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The Fuzzball Paradigm 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 ⇤ e2π⇥2⇥n1np Q 1 + 1 r2 Q 1 + p r2 e2π⇥2⇥n1n5 i(t+y) ikz w = e− − w˜(r, θ, ⇤) (10) (2) i(t+y) ikz ˜(2) BMN = e− − BMN (r, θ, ⇤) , (11) 2 1 i = dx[ F a F µ⇥a + ⇥¯⌅⇥ + . .] L −4 µ⇥ 2 ⇥ 2πn 2π(n n ) P = p = 1 p L LT 2πk p = LT knk = n1np 2-charge NS1-P extremal hole k √ 1 e2⇤4 2√n1np IIB: M9,1 M4,1 S T ! ⇥ ⇥ S = 2π⇤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, α, g, LS1 , VT 4 ] (102) R [n1, n5, np, α, 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 2πnp 2π(n1np) ⇥ P = = L LT 2πnp 2π(n1np) P = = 1 2πk i L LT = dx[ Fpa=F µ⇥a + ⇥¯⌅⇥ + . .] L −4 µ⇥ L 2 ⇥ T 2πk p = LT 2πnp 2π(n1np) P = knk== n1np L LT k knk = n1np √ 2πk k e2p⇤=2√n1np LT e2⇤√2√n1np S = k2πnk⇤=2⇤nn11nnpp k S = 2π⇤2⇤n1np 2⇤√2√n1np eLT = n1L LT = n1L S = 2π⇤L2⇤n1np L LT = n1L L ‘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 (α 1) (α 2−) . 0 (b) A mode ↵ (1 37)k maps to − i1 n1 − i2 n2 | ⇤ − (α 1) (α 2) . 0 a loop with winding (1k 37) − − | ⇤ and spin in the direction i (c) NS1-P: knk = n1np k D1-D5 X state D1-D5: knk = n10 n50 Xk 8 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 1σ2 ]( 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 Ψ = J nR ...J 1J nL ...J 1[σk] 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 ✓dφ2) − − r 1 r0 − r Make the analytic continuation t i ⌧ ! r dr2 ds2 =(1 0 )d⌧ 2 + + r2(d✓2 +sin2 ✓dφ2) − r 1 r0 − r Let the ⌧ direction be a circle 0 ⌧ < 4⇡r 0 Not part of spacetime ✓, φ r ⌧ This gives the 4-d Euclidean Schwarzschild geometry Dimensional reduction 4+1 dim 3+1 dim Dimensional reduction of the ⌧ circle 2 Φ p3 gives a scalar Φ in the 3+1 spacetime g⌧⌧ = e p3 r Φ = ln(1 0 ) 2 − r The 3+1 metric is defined as 1 Φ 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 ✓dφ ) − − r (1 0 ) 2 − r − r The action is 1 1 S = d4xp g R Φ Φ,µ 16⇡G − E − 2 ,µ Z ✓ ◆ ✓, φ r ⌧ The stress tensor is the standard one for a scalar field 1 T = Φ Φ gE Φ Φ,λ µ⌫ ,µ ,⌫ − 2 µ⌫ ,λ which turns out to be T µ = diag ⇢,p ,p ,p = diag f,f, f, f ⌫ {− r ✓ φ} {− − − } 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 Ψ = J nR ...J 1J nL ...J 1[σk] 0 R | − − − − | (b) We know its metric at strong coupling ..
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