Particle In Gravitational Field Research Articles

Relativistic quantum bouncing particles in a homogeneous gravitational field

In this paper, we study the relativistic effect on the wave functions for a bouncing particle in a gravitational field. Motivated by the equivalence principle, we investigate the Klein–Gordon and Dirac equations in Rindler coordinates with the boundary conditions mimicking a uniformly accelerated mirror in Minkowski space. In the nonrelativistic limit, all these models in the comoving frame reduce to the familiar eigenvalue problem for the Schrödinger equation with a fixed floor in a linear gravitational potential, as expected. We find that the transition frequency between two energy levels of a bouncing Dirac particle is greater than the counterpart of a Klein–Gordon particle, while both are greater than their nonrelativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wave functions around the mirror boundary.

Settlement behavior and stratification of waste printed circuit boards particles in gravitational field

Waste printed circuit board (WPCBs) is a kind of sustainable resource for providing economically valuable metals due to its huge output and excellent metal grade. Gravity separation is an effective pre-concentration technology to separate WPCBs particles with significant density difference. This study focuses on revealing the mechanism of gravity concentration from the perspective of dynamics and statics. The settling velocity of three kinds of particles in WPCBs (organic matter, glass fiber and metal particles) are calculated, which can be expressed by traditional settlement velocity model with correction coefficient. Subsequently, the equal-sedimentation model of the particles was established, and the calculation results show that the particle size ratio should be within 5.58. The lowest potential energy model was proposed to explain the stratification process of particles from the static point of view. The potential energy calculation results show that the stratification process dominated by density difference is spontaneous, which can achieve the lowest potential energy. Shaking table test shows that the metal particles with different sizes can be effectively enriched in the concentrates, and the metal grade increases with the decrease of particle size, from 56.5% to 68.2%, and the metal recovery decreases from 86.41% to 83.04%.

Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle

We study the influence of noncommutativity of coordinates and noncommutativity of momenta on the motion of a particle (macroscopic body) in uniform and nonuniform gravitational fields in noncommutative phase space of canonical type with preserved rotational symmetry. It is shown that because of noncommutativity the motion of a particle in a gravitational field is determined by its mass. The trajectory of motion of a particle in a uniform gravitational field corresponds to the trajectory of a harmonic oscillator with frequency determined by the value of the parameter of momentum noncommutativity and mass of the particle. The equations of motion of a macroscopic body in a gravitational field depend on its mass and composition. From this follows a violation of the weak equivalence principle caused by noncommutativity. We conclude that the weak equivalence principle is recovered in rotationally invariant noncommutative phase space if we consider the tensors of noncommutativity to be dependent on mass. So, finally we construct noncommutative algebra which is rotationally invariant, equivalent to noncommutative algebra of canonical type, and does not lead to violation of the weak equivalence principle.

Maximal acceleration and radiative processes

We derive the radiation characteristics of an accelerated, charged particle in a model due to Caianiello in which the proper acceleration of a particle of mass [Formula: see text] has the upper limit [Formula: see text]. We find two power laws, one applicable to lower accelerations, the other more suitable for accelerations closer to [Formula: see text] and to the related physical singularity in the Ricci scalar. Geometrical constraints and power spectra are also discussed. By comparing the power laws due to the maximal acceleration (MA) with that for particles in gravitational fields, we find that the model of Caianiello allows, in principle, the use of charged particles as tools to distinguish inertial from gravitational fields locally.

Spinning Particle in Gravitational Field of Black Hole Involving Global Monopole

In this paper we study the dynamics of trajectory of a spinning particle in a Schwarzschild spacetime involving a global monopole. We set up the equations of motion and find three types of trajectories. We study the conditions that a spinning particle, originally moving in the innermost stable circular orbit around the black hole involving a global monopole, will escape to infinity after it is kicked by another particle or photon. Three types of trajectories of a spinning particle in a Schwarzschild spacetime involving a global monopole are simulated in detail and the escaping energy and velocity of the spinning particle is also obtained in the present paper.

