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Georgiy (George) Albertovich Krasinsky was one of the scientists who made Soviet and later Russian science both advanced and competitive by arousing interest in it with the leading researchers around the world. Having received his education in the Department of Mathematics and Mechanics at Leningrad State University, for a long time he was engaged in research in the field of classical celestial mechanics, where he performed a number of first-class studies whose results are still in demand at present. The onset of the Space Age, with its sharp jump in the accuracy of astronomical observations, caused by the rapid development of technology, made Krasinsky realize that the classical methods of celestial mechanics and astrometry could no longer cope with the processing of up-to-date measurements, as these were unable to provide the accuracy of the ephemerides of the Solar System bodies at a level demanded by the new requirements. As a result, under his expert guidance and with his active participation, the ERA (Ephemeris Research in Astronomy) software system was developed, which became the basis for all of the later work at the Institute of Applied Astronomy of the Russian Academy of Sciences (IAA RAS) on processing modern high-precision measurements and formulating relativistic numerical theories of the orbital motion of the Solar System bodies. While working at the IAA RAS, George got deeply involved in such issues as the fundamental coordinate-time system, thus becoming one of those who were active in building up the quasar radio-interferometric network. During the last years of his life, George developed a numerical theory of the rotation of a deformable Earth with a liquid core, and theories of the evolution of both the rotational motion of the Earth and of the Moon's orbit. [ABSTRACT FROM AUTHOR]

The axiomatic structure of the κ -statistcal theory is proven. In addition to the first three standard Khinchin–Shannon axioms of continuity, maximality, and expansibility, two further axioms are identified, namely the self-duality axiom and the scaling axiom. It is shown that both the κ -entropy and its special limiting case, the classical Boltzmann–Gibbs–Shannon entropy, follow unambiguously from the above new set of five axioms. It has been emphasized that the statistical theory that can be built from κ -entropy has a validity that goes beyond physics and can be used to treat physical, natural, or artificial complex systems. The physical origin of the self-duality and scaling axioms has been investigated and traced back to the first principles of relativistic physics, i.e., the Galileo relativity principle and the Einstein principle of the constancy of the speed of light. It has been shown that the κ -formalism, which emerges from the κ -entropy, can treat both simple (few-body) and complex (statistical) systems in a unified way. Relativistic statistical mechanics based on κ -entropy is shown that preserves the main features of classical statistical mechanics (kinetic theory, molecular chaos hypothesis, maximum entropy principle, thermodynamic stability, H-theorem, and Lesche stability). The answers that the κ -statistical theory gives to the more-than-a-century-old open problems of relativistic physics, such as how thermodynamic quantities like temperature and entropy vary with the speed of the reference frame, have been emphasized. [ABSTRACT FROM AUTHOR]

A modification to the vis-viva equation that accounts for general relativistic effects is introduced to enhance the accuracy of predictions of orbital motion and precession. The updated equation reduces to the traditional vis-viva equation under Newtonian conditions and is a more accurate tool for astrodynamics than the traditional equation. Preliminary simulation results demonstrate the application potential of the modified vis-viva equation for more complex n-body systems. Spherical symmetry is assumed in this approach; however, this limitation could be removed in future research. This study is a pivotal step toward bridging classical and relativistic mechanics and thus makes an important contribution to the field of celestial dynamics. [ABSTRACT FROM AUTHOR]

A linear response relation between metric and fluid perturbations driven by a background internal noise source is used as a framework for addressing stochastic effects in order to establish a mesoscopic theory for dense matter relativistic stars. In this paper, nonradial polar perturbations are worked out, which are important from the point of view of detection in future. We present qualitative first results in this paper, numerical estimates have to await further progress in theoretical modeling. These perturbations carry a new generalized stochastic nature and are obtained as solutions of the classical Einstein–Langevin equation which has been recently proposed. The significance of these stochastic nonradial polar perturbations lies at probing the intermediate sub-hydro scales inside the dense fluid. This formalism extends towards a mesoscopic scale nonequilibrium/near-equilibrium statistical mechanics study for relativistic star interiors. The generalized stochastic noise originates as the remnant of collapse mechanism and dynamical effects at intermediate scales in isolated massive stars which drives these polar perturbations. More specifically, for cold dense matter which is our focus in this paper, it is either the interplay between the degeneracy pressure and the gravitational pressure, or the multiscale phenomena like turbulences giving rise to the seeds of stochastic effects in the gravitating body. Characterizing such stochastic effects can lead to an improved understanding of the nature of dense matter and help to probe multiple scales which are yet untouched. [ABSTRACT FROM AUTHOR]

The article focuses on the question of whether time passes, delving into the historical shift from presentism to eternalism in the philosophy of time during the twentieth century. Topics discussed include McTaggart's A-series and B-series, the impact of relativistic physics, and the development of pro-passage and no-passage interpretations.

