quantum bits from evaporating black holes...string theory string theory provides quantum models of...
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QUANTUM BITS
FROM
EVAPORATING
BLACK HOLES
Jose L. F. BarbonIFT UAM/CSIC
Madrid, January 2005.
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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 �
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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)
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AdS/CFT
AdS
Supergravity and closed-string gas: Thermal state of glue-balls in the CFT low temperature phase y z { |
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AdS/CFT
AdS
Small AdS black holes, � � � � are superheated unstableresonances of the CFT (unknown in detail)
They solve the unitarity puzzle in principle
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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
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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
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AdS/CFT
Energy
1/R
Temperature
graviton
gas
Hagedorn strings
small bhlarge bh
The Hawking–Page first-order phase transition
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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)
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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
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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.
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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
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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)
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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 «
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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
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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)
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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
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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
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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
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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
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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
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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
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