Massive Klein-Gordon Equation Research Articles

Ladder operators and quasinormal modes in Bañados-Teitelboim-Zanelli black holes

We study quasinormal modes (QNMs) of massive Klein-Gordon fields in static Ba\~nados-Teitelboim-Zanelli (BTZ) black holes in terms of ladder operators constructed from spacetime conformal symmetries. Because the BTZ spacetime is locally isometric to the three-dimensional anti-de Sitter spacetime, ladder operators, which map a solution of the massive Klein-Gordon equation into that with different mass squared, can be constructed from spacetime conformal symmetries. In this paper, we apply the ladder operators to the QNMs of the Klein-Gordon equations in the BTZ spacetime. We demonstrate that the ladder operators can change indices of QNM overtones, and all overtone modes can be generated from a fundamental mode when we impose the Dirichlet or Neumann boundary condition at infinity. We also discuss the case with the Robin boundary condition.

Hyperbolic metamaterials and massive Klein-Gordon equation in ( 2+1 )-dimensional de Sitter spacetime

The wave equation obeyed by the extraordinary component of the electric field in a hyperbolic metamaterial was shown to be a massless Klein-Gordon field living in a flat spacetime with two timelike and two spacelike dimensions. Such a wave equation, unexpectedly, allows dispersionless propagation albeit having two spatial dimensions. Here we show that the same equation can be naturally interpreted as a particular massive Klein-Gordon equation with the usual one timelike and two spacelike dimensions in a de Sitter (dS) background spacetime. The mass parameter of the scalar field is given in terms of the cosmological constant, Planck constant, and the speed of light as $m=\sqrt{\mathrm{\ensuremath{\Lambda}}}\frac{\ensuremath{\hbar}}{c}$ which corresponds to the point for which the left and right conformal weights of the boundary conformal field theory (CFT) (via the de Sitter/CFT correspondence) are equal. This particular mass corresponds to the gapless mode in the dS spacetime for which the dispersion relation is linear in the wave number.

Heavy quarkonium systems for the deformed unequal scalar and vector Coulomb–Hulthén potential within the deformed effective mass Klein–Gordon equation using the improved approximation of the centrifugal term and Bopp’s shift method in RNCQM symmetries

In this study, the analytical solutions of the Klein–Gordon equation for any [Formula: see text] states of the modified effective mass potential under the modified unequal scalar and vector Coulomb–Hulthén potential (MUSVCH-P) are derived by using an approximation method to the centrifugal potential term in the symmetries of relativistic noncommutative three-dimensional real space (RNC: 3D-RS). The new analytical expressions for eigenvalues of the energy spectrum and the new mass of mesons, such as charmonium and bottomonium that have the quark and antiquark flavor, have been estimated by using Bopp’s shift method, and perturbation theory. The energy state equation depends on the global parameters characterizing the noncommutativity space and the potential parameter [Formula: see text] in addition to the Gamma function and the discreet atomic quantum numbers [Formula: see text]. The expression for the new energy spectra is applied to obtain the new mass spectra of heavy quarkonium systems (charmonium and bottomonium) in the symmetries of (RNC: 3D-RS). The comparisons show that our theoretical results are in very good agreement with the reported works.

Acoustic black-hole bombs and scalar clouds in a photon-fluid model

Massive bosonic fields in the background of a Kerr black hole can either trigger superradiant instabilities (black-hole bombs) or form equilibrium configurations corresponding to pure bound states, known as stationary scalar clouds. Here, similar phenomena are shown to emerge in the fluctuation dynamics of a rotating photon-fluid model. In the presence of suitable vortex flows, the density fluctuations are governed by the massive Klein-Gordon equation on a ($2+1$) curved spacetime, possessing an ergoregion and an event horizon. We report on superradiant instabilities originating from quasibound phonon states trapped by the vortex background and, remarkably, on the existence of stationary modes in synchronous rotation with the horizon. These represent the acoustic counterpart of astrophysical scalar clouds. Our system offers a promising platform for analogue gravity experiments on superradiant instabilities of massive bosons and black-hole-field equilibrium configurations.

Propagation of Massive Scalar Fields in Pre-Big Bang Cosmologies

We investigate the linear and semilinear massive Klein-Gordon equations in geometrical frameworks of type "Conformal Cyclic Cosmology" of R. Penrose, or "Singular Bouncing Scenario" as well. We give sufficient conditions on the decay of the mass to the fields be able to propagate across the Big-Bang.

Plausible scenario for a generic violation of the weak cosmic censorship conjecture in asymptotically flat four dimensions

We present a plausible counterexample to the weak cosmic censorship conjecture in the four-dimensional Einstein-Scalar theory with asymptotically flat boundary conditions. Our setup stems from the analysis of the massive Klein-Gordon equation on a fixed Kerr black hole background. In particular, we construct the quasinormal spectrum numerically, and analytically in the WKB approximation, then go on to compute its backreaction on the Kerr geometry. In the regime of parameters where the analytic and numerical techniques overlap we find perfect agreement. We give strong evidence for the existence of a nonlinear instability at late times.

