Equations Of Relativistic Dynamics Research Articles

Relativistic massive and massless scalar fields bound to the Bumblebee gravity's black hole

In this letter, we examine massive and massless scalar quasibound states bound in the static spherically symmetric black hole solution of the Einstein-Bumblebee space-time. We start with constructing the governing relativistic fields' dynamical equation, i.e. the Klein-Gordon equation, component by component. With the help of the ansatz of separation of variables, we find the exact solution of the angular part in terms of Spherical Harmonics while the radial exact solution is discovered in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that reveals a complex-valued energy levels expression for the case of massive scalar, where the real part is the scalar particle's energy and the imaginary part represents the quasibound state's decay. And for a massless scalar, a pure imaginary energy levels is obtained. The quasibound states, thus, describe decaying relativistic states bound in the black hole's gravitational potential well. At the end, we investigate the Hawking radiation of the static Bumblebee black hole's horizon by deriving and investigating the radiation distribution function.

Group classification of one-dimensional equations of relativistic gas dynamics

Analysis of the one-dimensional relativistic gas dynamics equations is presented in the paper. The equations contain the internal energy as an arbitrary element. Depending on the form of this function the equations have different symmetries. Complete group classification of the one-dimensional relativistic gas dynamics equations with respect to an admitted Lie group is obtained in the present paper. The classification divides the equations into several classes. Analysis of invariant solutions is also provided.

Conservation laws of the one-dimensional equations of relativistic gas dynamics in Lagrangian coordinates

The present paper is focused on the analysis of the one-dimensional relativistic gas dynamics equations. The studied equations are considered in Lagrangian description, making it possible to find a Lagrangian such that the relativistic gas dynamics equations can be rewritten in a variational form. Such a Lagrangian is found in the paper. Complete group analysis of the Euler–Lagrange equation is performed. The found Lagrangian and the symmetries are used to derive conservation laws in Lagrangian variables by means of Noether’s theorem. The analogs of the newly found conservation laws in Eulerian coordinates are presented as well.

Symmetry and Special Relativity

We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of light. We also obtain the spacetime transformations between inertial frames depending on this speed. From experimental evidence, this universally preserved speed is c, the speed of light, and the transformations are the usual Lorentz transformations. The ball of relativistically admissible velocities is a bounded symmetric domain with respect to the group of affine automorphisms. The generators of velocity addition lead to a relativistic dynamics equation. To obtain explicit solutions for the important case of the motion of a charged particle in constant, uniform, and perpendicular electric and magnetic fields, one can take advantage of an additional symmetry—the symmetric velocities. The corresponding bounded domain is symmetric with respect to the conformal maps. This leads to explicit analytic solutions for the motion of the charged particle.

Remarks on Asymptotic Solutions of Linearized Equations of Relativistic Hydrodynamics

In the present note, we show that, for some special classes of external flows, the short-wave asymptotic behavior of the solution of linearized equations of relativistic gas dynamics can be described rather explicitly.

Relativistic Modeling of Ultra-Short Electron Pulse Propagation

The ultrafast electron microscopy, electron diffraction, electron crystallography, and nanocrystallography methods opened the possibility of studying the coherent structural dynamics of matter. The time resolution of the ultrafast electron microscopy and electron diffraction methods determined by the duration of electron pulses is the key parameter of experimental setups. This paper treats electron pulse dynamics in the field free drift region specifically for applications in atomic imaging. The electron beam is modeled as a system of particles (N) with N = 1000 and N = 10 000 electrons. The beam propagates for a certain period of time (1–4 ns); during its propagation, electron distribution parameters (over coordinates and velocities) are calculated to characterize the temporal profile and uncertainty in the electron wavelength at the sample. The results of applying relativistic dynamic equations show that nonrelativistic results are satisfactorily applicable (with 15 per cent or better accuracy) for modeling short electron pulse elongation and broadening at 30 keV and lower energies. However, the results of such modeling may be significantly in error for intermediate energies (300 keV), and for the fast relativistic beams (3 MeV) they become completely wrong. The relative reduction in Coulomb repulsion effects at higher energies is known, however; we give a comprehensive treatment that allows a quantitative picture. Using high-energy electron pulses results in almost complete elimination of the repulsive Coulomb effect. Dispersion of electron velocities becomes much lower at higher energies. For 3 MeV electrons, electron pulse duration as well as its radius does not noticeably change even after traveling for 4 ns (1.2 m). Even at 300 keV, the pulse duration increase is negligible until 1 ns (0.2 m). A simple mean-field model suggested in [13] has been extended to arbitrarily fast relativistic electron pulses with good correspondence to direct dynamic modeling.

