Relativistic Equations Research Articles

Towards Multifractality through an Ernst-Type Potential in Complex Systems Dynamics.

Some possible correspondences between the Scale Relativity Theory and the Space-Time Theory can be established. Since both the multifractal Schrödinger equation from the Scale Relativity Theory and the General Relativity equations for a gravitational field with axial symmetry accept the same SL(2R)-type invariance, an Ernst-type potential (from General Relativity) and also a multi-fractal tensor (from Scale Relativity) are highlighted in the description of complex systems dynamics. In this way, a non-differentiable description of complex systems dynamics can become functional, even in the case of standard theories (General Relativity and Quantum Mechanics).

Role of decoupling measure on the complexity factor and isotropization of the charged anisotropic spheres

An electromagnetic field is a consequential aspect of the space carried by moving electric charges. The contemporaneous interaction of Einstein–Maxwell field could be provoked by the dispersing electro-magnetic waves owing to the warping of the spacetime. Within the complete geometric deformation scenario, current work is devoted to exploring the substantial impact of an electromagnetic field on the anisotropic spherically symmetric structures that are taken to be the seed source, and then augment to the additional source. In this case, the gravitational decoupling strategy and embedding Class-I constraint will be used as two techniques to address the issue. We compute the Einstein–Maxwell field equations for the desired configuration. The transformed metric potentials corresponding to the temporal and radial components are then applied to formulate the two distinct sets of field equations. Through the Kuchowicz-ansatz for the seed metric potential, we enforce the Karmarkar constraint to grasp the entire gravitational aspect of a static compact sphere together with the effects of charged anisotropic pressure. The uniqueness of our study is that we used isotropization and disappearing complexity requirements to derive the deformation function within the bounds of a charged scenario. Thereby, two solutions to the associated relativistic equations are then presented for physical validity. We address the fact that the existence of charged pressure anisotropy in the bounded sphere has a consequential influence on governing its stability. Furthermore, it is identified that the decoupling-constant magnitude controls the energy flow within both the generically charged fluid and the fluid matter composition.

Spin Alignment of Vector Mesons in Heavy-Ion Collisions.

Polarized quarks and antiquarks in high-energy heavy-ion collisions can lead to the spin alignment of vector mesons formed by quark coalescence. Using the relativistic spin Boltzmann equation for vector mesons derived from Kadanoff-Baym equations with an effective quark-meson model for strong interaction and quark coalescence model for hadronizaton, we calculate the spin density matrix element ρ_{00} for ϕ mesons and show that anisotropies of local field correlations with respect to the spin quantization direction lead to ϕ meson's spin alignment. We propose that the local correlation or fluctuation of ϕ fields is the dominant mechanism for the observed ϕ meson's spin alignment and its strength can be extracted from experimental data as functions of collision energies. The calculated transverse momentum dependence of ρ_{00} agrees with STAR's data. We further predict the azimuthal angle dependence of ρ_{00} which can be tested in future experiments.

Propagation of anisotropic gravitational and electromagnetic waves at very high energies

We analyze the dispersion relation for an anisotropic gravity-electromagnetic theory at very high energies. In particular for photons of very high energy. We start by introducing the anisotropic gravity-gauge vector field model. It is invariant under spacelike diffeomorphisms, time parametrization, and U(1) gauge transformations. It includes high-order spacelike derivatives as well as polynomial expressions of the Riemann and field strength tensor fields. It is based on the Hořava-Lifshitz anisotropic proposal. We show its consistency, and the stability of the Minkowski ground state. Finally, we determine the exact zone at which the physical degrees of freedom, i.e. the transverse-traceless tensorial degrees of freedom and the transverse vectorial degrees of freedom propagate according to a linear wave equation. This is so, in spite of the fact that there exists in the zone a non-trivial Newtonian background of the same order. The wave equation contains spatial derivatives up to the sixth order, in the lowest order it exactly matches the relativistic wave equation. We then analyze the dispersion relation at very high energies in the context of recent experimental data. The qualitative predictions of the proposed model, concerning the propagation of highly energetic photons, are different from the ones obtained from the modified dispersion relation of the LIV models.

Entropic force for quantum particles

Entropic force has been drawing the attention of theoretical physicists following E Verlinde’s work in 2011 to derive Newton’s second law and Einstein’s field equations of general relativity. In this paper, we extend the idea of entropic force to the distribution of quantum particles. Starting from the definition of Shannon entropy for continuous variables, here we have derived quantum osmotic pressure as well as the consequent entropic forces for bosonic and fermionic particles. The entropic force is computed explicitly for a pair of bosons and fermions. The low temperature limit of this result show that the entropic force for bosons is similar to Hooke’s law of elasticity revealing the importance of this idea in the formation of a Bose–Einstein condensate. For fermions, the low temperature limit boils down to the well known Neumann’s radial force and also reveals the Pauli’s exclusion principle. The classical limit of the entropic force between quantum particles is then discussed. As a further example, the entropic force for quantum particles in noncommutative space is also computed. The result reveals a violation of the Pauli exclusion principle for fermions in noncommutative space.