The transition state theory allows the development of approximated models useful to study the non‐equilibrium evolution of systems undergoing transformations between two states (e.g., chemical reactions). In a simplified 1D setting, the characteristic rate constants are typically written in terms of a temperature‐dependent characteristic oscillation frequency νs$\nu _s$, describing the exploration of the phase space. As a particular case, this statistical oscillation frequency νs$\nu _s$ can be defined for an arbitrary convex potential energy well. This value is compared here with the deterministic oscillation frequency νd$\nu _d$ of the corresponding anharmonic oscillator. It is proved that there is a universal relationship between statistical and deterministic frequencies, which is the same for classical and relativistic mechanics. The independence of this relationship from the adopted physical laws gives it an interesting thermodynamic and pedagogical meaning. Several examples clarify the meaning of this relationship from both physical and mathematical viewpoints. [ABSTRACT FROM AUTHOR]

Relevance and formulation of the problem. The number of areas of scientific and practical activity in which it is necessary to consider relativistic corrections is steadily growing. In many cases, two objects under study move towards one another. This takes place both in relation to astronomical objects and in relation to quantum particles, including in colliders—accelerators of charged particles in colliding beams. With counter relativistic motions, the relative velocity does not coincide with the approach velocity. However, considering relative velocity alone limits the arsenal of research tools and methods. As opposed to relative velocity, which is determined in accordance with the relativistic formula for velocity addition, the approach velocity of unaccelerated objects is defined as the ratio of the distance between them to the time it takes to cover it. The purpose of this work is to analyze the variety of the relativistic approach velocity of objects depending on the choice of inertial reference frames based on the data of the Large Hadron Collider. Results. At the Large Hadron Collider, the approach velocity of protons is almost twice as high as the speed of light in the laboratory reference frame. In frames of reference associated with moving protons, depending on the options of relativistic transformation of segments of lengths and time intervals, the maximum approach velocity of protons is 1.1 × 108с, and the minimum is 1.2 m/s. In accordance with the technique based on the relativistic velocity addition formula, the approach velocity in reference systems associated with moving protons is almost equal to the speed of light. In this case, the approach velocity becomes equal to the relative velocity, which should not be considered as a generalization of the classical mechanics rule on the indistinguishability of these velocities to relativistic mechanics. Practical significance. The results obtained may be of interest in assessing the approach velocities of astronomical objects, including the Earth and asteroids, as well as significantly expand the variability of hypotheses when processing experimental data arrays obtained at elementary particle accelerators, including the Large Hadron Collider. [ABSTRACT FROM AUTHOR]

We present a general formulation of special relativity with generalized Lorentz transformations and Minkowski line element. Besides the Minkowski and Euclidean metrics associated, respectively, with the Lorentz and Euclidean transformations, the general formulation can also be used to establish new forms of coordinate transformations that also leave their associated metrics invariant. We also formulate relativistic mechanics under the framework of the generalized special relativity. [ABSTRACT FROM AUTHOR]

Two different versions of relativistic Langevin equation in curved spacetime background are constructed, both are manifestly general covariant. It is argued that, from the observer's point of view, the version which takes the proper time of the Brownian particle as evolution parameter contains some conceptual issues, while the one which makes use of the proper time of the observer is more physically sound. The two versions of the relativistic Langevin equation are connected by a reparametrization scheme. In spite of the issues contained in the first version of the relativistic Langevin equation, it still permits to extract the physical probability distributions of the Brownian particles, as is shown by Monte Carlo simulation in the example case of Brownian motion in (1 + 1) -dimensional Minkowski spacetime. [ABSTRACT FROM AUTHOR]

The article deals with a discussion of fundamental problems arising in relativistic mechanics (general relativity) in relation to the determination of stresses generated by gravity in a deformable solid. Three such problems are considered. The first one is associated with the incompleteness of Einstein's system of equations that includes six mutually independent equations with ten unknown coefficients of the metric tensor. The second one arises when determining the stresses in a solid generated by gravity. For a static problem, three equations of the law of conservation of theory (equilibrium equations) include six unknown stresses, which, unlike Newton's theory, does not allow determining gravitational stresses. The third problem is related to the reduction of linearized Einstein equations to the equations of Newton's gravitational theory. Such a reduction turns out to be possible only for empty space and is not valid for a solid body. The noted contradictions can be eliminated by limiting the scope of the theory to a special space, which is Euclidean with respect to spatial coordinates and Riemannian only with respect to time. The discussion is illustrated by a spherically symmetric problem that reduces to ordinary differential equations. [ABSTRACT FROM AUTHOR]

