The Fuzzball Paradigm

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The fuzzball paradigm

Lecture III

Samir D. Mathur

The Ohio State University Previous lecture total ,total n1n5 ,total n1n5 ,total n1n5 total n = (J (2n 2)) (J (2n 4)) . . . (J 2 ) 1 (5) | ⇧ | ⇧ A = S = 2⇥⌥n n n 4G 1 2 3 1 1 2 E = + = nR nR nR 2 E = nR The sizeS = ofln(1) bound= 0 states grows with the number of(6 branes)

S = 2⌥2⇥⌥n1n2 (7)

S = 2⇥⌥n1n2n3 (8)

S = 2⇥⌥n1n2n3n4 (9)

n1 n5 n ⇤ 1 ⇤ n 4 l ⇤ p 1 n 2 lp D R ⇤ ⇠ H n l ⇤ p Horizon will M M K3 S1 not form 9,1 ⌅ 4,1 ⇥ ⇥ A n n J S 4G ⇤ 1 5 ⇤ FractionationA generates low tension objects that can stretch far ⌥n n S 4G ⇤ 1 5 ⇤

e2⇥2⇥n1np

Q 1 + 1 r2 Q 1 + p r2

e2⇥2⇥n1n5

i(t+y) ikz w = e w˜(r, , ⇤) (10) (2) i(t+y) ikz ˜(2) BMN = e BMN (r, , ⇤) , (11)

2 1 i = dx[ F a F µ⇥a + ⇥¯⌅⇥ + . . .] L 4 µ⇥ 2 ⇥ 2n 2(n n ) P = p = 1 p L LT

2k p = LT

knk = n1np 2-charge NS1-P extremal hole k

1 e2⇤4 2n1np IIB: M9,1 M4,1 S T ! ⇥ ⇥

S = 2⇤2⇤n1np

L L = n L S E ⇤SE⇤EE ⇤E⇤E T 1 (81) (81) ⇥ ⇥ ⇥ ⇥ n1 n¯1 n1 np n¯1 n¯p np n¯p (82) (82) L

S = 2⇥⇤2(S⇤=n12⇥+⇤⇤2(n¯1⇤)(n⇤1n+p +⇤n¯1)(n¯p⇤) np⇤+E⇤nE¯p) E⇤E⇤E E (83) (83) ⇥ ⇥⇥ ⇥ 3 3 S = 2⇥(⇤Sn1=+2⇤⇥n¯(⇤1)(n⇤1 n+5 ⇤+n¯⇤1n¯)(5⇤⇤)(⇤n5n+p +⇤n¯5⇤n¯)(p)⇤npE+2 n¯p) E 2 (84) (84) ⇥ ⇥covering space 2 2 S = 2⇥(⇤Sn1=+2⇤⇥(n¯⇤1)(n⇤1 +n2⇤+n¯⇤1)(n¯2⇤)(n⇤2n+3 +⇤⇤n¯2n¯)(3)(⇤⇤⇤nn3 4++⇤⇤n¯n¯34)()⇤⇤nE4 + ⇤n¯4)(85)E (85) ⇥ ⇥ A NS1 stringN carryingN the momentum P in the form of travelling waves N N S = AN S(⇤=nAi + ⇤n¯i()⇤nE+2 ⇤n¯ ) E 2 (86) (86) N ⇥ i i ⇥ i=1 i=1 ⇥ ⇥ 2 2 2 2 2 2 ds = dt ds+ = aidt(t)dx+idxi a (t)dx dx (87) (87) i i i i i S = 2⇥( n + ⇤n¯ )( n + ⇤n¯ )( n + ⇤n¯ )( n + ⇤n¯ ) (88) ⇤S 1= 2⇥(⇤1 n⇤1 +2 ⇤n¯1)(2⇤n⇤2 +3 ⇤n¯2)(3 ⇤⇤n3 4+ ⇤n¯34)(⇤n4 + ⇤n¯4) (88) S = 2⇥(⇤n + ⇤n¯ )(⇤n + ⇤n¯ )(⇤n + ⇤n¯ ) (89) S1= 2⇥(1⇤n1 +2 ⇤n¯1)(2⇤n2 +3 ⇤n¯2)(3 ⇤n3 + ⇤n¯3) (89)

n4 = n¯4 1 (90) n4 = n¯4 1 (90) Smicro = 2⇥⇤2⇤n1np = Sbek (91) Smicro = 2⇥⇤2⇤n1np = Sbek (91)

Smicro = 2⇥⇤n1n5np = Sbek (92) Smicro = 2⇥⇤n1n5np = Sbek 1 (92) Smicro = 2⇥⇤n1n5npnkk = Sbek (93) Smicro = 2⇥⇤n1n5npnkk = Sbek (93) Smicro = 2⇥⇤n1n5(⇤np + n¯p) = Sbek (94) Smicro = 2⇥⇤n1n5(⇤np + n¯p) = Sbek (94)

Smicro = 2⇥⇤n5(⇤n1 + ⇤n¯1)(⇤⇤np + n¯p) = Sbek (95) Smicro = 2⇥⇤n5(⇤n1 + ⇤n¯1)(⇤⇤np + n¯p) = Sbek (95) Smicro = 2⇥(⇤n5 + ⇤n¯5)(⇤n1 + ⇤n¯1)(⇤⇤np + n¯p) (96) Smicro = 2⇥(⇤n5 + ⇤n¯5)(⇤n1 + ⇤n¯1)(⇤⇤np + n¯p) (96) Smicro = 2⇥(⇤n1 + ⇤n¯1)(⇤n2 + ⇤n¯2)(⇤n3 + ⇤n¯3)(⇤⇤n4 + ⇤n¯4) (97) Smicro = 2⇥(⇤n1 + ⇤n¯1)(⇤n2 + ⇤n¯2)(⇤n3 + ⇤n¯3)(⇤⇤n4 + ⇤n¯4) (97) nˆi = ni n¯i (98) nˆi = ni n¯i (98) E = (ni + n¯i) mi (99) E = (ni + n¯i) mi (99) i i N S = C (⇤n + ⇤Nn¯ ) (100) i i ⇤ i=1 S = C (⇤ni + n¯i) (100) ⇥ i=1 ⇥i Pa = (ni + n¯i) pa i (101) i Pa = (ni + n¯i) pa (101) i R [n1, n5, np, , g, LS1 , VT 4 ] (102) R [n1, n5, np, , g, LS1 , VT 4 ] (102) Rs (103) ⇥ Rs (103) 2⇥np ⇥ P = 2⇥n (104) L P = p (104) L

6 6 1 i = dx[ F a F µ⇥a + ⇥¯⌅⇥ + . . .] L 4 µ⇥ 2 ⇥ 1 i = dx[ F a F µ⇥a + ⇥¯⌅⇥ + . . .] L 4 µ⇥ 2 2np 2(n1np) ⇥ P = = L LT

2np 2(n1np) P = = 1 2k i L LT = dx[ Fpa=F µ⇥a + ⇥¯⌅⇥ + . . .] L 4 µ⇥ L 2 ⇥ T 2k p = LT 2np 2(n1np) P = knk== n1np L LT k knk = n1np 2k k e2p⇤=2n1np LT e2⇤2n1np