Solution of the problem of uniqueness and Hermiticity of Hamiltonians for Dirac particles in gravitational fields

The authors prove that the dynamics of spin 1/2 particles in stationary gravitational fields can be described using an approach, which builds upon the formalism of pseudo-Hermitian Hamiltonians. The proof consists in the analysis of three expressions for Hamiltonians, which are derived from the Dirac equation and describe the dynamics of spin 1/2 particles in the gravitational field of the Kerr solution. The Hamiltonians correspond to different choices of tetrad vectors and differ from each other. The differences between the Hamiltonians confirm the conclusion known from many studies that the Hamiltonians derived from the Dirac equation are non-unique. Application of standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians produces the same Hermitian Hamiltonian. The eigenvalue spectrum of the resulting Hamiltonian is the same as that of the Hamiltonians derived from the Dirac equation with any chosen system of tetrad vectors. For description of the dynamics of spin 1/2 particles in stationary gravitational fields can be used not only the formalism of pseudo-Hermitian Hamiltonians, but also an alternative approach, which employs the Parker scalar product. The authors show that the alternative approach is equivalent to the formalism of pseudo-Hermitian Hamiltonians.

Revised research about chaotic dynamics in Mankoet al.spacetime

A recent work by Dubeibe et al. [Phys. Rev. D 75, 023008 (2007)] stated that chaos phenomenon of test particles in gravitational field of rotating neutron stars which are described by Manko, Sanabria-Gomez, and Manko (Manko et al.) metric can only occur when the stars have oblate deformation. But the chaotic motions they found are limited in a very narrow zone which is very close to the center of the massive bodies. This paper argues that this is impossible because the region is actually inside of the stars, so the motions cannot exist at this place. In this paper, we scan all parameters space and find chaos and unstable fixed points outside of stars with big mass-quadrupole moments. The calculations show that chaos can only occur when the stars have prolate deformation. Because real deformation of stars should be oblate, all orbits of test particles around the rotating neutron stars described by Manko et al. solutions are regular. The case of nonzero dipolar magnetic moment has also been taken into account in this study.

Spin-Spin Interactions in Gauge Theory of Gravity, Violation of Weak Equivalence Principle and New Classical Test of General Relativity

For a long time, it has been generally believed that spin-spin interactions can only exist in a theory where Lorentz symmetry is gauged, and a theory with spin-spin interactions is not perturbatively renormalizable. But this is not true. By studying the motion of a spinning particle in gravitational field, it is found that there exist spin-spin interactions in gauge theory of gravity. Its mechanism is that a spinning particle will generate gravitomagnetic field in space-time, and this gravitomagnetic field will interact with the spin of another particle, which will cause spin-spin interactions. So, spin-spin interactions are transmitted by gravitational field. The form of spin-spin interactions in post Newtonian approximations is deduced. This result can also be deduced from the Papapetrou equation. This kind of interaction will not affect the renormalizability of the theory. The spin-spin interactions will violate the weak equivalence principle, and the violation effects are detectable. An experiment is proposed to detect the effects of the violation of the weak equivalence principle.

Equation of Motion of a Spinning Test Particle in Gravitational Field

Based on the coupling between the spin of a particle and gravitoelectromagnetic field, the equation of motion of a spinning test particle in gravitational field is deduced. From this equation of motion, it is found that the motion of a spinning particle deviates from the geodesic trajectory, and this deviation originates from the coupling between the spin of the particle and gravitoelectromagnetic field, which is also the origin of Lense–Thirring effects. In post-Newtonian approximations, this equation gives the same results as those of Mathisson–Papapetrou equation. Effect of the deviation of geodesic trajectory is detectable.

