preamble
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preamble Black holes in modern gravity theories 7 сентября 2010 г. 1 / 34 .History Intro of ’Black holes’ Experimental evidence BH thermodynamics BH thermodynamics Information paradox Fuzzball paradigm Intro String—Black hole correspondence Not a Black hole 2 / 34 History of ’Black holes’ Intro History of ’Black holes’ Experimental evidence BH thermodynamics BH thermodynamics Information paradox . Fuzzball paradigm String—Black hole correspondence Not a Black hole 1. Black body 1783 — Light can’t escape Big object with normal density. 2. Some strange solution in GR 1915-1933 — Schwarzschild surface. 3. Black hole 1958-1974 — The term ’Black hole’, new solutions (rotating, charged), black hole mechanics. 4. Nowadays Resolving paradoxes (thermodynamical interpretation, information paradox, initial singularity...) 3 / 34 Experimental evidence Intro History of ’Black holes’ Experimental evidence BH thermodynamics BH thermodynamics Information paradox . Fuzzball paradigm String—Black hole correspondence Not a Black hole 1. Middle-size BH Matter accretion, Binary systems. Several candidates. 2. Small BH Hawking radiation. Evaporation of primordial BHs. Yet no results. 3. Large BH Stellar orbits near the center of our galaxy. Large dark object is found. 4 / 34 BH thermodynamics Intro History of ’Black holes’ Experimental evidence BH thermodynamics BH thermodynamics Information paradox . Fuzzball paradigm String—Black hole correspondence Not a Black hole 0 — Surface gravity is constant on the event horizon (temperature). 1 — Dynamics: κS δA + ω δJ + φ δq, δM = 8π 2 — Evolution: Area of event horizon can’t decrease (entropy). 3 — Surface gravity can’t reach zero Yet it is always zero for extremal BHs. (temperature) 5 / 34 BH thermodynamics Intro History of ’Black holes’ Experimental evidence BH thermodynamics BH thermodynamics Information paradox . Fuzzball paradigm String—Black hole correspondence Entropy-area correspondence S = A/(4π). In case of radiation: Area+Radiation entropy can’t decrease. Surface gravity-temperature correspondence Not a Black hole θ = κS /(2π). Thermodynamical systems have statistical interpretation. What is the statistical entropy of the event horizon? 6 / 34 Information paradox Intro History of ’Black holes’ Experimental evidence BH thermodynamics BH thermodynamics Information paradox . During accretion: Whatever1 Whatever2 Fuzzball paradigm String—Black hole correspondence Not a Black hole Distant observer: M,J,Q coincide Information is lost (Whatever1-Whatever2) 7 / 34 Intro Fuzzball paradigm History Fuzzball Advantages Open questions . String—Black hole correspondence Not a Black hole Fuzzball paradigm 8 / 34 History Intro Fuzzball paradigm . History Fuzzball Advantages Open questions String—Black hole correspondence Not a Black hole 1931 — Chandrasekhar: plasma collapses to BH. Others: something should stop the collapse. White dwarf collapses to a neutron star. 1939 — Oppenheimer: neutron star collapses to BH. Can something else stop the collapse? 2002 — Mathur, Lunin: collapse stops exactly at the event horizon. Stringy fuzzball. 9 / 34 Fuzzball Intro Black hole Fuzzball Fuzzball paradigm History . Fuzzball Advantages Open questions String—Black hole correspondence Not a Black hole Hawking Hawking ? radiation radiation vacuum state M, J, Q and ‘no hair’ complex state Quantum ‘hair’ 10 / 34 Advantages Intro Fuzzball paradigm History Fuzzball Advantages Open questions . String—Black hole correspondence Not a Black hole Solves information paradox, solves singularity paradox. Small density of large BHs: −2 3 . , ρ = M/V ∼ RSch RSch ∼ M, V ∼ RSch Supermassive BHs have density of water or air! During matter accretion strings recombine into very looooong strings. Their tension (and density) decreases exactly as for classical BHs! 