Space-time Equation Research Articles

An efficient new perturbative Laplace method for space-time fractional telegraph equations

In this paper, we propose a new technique for solving space-time fractional telegraph equations. This method is based on perturbation theory and the Laplace transformation. Fractional Taylor series and fractional initial conditions have been introduced. However, all the previous works avoid the term of fractional initial conditions in the space-time telegraph equations. The results of introducing fractional order initial conditions and the Laplace transform for the studied cases show the high accuracy, simplicity and efficiency of the approach.

Symmetry and special relativity in Finsler spacetime with constant curvature

Within the framework of projective geometry, we investigate kinematics and symmetry in (α,β) spacetime—one special types of Finsler spacetime. The projectively flat (α,β) spacetime with constant flag curvature is divided into four types. The symmetry in type A—Riemann spacetime with constant sectional curvature—is just the one in de Sitter special relativity. The symmetry in type B—locally Minkowski spacetime—is just the one in very special relativity. It is found that type C—Funk spacetime and type D—scaled Berwaldʼs metric spacetime both possess the Lorentz group as its isometric group. The geodesic equation, algebra and dispersion relation in the (α,β) spacetime are given. The corresponding invariant special relativity in the four types of (α,β) spacetime contain two parameters, the speed of light and a geometrical parameter, which may relate to the new physical scale. They all reduce to Einsteinʼs special relativity while the geometrical parameter vanishes.

Stability study of a model for the Klein–Gordon equation in Kerr space-time

The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field of mass $\mu$ in the background of a rotating black hole. Rigorous results proof the stability of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, field for sufficiently large masses. Some, but not all, numerical investigations find instability of the reduced field for rotational parameters $a$ extremely close to 1. Among others, the paper derives a model problem for the equation which supports the instability of the field down to $a/M \approx 0.97$.

The Lorentz-Dirac equation in complex space-time

A hypothetical equation of motion is proposed for Kerr-Newman particles. It is obtained by analytic continuation of the Lorentz-Dirac equation into complex space-time. A new class of "runaway" solutions are found which are similar to Zitterbewegung. Electromagnetic fields generated by these motions are studied, and it is found that the retarded times are multi-sheeted functions of the field points. This leads to non-uniqueness for the generated fields. It is found that with fixed weighting factors for these multiple roots, the solutions radiate electromagnetic energy. If the weighting factors are allowed to be position dependent, however, then it is possible to suppress radiation by solving a differential equation for these factors which ensures that the solution is a non-radiating electromagnetic source. Motion in response to external forces are also considered. Radiation suppression is possible here too in some cases. These results are relevant for the idea that Kerr-Newman solutions provide insight into elementary particles and into emergent quantum mechanics.

Some Blow-Up and Smooth Solutions about Landau-Lifshitz Equation and Their Spatial Curvature Behavior

We construct the exact solution of (2+1)-dimensional space-time Landau-Lifshitz equation (LLE) without the Gilbert term. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and nonsymmetric coordinates. The type of solutions cover the finite-time blow-up solution, smooth solution in time and vortex solution. At the end, some properties about these solutions and their spatial curvature are illustrated by the graphs.

Supermembrane interaction with dynamical D = 4 N = 1 supergravity. Superfield Lagrangian description and spacetime equations of motion

Abstract We obtain the complete set of equations of motion for the interacting system of supermembrane and dynamical D = 4 $ \mathcal{N} $ = 1 supergravity by varying its complete superfield action and writing the resulting superfield equations in the special $ ``{\text{W}}{{\text{Z}}_{{\widehat{\theta } = 0}}}'' $ gauge where the supermembrane Goldstone field is set to zero $ \left( {\widehat{\theta } = 0} \right) $ . We solve the equations for auxiliary fields and discuss the effect of dynamical generation of cosmological constant in the Einstein equation of interacting system and its renormalization due to some regular contributions from supermembrane. These two effects (discussed in late 70th and 80th, in the bosonic perspective and in the supergravity literature) result in that, generically, the cosmological constant has different values in the branches of the spacetime separated by the supermembrane worldvolume.

Fractional backward Kolmogorov equations

This paper derives the fractional backward Kolmogorov equations in fractal space-time based on the construction of a model for dynamic trajectories. It shows that for the type of fractional backward Kolmogorov equation in the fractal time whose coefficient functions are independent of time, its solution is equal to the transfer probability density function of the subordinated process X(Sα(t)), the subordinator Sα(t) is termed as the inverse-time α-stable subordinator and the process X(τ) satisfies the corresponding time homogeneous Itô stochastic differential equation.