Massive phonons and gravitational dynamics in a photon-fluid model

We theoretically investigate the excitation dynamics in a photon-fluid with both local and nonlocal interactions. We show that the interplay between locality and an infinite-range nonlocality gives rise to a gapped Bogoliubov spectrum of elementary excitations which, at lower momenta, correspond to massive particles (phonons) with a relativistic energy-momentum relation. In this regime and in the presence of an inhomogeneous flow the density fluctuations are governed by the massive Klein-Gordon equation on the acoustic metric and thus propagate as massive scalar fields on a curved spacetime. We finally demonstrate that in the non-relativistic limit the phonon modes behave as self-gravitating quantum particles with an effective Schr\"{o}dinger-Newton dynamics, although with a finite-range gravitational interaction and a non-zero cosmological constant. Our photon-fluid represents a viable alternative to BEC models for "emergent-gravity" scenarios and offers a promising setting for analogue simulations of quantum gravity phenomenology and semiclassical gravity.

The Massive Feynman Propagator on Asymptotically Minkowski Spacetimes II

Abstract We consider the massive Klein–Gordon equation on short-range asymptotically Minkowski spacetimes. Extending our results in [7], we show that the Klein–Gordon operator with Feynman-type boundary conditions at infinite times is invertible and that its inverse, called the Feynman inverse, satisfies the microlocal conditions of Feynman parametrices in the sense of Duistermaat and Hörmander. This supplements the recent work of Vasy [10] with more explicit techniques.

Conformal extension of the Bunch-Davies state across the de Sitter boundary

In the setting of the massive Klein-Gordon equation on de Sitter space, we discuss Vasy’s asymptotic data at conformal infinity in terms of plane waves. In particular, we derive a short-hand formula for reconstructing solutions from their asymptotic data. Furthermore, we show that the natural Hadamard state induced from future (or past) conformal infinity coincides with the Bunch-Davies state.

The massive Feynman propagator on asymptotically Minkowski spacetimes

We consider the massive Klein-Gordon equation on asymptoti-cally Minkowski spacetimes, in the sense that the manifold is R 1+d and the metric approaches that of Minkowski space at infinity in a short-range way (jointly in time and space variables). In this setup we define Feynman and anti-Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer type boundary conditions at infinite times. We then construct a parametrix (with compact remainder terms) for the Fredholm problem and prove that it is also a Feynman parametrix in the sense of Duistermaat and Hormander.

Absorption Cross Section and Decay Rate of Dilatonic Black Strings

We studied in detail the propagation of a massive tachyonic scalar field in the background of a five-dimensional (5D) Einstein–Yang–Mills–Born–Infeld–dilaton black string: the massive Klein–Gordon equation was solved, exactly. Next we obtained complete analytical expressions for the greybody factor, absorption cross section, and decay rate for the tachyonic scalar field in the geometry under consideration. The behaviors of the obtained results are graphically represented for different values of the theory’s free parameters. We also discuss why tachyons should be used instead of ordinary particles for the analytical derivation of the greybody factor of the dilatonic 5D black string.

Analytical solutions in a cosmic string Born–Infeld-dilaton black hole geometry: quasinormal modes and quantization

Chargeless massive scalar fields are studied in the spacetime of Born-Infeld dilaton black holes (BIDBHs). We first separate the massive covariant Klein-Gordon equation into radial and angular parts and obtain the exact solution of the radial equation in terms of the confluent Heun functions. Using the obtained radial solution, we show how one gets the exact quasinormal modes (QNMs) for some particular cases. We also solve the Klein-Gordon equation in the spacetime of BIDBH with a cosmic string in which we point out the effect of the conical deficit on the BIDBH. The analytical solutions of the radial and angular parts are obtained in terms of the confluent Heun functions. Finally, we study the quantization of the BIDBH. While doing this, we also discuss the Hawking radiation in terms of the effective temperature.

Approximate Analytical Solutions of the Effective Mass Klein-Gordon Equation for Two Interacting Potentials

In this study, the approximate analytical solutions of the relativistic Klein-Gordon equation in the spatial dimensions with unequal Coulomb-inverse Trigonometry scarf scalar and vector potentials for an effective mass function is investigated in the framework of supersymmetric and shape invariance method by employing a suitable approximation scheme to the centrifugal term. The energy equation for some special cases such as the Coulomb potential and inverse Trigonometry scarf potential are obtained. Using a certain transformation, the non-relativistic energy equation is obtained which is identical to the energy equation of the Hellmann potential.

