Download Remnants, Fuzzballs or Wormholes

Remnants, Fuzzballs or Wormholes ?
Samir D. Mathur
The Ohio State University
The black hole information paradox
General Relativity:
Black holes form
1
p (00 + 11)
2
Quantum mechanics:
entangled pairs are created
There is a problem near the
endpoint of evaporation
Remnant
A remnant would have an unbounded number of states for given mass and volume ….
baby Universes?
...
In string theory we believe we understand all the states at the planck scale …
The dual CFT does not allow an unbounded number of states for a given energy …
Thus the remnant solution to the information paradox does not look
correct for string theory …
Two options:
(1) Violate the no hair “theorem”, so we do not have the vacuum at the horizon
Fuzzballs
(2) Alter some quantum mechanical principle; e.g.
(a) Nonlocal identification of Hilbert spaces:
Wormholes (Maldacena+Susskind)
( Conflicts with AdS/CFT ? (SDM 2014) )
(b) Final state boundary condition (Horowitz+Maldacena)
(c) Nonlinear quantum mechanics (Papadodimas+Raju …)
(d) Computability limits in quantum theory (Hayden-Harlow)
(e) Others …
This talk:
(1) Summarize the fuzzball program
(2) How is the semiclassical approximation violated ?
(Use the Kraus-Wilczek-Parikh computation to get an exact cancellation)
(3) The idea of Fuzzball complementarity
The hole in the AMPS argument
(4) A problem with wormholes: Conflict with AdS/CFT?
The no hair “theorem” :
⇤
=0
= Ylm (✓, ) ˜(r)
If we found good functions
˜(r) , then we could populate these mode functions with quanta
to make quantum states of the form
â†l1 m1 â†l2 m2 . . . â†lk mk |0i
Such states would give the correct order of entropy
For
R
0  l .
lp
Sbek ⇠ (R/lp )2
l ⇠ R/lp we cannot do any easy calculation, but if we had found
normalizable
˜(r) for l = 0, 1, 2, 3, … then there would be no information paradox ….
At least the Hawking argument for
a paradox would have to be rethought …
1
p (00 + 11)
2
??
nR
S =: ln(1) = 0
String theory
⌥ ⌥
S = 2 2⇥ n1 n2
⌥
S = 2⇥ n1 n2 n3
⌥
S = 2⇥ n1 n2 n3 n4
weak coupling
(6)
(7)
(8)
(9)
n1 ⇤ n5 ⇤ n
1
4
⇤ n lp
1
2
??
horizon
⇤ n lp
⇤ n lp
M9,1 ⌅ M4,1 ⇥ K3 ⇥ S 1
singularity
A
⇤ n1 n5 J ⇤ S
4G
⌥
A
⇤
n1 n5Witten’s
⇤S
Rough4Ganalogy:
bubble of nothing (1982)
⇥ ⇥
2 n1 npwith an extra compact circle is unstable to
Minkowski
e2 space
tunneling into a ‘bubble of nothing’
Q1
r2
compact
Qp circle pinches off
1+ 2
r
1+
2
e
⇥ ⇥
2 n1 n5
not part of spacetime
i(t+y) ikz
strong coupling
NO !!
np
knk = n1 np
k
√
√
√
√
weak
coupling:S = 2π n5 ( n1 + n̄1 )( np +
e2⇤ 2 n1 np
“supertube”
S 1 → y y : (0, 2πR)√
E
low redshift
)
= 2π n5 ( √
⇧ ⇧C V̂ [l]
S = 2 2 n1lnp
!
KK(57)
monopole
n̄p )
(175)
(58)
strings
(59)
m11mp
(176)
y
:
(0,
2πR)
√
S
=
2π
n1 n5 np nkk
N = n1 n5
(177)
√
V̂
√ Cl V̂![l]
fluxes
S = 2π ndn
n̄p )
(60)
1 n5 nkk ( np +
√
√
LT = n1 L
∝ (n + 1) N = nn1 n5
(178)
≈ N
n+1
11np
√dt
E
) √
(61)
= 2π n1 n5 ( √
√
√
√
dn
m
m
1
redshift
∝ (n + 1)
n
nψ m +
nm
+φ1n] = ω gravity
≈p kkN
n
+1
ωR = [−l
mhigh
(179)
L − 2N−−
!
