Gravitational Perturbations of a Radiating Spacetime

This paper discusses the problem of gravitational perturbations of radiating spacetimes. We lay out the theoretical framework for describing the interaction of external gravitational fields with a radiating spacetime. This is done by deriving the field perturbation equations for a radiating metric. The equations are then specialized to a Vaidya spacetime. For the Hiscock ansatz of a linear mass model of a radiating blackhole the equations are found separable. Further, the resulting ordinary differential equations are found to admit analytic solutions. We obtain the solutions and discuss their characteristics.     

Perturbations of the Plebanski metric in general relativity. I. Equations for gravitational perturbations

The Plebanski metric is analyzed by means of the Newman-Penrose formalism. The components of the Weyl and Ricci tensors and the spin coefficients are calculated. These results are applied to obtain the equations of gravitational perturbations around the Plebanski metric.     

Perturbations of spherically symmetric black holes

The linearized Einstein-Maxwell equations on a Reissner-Nördstrom background are reduced to two (decoupled) first order ordinary differential equations which determine the even and the odd parity perturbations via quadratures.     

Energy Multipliers for Perturbations of the Schwarzschild Metric

We consider the wave equations associated to metrics close to the Schwarzschild metric. We investigate spacelike energy multipliers likely to yield local decay of solutions to these wave equations, in the spirit of Morawetz. For rotationally invariant metrics, we obtain multipliers giving a control of the solutions having finitely many vanishing spherical harmonics. The structure of these multipliers is closely related to the photosphere of the metric. For Kerr metrics, in contrast, we display a region, which we call the intersphere region, where no energy inequality with the required properties can exist.     

Vacuum Rank 1 Space-Time Perturbations

We study Kerr-Schild type perturbations with a non-null perturbation vector in the vacuum case. The perturbation equations are derived and it is shown that they lead to constraints on the background space-time which can be interpreted in terms of the curvature of 3-spaces. The first order perturbation equations are used to construct new Petrov type D solutions tangent to the Schwarzschild metric.     

Aspects of Scalar Brane-World Cosmological Perturbations

We summarize an approach to deal with scalar brane-world cosmological perturbations based on Mukohyama's master equation. We also give its relation to one based on perturbing the effective Einstein's equations on the brane (involving the Weyl fluid).     

Second Order Perturbations of a Schwarzschild Black Hole: Inclusion of Odd Parity Perturbations

We consider perturbations of a Schwarzschild black hole that can be of both even and odd parity, keeping terms up to second order in perturbation theory, for the l = 2 axisymmetric case. We develop explicit formulae for the evolution equations and radiated energies and waveforms using the Regge–Wheeler–Zerilli approach. This formulation is useful, for instance, for the treatment in the "close limit approximation" of the collision of counterrotating black holes.     

Complex relativity and real solutions. IV. Perturbations of vacuum Kerr-Schild spaces

In this paper the theory of integrable double Kerr-Schild (IDKS) spaces is examined. The vacuum field equations are shown to reduce to the single equation of Plebaski and Robinson [20]. These metrics are given essentially in terms of one potentialH. First-order perturbations ofH lead to metric (gravitational) perturbations of vacuum algebraically degenerate spaces in a direct manner and give results in agreement with those of Cohen and Kegeles [6, 7, 8], Stewart [9], Teukolsky [5], Torres del Castillo [12, 13], and others. Higher-order perturbations ofH are also obtained with the view that, in the limit, these solutions should yield (new) exact vacuum solutions. The success of this construction lies in the (complex) geometric structure of IDKS spaces. This structure induces a natural splitting of the field equations which allows a potentialization of the perturbation (as well as the vacuum metric itself). It also allows massless spin 1/2 and 1 fields to be examined on the IDKS background in a similar manner.     

Investigation of Vector Perturbations in a Radiation-Dominated Cosmological Plasma

Based on the Maxwell–Einstein equations together with hydrodynamic plasma equations derived with allowance for Compton scattering, vector perturbations are examined for a homogeneous isotropic spatially-flat model of the Universe. The magnetic field strength generated by these vector perturbations in the radiation-dominated stage of evolution of the Universe is calculated.     

Scattering Problem for Local Perturbations\newline of the Free Quantum Gas

A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern–Simons Higgs model and have been studied recently by various authors. Received: 30 September 1997 / Accepted: 21 July 1998     

Letter: Wave Equations for the Perturbations of a Charged Black Hole

A pair of simple wave equations is presented for the symmetric gravitational and electromagnetic perturbations of a charged black hole. One of the equations is uncoupled, and the other has a source term given by the solution of the first equation. The derivation is presented in full detail for either axially symmetric or stationary perturbations, and is quite straightforward. This result is expected to have important applications in astrophysical models.     

