A State-Dependent Construction of the Black Hole
Interior
Suvrat Raju
International Centre for Theoretical Sciences
Strings 2015
References
Based on work with Kyriakos Papadodimas (CERN & Groningen)
1
“Comments on the Necessity and Implications of
State-Dependence in the Black Hole Interior”, arXiv: 1503.08825.
2
“Local Operators in the Eternal Black Hole”, arXiv:1502.06692.
3
“State-Dependent Bulk-Boundary Maps and Black Hole
Complementarity”, arXiv: 1310.6335.
4
“The Black Hole Interior in AdS/CFT and the Information
Paradox”, arXiv:1310.6335.
5
“The unreasonable effectiveness of exponentially suppressed
corrections in preserving information”
6
“An Infalling Observer in AdS/CFT”,arXiv:1211.6767.
and work in progress with Souvik Banerjee (Groningen), Prashant
Samantray (IIT-Indore), Sudip Ghosh (ICTS-TIFR).
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
2 / 40
Context
The context for this talk is the Information Paradox. In its modern
avatar, this turns into the question:
“Can AdS/CFT describe the BH Interior?”
[Mathur, Almheiri, Marolf, Polchinski, Sully, Stanford, 2009-2015]
This version is not restricted to evaporating BHs.
Paradox extends to the Eternal Black Hole.
[Kyriakos Papadodimas, S.R., 2015]
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
3 / 40
Overview
Resolution: Paradox can be complete resolved using a
state-dependent map between interior bulk observables and
boundary observables.
[K.P., S.R, 2013–15]
New Consequences: This construction of the interior leads to a
precise version of ER=EPR conjecture of Maldacena and
Susskind.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
4 / 40
Outline
1
The Old Information Paradox
2
The Modern Information Paradox for the Eternal BH
3
State-Dependent Resolution of the Modern Information Paradox
4
ER=EPR
5
Significance of State-Dependence
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
5 / 40
Outline
1
The Old Information Paradox
2
The Modern Information Paradox for the Eternal BH
3
State-Dependent Resolution of the Modern Information Paradox
4
ER=EPR
5
Significance of State-Dependence
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
6 / 40
The Old Information Paradox
111111
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P
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In the shaded patch, physics is independent of details of collapse.
haω† aω0 i =
e−βω
δ(ω − ω 0 )
1 − e−βω
Suggests that for different inputs, we get the same output.
Suvrat Raju (ICTS-TIFR)
Input A
Black
Hole
Input B
Black
Hole
Black Body
Radiation
Black Body
Radiation
Information Paradox and BH Interior
Strings 2015
7 / 40
Resolution to the Old Information Paradox
Very small corrections of the order of e−S can restore unitarity.
[Maldacena, 2001]
Pure density matrix in a very large system can mimic a thermal
density matrix to extreme accuracy
S
1 Tr (ρpure Aα ) = Tr e−βH Aα + O e− 2 ,
Z
for a large class of observables Aα .
Another way to state this is
ρpure =
1 −βH
e
+ e−S ρcorr ;
Z
ρ2pure = ρpure
Information can be encoded in tiny correlations between the Hawking
quanta.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
8 / 40
Path Integral Perspective
Effective field theory insufficient to control such corrections.
A semi-classical spacetime is a saddle point of the QG
path-integral.
Z=
Z
e−S Dgµν
Perturbative effective field theory (used to derive the Hawking
answer) is an asymptotic series expansion of this path-integral.
Non-perturbatively, the notion of local spacetime breaks down.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
9 / 40
Outline
1
The Old Information Paradox
2
The Modern Information Paradox for the Eternal BH
3
State-Dependent Resolution of the Modern Information Paradox
4
ER=EPR
5
Significance of State-Dependence
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
10 / 40
Revisiting the Information Paradox
Recent developments have challenged the notion that small
corrections can resolve the information paradox.
[Mathur, Almheiri, Marolf, Polchinski, Sully, Stanford, 2009–13]
We will review an extension of these arguments to the eternal
black hole.
Resolution of the paradox in the eternal black hole provides strong
evidence that a similar resolution applies to the single-sided case.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
11 / 40
Thermofield Doubled State
U
V
II
III
I
IV
Eternal black hole is dual to an entangled state of two CFTs
X βE
1
|Ψtfd i = p
e− 2 |E, Ei
Z (β) E
[Maldacena, 2001]
On the left boundary, we identify
tsch = −tCFT ,L
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
12 / 40
Time Shifted States
We now consider time-shifted states
X
βE
1
|ΨT i = eiHL T |Ψtfd i = p
eiφE − 2 |E, Ei
Z (β)
Geometrically, this corresponds to a large diffeomorphism that
dies off at the right-boundary but not left-boundary.
