A Note on the Gravitoelectromagnetic Analogy

We discuss the linear gravitoelectromagnetic approach used to solve Einstein’s equations in the weak-field and slow-motion approximation, which is a powerful tool to explain, by analogy with electromagnetism, several gravitational effects in the solar system, where the approximation holds true. In particular, we discuss the analogy, according to which Einstein’s equations can be written as Maxwell-like equations, and focus on the definition of the gravitoelectromagnetic fields in non-stationary conditions. Furthermore, we examine to what extent, starting from a given solution of Einstein’s equations, gravitoelectromagnetic fields can be used to describe the motion of test particles using a Lorentz-like force equation.

Stability and evolution of electromagnetic solitons in relativistic degenerate laser plasmas

The dynamical behaviours of electromagnetic (EM) solitons formed due to nonlinear interaction of linearly polarized intense laser light and relativistic degenerate plasmas are studied. In the slow-motion approximation of relativistic dynamics, the evolution of weakly nonlinear EM envelope is described by the generalized nonlinear Schrödinger (GNLS) equation with local and nonlocal nonlinearities. Using the Vakhitov–Kolokolov criterion, the stability of an EM soliton solution of the GNLS equation is studied. Different stable and unstable regions are demonstrated with the effects of soliton velocity, soliton eigenfrequency, as well as the degeneracy parameter $R=p_{Fe}/m_ec$ , where $p_{Fe}$ is the Fermi momentum and $m_e$ the electron mass and $c$ is the speed of light in vacuum. It is found that the stability region shifts to an unstable one and is significantly reduced as one enters from the regimes of weakly relativistic $(R\ll 1)$ to ultrarelativistic $(R\gg 1)$ degeneracy of electrons. The analytically predicted results are in good agreement with the simulation results of the GNLS equation. It is shown that the standing EM soliton solutions are stable. However, the moving solitons can be stable or unstable depending on the values of soliton velocity, the eigenfrequency or the degeneracy parameter. The latter with strong degeneracy $(R>1)$ can eventually lead to soliton collapse.

Nonlocal Gravitomagnetism

We briefly review the current status of nonlocal gravity (NLG), which is a classical nonlocalgeneralization of Einstein’s theory of gravitation based on a certain analogy with the nonlocalelectrodynamics of media. Nonlocal gravity thus involves integro-differential field equationsand a causal constitutive kernel that should ultimately be determined from observational data.We consider the stationary gravitational field of an isolated rotating astronomical source in the linearapproximation of nonlocal gravity. In this weak-field and slow-motion approximation of NLG,we describe the gravitomagnetic field associated with the rotating source and compare our resultswith gravitoelectromagnetism (GEM) of the standard general relativity theory. Moreover, we brieflystudy the energy-momentum content of the GEM field in nonlocal gravity.

Analytic approximations in GR and gravitational waves

Analytic approximation methods in general relativity play a very important role when analyzing the gravitational wave signals recently discovered by the LIGO and Virgo detectors. In this contribution, we present the state of the art and some recent developments in the famous post-Newtonian (PN) or slow-motion approximation, which has successfully computed the equations of motion and the early inspiral phase of compact binary systems. We discuss also some interesting interfaces between the PN and the gravitational self-force (GSF) approach based on black-hole perturbation theory, and between PN and the post-Minkowskian (PM) approximation, namely a nonlinearity expansion valid for weak field and possibly fast-moving sources.

The Gravitomagnetism in the Solar System

In 1918, Joseph Lense and Hans Thirring discovered the gravitomagnetic (GM) effect of Einstein field equations in weak field and slow motion approximation. They showed that Einstein equations in this approximation can be written as in the same form as Maxwell’s equation for electromagnetism. In these equations the charge and electric current are replaced by the mass density and the mass current. Thus, the gravitomagnetism formalism in astrophysical system is used with the mass assuming the role of the charge. In this work, we present the deduction of gravitoelectromagnetic equations and the analogue of the Lorentz force in the gravitomagnetism. We also discuss the problem of Mercury’s perihelion advance orbit, we propose solutions using GM formalism using a dipole-dipole potential for the Sun-Planet interaction.

A Fourier Descriptor-Based Approach to Design Space Decomposition for Planar Motion Approximation

In an earlier work, we have combined a curve fitting scheme with a type of shape descriptor, Fourier descriptor (FD), to develop a unified method to the synthesis of planar four-bar linkages for generation of both open and closed paths. In this paper, we aim to extend the approach to the synthesis of planar four-bar linkages for motion generation in an FD-based motion fitting scheme. Using FDs, a given motion is represented by two finite harmonic series, one for translational component of the motion and the other for rotational component. It is shown that there is a simple linear relationship between harmonic content of the rotational component and that of the translational component for a planar four-bar coupler motion. Furthermore, it is shown that the rotational component of the given motion identifies a subset of design parameters of a four-bar linkage including link ratios, while the translational component determines the rest of the design parameters such as locations of the fixed pivots. This leads naturally to a decomposed design space for four-bar mechanism synthesis for approximate motion generation.

Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization Via Kinematic Mapping

It has been well established that kinematic mapping theory could be applied to mechanism synthesis, where discrete motion approximation problem could be converted to a surface fitting problem for a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach is presented, aiming at the problem of mixed exact and approximate motion realization with three types of planar dyad chains (RR, RP, and PR). A two-step unified strategy is established: first N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, then up to four exact poses could be imposed as pose-constraint equations to this surface fitting system such that they could be strictly satisfied. The former step, the surface fitting problem, is converted to a linear system with two quadratic constraint equations, which could be solved by a null-space analysis technique. On the other hand, the given exact poses in the latter step are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.