Relativistic Two-body Problem Research Articles

Classification of orbits of Fokker’s time-asymmetric relativistic two-body problem

The requirements that the proper velocities of the particles in Fokker’s time-asymmetric relativistic two-body problem be timelike and future pointing restrict the variables ρ1 and ρ2 associated with the distances between the particles as measured in their rest frames. Employing these restrictions in an algebraic equation relating ρ1 and ρ2 to the total angular momentum and the total four-momentum, assumed timelike and future pointing, classifies the physical orbits of the system. The results include orbits similar to those of the nonrelativistic Kepler problem and several new types in which the angular velocity is opposite to the total angular momentum. This information is required for the integration of the equations of motion to determine the orbits in four-space.

Example of relativistic two-body problem

Example of relativistic two-body problem

Example of relativistic two-body problem. I. Boundary-value problem for minimal surface

Example of relativistic two-body problem. I. Boundary-value problem for minimal surface

On classical and quantum relativistic dynamics

A canonical formalism for the relativistic classical mechanics of many particles is proposed. The evolution equations for a charged particle in an electromagnetic field are obtained and the relativistic two-body problem with an invariant interaction is treated. Along the same line a quantum formalism for the spinless relativistic particle is obtained by means of imprimitivity systems according to Mackey theory. A quantum formalism for the spin-1/2 particle is constructed and a new definition of spin1/2 in relativity is proposed. An evolution equation for the spin-1/2 particle in an external electromagnetic field is given. The Bargmann Michel, and Telegdi equation follows from this formalism as a quasiclassical approximation. Finally, a new relativistic model for hydrogenlike atoms is proposed. The spectrum predicted is in agreement with Dirac's when radiative corrections have been added.

Orbital equations in the relativistic two-body problem

Synge's approximation method is applied to derive the orbital equations for a binary system consisting of two rotating, spherical, rigid bodies of comparable mass and radius. Approximations are based on the weakness of the field and on the distance between the bodies being considered large by comparison with their radii.

Spin precession in the relativistic two-body problem

Synge's approximation procedure is applied to calculate the spin precession of either body in a binary system consisting of two rotating, spherical, rigid bodies of comparable mass and radius. The calculations are valid for the case in which the mass-radius ratio of each body, as well as the ratio of the radius of either body to the distance between their centers, is small. The results agree with those of earlier authors, who use different techniques, except for a term that arises from the effect of the rotation on the stress within the bodies. This term is similar in form to the quadrupole term of Barker and O'Connell, which they obtain when they allow the bodies to become distorted under the influence of the rotation.

Constraint formalism of classical mechanics

The constraint formalism for classical mechanics is developed with an eye toward facilitating a manifestly covariant relativistic quantization of classical mechanics. The very close relationship of this formalism to the more familiar Hamiltonian formalism has long obscured its independent status. In this paper, after defining the concept of a classical trajectory in a particularly natural and intrinsic way (that is, without reference to a special parameter such as the time), we prove that the trajectories satisfy the Hamilton equations of motion. In the following paper, the power of this formalism is demonstrated, when as an illustrative example it is employed to solve the nontrivial relativistic two-body problem, both classically and quantum mechanically.

Some nonlinear effects in the relativistic two-body problem

The test-particle motion in the centrally symmetric gravitational field can be described by the equation in the form appropriate for a nonlinear oscillator — the nonlinear terms being due to the nonrelativistic effects. This enables us to apply to this equation the well-known asymptotic methods of the theory of nonlinear oscillations. Typical nonlinear oscillation phenomena arising from the action of external forces are shown to take place. The form of equations and the main results remain valid in the problem of two bodies of comparable mass in the post-Newtonian approximation.