S = k2nk⇤=2⇤nn11nnpp k S = 2⇤2⇤n1np 2⇤2n1np eLT = n1L

LT = n1L

S = 2⇤L2⇤n1np

LT = n1L

‘Naive An ‘actual geometry’ geometry’ ‘singular horizon’ 1 1

1 We do not ﬁnd a horizon

The states are not spherically symmetric

Generic states will have structure at the string scale, but we can estimate the size of the region over which the metric deformations are nontrivial

A pn1np Smicro G ⇠ ⇠

D R ⇠ h Horizon

r =0 Simple fuzzball More complicated Traditional picture state fuzzball state

Modes evolve like in a piece of coal, so there is no information problem D1-D5-P states

Avery, Balasubramanian, Bena, Carson, Chowdhury, de Boer, Gimon, Giusto, Guo, Hampton, Keski-Vakkuri, Levi, Lunin, Maldacena, Maoz, Martinec, Niehoff, Park, Peet, Potvin, Puhm, Ross, Ruef, Russo, Saxena, Shigemori, Simon, Skenderis, Srivastava, Taylor, Turton, Vasilakis, Virmani, Warner ... NS1 D5

P D1

NS1-P bound state D1-D5 bound state

n 10 n 50 fractional We can join up these fractional D1 branes strings in different ways r = 0 (129) rr==02M (129) (130) r = 2M (130) t (131) t (131) V V V V V (132) V ⇥ G 2G (132) ⇥ G 2G 2 E = 2 (133) E = n n R (133) NS1-P D1-D5 n1n51R5 1 1 l l 1 1 (a) n n0 hh== ++( ( m)m) 2mn2mn 1 5(134) (134) k k 2 2 k k ! np n0 ¯ 1 1 l l 1 1 1 hh¯== + +( ( m¯ )m¯ ) ! (135) (135) k k 2 2 k k F✓ (y ct) = (136) NS1-P F✓ (y ct) = i (136) n1 n2 state ( 1) ( 2) . . . 0 (b) A mode ↵ (1 37)k maps to i1 n1 i2 n2 | ⇤ ( 1) ( 2) . . . 0 a loop with winding (1k 37) | ⇤ and spin in the direction i

knk = n1np k D1-D5 X state D1-D5:

knk = n10 n50 Xk

8 Dualize the geometries

NS1-P D1-D5

Thus these solutions have no source, just a novel topology

NS1 string source KK monopole tube (Lunin+SDM 01)

The family of D1D5 geometries can be quantized (Lunin, Maldacena, Maoz 03) and their number gives the correct entropy (Rhychkov 06) Elementary objects in IIB string theory

graviton string (NS1) NS 5-brane

Kaluza-Klein D1, D3, D5, D7, D9 monopole branes

Any one of these objects can be mapped to any other by S,T dualities, which are exact symmetries of the theory General states:

D1-D5-P extremal D1-D5-P near-extremal

+ + n1n5 A supergravity [J 12 ]( 0 R) | i quantum localized (SDM,Saxena, in the 'cap' Srivastava 03) + + + n1n5 [J 2n+1 ...J 1 0 R] | i (Giusto, SDM, Saxena 04)

string (using the pp-wave formalism) (Gava+Narian 02. Lunin+SDM 03) ergoregion

+ + + + N ¯ ¯ k = J nR ...J 1J nL ...J 1[k] 0 R | | (Jejalla, Madden, Ross Titchener ’05)

KK monopoles General program pioneered by Bena+Warner: ﬁnd large families of solutions.

Several different innovative spheres techniques used carrying ﬂuxes A toy model Start with the 3+1 dimensional Schwarzschild metric

r dr2 ds2 = (1 0 )dt2 + + r2(d✓2 +sin2 ✓d2) r 1 r0 r Make the analytic continuation t i ⌧ !

r dr2 ds2 =(1 0 )d⌧ 2 + + r2(d✓2 +sin2 ✓d2) r 1 r0 r Let the ⌧ direction be a circle 0 ⌧ < 4⇡r  0 Not part of spacetime ✓,

r ⌧

This gives the 4-d Euclidean Schwarzschild geometry Dimensional reduction

4+1 dim

3+1 dim

Dimensional reduction of the ⌧ circle 2 p3 gives a scalar in the 3+1 spacetime g⌧⌧ = e p3 r = ln(1 0 ) 2 r The 3+1 metric is deﬁned as

1 E p3 gµ⌫ = e gµ⌫

2 r0 1 dr r0 1 2 2 2 2 2 2 2 2 dsE = (1 ) dt + r 1 + r (1 ) (d✓ +sin ✓d ) r (1 0 ) 2 r r The action is 1 1 S = d4xp g R ,µ 16⇡G E 2 ,µ Z ✓ ◆

✓,

r ⌧ The stress tensor is the standard one for a scalar ﬁeld 1 T = gE , µ⌫ ,µ ,⌫ 2 µ⌫ , which turns out to be

T µ = diag ⇢,p ,p ,p = diag f,f, f, f ⌫ { r ✓ } { } 2 3r0 f = r 3 8r4(1 0 ) 2 r

(a) We see that the energy density and radial pressure are positive. The tangential pressures are negative

(b) All these quantities diverge as we reach the tip of the cigar.

g never changes sign, so there is no horizon (c) tt (SDM 16) So what happened to Buchdahl’s theorem? p =0 R =2M pressure will diverge somewhere 9 if radius of ball is R< M 4

Because the radial pressure diverged, Buchdahl would have discarded this solution as unphysical.

But we see that the problem is with the dimensional reduction: the full spacetime is completely smooth Hawking radiation from fuzzballs Recall our difﬁculty with the information paradox …

(weak coupling) Radiates like a normal body; no problem of growing entanglement (Strong coupling) Entangled pairs; Entanglement keeps growing

The average rate of radiation is the same in both cases, but the detailed mechanism of radiation is very different (a) We take a simple CFT state at weak coupling

+ + + + N ¯ ¯ k = J nR ...J 1J nL ...J 1[k] 0 R | |

(b) We know its metric at strong coupling ... it does not have a horizon as naively expected, but a ‘cap’

ergoregion where

˜ 2 2 2 2 2 2 Hi = f + M sinh δi, f = r + a1 sin θ + a2 cos θ, (2.4)

The D1 and D5 charges of the solution produce a RR 2-form gauge ﬁeld given by [6]