T
For example
U → U γ θ̂(−X ) + θ̂(X ) ; X = V − U
2πT
V → Vγ θ̂(−X ) + γ θ̂(X ) ;
γ=e β .
with θ̂ a smooth version of the theta function.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
13 / 40
Smoothness of Time Shifted States
Large diffs that differ by trivial diffs are equivalent. Use this to undo
the diff everywhere, except infinitesimally close to the boundary.
T
Makes it clear that the large diffs leaves intrinsic properties of the
geometry invariant, but just slides the boundary.
Therefore, the states |ΨT i = eiHL T |Ψtfd i are also smooth but glued
differently to the boundary
tsch = −tCFT ,L + T .
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
14 / 40
Long Time Shifts
The states |ΨT i = eiHL T |Ψtfd i are smooth, even for T ∼ eN
2
Strong conclusion relies only on equivalence of Hamiltonian
evolution and diffeomorphisms; not on eom.
Equivalent to statement that an infalling observer from the right
observes the same geometry at arbitrarily late time.
Equivalent to statement that there is no natural common origin of
time on the two sides.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
15 / 40
Relational Observables
AdS boundary allows us to define quasi-local diffeomorphism
invariant observables.
Consider the following process: jump from the boundary, wait for a
certain proper time, measure the field.
(t1 + τ2 , Ω3 )
Pλ (t1 , λ, Ω1 )
(t1 + τ1 , Ω2 )
λ=1
λ=0
(t1 , Ω1 )
Behind the horizon, we write
X
(1)
(2)
eω,m eiωt gω,m
φ(t, Ω, λ) =
Oω,m e−iωt gω,m (Ω, λ) + O
(Ω, λ) + h.c,
ω,m
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
16 / 40
Central Question
Can we find a CFT operator φ(x), so that for generic T
hΨT |φ(x1 ) . . . φ(xn )|ΨT i = G(x1 . . . xn )
with |ΨT i = eiHL T |Ψtfd i.
On the RHS, G is the Green function as computed by
semi-classical effective field theory in the eternal black hole
background.
Well-posed question about existence of operators in the CFT.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
17 / 40
Modern Information Paradox
By extending work of AMPSS, can show that @φ(x) s.t. ∀T
hΨT |φ(x1 ) . . . φ(xn )|ΨT i = G(x1 . . . xn ).
[K.P,S.R., 2015]
Reason is, roughly, that states |ΨT i are overcomplete.
Seems different from the information paradox, but principal
paradox is still
Unitarity vs Effective Field Theory
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
18 / 40
Some Possibilities
AdS 6= CFT (a.k.a. superselection
sectors)?
[Marolf, Wall, 2012]
Eternal Black Hole also has firewalls?
But both possibilities above contradict explicit computations that look
inside the horizon of the eternal BH.
[Hartman, Maldacena, Kraus, Ooguri, Shenker]
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
19 / 40
Proposed Resolution: State Dependence
No need to use the “same operator” for all T : state-dependence.
We use one operator φ{0} in a range of states about T = 0.
After exponentially long T , we switch to another operator φ{T }
T = −eN
2
Suvrat Raju (ICTS-TIFR)
T =0
Information Paradox and BH Interior
T = eN
Strings 2015
2
20 / 40
Outline
1
The Old Information Paradox
2
The Modern Information Paradox for the Eternal BH
3
State-Dependent Resolution of the Modern Information Paradox
4
ER=EPR
5
Significance of State-Dependence
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
21 / 40
The Little Hilbert Space
|ΨT i ≡ Black Hole Microstate
Little Hilbert Space: all possible effective field theory excitations of
|ΨT i
HΨT = A|ΨT i,
A = span{Oω1 , Oω1 Oω2 , . . . , Oω1 Oω2 . . . OωK }.
with
H
T = −eN
2
ωm N ,
T =0
K N
H ΨT
T
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
T = eN
2
|ΨT i
Strings 2015
22 / 40
eω
Definition of O
eω precisely within HΨ
Define O
SAα |Ψi = A†α |Ψi
and
1
eω = S∆ −1
2 O ∆2 S
O
ω
[KP, SR, 2013]
This is closely related to the isomorphism used in Tomita-Takesaki
theory.
eω is a linear operator on HΨ and
φ(t, Ω, λ) constructed using this O
has the correct effective field theory correlators
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
23 / 40
Obtaining a Smooth Horizon
In eternal BH, can carry out this program explicitly
s
Z Tcut
C
e
OLω (Ti )PHΨT dTi ,
Oω =
i
πβ 2 −Tcut
Smear with appropriate mode functions
X
eω g (2) (x) + h.c.
φ(x) =
Oω gω(1) (x) + O
ω
ω
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
24 / 40
Obtaining a Smooth Horizon
We can probe the interior with correlators of these operators.
hΨT |φ(x1 ) . . . φ(xn )|ΨT i = G(x1 , . . . xn )
Explicitly consistent with perturbative fields propagating on a
weakly curved spacetime near the horizon.
