòñ èÅó ìThe outgoing energy flux è1ô ìeõföÀ÷ is balanced by a NEGATIVEinfalling energy flux carried by the entangled partners è1ô ìùøEúrûzûBlack hole loses mass at FIXED èêé ë ì , by absorbing anincreasing cloud of negative energy Hawking anti-particles

January 2005 ü ýCþEÿ��Hý���������� ��������Xþ������\þ����^ÿ������^þ���� �!����"# $ 4 Jose L. F. Barbon

INTRODUCTION

OUT

IN

Is the process of formation/evaporation unitary?Tracing over UNOBSERVABLE infalling particles yields athermal density matrix detectable at % & ' with temper-ature

(*)

+-,/. 0 132 46587:9 ;<

+�= 0?>8@BADCFE:G < H = +

If infinitely hairy remnants do not exist, then complete evap-oration seems to violate quantum coherence

January 2005 IKJML�N�OJ�P�Q�R�OS�T UV�P�W�X�L�Y�V�U�L�O�RZN�[�Q�\ZL�]�^ _!V�\�`#S a 5 Jose L. F. Barbon

THE PARADOX

OUT

IN

External observers only need EXTERIOR

bdcfe8gih jlknmoeqp r s t bvudw jyx z { | } s

Black hole never really forms!!!

Everything is thermalized at horizon and emittedback, easily compatible with unitarity.

January 2005 ~K�M����������������� ����������������������Z�������Z����� �!�����#� � 6 Jose L. F. Barbon

THE PARADOX

YET, the Equivalence Principle ensures that thereis (long) life for infalling observers

Can information be duplicated at horizon cross-ing??

Duplication exotic, and most likely IMPOSSIBLE

��� � � ��� � � ��� ��� ��� � � ��� � � ��� �

Such operator violates linearity of QM

��� �¡  ��¢ �¤£ �¥� �§¦ ��� �§  �¥¢ �§¦ ��¢ �¡  ¨ª©¬«®­°¯²±�­³¯´­³©Fµ ­ «®­³¯²¶ ·

January 2005 ¸K¹Mº�»�¼¹�½�¾�¿�¼À�Á ÂÃ�½�Ä�Å�º�Æ�Ã�Â�º�¼�¿Z»�Ç�¾�ÈZº�É�Ê Ë!Ã�È�Ì#À Í 7 Jose L. F. Barbon

THE PARADOX

A

A’

B

C

A Way Out: BLACK HOLE COMPLEMENTARITY

External and Infalling stories refer to sets of NONCOMMUT-ING operators, despite the fact that both have semiclassicallimits (’t Hooft 1990, Susskind 1993)

In fact, attempts at operationally verifying the independenceof inside/outside local measurements run into problems ata heuristic level

January 2005 ÎKÏMÐ�Ñ�ÒÏ�Ó�Ô�Õ�ÒÖ�× ØÙ�Ó�Ú�Û�Ð�Ü�Ù�Ø�Ð�Ò�ÕZÑ�Ý�Ô�ÞZÐ�ß�à á!Ù�Þ�â#Ö ã 8 Jose L. F. Barbon

THE PARADOX

OUT

IN

BH

U

YM

Perhaps a final state condition? (Horowitz-Maldacena2003)

Information is Quantum-teleported from the singularity,where a fixed state ä�å æ ç is POST-SELECTED

January 2005 èKéMê�ë�ìé�í�î�ï�ìð�ñ òó�í�ô�õ�ê�ö�ó�ò�ê�ì�ïZë�÷�î�øZê�ù�ú û!ó�ø�ü#ð ý 9 Jose L. F. Barbon

STRING THEORY

String Theory provides quantum models of black holes inHAMILTONIAN form, with respect to asymptotic time frames

Hence, it furnishes a fully unitary description for EXTER-NAL OBSERVERS

It shows little light on the issue of Black Hole Complemen-tarity, or the fate of infalling observers/observables

BASIC MODEL

The AdS/CFT Correspondence that DEFINES quantumgravity on asymptotically þ ÿ�������� spaces with curvature ra-dius � in terms of the quantum mechanics of a �� on a� �� �� sphere of radius �

January 2005 ����������������� ��!�"$#�%��'&)(*�,+-%�#.�*�/�0��12��30��4.5768%�309:! ; 10 Jose L. F. Barbon

