Strings and Black Holes
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1 Strings and Black Holes Erik Verlinde Institute for Theoretical Physics University of Amsterdam
2 General Relativity R Rg GT µν µν = 8π µν Gravity = geometry Einstein: geometry => physics Strings: physics => geometry
3 A Brief History of Black Holes 96 Schwarzschild solution. 939 Einstein: ``I had not expected that the exact solution to the problem could be formulated. Your analytical treatment to the problem appears to me splendid.'' Oppenheimer-Volkov Gravitational collapse: frozen or black stars. 959~ Global structure, exact solutions. Kruskal, Kerr, Newman Black hole singularity theorems. Black hole thermodynamics. Holographic principle. Penrose-Hawking Bekenstein-Hawking t Hooft- Susskind. 996 Microscopic origin of entropy. Strominger-Vafa
4 A Brief History of String Theory Veneziano amplitude, dual models Bosonic and fermionic string theory Superstrings as theory of quantum gravity Nambu-Goto Neveu-Schwarz-Ramond Scherk-Schwarz Anomaly cancellation, heterotic strings String dualities, D-branes Microscopic origin of black hole entropy Green-Schwarz Witten, Polchinski Strominger-Vafa 997 AdS-CFT correspondence and holography. Maldacena
5 Black Holes Outline Hawking Radiation Information Paradox The Holographic Principle Black Holes in String Theory Strings and D-branes Microscopic Origin of Black Hole Entropy Holography in String Theory
6 Hawking Radiation Black Holes emit thermal radiation with temperature Hawking 975 T = 3 hc 8π kgm Note: depends on 4 fundamental constants of Nature!! Bekenstein-Hawking Entropy Boltzman factor determined by acceleration at horizon ca S =, A = 4π R hω 4hG πωc exp = exp kt a What is the origin of this entropy? GM a = R
7 Black Hole Thermodynamics Charged Black Holes: Reisner-Nordstrom Inner and outer horizon dr dr ( ds ds = H( r( r) dt ) dt rr d Ωθ + sin θdφ ) H( r( r) ) r = M ± M ± Q M Q Q H( r) = + r r r Electric Potential, Entropy and Temperature Extremal Black Holes M = Q Φ= Q r + S = π r + obey the first law of thermodynamics T = r r 4π r + ( ) + T TdS = dm +ΦdQ + Ω dj
8 Einstein equation as equation of state Raychaudhuri equation (no shear or vorticity) Ted Jacobson d d θ µ v θ Rµ vk k λ = θ = k µ ; µ µ k k = µ can be integrated using the Einstein equation µ v θ = λtµ vk k + O λ 8π G ( ) = local version of st law of thermodynamics θ = horizon Tds = ρ
9 The Information Paradox (early 9 s) Lkjoiwjojjdndfldfl Fkdhshwofekfew Kdlfsflsdfjlsdfsldf Lkjoiwjojjdndfldfl Lkjoiwjojjdndfldfl Lkjoiwjojjdndfldfl Fkdhshwofekfew Fkdhshwofekfew Fkdhshwofekfew Kdlfsflsdfjlsdfsldf Kdlfsflsdfjlsdfsldf Kdlfsflsdfjlsdfsldf Skdjfkjshkj Skdjfkjshkj Skdjfkjshkj Sdjflsjlfsl Sdjflsjlfsl Sdjflsjlfsl Sdd,f Sdd,f Sdd,f Skdjfkjshkj Sdjflsjlfsl Sdd,f Where did the information go? Lkjoiwjojjdndfldfl Lkjoiwjojjdndfldfl Fkdhshwofekfew Fkdhshwofekfew Kdlfsflsdfjlsdfsldf Kdlfsflsdfjlsdfsldf Skdjfkjshkj Skdjfkjshkj Sdjflsjlfsl Sdjflsjlfsl Sdd,f Sdd,f Lkjoiwjojjdndfldfl Fkdhshwofekfew Kdlfsflsdfjlsdfsldf Skdjfkjshkj Sdjflsjlfsl Sdd,f Lkjoiwjojjdndfldfl Fkdhshwofekfew Kdlfsflsdfjlsdfsldf Skdjfkjshkj Sdjflsjlfsl Sdd,f
10 #States exp 4 A G = The Holographic Principle (994) Planck Area t Hooft Susskind
11 Strings, D-branes and Gravity Open strings: gauge interactions A Closed strings: gravity µ, X I g µν, Bµν, Φ D-branes: gauge theory on worldvolume. source for gravity: gravity induced by open string loops
12 D-branes as Black Holes D-branes wrap certain cycles inside compactification manifold. They become charged massive objects: (extremal) black holes. Their world-volume theory gives a microcopic description of the states associated with the black holes. ds = dt + H ( r) dr + r dω H( r) ( ) H( r) = + Q r i Q = φ Q i
13 Microscopic Origin of Entropy Excited string states have high degeneracy M = Nl s #states expπ cn But not enough to explain black hole entropy exp πml expπm s l p For extremal black holes #states expπq D-brane described microscopically by gas of strings with c = Q N = Q
14 Extremal dyonic black holes: Exact counting SQP (, ) A = =π 4 PQ SQP DQ P (, ) = log (, ) Generating function: with: DNMe (, ) = N, M nm, ( e ) tn sm nt ms c( nm) n cnq ( ) N n + q = n n q 4
15 AdS/CFT Correspondence Anti-de Sitter- Conformal FieldTheory Near horizon geometry of a D3-brane AdS AdSmetric black hole = thermal CFT dr ds ds Hrr dt rr dd H+ ( r ) = = ( + dr ( ) + + Ω H( r) = M + r r Strings on AdS5 S 5 = dual to a CFT: N=4 SuperYang-Mills Z ( g, φ ) = Z ( g, φ ) CFT string
16 strings Bulk: 5D antide Sitter space Boundary: 4D Minkowski space Black holes Wilson loops quarks Hawking radiation
17 4D Attractor Black Holes and Entropy Semiclassical entropy S Q, P = X F X Λ F ( ) Λ, Λ, Λ Q, P Re Re ( Λ X ) ( F ), Λ = Q = P Λ Λ Entropy as Legendre transform Ω ( PQ, ) = dφψpq, ( Φ) ΨPQ, ( Φ) Λ F (, ) = (, Φ) Φ (, Φ Λ ) iφp Φ ΨPQ, ( Φ ) = e Ψ ( Q+Φ) F P = ( Q, Φ) S Q P F Q Q Λ Φ Λ Ooguri, Strominger, Vafa i ( Φ ) = F Q F Q, Im ( + Φ) SQP (, ) = log DQP (, ) Mixed partition function factorizes as Ω = Ψ + Φ p ( ) P Q, P e Φ ( Q i ) Ψ X = expif X ( ) ( )
18 The Entropic Principle Flux Vacua Moduli fixed by fluxes : discrete points. Flux Wave Functions X PQ, Flux vacua as wave functions on moduli space Relative probability determined by entropy ΨPQ, ΨPQ, exp SPQ (, ), ( ) Ψ PQ X Entropic Principle Nature is (most likely) described by state of maximal entropy Constructive way to select vacua (in contrast with Anthropic Principle )
19
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