The Generalized Jacobi Equation

The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analyzed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed $2^{-1/2}c\approx 0.7 c$ is pointed out. The astrophysical implications of this result for the terminal speed of a relativistic jet is briefly explored.Comment: LaTeX file, 4 PS figures, 28 pages, revised version, accepted for publication in Classical and Quantum Gravit

Resonant Metalenses for Breaking the Diffraction Barrier

We introduce the resonant metalens, a cluster of coupled subwavelength resonators. Dispersion allows the conversion of subwavelength wavefields into temporal signatures while the Purcell effect permits an efficient radiation of this information in the far-field. The study of an array of resonant wires using microwaves provides a physical understanding of the underlying mechanism. We experimentally demonstrate imaging and focusing from the far-field with resolutions far below the diffraction limit. This concept is realizable at any frequency where subwavelength resonators can be designed.Comment: 4 pages, 3 figure

(2,2)-Formalism of General Relativity: An Exact Solution

I discuss the (2,2)-formalism of general relativity based on the (2,2)-fibration of a generic 4-dimensional spacetime of the Lorentzian signature. In this formalism general relativity is describable as a Yang-Mills gauge theory defined on the (1+1)-dimensional base manifold, whose local gauge symmetry is the group of the diffeomorphisms of the 2-dimensional fibre manifold. After presenting the Einstein's field equations in this formalism, I solve them for spherically symmetric case to obtain the Schwarzschild solution. Then I discuss possible applications of this formalism.Comment: 2 figures included, IOP style file neede

An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity

In this paper we treat the so called clock paradox in an analytical way by assuming that a constant and uniform force F of finite magnitude acts continuously on the moving clock along the direction of its motion assumed to be rectilinear. No inertial motion steps are considered. The rest clock is denoted as (1), the to-and-fro moving clock is (2), the inertial frame in which (1) is at rest in its origin and (2) is seen moving is I and, finally, the accelerated frame in which (2) is at rest in its origin and (1) moves forward and backward is A. We deal with the following questions: I) What is the effect of the finite force acting on (2) on the proper time intervals measured by the two clocks when they reunite? Does a differential aging between the two clocks occur, as it happens when inertial motion and infinite values of the accelerating force is considered? The Special Theory of Relativity is used in order to describe the hyperbolic motion of (2) in the frame I II) Is this effect an absolute one, i.e. does the accelerated observer A comoving with (2) obtain the same results as that in I, both qualitatively and quantitatively, as it is expected? We use the General Theory of Relativity in order to answer this question.Comment: LaTex2e, 19 pages, no tables, no figures. Rewritten version, it amends the previous one whose results about the treatment with General Relativity were wrong. References added. Eq. (55) corrected. More refined version. Comments and suggestions are warmly welcom

Ultra-relativistic electrostatic Bernstein waves

A new general form of the dispersion relation for electrostatic Bernstein waves in ultra-relativistic pair plasmas, characterized by a−1 = kBT/(mec2)  1, is derived in this paper. The parameter Sp = aΩ0/ωp, where Ω0 is the rest cyclotron frequency for electrons or positrons and ωp is the electron (or positron) plasma frequency, plays a crucial role in characterizing these waves. In particular, Sp has a restricted range for permitted wave solutions; this range is effectively unlimited for classical plasmas, but is significant for the ultra-relativistic case. The characterization of these waves is applied in particular to the presence of such plasmas in pulsar atmospheres

Explicit Fermi Coordinates and Tidal Dynamics in de Sitter and Goedel Spacetimes

Fermi coordinates are directly constructed in de Sitter and Goedel spacetimes and the corresponding exact coordinate transformations are given explicitly. The quasi-inertial Fermi coordinates are then employed to discuss the dynamics of a free test particle in these spacetimes and the results are compared to the corresponding generalized Jacobi equations that contain only the lowest-order tidal terms. The domain of validity of the generalized Jacobi equation is thus examined in these cases. Furthermore, the difficulty of constructing explicit Fermi coordinates in black-hole spacetimes is demonstrated.Comment: 23 pages, 3 figures; v2: expanded version (27 pages, 3 figures

Newtonian and Post-Newtonian approximations of the k = 0 Friedmann Robertson Walker Cosmology

In a previous paper we derived a post-Newtonian approximation to cosmology which, in contrast to former Newtonian and post-Newtonian cosmological theories, has a well-posed initial value problem. In this paper, this new post-Newtonian theory is compared with the fully general relativistic theory, in the context of the k = 0 Friedmann Robertson Walker cosmologies. It is found that the post-Newtonian theory reproduces the results of its general relativistic counterpart, whilst the Newtonian theory does not.Comment: 11 pages, Latex, corrected typo

Dynamics of test bodies with spin in de Sitter spacetime

We study the motion of spinning test bodies in the de Sitter spacetime of constant positive curvature. With the help of the 10 Killing vectors, we derive the 4-momentum and the tensor of spin explicitly in terms of the spacetime coordinates. However, in order to find the actual trajectories, one needs to impose the so-called supplementary condition. We discuss the dynamics of spinning test bodies for the cases of the Frenkel and Tulczyjew conditions.Comment: 11 pages, RevTex forma

Identifying the development in phase and amplitude of dipole and multipole radiation

The spatial variation in phase and the propagating wave-front of plane wave electromagnetic radiation are widely familiar text-book territory. In contrast, the developing amplitude and phase of radiation emitted by a dipole or multipole source generally receive less attention, despite the prevalence of these systems. There is additional complexity in such cases where, in consequence of retardation, the character and features significantly and progressively change as radiation propagates onwards, from the near-field and out towards the wave-zone. Readily developed analytical representations of the electric field, cast as a function of distance from the source, provide illuminating insights into the most prominent and distinctive properties of radiant electromagnetic emission. Graphical implementations and animations of the results prove particularly instructive in revealing the spatial form and temporal evolution of the emergent electromagnetic fields

Relativistic contraction and related effects in noninertial frames

Although there is no relative motion among different points on a rotating disc, each point belongs to a different noninertial frame. This fact, not recognized in previous approaches to the Ehrenfest paradox and related problems, is exploited to give a correct treatment of a rotating ring and a rotating disc. Tensile stresses are recovered, but, contrary to the prediction of the standard approach, it is found that an observer on the rim of the disc will see equal lengths of other differently moving objects as an inertial observer whose instantaneous position and velocity are equal to that of the observer on the rim. The rate of clocks at various positions, as seen by various observers, is also discussed. Some results are generalized for observers arbitrarily moving in a flat or a curved spacetime. The generally accepted formula for the space line element in a non-time-orthogonal frame is found inappropriate in some cases. Use of Fermi coordinates leads to the result that for any observer the velocity of light is isotropic and is equal to $c$, providing that it is measured by propagating a light beam in a small neighborhood of the observer.Comment: 15 pages, significantly revised version, title changed, to appear in Phys. Rev.