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Microstate Geometries Resolving the Structure of Black Holes IHES September 19, 2013 Recent work with: Iosif Bena, Gary Gibbons Based on Collaborations with: N. Bobev, G. Dall’Agata, J. de Boer, S. Giusto, Ben Niehoff, M. Shigemori, A. Puhm, C. Ruef, O. Vasilakis, C.-W. Wang Outline • Motivation: Solitons and Microstate Geometries • Smarr Formula: “No Solitons without Horizons” • Topological stabilization: “No Solitons without Topology” • Microstates and fluctuations of microstate geometries • New scales in microstate geometries/black holes • Conclusions Solitons versus Particles Electromagnetism: ‣ Divergent self-energy of point particles ... ‣ Self-consistency/Completeness: Motion of particles should follow from action of electromagnetism ... ★ Replace point sources by smooth “lumps” of classical fields ⇒ Mie, Born-Infeld: Non-linear electrodynamics Yang-Mills • Non-abelian monopoles and Instantons General Relativity Non-linearities ⇒ new classes of solitons? Four dimensional GR, electromagnetism + asymptotically flat: “No Solitons without horizons” Nearest thing: Extreme, supersymmetric multi-black-hole solutions Hawking Radiation versus Unitary Evolution of Black Holes Black hole uniqueness ⇒ Universality of Hawking Radiation Independent of details and states of matter that made the black hole Complete evaporation of the black hole ⇒ Loss of information about the states of matter that made the black hole Entangled State of Hawking Radiation H 0 0 1 0 0 1 1 √2 ( + ) Evaporation of the black hole: ⇒ 1 1 Sum over internal states H Pure state → Density matrix Entanglement of N Hawking quanta with internal black hole state = N ln 2 Complete evaporation + Entanglement ⇒ Hawking radiation cannot be described by a simple wave function Tension of Black hole uniqueness and Unitarity of Quantum Mechanics Fix with small corrections to GR? Entangled State of Hawking Radiation 1 1 0 0 0 1 1 + ε ( 0 0 - 1 1 ) + (ε 0 1 + ε2 ) √2 ( + ) 1 Restore the pure state over vast time period for evaporation? Mathur (2009): Corrections cannot be small for information recovery ⇒ There must be O(1) to the Hawking states at the horizon. New physics at the horizon scale? • Is there a way to avoid black holes and horizons in the low energy (massless) limit of string theory = supergravity? • Can it be done in a manner that looks like a black hole on large scales in four dimensions? Are there horizonless solitons? Microstate Geometries: Definition ‣ Solution to the bosonic sector of supergravity as a low energy limit of string theory ‣ Smooth, horizonless solutions with the same asymptotic structure as a given black hole or black ring Singularity resolved; Horizon removed Simplifying assumption: ‣ Time independent metric (stationary) and time independent