Strings and Black Holes Erik Verlinde Institute for Theoretical Physics University of Amsterdam
General Relativity R Rg GT µν µν = 8π µν Gravity = geometry Einstein: geometry => physics Strings: physics => geometry
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 967 975 994 Black hole singularity theorems. Black hole thermodynamics. Holographic principle. Penrose-Hawking Bekenstein-Hawking t Hooft- Susskind. 996 Microscopic origin of entropy. Strominger-Vafa
A Brief History of String Theory 968 97 976 Veneziano amplitude, dual models Bosonic and fermionic string theory Superstrings as theory of quantum gravity Nambu-Goto Neveu-Schwarz-Ramond Scherk-Schwarz 984 995 996 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
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
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
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
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
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#States exp 4 A G = The Holographic Principle (994) Planck Area t Hooft Susskind
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
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
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
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
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
strings Bulk: 5D antide Sitter space Boundary: 4D Minkowski space Black holes Wilson loops quarks Hawking radiation
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 ( ) ( )
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 )