Relativistic Correction Terms Research Articles

A Special Relativistic Exploitation of the Second Law of Thermodynamics and Its Non-Relativistic Limit.

A thermodynamic process is a solution of the balance equations fulfilling the second law of thermodynamics. This implies restrictions on the constitutive relations. The most general way to exploit these restrictions is the method introduced by Liu. This method is applied here, in contrast to most of the literature on relativistic thermodynamic constitutive theory, which goes back to a relativistic extension of the Thermodynamics of Irreversible Processes. In the present work, the balance equations and the entropy inequality are formulated in the special relativistic four-dimensional form for an observer with four-velocity parallel to the particle current. The restrictions on constitutive functions are exploited in the relativistic formulation. The domain of the constitutive functions, the state space, is chosen to include the particle number density, the internal energy density, the space derivatives of these quantities, and the space derivative of the material velocity for a chosen observer. The resulting restrictions on constitutive functions, as well as the resulting entropy production are investigated in the non-relativistic limit, and relativistic correction terms of the lowest order are derived. The restrictions on constitutive functions and the entropy production in the low energy limit are compared to the results of an exploitation of the non-relativistic balance equations and entropy inequality. In the next order of approximation our results are compared to the Thermodynamics of Irreversible Processes.

Quantum mechanics of a fermion confined to a curved surface in Foldy-Wouthuysen representation

In Foldy-Wouthuysen representation, this paper deduces the effective quantum mechanics for a relativistic particle confined to a curved surface in an external field by using the thin-layer quantization scheme. It is found that the quantum effects caused by introducing the confining potential can be taken as the results of relativistic correction in the nonrelativistic limit. Meanwhile, the spin connection of the curved surface can produce a Zeeman-like gap through the relativistic correction term. In addition, the confining potential can provide a curvature-independent energy shift resulting from the Zitterbewegung effect. As an example, this paper applies the effective Hamiltonian to a torus surface, in which the spin effects related to the introduced confining potential are obtained. These results demonstrate the scaling of the noncommutation of the nonrelativistic limit and the thin-layer quantization procedure.

Mass-spectroscopy of [$$bb][{\bar{b}}{\bar{b}}$$] and [$$bq][{\bar{b}}{\bar{q}}$$] tetraquark states in a diquark–antidiquark formalism

In this article, we utilise the non-relativistic potential model to calculate the mass-spectra of all bottom [ $$bb][{\bar{b}}{\bar{b}}$$ ] and heavy-light bottom [ $$bq][{\bar{b}}{\bar{q}}$$ ] (q = u,d) tetraquark states in diquark–antidiquark approximation. The four-body problem is reduced into two body problem by numerically solving the $$Schr\ddot{o}dinger$$ equation using a Cornell-inspired potential along with relativistic correction term. The splitting structure of the tetraquark spectrum is described using spin-dependent terms (spin-spin, spin-orbit, and tensor). We have successfully calculated and predicted the masses of bottom mesons, diquarks and tetraquarks. The masses of S and P-wave tetraquark states [ $$bb][{\bar{b}}{\bar{b}}$$ ] and [ $$bq][{\bar{b}}{\bar{q}}$$ ], respectively, are found to be between 18.7–19.4 GeV and 10.4–11.3 GeV, in which the masses of S-wave [ $$bb][{\bar{b}}{\bar{b}}$$ ] states are less than the 2 $$\eta _{b}$$ , $$\eta _{b}\Upsilon $$ , and 2 $$\Upsilon $$ threshold. Additionally, we have investigated the $$Z_b(10610)$$ and $$Z_ b(10650)$$ states in the current model and found that they are 150 MeV below the $$BB^{*}$$ and $$B^{*}B^{*}$$ thresholds.

Analytical Calculation of Cl$$^{15+}$$ Ion Immersed in Dense Plasmas

A method of analytically calculating energy levels of He-like ions in an environment of dense plasma is given by using the angular momentum coupling theory and irreducible tensor theory under Hartree–Fock approximation. In order to obtain higher calculation precision, relativistic correction terms of the non-relativistic energy including corrections caused by relativistic mass, one- and two-body Darwin effect, spin–spin contact interaction and orbit–orbit interaction, are calculated. The binding energies of the ground state $$1s^2$$ $$^1S$$ and excited state 1snp $$^1P$$ (n = 1 − 4) of He-like Cl ions and the transition energy between two energy levels in an environment of dense plasma are calculated. The scaling relationship between the energy shift of plasma and its temperature and density is given. According to our study, the energy shift of plasma conforms very well with the recent high-precision experimental result (Phys Rev A 100:012511, 2019).