Positive Classical Solutions of Dirichlet Problem for the Steady Relativistic Heat Equation

Positive Classical Solutions of Dirichlet Problem for the Steady Relativistic Heat Equation

A discussion on the relativistic corrections of the electronic structures of multi‐electron atoms

AbstractThe theory of special relativity is a topic not addressed in theoretical chemistry courses. Einstein's theory of relativity is mainly applied to high‐energy phenomena governed by physical laws not used to describe the chemical bond. Nevertheless, the electrons of heavy elements orbit around the nucleus with relativistic velocities. Here it is, then, the laws that appeared so unusual for chemistry became fundamental tools for investigating the properties of these atoms. Studying the electronic structure of heavy elements is the best place for chemists to learn to tame the pitfalls of a little‐known subject, such as Einstein's theory, and become familiar with its elegant formalism. In this paper, the main relativistic approaches to quantum chemistry are revisited. The problem of the finite nuclear size, the approximation methods used to solve the relativistic equations for k‐shell electrons, and the problem of the negative energy continuum are investigated. Some ansatzes to solve divergence's problem of relativistic corrections near the nucleus are proposed. Finally, a possible connection between the self‐consistent field method and bondonic chemistry is also investigated.

The Influence of Deformation Space-Space on High and Low Energy Spectra of Fermionic Particles and Spectra of Heavy Quarkonia with Improved Hulthén Plus Hyperbolic Exponential Inversely Quadratic Potential

In this work, the modified approximation to the centrifugal barrier term is applied to find approximate bound-state solutions of the deformed Dirac equation for the spin and pseudospin symmetries in a model with the improved hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials (IHHEIQPs) using the parametric method of Bopp’s shift and the standard perturbation theory in the extended relativistic quantum mechanics (ERQM). Our results indicate that the new energy eigenvalues are highly sensitive to the potential parameters (ν1, A) and to the values of quantum atomicnumbers (j, k, l, m,̃︀ l, ̃︀ m, s, ̃︀s), range of the potential ν, and noncommutativity parameters (θ, σ, β). We found that the effect of a space-space deformation gives a correction in the energy spectrum, where the main energy term remains due to the effect of the hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials known in the literature. The new nonrelativistic energies are obtained by applying the nonrelativistic limit to the relativistic spin-energy equation in the extended nonrelativistic quantum mechanics (ENRQM). The proposed potential model reduces to the improved Hulth´en and exponential inversely quadratic potentials as special cases in ERQM. The present results are applied for calculating the new mass spectra Mhiqpnc−nl of heavy mesons such as cc, bb, bc, bs, cs, and bq,q = (u, d) in ENRQM. It turns out that the values of masses come from the contribution of the mass spectra Mhiqpnl in NRQM, while the effect of a space-space deformation δMhiqpnc−nl is an infinitesimal correction as compared with Mhiqpnl. Our results seem to be significant and agree perfectly with the ones in the literature.

Concentration and cavitation in the vanishing pressure limit of solutions to the relativistic Euler equations with the logarithmic equation of state

In this paper, we constructively solve the Riemann problem for the relativistic Euler equations with the logarithmic equation of state. The concentration and cavitation phenomena are observed and analyzed during the process of vanishing pressure in the Riemann solutions. It is rigorously proved that, as the pressure vanishes, they tend to the two kinds of Riemann solutions to the zero-pressure relativistic Euler equations, which include a delta shock formed by a weighted δ-measure and a vacuum state.

Global high-order numerical schemes for the time evolution of the general relativistic radiation magneto-hydrodynamics equations

Modeling correctly the transport of neutrinos is crucial in some astrophysical scenarios such as core-collapse supernovae and binary neutron star mergers. In this paper, we focus on the truncated-moment formalism, considering only the first two moments (M1 scheme) within the grey approximation, which reduces Boltzmann seven-dimensional equation to a system of equations closely resembling the hydrodynamic ones. Solving the M1 scheme is still mathematically challenging, since it is necessary to model the radiation-matter interaction in regimes where the evolution equations become stiff and behave as an advection-diffusion problem. Here, we present different global, high-order time integration schemes based on Implicit-Explicit Runge-Kutta methods designed to overcome the time-step restriction caused by such behavior while allowing us to use the explicit Runge-Kutta commonly employed for the magneto-hydrodynamics and Einstein equations. Finally, we analyze their performance in several numerical tests.