11 / 34 Open questions Intro Fuzzball paradigm History Fuzzball Advantages . Open questions String—Black hole correspondence Collapse into BH happens at the scale tcross = RSch /c. Formation of the fuzzball happens at the scale Not a Black hole tevap = tcross (M/mpl )2 tcross . How can they do it? 12 / 34 Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results . String—Black hole correspondence Not a Black hole 13 / 34 BH evaporation Intro Fuzzball paradigm String—Black hole correspondence . BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results Not a Black hole < m > g Beautiful coincidences Resolves ’singularity evaporation’. Stores information in the stringy state. 14 / 34 String action Intro Fuzzball paradigm String—Black hole correspondence BH evaporation . String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results Not a Black hole Effective bosonic action for the heterotic string: Z 2 p H 1 2 −2Φ 10 (10) R + 4(∂Φ) − d x −g e , (10) 12 16πG where H = dB is a field strength of the NS 2-form gauge potential B. The ansatz for the compactification on S 1 × T 5 reads: ds210 = ds2 + e2λ (dx4 + Aµ dxµ ) + e2ν d`2 (T 5 ), 2Φ = 2φ + λ + 5ν, B = Bµ dxµ ∧ dx4 . 15 / 34 Classical action Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action . Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results Dilatonic black hole can be described by the action Z 4 √ 1 −2 2 2 2 R + S (∂S) − S F d x −g S= 16πG with the dilatonic exponent S = e−2φ from the string theory and with the metrics ds2 = wdt2 − w−1 dr + ρ2 dΩ2 . Not a Black hole 16 / 34 Classical solution Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results . Not a Black hole The well-known Gibbons–Maeda solution can be written as p Q(r + D) 2 2 , ρ= r −D , S = P (r − D) (r − M )2 − (M 2 + D2 − Q2 − P 2 ) w= , 2 ρ It has two horizons and singular dilaton. Extremal limit: S = Q/P ⇒ constant dilaton and Reissner–Nordström solution. Non-extremal BH: diverging dilaton (no-hair theorem). 17 / 34 Semi-classical action Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results . Not a Black hole Four-dimensional action with stringy corrections: √ −2 2 2 √ −g + ∆L −g, L ∼ S R + S (∂S) − F where the correction term is second-order by curvature: α ∆L = ψ(S) LGB , 16π where LGB = R2 − 4Rµν Rµν + Rαβµν Rαβµν . 18 / 34 Hair Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results In the extremal limit the dilaton is not diverging and not vanishes: dilatonic hair. BH in higher curvature gravity . Not a Black hole ? Just hair 19 / 34 Different corrections Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results What is the correction function ψ(S) here? From the S-duality symmetry: 3 ψ(S) = − ln(2S|η(iS)|4 ) π . with the Dedekind η−function: η(τ ) ≡ e2πiτ /24 Not a Black hole ∞ Y (1 − e2πinτ ). n=1 Simple choice ψ(S) = S. 20 / 34 Decouple Maxwell field Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results . Not a Black hole The Maxwell field is given by A = −f (r)dt − m cos θdϕ with the only function f easily obtained from the equations of motion: g f = 2 . ρS 0 Here g and m are charges ’on horizon’ and the real value of the electric charge depends on the dilatonic asymptotic. 21 / 34 Find symmetries Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results . The shift of the dilaton (only the EMD part) S → βS 4 w → β w, g → g, ρ ρ→ , β r → βr, m m→ . β The charge rescaling g → γg, m → γm, Not a Black hole ρ → γρ, w w → 2, γ α → γ 2 α. 22 / 34 Formulate ICs Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries . Formulate ICs Find asymptotics Find entropy Obtain results Looking for the extremal black hole solutions: w(r0 ) = w0 (r0 ) = 0, ρ(r0) = ρ0 > 0, the asymptotic of the metrics must be of the flat space w(r) = const, ρ0 (r) = const as r → ∞. Non-singular series expansion on horizon: Not a Black hole w= ∞ X n=2 wn xn , ρ = ∞ X n=0 ρ n xn , S = ∞ X Sn xn . n=0 23 / 34 Find asymptotics Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy Obtain results . Not a Black hole The AdS2 × S 2 metrics on horizon: 2 dx 2 2 ds2H = −w2 x2 dt2 + + w dΩ 2 2. 