M cos2 θ C2 = [(a2c1s5cp a1s1c5sp)dt + (a1s1c5cp a2c1s5sp)dy] dψ H˜1 − − ∧ M sin2 θ + [(a1c1s5cp a2s1c5sp)dt + (a2s1c5cp a1c1s5sp)dy] dφ H˜1 − − ∧ Ms1c1 Ms5c5 2 2 2 2 dt dy (r + a2 + Ms1) cos θdψ dφ. (2.5) − H˜1 ∧ − H˜1 ∧ The angular momenta are given by πM J = (a c c c a s s s ) (2.6) ψ (5) 1 1 5 p 2 1 5 p 2 The non-extremal microstate geometries: Review −4G − πM J = (a2c1c5cp a1s1s5sp) (2.7) In this section we recall the microstate geometries that we wish to study, aφnd explain−h4owG(a5) − suitable limit can be taken in which the physics can be described by a dual CFT. and the mass is given by 2.1 General nonextremal geometries πM 2 2 2 3 MADM = (s + s + s + ) (2.8) Let us recall the setting for the geometries of [13]. Take type IIB string theory, and compact4ifGy (5) 1 5 p 2 10-dimensional spacetime as IMt is coMnvenTie4ntSt1o define (2.1) 9,1 → 4,1 × × The volume of T 4 is (2π)4V and the length of S1 is (2π)R. The T 4 is described by coordinates 1 zi and the S by a coordinate y. The noncompactQM14,1=is Mdescsriinbehd δb1y caotsimheδc1o,ordQina5te=t, Ma sinh δ5 cosh δ5, Qp = M sinh δp cosh δp (2.9) radial coordinate r, and angular S3 coordinates θ, ψ, φ. The solution will have angular momenta along ψ, φ, captured by two parameEtexrstrae1,ma2a. lTshoe lsuoltuitoionns wairllecarreryacthhreeedkinndstohfechlaimrgeist. We have n units of D1 charge along S1, n units of D5 charge wrapped on T 4 S1, and n 1 5 × p units of momentum charge (P) along S1. These charges will be described in the solution by whetrheree parameters δ , δ , δ . We will use the abbreviations M 0, δ , Q fixed (2.10) 1 5 p → i → ∞ i ˜ si = sinh δi, ci =2cosh δi, (i = 1,25, p) 2 2 2 (2.2) Hi = f +wMhesrienuhpoδni,wefg=et rthe+BaP1 Ssinrelθat+iona2 cos θ, (2.4) The metrics are in general non-extremal, so the mass of the system is more than the minimum needed to carry these charges. The non-extremality is captured by a mass parameter M. π The D1 and D5 charges of the solution produce a RR 2-forMm gauge =field give[Qn 1b+y [Q6]5 + Q5] (2.11) With these preliminaries, we can write down the solutions of interest. Theexgterneemraallnon- 4G(5) extremal 3-charge metrics with rotation were given in [23] M cos2 θ f The inteMger charges of the solution are related to the Qi through Cd2s2 = (dt2 [(ady22c)1+s5cp a(s1sd1yc5cspd)t)d2t + (a1s1c5cp a2c1s5sp)dy] dψ − ˜H˜˜ − ˜ −˜ p − p − ∧ H1H15 H1H5 3 2 2 2 gα′ !M sin θ r d!r 2 + H˜1H˜5 + dθ Q1 = n1 (2.12) + (r2 +[(aa2)1(rc21+s5ac2)p Mar2s1c5sp)dt + (a2s1c5cp a1c1s5sp)Vdy] dφ " H˜# 1 2 −− $ − ∧ 1 (H˜ + H˜ f) cos2 θ Q5 = gα′n5 (2.13) + H˜ H˜ (a2 a2) 1 5 − cos2 θdψ2 Ms1 1c5 1− 2 − 1 Ms5˜c5˜ 2 4 % H1H5 2 & 2 2 2 g α " dt dy (r + a2 + Ms1) cos θdψ dφ. ′ (2.5) − H˜ ∧ −˜ H˜˜ 2 Qp∧ = 2 np (2.14) ˜ 1˜ 2 2 (H1 +!H51 f) sin θ 2 2 V R + H1H5 + (a2 a1) − sin θdφ % − H˜ H˜ & The angular momen"ta are given by 1 5 M 2 !2 2 + (a1 cos2.θ2dψ + Ca2 soinnθsdtφr)ucting regular microstate geometries H˜1H˜5 (Jejalla, Madden, Ross 2 πM Titchener ’05) 2!M cos θ TJhe s=olutions (2(.3a) cincgceneraal shasvesh)orizons and singularities.(2O.6n)e can take careful limits of + [(a1c1cψ5cp a2s1s5sp)dt(5+)(a2s11s15cp5 pa1c1c5s2p)d1y]d5ψ p H˜1H˜5 the p−aram−e4tGers in the s−olu−tion and find solutions which have no horizons or singularities. In 2M sin2 θ πM + ! [(a c[2c4c] reagsulsasr )2d-tc+h(aarsges cextraecmcasl )gdeyo]dmφ etries were found while in [6, 7] regular 3-charge extremal ˜ ˜ 2 1Jφ5 p −=1 1 5 p (a1 21c51cp5−cp2 1 5a1ps1s5sp) (2.7) H1H5 −4G(5) − 4 !H˜ + 1 dz2 (2.3) and the mass is given˜ by i 5 'H5 i=1 ( πM 2 2 2 3 where MADM = (s + s + s + ) (2.8) 4G(5) 1 5 p 2 ˜ 2 4 2 2 2 2 2 It is conveniHeni t=tof +deMfinseinh δi, f = r + a1 sin θ + a2 cos θ, (2.4)

The D1 and D5 charges of the solution produce a RR 2-form gauge field given by [6] Q1 = M sinh δ1 cosh δ1, Q5 = M sinh δ5 cosh δ5, Qp = M sinh δp cosh δp (2.9) M cos2 θ ExCtr2em=al solutions [a(rae2cr1esa5cchped ain1st1hce5slpim)ditt+ (a1s1c5cp a2c1s5sp)dy] dψ H˜1 − − ∧ 2 M sin θ M 0, δi , Qi fixed (2.10) + [(a1c1s5cp a2s→1c5sp)dt +→(a∞2s1c5cp a1c1s5sp)dy] dφ H˜1 − − ∧ whereupon we get the BPS relation Ms1c1 Ms5c5 2 2 2 2 dt dy (r + a2 + Ms1) cos θdψ dφ. (2.5) − H˜1 ∧ − H˜1 π ∧ M = [Q + Q + Q ] (2.11) extremal (5) 1 5 5 The angular momenta are given by 4G The integer charges of the solutπioMn are related to the Qi through Jψ = (a1c1c5cp a2s1s5sp) (2.6) −4G(5) − 3 gα′ πM Q1 = n1 (2.12) Jφ = (a2c1c5cp a1Vs1s5sp) (2.7) −4G(5) − Q5 = gα′n5 (2.13) and the mass is given by 2 4 g α′ πM Q2 p =2 2 3np (2.14) MADM = (s + s + s +2 ) (2.8) 4G(5) 1 5 Vp R 2 It is con2v.e2nienCt otondstefirnuecting regular microstate geometries

TheQs1ol=utMionssin(h2.δ31)coinshgδe1n, erQal5 h=avMe shionrhizδo5ncsosahnδd5,sinQgpu=larMitiseisn.hOδpnceoschanδp take c(a2r.e9f)ul limits of the parameters in the solution and ﬁnd solutions which have no horizons or singularities. In Extremal solutions are reached in the limit [24] regular 2-charge extremal geometries were found while in [6, 7] regular 3-charge extremal M 0, δ , Q ﬁxed (2.10) → i → ∞ i 5 whereupon we get the BPS relation π M = [Q1 + Q5 + Q5] (2.11) extremal 4G(5)

The integer charges of the solution are related to the Qi through

gα 3 Q = ′ n (2.12) 1 V 1 Q5 = gα′n5 (2.13) g2α 4 Q = ′ n (2.14) p V R2 p

2.2 Constructing regular microstate geometries The solutions (2.3) in general have horizons and singularities. One can take careful limits of the parameters in the solution and ﬁnd solutions which have no horizons or singularities. In [24] regular 2-charge extremal geometries were found while in [6, 7] regular 3-charge extremal