A study of the Lorentz-Dirac equation in complex space-time for clues to emergent quantum mechanics

A hypothetical equation of motion is proposed for Kerr-Newman particles. Obtained by analytic continuation of the Lorentz-Dirac equation into complex space-time, the goal is to study the implications of this theory for emergent quantum mechanics. A new class of "runaway" solutions is found which bear similarity to the quantum phenomenon of Zitterbewegung. Other solutions incorporating external forces are also presented. The electromagnetic fields generated by these motions are studied. It is found that the retarded times are multi-sheeted functions of the field points. As a consequence, the solutions are not unique. A cloaking mechanism is found which inhibits electromagnetic radiation by combining the fields from several Riemann sheets for the retarded time. These results reinforce to some extent, but also pose additional conceptual questions for the idea that Kerr-Newman solutions provide insight into elementary particles and into emergent quantum mechanics.

Black hole perturbation in the most general scalar-tensor theory with second-order field equations: The odd-parity sector

We perform a fully relativistic analysis of odd-type linear perturbations around a static and spherically symmetric solution in the most general scalar-tensor theory with second-order field equations in four-dimensional spacetime. It is shown that, as in the case of general relativity, the quadratic action for the perturbations reduces to the one having only a single dynamical variable, from which concise formulas for no-ghost and no-gradient instability conditions are derived. Our result is applicable to all the theories of gravity with an extra scalar degree of freedom. We demonstrate how the generic formulas can be applied to some particular examples such as the Brans-Dicke theory, $f(R)$ models, and Galileon gravity.

Space-time finite element approximation of parabolic optimal control problems

In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. Furthermore, we also propose a space-time mixed finite element discretization scheme to approximate the space-time elliptic equations, and derive a priori error estimates for the state and the adjoint state. Numerical examples are presented to illustrate our theoretical findings and the performance of our approach.

Null Infinity Waveforms from Extreme-Mass-Ratio Inspirals in Kerr Spacetime

We describe the hyperboloidal compactification for Teukolsky equations in Kerr spacetime. We include null infinity on the numerical grid by attaching a hyperboloidal layer to a compact domain surrounding the rotating black hole and the orbit of an inspiralling point particle. This technique allows us to study, for the first time, gravitational waveforms from large- and extreme-mass-ratio inspirals in Kerr spacetime extracted at null infinity. Tests and comparisons of our results with previous calculations establish the accuracy and efficiency of the hyperboloidal layer method.

Well-posedness and the energy and charge conservation for nonlinear wave equations in discrete space-time

We consider the discretization problem for U(1)-invariant nonlinear wave equations in any dimension. We show that the classical finite-difference scheme used by Strauss and Vazquez (in J. Comput. Phys. 28, 271–278 (1978)) conserves the positive-definite discrete analog of the energy if the grid ratio satisfies $dt/dx \leqslant 1/\sqrt n $ , where dt and dx are the mesh sizes of the time and space variables and n is the spatial dimension. We also show that, if the grid ratio is $dt/dx \leqslant 1/\sqrt n $ , then there is a discrete analog of charge, and this discrete analog is conserved. We prove the existence and uniqueness of solutions to the discrete Cauchy problem. We use energy conservation to obtain a priori bounds for finite energy solutions, thus showing that the Strauss-Vazquez finite-difference scheme for the nonlinear Klein-Gordon equation with positive nonlinear term in the Hamiltonian is conditionally stable.

Formal solution of the fourth order Killing equations for stationary axisymmetric vacuum spacetimes

An analytic understanding of the geodesic structure around non-Kerr spacetimes will result in a powerful tool that could make the mapping of spacetime around massive quiescent compact objects possible. To this end, I present an analytic closed form expression for the components of a the fourth order Killing tensor for stationary axisymmetric vacuum (SAV) spacetimes. It is as yet unclear what subset of SAV spacetimes admit this solution. The solution is written in terms of an integral expression involving the metric functions and two specific Green's functions. A second integral expression has to vanish in order for the solution to be exact. In the event that the second integral does not vanish it is likely that the best fourth order approximation to the invariant has been found. This solution can be viewed as a generalized Carter constant providing an explicit expression for the fourth invariant, in addition to the energy, azimuthal angular momentum and rest mass, associated with geodesic motion in SAV spacetimes, be it exact or approximate. I further comment on the application of this result for the founding of a general algorithm for mapping the spacetime around compact objects using gravitational wave observatories.