The Boltzmann–Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars—classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism—and its foundations are grounded on the optimization of the BG (additive) entropic functional SBG=−k∑ipiln⁡pi. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional Sq=k((1−∑ipiq)/(q−1)) (q∈R; S1=SBG). The index q and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'. [ABSTRACT FROM AUTHOR]

This article presents alternative Hamiltonian and Lagrangian formalisms for relativistic mechanics using proper time and proper Lagrangian coordinates in 1 + 1 dimensions as parameters of evolution. The Lagrangian and Hamiltonian formalisms for a hypothetical particle with and without charge are considered based on the relativistic equation for the dynamics and integrals of particle motion. A relativistic invariant law for the conservation of energy and momentum in the Lorentz representation is given. To select various generalized coordinates and momenta, it is possible to modify the Lagrange equations of the second kind due to the relativistic laws of conservation of energy and momentum. An action function is obtained with an explicit dependence on the velocity of the relativistic particles. The angular integral of the particle motion is derived from Hamiltonian mechanics, and the displacement Hamiltonian is obtained from the Hamilton–Jacobi equation. The angular integral of the particle motion θ is an invariant form of the conservation law. It appears only at relativistic intensities and is constant only in a specific case. The Hamilton–Jacobi–Lagrange equation is derived from the Hamilton–Jacobi equation and the Lagrange equation of the second kind. Using relativistic Hamiltonian mechanics, the Euler–Hamilton equation is obtained by expressing the energy balance through the angular integral of the particle motion θ. The given conservation laws show that the angular integral of the particle motion reflects the relativistic Doppler effect for particles in 1 + 1 dimensions. The connection between the integrals of the particle motion and the doubly special theory of relativity is shown. As an example of the applicability of the proposed invariant method, analyses of the motion of relativistic particles in circularly polarized, monochromatic, spatially modulated electromagnetic plane waves and plane laser pulses are given, and comparisons are made with calculations based on the Landau and Lifshitz method. To allow for the analysis of the oscillation of a particle in various fields, a phase-plane method is presented. [ABSTRACT FROM AUTHOR]

In a series of recent papers we developed a formulation of general relativity in which spacetime and the dynamics of matter evolve with a Poincaré invariant parameter  τ. In this paper, we apply the formalism to derive the metric induced by a 'static' event evolving uniformly along its t-axis at the spatial origin  x = 0 . The metric is shown to vary with t and  τ , as well as spatial distance r, taking its maximum value for a test particle at the retarded time  τ = t − r / c . In the resulting picture, an event localized in space and time produces a metric field similarly localized, where both evolve in  τ. We first derive this metric as a solution to the wave equation in linearized field theory, and discuss its limitations by studying the geodesic motion it produces for an evolving event. By then examining this solution in the 4+1 formalism, which poses an initial value problem for the metric under  τ -evolution, we clarify these limitations and indicate how they may be overcome in a solution to the full nonlinear field equations. [ABSTRACT FROM AUTHOR]

A brief outline of the development of ideas and a review of some achievements in modern celestial mechanics are presented. The emphasis is on the fact that the classical definition of this science given by Laplace does not fully reflect the content of modern celestial mechanics, and the term dynamic astronomy is more capacious. Dynamic astronomy studies almost everything that moves and rotates in space: from dust particles to comets and asteroids, from artificial satellites, planets, and their satellites to stars and galaxies. This complex science includes problems of both classical and relativistic celestial mechanics; it includes the theory of equilibrium figures and various numerical and computer simulation methods. Qualitative methods are of great importance, the culmination of which was the creation of the KAM theory. The development of celestial mechanics went through the practice of various applications, and the range of problems in it is exceptionally wide. A striking stimulus for the development of dynamical astronomy was the discovery of exoplanets around other stars. The article traces a chain of ideas from Keplerian orbits to osculating Lagrangian ellipses, from two-body problems to many-body problems, and from Gaussian rings to models built on the basis of precessing analogues of these rings. [ABSTRACT FROM AUTHOR]