A theory of dark energy that matches dark matter

In this paper, a theory of dark energy is proposed that matches dark matter. The relativistic quantum mechanics equations reveal that free particles can have negative energies. We think that the negative energy is the dark energy, which behaves as dark photons with negative energies. In this work, the photon number states are extended to the cases where the photon number can be negative integers, called negative integer photon states, the physical meaning of which are that the photons in such a state are of negative energy, i.e., dark photons. The dark photons constitute dark radiation, also called negative radiation. The formalism of the statistical mechanics and thermodynamics of the dark radiation is presented. This version of dark energy is of negative temperature and negative pressure, the latter regarded as responsible for the accelerate expansion of the universe. It is believed that there is a symmetry of energy-dark energy in the universe. In our previous work, the theory of the motion of the matters with negative kinetic energy was presented. In our opinion, the negative kinetic energy matter is dark matter. In the present work, we demonstrate that the dark substances absorb and release dark energy. In this view, the dark matter and dark energy match. Therefore, there is a symmetry of matter-energy match and dark matter-dark energy match in the universe. We present the reasons why the negative kinetic energy systems and negative radiation are dark to us.

Erratum: Timelike orbits around accelerating black holes [Phys. Rev. D 106 , 044037 (2022)

Received 24 April 2023DOI:https://doi.org/10.1103/PhysRevD.107.129901© 2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasClassical black holesGeneral relativity equations & solutionsGravitation, Cosmology & Astrophysics

Study of energies spectra and thermodynamic properties of the relativistic Dirac equation using Feynman path integral method

In the current paper, the contribution of a Coulomb-like potential tensor interaction on the solution of the Dirac equation with a new generalized Morse-like potential is investigated using the Feynman path integral method. Relativistic and non-relativistic energy spectra were obtained. It has been established that the Coulomb-like potential eliminates the degeneracy of all pairs of spin doublets. Using the resulting non-relativistic energy eigenvalues equation, three diatomic molecules (H2, LiH, and HCl) were investigated and their thermodynamic properties, including mean energy, free energy, entropy, and specific heat capacity were shown. A comparison with the available literature shows that the thermodynamic plots obtained are consistent with previous work.

Groups of Coordinate Transformations between Accelerated Frames

The analysis of the present paper reveals that, besides the relativistic symmetry expressed by the Lorentz group of coordinate transformations which leave invariant the Minkowski metric of space-time of inertial frames, there exists one more relativistic symmetry expressed by a group of coordinate transformations leaving invariant the space-time metric of the frames with a constant proper-acceleration. It is remarkable that, in the flat space-time, only those two relativistic symmetries, corresponding to groups of continuous transformations leaving invariant the metric of space-time of extended rigid reference frames, exist. Therefore, the new relativistic symmetry should be considered on an equal footing with the Lorentz symmetry. The groups of transformations leaving invariant the metric of the space-time of constant proper-acceleration are determined using the Lie group analysis, supplemented by the requirement that the group include transformations to or from an inertial to an accelerated frame. Two-parameter groups of two-dimensional (1 + 1), three-dimensional (2 + 1), and four-dimensional (3 + 1) transformations, with the group parameters related to the ratio of accelerations of the frames and the relative velocity of the frame space origins at the initial moment, can be considered as counterparts of the Lorentz group of corresponding dimensions. Defining the form of the interval and the groups of coordinate transformations satisfying the relativity principle paves the way to defining the invariant forms of the laws of dynamics and electrodynamics in accelerated frames. Thus, the problem of extending the relativity principle from inertial to uniformly accelerated frames has been resolved without use of the equivalence principle and/or the general relativity equations. As an application of the transformations to purely kinematic phenomena, the problem of differential aging between accelerated twins is treated.

Relativistic modeling of stellar objects in a Schwarzschild’s coordinates with embedded class one spacetime

In this study, we investigate a new relativistic anisotropic Einstein field equations solution for compact stars under embedding class 1 conditions. In order to do this, we apply the Karmarkar condition and the embedding class one technique. By taking Buchdahl type metric potential g_{rr} into consideration, the precise analytical solution has been investigated. We have studied physical characteristics of various compact star using this analytical solution. Central singularities are absent from the solution. We have explored thermodynamic observables inside the stellar models, such as radial and tangential pressures, matter density, anisotropic factor, energy conditions, TOV, red-shift, and the speed of sound, etc., after establishing this space-time geometry for the stellar models. From the graphical representation of various physical characteristics, it is demonstrated that our model meets all the specification for ultra-high density compact bodies.