5 iωt Ψ = ψ(x)e− (201)

1 µ iωt L = ∂µφ∂ φ (202) Ψ = ψ(x)e− 2 (201) 1 τ (203) L = ∂ φ∂µφ (202) 2 µ 1 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) τ | ⟩ √2 | ⟩ ⊗ | (⟩203) | ⟩ ⊗ | ⟩ 1 GMm GM ψ 1 = (1.1 0 b1 0 c1 + 0.9 E1 b=1 m1c2c1 ) E = mc2 (204) E 0 r (205) | ⟩ √2 | ⟩ ⊗ | ⟩ | ⟩ ⊗ | ⟩ − r ∼ ∼ c2 GMm GM E = mc2 E = mc2 E 0 r (205) Ψ2 = [ 0 0 + 1 1 ] − r ∼ ∼ |c ⟩ | ⟩b1 | ⟩c1 | ⟩b1 | ⟩c1 [ 0 b2 0 c2 + 1 b2 1 c2 ] Ψ = [ 0 0 + 1 1 ] ⊗ | ⟩ | ⟩ | ⟩ | ⟩ | ⟩ | ⟩b1 | ⟩c1 | ⟩b1 | ⟩c1 . . . [ 0 b2 0 c2 + 1 b2 1 c2 ] ⊗ | ⟩ | ⟩ | ⟩ | ⟩ [ 0 b1 0 c1 + 1 b1 1 c1 ] (206) . . . ⊗ | ⟩ | ⟩ | ⟩ | ⟩ iθ iθ [ 0 0 + 1 1 ] e (20e6−) (207) ⊗ | ⟩b1 | ⟩c1 | ⟩b1 | ⟩c1 iθ iθ c, !, G (208) e e− (207) ! A c, , G √n1np(208S)micro (209) G ∼ ∼ A √n1np Smicro (209) G ∼ ∼ R R + R2 (210)

2 A R R + R S = = 4π√(2n10n) = S (211) bw 2G 1 2 micro A 2 Sbw =As in any= 4statisticalπ√n1n 2system,= Sm ieachcro microstate radiatesK a 3little(T 2differently11) (212) 2G × 2 K3 T S = 2√2π(√21n21)n5 (213) × S = 2√2π√n n AdS S(3213)T 4 (214) 1 5 3 × × 3 4 1 AdS3 S T (n n(2)164l) (215) × × 1 5 p 1 ∼ 6 (n1n5) lp Γ = V(2ρ15)ρ (216) ∼ CF T L R ΓCF T = V ρL ρR ΓCF T = V(2ρ¯1L6)ρ¯R (217)

ΓCF T = V ρ¯L ρ¯R (217) Emission Occupation numbers Occupation numbers vertex of left, right excitations for this particular Bose, Fermi distributions microstate for generic state

Emission from the special microstate is peaked at deﬁnite frequencies and grows exponentially, like a laser .....

11 11 iωt Ψ = ψ(x)e− (201) 1 L = ∂ φ∂µφ (202) 2 µ τ (203) 1 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) | ⟩ √2 | ⟩ ⊗ | ⟩ | ⟩ ⊗ | ⟩ GMm GM E = mc2 E = mc2 E 0 r (205) − r ∼ ∼ c2

Ψ = [ 0 0 + 1 1 ] | ⟩ | ⟩b1 | ⟩c1 | ⟩b1 | ⟩c1 [ 0 0 + 1 1 ] ⊗ | ⟩b2 | ⟩c2 | ⟩b2 | ⟩c2 . . . [ 0 0 + 1 1 ] (206) ⊗ | ⟩b1 | ⟩c1 | ⟩b1 | ⟩c1 iθ iθ e e− (207) c, !, G (208) A √n1np Smicro (209) G ∼ ∼

R R + R2 (210) A S = = 4π√n n = S (211) bw 2G 1 2 micro K3 T 2 (212) where there is no inco×ming wave, but we still have an outgoing wave carrying energy out to infinity. TheSse=ins2ta√bi2liπty√frneq1une5ncies are given by solutions to the tran(s2ce1n3d)ental equation 2ν 1 1 iνRadiationπ Γ(1 ν )from3κ the 4specialΓ(ν) microstate’Γ( 2 (1 + ζ + ξ ν))Γ( 2 (1 + ζ ξ ν)) e− AdS3− S T= | | − | (| 2−14−) (3.50) − Γ(1 +×ν) 2× Γ( ν) Γ( 1 (1 + ζ + ξ + ν))Γ( 1 (1 + ζ ξ + ν)) ! 1" − 2 | | 2 | | − 6 We reproduce the(snol1unti5o)n tlop this equation, found in [14], in appendix(B2.1I5n)the large R limit (A) The emitted frequencies are peaked at (2.27) the insta∼bility frequencies are real to leading order ΓCF T = V ρL ρR (216) 1 ω ωR = ( l mψm + mφn λ mψn + mφm 2(N + 1)) (3.51) ΓC≃F T = VRρ¯L− ρ¯−R − | − − | −(217) where N 0 is an integer. The imaginary part of the frequency is found by iterating to a ≥ higher order(B); th Emissione result rateis grows exponentially with time because after n 1 R de-excitedS l, stringsmφ, m haveψ beenλ created,2 thel +probability1 for creating (218) 1 2π 2 λ Q1Q5 l+1+N l+1+N+ ζ theω next= one is Bose( enhancedω )by (n+1) C p | |C (3.52) CF T graviIty R [l!]2CF T − R2 gr4aRvi2ty aˆ† n l+=1 n +1nl++11 ωR = ωR # ωI $ = ωI % | i (2|19) &i gravity gravity Note thaωt =ωI ω> 0, so w+e ihωave an exponentially growing perturbation(.22O0u)r task will be to reproduce (3.51)R,(3.52) from tIhe microscopic computation. CF T 1 ω = [ l 2 mψm + mφn] (221) 4 R The RM−icro−sco−pic Model: the D1-D5 CFT m = n = n + R +Emission1, n rate= n grows nas Exp[ωCF T t] (222) LIn this section we discuss thLe−CFTRduals of theI geometries of [13]. Recall that we are working with IIB string theory compactified to M S1 T 4. The S1 is parameterized by a coordinate 4,1 × × y with 11 0 y < 2πR (4.53) ≤ 4 The T is described by 4 coordinates z1, z2, z3, z4. Let the M4,1 be spanned by t, x1, x2, x3, x4. We have n D1 branes on S1, and n D5 branes on S1 T 4. The bound state of these branes 1 5 × is described by a 1+1 dimensional sigma model, with base space (y, t) and target space a 4 n1n5 4 deformation of the orbifold (T ) /Sn1n5 (the symmetric product of n1n5 copies of T ). The CFT has = 4 supersymmetry, and a moduli space which preserves this supersymmetry. It N is conjectured that in this moduli space we have an ‘orbifold point’ where the target space is 4 n1n5 just the orbifold (T ) /Sn1n5 [28]. The CFT with target space just one copy of T 4 is described by 4 real bosons X1, X2, X3, 4 1 2 3 4 X (which arise from the 4 directions z1, z2, z3, z4), 4 real left moving fermions ψ , ψ , ψ , ψ and 4 real right moving fermions ψ¯1, ψ¯2, ψ¯3, ψ¯4. The central charge is c = 6. The complete 4 n1n5 theory with target space (T ) /Sn1n5 has n1n5 copies of this c = 6 CFT, with states that are symmetrized between the n1n5 copies. The orbifolding also generates ‘twist’ sectors, which are created by twist operators σk. A detailed construction of the twist operators is given in [19, 20], but we summarize here the properties that will be relevant to us. The twist operator of order k links together k copies of the c = 6 CFT so that the Xi, ψi, ψ¯i act as free fields living on a circle of length k(2πR). Thus we end up with a c = 6 CFT on a circle of length k(2πR). We term each separate c = 6 CFT a component string. Thus if we are in the completely untwisted sector, then we have n1n5 component strings, each giving a c = 6 CFT living on a circle of length 2πR. If we twist k of these component strings together by a twist operator, then they turn into one component string of length k(2πR). In a generic CFT state there will be component strings of many different twist orders ki with i ki = n1n5. ' 10 The gravity solution

spinning star

light cones tilt so much that every object must rotate Ergoregions produce particle pairs

