Massless Wave Equation Research Articles

Quasinormal modes and universality of the Penrose limit of black hole photon rings

We study the physics of photon rings in a wide range of axisymmetric black holes admitting a separable Hamilton-Jacobi equation for the geodesics. Utilizing the Killing-Yano tensor, we derive the Penrose limit of the black holes, which describes the physics near the photon ring. The obtained plane wave geometry is directly linked to the frequency matrix of the massless wave equation, as well as the instabilities and Lyapunov exponents of the null geodesics. Consequently, the Lyapunov exponents and frequencies of the photon geodesics, along with the quasinormal modes, can be all extracted from a Hamiltonian in the Penrose limit plane wave metric. Additionally, we explore potential bounds on the Lyapunov exponent, the orbital and precession frequencies, in connection with the corresponding inverted harmonic oscillators and we discuss the possibility of photon rings serving as effective holographic horizons in a holographic duality framework for astrophysical black holes. Our formalism is applicable to spacetimes encompassing various types of black holes, including stationary ones like Kerr, Kerr-Newman, as well as static black holes such as Schwarzschild, Reissner-Nordström, among others.

Theory of sounds in He II

A dynamical model for Landau’s original approach to superfluid helium is presented, with two velocities but only one mass density. The second sound is an adiabatic perturbation that involves the temperature and the roton, aka the notoph. The action incorporates all the conservation laws, including the equation of continuity. With only 4 canonical variables it has a higher power of prediction than Landau’s later, more complicated model, with its 8 degrees of freedom. The roton is identified with the massless notoph. This theory gives a very satisfactory account of second and fourth sounds. Second sound is an adiabatic oscillation of the temperature and both vector fields, with no net material motion. Fourth sound involves the roton, the temperature, and the density. With the experimental confirmation of gravitational waves, the relations between Hydrodynamics and Relativity and particle physics have become more clear, and urgent. The appearance of the Newtonian potential in irrotational hydrodynamics comes directly from Einstein’s equations for the metric. The density factor ρ is essential; it is time to acknowledge the role that it plays in particle theory. To complete the 2-vector theory we include the massless roton mode. Although this mode too is affected by the mass density, it turns out that the wave function of the unique notoph propagating mode N satisfies the normal massless wave equation N=0; the roton propagates as a free particle in the bulk of the superfluid without meeting resistance. In this circumstance, we may have discovered the mechanism that lies behind the flow of He II through very thin pores.

Critical exponent of Fujita type for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

Positive Energy Functional for Massless Scalars in Rotating Black Hole Backgrounds of Maximal Ungauged Supergravity.

We outline a proof of the stability of a massless neutral scalar field ψ in the background of a wide class of four dimensional asymptotically flat rotating and "electrically charged" solutions of supergravity, and the low energy limit of string theory, known as STU metrics. Despite their complexity, we find it possible to circumvent the difficulties presented by the existence of ergo regions and the related phenomenon of superradiance in the original metrics by following a strategy due to Whiting, and passing to an auxiliary metric admitting an everywhere lightlike Killing field and constructing a scalar field ψ (related to a possible unstable mode ψ by a nonlocal transformation) which satisfies the massless wave equation with respect to the auxiliary metric. By contrast with the case for ψ, the associated energy density of ψ is not only conserved but is also non-negative.

Synchrotron radiation in odd dimensions

In odd space-time dimensions, the retarded solution of the massless wave equation has support not only on the light cone, but also inside it. At the same time, a free massless field should propagate at the speed of light. The apparent contradiction of these two features is resolved by the fact that the emitted part of the field in the wave zone depends on the history of motion up to the retarded moment of proper time. It is shown that in the case of circular motion with ultrarelativistic velocity, the main contribution to the radiation amplitude is made by a small interval of proper time preceding the retarded time, and thus the tail term is effectively localized. We obtain a tentative formula for scalar synchrotron radiation in $D$ dimensions: $P =g^2(\omega_0\gamma^2/\sqrt{3})^{D-2}$, which is explicitly verified in $D=3,\,4,\,5$.

Trajectory-state theory of the Klein-Gordon field

.We develop a trajectory construction of solutions to the massless wave equation in n + 1 dimensions and hence show that the quantum state of a massive relativistic system in 3 + 1 dimensions may be represented by a stand-alone four-dimensional congruence comprising a continuum of 3-trajectories coupled to an internal scalar time coordinate. A real Klein-Gordon amplitude is the current density generated by the temporal gradient of the internal time. Complex amplitudes are generated by a two-phase flow. The Lorentz covariance of the trajectory model is established.

