Narrow Results

The Lorentz transformation equations, which are considered the cornerstone of the special theory of relativity, are fundamental theoretical tools to describe the transformation of spacetime coordinates between frames of reference moving at a constant velocity relative to each other. Here we obtain the Lorentz transformation equations between frames of reference moving in the three dimensions of space and further we present the matrix form of the obtained transformation equations with the help of the position six-vector. Moreover, we discuss the notion of six-velocity and six-momentum, which appear to be unknown in the literature, and we also investigate its main consequences, including demonstrating that it is completely consistent with many of the familiar outcomes of four-vector and deriving the transformation of six-momentum and energy that are completely compatible with the Lorentz invariant energy-momentum relation.

The mathematical treatment and graphical representation of Special Relativity (SR) are well established, yet carry deep implications that remain hard to visualize. This paper presents a new graphical interpretation of the geometry of SR that may, by complementing the standard works, aid the understanding of SR and its fundamental principles in a more intuitive way. From the axiom that the velocity of light remains constant to any inertial observer, the geodesic is presented as a line of constant angle on the complex plane across a set of diverging reference frames. The resultant curve is a logarithmic spiral, and this view of the geodesic is extended to illustrate the relativistic Doppler effect, time dilation, length contraction, the twin paradox, and relativistic radar distance in an original way, whilst retaining the essential mathematical relationships of SR. Using a computer-generated graphical representation of photon trajectories allows a visual comparison between the relativistic relationships and their classical counterparts, to visualize the consequences of SR as velocities become relativistic. The model may readily be extended to other situations, and may be found useful in presenting a fresh understanding of SR through geometric visualization.

A review of experimental tests of time dilation in special relativity is given, with an emphasis on the recent heavy-ion storage-ring work, currently the most sensitive method. The experimental results are evaluated in the context of the kinematic test theories of Robertson and Mansouri and Sexl, which assume the existence of a preferred inertial reference frame. Prospects for future improvements and the sensitivity of time dilation experiments to Lorentz and CPT violating extensions of the standard model are discussed.

The gravitomagnetic field is the force exerted by a moving body on the basis of the intriguing interplay between geometry and dynamics which is the analog to the magnetic field of a moving charged body in electromagnetism. The existence of such a field has been demonstrated based on special relativity approach and also by special relativity plus the gravitational time dilation for two different cases, a moving infinite line and a uniformly moving point mass, respectively. We treat these two approaches when the applied cases are switched while appropriate key points are employed. Thus, we demonstrate that the strength of the resulted gravitomagnetic field in the latter approach is twice the former. Then, we also discuss the full linearized general relativity and show that it should give the same strength for gravitomagnetic field as the latter approach. Hence, through an exact analogy with the electrodynamic equations, we present an argument in order to indicate the best definition amongst those considered in this issue in the literature. Finally, we investigate the gravitomagnetic effects and consequences of different definitions on the geodesic equation including the second order approximation terms.

All the Doubly Special Relativity (DSR) models studied in literature so far involve a deformation of the energy conservation rule that forces us to release the hypothesis of the additivity of the energy for composite systems. In view of the importance of the issue for a consistent formulation of a DSR statistical mechanics and a DSR thermodynamics, we show that DSR models preserving the usual (i.e. additive) energy conservation rule can be found. These models allow the construction of a DSR-covariant extensive energy. The implications of the analysis for the dynamics of DSR-covariant multiparticle systems are also briefly discussed.

Glashow and Cohen make the interesting observation that certain proper subgroups of the Lorentz group like HOM(2) or SIM(2) can explain many results of special relativity like time dilation, relativistic velocity addition and a maximal isotropic speed of light. We show here that such SIM(2) and HOM(2) based VSR theories predict an incorrect value for the Thomas precession and are therefore ruled out by observations. In VSR theories the spin-orbital coupling in atoms turn out to be too large by a factor of 2. The Thomas–BMT equation derived from VSR predicts a precession of electrons and muons in storage rings which is too large by a factor of 10³. VSR theories are therefore ruled out by observations.