√
√
√
R
√
dt
R
S = 2π n1 n5 ( np + n̄p )( nkk + n̄kk )
(62)
1
gravity
m = nL + nR + 1, n = nωLR−=∼nRl [−l − 2 − mψ m + mφ n]
(180)
=
ω
(63)
R
Rp
M9,1 ⇤ M4,1 ⇥ T 4 ⇥ S 1
|λ − mψ n + mφ m| = 0, N ∼=n016 l
(181)
(64)
m = pnL + nR + 1, n = nL − nR
λD1D5
= 0, system
mψ = :−l, n M
=9,1
0, → M
N4,1=×0 T 4 × S 1
(182)
(65)
(Strominger-Vafa
96,
|λ
−
m
n
+
m
m|
=
0,
N =0
D5
P
4
1
ψ
φ
T ⇥S
Callan-Maldacena 96,
(183)
ωI = ωIgravity E/(2m
)
=
0.5
(66) 96)
kk
1
λ = 0, 16 mψ = −l, n = 0, Maldacena-Susskind
N =0
3
D ∼ G5 (n11n5 np ) ∼ RS
(81)
4
|0⟩
|ψ⟩
⟨0|ψ⟩
≈
0
(184)
E/(2m
)
=
1.2
(67)
gravity
T
S kk
P n1 n5
ω
=
ω
I
I
2 ′4 √
g
α
n
n
n
n1 , n2 , n3
n
(185)
α 1 5 p 1
lp
(82)
⟨0|ψ⟩ ≈ 0
Lz ∼ [4 ∼ N |0⟩
] 3 ∼|ψ⟩
Rs
(68)
V
R
1
1/n
(186)
T4
S
1 n2 n3
+
eS∆S
(83)
n1 , n2 , n3
n4
(69)
√
√
α
(n1 n5 ) lp S ∼ E ∼ SE E
(187)
(84)
1/n
n
n
e
(70)
1
2
3
!
α
n1 n5
knkn=
nn5 p
(188)
n̄(n
(85)
1 n1 n5n̄1
p ‘Effective
eS+∆S
(71)
1 n5 ) lp string’ with
√winding number
√D1
! !√
"√branes !
√ D5
√ branes
total
!
√E
√
′
′
′
′
+
2π2π
2(
n
+
n̄
)(
n
n̄
)
∼
E
E
∼
(86)
p
1
1
p
np = n1 S S=n=
=
n
,
kn
=
n
n
)n
+12π
( k n=k +
)
(72)
n5p 1n1 n5 np fkn
n1 n5n̄k(189)
n5
5n1 n5 np (1 − f k
1
!
√
√
√
3
√
√
n̄5 )( np1+ n̄p ) ∼ !
E2
(87)
S = 2π( n1 + n̄1 )( n5 +1 ∆E
1
V̂
S →y
′
′
′
′
(175)
(176)
(177)
(178)
(179)
(180)
(181)
(182)
(183)
(184)
(185)
(186)
(187)
(188)
λ = 0,
In the CFT, all extremal and
near extremal states are described by
different choices of
windings, spins, and
left/right excitations
( P, P̄ )
mψ = −l,
n = 0,
N =0
ωI = ωIgravity
|0⟩
|ψ⟩
n1 , n2 , n3
⟨0|ψ⟩ ≈ 0
n4
1/n1 n2 n3
n1 n5
n′p = n1
(n1 n5 )α lp
!
k mk = n1 n5
n5
!
′
n1 = n5 ,
k mk = n′p n′1
All 2-charge extremal states:
(Lunin+SDM 01,
Lunin+Maldacena+Maoz 02,
Skenderis+Taylor 07)
All strings equal, one excited by a pair of excitations
Wavefunction for supergravity quantum
in “cap”
(Maldacena+Strominger 98,
SDM+Saxena+Srivastava 03,
Giusto+Russo 13)
One string highly twisted, excited by several excitations,
String in cap (pp-wave limit)
(Lunin+SDM 02)
2-charge extremal + integer moded chiral algebra generators:
Jan . . . L
m
. . . | i2
charge
Brown-Henneaux: States should
lie at boundary of AdS
Sen: Calls these states at neck as “hair”
(SDM+Turton 11,
Lunin+SDM+Turton 12)
All states are captured by applying arbitrary fractional mode excitations to different cycles
J
a(i)
k
n
...L
(j)
l
m
. . . | i2
charge
All strings excited the same way, “filled fermi sea” of fractional excitations
⇣
J
+(1)
2
n
J
+(1)
4
n
...J
+(1)
2k
n
⌘⇣
⌘
+(2) +(2)
+(2)
J 2 J 4 . . . J 2k . . . | i
n
n
n
States not in Brown-Henneaux class of
“neck excitations”; thus they are give
structure at the location where the horizon
would have been
(Giusto+Lunin+SDM+Turton 12)
Different strings excited differently, integer chiral modes on strings, but not in the combination
to give neck modes
(Bena+Giusto+Russo +
Shigemori+Warner 15)
CF T
= ⇢thermal
⇢thermal
V
L
R
=
non-unitary !!