This work is an attempt to modernise Weber's electrodynamics to make it compatible with the high-velocity regime, and with the existence of a limiting velocity, c. For this purpose, starting from the law of energy conservation and the mass–energy equivalence, new expressions for potential energy and for kinetic energy are derived jointly which are consistent with an ultimate velocity of the value of c. The new potential energy, already reported by Phipps, becomes Weber's expression in the limit of low velocities. The new kinetic energy differs from the relativistic expression, but, like the latter, it also becomes the Newtonian expression in the limit of low velocities. New expressions for force and linear momentum are also derived which complete a new mechanics. Phipps' potential energy and new kinetic energy are applied to the problem of two interacting charges in a radial motion and orbital motion. The new framework is also applied to the problem of a charge moving between the two plates of a charged capacitor, obtaining a result similar to that obtained by means of Maxwell–Lorentz electromagnetism and relativistic mechanics. The metaphysical considerations that clearly differentiate the conventional framework from the new framework proposed here are discussed. [ABSTRACT FROM AUTHOR]

We provide a Maupertuis-type principle for the following system of ODE, of interest in special relativity: d d t m x ˙ 1 - | x ˙ | 2 / c 2 = ∇ V (x) , x ∈ Ω ⊂ R n , where m , c > 0 and V : Ω → R is a function of class C 1 . As an application, we prove the existence of multiple periodic solutions with prescribed energy for a relativistic N-centre type problem in the plane. [ABSTRACT FROM AUTHOR]

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The paper proposes an amendment to the relativistic continuum mechanics which introduces the relationship between density tensors and the curvature of spacetime. The resulting formulation of a symmetric stress–energy tensor for a system with an electromagnetic field leads to the solution of Einstein Field Equations indicating a relationship between the electromagnetic field tensor and the metric tensor. In this EFE solution, the cosmological constant is related to the invariant of the electromagnetic field tensor, and additional pulls appear, dependent on the vacuum energy contained in the system. In flat Minkowski spacetime, the vanishing four-divergence of the proposed stress–energy tensor expresses relativistic Cauchy's momentum equation, leading to the emergence of force densities which can be developed and parameterized to obtain known interactions. Transformation equations were also obtained between spacetime with fields and forces, and a curved spacetime reproducing the motion resulting from the fields under consideration, which allows for the extension of the solution with new fields. [ABSTRACT FROM AUTHOR]

Prealigning nonpolar reacting molecules leads to large stereodynamical effects because of their weak steering interaction en route to the reaction barrier. However, experimental limitations in preparing aligned molecules efficiently have hindered the investigation of steric effects in bimolecular reactions involving hydrogen. Here, we report a high-resolution crossed-beam study of the reaction H + HD(v = 1, j = 2) → H2(v′, j′) + D at collision energies of 0.50, 1.20, and 2.07 electron volts in which the vibrationally excited hydrogen deuteride (HD) molecules were prepared in two collision configurations, with their bond preferentially aligned parallel and perpendicular to the relative velocity of collision partners. Notable stereodynamical effects in differential cross sections were observed. Quantum dynamics calculations revealed that strong constructive interference in the perpendicular configuration plays an important role in the stereodynamical effects observed. [ABSTRACT FROM AUTHOR]

In this paper we constructed thermo-hydrodynamics for relativistic fluid (taking into account the second order of deviation from equilibrium for dissipative heat and viscosity flows) on the basis of extended irreversible thermodynamics. EIT formalism, providing adequate modeling of systems close to the equilibrium state, goes beyond the local equilibrium hypothesis by expanding the number of basic independent variables (including dissipative flows), as well as by modifying such conceptual concepts as entropy, temperature and pressure. The evolutionary laws for the main nonequilibrium field quantities of the relativistic system are postulated: 4-vector particle flux, 4-vector energy-momentum and 4-vector entropy flux. In order to derive the constitutive equations, a nonlocal Gibbs covariance relation and a nonlocal form of the second principle of thermodynamics with a source of entropy due to additional variables-dissipative flowswere obtained. The defining equations of the hyperbolic type, forbidding superluminal velocities, modified by relaxation terms, have been obtained. The construction of relativistic thermodynamics is carried out using the hydrodynamic 4-speed defined by Eckart. The constructed relativistic hydrodynamics has its applications in such important fields of science as nuclear physics, astrophysics and cosmology. [ABSTRACT FROM AUTHOR]

