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Black Holes and Singularities in Higher Dimensions

Black Holes and Singularities in Higher Dimensions. YITP Hideo Kodama. Black Hole and Singularity Workshop at TIFR, 3 – 10 March 2006. Four Dimensions. Classical Problems. “To be naked or not to be naked, it is the problem!’’ Cosmic Censorship in the Gravitational Collapse

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Black Holes and Singularities in Higher Dimensions

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  1. Black Holes and Singularities in Higher Dimensions YITP Hideo Kodama Black Hole and Singularity Workshop at TIFR, 3 – 10 March 2006

  2. Four Dimensions

  3. Classical Problems “To be naked or not to be naked, it is the problem!’’ • Cosmic Censorship in the Gravitational Collapse Spherically symmetric ⇒Non-spherically symmetric ⇒ Non-Einstein theories • Black Hole Uniqueness and Stability Established for AF EMS ⇒extension of theories

  4. Classification of Regular AF BHs in Four Dimensions

  5. Classical Problems “To be naked or not to be naked, it is the problem!’’ • Cosmic Censorship in the Gravitational Collapse Spherically symmetric ⇒Non-spherically symmetric ⇒ Non-Einstein theories • Black Hole Uniqueness and Stability Established for AF EMS ⇒extension of theories cf. Positive Energy Theorems, topological censorship Extension to non-empty systems such as a ring-bh system • Structures and Classifications of (Naked) Singularities Massless, null, spherical ⇒ non-spherical, massive & time-like cf. TS & ZVW solutions [Kodama & Hikida 2003]

  6. Horizons and singularity of the TS2 Kodama & Hikida, Class.Quant.Grav.20:5121-5140,2003

  7. Zipoy-Voorhees Spacetime In the non-rotating limit, the TS family approaches the Zipoy-Voorhees family of static Weyl solutions.

  8. Classical Problems “To be naked or not to be naked, it is the problem!’’ • Cosmic Censorship in the Gravitational Collapse Spherically symmetric ⇒Non-spherically symmetric ⇒ Non-Einstein theories • Black Hole Uniqueness and Stability Established for AF EMS ⇒extension of theories cf. Positive Energy Theorems, topological censorship Extension to non-empty systems such as a ring-bh system • Structures and Classifications of (Naked) Singularities Massless, null, spherical ⇒ non-spherical, massive & time-like cf. TS & ZVW solutions [Kodama & Hikida 2003] • Stability and Smoothness of Various Horizons Strong (curvuture) singularities ⇒ weak singularities • Other Problems Hoop Conjecture, Causality Violation

  9. Quantum Problems “Does naked singularities produce macroscopic phenomena?“ • Black hole evaporation and information loss problem • Black hole thermodynamics • Quantum Gravity: LQG, superstring/M theory • Physical effects of naked singularities • Diverging flux?, back reaction? • Quantum particle creation • Cosmic singularity avoidance Big-bang, big crunch, big rip • Quantum gravity, higher-dimensions?

  10. Higher Dimensions

  11. Static AF Black Holes are Unique • Vacuum: unique (Tangherlini-Schwarzschild) [S. Hwang(1998), Rogatko(2003)] • Einstein-Maxwell: unique (HD RN or Majumdar-Papapetrou) [Gibbons, Ida & Shiromizu(2002), Rogatko(2003)] • Einstein-Maxwell-Dilaton system (non-degenerate): unique (Gibbons-Maeda sol)[Gibbons, Ida & Shiromizu(2002)] • Einstein-Harmonic scalar system (non-degenerate): unique (Tangherlini-Schwarzschild)[Rogatko (2002)]

  12. BH Horizon Can Be Non-Spherical • 5-dim Black Ring Solution[Emparan & Reall 2002] AF, rotating, regular vacuum solution with horizon • What kind of topologies can black hole horizons have? • General restrictions on AF bhs under the weak energy condition • Horizon of a static black hole has a positive scalar curvature or is Ricci flat. [Cai & Galloway 2001] • Horizon (and an outer apparent horizon) is of the positive Yamabe type [Galloway & Schoen 2005] • In 5 dims, horizon [Helfgot, Oz, Yanay 2005] • In 6 dims, horizon [ibid.] Cf. For the D1-D5-P supertube black solution in IIB sugra, horizon geometry is .[Elvang, Emparan, Mateos & Reall 2004] Is a black ring solution exists for D>5 ? Is there a more exotic black hole in spacetime with D>6 ?