Hawking
unitary
here is no incoming wave, but we still have an outgoing wave carrying energy out to
These
frequencies
by
to
the transcendental
equation
=an⇢state
⇢state
V carrying energy
here
is instability
no incoming
wave, butare
wegiven
stillCF
have
outgoing
out to
T solutions
L
R wave
1
"2ν
TheseΓ(1
instability
by solutions
to 1the
transcendental equation
ν))Γ(
− ν) ! κfrequencies
Γ(ν)areΓ(given
=ξ −emission
−iνπ
2 (1 + |ζ| +
2 (1 + |ζ| − ξ − ν))
−e
(3.50)
1
1
! 2κ "2ν = Γ(−ν)
Γ(1
+
ν)
Γ(
(1
+
|ζ|
+
ξ
+
ν))Γ(
(1
+
|ζ|
−
ξ
+
ν))
−
−
−
Γ(ν)
2
2
−e−iνπ
=
(3.50)
1
1
Γ(1
+
ν)
2
Γ(−ν)
Γ(
(1
+
|ζ|
+
ξ
+
ν))Γ(
(1
+
|ζ|
−
ξ
+
ν))
oduce the solution to this equation,
B. In the large R limit
2 found in [14], in appendix
2
he
instability
frequencies
real to leading
oduce
the solution
to thisare
equation,
found inorder
[14], in appendix B. In ergoregion
the large R limit
he instability frequencies
are real to leading order
1
ω ≃ ωR = (−l − mψ m + mφ n − | − λ − mψ n + mφ m| − 2(N + 1))
(3.51)
R
1
ω ≃ ω = (−l − mψ m + mφ n − | − λ − mψ n + mφ m| − 2(N + 1))
(3.51)
N ≥ 0 is anR integer.
The imaginary part of the frequency is found by iterating to a
R
result
is
Nrder;
≥ 0the
is an
integer.
The imaginary part of the frequency is found by iterating to a
&
#
$
%l+1
rder; the result
is
2
2π
λ Q1 Q5
1
l+1+N
(3.52)
Cl+1 l+1+N +|ζ| Cl+1 &
ωI = # 2 $(ω 2 − 2 )
%l+1
2
R
[l!]
R
4R
2π
λ Q1 Q5
1
l+1+N
2
(3.52)
)
Cl+1 l+1+N +|ζ| Cl+1
(ω
−
ωI =
2
2
2
R
[l!]
R
4R
at ω > 0, so we have an exponentially growing perturbation. Our task will be to
I
Titchener
05,
Cardoso,Dias,Hobvedo,Myers
05,task
Chowdhury
07)
ce
from
the an
microscopic
computation.
at (3.51),(3.52)
ωI (Jejjala,Madden,Ross
> 0, so we
have
exponentially
growing perturbation. Our
will be +SDM
to
unitary
Large families of states (with perhaps the correct order of the entropy of black holes), have been constructed
by Bena+Warner + many others …
These have not been indentified with corresponidng CFT states … it is possible that some of them are lifted
slightly (order 1/M ?) by quantum effects …
In that case they contribute to the count of near extremal states rather than extremal states …
(Bena+Warber 2005 + …)
In principle there are two possibilities:
(1) (Fuzzball conjecture): No microstate will have a vacuum horizon
(2) Some states will be fuzzballs, but most have a traditional horizon
But if (2) is true, then what is the solution to the information puzzle?
One suggestion was “Large cumulative effect from small corrections”
There can be a small correction (perhaps due to instanton effects), slightly modifying the
state at
each emissionof this hair, but many of them were still not worried
cently,
a construction
not have, until recently, a construction of this hair, but many of them were still not wo
paradox. The reason was based on the following misconception. Suppose the
about Hawking’s paradox. The reason
was based on the following
misconception. Suppos
1
1
p1 (|0
|0
+
|1
|1
)
p
p
b
c
b
c
ce with
‘normal
physics’,
and
let
us
include
a
small
correction,
order
⇧ c1 |1⇥b |1⇥c )
(|0⇥
|0⇥
+
|1⇥
|1⇥
)
+
✏
(|0⇥
b
c and let
b uscinclude
k a2 small
b |0⇥
2horizon was a place with ‘normal2physics’,
correction, order
ch created
number
of pair.
pairs N
very large,
so itNmight
that so it might be
to thepair.
state The
of each
created
Theis number
of pairs
is verybelarge,
( ✏k ⌧ 1 )
f thesesuitable
small corrections
would
leadcorrections
to a situation
Sent
does decrease
choices of these
small
wouldwhere
lead to
a situation
where Sent does dec
pected
a normal
body.