AdS/CFT

<=<>=>?=?@=@A=AB=B

C=CD=DE=EF=FG=GH=HE

N / R

1 / R

2

A standard Hamiltonian picture with discrete spectrum

Asymptotic density of states of I dimensional CFT with J Kconformal d.o.f. matches entropy of large L MONQP�R�S blackholes

T UWV X Y J Z[ UWV \ X [^]`_[

For I a b , modular invariance ensures that the coefficientmatches (succesful counting of Strominger and Vafa 1995)

January 2005 c�d�e�f�g�d�h�i�j g�k�l$m�n�h'o)p*e,q-n�m.e*g/j0f�r2i�s0e�t.u7v8n�s0w:k x 11 Jose L. F. Barbon

AdS/CFT

AdS

Supergravity and closed-string gas: Thermal state of glue-balls in the CFT low temperature phase y z { |

January 2005 }�~�������~������ �����$�����'�)�*�,�-���.�*�/�0���2���0���.�7�8���0�:� � 12 Jose L. F. Barbon

AdS/CFT

AdS

Small AdS black holes, � � � � are superheated unstableresonances of the CFT (unknown in detail)

They solve the unitarity puzzle in principle

January 2005 ����������������� �����$ �¡��'¢)£*�,¤-¡� .�*�/�0��¥2��¦0��§.¨7©8¡�¦0ª:� « 13 Jose L. F. Barbon

AdS/CFT

AdS

Large AdS black holes, ¬ ­ ® ¬ are stable canonical statesin the plasma phase of the CFT

Can be used to study relaxation in quasi-equilibrium

January 2005 ¯�°�±�²�³�°�´�µ�¶ ³�·�¸$¹�º�´'»)¼*±,½-º�¹.±*³/¶0²�¾2µ�¿0±�À.Á7Â8º�¿0Ã:· Ä 14 Jose L. F. Barbon

AdS/CFT

gs

Large

AdS

black holes

1/N

1

E RN 21

10 dim

Yang−Mills Yang−Mills

4 dim

4 dim

Hagedorn

strings

Schwarzschild

black holes

10 dim

10 dim

gravitons

O(1) states

constituents

N 2

The full phase diagram

January 2005 Å�Æ�Ç�È�É�Æ�Ê�Ë�Ì É�Í�Î$Ï�Ð�Ê'Ñ)Ò*Ç,Ó-Ð�Ï.Ç*É/Ì0È�Ô2Ë�Õ0Ç�Ö.×7Ø8Ð�Õ0Ù:Í Ú 15 Jose L. F. Barbon

AdS/CFT

Energy

1/R

Temperature

graviton

gas

Hagedorn strings

small bhlarge bh

The Hawking–Page first-order phase transition

January 2005 Û�Ü�Ý�Þ�ß�Ü�à�á�â ß�ã�ä$å�æ�à'ç)è*Ý,é-æ�å.Ý*ß/â0Þ�ê2á�ë0Ý�ì.í7î8æ�ë0ï:ã ð 16 Jose L. F. Barbon

TOPOLOGY AND UNITARITY

Interior

OUT

IN

Local QFT on standard Carter-Penrose diagram leads toviolation of quantum coherence if interior is truly inaccesibleto S-matrix ”out” data

S-matrix map is linear, conserving probability

ñóòõô ñ÷ö ñ ø ùúö ûýü ñ ô þ

But violating coherence as ûýü ñ�ÿ decreases (Hawking1976)

January 2005 ����������� ������������������������������ ��!"��# ��$�%'&(��# )*� + 17 Jose L. F. Barbon

TOPOLOGY AND UNITARITY

?

OUT

IN

Insisting on unitary S-matrix requires topologically trivialCarter-Penrose diagram

Hawking claims now that these contributions are enough torestore unitarity

Problem: for S-matrix observables, no classical trajectorieswith trivial topology, hence the claim is difficult to check

January 2005 ,�-�.�/�0-�1 2�3�04�5�67�1�8�9�.�:�7�6�.�0�3 /�;"2�< .�=�>'?(7�< @*4 A 18 Jose L. F. Barbon

RECURRENCES

FLUCTUATION

DISSIPATIONDISSIPATION

FLUCTUATION

We can study unitarity by looking at the details of dissipationat very long times (Maldacena)

“NO HAIR” arguments imply that perturbations on the clas-sical black holes die out as

BDCFEHGJI KMLONQP R ITS UInconsistent with Poincare recurrences of bounded sys-tems.