Linear waves on constant radius limits of cosmological black hole spacetimes

In this paper we consider the Klein-Gordon equation on spherically symmetric background spacetimes with a constant area radius. The spacetimes under consideration are Nariai and Plebanski-Hacyan, and can be considered constant radius limits of Reissner-Nordstrom-de Sitter spacetimes. We prove boundedness in the case of a non-negative Klein Gordon mass and decay unless the mass is zero. In the latter case we prove decay of solutions that are supported on all harmonic modes with angular momentum l greater or equal to 1. We show that the l=0 modes of solutions to the massless Klein-Gordon equation do not decay. They are subject to conservation laws along degenerate Killing horizons. We apply the estimates in Nariai to give decay of solutions to the massive Klein-Gordon equation on an n-dimensional de Sitter background, using only the vector field method and with no restrictions on the positive Klein-Gordon mass.

Mass ladder operators from spacetime conformal symmetry

Ladder operators can be useful constructs, allowing for unique insight and intuition. In fact, they have played a special role in the development of quantum mechanics and field theory. Here, we introduce a novel type of ladder operators, which map a scalar field onto another massive scalar field. We construct such operators, in arbitrary dimensions, from closed conformal Killing vector fields, eigenvectors of the Ricci tensor. As an example, we explicitly construct these objects in anti-de Sitter spacetime (AdS) and show that they exist for masses above the Breitenlohner-Freedman (BF) bound. Starting from a regular seed solution of the massive Klein-Gordon equation (KGE), mass ladder operators in AdS allow one to build a variety of regular solutions with varying boundary condition at spatial infinity. We also discuss mass ladder operator in the context of spherical harmonics, and the relation between supersymmetric quantum mechanics and so-called Aretakis constants in an extremal black hole.

Several solutions of the Klein–Gordon equation in Kerr–Newman spacetime and the BSW effect

We investigate the radial part of the charged massive Klein-Gordon equation in Kerr-Newman spacetime, and in several specific situations, obtain exact solutions by means of essentially hypergeometric functions or their confluent types. Using these global solutions and generally obtained local solutions, we calculate a sort of intensity of the collision of two field excitations, which is a slight generalization of the trace of the stress tensor. We find that when the black hole is nonextremal, the intensity of the collision of two ingoing modes is bounded. However, in the extremal limit, more precisely $\hbar \kappa_H \rightarrow 0$, the upper bound grows so that when the frequency of one of the two modes satisfies the critical relation, the intensity of the collision at the horizon becomes unboundedly large. Furthermore, the intensity of the collision of ingoing and outgoing modes is always unbounded, as well as in the classical particle theory. Our results suggest that the BSW effect is inherited by the quantum theory.

Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes

We consider the massive Klein-Gordon equation on a class of asymptotically static spacetimes (in the long range sense) with Cauchy surface of bounded geometry. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods.

Bound and Scattering State of Position Dependent Mass Klein–Gordon Equation with Hulthen Plus Deformed-Type Hyperbolic Potential

In this article we use supersymmetry quantum mechanics and factorization methods to study the bound and scattering state of Klein–Gordon equation with deformed Hulthen plus deformed hyperbolical potential for arbitrary state in D-dimensions. The analytic relativistic bound state eigenvalues and the scattering phase factor are found in closed form. We report on the numerical results for the bound state energy in D-dimensions.

Superradiant instability of the Kerr brane

We consider linear gravitational perturbations of the Kerr brane, an exact solution of vacuum Einstein's equations in dimensions higher than four and a low-energy solution of string theory. Decomposing the perturbations in tensor harmonics of the transverse Ricci-flat space, we show that tensor- and vector-type metric perturbations of the Kerr brane satisfy respectively a massive Klein-Gordon equation and a Proca equation on the four-dimensional Kerr space, where the mass term is proportional to the eigenvalue of the harmonics. Massive bosonic fields trigger a well-known superradiant instability on a Kerr black hole. We thus establish that Kerr branes in dimensions $D\geq6$ are gravitationally unstable due to superradiance. These solutions are also unstable against the Gregory-Laflamme instability and we discuss the conditions for either instability to occur and their rather different nature. When the transverse dimensions are compactified and much smaller than the Kerr horizon, only the superradiant instability is present, with a time scale much longer than the dynamical time scale. Our formalism can be also used to discuss other types of higher-dimensional black objects, taking advantage of recent progress in studying linear perturbations of four-dimensional black holes.

Analytical solutions of position-dependent mass Klein–Gordon equation for unequal scalar and vector Yukawa potentials

Analytical solutions of the position-dependent mass Klein–Gordon equation in the presence of unequal scalar and vector Yukawa potentials for arbitrary l-state are obtained by using the generalized parametric Nikiforov–Uvarov method. With an approximation scheme to deal with the centrifugal term, we get the bound state energy eigenvalues and the corresponding wave functions, expressed in terms of the Jacobi polynomials. Subsequently, we consider a special case for α = 0 and explicitly obtain the energy eigenvalues as well as the corresponding eigenfunctions in terms of the Laguerre polynomials. Some results are also compared with the previous studies.