Thisinof
creates
after N
steps
2N correction
the
manner
expected terms
of a normal
body.
not wrong
to think
might
cause
Sent to might
decrease.
A priori,
it isthat
not small
wrongcorrections
to think that
small
corrections
cause Sent to dec
1
⇥ (|0⌃
|1⌃b1is|1⌃⇥1c1 (|0⌃
). At
the
next
ngled
step ispair
c1 +step
b1 |0⌃
|1⌃b1step
|1⌃c1 we
). At the next ste
Suppose
thefirst
entangled
first
c1 +
b1 |0⌃
Firstpair
stepatofthe
emission
2 at the
2
e
can have the state
1
1
step
|
⌃
=
|0⌃c2b1+
|0⌃(1
⌃ Second
=
|0⌃b1of
|0⌃emission
[(1
+
)|0⌃
c1 [(1 +
1 )|0⌃
b2 |0⌃
c1
1
1 )|1⌃
c2 ]c2 + (1
b2 |0⌃
b2 |1⌃
2
2
⇥
+|1⌃b1 |1⌃c1 [(1 +
+|1⌃ |1⌃c1 [(1 +1 )|1⌃
b2 |0⌃
1 )|0⌃
c2 ]c2
b2 |1⌃
1 )|0⌃b2 |0⌃cb21 + (1
1 )|1⌃b2 |1⌃c2 ]
+ (1
(2.1)
b2 |1⌃c2 ]
1 )|1⌃
⇥
that step
the correction
at on
each
step can depend
on everything
in the
hole at all earlier s
rectionNote
at each
can depend
everything
in the hole
at all earlier
steps;
the
only
that
the
small:
| 1 | <⌅ ,2|N1 |correction
< resolve
. We have
2N corre
Thus
subleading
corrections
to
Hawkings’s
will
the ⌅
puzzle:
ent
is that
the requirement
correction
beis small:
| 1correction
| < leading
, | 1 | be
< order
. Wecomputation
have
M 2
terms
in
general
after
N
steps.
Since
N
⌅
(
) forblack
a 3+1
dimensional
black hole, it ap
M
2
m
after N steps. Since N ⌅ ( mp ) for a 3+1 dimensional
hole,
it appears
p
a priori
possibletofor
small
corrections
to decrease
pile up toafter
makethe
Sent
decrease
after
the halfway
(Maldacena
2000
or small
corrections
pile
up to
make Sent
halfway
point
entanglement
of evaporation.
entanglement
Hawking 2004)
In [?] it was proved, using strong subadditivity, that such small corrections cannot le
roved, using strong subadditivity,
such small corrections cannot lead to
?? that this
a decrease in Sent . AMPS invoked
argument in their analysis, so let us outline the ste
. AMPS invoked this argument in their analysis, so let us outline the steps in
[?]. Let {b1 , . . . bN } ⇥ {bi } be the quanta radiated in the first N steps, and {ci } their enta
But using strong subadditivity of entanglement entropy, it can be shown that this is not possible
SN +1 > SN + ln 2 + 2✏
(SDM 2009)
Entanglement keeps rising monotonically
entanglement
Hawking’s argument (1974)
Hawking’s ‘Theorem’:
If we assume
(i) The Niceness conditions N give local Hamiltonian evolution.
(ii) A traditional black hole (i.e. one with an information-free horizon) exists in the theory.
Then formation and evaporation of such a hole will lead to mixed states/remnants.
Corollary: If the state of Hawking radiation has to be a pure state with no entanglement with the rest
of the hole then the evolution of low energy modes at the horizon has to be altered by
order unity.
(SDM 2009)
What happens to a collapsing shell ?
What violates the semiclassical approximation?
First consider a toy model:
Consider a shell made of an ingoing gravitational wave with energy M
Let there be a large number N of massless scalars in the theory :
N
Radiated quanta
(M/mp )2
Collapsing shell
The shell evaporates to a collection of scalar quanta, without forming a horizon
We do not have a large number of massless scalars in string theory …
But we have
eSbek
massive fuzzball states
Emission rate without back reaction:
⇠e
!