  13. Rotating Black Holes Are Not Unique • For the 5-dim vacuum system, there exist two families of stationary 'axisymmetric' regular solutions: • Myers-Perry solution (1986): 3 params, horizon • Emparan-Reall solution (2002): 2 params, horizon

  14. Regular AF BH Solutions in Higher Dimensions M=Myers, R=Reall, E=Emparan, MP=M&Perry, ER=E&R, GG=Gauntlett&Gutowski, EEMR=E&Elvang&Mateos&R, BMPV=Breckenridge& M& Peet & Vafa, BLMPSV=BMPV&Lowe&Strominger

  15. Infinite Non-uniqueness Black Rings • For the Einstein-Maxwell(-Dilaton) system, there exists a continuous family of regular black ring solutions parametrized by a dipole charge Q for fixed mass and angular momenta [Emparan (2004)] • The dipole charge Q appears in the thermodyanamical formula: Supertubes • A configuration consisting of 3 M2-branes and 3 M5-branes wrapped on in M-theory gives 1/8-BPS solutions with 7 parameters. [EEMR 2005] • By dimensional reduction, this leads to 7 param solutions with 5 conserved charges in 5 dimensions.

  16. Black Objects Are Not Stable Black Brane Solutions Direct-product-type spacetime Vacuum Einstein equations • For D ≤ 4, possible solutions are locally • For D ≥ 5, there are infinitely many solutions if m≥ 4: e.g.

  17. Gregory-Laflamme Instability For the black brane solution the linearised Einstein equations for the S-mode perturbations can be reduced to the following two equations

  18. The effective potential V has a negative region for

  19. Black branes are unstable against S-mode perturbations with[Gregory & Laflamme 1993] • Black strings in the Λ<0 background are also unstable [Gregory 2000; Hirayama & Kang 2001] • Magnetically charged branes (NS5, Dp) in a system also exihibit similar instabilty. However, BPS branes are always stable [Gregory & Laflamme 1994; Hirayama, Kang & Lee 2003]

  20. Implication of Gregory-Laflamme Instability Non-uniqueness of black holes in spacetimes SxM [Kudoh & Wiseman 2003, 2004]

  21. Fate of Instability • Naked Singularities • Due to the famous theorem by Hawking and Ellis, a black hole horizon cannot bifurcate without formation of naked singularities. • Further, it was shown that even if naked singularities are allowed, a black string cannot be pinched off to localised black holes within a finite affine time. [Horowitz & Maeda 2001] • Nevertheless, some people argue that such a pinching off can be realised in a finite time with respect to some observers. • Kaluza-Klein Bubbles • In addition to the black string, non-uniform black string and caged black holes,there is a large family of solutions consisting of black holes and static Kaluza-Klein bubbles[Elvang & Horowitz 2003;Elvang, Harmark & Obers 2005]

  22. Cosmology Orders Naked Singularities Basic Problems in Higher-Dimensional Unification • Moduli stabilisation • Primordial inflation and dark energy Einstein Equations in IIB Sugra If is y-dependent. Hence, the 4D metricdepends not only on x but also on the internal coordinate y. Warped Compactification Moduli stabilisation requires non-vanishing flux and the warped compactification

  23. Conifold compactification • When Y is Ricci flat, 3-form flux vanishes and the 5-form flux takes the form the field equations are reduced to • For example, for the 5-form flux produced by D3 branes with flat X, we obtain a regular spacetime with full SUSY: • In general, the warp factor h is written

  24. Naked Singularities • For generic flux, the equation for h is written • From this, it follows that if Y is a compact manifold without boundary, h is smooth and , then h must be constant, i.e. no warp. • Hence, in order to get a warped solution with compact Y, we have to introduce some singular sources, such as orientifold O3with negative charge, to cancel the negative term in the right-hand side of the equation • Such a source gives rise to a singular negative contribution to the warp factor h in general, and produces naked singularity.

  25. Size Moduli Instability Assumptions • Metric • All moduli except for the size modulus are stabilised • Form fields General solution[Kodama & Uzawa, JHEP0507:061(2005),hep-th/0512104] where

  26. In the cosmological context, this instability may produce a Big-Rip singularity: Cf. In the case, this instability can be interpreted as representing D3 brane collision and associated naked singularity production[Gibbons, Lu & Pope 2004] • This size modulus instability can be stabilised only by some quantum effects and for rather special models. Cf. KKLT construction utilising instantons [Kachru, Kallosh, Linde & Trivedi 2003; Witten 1996; Denef, Douglas and Florea 2004; Denef et al 2005;Aspinwall & Kallosh 2005]

  27. NO-GO Theorem against Inflation Assume • The spacetime metric is of the warped product type • The internal space is static and compact without boundary. • The warp factor is regular and bounded everywhere. • The strong energy condition is satisfied in the full theory. Then, no accelerated expansion is allowed for the four-dimensional spacetime This theorem implies that if the string/M theory is the fundamental theory, the internal space has a boundary or is singular, or some quantum effects play crucial roles

  28. Proof For the geometry from the relation for any time-like unit vector V on X, we obtain Hence, if Y is a compact manifold without boundary, W is a smooth function on Y, and the strong energy condition is satsified in the (n+4)-dimensional theory, then the strong energy condition is satisfied on X. Cf. Raychaudhuri eq.

