A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions

Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S ² × S ¹. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).     

A Quantum Weak Energy Inequality¶for Dirac Fields in Curved Spacetime

Quantum fields are well known to violate the weak energy condition of general relativity: the renormalised energy density at any given point is unbounded from below as a function of the quantum state. By contrast, for the scalar and electromagnetic fields it has been shown that weighted averages of the energy density along timelike curves satisfy “quantum weak energy inequalities” (QWEIs) which constitute lower bounds on these quantities. Previously, Dirac QWEIs have been obtained only for massless fields in two-dimensional spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields of mass m≥ 0 on general four-dimensional globally hyperbolic spacetimes, averaging along arbitrary smooth timelike curves with respect to any of a large class of smooth compactly supported positive weights. Our proof makes essential use of the microlocal characterisation of the class of Hadamard states, for which the energy density may be defined by point-splitting. Received: 21 May 2001 / Accepted: 23 August 2001     

The global existence problem and cosmic censorship in general relativity

We formulate two global existence conjectures for the Einstein equations and discuss their relevance to the cosmic censorship conjecture. We argue that the reformulation of the cosmic censorship conjecture as a global existence problem renders it more amenable to direct analytical attack. To demonstrate the facilty of this approach we prove the cosmological version of our global existence conjecture for the Gowdy spacetimes onT ³×R.     

The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes

In the setting of asymptotically anti-de Sitter spacetimes, we consider Klein–Gordon fields subject to Dirichlet boundary conditions, with mass satisfying the Breitenlohner–Freedman bound. We introduce a condition on the \(\mathrm{b}\)-wave front set of two-point functions of quantum fields, which locally in the bulk amounts to the usual Hadamard condition, and which moreover allows to estimate wave front sets for the holographically induced theory on the boundary. We prove the existence of two-point functions satisfying this condition and show their uniqueness modulo terms that have smooth Schwartz kernel in the bulk and have smooth restriction to the boundary. Finally, using Vasy’s propagation of singularities theorem, we prove an analogue of Duistermaat and Hörmander’s theorem on distinguished parametrices.     

Moduli and (un)attractor black hole thermodynamics

We investigate four-dimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars non-minimally coupled to gauge fields. In the non-extremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of black hole thermodynamics. In the extremal limit, the near horizon geometry is AdS ₂ × S ² and the entropy does not depend on the values of moduli at infinity. We discuss the attractor behaviour by using Sen’s entropy function formalism as well as the effective potential approach and their relation with the results previously obtained through special geometry method. We also argue that the attractor mechanism is at the basis of the matching between the microscopic and macroscopic entropies for the extremal non-BPS Kaluza–Klein black hole.     

Comments on the Stress-Energy Tensor Operator in Curved Spacetime

 The technique based on a *-algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is strightforwardly generalized in order to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. Within the proposed formalism, there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress-energy tensor are assumed to be related to the interface between the quantum and classical formalism. The proposed stress-energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of the Klein-Gordon equation. Considering the averaged stress-energy tensor with respect to Hadamard quantum states, the presented definition turns out to be equivalent to an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is determined by the local geometry and the parameters which appear in the Klein-Gordon operator. In particular, no extra added-by-hand term g _αβQ and no arbitrary smooth part of the Hadamard parametrix (generated by some arbitrary smooth term ``ω ₀ ') are involved. The averaged stress-energy tensor obtained by the point-splitting procedure also coincides with that found by employing the local ζ-function approach whenever that technique can be implemented. Received: 24 September 2001/Accepted: 14 May 2002 Published online: 22 November 2002     

Quantum ’t Hooft Loops of SYM $${{\mathcal N}=4}$$ as Instantons of YM2 in Dual Groups SU(N) and SU(N)/ZN

A relation between circular 1/2 BPS ’t Hooft operators in 4d N=4{{\mathcal N}=4} SYM and instantonic solutions in 2D Yang-Mills theory (YM₂) has recently been conjectured. Localization indeed predicts that those ’t Hooft operators in a theory with gauge group G are captured by instanton contributions to the partition function of YM₂, belonging to representations of the dual group ^L G. This conjecture has been tested in the case G = U(N) =  ^L G and for fundamental representations. In this paper, we examine this conjecture for the case of the groups G = SU(N) and ^L G = SU(N)/Z _N and loops in different representations. Peculiarities when groups are not self-dual and representations not “minimal” are pointed out.     