T
⇠e
8⇡GM !
Emission rate for a spherical shell with back reaction (Kraus-Wilczek-Parikh)
⇠e
8⇡G(M
!
2 )!
This is a special case of the general expression
⇠e
Sbek
⇠ eSbek (M
!) Sbek (M )
This expression works for both massless and massive shells,
and can be used for any !
We set
!=M
⇠e
Sbek (M )
Overall rate of tunneling into fuzzballs :
⇠fe
Sbek (M )
⇠ eSbek e
Sbek
⇠1
Thus in a time of order unity the collapsing shell transitions to a linear combination of fuzzball
states.
Exp[Sbek ]
| i!
X
i=1
The data of the shell is encoded in the set
Ci |Fi i
{Ci }
Fuzzball complementarity
How is the AMPS argument against complementarity bypassed ?
Bena: If you fall towards the horizon,
you will crack you head and die …
So why did we not say right away that fuzzballs
behave like a “brick wall” or a “firewall” ?
Because there is another possibility :
“fuzzball complementarity”
To see the basic idea, consider AdS/CFT duality…
(Maldacena 97, Gubser+Klebanov+Polyakov 98, Witten 98)
This works because the open string excitations are not an arbitrary set of excitations …
The spectrum of excitations along the brane surface reproduces the spectrum of radial infall
In the fuzzball, the infalling graviton cannot go further inside; its energy excites the fuzzball surface
Could this spread of the excitations along the fuzzball surface map, approximately, to the dynamics of radial
infall ?
Conjecture of fuzzball complementarity: The dynamics of the fuzzball surface maps approximately
to the dynamics of infall into the interior of the traditional hole …
The word “approximately” is crucial here, since different fuzzball microstates have different
microstructure, and radiate differently … thus the E ⇠ T physics cannot be universal
T physics can be a universal “hydrodynamical excitation” of the fuzzball, and this is
But the E
conjectured to be dual to infall in the traditional hole
Traditional complementarity :
horizon
stretched horizon
Quantum state at the horizon is
the vacuum, since we have
found no hair in our theory
In the external observer’s frame,
the graviton is absorbed by a
stretched horizon and its
information returned to infinity
In the infalling observer’s frame,
the graviton just passes through
These two different descriptions are compatible since the two observers cannot communicate
before the infaller hits the singularity
AMPS rule out traditional complementarity, but fuzzball complementarity is quite different:
With fuzzballs, there are real degrees of freedom where the stretched horizon
would have been; these degrees of freedom differentiate microstates
AMPS: You will burn up if you try to fall through the horizon
Fuzzball: But we are not trying to fall through the horizon; there is no horizon to fall through.
We are hitting the fuzzball surface and obtaining the infall as a DUAL description of oscillations
of the fuzzball surface
AMPS: Our argument against complementarity should still work … you should get burnt by the
Hawking radiation near the fuzzball surface …
Fuzzball: There is something strange about one of your assumptions. You assume that the stretched horizon
does not respond before it is hit. But normal horizons do increase in size before they are hit.
The stretched horizon has one bit of data per planck area.
If the infalling bit lands on it, then you have
S=
A
A
+1>
4G
4G
Fuzzball :
fuzzballs tunnel out to this
radius due to the extra
energy brought in by the
graviton
Temperature is very low at the location where the graviton
is absorbed into the new fuzzball surface, if E
T …
Do not get burnt !
AMPS: The black hole is maximally entangled with radiation at infinity after its half-way evaporation point.
In this situation, we have argued that you cannot get complementarity
Fuzzball: When a quantum with E
T falls in, you create a large number of NEW states,
and these states are NOT entangled with infinity.
Fuzzball complementarity arises only in this limit, and is the dynamics of these new states
Nf
eS(M +E)
eS(M )+
= S(M ) =
Ni
e
eS(M )
S
=e
S
⇡e
E
T
1
(SDM+Turton 2013)
The notion of fuzzball complementarity provides a scenario where the following situation holds:
We fix our theory of the standard model, and also fix the infalling observer. If we now take the black hole
mass to be sufficiently large, we will have both of the following
(i) The evaporation is unitary
(ii) The infalling observer will feel no drama in the sense that the number of particles they intercept goes
to zero, as does the energy of each particle intercepted (as measured in the infalling frame).
