Black holes in String Theory Samir D. Mathur The Ohio State University There are two main ideas in black hole physics … The first is the notion that the entropy of a black hole is given by its surface area A S = bek 4G (Bekenstein 72) This is mysterious for two reasons: (a) The entropy is proportional to the area rather than the volume (b) By the ‘no-hair’ theorem, there is nothing at the horizon, so what states are being counted by this entropy? The other issue is the black hole information paradox … In quantum mechanics, the vacuum can have fluctuations which produce a particle-antiparticle pair ∆E ∆t ~ ⇠ But if a fluctuation happens near the horizon, the particles do not have to re- annihilate : The negative gravitational potential gives the inner particle negative energy ( E = mc 2 GMm ) − r ∆E =0 ∆t = ! 1 Thus real particle pairs are continuously created (Hawking 74) Hole shrinks to a small size The essential issue: Vacuum fluctuations produce entangled states 1 + + = e e− + e−e + | i p2 1 = (01 + 10) p2 So the state of the radiation is entangled with the state of the remaining hole … The radiation does not have a state by itself, the state can only be defined when the radiation and interior are considered together The amount of this entanglement is very large ... If N particles are emitted, then there are 2 N possible arrangements We can call an electron a 0 and a positron a 1 111111 000000 …. …. + 010011 101100 + 000000 111111 Possibility A: Information loss — The evaporation goes on till the remnant has zero mass. At this point the remnant simply vanishes vacuum 000000 The radiation is entangled, …. but there is nothing that it is entangled WITH 101100 111111 The radiation cannot be assigned ANY quantum state ... it can only be described by a density matrix ... this is a violation of quantum mechanics (Hawking 1975) Possibility B : Remnants: We assume the evaporation stops when we get to a planck sized remnant. The remnant must have at least 2 N internal states 111111 000000 …. …. + 010011 101100 + 000000 111111 But how can we hold an unbounded number of states in planck volume with energy limited by planck mass? (Baby Universe ?) The story of entropy 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 S = ln(1) = 0 (6) ⌥ S Consider= 2 2⇥⌥ na1 collectionn2 of strings and branes at(7 weak) coupling, and at strong Scoupling= 2⇥⌥n1n2n3 (8) S = 2⇥⌥n1n2n3n4 (9) n1 n5 n ⇤ 1 ⇤ n 4 l ⇤ p 1 n 2 lp ⇤ weak strong n l ⇤couplingp coupling M M K3 S1 9,1 ⌅ 4,1 ⇥ ⇥ A If wen aren lookingJ S at supersymmetric states, the number of states should not change 4G ⇤… (Vafa1 5 94,− Sen⇤ 95) A ⌥n1n5 S 4G ⇤ ⇤ Smicro = Sbek (Strominger+Vafa 96) e2π⇥2⇥n1np So it seems that the area entropy is indeed a count of states in some way. But Q the1 puzzle+ 1 remains: what is at the horizon ? 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 There was a general idea (Susskind …) that black holes have the maximal possible entropy allowed in a region … Black holes A S = 4G But in Cosmology … Homogenous cosmology Entropy is proportional to volume … A For a large enough volume, S> 4G In fact if we fix the volume, rather than the total energy, then we get much more entropy from a lattice of black holes … E E much more 1 2 entropy ! S1 S2 EV One then finds S ⇠ r G E Writing ⇢ = V we see that S is extensive ⇢ (Brustein+Veneziano, , Fischler+Susskind, S V ⇠ G Banks+Fischler, Masoumi+SDM) r A So is there any role for the area formula S = ? 