Bound states of Dirac particles in gravitational fields

We investigate the quantum motion of a neutral Dirac particle bouncing on a mirror in curved spacetime. We consider different geometries: Rindler, Kasner-Taub, and Schwarzschild, and show how to solve the Dirac equation by using geometrical methods. We discuss, in a first-quantized framework, the implementation of appropriate boundary conditions. This leads us to consider a Robin boundary condition that gives the quantization of the energy, the existence of bound states and of critical heights at which the Dirac particle bounces, extending the well-known results established from the Schr\"odinger equation. We also allow for a nonminimal coupling to a weak magnetic field. The problem is solved in an analytical way on the Rindler spacetime. In the other cases, we compute the energy spectrum up to the first relativistic corrections, exhibiting the contributions brought by both the geometry and the spin. These calculations are done in two different ways. On the one hand, using a relativistic expansion and, on the other hand, with Foldy-Wouthuysen transformations. Contrary to what is sometimes claimed in the literature, both methods are in agreement, as expected. Finally, we make contact with the GRANIT experiment. Relativistic effects and effects that go beyond the equivalence principle escape the sensitivity of such an experiment. However, we show that the influence of a weak magnetic field could lead to observable phenomena.

Coagulation kernel of particles in a turbulent gas flow

On the base of modern probability density functions approach turbulent coagulation of particles in gravitational field is investigated. Spectral presentation of second velocity moments of gas phase is used for calculation of intensity of particles relative chaotic motion. Closed diffusion equation for two-particle distribution in space is obtained. Boundary condition taking into account coefficients of new particle formation and momentum restitution during two particles collision is found. Formula for calculation of turbulent coagulation kernel of particles in gravity field is gain. Influence of cloud turbulence and turbulence in a pipe flow on intensity of droplets coagulation is studied. Strong effects of relative turbulent diffusion between droplets, droplets inertia and droplets gravitational settling on intensity of coagulation are found out. Connection between internal structure of turbulence type and coagulation rate is illustrated. Obtained results are right for polydisperse additions in turbulent flows.

Non-relativistic Limit of Dirac Equations in GravitationalField and Quantum Effects of Gravity

Based on unified theory of electromagnetic interactions andgravitational interactions, the non-relativistic limit of theequation of motion of a charged Dirac particle in gravitationalfield is studied. From the Schrödinger equation obtained fromthis non-relativistic limit, we can see that the classicalNewtonian gravitational potential appears as a part of the potentialin the Schrödinger equation, which can explain the gravitationalphase effects found in COW experiments. And because of thisNewtonian gravitational potential, a quantum particle in the earth'sgravitational field may form a gravitationally bound quantizedstate, which has already been detected in experiments. Three differentkinds of phase effects related to gravitational interactionsare studied in this paper, and these phase effects shouldbe observable in some astrophysical processes.Besides, there exists direct coupling between gravitomagneticfield and quantum spin, and radiation caused by this couplingcan be used to directly determine the gravitomagneticfield on the surface of a star.

A Generalization of the Bargmann’s Theory of Ray Representations

The paper contains a complete theory of factors for ray representations acting in a Hilbert bundle, which is a generalization of the known Bargmann’s theory. With its help, we have reformulated the standard quantum theory so that the gauge freedom emerges naturally from the very nature of quantum laws. The theory is of primary importance in the investigations of covariance (in contradistinction to symmetry) of a quantum theory which possesses a nontrivial gauge freedom. In that case the group in question is not any symmetry group but a covariance group only – the case not yet investigated in depth. It is shown that the factor of a covariance group representation depends on space and time when the system in question possesses gauge freedom. In nonrelativistic theories, the factor depends on time only. In relativistic theory, the Hilbert bundle is built over spacetime while in the nonrelativistic case-over time. We explain two applications of this generalization: in the theory of a quantum particle in gravitational field in the nonrelativistic limit, and in quantum electrodynamics.

Quantum mechanics and the equivalence principle

A quantum particle moving in a gravitational field may penetrate the classically forbidden region of the gravitational potential. This raises the question of whether the time of flight of a quantum particle in a gravitational field might deviate systematically from that of a classical particle due to tunnelling delay, representing a violation of the weak equivalence principle. I investigate this using a model quantum clock to measure the time of flight of a quantum particle in a uniform gravitational field, and show that a violation of the equivalence principle does not occur when the measurement is made far from the turning point of the classical trajectory. The results are then confirmed using the so-called dwell time definition of quantum tunnelling. I conclude with some remarks about the strong equivalence principle in quantum mechanics.