  29. Hidden Singularities 4-dim Braneworld Model SO(2) symmetric static regular bh solution is obtained from a half of the C-metric. The conic singularity associated with a string is hidden behind the brane.[Empran,Gregory,Santos 2001] 5-dim Braneworld Model SO(3) symmetric static regular bh solution yet to be found should have naked singularity or non-compact horizon back behind the brane, provided that a regular static AdS bh is unique. [Chamblin,Hawking,Reall 2000;Kodama 2002]

  30. Topology Change and Decompactification • Compactification produces richness in the black hole classification problem • Black hole spacetime in the real world has a (twisted) product structure in the infinity: • The black hole spacetime may decompactify near horizon. [Ishihara & Matsuno 2005] • In the universe, the structure of compactification may change spatially and/or dynamically.

  31. Supersymmetric light-like solution in M-theory The metric and 4-form flux gives a supersymmetric solution to the M-theory iff all functions depend only on u and satisfy [Ishino,Kodama,Ohta 2005] In particular, the metric gives a solution if f(u) and E(u) satisfy

  32. Dynamica decompactification? • When is constant or αu, the spacetime is flat (or a Rindler wedge), because the curvature tensor is proportional to . In the latter case, we can extend the spacetime across the null boundary to another flat region: • We can compactify this spacetime by the discrete isometry group , where

  33. In particular, if we compactify a solution such that by this method, the size of the internal space approaches constant at u=0 and diverges as u →∞. • Hence, this solution represent a dynamical decompactification transition. Unfortunately, however, we can show that there always exists a value of u for which F(u) vanishes and the transfromation becomes null boosts. At this null surface, the action of Γ become degenerate to produce a singularity.

  34. Open Problems Black Hole Classification • For each horizon topology, is there a single continuous family of black holes? • Near the Schwarzschild-Tanghelini solution, the MP solutions are unique regular AF stationary vacuum solutions [Kodama 2004] • Vacuum, stationary axisymmetric black holes with spherical horizon are unique [Morisawa & Ida 2004] • For the 5-dim N=1 minimal SUGRA, the answer is yes.[Gutowski 2004] • How large is the maximum number of parameters characterising a black hole/ring family? Cf. Horowitz & Reall(hep-th/0411286)’s critique against Bene & Warner, hep-th/0408106 • Is there a vacuum black ring solution with generic angular momenta?

  35. Classify all BPS objects in 11 & 10 dimensions. • The same classification is completed for minimum sugra in 4, 5 and 6 dimensions[Tod 1983, Gauntlett et al 2002, Gauntlett & Gutowski 2003] Cf. The analysis was extended to sugras coupled with abelian 1-multiplets and led to the multi-ring solutions and the EEMP solution. • General structure of supersymmetric solutions is determined for the M-theory and IIA/IIB sugras. [Gauntlett & Pakis 2003; Gutowski, Martelli & Reall 2003; Gauntlett, Gutowski & Pakis 2003; Gran, Papadopoulos & Gutowski 2005] Cf. Mathur Conjecture[e.g. Mathur, hep-th/0502050] • Are supersymmetric bh solutions are representative with regard to spacetime symmetry and horizon topology? A black hole is a quantum state or a fussy ball that can be described as an ensemble of microstates. Each microstate is represented by a regular classical solution without horizon.

  36. Black Hole/Brane Stability • Prove analytically that black branes are unstable only for S-modes. Cf. Seahra, Clarkson & Maartens 2005 • What the fate of the Gregory-Laflamme instability? • Are Myers-Perry solutions and black ring solutions stable? Cf. Emparan & Myers 2003; Marolf & Virmani 2005 • Develop a tractible formulation for perturbations of a rotating black hole/ring in higher dimensions. • Does the horizon area really provide a criteria for stability? • Is supersymmetry really a sufficient condition for stability in supergravity?

  37. Black Brane Perturbations Black Brane Solution Perturbation

  38. Tensor-Scalar, Vector-Vector Components • Tensor-Scalar where • Scalar-Tensor • Vector-Vector

  39. Vector-Scalar & Scalar-Vector Components • Vector-Scalar Component In terms of , let us define the vector Φ as Then, Φ obeys the coupled system of 2nd-order ODEs where • Scalar-Vector Component The equations for the scalar-vector component are obtained by the replacement from the above equations.

  40. Diagonalisable? • Special mode (l=1 ) For the special mode, by introducing the new variable Ψ defined by we obtain the decoupled 2nd-order ODEs: The first mode is also stable because

  41. Stability? From the equations for Ψ, we obtain where From this we see that there exists no mode with , but it is difficult to show that there is no unstable mode with .

  42. Generic modes ( l >1 ⇔ ) For generic modes, the above transformation yields where The eigenvalues of the matrix K are

  43. Scalar-Scalar Component • Gauge-Invariant Variables • Einstein Equations The Einstein equations lead to the following decoupled single ODE for : where

  44. The other variables obey the following intricate coupled system of ODEs:

  45. Canonical form Let Φ be the 3-component vector defined by Then, we obtain where And W is the 3-dim matrix with entries

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