A new approach to the semi-classical relativistic two-body problem for charged fermions

Generalizing from a recently developed hybrid formulation of classical electrodynamics with «direct (charge-field) action» structure we construct an analogous semi-classical Dirac formulation of the theory, which is capable of describing the semi-classical quantum mechanics of two identical spin-1/2 particles. This semi-classical formulation is to be used as a heuristic aid in searching for the theoretical structure of a fully «second quantized» theory. The Pauli exclusion principle is incorporated by making the interaction fields (in the action principle) antisymmetric with respect to «charge-field» labeling. In this manner, «position correlation» effects associated with «configuration interaction» can also be accounted for. By studying the nature of the stationary-state solutions, the formalism is compared with the conventional quantum-mechanical one (to understand the similarities and the differences between this approach and the usual correlated Hartree-Fock approximation of ordinary relativistic quantum theory).The stationarystate solutions to the semi-classical formalism are shown to closely approximate the usual quantum-mechanical solutions when the wave functions are represented as a superposition of Slater determinants of Dirac-Coulombic-type wave functions with radial parts having a form which extremizes the total Breit energy. The manner in which this semi-classical theory might be extended to a fully «second quantized» formalism is sketched.

Hard core in the nucleon-nucleon interaction as a relativistic effect

The author shows that a recent formulation of the relativistic two-body problem given by Suura (1977) leads to the existence of a hard core in the nucleon-nucleon interaction. The hard-core radius R is estimated to be R approximately=0.48 fm in the 1S0 partial wave. The results agree with those obtained from the Gross equation, although both approaches to the relativistic two-body problem are unrelated.

Three-dimensional formulation of the relativistic two-body problem in terms of rapidities

Three-dimensional formulation of the relativistic two-body problem in terms of rapidities

Critical mass in a relativistic two-body problem

We use the relativistic 2-body Dirac Hamiltonian with a square-well vector interaction, to investigate how the finite mass of the source of the external field changes the behavior of the bound-state spectrum from that of one Dirac particle in an attractive square well. Limiting ourselves for simplicity to states of total spin zero, we find that, as long as the massm2 of the heavier particle is smaller than some critical massmc, the total energyE of the system does not reach the particle-antiparticle continuum, so that no difficulty of interpretation arises. If, however,m2 is greater thanmc, there exists, as in the one-body Dirac equation, one value of the coupling strengthVc for whichE reaches the antiparticle continuum. Then, when the coupling strength exceedsVc, the bound-state level disappears. Furthermore, for some coupling strengthV>Vc, there is a 2-particle bound-state level emerging from the particle-antiparticle continuum, whose total energyE increases until it reaches the particle-particle continuum. This latter result shows that, form2>mc, a 2-body analysis actually further complicates the interpretation of deep bound states in relativistic quantum mechanics.

A new general covariant approach to the general relativistic two-body problem

A new general covariant approach to the general relativistic two-body problem

A New General Covariant Approach to the General Relativistic Two-Body Problem

A new general covariant approach to the general relativistic equations of motion is presented. It is stressed that our present understanding of the development of binaries due to general relativistic effects and of the power emitted by these systems in the form of gravitational radiation is highly unsatisfactory.

Two-Body Problem in Quantum Field Theory

We study an approach to the relativistic two-body problem that represents a considerable improvement of the Bethe-Salpeter equation. The main advantages are the improved properties of the simple ladder approximation. (a) It has the correct static limit and is in this sense equivalent to the sum of all crossed ladders of the Bethe-Salpeter equation, (b) in the case of massless exchange it is possible to solve analytically and obtain the wave functions and the T matrix in closed form, (c) the system is closed, that is, the two-particle T matrix is unitary. (d) The wave functions admit a complete quantum-mechanical interpretation without compromise of relativistic covariance, the current is conserved, and gauge invariance is respected. (e) The evaluation of physical amplitudes is greatly simplified, as illustrated by positronium decay and the Lamb shift. Both are carried out with full relativistic covariance, and the Lamb-shift calculation in particular is greatly simplified and clarified in comparison with other methods. Most of the results are applicable only to the case of spin-0 or spin-\textonehalf{} particles interacting through the exchange of a single scalar or vector boson. In particular, the calculations of positronium decay and the Lamb shift are carried out with spinless particles. It is shown, however, that the introduction of photon spin presents no problem and that the Lamb shift can be calculated in a fully gauge-invariant manner.