Modulation of electromagnetic waves in a relativistic degenerate plasma at finite temperature

We study the modulational instability (MI) of a linearly polarized electromagnetic (EM) wave envelope in an intermediate regime of relativistic degenerate plasmas at a finite temperature (T≠0) where the thermal energy (KBT) and the rest-mass energy (mec2) of electrons do not differ significantly, i.e., βe≡KBT/mec2≲ (or ≳) 1, but the Fermi energy (KBTF) and the chemical potential energy (μe) of electrons are still a bit higher than the thermal energy, i.e., TF>T and ξe=μe/KBT≳1. Starting from a set of relativistic fluid equations for degenerate electrons at finite temperature, coupled to the EM wave equation and using the multiple scale perturbation expansion scheme, a one-dimensional nonlinear Schödinger (NLS) equation is derived, which describes the evolution of slowly varying amplitudes of EM wave envelopes. Then, we study the MI of the latter in two different regimes, namely, βe<1 and βe>1. Like unmagnetized classical cold plasmas, the modulated EM envelope is always unstable in the region βe>4. However, for βe≲1 and 1<βe<4, the wave can be stable or unstable depending on the values of the EM wave frequency, ω, and the parameter ξe. We also obtain the instability growth rate for the modulated wave and find a significant reduction by increasing the values of either βe or ξe. Finally, we present the profiles of the traveling EM waves in the form of bright (envelope pulses) and dark (voids) solitons, as well as the profiles (other than traveling waves) of the Kuznetsov–Ma breather, the Akhmediev breather, and the Peregrine solitons as EM rogue (freak) waves, and discuss their characteristics in the regimes of βe≲1 and βe>1.

The Relativistic Equations of Motion for a Charged Particle in the Magnetic Field of an Infinite Current-Carrying Wire

We show how to produce the closed form solution for the motion of a charged particle in the magnetic field of an infinitely long, current-carrying wire in the relativistic range, thus extending the results produced recently (Asadi-Zeydabadi, M., & Zaidins, C. S., 2019). We outline the areas where the two solutions are similar aa well as where the two solutions are slightly different. We also extend the work in (Asadi-Zeydabadi, M., & Zaidins, C. S., 2019) to the more realistic case where the current generates both a magnetic and an electric field, as is the case in real life. We plan to extend the approach in the future to more complicated cases as the one of the finite length wire. The solution is of great interest for the design of particle accelerators, hence it is interesting for both theoretical physicists and engineers alike.

Effects of magnetic field on the evolution of energy density fluctuations

We study the effects of a static and uniform magnetic field on the evolution of energy density fluctuations present in a medium. By numerically solving the relativistic Boltzmann-Vlasov equation within the relaxation time approximation, we explicitly show that magnetic fields can affect the characteristics of energy density fluctuations at the timescale the system achieves local thermodynamic equilibrium. A detailed momentum mode analysis of fluctuations reveals that a magnetic field increases the damping of mode oscillations, especially for the low momentum modes. This leads to a reduction in the ultraviolet (high momentum) cutoff of fluctuations and also slows down the dissipation of relatively low momentum fluctuation modes. We discuss the phenomenological implications of our study on various sources of fluctuations in relativistic heavy-ion collisions.

Propagation of Moments for Large Data and Semiclassical Limit to the Relativistic Vlasov Equation

We investigate the semiclassical limit from the semirelativistic Hartree–Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form , , , and , with the convention if . For mixed states, we show convergence in the Schatten norms with an explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [E. Dietler, S. Rademacher, and B. Schlein, J. Stat. Phys., 172 (2018), pp. 398–433], where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions.

Topological effects on generalized Duffin-Kemmer-Petiau oscillator under Aharonov–Bohm flux field and Coulomb potential

In this paper, we study the relativistic quantum motions of the oscillator field of the wave equation under the influence of the Aharonov–Bohm (AB) flux field with a Coulomb vector potential in the background of the topological defects produced by a cosmic string and global monopole space-time. We derive the radial equation of the generalized Duffin-Kemmer-Petiau (DKP) oscillator in a static cosmic string space-time and solve it through the Heun function equation. Afterwards, we derive the radial equation of the same generalized DKP oscillator in a point-like global monopole background and obtain the eigenvalue solutions using the same procedure. The generalized oscillator field is studied by substituting the radial momentum operator ∂ r → (∂ r + i M ω η 0 f(r)), where f(r) is an arbitrary function other than linear and introduces a vector potential of Coulomb-types through a minimal substitution via ∂ μ → (∂ μ − i q A μ ) in the relativistic wave equation. It is shown that the eigenvalue solutions of the oscillator field are influenced by the topological defects of the cosmic string and point-like global monopole space-times and get them modified. Furthermore, we see that the eigenvalue solutions depend on the geometric quantum phase, and hence, shifted them more in addition to the topological defects that show the gravitational analogue to the Aharonov–Bohm effect for the bound-states.