Topological fluctuations to the rescue? (Maldacena 2001)January 2005 V�W�X�Y�ZW�[ \�]�Z^�_�`a�[�b�c�X�d�a�`�X�Z�] Y�e"\�f X�g�h'i(a�f j*^ k 19 Jose L. F. Barbon

RECURRENCES

Fixed Energy Surface ( P , Q )

W

U (W)t

Classical Poincare recurrences follow from compact phasespace (at finite energy) plus Liouville’s theorem

Time development of initial set l monqp r intersects itself

Quantum correspondences Compact phase space tTu Discrete energy spectrums Liouville theorem tvu Unitaritys w xzy nqp r tvu Purity of initial state

January 2005 {�|�}�~��|�� ������������������}�������}���� ~��"��� }����'�(��� �*� � 20 Jose L. F. Barbon

RECURRENCES

REFERENCES

The following is based on joint work with E. Rabinovici(Jerusalem)JHEP 0311 (2003) hep-th/0308063

J. Maldacena, JHEP 0304(2003) hep-th/0106112

L. Dyson, N. Goheer, M. Kleban, J. Lindesay, Susskind(2002)

D. Birmingham, I. Sachs, S. Solodukhin (2003)

M. Kleban, M. Porrati, R. Rabadan (2004)

S. Hawking (2004)

January 2005 ������������ ������������������������������� �� "��¡ ��¢�£'¤(��¡ ¥*� ¦ 21 Jose L. F. Barbon

RECURRENCES

GE

In finite-volume CFT, normalized correlation functions arequasiperiodic functions of time

§ ¨ª©¬« ­ ®°¯ ±³² ´ ¨ª©¬«µ´ ¨·¶¸«¸¹v­º¼»½

¾�¿ À º ¾�Á  À�ÃÅÄÇÆ'ÈzÄ É"Ê�Ë

Dissipation on time scale Ì ÈµÍ, followed by large fluctuations

on the Heisenberg time scale

©'Î ­ ÏÐ�Ñ º Ò Ñ ½ÔÓ ­ Õ ÖØ×Ì

Finally

© Î ­ Ì ÈFÍ ÙDÚFÛ Ü ¨ÞÝ «

January 2005 ß�à�á�â�ãà�ä å�æ�ãç�è�éê�ä�ë�ì�á�í�ê�é�á�ã�æ â�î"å�ï á�ð�ñ'ò(ê�ï ó*ç ô 22 Jose L. F. Barbon

RECURRENCES

t

1L(t)

t H

exp (− t )G

t H2G-1

Recurrence index

õ ö ÷ùøùúûzü ýû

þ ÿ�� õ � ��� �� õ �� �� � ��

In the ����� expansion of the CFT, this index is nonpertur-bative

õ � ����� ��� � � �Gravity perturbation theory is the ����� power expansion.So we expect

õ � �in gravity perturbation theory around a

classical black hole

January 2005 �! #"%$'&( *),+.-/&(0.132(4%)65879";:<4%2="9&>-?$*@A+.B?"'C=DFEG4%B?HI0 J 23 Jose L. F. Barbon

RECURRENCES

t

r

X

r=RS

Euclidean black hole manifold K L MONPRQTS U VXWZY�[\P K S^] P_Y S

VXWZY`[ ] Y6S_PRa Sbdc�e

withV�fgWZY L h i [ L j6k lnm

Has non-standard topology o L p S q r bdc�eUnlike standard thermal topology in QFT s L r e q p bEuclidean saddle points determine partition function inWKB approximation (Hawking-Gibbons 1977)

January 2005 t!u#v%w'x(u*y,z.{/x(|.}3~(�%y6�8�9v;�<�%~=v9x>{?w*�Az.�?v'�=�F�G�%�?�I| � 24 Jose L. F. Barbon

RECURRENCES

t = constant

r = constant

Reason for � ��� � � is topological

Despite Green’s functions being smooth on Euclidean sec-tion at � � � � , Hamiltonian foliation � � ��������������� is sin-gular at � � � �Rendering the spectrum of the frequency operator