Fick's diffusion equation represents physical reality that has been interpreted by Einstein and Smoluchowski. In this way, the question of interpretation of diffusion is answered in the affirmative. It gives rise to a new question critical for the understanding of our world: how broad is the spectrum of physical reality that diffusion could in principle give a complete account. The answer in this work is based on the elegant mathematical foundations formulated three decades before Fick by French mathematician Augustin Cauchy (~ 1822). It will be shown that the diffusion equation is a consequence of his model of the ideal elastic continuum. Namely, a product of the classical energy and momentum balance equations and their solutions. This demonstrates that the complete ontological construal of the diffusion theory exists. Explicitly, the interpretation of both, the diffusion equation and the flux constative formulae exist. The two terms in the flux equations, the driving forces defined by the potential gradients and the kinetic coefficients in front of the driving forces, are derived in this paper. Some fundamental consequences of all derived equations and relations for physics, chemistry and the prospects are presented. The ontological interpretation of the diffusion equation presented here provides evidence of the common roots of the chemistry and physics. [ABSTRACT FROM AUTHOR]

We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories. • This study extends Bohmian mechanics into special relativity via derived conserved currents of the Dirac equation. • We derive a novel covariant relativistic guidance formula incorporating space-time derivatives of Dirac spinor phases. • This model shows the interplay between particle mass and relativistic quantum effects via the quantum potential. • Applying the theory to the double-slit experiment generalizes the equivariance theorem and reveals zitterbewegung. • This work offers new perspectives on quantum questions like causality, nonlocality, and the Born rule in curved space-time. [ABSTRACT FROM AUTHOR]

Symmetric trace-free tensors are used in many areas of physics, including electromagnetism, relativistic celestial mechanics and geodesy, as well as in the study of gravitational radiation and gravitational lensing. Their use allows integration of the relevant wave propagation equations to arbitrary order. We present an improved iterative method for the trace-free decomposition of symmetric tensors of arbitrary rank. The method can be used both in coordinate-free symbolic derivations using a computer algebra system and in numerical modeling. We obtain a closed-form representation of the trace-free decomposition in arbitrary dimensions. To demonstrate the results, we compute the coordinate combinations representing the symmetric trace-free (STF) mass multipole moments for rank 5 through 8, not readily available in the literature. [ABSTRACT FROM AUTHOR]

High‐precision tests of local Lorentz invariance, via monitoring of the sidereal time variation of the photon energies emitted by ultrarelativistic heavy‐ion beams and of the beam momentum, are proposed. This paper includes descriptions of the physics ideas and the concept for the detector. The experiment results will allow high‐precision tests of LLI via anisotropy of the maximum attainable speed of a photon and an ion. The projected accuracy for the asymmetries interpreted in the framework of the anisotropic relativistic mechanics corresponds to the limit on sidereal time variation of the one‐way maximum attainable speed at the levels between 10−14 and 10−17. [ABSTRACT FROM AUTHOR]

We have studied the strength of the Bardeen–Cooper–Schrieffer (BCS) pairing force, used as a residual interaction to the relativistic mean-field approach, to reproduce the height of the inner fission barriers for actinide nuclei. It was found that increasing the pairing strength by about |$13\%$| makes the reproduction of the inner fission barriers better over a wide range of actinide nuclei. This result was verified by using the moment of inertia of the pairing rotational energy, which was introduced to avoid mean-field and odd-mass effects in the pairing interaction, to deduce purely the pairing strength. The pairing interaction thus determined could also improve the description of the binding energy of heavy nuclei. As a result, a consistent picture among inner fission barrier, binding energy, and pairing moment of inertia could be obtained in terms of the relativistic mean-field + BCS theory for a broad region of the actinide nuclei. [ABSTRACT FROM AUTHOR]

The parameterised post-Newtonian (PPN) formalism is a weak-field and slow-motion approximation for both General Relativity (GR) and some of its viable generalisations. Within this formalism, the motion can be approached using various parameterisations, among which are the Lagrange-type and Gauss-type orbital equations. Often, these equations are developed under the premise of the Lagrange constraint. This constraint makes the evolving orbital elements parameterise the instantaneous conics always tangent to the actual trajectory. Arbitrary mathematically, this choice of a constraint is convenient under perturbations dependent only on positions. However, under perturbations dependent also on velocities (like in the relativistic celestial mechanics) the Lagrange constraint unnecessarily complicates solutions that can be simplified by relaxing the constraint and introducing a freedom in the orbit parameterisation, which is analogous to the gauge freedom in electrodynamics and gauge field theories. Geometrically, this freedom is the freedom of non-osculation, i.e., of the degree to which the instantaneous conics are permitted to be non-tangent to the actual orbit. Under the same perturbation, all solutions with different degrees of non-osculation look mathematically different, though describe the same physical orbit. While non-intuitive, the modeling of an orbit with a sequence of nontangent instantaneous conics can at times simplify calculations. The appropriately generalised ("gauge-generalised") Lagrange-type equations, and their applications, appeared in the literature heretofore. We, in this paper, derive the gauge-generalised Gauss-type equations and apply them to the PPN two-body problem. Fixing the gauge freedom in three different ways (i.e., modeling an orbit with non-osculating elements of three different types), we find three parameterisations of the PPN two-body dynamics. These parameterisations render orbits with either a fixed non-osculating semimajor axis, or with a fixed non-osculating eccentricity, or with a fixed non-osculating argument of periastron. We also develop a transformation from non-osculating to classical osculating orbital elements, and illustrate the new solutions using numerical simulations. [ABSTRACT FROM AUTHOR]