matter Smooth, stable, end-states of stars in massless bosonic sector of string theory? This is supposed to be impossible because of many no-go theorems: “No Solitons without horizons” Intuition: Massless fields travel at the speed of light ... only a black hole can hold such things into a star. The Komar Mass Formula In a D-dimensional space-time with a Killing vector, K, that is time-like at infinity one has 1 (D 2) @ M = − dK K = − 16⇡GD (D 3) D 2 ⇤ @t − ZS − where SD-2 is (topologically) a sphere near spatial infinity in some hypersurface, Σ. SD-2 Σ 16⇡GD M g00 = 1+ D 3 + ... dK (@⇢g00) (dt d⇢) − (D 2) AD 2 ⇢ − − − ⇤ ⇡ ⇤ ^ More significantly µ ⌫ 1 d dK = 2 (K R dx ) Rµ⌫ =8⇡GD T µ⌫ T gµ⌫ ⇤ − ⇤ µ⌫ − (D 2) ⇣ − ⌘ If Σ is smooth with no interior boundaries: 1 (D 2) µ ⌫ 0 M = − K Rµ⌫ d⌃ T 00 d⌃ 8⇡GD (D 3) ⌃ ⇡ − Z linearized Z⌃ Smarr Formula I H1 H2 More generally, Σ will have interior boundaries that can be located at horizons, HJ. D-2 Excise horizon S ⌃ ⌃ interiors: ! Σ 0 8⇡GD (D 3) µ ⌫ 1 − Me = Rµ⌫ K d⌃ + 2 dK (D 2) ⌃ HJ ⇤ − Z XHJ Z e Null generators of Kerr-like horizons: ⇠ = K + ⌦~ L~ H · H Surface gravity ⇠a ⇠b = ⇠b ⇒ 1 d⇠ = of horizon, κ a 2 ⇤ A r ZH 1 ~ ~ ⇒ 2 dK = HI HI +8⇡GD ⌦HI JHI HJ ⇤ A · XHJ Z XHI h i Vacuum outside horizons: 8⇡GD (D 3) ~ ~ − M = HI +8⇡GD ⌦HI JHI (D 2) AHI · − XHI h i Smarr Formula II: No Solitons Without Horizons If Σ is smooth with no 8⇡G (D 3) D − M = R Kµ d⌃⌫ interior boundaries: (D 2) µ⌫ − Z⌃ µ ⌫ Goal: Show that Rµ⌫ K d⌃ = Boundary term Z⌃ (with no contribution at infinity) ✦ Not true for massive fields ... but (almost) true for massless fields If Σ is a smooth space-like hypersurface populated only by smooth solitons (no horizons) the one must have: SD-2 Σ M ≡ 0 Positive mass theorems with asymptotically flatness: ⇒ Space-time can only be globally flat, R4,1 ⇒ “No Solitons Without Horizons” .... It all comes down to: 1 (D 2) M = − (KµR dx⌫ ) 8⇡G (D 3) ⇤D µ⌫ D − Z⌃ and “No solitons without horizons” requires showing that µ ⌫ (K Rµ⌫ dx )=d(γD 2) ⇤ − for some global (D-2)-form, γ. Bosonic sector of a generic massless supergravity A K • Graviton, gμν • Scalars, Φ • Tensor gauge fields, F(p) Scalar matrices in kinetic terms: QJK(Φ), MAB(Φ) K Bianchi: d(F(p) ) = 0 K Equations of motion: d❋(QJK(Φ) F(p) ) = 0 K JK IK I Define: GJ,(D-p) ≡ ❋ (QJK(Φ) F(p) ) and Q by Q QKJ = δ J K then: d(F(p) ) = 0 and d(GJ,(D-p)) = 0 Einstein equations: ⇢1...⇢p 1 I J − I J ⇢1...⇢p Rµ⌫ = QIJ F µ⇢1...⇢p 1 F ⌫ cgµ⌫ F ⇢1...⇢p F − − h A B i + M AB @µΦ @⌫ Φ h i I J ⇢1...⇢p 1 A B − + M @ Φ @ Φ = a QIJ F µ⇢1...⇢p 1 F ⌫ AB µ ⌫ − IJ ⇢1...⇢D p 1 h i − − + b Q GI µ⇢1...⇢D p 1 GJ ⌫ − − for some constants a,b and c Time Independent Solutions Killing vector, K, is time-like at infinity I A Assume time-independent matter: F =0, K Φ =0 LK L ⇒ G =0 LK I 0 µ µ I J ⇢1...⇢p 1 K R = − µ A B µ⌫ a QIJ K F µ⇢1...