  � ¡¢¢ �

CONTINUOUS

January 2005 £!¤#¥%¦'§(¤*¨,©.ª/§(«.¬3­(®%¨6¯8°9¥;±<®%­=¥9§>ª?¦*²A©.³?¥'´=µF¶G®%³?·I« ¸ 25 Jose L. F. Barbon

TOPOLOGICAL DIVERSITY

¹º »¼½�½¾¿ÀÁÂ

Ã�ÃÄÅ�ÅÆ�Æ Ç�ÇÈ�È

É�ÉÊ�ÊËËÌÌ ÍÎ Ï�ÏÐ�Ð ÑÒ

Ó�ÓÔ Õ�ÕÖ×�×Ø�ØÙ�ÙÚÛÜ

ÝÞßà

áâ ã�ãã�ãää

åæ çèé�éê+

+

AdS AdS

X Y

bh thermal

Can smooth instanton corrections restore the recurrences?

ë ìîíðïòñôó õ÷öùøôú�û ü ý þRÿý þ � þ � û þ�� ó û � ú�� � ÿ� � ú�� �ÿ ó �

Time average correct in order of magnitude

ë ìIíðï ñ ó õ�ö�ø�ú�û � �ÿJanuary 2005 � ������������������������ ��"!$#%�'&( ��)�%�*�+��,-��.+��/)0213 �.+45� 6 26 Jose L. F. Barbon

TOPOLOGICAL DIVERSITY

L(t)

t

1

exp (− N )

c t + t (Y)c H

2

t

Not so for the long-time profile

7 8:9<;>=@?BADCFE 7 8:9<;HG I J KMLONQPSRT7 8:9<;>U

Natural time scale of7 8:9<;VU

is9XW 8�Y ; Z [ \

Critical time for topological fluctuations to affect7 8:9];

is

9_^ Z ` ab Z [ N

Also found by Kraus, Ooguri and Shenker, 2003

Kleban, Porrati and Rabadan, 2004 argue that the semi-classical approximation breaks down for

9 c 9 ^in the case

of 3-dimensional BTZ black holes

January 2005 d e�f�g�h�e�i�j�k�h�l�m�n�o�i"p$q%f'r(o�n)f%h*k+g�s-j�t+f�u)v2w3o�t+x5l y 27 Jose L. F. Barbon

TOPOLOGICAL DIVERSITY

L(t)

t

1

t (Y) tH H

e −N2

Very long time structure is not right, despite time averagecorrectly obtained by the instanton approximation

Recall z { |~}�|�����

z������B� {�

� ��� ��� � { �������_� � � �

z ¡@¢B£ � {�M¤ � �� ��� { �����¥�_� � �¦�

January 2005 § ¨�©�ª�«�¨�¬�­�®�«�¯�°�±�²�¬"³$´%©'µ(²�±)©%«*®+ª�¶-­�·+©�¸)¹2º3²�·+»5¯ ¼ 28 Jose L. F. Barbon

TOPOLOGICAL DIVERSITY

( P , Q )

graviton

gas

AdS Black Hole states

states

W

U (W)t

Energy 1/R

What is going on?

Only recurrences of the small corner of thermal gravitongases are correctly accounted for, since semiclassical 1/Nexpansion regards the black hole states as continuous

Time average is the same because of ergodic behaviour

January 2005 ½ ¾�¿�À�Á�¾�Â�Ã�Ä�Á�Å�Æ�Ç�È�Â"É$Ê%¿'Ë(È�Ç)¿%Á*Ä+À�Ì-Ã�Í+¿�Î)Ï2Ð3È�Í+Ñ5Å Ò 29 Jose L. F. Barbon

CONCLUSIONS

Ó Recurrences in AdS/CFT measure the fine structure ofunitarity on exponential time scales

Ó Topological fluctuations in WKB approximation restoreunitarity at the level of time averages. GR fundamentallythermodynamical?

Ó Detailed time structure of recurrences not accounted forby large, smooth topological fluctuations.

Ó Can we draw conclusions for S-matrix questions, asHawking claims?

Ó Holographic probes of black hole interior remain the mostimportant open question

January 2005 Ô Õ�Ö�×�Ø�Õ�Ù�Ú�Û�Ø�Ü�Ý�Þ�ß�Ù"à$á%Ö'â(ß�Þ)Ö%Ø*Û+×�ã-Ú�ä+Ö�å)æ2ç3ß�ä+è5Ü é 30 Jose L. F. Barbon