D1-D5-P superstrata in 5 and 6 dimensions: separable wave equations and prepotentials

We construct the most general single-mode superstrata in 5 dimensions with ambipolar, two centered Gibbons Hawking bases, via dimensional reduction of superstrata in 6 dimensions. Previously, asymptotically AdS3 × \U0001d54a2 5-dimensional superstrata have been produced, giving microstate geometries of black strings in 5 dimensions. Our construction produces asymptotically AdS2 × \U0001d54a3 geometries as well, the first instances of superstrata describing the microstate geometries of black holes in 5 dimensions. New examples of superstrata with separable massless wave equations in both 5 and 6 dimensions are uncovered. A ℤ2 symmetry which identifies distinct 6-dimensional superstrata when reduced to 5 dimensions is found. Finally we use the mathematical structure of the underlying hyper-Kähler bases to produce prepotentials for the superstrata fluxes in 5 dimensions and uplift them to apply in 6 dimensions as well.

Gravitational waves and degrees of freedom in higher derivative gravity

We study the degrees of freedom of the metric in a general class of higher derivative gravity models, which are interesting in the context of quantum gravity as they are (super)renormalizable. First, we linearize the theory for a flat background metric in Teyssandier gauge for an arbitrary number of spacetime dimensions $D$. The higher-order derivative field equations for the metric perturbation can be decomposed into tensorial and scalar field equations resembling massless and massive wave equations. For the massive tensor field in $D$-dimensions we demonstrate that the harmonic gauge condition is induced dynamically and only the transverse modes are excited in the presence of a matter source. For the special case of quadratic gravity in four-dimensional spacetime, we show that only the quadrupole moment contributes to the gravitational radiation from an idealized binary system.

Mapping superintegrable quantum mechanics to resonant spacetimes

We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass. This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and Anti-de Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in Anti-de Sitter spacetime and algebras of conserved quantities of the Higgs oscillator. Our conversion procedure ("Klein-Gordonization") reduces to a nonlinear elliptic equation closely reminiscent of the one emerging in relation to the celebrated Yamabe problem of differential geometry. As an illustration, we explicitly demonstrate how to apply this procedure to superintegrable Rosochatius systems, resulting in a large family of spacetimes with resonant spectra for massless wave equations.

Non-existence of isometry-invariant Hadamard states for a Kruskal black hole in a box and for massless fields on 1+1 Minkowski spacetime with a uniformly accelerating mirror

We conjecture that (when the notion of Hadamard state is suitably adapted to spacetimes with timelike boundaries) there is no isometry-invariant Hadamard state for the massive or massless covariant Klein–Gordon equation defined on the region of the Kruskal spacetime to the left of a surface of constant Schwarzschild radius in the right Schwarzschild wedge when Dirichlet boundary conditions are put on that surface. We also prove that, with a suitable definition for ‘boost-invariant Hadamard state’ (which we call ‘strongly boost-invariant globally Hadamard’) which takes into account both the existence of the timelike boundary and the special infra-red pathology of massless fields in 1+1 dimensions, there is no such state for the massless wave equation on the region of 1+1 Minkowski space to the left of an eternally uniformly accelerating mirror—with Dirichlet boundary conditions at the mirror. We argue that this result is significant because, as we point out, such a state does exist if there is also a symmetrically placed decelerating mirror in the left wedge (and the region to the left of this mirror is excluded from the spacetime). We expect a similar existence result to hold for Kruskal when there are symmetrically placed spherical boxes in both right and left Schwarzschild wedges. Our Kruskal no-go conjecture raises basic questions about the nature of the black holes in boxes considered in black hole thermodynamics. If true, it would lend further support to the conclusion of Kay (2015 Gen. Relativ. Gravit. 47 1–27) that the nearest thing to a description of a black hole in equilibrium in a box in terms of a classical spacetime with quantum fields propagating on it has, for the classical spacetime, the exterior Schwarzschild solution, with the classical spacetime picture breaking down near the horizon. Appendix to the paper points out the existence of, and partially fills, a gap in the proofs of the theorems in Kay and Wald (1991 Phys. Rep. 207 49–136).

Scattering for a massless critical nonlinear wave equation in two space dimensions

We prove scattering for a massless wave equation which is critical in two space dimensions. Our method combines conformal inversion with decay estimates from Struwe's previous work on global existence of a similar equation.

AdS waves of the four and six-drivative gravity models

In this paper we investigate AdS waves in the R 3 extension of new massive gravity. We show that R 3 − N M G admit the geometries with the Schrodinger isometry group. When we approach the critical point of the parameter space, solution with loga-rithmic fall-off arise. Then we extend our work to the three-dimensional tricritical gravity. The equation of motion of AdS wave in this case is a six-derivative differential equation. We decompose this equation as a coupled massive wave equation and a one massless wave equation. It is interesting that in tricritical point when both of the massive wave solutions become massless wave, we obtain the solution with logarithmic-squared behavior.