The recent OPERA experiment of superluminal neutrinos has deep consequences in cosmology. In cosmology a fundamental constant is the cosmological constant. From observations one can estimate the effective cosmological constant Λ_eff which is the sum of the quantum zero point energy Λ_{dark energy} and the geometric cosmological constant Λ. The OPERA experiment can be applied to determine the geometric cosmological constant Λ. It is the first study to distinguish the contributions of Λ and Λ_{dark energy} from each other by experiment. The determination is based on an explanation of the OPERA experiment in the framework of Special Relativity with de Sitter spacetime symmetry.

A general action for particles faster than light is presented. It is demonstrated that this action is invariant under reparametrizations. For several cases, it is shown that in the high velocity regime, the action is invariant under anisotropic spacetime transformation and at quantum level the system has fractal behavior. For those cases, it is shown that the action describes a particle in Finsler geometry and equivalent to one-dimensional field theory in a curved space, where the metric depends on temporal derivatives.

We explore the recent OPERA experiment of superluminal neutrinos in the framework of special relativity with de Sitter spacetime symmetry (dS-SR). According to Einstein, a photon is treated as a massless particle in the framework of special relativity. In special relativity (SR) we have the universal parameter c, the photon velocity c_photon and the phase velocity of a light wave in vacuum c_wave = λν. Due to the null experiments of Michelson–Morley we have c = c_wave. The parameter c_photon is determined by the Noether charges corresponding to the spacetime symmetries of SR. In Einstein's special relativity (E-SR) we have c = c_photon. In dS-SR, i.e. the special relativity with SO(4, 1) de Sitter spacetime symmetry, we have c_photon > c. In this paper, the OPERA datum are examined in the framework of dS-SR. We show that OPERA anomaly is in agreement with the prediction of dS-SR with R≃1.95×10¹² l.y. Based on the p-E relation of dS-SR, we also prove that the Cohen and Glashow's argument of possible superluminal neutrino's Cherenkov-like radiation is forbidden. We conclude that OPERA and ICARUS results are consistent and they are explained in the dS-SR framework.

In this paper, we propose to solve the relativistic Klein–Gordon and Dirac equations subjected to the action of a uniform electromagnetic field with a generalized uncertainty principle in the momentum space. In both cases, the energy eigenvalues and their corresponding eigenfunctions are obtained. The limit case is then deduced for a small parameter of deformation.

The action for a (3+1)-dimensional particle in very special relativity (VSR) is studied. It is proved that massless particles only travel in effective (2+1)-dimensional spacetime. It is remarkable that this action can be written as an action for a relativistic particle in a background gauge field and it is shown that this field causes the dimensional reduction. A new symmetry for this system is found. Furthermore, a general action with restored Lorentz symmetry is proposed for this system. It is shown that this new action contains VSR and two-time physics.

Under consideration of coordinate conditions, we get the Schwarzschild–Beltrami–de Sitter (S-BdS) metric solution of the Einstein field equations with a cosmological constant Λ. A brief review to the de Sitter invariant special relativity (dS-SR), and de Sitter general relativity (dS-GR, or GR with a Λ) is presented. The Beltrami metric B_μν provides inertial reference frame for the dS-spacetime. By examining the Schwarzschild–de Sitter (S-dS) metric existed in literatures since 1918, we find that the existed S-dS metric describes some mixing effects of gravity and inertial-force, instead of a pure gravity effect arisen from "solar mass" M in dS-GR. In this paper, we solve the vacuum Einstein equation of dS-GR, with the requirement of gravity-free metric . In this way we find S-BdS solution of dS-GR, written in inertial Beltrami coordinates. This is a new form of S-dS metric. Its physical meaning and possible applications are discussed.

