Relativistic Trajectory Research Articles

Classical-relativistic variability of inertia of photon displacement mass in the matter and the specific vacuum temperature

The effects of photon inertia on the determination of its trajectory were verified and the representation of a displacement mass characterized by the flow of the number of wavefronts and the decomposition of photon inertia into parts associated with translation and rotation motions was considered. It was found that with the relativistic increase of the photon's resistance to change its directional properties, it inhibits the relativistic trajectory of the second torque, so called Minkowski torque, in an angular range of incidence. After synchronizations, in the OAM inversions, there are reductions of the inertia associated to the translational part that assumes classical predominance, where the relativistic trajectory is allowed while the photon offers less resistance to changes in its directional properties. The classical-relativistic variability of the photon inertia characterizes the classical or relativistic profile of the energy distribution in forms of motion, where adjustments of the rotational and translational parts can be performed as a function of the refractive index rate, temperature and angle of incidence. It was found that with increasing temperature of the refringent medium, the synchronizations displacement in the sense of the normal incidence. A specific vacuum temperature for the refringent medium was characterized, where the photon exhibits a classical-relativistic synchronization under all angles of incidence, characteristic of its immaterial state in vacuum.

Experimental observation of polarization-resolved nonlinear Thomson scattering of elliptically polarized light

We report experimental results from a study of nonlinear Thomson scattering of elliptically polarized light. Polarization-resolved radiation patterns of the scattered light are measured as a function of the elliptical polarization state of the incident laser light. The relativistic electron trajectory in intense elliptically polarized fields leads to the formation of unique radiated polarization states, which are observed by our measurements and predicted by a theoretical model. The polarization of Thomson scattered light depends strongly on the intensity of the incident light due to nonlinearity. The results are relevant to high-field electrodynamics and to research and development of light sources with novel capabilities.

The relativistic and the hidden momentum of Minkowski and Abraham in relativistic energy wave

An analysis on the consistency of the Abraham and Minkowski momenta in the determination of the photon trajectory was carried out considering a new principle of conservation of the photon's mechanical energy. In this principle, the photon conserves the translational energy in orbital angular momentum when transiting between two media, introducing the relativistic energy wave (REW). The confrontation between REW and the recent theory of space-time waves (ST) was considered, pondering your differences. Throughout this study it was possible to verify that the Abraham momentum appears a relativistic photon ignition device in the transition between two media, acting as the hidden momentum of the Minkowski’s relativistic momentum. The wavy behavior in the matter is relativistic, and the relativistic trajectory appears with delays and advances, with points of synchronization between source-observer. The classical or relativistic trajectories are determined as a function of the angle of incidence and the relative refractive index, by one of two distinct non-additive torques, the classic by Abraham or the relativistic by Minkowski. It was found that the same analysis conducted under the principle of conservation of the mechanical energy of the photon can be treated by an new Doppler, Relativistic Apparent, that can be confused with other Dopplers in the treatment of redshift from distant sources. It was found that the conservation of energy in Orbital Angular Momentum (OAM), in the interaction with matter, explains that the synchronization instants are found in the inversion of the OAM, where the advances and delays of REW occur under negligible variations of the OAM, however, opposites.

Relativistic Motion with Viscosity. I Newton’s Law of Resistance

The equation of motion for a relativistic neutral particle that moves in a medium characterized by a friction proportional to the square of the velocity is analyzed. The relativistic trajectory is derived in a numerical way and in the form of a Taylor series. The astrophysical applications cover the trajectory of SN 1993J and the light curve of gamma ray bursts.

Energy Conservation in the Thin Layer Approximation: IV. The Light Curve for Supernovae

The light curves (LC) for Supernova (SN) can be modeled adopting the conversion of the flux of kinetic energy into radiation. This conversion requires an analytical or a numerical law of motion for the expanding radius of the SN. In the framework of conservation of energy for the thin layer approximation we present a classical trajectory based on a power law profile for the density, a relativistic trajectory based on the Navarro--Frenk--White profile for the density, and a relativistic trajectory based on a power law behaviour for the swept mass. A detailed simulation of the LC requires the evaluation of the optical depth as a function of time. We modeled the LC of SN~1993J in different astronomical bands, the LC of GRB 050814 and the LC GRB 060729 in the keV region. The time dependence of the magnetic field of equipartition is derived from the theoretical formula for the luminosity.