Nevertheless, the density of energy quanta remains much larger than that predicted by the Hawking
state.
mp , but it needs to be checked if astrophysical
Caveat (Marolf) : The term “sufficiently large” means M
holes are large enough to satisfy this condition, in a theory with large gauge hierarchies.
Can wormholes be the correct solution ?
Maldacena 2000: Consider the eternal hole in AdS
There are two asymptotically AdS boundaries
Thus the spacetime is dual to two uncoupled CFTs, in an entangled state
?
|
X
=
P
i
e
Ei
2T
|Ei
L |Ei R
?
⌦
=
(Van Raamsdonk 2010)
Hawking radiation is entangled with its partners inside the hole …
1
p (00 + 11)
2
Entangled quanta create wormholes …
wormhole
This can change the physics at the horizon, in such a way that we have the local vacuum, and while being
able to get the information out …
(Maldacena+Susskind 2013)
But there is a problem with such a postulate … (SDM 2014)
(a) Suppose two entangled CFTs are indeed dual to the eternal hole
?
|
=
P
i
e
Ei
2T
|Ei
L |Ei R
(b) We can couple the CFTs to external spaces, from which we can throw in or extract
particles into the AdS
X
c
b
O
(c) We wish to maintain a smooth horizon, which means that we cannot stop particle creation there.
The eternal hole maintains its mass only because we throw in particles at the same rate that they are
emitted.
We choose to not throw in these particles, so that both horizons shrink in area as the holes radiates
X
c
O
b
We do not say how information is recovered, but we assume that the radiation is unitary due to some
mechanism …
(d) In the CFT, if we throw in something into the right CFT, we assume that its information will be
emitted only on the right (entanglement cannot transfer information) …
p
|
=
P
i
e
Ei
2T
|Ei
L |Ei R
(e) In the gravity dual, the particle send in from the right, can cross the central line,
though it cannot emerge from the left horizon
p
We ask: which side will the information emerge ?
(f) We may say that left movers send their information out to the right, but we can collide particles and
confuse what is right moving and what is left moving …
p
Key point: Even though classical particles cannot traverse from right to left, Hawking radiation
“tunnels” out of the horizon, and so we do have to consider what is right and left …
If the dual of two entangled CFTs creates a connected manifold with a “forward wedge”,
then there seems to be no obvious way to avoid information transfer from right to left through
the tunneling process of Hawking radiation
(g) Conclusion: The dual of two entangles CFTs is NOT a connected spacetime …
NO !
|
|
=
=
P
P
ie
Ei
2T
|Ei
L |Ei R
X
ie
Ei
2T
|Ei
⌦
Fuzzballs
L |Ei R
(h) But what spacetime do we generate if
we start with the wormhole?
wormhole
If we get the eternal hole spacetime, then it is dual to two entangled CFTs, and the problem starts …
Conjecture: If the neck is narrower than r = 2M then the spacetime slice is unstable to tunneling
quickly into an entangled pair of fuzzballs
r < 2M
r = 2M
Thus the eternal hole spacetime does not exist, and we avoid the contradiction with AdS/CFT duality …
But then we also lose the wormhole solution to the information paradox …
Summary
(a) The small corrections theorem
SN +1 > SN + ln 2 + 2✏
tells us that there are three categories of solutions to the information paradox:
...
(1) Remnants
(2) Fuzzballs
(3) Wormholes (or other new quantum effects)
(b) If we trust AdS/CFT, then we cannot get remnants …
We have also argued that there is a problem with AdS/CFT and the wormhole solution
p
(c) The fuzzball construction has captured many corners of the space of all
states …
The small corrections theorem then suggests that all states are fuzzballs …
(d) The conjecture of fuzzball complementarity says that we can obtain the traditional black hole as an
effective geometry for E
T processes …
Conjecture of fuzzball complementarity: The dynamics of the fuzzball surface maps approximately
to the dynamics of infall into the interior of the traditional hole …
(e) The AMPS argument has a hole so that it does not rule out fuzzball complementarity …
(SDM+Turton 2013)
(1) They do not let the stretched horizon respond before the quantum hits it …
(2) They do not consider the approximation E
T to obtain a complementarity picture
Nf
eS(M +E)
eS(M )+
= S(M ) =
Ni
e
eS(M )
S
=e
S
E
⇡ eT
1
(Most of the newly created degrees of freedom are unentangled with infinity)
(f) The underlying theme of black hole physics seems to be that the large number Exp[S] of fuzzball states
violates the semiclassical approximation …
The path integral measure is comparable to the classical action …
⇠fe
Sbek (M )
⇠ eSbek e
Sbek
⇠1