Hilbert's forgotten equation, the equivalence principle and velocity dependence of free fall

Referring to the behavior of accelerating objects in special relativity, and applying the principle of equivalence, one expects that the coordinate acceleration of point masses under gravity will be velocity dependent. Also, using the Schwarzschild solution, we analyze the similar case of masses moving on timelike geodesics, which reproduces a little-known result by Hilbert from 1917, describing this dependence. We find that the relativistic correction term for the acceleration based on general relativity differs by a factor of two from the simpler acceleration arguments in flat space. As we might expect from the general theory, the velocity dependence can be removed by a suitable coordinate transformation, such as the Painlevé–Gullstrand coordinate system. The validity of this approach is supported by previous authors who have demonstrated vacuum solutions to general relativity producing true flat space metrics for uniform gravitational fields. We suggest explicit experiments could be undertaken to test the property of velocity dependence.

Relativistic effect of J/\U0001d713 hadroproduction in large pT region

By combining NRQCD factorization and collinear factorization, we compute a series of relativistic corrections for J/\U0001d713 hadroproduction to all orders in \U0001d7102 at large pT limit. The \U0001d7102 expansion converges well for all channels. We find that the ratio of relativistic correction term to the corresponding leading term is independent of kinematic variables for any channel, which generalizes the proportional relations found in previous works to all orders.

Asymptotics of the ground state energy in the relativistic settings

The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.

Generalisation of Gilbert damping and magnetic inertia parameter as a series of higher-order relativistic terms

The phenomenological Landau–Lifshitz–Gilbert (LLG) equation of motion remains as the cornerstone of contemporary magnetisation dynamics studies, wherein the Gilbert damping parameter has been attributed to first-order relativistic effects. To include magnetic inertial effects the LLG equation has previously been extended with a supplemental inertia term; the arising inertial dynamics has been related to second-order relativistic effects. Here we start from the relativistic Dirac equation and, performing a Foldy–Wouthuysen transformation, derive a generalised Pauli spin Hamiltonian that contains relativistic correction terms to any higher order. Using the Heisenberg equation of spin motion we derive general relativistic expressions for the tensorial Gilbert damping and magnetic inertia parameters, and show that these tensors can be expressed as series of higher-order relativistic correction terms. We further show that, in the case of a harmonic external driving field, these series can be summed and we provide closed analytical expressions for the Gilbert and inertial parameters that are functions of the frequency of the driving field.

Mass spectra of bottomonia using relativistic corrections to the potential

We have calculated the masses of different states of bottomonia in the framework of potential non-relativistic QCD, non-perturbatively. The potential consists of two terms, a static term incorporating Coulombic plus confinement part along with a constant and a relativistic correction term classified in powers of the inverse of heavy quark mass O(1/m). Spin-spin interaction has been considered to calculate the masses of the singlet and the triplet states of bottomonia. The Schrödinger equation is solved numerically to obtain the masses from 1S to 6S states, the calculated masses for ηb(1S) is 9.390 GeV and for ϒ(1S) is 9.459 GeV and is found to be in good accordance with the experimental results.

Atomic structure under external confinement: effect of plasma on the spin orbit splitting, relativistic mass correction and Darwin term for hydrogen-like ions

The effect of Debye and quantum plasma environment on the structural properties such as spin orbit splitting, relativistic mass correction and Darwin term for a few iso-electronic members of hydrogen viz. C5 +, O7 +, Ne9 +, Mg11 +, Si13 +, S15 +, Ar17 +, Ca19 + and Ti21 + has been analysed systematically for the first time for a range of coupling strengths of the plasma. The Debye plasma environment has been treated under a standard screened Coulomb potential (SCP) while the quantum plasma has been treated under an exponential cosine screened Coulomb potential (ECSCP). Estimation of the spin orbit splitting under SCP and ECSCP plasma is restricted to the lowest two dipole allowed states while for the other two properties, the ground state as well as the first two excited states have been chosen. Calculations have been extended to nuclear charges for which appreciable relativistic corrections are noted. In all cases calculations have been extended up to such screening parameters for which the respective excitation energies tend towards their stability limit determined by the ionisation potential at that screening parameter. Interesting behavior of the respective properties with respect to the plasma coupling strength has been noted.

Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime

We consider the relativistic transfer equations for photons interacting via emission absorption and scattering with a moving fluid. We prove a comparison principle and we study the non-equilibrium regime: the relativistic correction terms in the scattering operator lead to a frequency drift term modeling the Doppler effects. We prove that the solution of the relativistic transfer equations converges toward the solution of this drift diffusion equation.

Non-linear power spectra in the synchronous gauge

We study the non-linear corrections to the matter and velocity power spectra in the synchronous gauge (SG). For the leading correction to the non-linear power spectra, we consider the perturbations up to third order in a zero-pressure fluid in a flat cosmological background. Although the equations in the SG happen to coincide with those in the comoving gauge (CG) to linear order, they differ from second order. In particular, the second order hydrodynamic equations in the SG are apparently in the Lagrangian form, whereas those in the CG are in the Eulerian form. The non-linear power spectra naively presented in the original SG show rather pathological behavior quite different from the result of the Newtonian theory even on sub-horizon scales. We show that the pathology in the nonlinear power spectra is due to the absence of the convective terms in, thus the Lagrangian nature of, the SG. We show that there are many different ways of introducing the corrective convective terms in the SG equations. However, the convective terms (Eulerian modification) can be introduced only through gauge transformations to other gauges which should be the same as the CG to the second order. In our previous works we have shown that the density and velocity perturbation equations in the CG exactly coincide with the Newtonian equations to the second order, and the pure general relativistic correction terms starting to appear from the third order are substantially suppressed compared with the relativistic/Newtonian terms in the power spectra. As a result, we conclude that the SG per se is an inappropriate coordinate choice in handling the non-linear matter and velocity power spectra of the large-scale structure where observations meet with theories.

Core-dependent and ligand-dependent relativistic corrections to the nuclear magnetic shieldings in MH4-n Y n (n = 0-4; M = Si, Ge, Sn, and Y = H, F, Cl, Br, I) model compounds.

The nuclear magnetic shieldings of Si, Ge, and Sn in MH(4-n) Y(n) (M = Si, Ge, Sn; Y = F, Cl, Br, I and n = 1-4) molecular systems are highly influenced by the substitution of one or more hydrogens by heavy-halogen atoms. We applied the linear response elimination of small components (LRESC) formalism to calculate those shieldings and learn whether including only a few of the leading relativistic correction terms is sufficient to be able to quantitatively reproduce the full relativistic value. It was observed that the nuclear magnetic shieldings change as the number of heavy halogen substituents and their weights vary, and the pattern of σ(M) generally does not exhibit the normal halogen dependence (NHD) behavior that can be seen in similar molecular systems containing carbon atoms. We also analyzed each relativistic correction afforded by the LRESC method and split them in two: core-dependent and ligand-dependent contributions; we then looked for the electronic mechanisms involved in the different relativistic effects and in the total relativistic value. Based on this analysis, we were able to study the electronic mechanism involved in a recently proposed relativistic effect, the "heavy atom effect on vicinal heavy atom" (HAVHA), in more detail. We found that the main electronic mechanism is the spin-orbit or σ p (T(3)) correction, although other corrections such as σ p (S(1)) and σ p (S(3)) are also important. Finally, we analyzed proton magnetic shieldings and found that, for molecules containing Sn as the central atom, σ(H) decreases as the number of heavy halogen substituents (of the same type: either F, Cl, or Br) increases, albeit at different rates for different halogens. σ(H) only increase as the number of halogen substituents increases if the halogen is iodine.

Newtonian, Post-Newtonian and Relativistic Cosmological Perturbation Theory

Newtonian cosmological perturbation equations valid to full nonlinear order are well known in the literature. Assuming the absence of the transverse-tracefree part of the metric, we present the general relativistic counterpart valid to full nonlinear order. The relativistic equations are presented without taking the slicing (temporal gauge) condition. The equations do have the proper Newtonian and first post-Newtonian limits. We also present the relativistic pressure correction terms in the Newtonian hydrodynamic equations.