A quantum rotating opposite to the ergoregion slows down rotation, and so decreases energy: Net negative energy as seen from inﬁnity

Ergoregion instability: the produced waves grow exponentially Smicro = 2π√n1n5np (152) A SmicroS=bek2=π√n1n=5n2pπ√n1n5np = Smicro (152) (153) 4G The mass ofAthe extremal D1-D5 system is Sbek = = 2π√n1n5npn=i S=min¯crio (153) (154) 4G Smicro = 2π√n1n5np (152) i Smicro = 2π√n1n5np πM 2 2 (152) X N Mpeixtr=emawl =i ρ ρ(s1 +Nsi5 + 1) (155) (2.37) ni = n¯iA 4G(5) (154) 1 2Sbek =A 3 = 2π√n1n5np = Smicro (153) i X XSbek = X4G= 2π√nw1n=5np1=, .5S,m.i5cro N = 2 (153) (156) XFrom (2.8)Nwe see thpait t4=hGe ewnierρgy of tρhe{syNstiem a}bove the energy (o1f 5th5e) extremal D1-D5 system 1 is 2 3 ni = n¯i (154) X X X w =a1 1n, i.5,=a.52n¯i Na3 = 2 (156) (154) (157) i X N{ πMpi} = wi ρ ρ Nπ i Q Q (155) i L1 L2 2 1 5 2 2 (158) a1X ∆aM2 NADM ap3i =(1w+i ρ2sp) ρ Ni nm(1(s5−7) +(1s55)) 1 2 ≃3 8G(5) ≃ 8G(5) R2 1X 2X 3X w = 1, .5, .5 N = 2 (156) X XL XPL = w(=ni +1,{n.¯5i,).5pi } N π= Q2 Q (158) (156) (159) 1 2 { } = 1 5 (m2 + n2 1) a i a a (5) 2 (157) a 1 a 2 a 3 8G R (1−57) P = (ni +1 n¯i)!p2i 3 (159) N 1 2 2 L1 L2 = (m + n 1)n n (158) (2.38) ˜ i L1 L2 1 (5158) S = S λ(Ebr!anes E) = AN (√ni + √n¯i)2R λ(2mini − E) (160) − −N − − P = (in=i1+ n¯i) pi (159) where we used (2.32),(2P.12=),(2.1(3n)i a+ndn¯i()2p.2i 5). Note that this result is co(n1s5is9t)ent with our initial S˜ = S λ(Ebranes E) = AN (√ni +i √"n¯i) λ(2mini E) (160) abeckw−[email protected]−piornoj(e2c.t2b8)ectkhwatitMh1@beyica!ohmoeos.csomm−all for large−R. i=1 ! N In the large R lim"it that we hNave taken we also have, using (2.21) and (2.36) D−1 [email protected] project˜beS˜ck=wSith1λ@(Eyabrhaoneos.coEm) = AN (√ni + √n¯i) λ(2mini E) (160) S = S λ−(Ebranes E−) = ASN (E√nDi + √n¯i) λ−(2mini E−) (160) (161) − − ∼ i=1 − − S = 2⇥⇧n−1(⇧n i=+1 ⇧n¯ )( n + n¯ ) (57) D 5 1"" 1 ⇧ pQ Q p s2 [email protected] [email protected] 1 5 (161) [email protected] [email protected]+ 2 2 2 ∼ S EE≈ − R s− s (162) = 2⇥⇧n5( −1 ) − (58) ∼D−1D 2 ⇧mD1mp Q1Q5 s− S S ErE2D (16(11)61) (2.39) S E ∼ 2 2 2 (162) S = ∼2⇥ n −n n≈n − R s− s (59) S = A(√n1 + √n¯1∼)(√n2 + √⇧n¯2)1(√5 np3 +kk √n¯3) . . . (√−nN + √n¯N ) (163) which gives N S = 2⇥⇧Sn1nS5nkEk(2⇧Enp + n¯p) (16((2160)62)) S = A(√n1 + √n¯1)(√n2 + √n¯2)(√n3 + √∼n¯∼E3) . . .2(√n2N +Q√1nQ¯N5 ) (163) (164) ∼ r+ r 2 (2.40) N E − − ≈ R E =2 2⇥⇧n1n5( ) (164) (61) S =S A=(A√(n√n+1 √+n¯√)n¯(1√)(n√n+2 √+n¯√)n¯(2√)(n√mn+3m√+n¯√)n¯.3.)..(.√. (n√n+N √+n¯√n¯)N ) (16(31)63) 1 ∼1 SO2(4) 2SU(⇧23) p SkUk3(2) N N (165) N N 3 The instability o≈f th2e 2 g×eometries: Review S = 2⇥⇧n1n5(⇧npE+1 E1 n¯p)(⇧nkk + ⇧n¯kk) (16((4162))64) SO(4) SU(2) S∼U(∼(2,) ) (165) (166) Shortly after th≈e construc×tion of2tlh2e above 3-charge regular geometries i(6t 3)was shown in [14] 1 1 p that these geometriesRadiation:suffered f rThe⇤om agravityn inst acalculationbility. Th+is was a classical ergoregion instability 1 2 3 ( 4S,OS()O4)(4)+SU˜S(+U2)1(2)SU˜S(U2)(2) ¯+ ¯˜ ¯ ¯˜−(166) (16(51)65) whicXh is,aXge,nXeric, Xfe2atu2re ofψ≈ro≈t,aψtinn,g6ψ×lnp−o,×nψ-e−xtremaψl ge,oψmet,rψie−s., ψIn this sectio(6n4)w(1e6w7i)ll reproduce 1⇤1 1 + 1 2 th3e com4 putatio+ns o˜f+[14] to˜fin(d ,(the,)1c¯)o+m4p¯˜le1x e1¯igen¯˜fr−equencies for this in(s1t6a(6b1)6il6it)y in the large R X , X , X , X ψ , ψ ,(ψ0M−, 09,,)1ψ− 2M(024,,1 ψ) T, ψ( ,S,ψ0−), ψ (167) (65)(168) limit. ⌅ 2 2⇥ ⇥ 2 2 + 1 2 3 4 1 +E/(21+mkk) = 0.5 + ¯ + ¯ − (66) X , X1 , X2 , X3 4 ψ , ψ+˜ ,˜ψ+−, ψ˜−˜ ψ¯ ,¯ψ+˜ ¯,˜ψ¯−,¯ψ˜ ¯˜− (167) X(0,,X0), X (,0X, Graviton) nψ(L ,with,ψ0n)R, indicesψ−, ψ− on the ψtorus, ψ is, aψ scalar−, ψ in (6-d168) (167()169) 3.1 The wave equ2atioEn/(22fomrkkm) =in1im.{2 ally coupled scalars (67) h Ψ1 1 !Ψ1 =1 0 (170) nL n(R0,(00),102)2 (04,(0,) ) ( ,(0), 0) (169) (16(81)68) We consider a minimally cogu{≡ple d⇧2snc1a2nla5rnfip2e1l2d in the 6-dimensional geometry obtained by dimen- Lz M[ t, r, θ, ψ] 3, φ Rs (68)(171) h12 Ψ !Ψn4 4,=1 n0 (170) (169) sional reduction on the⇤T .LnSLu→cRVhnRRa scalar ⇤arises for instance from hij, w(h16ic9h) is the graviton ≡ 1 { with both indices alongSthe Ty4. Tyhe: w(a0v,{e2eπqRu)ation for the scalar is (172) M4,1 t, rh,1θ2, ψ,φφ !Sφ = 0 (171) (17(60)9) → h→≡12 φ !φ = 0 (170) 1 C Vˆ≡[l] S Vˆ (173) S y y :M(40,1, l2πRt,)re, θ, ψ, φ ✷Ψ = 0 (172) (17(71)0) (3.41) M4→,1 t, r, θ, ψ, φ (171) → →S+S ˆ 1 ˆNe= n1n5 (71)(174) We can separCatleVv[la]riaSblSes1 wyVithy yth:ey(a:0n,(s20aπ,tR2zπ)[2R7), 13, 14]5 (173) (17(21)72) →→ dn S√=N2⇥ Nnn=1√n5nn1pn(15 f) + 2⇥√Nn1n√5ynnpf(⇧nk + ⇧n¯kn) (174) (72)(175) Ψ =exp( iωt + iλ + imψψ + imφφ)χ(θ)h(r) (3.42) − −≈ dn R dt ∝ √N n √n ⇥ √N √n1 E 1 n (175) 5 n = n¯ = = (73) − Our conventions are≈slikghtly kdiffer2enmt fromdtt2hDos∝emin [14]: we have the opposite sign of λ, for us positive ω will correspond to positive energy qua9nta, aknd for us ωkhas dimensions of inverse length. (Solve by matching99 2 4 inner and outer region solutions) ⇧n1n5npg 1 D9 [ ] 3 RS (74) ⇤ V Ry ⇤ 8