⇢p 1 F ⌫ + M AB K @µΦ @⌫ Φ − IJ µ ⇢1...⇢D p 1 h i − − + b Q K GI µ⇢1...⇢D p 1 GJ ⌫ − − A µ A µ • Φ =0 ⇔ K @µΦ =0 ⇒ Scalars drop out of Rµ⌫ K LK 0 0 • Cartan formula for forms: K ! = d(iK (!)) + iK (d!) L I I d(F(p) ) = 0, d(GJ,(D-p)) = 0 ⇒ d(iK(F(p) )) = 0, d(iK(GJ,(D-p))) = 0 I I • Ignore topology: iK(F(p) ) = dα(p-2) , iK(GJ,(D-p)) = dβJ,(D-p-2) J J • Define (D-2)-form, γD-2 = a α(p-2) ∧ GJ,(D-p) + b βJ,(D-p-2) ∧ F(p) µ ⌫ Then: (K Rµ⌫ dx )=d(γD 2) ⇤ − ⇒ M = 0 ⇒ “No Solitons Without Horizons” .... Omissions: • Topology • Chern-Simons terms Equations of motion in generic massless supergravity: K d❋(QJK(Φ) F(p) ) = Chern-Simons terms ⇒ d(GJ,(D-p)) = Chern-Simons terms µ ⌫ ⇒ (K Rµ⌫ dx )=d(γD 2) + Chern-Simons terms ⇤ − ⇒ M ~ Topological contributions + Chern-Simons terms Five Dimensional Supergravity N=2 Supergravity coupled to two vector multiplets Three Maxwell Fields, F I, two scalars, X I, X 1X 2X 3 = 1 S = p gd5x R 1 Q F I F Jµ⌫ Q @ XI @µXJ 1 C F I F J AK ✏¯µ⌫⇢σλ − − 2 IJ µ⌫ − IJ µ − 24 IJK µ⌫ ⇢ λ Z ⇣ ⌘ 1 1 2 2 2 3 2 Q = diag (X )− , (X )− , (X )− IJ 2 Einstein Equations: ⇢ R = Q F I F J 1 g F I F J ⇢ + @ XI @ XJ µ⌫ IJ µ⇢ ⌫ − 6 µ⌫ ⇢ µ ⌫ h 2 2 2 i 2 Generic stationary metric: ds = Z− (dt + k) + Zds 5 − 4 Four-dimensional spatial base slices, Σ: • Assume simply connected • Topology of interest: H2(Σ,Z) ≠ 0 Simple connectivity I I I I d(iK (F )) = 0 ⇒ iK (F )=dλ for some functions, λ ⇢ ⇢⌫ Kµ Q F I F J = Q λI F J + 1 C ✏⌫↵βγδ λI F J F K IJ µ⇢ ⌫ r⇢ IJ 16 IJK ↵ γδ = Boundary term + Chern-Simons contribution J Dual 3-forms: GI = *5 QIJ F Cartan: d(iK (GI )) = iK (d(GI )) − d(G )=d (Q F J ) C F J F K ≠ 0 I ⇤ IJ ⇠ IJK ^ J J L M L M Use i F = dλ ⇒ iK CILM F F CILM d λ F K ^ ⇠ 1 J K Therefore d iK (GI )+ 2 CIJK λ F =0 ⇣ ⌘ ⇒ i (G )=dβ 1 C λJ F K + H K I I − 2 IJK I where βI are global one-forms and HI are closed but not exact two forms ... Kµ QIJ G G ⌫⇢σ = 2 QIJ β G ⇢⌫σ 1 C ✏⌫↵βγδ λI F J F K Iµ⇢ J − r⇢ I σ J − 4 IJK ↵ γδ IJ ⇢ + Q HI GJ ⇢⌫ Generalized Smarr Formula 2 I J ⇢ I J 1 IJ ⇢ Rµ⌫ = QIJ 3 F µ⇢ F ⌫ + @µX @⌫ X + 6 Q GIµ⇢ GJ ⌫ h i µ I K @µX =0 ⇢ ⇢⌫ Kµ Q F I F J = Q λI F J + 1 C ✏⌫↵βγδ λI F J F K IJ µ⇢ ⌫ r⇢ IJ 16 IJK ↵ γδ µ IJ ⌫⇢σ IJ ⇢⌫σ 1 ⌫↵βγδ I J K K Q GIµ⇢ GJ = 2 ⇢ Q β GJ CIJK ✏ λ F ↵ F γδ − r I σ − 4 IJ ⇢ + Q HI GJ ⇢⌫ ⇒ KµR = 1 µ 2 Q λI F J + QIJ β σ G + 1 QIJ H⇢ G µ⌫ − 3 r IJ µ⌫ I J µ⌫ 6 I J ⇢⌫ ⇥ boundary terms ⇤ cohomology Chern-Simons contributions cancel! 3 µ ⌫ 1 J ⇒ M = K Rµ⌫ d⌃ = HJ F 16⇡G5 ⌃ 16⇡G ^ Z 5 Z⌃ i (G )=dβ 1 C λJ F K + H K I I − 2 IJK I No Solitons without Topology If Σ is a smooth hypersurface with no interior boundaries SD-2 Σ 3 µ ⌫ 1 J M = K Rµ⌫ d⌃ = HJ F 16⇡G5 ⌃ 16⇡G ^ Z 5 Z⌃ The mass can topologically supported by the cohomology H2(Σ,R) Stationary end-state of star held up by topological flux ..
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