Three-Dimensional Formulation of the Relativistic Two-Body Problem and Infinite-Component Wave Equations

A relativistic quasipotential equation is derived from the conventional Hamiltonian formalism and old-fashioned "noncovariant" off-energy-shell perturbation theory in a similar way to that by which the four-dimensional Bathe-Salpeter equation is obtained from the off-mass-shell Feynman rules. The three-dimensional equation for the (off-energy-shell) scattering amplitude appears as a straightforward generalization of the nonrelativistic Lippmann-Schwinger equation. The corresponding homogeneous equation for the bound-state wave function and the normalization condition for its solutions are derived from the equation for the complete four-point Green's function. In order to obtain a solvable model, we consider a simplified version of the quasipotential equation which still reproduces correctly the on-shell scattering amplitude and is consistent with the elastic unitarity condition. It involves a "local" approximation to the potential $V(p\ensuremath{-}q)$ which defines the kernel of our integral equation (the integration being carried over a two-sheeted hyperboloid in the energy-momentum space). It is shown that for the scalar Coulomb potential $V(p\ensuremath{-}q)=\frac{\ensuremath{\alpha}}{{(p\ensuremath{-}q)}^{2}}$, our model equation is equivalent to a simple infinite-component wave equation of the type considered by Nambu, Barut, and Fronsdal. The energy eigenvalues for the bound-state problem are calculated explicitly in this case and are found to be $O(4)$ degenerate (just as in the nonrelativistic Coulomb problem and in Wick and Cutkosky's treatment of the Bethe-Salpeter equation in the same approximation).

Effects of Gravitational Radiation Reaction in the General Relativistic Two-Body Problem by a Lorentz-Invariant Approximation Method

A Lorentz-invariant approximation method for obtaining the equations of motion of mass points in the general theory of relativity was developed recently. If retarded interactions are assumed, the first-order equations contain radiation reaction terms similar to those appearing in electrodynamics. These equations are used to investigate the question of the existence of gravitational radiation through the examination of selected approximate solutions of the two-body problem, without recourse to the assumption of slow motion. For the case of equal masses, the motion is almost circular; the particles are found to spiral outward in time. This result is in disagreement with Einstein's calculation of the distant field and the resulting energy loss due to quadrupole radiation based on the linear approximation, which, however, did not take proper account of the motion of the particles consistent with the approximate field equations used. These equations lead in a natural manner to an energy-momentum pseudotensor of the field consistent with the approximate equations of motion employed here. Using this pseudotensor, the distant outgoing retarded field due to the circular motion of two equal masses is found to correspond to an energy influx of quadrupole type; similar results are found for elliptic motion of unequal masses in slow-motion approximation. On the other hand, the assumption of advanced interactions would lead to inward spiraling and an incoming field with energy outflow. The significance of these discomforting results is discussed; possible objections against the equations of motion used, the method of solution of these equations, or the interpretation of the solutions obtained are examined. The possibility that the calculation of radiation effects on the basis of the higher order equations of motion will yield results differing qualitatively from those obtained here in first order cannot be excluded.

Relativistic Potential Scattering

The scattering properties of the relativistic two-body problem, governed by the Dirac equation, are investigated. It is shown rigorously that the associated Hamiltonians are self-adjoint, that the associated wave operators exist, and that the scattering operator exists and is unitary, all under suitable conditions on the potential. These conditions on the potential are analogues of those required for the nonrelativistic two-body problem governed by the Schrödinger equation.

Relativistic two-body problem in the classical theory of meson field

The present article deals with the motion of two particles of equal masses in a scalar meson field. The relativistic equation of motion, in which the retardation was also taken into account, was integrated numerically. At the point where the absolute value of the meson potential equals the value of the rest energymc 2 of the particle the acceleration of the two particles attracting each other becomes zero and within this short distance changes its sign: the “relativistic repulsion” arises.

One-time formulation of the relativistic two-body problem. Separation of angular variables

The separation of angular variables in the relativistic one-time equation for the two-fermion problem in Quantum Electrodynamics is carried out for the case of one-quantum interaction. The momentum representation is used. A system of sixteen integral equations with respect to one variable is obtained. This system reduces to four equations in the casej=0 and to eight equations forj≠0.