On the effects of interaction of a spinor field with a vortex gravitational field

We consider the problem of interaction of a self-gravitating spinor field and a vortex gravitational field in a stationary space-time with cylindrical symmetry with a possible existence of a vortex gravitational field. For this purpose, we solve the set of the Einstein-Dirac equations in such space-time. We find exact solutions to these equations, and some of them describe a wormhole geometry.

An infinite family of relativistic magnetized finite thin disks

An infinite family of relativistic finite thin disk model with magnetic field is presented. The model is obtained for solving the Einstein-Maxwell equations for static spacetimes by means of the Horský-Mitskievitch generating conjecture. The vacuum limit of these obtained solutions is the well known Morgan and Morgan solution. The obtained expressions are simply written in terms of oblate spheroidal coordinates. The mass of the disks are finite and the energy-momentum tensor agrees with all the energy conditions. The magnetic field and the circular velocity are evaluated explicitly. All the physical quantities obtained shown an acceptable behavior

Local well-posedness for the space–time Monopole equation in Lorenz gauge

It is known from the work of Czubak that the space-time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in $H^s(\mathbb{R}^2)$ for $s>1/4$. Here we prove local well-posedness for arbitrary initial data in $H^s(\mathbb{R^2})$ with $s>1/4$ in the Lorenz gauge.

κ-DEFORMED DIRAC EQUATION

We construct a Dirac equation in κ-Minkowski spacetime and analyze its implications. This κ-deformed Dirac equation is expanded as a power series involving derivatives with respect to commutative coordinates and the deformation parameter, a. We show that the κ-deformation breaks the charge conjugation invariance but preserves parity and time reversal. We then study how the hydrogen atom spectrum is modified due to the κ-deformation, applying perturbation theory. Using this, we obtain bounds on the deformation parameter a, which are a few orders higher than the Planck length. We also show that the effects of deformation on the spectrum are distinct from that of Moyal deformation and generalized uncertainty principle.

Effects of nonlinearity on the optical diffraction of Bose-Einstein condensates: Direct integration of optically coupled multicomponent Gross-Pitaevskii equation

We investigate in detail the effects of nonlinearity on optical diffraction of Bose-Einstein condensates (BECs). By directly integrating the optically coupled two-component Gross-Pitaevskii equation in real space-time, comprehensive analyses of BEC optical diffraction phenomena are done under various conditions of light-pulse irradiation, total number of BEC atoms, etc., without using the adiabatic elimination approximation for an atomic excited state. Calculation results for the optical diffraction of {sup 87}Rb BECs revealed that (1) the effect of nonlinearity on the atomic states causes the ''nonkinetic'' nonlinear effect in the Raman-Nath regime of diffraction, while the dynamics of BEC atoms due to the nonlinearity-induced repulsive forces works dominantly to produce the ''kinetic'' nonlinear effect in the Bragg regime of diffraction; (2) nonlinearity reduces the amplitude and frequency of the two-photon Rabi oscillation between BEC stationary and moving states, suggesting limitations in implementing the BEC Mach-Zehnder interferometer; and (3) the observed nonlinear effects are free from kinetic effects of the atomic excited state and not responsible for the optical transition process.

Dynamical Symmetry Breaking in RN Quantum Gravity

We show that in the RN gravitation model, there is no dynamical symmetry breaking effect in the formalism of the Schwinger-Dyson equation (in flat background space-time). A general formula for the second variation of the gravitational action is obtained from the quantum corrections hμν (in arbitrary background metrics).

Second-order hyperbolic Fuchsian systems and applications

We introduce a new class of singular partial differential equations, referred to as the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First, we establish a general existence theory of solutions with asymptotic behavior prescribed on the singularity, which relies on a new approximation scheme, suitable also for numerical purposes. Second, this theory is applied to the (vacuum) Einstein equations for Gowdy spacetimes, and allows us to recover, by more direct arguments, well-posedness results established earlier by Rendall and collaborators. Another main contribution in this paper is the proposed approximation scheme, which we refer to as the Fuchsian numerical algorithm and is shown to provide highly accurate numerical approximations to the singular initial value problem. For the class of Gowdy spacetimes, the numerical experiments presented here show the interest and efficiency of the proposed method and demonstrate the existence of a class of Gowdy spacetimes containing a smooth, incomplete and non-compact Cauchy horizon.