In search of photon’s elementary axial magnetostatic field

 Experimental investigations of the photon’s B ⁽³⁾-field (third longitudinal polarization) are reported. The existence of an “axial magnetostatic field of photon” has been predicted in B _π or B ⁽³⁾-theory as the fundamental property of the circularly polarized light, and reported in numerous papers and monographs. High-sensitivity detection has been employed in photomagnetic induction, Faraday, and inverse Faraday effects (IFE) originating from such a field. The results of all three experiments clearly disprove the claims of B _π-theory. Putting together these results and theoretical calculations in perspective, it is concluded that such fields are non-existent. Received: 25 January 1996 / Revised version: 24 May 1996     

Influence of anisotropy on the dispersion of surface plasmon-phonon polaritons in silicon carbide

Surface plasmon-phonon polaritons (SPPP’s) of types 3 and 4 are investigated in doped anisotropic single crystals of hexagonal silicon carbide (6H-SiC) in the orientation corresponding to K⊥C and xy⊥C. It is shown that a dispersion dependence of the type-3 SPPP’s bounded by K appears in 6H-SiC when the plasmon frequency increases to ν _p⊥⩾350 cm⁻¹. At ν _p⊥⩾400 cm⁻¹, ν _s(K) exists for type-4 SPPP’s in the frequency range Ω _∥ ⁺ <ν< _⊥ ⁺ . When the concentration of free charge carriers is increased, the dispersion curves are displaced toward higher frequencies. The conditions for the existence of type-3 and type-4 SPPP’s in 6H-SiC are determined. Fiz. Tverd. Tela (St. Petersburg) 40, 636–639 (April 1998)     

A Mirror Symmetric Solution to the Quantum Toda Lattice

We give a representation-theoretic proof of a conjecture from Rietsch (Adv Math 217:2401–2442, 2008) providing integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SL _n /B in a uniform way to arbitrary type, and can be interpreted as a kind of mirror theorem for the full flag variety G/B. We also prove the existence of a totally positive and totally negative critical point of the ‘superpotential’ in every mirror fiber.     

Black holes with metric and fields of the same form

We present two rotating black hole solutions with axion ξ, dilaton f{\phi} and two U(1) vector fields. Starting from a non-rotating metric with three arbitrary parameters, which we have found previously, and applying the “Newman–Janis complex coordinate trick” we get a rotating metric g _μν with four arbitrary parameters namely the mass M, the rotation parameter a and the charges electric Q _E and magnetic Q _M . Then we find a solution of the equations of motion having this g _μν as metric. Our solution is asymptotically flat and has angular momentum J = M a, gyromagnetic ratio g = 2, two horizons, the singularities of the solution of Kerr, axion and dilaton singular only when r = a cos θ = 0 etc. By applying to our solution the S-duality transformation we get a new solution, whose axion, dilaton and vector fields have one more parameter. The metrics, the vector fields and the quantity l = x+ie^-2f{\lambda=\xi+ie^{-2\phi}} of our solutions and the solution of: Sen for Q _E , Sen for Q _E and Q _M , Kerr–Newman for Q _E and Q _M , Kerr, Reference Kyriakopoulos [Class. Quantum Grav. 23:7591, 2006, Eqs. (54–57)], Shapere, Trivedi and Wilczek, Gibbons–Maeda–Garfinkle–Horowitz–Strominger, Reissner–Nordstr?m, Schwarzschild are the same function of a, and two functions ρ ² = r(r + b) + a ² cos² θ and Δ = r(r + b) − 2Mr + a ² + c, of a, b and two functions for each vector field, and of a, b and d respectively, where a, b, c and d are constants. From our solutions several known solutions can be obtained for certain values of their parameters. It is shown that our two solutions satisfy the weak the dominant and the strong energy conditions outside and on the outer horizon and that all solutions with a metric of our form, whose parameters satisfy some relations satisfy also these energy conditions outside and on the outer horizon. This happens to all solutions given in the “Appendix”. Mass formulae for our solutions and for all solutions which are mentioned in the paper are given. One mass formula for each solution is of Smarr’s type and another a differential mass formula. Many solutions with metric, vector fields and λ of the same functional form, which include most physically interesting and well known solutions, are listed in an “Appendix”.     