S = S 2⇥⇧n1n5np = 1 (75) A S = (76) 4G 2 G5 D mk 2 (77) ⇤ G4 ⇤ G5

1 1 D G 3 (n n n ) 6 R (78) ⇤ 5 1 5 p ⇤ S

N lp (79) eS (80)

5 iωt Ψ = ψ(x)e− (201) 1 L = ∂ φ∂µφ (202) 2 µ τ (203) 1 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) | ⟩ √2 | ⟩ ⊗ | ⟩ | ⟩ ⊗ | ⟩ GMm GM E = mc2 E = mc2 E 0 r (205) − r ∼ ∼ c2

R R + R2 (210) A S = = 4π√n n = S (211) bw 2G 1 2 micro K3 T 2 (212) × S = 2√2π√n1n5 (213) AdS S3 T 4 (214) 3 × × 1 (n n ) 6 l (215) ∼ 1 5 p ΓCF T = V ρL ρR (216)

ΓCF T = V ρ¯L ρ¯R (217)

1 R S l, mφ, mψ λ (218) CF T gravity CF T gravity ωR = ωR ωI = ωI (219) ω = ωgravity + iωgravity (220) wwhheerree tthheerree is nnoo iinnccoommininggwwavaev,eb, ubtuwtewsetilslthilalvheaavRne oauntgoouintgowinaIgvewcaavrreyicnagrreynienrggyenouertgtyo out to ininﬁﬁnnitityy.. TThheese iinnssttaabbiliiltiytyfrferqeuqeunecniecsieasreargeivgeinvebny bsoylustoioluntsiotonsthteottrhanes(Cardoso,tcreanndsecnet Dias,naldeeq nJordan,utatlioe nquation 1 1 Γ(1 ν) κ 2ν2ν Γ(ν) Γ( (11+ ζ + ξ ν))Γ( (1 +1 ζ Hovdebo,ξ ν Myers,)) ’06) iiννπ Γ(1 ν) κ Γ(ν) Γ2 ( 2 (1|+| ζ +− ξ ν)2)Γ( 2 (|1|+− ζ− ξ ν)) ee−− −− == 1 | | − 1 | | − − (3.50) (3.50) − Γ(1 + ν) 2 Γ( ν) Γ( 2 (11+ ζ + ξ + ν))Γ( 2 (1 +1 ζ ξ + ν)) − Γ(1 + ν) 2 Γ−( ν) Γ( 2 (1|+| ζ + ξ + ν))Γ( 2 (|1|+− ζ ξ + ν)) !! "" − | | | | − WWeerreepprroodduucce tthhee ssooluluttioionntotothtihsiesqeuqautiaotni,onfo,ufnodunind[1in4],[1in4],apinpeanpdpixenBd.ixInBth. eInlartgheeRlalrigmeitR limit (2.27) the instability frequencies are real to leading order (2.27) the instability frequencies are real to leading order 1 11 ω ωR = 1( l mψm + mφn λ mψn + mφm 2(N + 1)) (3.51) ω ≃ ωR =R −( −l mψm + mφ−n| − −λ mψn + m| −φm 2(N + 1)) (3.51) where N 0 is≃an integRer. −Th−e imaginary par−t o|f−the−frequency is fou|n−d by iterating to a where N ≥ 0 is an integer. The imaginary part of the frequency is found by iterating to a higher ord≥er; the result is higher order; the result is 2 l+1 1 2π λ Q1Q5 2 l+l+11+N l+1+N+ ζ ωI = 2 (ω 2 )2 2 Cl+1 | |Cl+1 (3.52) R1 [l!2] π −2 R λ 4RQ1Q5 l+1+N l+1+N+ ζ ω = # $ (ω ) % C | &|C (3.52) I R [l!]2 − R2 4R2 l+1 l+1 Note that ωI > 0, so w#e have$ an exponentially %growing perturbation. Our task &will be to reproduce (3.51),(3.52) from the microscopic computation. Note that ωIThese> 0 ,exactso w matche hav ethean CFTexp valuesonentially growing perturbation. Our task will be to reproduce (3.51),(3.52) from the microscopic com(Chowdhury+SDMputation. '07) 4 The Microscopic Model: the D1-D5 CFT

4In thTis hseectioMn wice rdoiscsucssotpheicCFMT doudales lo:f thtehgeeomDet1ri-eDs o5f [1C3].FRTecall that we are working with IIB string theory compactiﬁed to M S1 T 4. The S1 is parameterized by a coordinate 4,1 × × Iny twhitish section we discuss the CFT duals of the geometries of [13]. Recall that we are working 1 4 1 with IIB string theory compactiﬁed to M0 4,1y