This construction explicitly resolves the information paradox for the
eternal black hole. Problem of overcompleteness is resolved by
state-dependence.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
25 / 40
Outline
1
The Old Information Paradox
2
The Modern Information Paradox for the Eternal BH
3
State-Dependent Resolution of the Modern Information Paradox
4
ER=EPR
5
Significance of State-Dependence
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
26 / 40
New Applications
Apart from resolving the information paradox, we can apply this
construction of the interior to other entangled systems.
Leads to a natural formulation of “ER=EPR”. (Originally proposed
as relation between entanglement and wormholes by Maldacena
and Susskind.)
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
27 / 40
Simple Operators and the Little Hilbert Space for
Entangled Systems
Consider an entangled state
|Ψen i =
X
e i i ⊗ |Ψi i
αi |Ψ
Expand the set of simple operators to include both left and right
operators.
A, AL , Aprod = AL ⊗ A
Therefore the little Hilbert space is
HΨen = Aprod |Ψen i
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
28 / 40
Mirrors for Entangled States
HΨen = Aprod |Ψjen i =
M
j
A|Ψjen i
We may have null relations due to entanglement. For example, in
−βω
the thermofield OL,ω |Ψtfd i = e 2 Oω† |Ψtfd i.
HΨen Aprod
Number of terms in the sum depends on number of null relations
involving both left and right operators acting on the state.
Define
S=
X
Sj
j
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
29 / 40
ER=EPR from Mirror Operators
Are correlators of right-relational observables affected by
simple unitaries on the left?
With UL = eiAL , compare
hΨen |UL† φ(x1 )φ(x2 ) . . . φ(xn )UL |Ψen i
and
hΨen |φ(x1 )φ(x2 ) . . . φ(xn )|Ψen i
Equivalent diagnostic is
Commutator
Two-Pt Function
eω , O† ]|Ψen i
[O
L,ω
eω O† |Ψen i
hΨen |O
L,ω
We can apply this diagnostic to several examples.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
30 / 40
Standard Wormhole for the Thermofield
Consider the thermofield double state
1 X − βE
|Ψtfd i = √
e 2 |E, Ei
Z E
We find the commutator and two point function
.
†
eω ] =
[OLω
,O
Cω ,
eω O† |Ψtfd i = Gβ,ω .
hΨtfd |O
Lω
By computing these and other correlators we can obtain a picture
of the dual geometry.
OR (t)
OL (−t)
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
31 / 40
Generic Entanglement: “Long Wormhole”
Consider a more “generic” entangled state
|Ψgen i = ULarb |Ψtfd i
Now we find that
S
.
eω , O† ] =
[O
O
e− 2 ,
Lω
eω O† |Ψgen i = O e− S2 .
hΨgen |O
Lω
But for some generic CFT operator on the left
eω ]|2 |Ψgen i = O (1) .
hΨgen |[YL , O
These correlators suggest a dual geometric picture.
OL (t)
Suvrat Raju (ICTS-TIFR)
eL (t)
O
eR (t)
O
Information Paradox and BH Interior
OR (t)
Strings 2015
32 / 40
Microcanonical Double: Low Band-Pass Wormhole
Consider the microcanonical doubled state.
1
|Ψmd i = √
D
Ei =E+∆
X
Ei =E−∆
|Ei , Ei i.
Take β∆ 1. This leads to a low pass wormhole!
For low frequencies, ω ∆,
eω , O† ] = Cβ (ω),
[O
Lω
eω O† |Ψmd i = Gβ (ω).
hΨmd |O
Lω
But for high frequencies, ω ∆,
eω , O† ] = 0 + O ω , hΨmd |O
eω O† |Ψmd i = 0 + O ω .
[O
Lω
Lω
∆
∆
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
33 / 40
Outline
1
The Old Information Paradox
2
The Modern Information Paradox for the Eternal BH
3
State-Dependent Resolution of the Modern Information Paradox
4
ER=EPR
5
Significance of State-Dependence
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
34 / 40
Why is State-Dependence Surprising?
H
T = −eN
2
T =0
T
H ΨT
T = eN
2
|ΨT i
Canonical gravity suggests that relationally defined observables
are linear operators on the H-space.
Canonical gravity intuition valid for the little Hilbert space ⇒ no
infalling observer can detect state-dependence within EFT.
Violations of Born Rule ↔ conceptually defined local bulk
observables correspond to non-linear CFT operators when
considered on the full H-space
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
35 / 40
Failure of Canonical Intuition in TFD
CFT inner-product has a fat tail invisible to canonical gravity.
|hΨtfd |ΨT i| = e
−CT 2
2β 2
|hΨtfd |ΨT i| = O e
eω =
O
s
C
πβ 2
Z
,
−S
2
T 1
,
T 1
Tcut
−Tcut
OLω (Ti )PHΨT dTi ,
i
State-independence ⇔ Tcut → ∞; Prevented by fat-tail.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
36 / 40
Marolf-Polchinski Shells
Shell
Horizon
For every black hole with empty interior at late times, consider
configuration with a ultra-relativistic shell just inside the horizon.