A quantum solution of Coulomb type of the classical equation of relativistic mechanics has been found. The solution describes a relativistic oscillator depending on a spatial variable with quantization condition determined by time dependence of the elasticity coefficient. Based on this, transformation of the acceleration energy of a charged dust plasma particle with Planck mass into emitted radiation has been investigated. The synthesis of classical and quantum physics can provide the basic formalism for the second quantum revolution since the existence of quantum solutions of all equations of classical physics implies that macroscopic bodies of inanimate and living matter can be quantum objects under certain conditions. [ABSTRACT FROM AUTHOR]

The article presents a simple, rigorous approach to introduce electromagnetic theory derived from a new, fundamental form of Gauss’s law using only the basic space–time transformation relations of special relativity. More advanced concepts of relativistic mechanics, such as the transformation of mass, momentum, and force, are not necessary in this development, resulting in a simple theory. The material fills an educational need to introduce the modern theory of relativity into the teaching of engineering electromagnetics. The approach is suitable for a senior (even a junior) undergraduate and an introductory graduate engineering class, depending on the level of depth and rigor. [ABSTRACT FROM AUTHOR]

The purpose of this study is to develop an original theory of a relativistic ideal gas and to prove the validity of the postulate of the special theory of relativity for the characteristic (i.e., arithmetic mean, root-mean-square) velocities of particles of a relativistic ideal gas even in the massless limit. In this work, the following original methods are used for the first time in the theory of a relativistic ideal gas: the method of nonlinear transformation to prove of the distribution function to find the distribution function of the velocities of particles of a relativistic ideal gas; the equation of state of a relativistic ideal gas was first obtained by averaging the relativistic - invariant components of the energy - momentum tensor of a system of noninteracting particles, i.e. ideal gas by the distribution function of the velocities of their particles. The uniqueness and definiteness of the distribution function of the velocities of the particles of a relativistic ideal gas are proved on the basis of the well-known relativistic invariance of the distribution function. For the first time, expressions were obtained for the arithmetic mean and mean square velocities of particles of a relativistic ideal gas. For the first time, a fundamental conclusion is made about the validity of the postulates of the special theory of relativity for the characteristic velocities of particles of a relativistic ideal gas. An equation of state for a relativistic ideal gas is obtained, which relates its pressure, average energy density and temperature. [ABSTRACT FROM AUTHOR]

Quantum solutions of the classical equation of relativistic mechanics have been found. A synthesis of classical and quantum physics can become a basic formalism for a second quantum revolution since the existence of quantum solutions of all of the equations of classical physics implies that macroscopic bodies, both of nonliving and living matter, under certain conditions can be quantum objects. This new direction of physics can find application in the development of nature-like technologies. [ABSTRACT FROM AUTHOR]

A covariant four-tensor rotation equation—for bi-dimensional composite body—, by generalising cross product to four-vectors, is obtained. From it, a relativistic angular impulse-angular momentum variation equation (Poinsot–Euler rotation equation) and its pseudo-work-rotational kinetic energy variation equation (Poinsot–Euler pseudo-work equation), are obtained for a body spinning—with constant angular momentum direction—by external torques. Two rotational processes are analysed by using this four-tensor formalism—completed by a four-vector fundamental equation (Newton's second law and thermodynamics first law)—: a rotating body subjected to conservative and friction forces torque—a mechanical energy dissipation process—, and a device spinning by torque produced by chemical reactions—a mechanical energy production process. [ABSTRACT FROM AUTHOR]