Thus the microstate radiates like a piece of coal; there is no information problem How is the semiclassical approximation violated at the horizon? In 1972, Bekenstein taught us that black holes have an entropy

c3 A A S = 2 ~ 4G ⇠ lp

144 This means that a solar mass black hole has 10 10 states ⇠

This is far larger than the number of states of normal matter with the same energy Consider a collapsing shell

As it approaches the horizon radius, there is a small amplitude for it to tunnel into a fuzzball state Amplitude for tunneling

1 S d4xp g tunnel ⇠ 16⇡G R Z Let us set all length scales to L GM ⇠

1 d4xp g (GM)4 ⇠ R ⇠ (GM)2 Z

S GM 2 tunnel ⇠ The probability of tunneling into the chosen fuzzball state is

2 2S P = e tunnel tunnel |Atunnel| ⇠ Using our estimate ↵GM 2 P e tunnel ⇠ We should now multiply thus by the number of fuzzball states we can tunnel to

2 S A 4⇡(GM) 4⇡GM 2 e bek e 4G e G e N ⇠ ⇠ ⇠ ⇠

We thus see that it is possible for the total probability for tunneling into fuzzballs can be order unity

1 PN ⇠ (SDM 0805.3716) Toy model

Small amplitude to tunnel to a neighboring well, but there are a correspondingly large number of adjacent wells

In a time of order unity, the wavefunction in the central well becomes a linear combination of states in all wells We call this phenomenon 'Entropy enhanced tunneling'

An argument can be made using Kraus-Wilczek tunneling from black holes that the exponentials exactly cancel (Kraus+SDM 15)

For simple families of fuzzball microstates the entropy enhanced tunneling has been explicitly calculated

(Bena, Mayerson, Puhm, Vernocke 15) ⇥ R (15) ⇥ R t (16) ⇥ c G2M 3 ⇥ R (15) t ⇥ (17) evap ⇥ c2 R t (16) 1 dE ⇥ c T = = (18) 8⇤GM dS G2M 3 t (17) 63 evap 2 tevap 10 years ⇥ c (19) ⇥ 1 dE eS T = = (20) (18) 8⇤GM dS S = ln(# statest )1063 years (21) (19) evap ⇥ 1077 10 eS (22) (20) 3 c A A S = ln(# states) (21) S = (c = = 1) (23) 4G ⇤ 4G 77 1010 (22) S = ln 1 = 0 (24) c3 A A ⇥ S⌅= (c = = 1)(25) (23) 4G ⇤ 4G Sgrav 1 4 2 A e ,S S =Rd lnx 1GM = 0(26) (24) ⇥ grav ⇥ 16⇤G ⇥ ⇥ ⌅ (25) # states eS,SGM 2 (27) ⇥ Sgrav ⇥ 1 4 2 AA e ,Sgrav Rd x GM (26) A =0 S = ⇥=0 S ⇥= ln(1)16⇤G = 0⇥ (28) bek 4G micro # states eS,SGM 2 (27) F⇧ (y ct)⇥ ⇥ (29) A A =0 Sbek4=⇥n1np=0 Smicro = ln(1) = 0 (28) n1 np e 4G (30)

Smicro =4⇤⌅n1np Sbek wald =4⇤⌅n1npF⇧ (y ctS)micro = Sbek (31) (29) iHt iHt 4⇥n1np ⇥f = e ⇥i ⇥ni1 = e np⇥f e (32) (30)

3 2 9 S ES4micro S=4⇤ES⌅n1np ESbek waldS =4E⇤2⌅n1np Smicro(33)= Sbek (31) ⇥ ⇥ ⇥ ⇥ iHt iHt H = Hvacuum ⇥+fO=(e) ⇥i ⇥i = e ⇥f (34) (32) 3 2 9 Path integral S E 4 1SS[g] ESE S E 2 (33) Z = ⇥ D[g]e ⇥ ⇥ ⇥ (35) H = Hvacuum + O() (34) Measure has Action determines 1 S[g] degeneracy of states Z = classicalD[g ]trajectorye (35) For traditional macroscopic objects the measure is order while the (36) action is order unity

2 But for black holes the entropy is so large that the two are comparable …

Thus the black hole is not a semiclassical object 2 A pictorial description of ‘entropy enhanced tunneling’ A pictorial description of ‘entropy enhanced tunneling’

(1) Shell far outside horizon, (2) As shell approaches its semiclassical collapse horizon, there is a nucleation of Euclidean Schwarzschild ‘bubbles’ just outside the shell (3) The bubbles cost energy, which is (4) As the shell reaches close drawn from the energy of the shell. to its new horizon, more The shell now has a lower energy, bubbles nucleate, and so on. which corresponds to a horizon radius that is smaller. The shell thus moves inwards without forming a horizon (5) Instead of a black hole with horizon, we end up with a horizon sized structure which has no horizon or singularity The causality paradox The causality problem

Light cones point inwards

So information cannot come out

When the hole evaporates away, what happens to the information in the shell? In quantum gravity do we have to stay inside the light cone?

(A) Perturbative quantum gravity

Light cones in ﬂat spacetime ⌘µ⌫ Light cones ﬂuctuate under

⌘µ⌫ + hµ⌫

Can we have small violations of causality?

No: We can quantize h µ ⌫ and we will ﬁnd that its causal propagators vanish outside the light cone (B) Nonperturbative quantum gravity:

Bubble nucleation: false vacuum (red) changes to true vacuum (brown)

bubble surface does not move faster than light General states

[(3)g]

A general state is a superposition of many 3-manifolds

So which light cones should we choose to determine causality ?

For a general state there is no well deﬁned notion of causality But consider a space with a maximal symmetry group like Minkowki space, de-Sitter space etc.

Assume that the quantum vacuum state 0 also satisﬁes these symmetries | i Then we conjecture that in our full quantum gravity theory there is a deﬁnition of local operators, such that causality is maintained using the light cones of the maximally symmetric background

We also assume that for gently curved space, we get 'approximate locality' The region used in the Hawking argument is gently curved

Thus the leakage of the wavefunction outside the light cone must be small

The small corrections theorem then tells us that these small violations of nonlocality will not help

If we assume order unity effect of nonlocal physics, then there was no black hole puzzle in the ﬁrst place ...

We can just take the stuff inside the hole and place it outside, and then there is no paradox Many of the alternatives to fuzzballs invoke some kind of nonlocal physics:

(A) Nonlocality on scale M for low energy modes (Giddings)

(B) Nonlocal effects on scales M 3

Wormholes between the hole and its radiation (Maldacena and Susskind 2013)

Bits describing the radiation are not independent of bits describing the remaining hole (Papadodimas and Raju)

(C) Nonlocal effects on inﬁnitely long length scales Gauge modes arising from diffeomorphisms at inﬁnity (Hawking,Perry Strominger 2015) Q: Can we show a concrete computation in string theory which gives effects outside the light cone?

We will assume that there is no signiﬁcant leakage outside the light cone ... Resolving the causality paradox We can ask how the causality problem is avoided in the fuzzball paradigm …

Shell falls in at (1) speed of light M M 0 Sees only normal physics when far r =2M

r = 2(M + M 0) When shell approaches (2) M horizon, it tunnels into fuzzballs …

r =2M We have seen that at this location a tunneling into fuzzballs becomes possible …

But we can still ask: What local property tells the infalling shell that it should start tunneling into fuzzballs at this location?