Binding energy with BH cancels rest+kinetic energy of shell ⇒
increase in AdM energy is small.
So, Marolf-Polchinski claim that as many states with firewalls as
states with smooth interiors ⇒ mirror operators must be singular.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
37 / 40
Back-Reaction
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However, the M-P states have singularities in the past.
So, gravitational back-reaction important in out-of-time order
correlator
†
hΨ|UMP
φ(xout )φ(xin )UMP |Ψi = hΨ|φ(xout )φ(xin )|Ψi?
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
38 / 40
Other Examples of State-Dependence
The Ryu-Takayanagi formula, and other attempts to relate
geometry to entanglement are state-dependent.
1
hA(R)i = Sent (R)
4GN
On the LHS, A(R) is an operator in canonical gravity. But, easy to
show that
@X , s.t.hX i = Sent (R)
Similar comments hold for ER=EPR, and other relations involving
complexity and geometry etc.
This does not prove that geometric quantities are state
dependent, but is suggestive.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
39 / 40
Summary
Modern Information Paradox can be rephrased as a question
about the existence of CFT operators dual to bulk fields.
Paradox extends to the eternal black hole.
Can be resolved using a state-dependent map between boundary
and bulk fields.
This map can be written down precisely.
Leads naturally to ER=EPR.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
40 / 40
Appendix
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
41 / 40
More on back-reaction
Naive MP prediction is that
−βω
eω0 aω eiNω θ |Ψi =
hΨ|e−iNω θ a
e 2 eiθω
δ(ω − ω 0 )
1 − e−βω
Naively, one may also have thought that
−βω
eω0 aω eiHL T |Ψi
hΨ|e−iHL T a
= e
iωT
e 2
δ(ω − ω 0 )
1 − e−βω
But this is incorrect.
Phase operator is more non-trivial, but note that the naive
prediction is clearly suspect in the presence of back-reaction.
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
42 / 40
Properties of Mirror Modes
From analysis of large diffeomorphisms, we find
eω ]|ΨT i = ωhΨT |AR ω O
eω |ΨT i.
hΨT |AR [H, O
Demanding that two-pt function of φ agrees with effective field
theory
eω O
eω† |ΨT i = eβω hΨT |O
e† O
e
hΨT |O
ω ω |ΨT i
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
43 / 40
Long Time Average
Long Time Average ⇒ cross-terms drop out.
1
Tav
Z
Tav
−Tav
hΨT |X |ΨT idT =
β(E+F )
e− 2
+
Z Tav
1 −βE
e
hE, E|X |E, Ei
Z
X
hE, E|X |F , F i
sin (E − F )Tav
E,F
From a Laplace Transform
eω O
eω† |E, Ei = e−βω hE, E|O
eω† O
eω |E, Ei
hE, E|O
eω ]|E, Ei = ωhE, E|AR O
eω |E, Ei.
hE, E|AR [H, O
Suvrat Raju (ICTS-TIFR)
Information Paradox and BH Interior
Strings 2015
44 / 40
No Wormhole in Eigenstate Pairs
QUESTION: How can state-independent operators describe
smooth effective field theory in eigenstate pairs — which have no
entanglement — if they cannot do so in single eigenstates.
To sharpen this: assume that there is no “wormhole” in eigenstate
pairs.
hE, E|UL† φ(x1 ) . . . φ(xn )UL |E, Ei = hE, E|φ(x1 ) . . . φ(xn )|E, Ei, ∀UL
|ELi
Suvrat Raju (ICTS-TIFR)
|ER i
Information Paradox and BH Interior
Strings 2015
45 / 40
The “Occupancy” Paradox for the Eternal Black Hole
Now, we can set up a version of the AMPSS paradox.
eω O
e† |Ω, Ei ≈
hΩ, E|O
ω
=
1
eω O
e† )
TrR (e−βHR O
ω
Z (β)
1 −βω
−βω
e† O
e
e† O
e
e
Tr(e−βH O
hΩ, E|O
ω ω) ≈ e
ω ω |Ω, Ei?
Z (β)
Here, we use equivalence of microcanonical and canonical
ensembles, cyclicity of trace and commutator with Hamiltonian.
If we also use
then this suggests
Suvrat Raju (ICTS-TIFR)
eω , O
e† ]|Ω, Ei > 0
hΩ, E|[O
ω
eω O
e† |Ω, Ei < 0?
hΩ, E|O
ω
Information Paradox and BH Interior
Strings 2015
46 / 40