Our ab initio all-electron fully relativistic Dirac-Fock (DF) and nonrelativistic (NR) Hartree-Fock calculations predict the DF relativistic and NR energies for the reaction: Sg + 6 CO → Sg(CO)6 as -7.39 and -6.96 eV, respectively, i.e., our calculated ground state total DF relativistic and NR energies for the reaction product Sg(CO)6 are lower by 7.39 and 6.96 eV than the total DF and NR ground state energies of the reactants, viz., one Sg atom plus six CO molecules, respectively. Our calculated DF relativistic and NR atomization energies (Ae) are 65.23 and 64.82 eV, respectively, and so the contribution of relativistic effects to the Ae of ~0.40 eV is marginal. The Sg-C and C-O optimized bond distances for the octahedral geometry as calculated in our DF (NR) calculations are 2.151 (2.318 Å) and 1.119 (1.114 Å), respectively. The BSSE correction calculated using the DIRAC code ~14 kcal/mol. The relativistic DF and NR mean energies predicted by us are 118.8 and 111.9 kJ/mol, respectively, and the contribution of ~7 kJ/mol due to relativistic effects to the mean energy of Sg(CO)6 is negligible. Ours are the first calculations of the relativistic effects for the atomization energy, mean bond energy, and energy of the reaction for possible formation of Sg(CO)6, and both our relativistic DF and the NR treatments clearly predict for the first time the existence of hexacarbonyl of the transactinide superheavy element seaborgium Sg. In conclusion, relativistic effects are not significant for Sg(CO)6. [ABSTRACT FROM AUTHOR]

A recently published correlated electron pseudopotentials (CEPPs) method has been adapted for application to the 3d-transition metals, and to include relativistic effects. New CEPPs are reported for the atoms Sc - Fe, constructed from atomic quantum chemical calculations that include an accurate description of correlated electrons. Dissociation energies, molecular geometries, and zero-point vibrational energies of small molecules are compared with all electron results, with all quantities evaluated using coupled cluster singles doubles and triples calculations. The CEPPs give better results in the correlated-electron calculations than Hartree-Fock-based pseudopotentials available in the literature. [ABSTRACT FROM AUTHOR]

Our ab initio all-electron fully relativistic Dirac-Fock (DF) and nonrelativistic (NR) Hartree-Fock calculations predict the DF relativistic and NR energies for the reaction: Sg + 6 CO → Sg(CO)6 as -7.39 and -6.96 eV, respectively, i.e., our calculated ground state total DF relativistic and NR energies for the reaction product Sg(CO)6 are lower by 7.39 and 6.96 eV than the total DF and NR ground state energies of the reactants, viz., one Sg atom plus six CO molecules, respectively. Our calculated DF relativistic and NR atomization energies (Ae) are 65.23 and 64.82 eV, respectively, and so the contribution of relativistic effects to the Ae of ~0.40 eV is marginal. The Sg-C and C-O optimized bond distances for the octahedral geometry as calculated in our DF (NR) calculations are 2.151 (2.318 Å) and 1.119 (1.114 Å), respectively. The BSSE correction calculated using the DIRAC code ~14 kcal/mol. The relativistic DF and NR mean energies predicted by us are 118.8 and 111.9 kJ/mol, respectively, and the contribution of ~7 kJ/mol due to relativistic effects to the mean energy of Sg(CO)6 is negligible. Ours are the first calculations of the relativistic effects for the atomization energy, mean bond energy, and energy of the reaction for possible formation of Sg(CO)6, and both our relativistic DF and the NR treatments clearly predict for the first time the existence of hexacarbonyl of the transactinide superheavy element seaborgium Sg. In conclusion, relativistic effects are not significant for Sg(CO)6. [ABSTRACT FROM AUTHOR]

By using two-dimensional particle-in-cell simulations, we propose a scheme for strong coupling of a petawatt laser with an opening gold cone filled with near-critical-density plasmas. When relevant parameters are properly chosen, most laser energy can be fully deposited inside the cone with only 10% leaving the tip opening. Due to the asymmetric ponderomotive acceleration by the strongly decayed laser pulse, high-energy-density electrons with net laser energy gain are accumulated inside the cone, which then stream out of the tip opening continuously, like a jet. The jet electrons are fully relativistic, with speeds around 0.98 - 0.998 c and densities at 1020/cm3 level. The jet can keep for a long time over 200 fs, which may have diverse applications in practice. [ABSTRACT FROM AUTHOR]

