Entropy function approach to charged BTZ black hole

We find solution to the metric function f(r) = 0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M _e (Q) = Q ²(1 − ln Q ²). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS₂ × S¹ represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald’s Noether formalism to reproduce the entropy of the extremal charged BTZ black hole without normalization when using the dilaton gravity approach.     

Gravitational collapse with charge and small asymmetries. II. Interacting electromagnetic and gravitational perturbations

Paper I analyzed the evolution of nonspherical scalar-field perturbations of an electrically charged, collapsing star; this paper treats coupled electromagnetic and gravitational perturbations. It employs the results of recent detailed work in which coupled perturbations were studied in a gauge-invariant manner by using the Hamiltonian (Moncrief s) approach and the Newman-Penrose formalism, and the relations between the fundamental quantities of these two methods were obtained.It is shown that scalar-field perturbations are a prototype for coupled perturbations. The collapse produces a Reissner-Nordström black hole, and the perturbations are radiated away completely. Alll-pole parts of the perturbations of the metric and the electromagnetic field decay according to power laws; in the extreme case (e ² =M ²), the interaction causes the quadrupole perturbations to die out more slowly than the dipole perturbations.     

Relativistic interactions without fields or potentials

Employing Poincaré degrees of freedomM ^jk=(¯K,¯J) andP ^k=(E,¯p) transforming linearly (but inhomogeneously) under the action of the Poincaré group we define a number of quantities which we later identify with physical observables. The identifications are consistent with the nonrelativistic limit and with other requirements following from the Poincaré covariance. Next, we treat a free relativistic particle as composed of two interacting parts. Relativistic quantum commutation relations for their Poincaré algebras and a kind of (inverse) relativistic correspondence principle are used to generate (quasi-) classical equations of their relative motion. A simple example based on these ideas is explicitly solved.I am indebted to Prof. B. Laurent, Dr. S. Flodmark, and Prof. I. Fischer-Hjalmars for pointing this out to me.     

Gravitational collapse with charge and small asymmetries I. Scalar perturbations

A recent analysis by Richard Price of spherical collapse with small nonspherical perturbations is here generalized to the case of an electrically charged collapsing star (0¦Q¦-M). The perturbations are confined to a scalar field generated by a nonspherical distribution of scalar charge in the star. As in the electrically neutral case, the scalar perturbations are probably a prototype for all others — electromagnetic, gravitational, and higherspin. The collapse is shown to produce a Reissner-Nordström black hole, and the scalar-field perturbations are shown to radiate completely away; but they die out more slowly the larger is the star's electric charge. For charge ¦Q¦M, the -pole part of the perturbation at fixedr and late times is dominated by a tail that dies out ast ^{–(2+ 2)}. But for ¦Q¦=M, the primary outgoing waves emitted from the star's surface are everywhere larger than the tail. At fixedr and late times they die as t^–(+2). Also, a calculation of the redshift shows that a collapsing star becomes black more slowly the larger is the star's electric charge.Supported in part by the National Science Foundation (GP-27304, GP-28027, GP-19887).     

Bäcklund transformations for harmonic maps in two independent variables

Bäcklund transformations for harmonic maps are described as the action of the structure group on harmonic one-forms or as gauge transformations of the soliton connection constructed via embedding the configuration manifold into a flat space. As an illustration, Bäcklund transformations for maps fromM ² to the Poincaré upper half-plane and for maps determining stationary vacuum gravitational fields with axial symmetry are obtained.     

Additive conservation laws in local relativistic quantum field theory

We consider generatorsQ of symmetry transformations acting additively on asymptotic particle states according to (1.1). [This equation can be derived forQ defined as integral over a conserved local current!]. For simplicity, we consider only the case that all asymptotic fields are scalar. Assuming that elastic scattering occurs at least in an open subset of the scattering manifold we show thatQ is at most alinear combination of generators of the Poincaré group and internal symmetries.     

Magnetic charge,black holes,and cosmic censorship

The possibility of converting a Reissner-Nordström black hole into a naked singularity by means of test particle accretion is considered. The dually charged Reissner-Nordström metric describes a black hole only when M² > Q³ + P². The test particle equations of motion are shown to allow test particles with arbitrarily large magnetic charge/mass ratios to fall radially into electrically charged black holes. To determine the nature of the final state (black hole or naked singularity) an exact solution of Einstein's equations representing a spherical shell of magnetically charged dust falling into an electrically charged black hole is studied. Naked singularities are never formed so long as the weak energy condition is obeyed by the infalling matter. The differences between the spherical shell model and an infalling point test particle are examined and discussed.     

The cauchy problem for non-linear Klein-Gordon equations

We consider in ⁿ⁺¹,n2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincaré covariant then the non-linear representation of the Poincaré Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincaré group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincaré group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.     