Partially conserved axial-vector current inS-matrix theory

Mandelstam’s argument that PCAC follows from assigning Lorentz quantum numberM=1 to the massless pion is examined in the context of multiparticle dual resonance model. We construct a factorisable dual model for pions which is formulated operatorially on the harmonic oscillator Fock space along the lines of Neveu-Schwarz model. The model has bothm _π andm _ϱ as arbitrary parameters unconstrained by the duality requirement. Adler self-consistency condition is satisfied if and only if the conditionmϱ²−mπ²=1/2 is imposed, in which case the model reduces to the chiral dual pion model of Neveu and Thorn, and Schwarz. The Lorentz quantum number of the pion in the dual model is shown to beM=0.     

Boost-rotation symmetric gravitational null cone data

The potential role of boost-rotation symmetric vacuum spacetimes as test beds for numerical studies of gravitational radiation is discussed. For application to null cone evolution codes, these spacetimes are analyzed in terms of their data on the preferred null cone left invariant by the symmetry group. On this cone, an explicit solution of the Bondi hypersurface and evolution equations is found. This solution has a smooth vertex, a smooth interior, and, except for polar singularities, admits a well-defined ⁺.This work was supported in part by NSF grant No. PHY800823 to the University of Pittsburgh.     

Entanglement and relaxation in exactly solvable models

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρ^ℐ(t) approaches an equilibrium state ρ_eq^ℐ, the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ_eq^ℐ, which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.     

Nuclearity and Thermal States in Conformal Field Theory

We introduce a new type of spectral density condition, that we call L ²- nuclearity. One formulation concerns lowest weight unitary representations of and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the conformal Hamiltonian L ₀, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L ₀ is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a β-KMS state for the translation dynamics on the net of C^*-algebras for every inverse temperature β > 0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L ²-nuclearity is satisfied for the scalar, massless Klein-Gordon field. Dedicated to László Zsidó on the occasion of his sixtieth birthday Supported by MIUR, GNAMPA-INDAM and EU network “Quantum Spaces–Non Commutative Geometry” HPRN-CT-2002-00280     

Chiral phase structure of QCD with many flavors

We investigate QCD with a large number of massless flavors with the aid of renormalization group flow equations. We determine the critical number of flavors separating the phases with and without chiral symmetry breaking in SU(N_c) gauge theory with many fermion flavors. Our analysis includes all possible fermionic interaction channels in the pointlike four-fermion limit. Constraints from gauge invariance are resolved explicitly and regulator-scheme dependencies are studied. Our findings confirm the existence of an N_f window where the system is asymptotically free in the ultraviolet, but remains massless and chirally invariant on all scales, approaching a conformal fixed point in the infrared. Our prediction for the critical number of flavors of the zero-temperature chiral phase transition in SU(3) is N_f^cr=10.0±0.29 (fermion)^+1.55_-0.63 (gluon), with the errors arising from approximations in the fermionic and gluonic sectors, respectively. PACS 11.10.Hi, 11.15.Tk, 11.30.Rd     

Decay of Solutions of the Wave Equation in the Kerr Geometry

We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in L ^∞ _loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable ω on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. Research supported in part by the Deutsche Forschungsgemeinschaft. Research supported by NSERC grant #RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30. An erratum to this article is available at .     

Homothetic and conformal symmetries of solutions to Einstein's equations

We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.     

On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations

We show existence and uniqueness of asymptotically flat solutions to the stationary Einstein equations inS=³–B _r , whereB _r is a ball of radiousr>0, when a small enough continuous complex function û on S is given. Regularity and decay estimates imply that these solutions are analytic in the interior ofS and also at infinity, when suitably conformally rescaled.