Deflection of highly relativistic particles in a gravitational field

A novel approach to the calculation of the deflection of highly relativistic test particles in gravitational fields is described. We make use of the light-like boosts of the gravitational fields of the sources. Examples are given of the deflection of highly relativistic particles in the Schwarzschild and Kerr gravitational fields, in the field of a static, axially symmetric, multipole source and in the field of a cosmic string. The deflection of spinning particles is also discussed.

The Effect Of Spin-Gravity Interaction On The Wave Lengths Of Photons Moving In The Earth's Gravitational Field

It is pointed out that there exist real discrepancies between the used theories and experimental measurements in two phenomena on astronomical and laboratory scales. Those two phenomena are concerned with the motion of spinning particles in gravitational fields. A thurough review is presented on different theoretical possibilities of searching for a feasable solution for the existing puzzling discrepancies. One theoretical possibility; in which one may find an interpretation for these discrepancies; is exposed. This possibility is based upon taking into considerations, spin- gravity interaction when spinning particles are moving in gravitational fields. The obtained results; due to such possibility; are applied to calculate the effect of the Earth's gravitational field on the wave lengths of photons carrying information between two points having different gravitational potential.

Fractal escapes in Newtonian and relativistic multipole gravitational fields

We study the planar motion of test particles in gravitational fields produced by an external material halo. Both the Newtonian and the general-relativistic dynamics are examined, and in the relativistic case the dynamics of both massive and massless particles are investigated. The halo field is given in general by a multipole expansion; we restrict ourselves to multipole fields of pure order, whose Newtonian potentials are homogeneous polynomials in cartesian coordinates. A pure n-pole field has n different escapes, one of which is chosen by the particle according to its initial conditions. We find that the escape has a fractal dependency on the initial conditions for n>2 both in the Newtonian and the relativistic cases for massive test particles, but with important differences between them. The relativistic motion of massless particles, however, was found to be regular for all the fields we could study. The box-counting dimension was used in each case to quantify the sensitivity to initial conditions which arises from the fractality of the escape route.

THE EFFECT OF THE EARTH'S GRAVITATIONAL FIELD ON PHOTONS WAVE LENGTHS OF UP- DOWN- LINK INFORMATION

It is pointed out that there exist real discrepancies between the used the-ories and experimental measurements in two phenomena on astronomi-cal and laboratory scales. Those two phenomena are concerned with the motion of spinning particles in gravitational fields. A thurough study is presented on different theoretical possibilities of searching for a feasable solution for the existing puzzling discrepancies. One theoretical possibility; in which one may find an interpretation for these discrepancies; is exposed. This possibility is based upon taking into considerations, spin- gravity interaction when spinning particles are moving in gravitational fields. The obtained results; due to such possibility; are applied to calculate the effect of the Earth's gravitational field on photons wave lengths of up- down- link information.

A BOHM QUANTUM-POTENTIAL DESCRIPTION FOR THE WAVE EQUATION OF SCALAR PARTICLE IN GRAVITATIONAL FIELD

Introducing Bohm quantum-potential, from Klein-Gordon equation for scalar particle in gravitational field, an equation of trajectory-motion equation can be derived, it is similar to the classical one and it can be reduced to a geodesic equation in a redefined metric space.

Equilibrium points of spin particles in static gravitational fields

General conditions are formulated within the framework of general relativity for “suspension” of rotating particles in static gravitational fields having the form of systems of differential equations for the components of the spin tensor. In the special case of suspension of particles in gravitational fields created by nonrotating objects, the spin components are a superposition of quantities which vary harmonically over time. As an example the effect of suspension of a spin particle within a gravitating liquid sphere is considered.