A relativistic Hamiltonian–Lagrangian formalism for a composite system submitted to conservative and non-conservative forces is developed. A block descending an incline with a frictional force, mechanical energy dissipation process, is described, obtaining an Euler–Lagrange equation including a Rayleigh's dissipation function. A cannonball rising on an incline, process evolving with mechanical energy production, is described by an Euler–Lagrange equation including a Gibbs' production function, with a chemical origin force. A matrix four-vector mechanical equation, considering processes' mechanical and phenomenological aspects, is postulated. This relativistic Hamiltonian–Lagrangian four-vector formalism complements the Einstein–Minkowski–Lorentz four-vector fundamental equation formalism. By considering a process' mechanical and thermodynamic description, temporal evolution equations, relating process' Hamiltonian (mechanical energy) evolution and the involved thermodynamic potentials (entropy of the universe, Helmholtz free energy, Gibbs free enthalpy) variations, are obtained. [ABSTRACT FROM AUTHOR]

To integrate traditional classical mechanics topics into a broader view, that some authors associate to drawing from I the theory of complex systems i [[13]], would be a great contribution to new generations of physics students. The new book constitutes a new modern approach to teaching mechanics, in particular under the context of facing classical mechanics versus quantum mechanics, and showing connections to quantum mechanics, classical field theory, gravitation and current ideas on astrophysics. And finally, how classical mechanics is related to modern physics, especially quantum mechanics and general relativity. A modern approach to teaching classical mechanics: A review of Modern classical mechanics by T. M. Helliwell and V. V. Sahakian, Cambridge, Cambridge University Press, 2020, 704 pp., £56.00, (hardback), ISBN 978-1-10-883497-1. [Extracted from the article]

The equations of motion for the parametrized post-Newtonian (PPN) formalism for several S-stars close to the relativistic compact object Sgr A* at the center of the galaxy are examined. The effect of the difference in the periods of Newtonian and post-Newtonian orbits is taken into account. The period of the best approximation to the PN-orbit of the star S2 exceeds the period of the Newtonian orbit by 16 days. The PPN parameters βPPN and γPPN are measured. Bayesian methods are used for fitting the trajectories of the PPN motion. Posterior estimates of βPPN and γPPN 0.97 − 0.65 + 0.42 and 0.81 − 0.60 + 0.46 are obtained, respectively. This result confirms the prediction of the general theory of relativity for the post-Newtonian equations of motion under conditions close to Sgr A*. [ABSTRACT FROM AUTHOR]

A systematic examination of the basic theory of general relativity is made, the meaning of coordinates again is emphasized, the confusion caused by unclear meaning of coordinates in the past is corrected, and the expression of the theory is made more accurate. Firstly, the equation of Einstein's gravitational field is solved in the usual coordinate system, the existence of light speed invariant solution in the spherically symmetric gravitational field is proved, and in the same time, the solution is determined. It turns out that black holes are not an inevitable prediction of general relativity. The more exact formulas for calculating the curvature of light on the surface of the Sun and the precession angle of the orbit of Mercury are given, and the convergence of general relativistic gravity and special relativistic mechanics under the weak field approximation is realized. Finally, it is shown that the coupling coefficient of the gravitational field equation is not unique. Modifying this coefficient is an ideal project to eliminate the singularities of general relativity on the condition keeping the field equation concise and elegant, and moreover, it reveals that dark matter and dark energy are the negative energy field in the matter, the expansion of the universe is the appearance of the gradual formation of galaxies in accordance with fractal rules, not only the space between galaxies is expanding but also the galaxies themselves are also expanding, new matter is continuously generated in the celestial bodies, for the first time, the unity of fractal geometry and cosmic dynamics of general relativity is realized, and the formation and evolution of galaxies are brought into the fractal generation mode. This is a living and vivacious universe in which all aspects are gradually strengthening, in sharp contrast to the dying universe under the current cosmological framework. [ABSTRACT FROM AUTHOR]

Einstein's special theory of relativity is presented in a Minkowski four-vector formalism integrating mechanics and thermodynamics at a sophomore level, allowing the solution of undergraduate exercises in linear translation requiring both. This relativistic formalism directly incorporates the mechanics (Newton's second law) and the thermodynamics (first law of thermodynamics) of a process in a four-vector fundamental matrix equation. This four-vector formalism is used to analyse two processes: a block descending an inclined plane with friction (a mechanical energy dissipation process increasing the entropy of the universe—if frictionless, a mechanical energy conservation process), and a cannonball ascending an incline, moved by a force exerted by a chemical reaction (a mechanical energy production process), with Gibbs' free enthalpy function decreasing. [ABSTRACT FROM AUTHOR]