Convergence of an Exact Quantization Scheme

It has been shown by Voros [V1] that the spectrum of the one-dimensional homogeneous anharmonic oscillator (Schrödinger operator with potential q^2M, M=2,3,...) is a fixed point of an explicit non-linear transformation. We show that this fixed point is globally and exponentially attractive in spaces of properly normalized sequences.Partially supported by Faperj and CNPq, BrazilCurrent address:Laboratoire de Probabilités et Modéles aléatoires, Université Pierre et Marie Curie–Boi te courrier 188, 75252–Paris Cedex 05, France. E-mail: artur@ccr.jussieu.fr     

Spin flip of neutrinos with magnetic moment in core-collapse supernova

Neutrinos with magnetic moment experience chirality flips while scattering off charged particles. It is known that if neutrino is a Dirac fermion, then such chirality flips lead to the production of sterile right-handed neutrinos inside the core of a star during the stellar collapse, which may facilitate the supernova explosion and modify the supernova neutrino signal. In the present paper we reexamine the production of right-handed neutrinos during the collapse using a dynamical model of the collapse. We refine the estimates of the values of the Dirac magnetic moment which are necessary to substantially alter the supernova dynamics and neutrno signal. It is argued in particular that Super-Kamiokande will be sensitive at least to μ _{ν Dirac} = 10⁻¹³μ_B in case of a galactic supernova explosion. Also we briefly discuss the case of Majorana neutrino magnetic moment. It is pointed out that in the inner supernova core spin flips may quickly equilibrate electron neutrinos with nonelectron antineutrinos if μ _{ν Majorana} ≳ 10⁻¹²μ_B. This may lead to various consequences for supernova physics.     

Contractions of bialgebras and super-bialgebras

The contraction scheme of Z ₂ graded super-bialgebras is discused. The existence of the contraction limit and the classical r-matrix with and without the coproduct rescaling is investigated. As an example the contraction of D = 4 Poincaré N = 2 super-bialgebra to D = 2 Poincaré N = 2 super-bialgebra is given.     

A comparison of high statistics measurements of the proton and deuteron structure functions

We present a detailed comparison of high statistics measurements of the structure functionF ₂(x, Q ²) from deep inelastic scattering of charged leptons on hydrogen and deuterium targets. It is found that the CERN results from the EMC and BCDMS Collaborations are both compatible with the lowQ ² electron scattering results from SLAC after an adjustment of the relative normalizations. We discuss the residual discrepancy between EMC and BCDMS which persists after this renormalization and find that the SLAC data are in better agreement with the steeperx dependence of the BCDMS results. A phenomenological parametrization ofF ₂(x, Q ²) is given.     

Exact solution of Poincaré gauge field equations of gravity with torsion

Under empty, static, and spherically symmetric conditions we find an exact metric solution of the Poincaré gauge field equations. The Schwarzschild metric solution is contained in the solution and we also obtain new gauge correction termsr ^–1 andr ² lnr.     

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Measurement of the Q and energy dependence of diffractive interactions at HERA

Diffractive dissociation of virtual photons, , has been studied in ep interactions with the ZEUS detector at HERA. The data cover photon virtualities 0.17 < Q ² < 0.70 GeV² and 3 < Q ² < 80 GeV² with 3 < M_X < 38 GeV, where M_X is the mass of the hadronic final state. Diffractive events were selected by two methods: the first required the detection of the scattered proton in the ZEUS leading proton spectrometer (LPS); the second was based on the distribution of M_X. The integrated luminosities of the low- and high-Q² samples used in the LPS-based analysis are 0.9 pb^-1 and 3.3 pb^-1, respectively. The sample used for the M_X-based analysis corresponds to an integrated luminosity of 6.2 pb^-1. The dependence of the diffractive cross section on W, the virtual photon-proton centre-of-mass energy, and on Q² is studied. In the low-Q² range, the energy dependence is compatible with Regge theory and is used to determine the intercept of the Pomeron trajectory. The W dependence of the diffractive cross section exhibits no significant change from the low-Q² to the high-Q² region. In the low-Q² range, little Q² dependence is found, a significantly different behaviour from the rapidly falling cross section measured for Q ² > 3 GeV². The ratio of the diffractive to the virtual photon-proton total cross section is studied as a function of W and Q². Comparisons are made with a model based on perturbative QCD. Received: 27 March 2002 / Published online: 9 August 2002     

An exact solution of gauge field equations of Poincaré gravity with torsion and spin current

Subject to the static spherical-symmetric condition, we found an exact metric solution and spin current of the Poincaré gauge field equations derived in Ref. 1. In the solution there are the terms of the Schwarzschild metric solution and new gauge termsr ^–1 andr ² lnr.     

Poincaré Algebra and Space-Time Critical Dimensions for Parabosonic Strings

We construct the parabosonic string formalism based on the paraquantization of both the center of mass variables and the excitation modes of the string. A critical study of the different commutators of the Poincaré algebra, based on the redefinition of its generators and the direct treatment using trilinear relations, is done. Space-time critical dimensions D as functions of the paraquantization order Q are obtained.     

Towards a unified dynamical theory of the Brownian particle in an ideal gas

We consider the trajectoryQ ^M(t) of a Brownian particle of massM in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 (the dynamics is uniform motion plus elastic collisions with the Brownian particle). Our theory, in dimension one, describes a variety of limiting processes — containing the Wiener process and the Ornstein-Uhlenbeck process — forA ^–1/2 Q ^M(A)(At) depending on the asymptotic behaviour ofM(A). Part of the theory is hypothetical while another part relies upon known results. We also prove that, ifA ^1/2+M(A)A, thenA ^–1/2 Q ^M(A) (At) converges to a Wiener process whose variance is known from papers of Sinai-Soloveichik and of the present authors.     

Poincaré stress and the Reissner-Nordström repulsion

The Reissner-Nordström repulsion is shown to be a consequence of Poincaré stresses in a static, charged object. The strong energy condition implies that the Reissner-Nordström repulsions do not stop a neutral test particle, falling freely from rest at infinity in the fields of a charged body, before it hits the surface of the body. However, if the particle falls from rest at a sufficiently small height above the surface of the body, it will not reach the surface due to the Reissner-Nordström repulsion.