Resolving the Structure of Black Holes IHES September 19, 2013 - PowerPoint PPT Presentation

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: Sum over internal states ⇒ 1 1 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 ( ) + ( ε 1 ) + ε 2 ) + ε ( - 1 0 + 0 1 0 0 1 1 0 0 1 1 √ 2 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) Z ∂ M = S D − 2 ∗ dK K = − 16 π G D ( D − 3) ∂ t where S D-2 is (topologically) a sphere near spatial infinity in some hypersurface, Σ . Σ S D-2 16 π G D M g 00 = − 1 + ρ D − 3 + . . . − ( ∂ ρ g 00 ) ∗ ( dt ∧ d ρ ) ∗ dK ≈ ( D − 2) A D − 2 More significantly 1 ⇣ ⌘ d ∗ dK = − 2 ∗ ( K µ R µ ν dx ν ) R µ ν = 8 π G D T µ ν − ( D − 2) T g µ ν If Σ is smooth with no interior boundaries : Z 1 ( D − 2) Z T 00 d Σ 0 K µ R µ ν d Σ ν M = ≈ 8 π G D ( D − 3) linearized Σ Σ

⇒ ⇒ Smarr Formula I H 2 H 1 More generally, Σ will have interior boundaries that can be located at horizons, H J . S D-2 Excise horizon Σ → e Σ Σ interiors: 0 8 π G D ( D − 3) Z Z R µ ν K µ d Σ ν + X 1 M = ∗ dK 2 ( D − 2) e Σ H J H J ⇠ = K + ~ Ω H · ~ L H Null generators of Kerr-like horizons: Z Surface gravity ξ a r a ξ b = κ ξ b 1 ∗ d ξ = κ A of horizon, κ 2 H Z h i X X  H I A H I + 8 ⇡ G D ~ Ω H I · ~ 1 ∗ dK = J H I 2 H J H J H I Vacuum outside horizons: 8 ⇡ G D ( D − 3) h i X  H I A H I + 8 ⇡ G D ~ Ω H I · ~ M = J H I ( D − 2) H I

Smarr Formula II: No Solitons Without Horizons If Σ is smooth with no 8 π G D ( D − 3) Z R µ ν K µ d Σ ν M = interior boundaries : ( D − 2) Σ Z R µ ν K µ d Σ ν Goal : Show that = Boundary term (with no contribution at infinity) Σ ✦ 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: S D-2 S D-2 Σ Σ M ≡ 0 Positive mass theorems with asymptotically flatness: ⇒ Space-time can only be globally flat, R 4,1 ⇒ “No Solitons Without Horizons” ....

It all comes down to: 1 ( D − 2) Z ∗ D ( K µ R µ ν dx ν ) M = 8 π G D ( D − 3) Σ and “No solitons without horizons” requires showing that ∗ ( K µ R µ ν dx ν ) = d ( γ D − 2 ) for some global (D-2) -form, γ .

Bosonic sector of a generic massless supergravity • Tensor gauge fields, F (p)K • Graviton, g μν • Scalars, Φ A Scalar matrices in kinetic terms: Q JK ( Φ ), M AB ( Φ ) Bianchi: d( F (p)K ) = 0 Equations of motion: d ❋ ( Q JK ( Φ ) F (p)K ) = 0 Define: G J,(D-p) ≡ ❋ ( Q JK ( Φ ) F (p)K ) and Q JK by Q IK Q KJ = δ IJ then: d( F (p)K ) = 0 and d( G J,(D-p) ) = 0 Einstein equations: h ρ 1 ... ρ p F J ρ 1 ... ρ p i ρ 1 ... ρ p − 1 − c g µ ν F I F I µ ρ 1 ... ρ p − 1 F J R µ ν = Q IJ ν h ∂ µ Φ A ∂ ν Φ B i + M AB ρ 1 ... ρ p − 1 h ∂ µ Φ A ∂ ν Φ B i = a Q IJ F I µ ρ 1 ... ρ p − 1 F J + M AB ν + b Q IJ G I µ ρ 1 ... ρ D − p − 1 G J ν ρ 1 ... ρ D − p − 1 for some constants a,b and c