A smooth trip to Alpha Centauri: Comment on “The least uncomfortable journey from A to B” [Am. J. Phys. 84(9) 690–695

We consider the relativistic generalization of the problem of the “least uncomfortable” linear trajectory from point A to point B. The traditional problem minimizes the time-integrated squared acceleration (termed the “discomfort”), and there is a universal solution for all distances and durations. This universality fails when the maximum speed of the trajectory becomes relativistic, and we consider the more general case of minimizing the squared proper acceleration over a proper time. The least uncomfortable relativistic trajectory has a rapidity that evolves like the motion of a particle in a hyperbolic sine potential, agreeing with the classical solution at low velocities. We consider the special case of a hypothetical trip to Alpha Centauri and compare the minimal-discomfort trajectory to the one with uniform Earth-like acceleration.

Acceleration toward polarization singularity inspired by relativistic E×B drift.

The relativistic trajectory of a charged particle driven by the Lorentz force is different from the classical one, by velocity-dependent relativistic acceleration term. Here we show that the evolution of optical polarization states near the polarization singularity can be described in analogy to the relativistic dynamics of charged particles. A phase transition in parity-time symmetric potentials is then interpreted in terms of the competition between electric and magnetic ‘pseudo’-fields applied to polarization states. Based on this Lorentz pseudo-force representation, we reveal that zero Lorentz pseudo-force is the origin of recently reported strong polarization convergence to the singular state at the exceptional point. We also demonstrate the deterministic design of achiral and directional eigenstates at the exceptional point, allowing an anomalous linear polarizer which operates orthogonal to forward and backward waves. Our results linking parity-time symmetry and relativistic electrodynamics show that previous PT-symmetric potentials for the polarization singularity with a chiral eigenstate are the subset of optical potentials for the E×B “polarization” drift.

Initial conditions for cold dark matter particles and general relativity

We describe the irrotational dust component of the Universe in terms of a relativistic gradient expansion and transform the resulting synchronous metric to a Newtonian coordinate system. The two metrics are connected via a spacelike displacement field and a timelike perturbation, providing a relativistic generalization of the transformation from Lagrangian to Eulerian coordinates. The relativistic part of the displacement field generates already at initial time a nonlocal density perturbation at second order. This is a purely relativistic effect since it originates from space-time mixing. We give two options, the passive and the active approach, on how to include the relativistic corrections for example in $N$-body simulations. In the passive approach we treat the corrections as a non-Gaussian modification of the initial Gaussian field (primordial non-Gaussianity could be incorporated as well). The induced non-Gaussianity depends on scale and the redshift at which initial conditions are set, with ${f}_{\mathrm{NL}}\ensuremath{\sim}\mathrm{\text{few}}$ for small enough scales and redshifts. In the active approach we show how to use the relativistic trajectory to obtain the initial displacement and velocity of particles for $N$-body simulations without modifying the initial Gaussian field.

Implications of the disagreement between newtonian and relativistic low-speed trajectories

It was shown recently for a chaotic and a nonchaotic slow- moving dynamical system that the Newtonian and special relativistic trajectories eventually disagree completely. Here, it is argued that once the Newtonian trajectory is completely different from the relativistic trajectory, Newtonian mechanics can no longer be used to study the trajectory of the slow- moving system. Special relativistic mechanics must be used instead. For a nonchaotic system, Newtonian mechanics can still be used for a very long time. However, for a strongly chaotic system, Newtonian mechanics cannot be used essentially from the outset.

Coulomb excitation at intermediate energies

Straight line trajectories are commonly used in semiclassical calculations of the first-order Coulomb excitation cross section at intermediate energies, and simple corrections are often made for the distortion of the trajectories that is caused by the Coulomb field. These approximations are tested by comparing to numerical calculations that use exact Coulomb trajectories. In this paper a model is devised for including relativistic effects in the calculations. It converges at high energies toward the relativistic straight-line trajectory approximation and approaches the non-relativistic Coulomb trajectory calculation at low energies. The model is tested against a number of measurements and analyses that have been performed at beam energies between 30 and 70 MeV/nucleon, primarily of quadrupole excitations. Remarkably good agreement is achieved with the previous analyses, and good agreement is also achieved in the few cases, where the $B$($E\ensuremath{\lambda}$) value is known from other methods. The magnitudes of the relativistic and Coulomb distortion effects are discussed.

Transition rates in atoms constrained to move relativistically

We consider the bound state of a scalar nucleus and a spin 1/2 electron interacting covariantly with a scalar radiation field. The nucleus follows a prescribed relativistic trajectory. The description is wholly in terms of laboratory quantities. The aim of the paper is twofold. We show from first principles how electronic wavefunctions and atomic transition rates transform in accordance with relativistic requirements in the case of uniform nuclear motion. We also present a general expression for the transition probability when the nucleus performs a prescribed accelerated motion.