Capture and acceleration of electrons by the ultrarelativistic wakefield

Modeling results of the external injection of electrons into wakefield excited by the ultrarelativistic ion driving bunch in plasma are presented. Capture and acceleration of the injected electron bunch in the wakefield are described by the system of relativistic dynamic equations. Initial parameters of the plasma and the ion bunch are similar to that in the experiment AWAKE which is currently carried out at CERN. Characteristics of the accelerated electron bunch (percentage of captured electrons and energy dispersion) are studied at small angle of injection relative to the direction of ion driver propagation and far enough from the driver where the wakefield structure is strongly nonlinear.

Relativistic rotation dynamics —Formalism and examples

We obtain the relativistic rotation dynamical equations by using the Hamiltonian-Lagrangian formalism. After establishing the dynamical equation for the relativistic rotation, we derive the relation between the rotational-kinetic-energy variation and the angular-momentum variation. Then, we discuss three examples that illustrate how this special relativity formalism deals with rotational processes of systems acted upon by external torques, including cases in which dissipative forces are present.

Relativistic dynamics of superfluid-superconducting mixtures in the presence of topological defects and an electromagnetic field with application to neutron stars

The relativistic dynamic equations are derived for a superfluid-superconducting mixture coupled to the electromagnetic field. For definiteness, and bearing in mind possible applications of our results to neutron stars, it is assumed that the mixture is composed of superfluid neutrons, superconducting protons, and normal electrons. Proton superconductivity of both I and II types is analysed, and possible presence of neutron and proton vortices (or magnetic domains in the case of type-I proton superconductivity) is allowed for. The derived equations neglect all dissipative effects except for the mutual friction dissipation and are valid for arbitrary temperatures (i.e. they do not imply that all nucleons are paired), which is especially important for magnetar conditions. It is demonstrated that these general equations can be substantially simplified for typical neutron stars, for which a kind of magnetohydrodynamic approximation is justified. Our results are compared to the nonrelativistic formulations existing in the literature and a number of discrepancies are found. In particular, it is shown that, generally, the electric displacement ${\pmb D}$ does not coincide with the electric field ${\pmb E}$, contrary to what is stated in the previous works. The relativistic framework developed here is easily extendable to account for more sophisticated microphysics models and it provides the necessary basis for realistic modelling of neutron stars.

Wigner function of a relativistic particle in a time-dependent linear potential

We construct phase-space representations for a relativistic particle in both a constant and a time-dependent linear potential. We obtain explicit expressions for the Wigner distribution functions for these systems and find the correct nonrelativistic limit and free-particle limit for these functions. We derive the relativistic dynamical equation governing the time development of the Wigner distribution function and relativistic equation for the Wigner distribution function of stationary states and also calculate the amplitudes of transitions between energy states.

Physical Determination of the Action

Physical theories ought to be built up from colloquial notions such as 'long bodies', 'energetic sources' etc. in terms of which one can define pre-theoretic ordering relations such as 'longer than', 'more energetic than'. One of the questions addressed in previous work is how to make the transition from these pre-theoretic notions to quantification, such as making the transition from the ordering relation of 'longer than' (if one body covers the other) to the notion of how much longer. In similar way we introduce dynamical notions 'more impulse' (if in a collision one object overruns the other) and 'more energetic' (if the effect of one source exceeds the effect of the other). In a physical model - built by coupling congruent standard actions - those basic pre-theoretic notions become measurable. We uncover the origin of (basic) physical quantities of Energy, Momentum and Inertial Mass. From physical and methodical principles - without mathematical presuppositions - we derive all equations of (classical and relativistic) Dynamics and ultimately the principle of least action.

Modulational instability of ion-acoustic waves in fully relativistic two-component plasma

Nonlinear amplitude modulation of ion-acoustic waves (IAWs) in a fully relativistic unmagnetized two-fluid plasma has been theoretically studied by using complete set of fully relativistic dynamic equations. To describe the nonlinear evolution of the wave envelope a nonlinear Schrödinger (NLS) equation is derived by using standard multiple scale perturbation technique. Using this equation it is shown that the wave becomes modulationally unstable as the wavenumber exceeds certain critical value. This critical wavenumber is found to decrease with increase in relativistic effect. The instability growth rate has also been calculated numerically for different values of plasma drift velocity. The growth rate is shown to decrease with increase in the relativistic effect.