If the local spacetime is the vacuum (or close to the vacuum), then one might try to use the equivalence principle and say that the shell can feel nothing as it passes this location

r = 2(M + M 0)

r =2M Is there a picture of spacetime where low energy matter sees nothing special, but matter with energy more than a given threshold sees signiﬁcantly altered physics ?

Toy model: The c =1 Matrix model

The essential point is that spacetime is not just a manifold, but has an additional property that we can call the “depth” or “thickness” Is there a picture of spacetime where low energy matter sees nothing special, but matter with energy more than a given threshold sees signiﬁcantly altered physics ?

Toy model: The c =1 Matrix model (Das+Jevicki ….)

M : N N matrix ⇥ 1 L = Tr M˙ (t)2 V (M) Z = dM dM eiL 2 i,j ij ij⇤ ⇣ ⌘ R Q Eigenvalues behave like fermions, so the lowest energy state has energy levels ﬁlled upto a fermi surface … Small deformation on the fermi sea travel as massless bosons

But large deformation on the fermi sea suffer a distortion

The waveform can ‘fold’ over, after which it is no longer described by a classical scalar ﬁeld … (Das+SDM 95) In fact the matrix model gives a fermi sea of varying depth

wave is distorted by wave does not touching bottom of sea notice depth of sea

The matrix model does not actually have a black hole …

Also, in our actual fuzzball paradigm, what effect provides the analogue of the varying depth fermi sea? Conjecture: The Rindler region outside the hole has a different set of quantum ﬂuctuations from those in a patch of empty Minkowski space (‘pseudo-Rindler’)

Quantum ﬂuctuations will be different near the surface of the fuzzball since there is a nontrivial structure there …

(a) What is the nature of these ﬂuctuations?

(b) Why should they be important ? (a) The ﬂuctuations are the ﬂuctuations to larger fuzzballs

M M + M !

Our energy is still M so this is a virtual excitations (vacuum ﬂuctuation) (b) The reason these ﬂuctuations are important is because they are ‘entropy enhanced’ (there are a lot of fuzzballs with that larger mass)

Exp[Sbek(M + M)] states Rindler space

pseudo-Rindler space

(Quantum ﬂuctuations are different from empty space)

At a location depending on the energy of the quantum, there will be a tunneling into fuzzballs …

This resolves the causality paradox (SDM 17) Complementarity and fuzzball complementarity 't Hooft, Susskind .... Is it possible that the interior of a black hole is a manifestation of some new physics (different from the physics outside) ?

information reﬂects from stretched horizon for the purpose of the outside observer

infalling Normally we cannot do object 'quantum cloning'. A second copy of the But here it is allowed since we information continues to cannot compare the two fall in copies easily (a) But what reﬂects the information off the horizon?

(b) Also, if we look at good slices, what special physics separates the interior from the exterior?

r = 1

r =0

The Firewall argument tries to gives a rigorous proof that this idea does not work But with fuzzballs things are quite different

We have a surface rather than a smooth horizon, so there is no difﬁculty in radiating the information back

There is no interior, so can we have any notion of smooth infall?

Should we not already say that the surface of a fuzzball will have to behave like a ﬁrewall?

No, because there is a possibility which we will call Fuzzball complementarity A collapsing shell tunnels into a linear combination of fuzzball states

There is a vast space of fuzzball states

eSbek(M) N ⇠ When the collapsing shell tunnels into fuzzballs, then its state keeps evolving in this large space of fuzzball states …

…. Thus we must study dynamics is SUPERSPACE, the space of all gravity conﬁgurations

…

Superspace, the space of all fuzzball conﬁgurations

The full quantum gravity state is a wavefunctional over superspace Conjecture: This evolution in superspace can be approximately mapped to infall in the classical black hole

Fuzzball complementarity E T This may seem strange, but something like this happened with AdS/CFT duality (Maldacena 1997) …

Create random excitations A complicated set of We can map this gluon excitations spreads complicated evolution to on the D-branes free infall of the graviton

The difference is that AdS/CFT duality is exact, while fuzzball complementarity is an approximate map Fuzzball complementarity

Different fuzzballs radiate different at energies E T ⇠

Free infall onto the fuzzball is a hard ⇡ impact process with E T

For these hard impact processes the evolution in the space of fuzzballs map to the ‘vibrations’ of empty space Causality and the ﬁrewall argument What is the ﬁrewall argument?

Hawking (1975) showed that if we have the vacuum at the horizon (no hair) then there will be a problem with growing entanglement

This is equivalent to the statement: If we assume that

Ass:1 The entanglement does not keep rising, but instead drops down after some point like a normal body

Then the horizon cannot be a vacuum region.

AMPS use the same argument of bits and strong subadditivity that we used to make the Hawking argument rigorous (the small corrections theorem)

So what is the difference between the Hawking paradox and the ﬁrewall argument? AMPS tried to make the a stronger statement, by adding an extra assumption

Ass2: Outside the stretched horizon, we have ‘normal physics’ (Effective ﬁeld theory). In particular, a shell coming in at the speed of light will encounter no new physics till it hits the stretched horizon (causality)

AMPS claim: Given assumptions Ass:1 and Ass:2, an infalling object will encounter quanta with energies reaching planck scale as it approaches the horizon (Firewall)

infalling shell only sees black hole effective ﬁeld theory (no new physics) outside the hole stretched horizon The intuition behind the AMPS argument is simple:

In the Hawking computation with vacuum horizon, the particles do not actually materialize until they are long wavelength (low energy) excitations

If we replace the black hole by a normal hot body, then we will have no entanglement problem (by deﬁnition)

But we can now follow the radiation quanta back to the emitting surface, where they will be high energy real quanta high energy quanta But there is a problem with the AMPS argument:

The two assumptions Amp:1 and Amp:2 are in conﬂict with each other …

Shell falls in at (1) speed of light M M 0 Sees only normal physics (Ass:2) r =2M

r = 2(M + M 0) (2) M Shell passes through its own horizon without drama, since by causality r =2M it has not seen the hole (3) r = 2(M + M 0) Light cones point inwards inside the new horizon, so the information of the shell cannot be sent to inﬁnity without violating causality.

This contradicts assumption Ass:1

Thus there is a conﬂict between Assumptions Ass:1 and Ass:2 made in the ﬁrewall argument.

If we drop the new assumption Ass:2, then we cannot argue that there is a ﬁrewall: in fact we can construct a bit model where high energy quanta feel ‘no drama’ at the horizon (fuzzball complementarity)

(SDM+Turton 2013, SDM 2015) Suppose an object of energy E>>kT falls in

Now there are e S ( M + E ) possible states of the hole

S(M+E) S(M)+S Nf E e e S kT = S(M) = S(M) = e e 1 Ni e e ⇡

So most of the new states created after impact are not entangled with the radiation at inﬁnity

(This is just like the entanglement before the halfway evaporation point)

Complementarity is the dynamics of these newly created degrees of freedom, and says that this dynamics is captured by the physics of the black hole interior

AMPS worry only about experiments with Hawking modes b, c, but these have E~kT SUMMARY

In string theory we ﬁnd the fuzzball paradigm, where black holes do not have the traditional structure, and radiate like normal bodies.

The lesson: The scale of quantum gravity excitations increases with the number of particles involved, and always prevents horizon formation

l N ↵l p ! p