⇒ ⇔ ⇒ Time Independent Solutions Killing vector, K, is time-like at infinity L K F I = 0 , L K Φ A = 0 Assume time-independent matter: L K G I = 0 0 a Q IJ K µ F I ρ 1 ... ρ p − 1 h K µ ∂ µ Φ A ∂ ν Φ B i K µ R µ ν = µ ρ 1 ... ρ p − 1 F J + M AB ν + b Q IJ K µ G I µ ρ 1 ... ρ D − p − 1 G J ν ρ 1 ... ρ D − p − 1 • K µ ∂ µ Φ A = 0 L K Φ A = 0 Scalars drop out of R µ ν K µ 0 0 • Cartan formula for forms: L K ω = d ( i K ( ω )) + i K ( d ω ) d( F (p)I ) = 0 , d( G J,(D-p) ) = 0 ⇒ d(i K ( F (p)I )) = 0 , d(i K ( G J,(D-p) )) = 0 • Ignore topology: i K ( F (p)I ) = d α ( p-2 )I , i K ( G J,(D-p) ) = d β J,( D-p-2 ) • Define (D-2)-form, γ D-2 = a α ( p-2 )J ∧ G J,(D-p) + b β J,( D-p-2 ) ∧ F (p)J ∗ ( K µ R µ ν dx ν ) Then: = d ( γ D − 2 ) ⇒ “No Solitons Without Horizons” .... ⇒ M = 0

⇒ Omissions: • Topology • Chern-Simons terms Equations of motion in generic massless supergravity: d ❋ ( Q JK ( Φ ) F (p)K ) = Chern-Simons terms ⇒ d( G J,(D-p) ) = Chern-Simons terms ∗ ( K µ R µ ν dx ν ) + Chern-Simons terms = d ( γ D − 2 ) ⇒ 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 1 X 2 X 3 = 1 Z √− g d 5 x ⇣ ✏ µ νρσλ ⌘ 2 Q IJ F I µ ν F Jµ ν − Q IJ @ µ X I @ µ X J − 1 24 C IJK F I µ ν F J ρσ A K R − 1 S = λ ¯ Q IJ = 1 ( X 1 ) − 2 , ( X 2 ) − 2 , ( X 3 ) − 2 � � 2 diag Einstein Equations: h ρσ F J ρσ + ∂ µ X I ∂ ν X J i ρ − 1 F I µ ρ F J 6 g µ ν F I R µ ν = Q IJ ν − Z − 2 ( dt + k ) 2 + Z ds 2 ds 2 5 = Generic stationary metric: 4 Four-dimensional spatial base slices, Σ : • Assume simply connected • Topology of interest: H 2 ( Σ ,Z) ≠ 0

⇒ ⇒ ⇒ Simple connectivity for some functions, λ I i K ( F I ) = d λ I d ( i K ( F I )) = 0 ρ � Q IJ � I F J ρν � 16 C IJK ✏ ναβγδ � I F J K µ � Q IJ F I µ ρ F J αβ F K � 1 � r ρ = + ν γδ = Boundary term + Chern-Simons contribution Cartan: Dual 3-forms: G I = * 5 Q IJ F J d ( i K ( G I )) = − i K ( d ( G I )) d ( G I ) = d ∗ ( Q IJ F J ) ∼ C IJK F J ∧ F K ≠ 0 C ILM F L ∧ F M � λ L F M � � � Use i K F J = d λ J i K ∼ C ILM d ⇣ 2 C IJK λ J F K ⌘ 1 Therefore i K ( G I ) + = 0 d 2 C IJK λ J F K + H I 1 i K ( G I ) = d β I − where β I are global one-forms and H I are closed but not exact two forms ... Q IJ G I µ ρσ G J Q IJ � I σ G J 4 C IJK ✏ ναβγδ � I F J K µ � νρσ � ρνσ � αβ F K 1 � = � 2 r ρ � γδ + Q IJ H ρσ I G J ρσν

⇒ ⇒ Generalized Smarr Formula ρ + ∂ µ X I ∂ ν X J i h 6 Q IJ G I µ ρσ G J ν 3 F I µ ρ F J 2 1 ρσ R µ ν = Q IJ + ν K µ ∂ µ X I = 0 ρ � Q IJ � I F J ρν � 16 C IJK ✏ ναβγδ � I F J K µ � Q IJ F I µ ρ F J αβ F K � 1 � r ρ = + ν γδ Q IJ G I µ ρσ G J Q IJ � I σ G J 4 C IJK ✏ ναβγδ � I F J K µ � νρσ � ρνσ � αβ F K 1 � = � 2 r ρ � γδ + Q IJ H ρσ I G J ρσν 2 Q IJ λ I F J σ G J µ νσ µ ν + Q IJ β I 6 Q IJ H ρσ K µ R µ ν = 3 r µ ⇥ � 1 ⇤ 1 + I G J ρσν boundary terms cohomology Chern-Simons contributions cancel! Z 1 Z 3 H J ∧ F J M = K µ R µ ν d Σ ν = 16 π G 5 16 π G 5 Σ Σ 2 C IJK λ J F K + H I 1 i K ( G I ) = d β I −