Newtonian hydrodynamics with general relativistic pressure

We present the general relativistic pressure correction terms inNewtonian hydrodynamic equations to the nonlinear order: these are equations (1.1)–(1.3). Thederivation is made in the zero-shear gauge based on the fully nonlinear formulation of cosmological perturbation in Einstein's gravity. The correction terms differ from many of the previously suggested forms in the literature based on hand-waving manners. We confirm our results by comparing with (i) the nonlinear perturbation theory, (ii) the first order post-Newtonian approximation, and (iii) the special relativistic limit, and by checking (iv) the consistency with full Einstein's equation.

Nonrelativistic limit of the Dirac-Schwarzschild Hamiltonian: GravitationalZitterbewegungand gravitational spin-orbit coupling

We investigate the nonrelativistic limit of the gravitationally coupled Dirac equation via a Foldy-Wouthuysen transformation. The relativistic correction terms have immediate and obvious physical interpretations in terms of a gravitational Zitterbewegung and a gravitational spin-orbit coupling. We find no direct coupling of the spin vector to the gravitational force, which would otherwise violate parity. The particle-antiparticle symmetry described recently by one of us [Jentschura, Phys. Rev. A 87, 032101 (2013)] is verified on the level of the perturbative corrections accessed by the Foldy-Wouthuysen transformation. The gravitational corrections to the electromagnetic transition current are calculated.

Qualitative analysis of the anisotropic two-body problem with relativistic potential

In this paper we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of a Newtonian force-law with two relativistic correction terms. We will show that the set of initial conditions leading to collisions and ejections have positive measure and study the capture and escape solutions in the zero-energy case using the infinity manifold. We will also apply the Melnikov method to show that the flow on the zero-energy manifold of another potential which is the sum of the classical Keplerian potential and two anisotropic perturbation which also take into account two relativistic correction terms is chaotic.

Relativistic neoclassical radial fluxes in the 1/ν regime

The radial neoclassical fluxes of electrons in the 1/ν-regime are calculated with relativistic effects taken into account and compared with those in the non-relativistic approach. The treatment is based on the relativistic drift-kinetic equation with the thermodynamic equilibrium given by the relativistic Maxwell–Jüttner distribution function. It is found that for the range of fusion temperatures, Te < 100 keV, the relativistic effects produce a reduction of the radial fluxes which does not exceed 10%. This rather small effect is a consequence of the non-monotonic temperature dependence of the relativistic correction caused by two counteracting factors: a reduction of the contribution from the bulk and a significant broadening with the temperature growth of the energy range of electrons contributing to transport.The relativistic formulation for the radial fluxes given in this paper is expressed in terms of a set of relativistic thermodynamic forces which is not identical to the canonical set since it contains an additional relativistic correction term dependent on the temperature. At the same time, this formulation allows application of the non-relativistic solvers currently used for calculation of mono-energetic transport coefficients.

The role of relativity and dispersion controlled inter-chain interaction on the band gap of thiophene, selenophene, and tellurophene oligomers

Ab initio relativistic density functional theoretical calculations have been carried out on π-conjugated oligomers of increasing length with S, Se, and Te as heteroatoms. The band gap of the corresponding polymers has been obtained by plotting lowest unoccupied molecular orbital (LUMO)-highest occupied molecular orbital (HOMO)gap against the reciprocal of the number of monomer units (1/N) and extrapolating the curve to 1/N = 0. With B3LYP functional, we predict that role of relativistic correction terms is not very significant in the determination of final band gap of thiophene, selenophene, and tellurophene polymer. The origin of this observation is provided through the density of states (DOS) analysis which manifests that DOS contribution across the Fermi level of these polymers is mostly governed by C atoms and as a consequence relativistic correction terms due to heavy heteroatom remain insignificant to the band gap modification. We also inspected the role of inter-chain interaction in determining the net LUMO-HOMO gap of π-stacked double chain oligomers of increasing length. We have found that due to the exciton splitting in the stacked configurations, the LUMO-HOMO gap decreases steadily. Furthermore, we have noticed that dispersion force has important role in the reduction of the LUMO-HOMO gap of the oligomers studied.

ON THE ANISOTROPIC POTENTIALS OF MANEV–SCHWARZSCHILD TYPE

In this paper, we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of a Newtonian force-law with two relativistic correction terms. We will show that the set of initial conditions leading to collisions and ejections and leading to escapes and captures have positive measure. Using the infinity manifold, we study capture and escape solutions in the zero-energy case. We also show that the flow on the zero energy manifold of a two-body problem given by the sum of the Newtonian potential and three anisotropic perturbations (corresponding to three relativistic correction terms) is chaotic.