Consistent description of Klein-Gordon dipole matrix elements

Relativistic semiclassical radial dipole matrix elements are derived analytically in the length and velocity forms using Klein-Gordon wavefunctions. In contrast to our previous works in which KGWKB matrix elements were derived only in the length form, neglecting the angular displacement of the relativistic trajectory, we have obtained these matrix elements in both length and velocity forms without the above-mentioned approximation. In the latter form, an effective velocity operator has been used. The present approaches lead to two new intermediate states which are discussed. Detailed numerical results obtained through a systematic study of lithium, sodium and copper isoelectronic sequences, starting with the nonrelativistic WKB approaches, enable us to draw inferences as to the improvements and also the limitations of the new relativistic formulae.

Calculation of ionization of helium by relativistic heavy ions with a relativistic CTMC-method

We have used a relativistic classical trajectory Monte Carlo (CTMC)-method to calculate cross sections for single and double ionization of helium by relativistic heavy ions. We compare our results with experimentally obtained cross sections. We comment on the limits of the validity of relativistic classical calculations.

Derivation of the trajectory equations of achromatic quadrupoles with Hamiltonian theory

Hamiltonian theory is used to develop a general method to derive the trajectory equations for charged particles in magnetic, electrostatic and combined multipoles with a straight ion optical axis. The effect of the fringing fields of the multipoles, the influence of the momentum spread of the particles and parasitic aberrations are easily incorporated into the trajectory equations. With this method both the relativistic and non-relativistic trajectory equations for a combined quadrupole are derived. A combined quadrupole is the combination of a magnetic quadrupole and an electrostatic quadrupole. The correct ratio of the magnetic and electrostatic field in a combined quadrupole results in an achromatic quadrupole. This is proved by the incorporation of the momentum spread into the trajectory equations.

Electrostatic lens potentials with small relativistic spherical aberration

We discuss a computerized systematic investigation to find axially symmetric electrostatic lens potentials with acceptable first-order properties and small spherical aberrations. The calculation of the spherical aberration is based on the direct solution of the relativistic trajectory equation in paraxial and nonparaxial approximation. The potentials are characterized by their properties in infinite magnification mode. Conditions for the computations and constraints to satisfy practical requirements are prescribed. A successful axial potential distribution and its relativistic spherical aberration coefficient, the spherical disk radius, and some first-order optical parameters, are also presented in the paper as an example.

Relativistic trajectory equations for charged particles in general orthogonal curvilinear coordinates

The relativistic trajectory equations are given in orthogonal curvilinear coordinates for charged particles moving in arbitrary electrostatic and magnetic fields without restrictions. The trajectory equations for Cartesian and cylindrical coordinates are presented as special cases.

Application of the relativistic trajectory Monte Carlo method to the K vacancy production with12C projectiles at 250 MeV amu-1

The K vacancy production cross section for 12C projectilesat Elab=250 MeV amu-1 on various targets (Z=22, 47, 73, 92) is calculated by means of the classical relativistic Monte Carlo trajectory method. The theoretical results are compared with the measurements of Anholt et al. (1977).

Classical relativistic trajectory method and application to the K-shell ionisation in U-U collisions

The classical trajectory method of Abrines and Percival (1966) has been extended to relativistic velocities of the charged particles. The method is based on the solution of the classical relativistic Hamilton equations of motion for a three-body system consisting of a target nucleus, a charged projectile and a single electron. The electromagnetic fields are taken to be of Lienard-Wiechert type with extended charges of the nuclei. The theory has been applied to calculate the cross section, mean impact parameter and angular distribution for the ionisation of a singly occupied K shell in U-U collisions at Elab=1 and 10 GeV/n.

Digital Computer Simulation of Charged Particle Beams and Electrostatic Lenses

A digital computer program for the simulation of charged particle beams in complex electrostatic lens systems is described. A large matrix is employed to simulate the potentials in rectangular or axisymmetric configurations, and relativistic trajectory computations are employed to simulate beams or charged particles of any energy. The beam may be injected with any energy or spatial distribution, including point sources. The program is intended to be an engineering tool which will produce results with an accuracy of one or two percent with modest effort in setting up the problem. Simulations of a CDC 6600 typically require 2 min or less, and complete solutions including plots of trajectories and equipotentials are frequently obtained in less than 30 sec. Results for some specific cases are shown to be within 2% of experimental or exact analytical results.

Nonweak-Coupling Regge Trajectories from Field-Theory Models

Perturbation-theory techniques are used to generate an approximate transcendental equation for the leading Regge pole produced by an infinite sum of Feynman ladder diagrams. This equation is not restricted to the weak-coupling domain and can be used to produce trajectories with $\ensuremath{\alpha}(0)>0$. The equation is solved both below and above the elastic and inelastic thresholds to provide the first complete description of a relativistic Regge trajectory in the ladder-diagram model. The same technique is used to generate potential-theory trajectories which agree with the exact solutions obtained by more complicated methods.