2 w2 x The flat asymptotic (Einstein frame): 2M wE = 1 − + O(r̂ −2), r̂ 2S∞ D + O(r̂ −2). S = S∞ + r̂ 24 / 34 Find entropy Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics . Find entropy Obtain results Not a Black hole For the metrics exactly of AdS2 × S 2 type write the Sen’s ’entropy function’: Z √ f (g, m, S0 ) ≡ dθdϕ −gL. The next step is a Legendre transformation of f (g, m, S0 ) to F (∂g f, ∂m f, ∂S f ). Entropy is the extremal value of F : S = πρ2E + 4παS0 . 25 / 34 Obtain results Intro Fuzzball paradigm String—Black hole correspondence BH evaporation String action Classical action Classical solution Semi-classical action Hair Different corrections Decouple Maxwell field Find symmetries Formulate ICs Find asymptotics Find entropy . Obtain results Not a Black hole α1/4 p 1 + m2/g 2 Growing string corrections produce the BH, or Evaporating BH will produce free string. 26 / 34 Intro Fuzzball paradigm String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History Formulate ICs Various cusps . Not a Black hole 27 / 34 Motivation Intro Fuzzball paradigm String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History Formulate ICs Various cusps ‘Naked’ singularities are unpopular — at least shielded by horizon. Appears for classical BHs: Q2 + (J/M )2 ≤ M 2 . . Modifications of the EH theory are popular. But what is a corrected gravity? Corrected gravity = First order of corrections Singular solutions => Smoothed by ‘full theory’ (quantum gravity, string theory). 28 / 34 GB Cusp Intro Fuzzball paradigm Gauss–Bonnet gravity in 4D: String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History Formulate ICs Various cusps . (EH action) + (GB term) ∗ dilaton Why GB-corrections? Simple R2 correction Comes from the string theory Cusp is: Finite non-vanishing metric components Diverging second derivatives of metrics 29 / 34 Different corrections Intro Fuzzball paradigm String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History Formulate ICs Various cusps . Simple R2 correction (EDGB): 2 √ (∂µ ln S) (E) 2 L = R− + αSRGB −g. 2 2a String-theory variant (SEDGB): L (str) 2 = R + (∂µ ln S) + αR2GB √ S −g. Dilaton comes as S = e2aφ , a = 1 in string action. 30 / 34 What is the difference? Intro Fuzzball paradigm String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History Formulate ICs Various cusps . From SEDGB to EDGB (with a = 1): conformal transformation (str) (E) gµν = S −1 gµν leads to (E) ∆SGB Z X 4 1 0n √ Λn (ln S) . −g d4 x. = 16π n=2 When R2GB -correction is not small, ∆SGB is not small too! 31 / 34 History Intro Fuzzball paradigm String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History Formulate ICs Various cusps EH system without GB-correction: Schwarzschild BH with constant dilaton. EDGB system: P. Kanti et al, S.O. Alexeyev and M.V. Pomazanov . BH-solution with inner x1/2 singularity (x = |r − rs |); Naked x1/2 singularity. SEDGB system: K. Maeda et al. 32 / 34 Formulate ICs Intro Fuzzball paradigm String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History For spherically symmetric metrics 2 dr ds2 = −w(r)σ(r)2dt2 + + ρ(r)2dΩ22 , w(r) in the gauge σ = 1 the cusp ansatz will be . Formulate ICs Various cusps w= ∞ X wn/z xn/z , ρ = ∞ X ρn/z xn/z , n=0 n=0 S= ∞ X pn/z xn/z . n=0 33 / 34 Various cusps Intro Fuzzball paradigm String—Black hole correspondence Not a Black hole Motivation GB Cusp Different corrections What is the difference? History Formulate ICs Various cusps . x1/2 case: From cusp to Minkowski asymptotic; From cusp at x1 to cusp at x2 . x1/3 case: From cusp at x1 to cusp at x2 . Minkowski transition area w ∼ const, ρ ∼ x, S ∼ x. 34 / 34