Conformal symmetry of a black hole as a scaling limit: a black hole in an asymptotically conical box

We show that the previously obtained subtracted geometry of four-dimensional asymptotically flat multi-charged rotating black holes, whose massless wave equation exhibit SL(2, $ \mathbb{R} $ ) × SL(2, $ \mathbb{R} $ ) × SO(3) symmetry may be obtained by a suitable scaling limit of certain asymptotically flat multi-charged rotating black holes, which is reminiscent of near-extreme black holes in the dilute gas approximation. The co-homogeneity-two geometry is supported by a dilation field and two gauge-field strengths. We also point out that these subtracted geometries can be obtained as a particular Harrison transformation of the original black holes. Furthermore the subtracted metrics are asymptotically conical (AC), like global monopoles, thus describing “a black hole in an AC box”. Finally we account for the the emergence of the SL(2, $ \mathbb{R} $ ) × SL(2, $ \mathbb{R} $ ) × SO(3) symmetry as a consequence of the subtracted metrics being Kaluza-Klein type quotients of 4AdS 3 × S 2. We demonstrate that similar properties hold for five-dimensional black holes.

Scaling dimensions in hidden Kerr/CFT correspondence

It has been proposed that a hidden conformal field theory (CFT) governs the dynamics of low frequency scattering in a general Kerr black hole background. We further investigate this correspondence by mapping higher order corrections to the massless wave equations in a Kerr background to an expansion within the CFT in terms of higher dimension operators. This implies the presence of infinite towers of CFT primary operators with positive conformal dimensions compatible with unitarity. The exact Kerr background softly breaks the conformal symmetry and the scaling dimensions of these operators run with frequency. The scale-invariant fixed point is dual to a degenerate case of flat spacetime.

Trace anomaly and massless scalar degrees of freedom in gravity

The trace anomaly of quantum fields in electromagnetic or gravitational backgrounds implies the existence of massless scalar poles in physical amplitudes involving the stress-energy tensor. Considering first the axial anomaly and using QED as an example, we compute the full one-loop triangle amplitude of the fermionic stress tensor with two current vertices, <T^{mn} J^a J^b>, and exhibit the scalar pole in this amplitude associated with the trace anomaly, in the limit of zero electron mass m -> 0. To emphasize the infrared aspect of the anomaly, we use a dispersive approach and show that this amplitude and the existence of the massless scalar pole is determined completely by its ultraviolet finite terms, together with the requirements of Poincare invariance of the vacuum, Bose symmetry under interchange of J^a and J^b, and vector current and stress tensor conservation. We derive a sum rule for the appropriate positive spectral function corresponding to the discontinuity of the triangle amplitude, showing that it becomes proportional to delta function of k^2, and therefore contains a massless scalar intermediate state in the conformal limit of zero electron mass. The effective action corresponding to the trace of the triangle amplitude can be expressed in local form by the introduction of two scalar auxiliary fields which satisfy massless wave equations. These massless scalar degrees of freedom couple to classical sources, contribute to gravitational scattering processes, and can have long range gravitational effects.

Massless Relativistic Wave Equations and Quantum Field Theory

We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincare group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fiber degrees of freedom of the covariant representation. They are characterized by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between \( {\mathcal E}(2) \) (the two-fold cover of the 2-dimensional Euclidean group) and \( \overline{{\mathcal E}(2)}. \)

Collective excitations at the boundary of a four-dimensional quantum Hall droplet

In this work we investigate collective excitations at the boundary of a recently constructed four-dimensional (4D) quantum Hall state. Local bosonic operators for creating these collective excitations can be constructed explicitly. Massless relativistic wave equations with helicity S can be derived exactly for these operators from their Heisenberg equation of motion. For the $S=1$ and $S=2$ cases these equations reduce to the free Maxwell and linearized Einstein equations respectively. These collective excitations can be interpreted as hydrodynamical modes at the boundary of the 4D quantum Hall effect droplet. Outstanding issues are critically discussed.

MASSLESS FIELDS OVER R1×H3 SPACE–TIME AND COHERENT STATES FOR THE LORENTZ GROUP

The solutions of the arbitrary-spin massless wave equations over R1×H3 space are obtained using the generalized coherent states for the Lorentz group. The use of these solutions for the construction of invariant propagators of quantized massless fields with an arbitrary spin over the R1×H3 space is considered. The expression for the scalar propagator is obtained in the explicit form.

Existence of global solutions to nonlinear massless Dirac system and wave equation with small data

We prove existence of global solutions to a semilinear massless Dirac system with small initial data. We study solutions in generalised Sobolev spaces suggested by S. Klainerman. Our approach is based on using conservation law of charge together with Sobolev type weighted estimates for the spinor field. Our result seems to be sharp in a view of blowing-up results obtained by F. John (see [7]). We also study decay properties of the spinor field. With similar methods we prove global existence for a nonlinear wave equation in three space dimension. The same equation was studied by T. Sideris [14] and H. Takamura [15]. They proved global existence for spherically symmetrical initial data. In this work we remove this condition on the initial data.

Local symmetries of massless wave equations

The structure of a wave-equation symmetry algebra for massless particles is studied. It is shown that the local-symmetry algebra of massless wave equations belongs to the enveloping algebra of a conformal-group Lie algebra. It is also proven that the space of nontrivial symmetries of fixed order is irreducible under adjoint representation of the conformal algebra, and its dimension is indicated.