Mode Coupling in Single-Mode Helical Fibres Under Perturbations

The mode coupling in a helical fibre with a circular cross-section undergoing perturbations is investigated in the Serret-Frenet frame. The coupled-mode equations have been derived and are compared with the conventional ones. Applying the coupled-mode equations to a helical fibre with an elliptical core deformation, we obtained, for the first time, a rigorous analysis of the transmission characteristics of the fibre. The numerical examples show that the mode ellipticity and elliptical birefringence increase as the core ellipticity increases and also vary as the pitch or the core offset change.     

Relativistic Hydrodynamic Cosmological Perturbations

Relativistic cosmological perturbation analysescan be made based on several different fundamental gaugeconditions. In the pressureless limit the variables incertain gauge conditions show the correct Newtonian behaviors. Considering the generalcurvature (K) and the cosmological constant ()in the background medium, the perturbed density in thecomoving gauge, and the perturbed velocity and the perturbed potential in the zero-shear gaugeshow the same behavior as the Newtonian ones in generalscales. In the first part, we elaborate these Newtoniancorrespondences. In the second part, using the identified gauge-in variant variables withcorrect Newtonian correspondences, we present therelativistic results with general pressures in thebackground and perturbation. We present the generalsuper-sound-horizon scale solutions of the above mentionedvariables valid for general K, Lambda, and generallyevolving equation of state. We show that, for vanishingK, the super-sound-horizon scale evolution ischaracterised by a conserved variable which is the perturbedthree-space curvature in the comoving gauge. We alsopresent equations for the multi-component hydrodynamicsituation and for the rotation and gravitational wave.     

Optimal Damping of Perturbations of Moving Thermoelastic Panel

The translational motion of a thermoelastic web subject to transverse vibrations caused by initial perturbations is considered. It is assumed that a web moving with a constant translational velocity is described by the model of a thermoelastic panel simply supported at its ends. The problem of optimal damping of vibrations when applying active transverse actions is formulated. For solving the optimization problem, modern methods developed in control theory for systems with distributed parameters described by partial differential equations are used.     

Perturbations of Multipliers of Systems of Periodic Ordinary Differential Equations

The paper deals with periodic systems of ordinary differential equations (ODEs). A new approach to the investigation of variations of multipliers under perturbations is suggested. It enables us to establish explicit conditions for the stability and instability of perturbed systems.     

Perturbations of unitary representations of the Galilei group: Mass operators with continuous spectra

We present a systematic procedure for constructing mass operators with continuous spectra for a system of particles in a manner consistent with Galilean relativity. These mass operators can be used to construct what may be called point-form Galilean dynamics. As in the relativistic case introduced by Dirac, the point-form dynamics for the Galilean case is characterized by both the Hamiltonian and momenta being altered by interactions. An interesting property of such perturbative terms to the Hamiltonian and momentum operators is that, while having well-defined transformation properties under the Galilei group, they also satisfy Maxwell’s equations. This result is an alternative to the well-known Feynman-Dyson derivation of Maxwell’s equations from non-relativistic quantum physics.     

On the Quasinormal Modes for Gravitational Perturbations of the Bardeen Black Hole

We investigate gravitational perturbations on a regular black hole described by the Bardeen solution. Bardeen’s black hole is a solution of Einstein’s equations with no singularity at the origin of the radially symmetric system. Notwithstanding this regularity, the Bardeen solution still has event horizons dependent on its characteristic parameters. When a black hole is perturbed, it oscillates and gives rise to damped vibrational modes known as quasinormal modes. Here, we compute the quasinormal frequencies of a regular black hole to third order in the WKB approximation for gravitational perturbations.     

Numerical Simulation of Evolution of Multimode Initial Perturbations in the Development of Richtmyer-Meshkov Instabilities

Results of two-dimensional and three-dimensional numerical calculations simulating development of multimode initial perturbations of the contact interface between two gases when shock waves pass through it are presented. A method of solving gas-dynamic nonlinear equations is described. The calculations are performed using an HP-712 workstation and a PARSYTEC CC multiprocessing computer system based on 12 processors.     

Renormalization Group Analysis of Nonlinear Diffusion Equations with Time Dependent Coefficients and Marginal Perturbations

In this paper we use a Renormalization Group (RG) method to study the long-time asymptotics of nonlinear diffusion equations with time-dependent diffusion coefficients and nonlinearities which are marginal (or critical) with respect to the RG operator. These equations describe the time evolution of the average concentration of a passive scalar being advected by a random velocity field. We prove that, besides the expected diffusive behavior, there is an extra logarithmic correction which is the imprint of the critical nonlinearity.     

Perturbations of Functions of Operators in a Banach Space

We consider an analytic function f of bounded operators A and [(A)\tilde]\tilde A represented by infinite matrices in a Banach space with a Schauder basis. Sharp inequalities for the norm of f(A)-f([(A)\tilde])f(A)-f(\tilde A) are established. Applications to differential equations are also discussed.