The theory of special relativity derives from the Lorentz transformation. The Lorentz transformation implies differential simultaneity and light speed isotropy. Experiments to probe differential simultaneity should be able to distinguish the Lorentz transformation from a kinematically-similar alternate transformation that predicts absolute simultaneity, the absolute Lorentz transformation. Here, we describe how published optical tests of light speed isotropy/anisotropy cannot distinguish between the two transformations. We show that the shared equations of the two transformations, from the perspective of the “stationary” observer, are sufficient to predict null results in optical resonator experiments and in tests of frequency changes in one-way light paths. In an influential 1910 exposition on differential simultaneity, Comstock described how a “stationary” observer would observe different clock readings for spatially-separated “moving” clocks. The difference in clock readings is an integral aspect of differential simultaneity. We derive the equation for the difference in clock readings and show that it is equivalent to the Sagnac correction that describes light speed anisotropies in satellite communications. We describe an experimental strategy that can measure the differences in spatially-separated clock times to allow a direct probe of the nature of simultaneity.

In this short paper, we comment on the possibility of superluminal velocities and present–future correlations with reference to a paper recently published.

We conjecture that the random walk and the corresponding diffusion in the relativistic velocity space is an adequate method for describing the acceleration process in relativistic jets. Considering a simple toy model, the main features of diffusion in the velocity space are demonstrated in both non-relativistic and relativistic regimes.

In this paper, we demonstrate that Robb-Geroch’s definition of a relativistic interval admits a simple and fairly natural generalization leading to a Finsler extension of special relativity. Another justification for such an extension goes back to the works of Lalan and Alway and, finally, was put on a solid basis and systematically investigated by Bogoslovsky under the name “Special-relativistic theory of locally anisotropic spacetime”. The isometry group of this spacetime, DISIM${}_b(2)$, is a deformation of the Cohen and Glashow’s very special relativity symmetry group ISIM(2). Thus, the deformation parameter $b$ can be regarded as an analog of the cosmological constant characterizing the deformation of the Poincaré group into the de Sitter (anti-de Sitter) group. The simplicity and naturalness of Finslerian extension in the context of this paper adds weight to the argument that the possibility of a nonzero value of $b$ should be carefully considered.

We know that space and time are treated almost equally in classical physics, but we also know that this is not the case for quantum mechanics. A quantum description of both space and time is important to really understand the quantum nature of reality. The Page–Wootters mechanism of quantum time is a promising starting point, according to which the evolution of the quantum system is described by the entanglement between it and quantum temporal degrees of freedom. In this paper, we consider a qubit clock that is entangled with a quantum system due to the Wigner rotation induced by Lorentz transformation. We study how this time-system entanglement depends on the rapidity of the Lorentz boost. We consider the case of a spin-1/2 particle with Gaussian momentum distribution as a concrete example. We also compare the time-system entanglement entropy with the spin-momentum entanglement entropy and find that the former is smaller than the latter.

We study the rest-frame instant form of a new formulation of relativistic perfect fluids in terms of new generalized Eulerian configuration coordinates. After the separation of the relativistic center of mass from the relative variables on the Wigner hyper-planes, we define orientational and shape variables for the fluid, viewed as a relativistic extended deformable body, by introducing dynamical body frames. Finally we define Dixon's multipoles for the fluid.

Standard formulae of classical electromagnetism for the forces between electric charges in motion derived from retarded potentials are compared with those obtained from a recently developed relativistic classical electrodynamic theory with an instantaneous intercharge force. Problems discussed include small angle Rutherford scattering, Jackson's recent "torque paradox" and circular Keplerian orbits. Results consistent with special relativity are obtained only with an instantaneous interaction. The impossibility of stable circular motion with retarded fields in either classical electromagnetism or Newtonian gravitation is demonstrated.

In this paper, we compute the corrections to the Cardy–Verlinde formula of four-dimensional Kerr black hole. These corrections are considered within the context of KKW analysis and arise as a result of the self-gravitational effect. Then we show that one can take into account the semiclassical corrections of the Cardy–Verlinde entropy formula by only redefining the Virasoro operator L₀ and the central charge c.