About inclusion of Maxwell’s equations in systems relativistic of the invariant equations

This paper describes the author’s thoughts and reflection inspired after publishing paper [1] and com� paring it with the lecture [2] delivered at a conference in Lyon. This lecture was based on coauthored work [3]. In [1] the equations of relativistic fluid dynamics of a charged liquid dielectric were supplemented with Maxwell’s equations for the electromagnetic field produced by this charge. Primary attention was given to the overcoming of the difficulties associated with the relativistically invariant incorporation of the param� eters μ and e depending on hydrodynamic variables, specifically, on the velocity. The study was a develop� ment of Minkowski’s classical work [4]. More specifically, a Lorentz invariant W(E, B, μ, e) that is a qua� dratic form in B i and E k was used to write Maxwell’s equations as

Covariant Uniform Acceleration

We derive a 4D covariant Relativistic Dynamics Equation. This equation canonically extends the 3D relativistic dynamics equation , where F is the 3D force and p = m0γv is the 3D relativistic momentum. The standard 4D equation is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. This solves a problem of Einstein and Planck.We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant types: null, linear, rotational, and general. For null acceleration, the worldline is cubic in the time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion.We use Generalized Fermi-Walker transport to construct a uniformly accelerated family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K' to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame.We obtain velocity and acceleration transformations from a uniformly accelerated system K' to an inertial frame K. We introduce the 4D velocity, an adaptation of Horwitz and Piron s notion of "off-shell." We derive the general formula for the time dilation between accelerated clocks. We obtain a formula for the angular velocity of a uniformly accelerated object. Every rest point of K' is uniformly accelerated, and its acceleration is a function of the observer's acceleration and its position. We obtain an interpretation of the Lorentz-Abraham-Dirac equation as an acceleration transformation from K' to K.

Solution of a class of rotational relativistic rotation dynamic equation using the generalized variational iteration theory

A class of relational relativistic rotation nonlinear disturbed dynamical equations possessing nonlinear damping force and forcing periodic force is investigated. Firstly, by using the variational principle, the generalized variational iteration is constructed. Then the initial approximate solution is determined. Finally, by using the iteration, the approximation to an arbitray degree for corresponding equation is found.

Making the relativistic dynamics equation covariant: explicit solutions for motion under a constant force

We derive a four-dimensional (4D) covariant relativistic dynamics equation. This equation canonically extends the 3D relativistic dynamics equation , where F is the 3D force and p = m0γv is the 3D relativistic momentum. The standard 4D equation is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. This solves a problem of Einstein and Planck. We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant types: null, linear, rotational and general. For null acceleration, the worldline is cubic in time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion.

Non-extensive statistics, relativistic kinetic theory and fluid dynamics

Experimental particle spectra can be successfully described by power law tailed energy distributions characteristic to canonical equilibrium distributions associated to Rényi’s or Tsallis’ entropy formula --over a wide range of energies, colliding system sizes, and produced hadron sorts. In order to derive its evolution one needs a corresponding dynamical description of the system which results in such final state observables. The equations of relativistic fluid dynamics are obtained from a non-extensive Boltzmann equation consistent with Tsallis’ non-extensive q-entropy formula. The transport coefficients like shear viscosity, bulk viscosity, and heat conductivity are evaluate based on a linearized collision integral.

Causal relativistic fluid dynamics

We derive causal relativistic fluid dynamical equations from the relaxation model of kinetic theory as in a procedure previously applied in the case of non-relativistic rarefied gases in Chen et al (2000 Phys. Lett. A 271 87). By treating space and time on an equal footing and avoiding the iterative steps of the conventional Chapman–Enskog (CE) method, we are able to derive causal equations in the first order of the expansion in terms of the mean flight time of the particles. This is in contrast to what is found using the CE approach. We illustrate the general results with the example of a gas of identical ultrarelativistic particles such as photons under the assumptions of homogeneity and isotropy. When we couple the fluid dynamical equations to Einstein's equation we find, in addition to the geometry-driven expanding solution of the FRW model, a second, matter-driven nonequilibrium solution to the equations. In only the second solution, entropy is produced at a significant rate.

Conformal hydrodynamics in Minkowski and de Sitter spacetimes

We show how to generate non-trivial solutions to the conformally invariant, relativistic fluid dynamic equations by appealing to the Weyl covariance of the stress tensor. We use this technique to show that a recently studied solution of the relativistic conformally invariant Navier–Stokes equations in four-dimensional Minkowski space can be recast as a static flow in three-dimensional de Sitter space times a line. The simplicity of the de Sitter form of the flow enables us to consider several generalizations of it, including flows in other spacetime dimensions, second order viscous corrections, and linearized perturbations. We also construct the anti-de Sitter dual of the original four-dimensional flow. Finally, we discuss possible applications to nuclear physics.

A New Self-Consistent Unbounded Magnetic Field 3-D FE Computation for Electron Guns

An innovative 3-D Finite Element approach for the computation of the self-consistent unbounded magnetic field of electron beams is presented for the analysis of electron guns. The solution of the whole problem is obtained by solving iteratively a Vlasov-Maxwell equation assuming stationary conditions together with the relativistic dynamical equations of motion. The self-consistent unbounded magnetostatic field is computed by using an edge elements curl-curl formulation for the magnetic vector potential. A fictitious boundary iterative algorithm is adopted to overcome the problem of boundary conditions assignment. An accuracy test concerning the spread of a relativistic beam and the simulation of a complex gridded electron gun are shown in order to validate and better illustrate the procedure.