4G We should look only inside a region of size the cosmological horizon A Then we must have S 4G (Fischler+Susskind 98) Big Bang (t=0) This idea were made precise by Bousso, and encoded in a principle called the ‘covariant entropy bound’ Area A of surface bounding the light sheet Covariant entropy bound Light sheet, Entropy S sheet (Bousso 99) made of light rays flows through the orthogonal to light sheet the bounding surface A S sheet 4G A ‘proof’ of this conjecture was given recently, both for free and interacting matter theories … (Bousso, Cassini, Fisher, Maldacena) ⇢ But in string theory it seems that we can realize the entropy S V ⇠ G as explicit string states ,… r It turns out that if we take an asymmetrically expansing Cosmology, this equation of state violates the Bousso bound … (Masoumi+SDM) So, either the area bound is incorrect, or for some reason we cannot have asymmetric expansion with this equation of state (Banks) A further puzzle is that we can get the area formula in situations where there is no black hole at all : The Ryu-Takayanagi conjecture: CFT on boundary Minimal area surface The entanglement entropy of subset A with subset B is given by Subset Subset A AdS Space 4G where A is the area of the minimal area surface in AdS that bounds the subset A Here an entropy is given by an area, but there is no Black Hole anywhere ! We can create an artificial horizon by considering accelerated observers; such observers cannot see some part of spacetime In this case the entropy describing the ‘Information’ which cannot be seen is related to the area of the horizon (Balasubramanian, Chowdhury, Czeck, de Boer) Thus the relation of entropy to area remains mysterious …. Resolving the information paradox in string theory: Fuzzballs Avery, Balasubramanian, Bena, Carson, Chowdhury, de Boer, Gimon, Giusto, Halmagyi, Keski-Vakkuri, Levi, Lunin, Maldacena, Maoz, Niehoff, Park, Peet, Potvin, Puhm,Ross, Ruef, Saxena, Simon, Skenderis, Srivastava, Taylor, Turton, Vasilakis, Warner ... Recall that the problem was created by the production of entangled pairs … 1 + + = e e− + e−e + | i p2 1 = (01 + 10) p2 String theory cannot have information loss, since it is based on quantum mechanics String theory also does not allow remnants: There are a finite number of states in a finite volume with a given energy, if AdS/CFT is to be correct First consider a rough analogy … Witten 1982: ‘Bubble of nothing’ Consider Minkowski space with an extra compact circle This space-time is unstable to tunneling into a ‘bubble of nothing’ not part of spacetime In more dimensions : not part of spacetime People did not worry about this instability too much, since it turns out that fermions cannot live on this new topology without having a singularity in their wave function … But now consider the black hole … 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 total ,total n1n5 ,total n1n5 ,total n1n5 total n = (J −(2n 2)) S(J=−(2ln(1)n 4))= 0 . (J −2 ) 1 (6(5) | ⇧ − − Black− −holes: − | ⇧ ⌥ AS = 2 2⇥⌥n1n2 (7) = S = 2⇥⌥n1n2n3 4GS The= 2⇥ ⌥traditionaln1n2n3 expectation ... (8) S = 21⇥⌥n1n12n3n4 2 (9) ∆E = + = nR nR nR strong n n 2 n weak ∆1 E⇤ =5 ⇤ coupling 1 nR coupling n 4 lp S =⇤ln(1) = 0 (6) 1 n 2 lp S = 2⇤⌥2⇥⌥n1n2 (7) 1-d spacetimen lp + 1 compact direction S = 2⇤⇥⌥n1n2n3 (8) 1 M9,1 M4,1 K3 S S =⌅2⇥⌥n⇥1n2n3n⇥4 (9) A n n J S 4G ⇤ 1 5 − ⇤ n n n 1 5 A ⇤ 1 ⇤ ⌥nn41nl 5 S 4G ⇤⇤ p ⇤ horizon 1 n 2 lp e2⇤π⇥2⇥n1np n l ⇤ p Q 1 M9,1 M1 +4,1 1K3 S ⌅ r⇥2 ⇥ A n1nQ5 J S 4G ⇤ 1 + p− ⇤ r2 A ⌥n1n5 S 4G e⇤2π⇥2⇥n1n5⇤ e2π⇥2⇥n1np i(t+y) ikz w = e− − w˜(r, θ, ⇤) (10) (2) i(t+y) ikzQ˜1 (2) BMN = e− 1−+ BMN (r, θ, ⇤) , (11) r2 Q 1 + p 2 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 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 S = ln(1) = 0 (6) S = 2⌥2⇥⌥n1n2 (7) S = 2⇥⌥n1n2n3 (8) SBut= 2one⇥⌥n finds1n2n 3thatn4 something different happens (9...) n1 n5 n + - ⇤ 1 ⇤ n 4 lp ⇤ Mass comes from 1 The geometry ‘caps off’ n 2 lp curvature, fluxes, ⇤ just outside the horizon n l (KK monopoles in simplest strings, branes etc .. ⇤ p (spacetime ‘ends’ 1 duality frame) M9,1 M4,1 K3 S consistently in a set ⌅ ⇥ ⇥ A of valid sources in n n J S 4G ⇤ 1 5 − ⇤ string theory) A ⌥n n S 4G ⇤ 1 5 ⇤ e2π⇥2⇥n1np Fuzzball proposal: - + Q All states of the hole 1 + 1 r2 + - are of this topology … No state has a smooth Qp 1 + Not part of space-time horizon with an ‘interior’ r2 (no horizon forms) 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 total ,total n1n5 ,total n1n5 ,total n1n5 total n = (J −(2n 2)) (J −(2n 4)) .