Relativistic Kepler Problem Research Articles

Relativistic Kepler problem. II. Asymptotic behavior of the field in the infinite past

The standard weak-field, slow-motion approximation to Einstein's relativistic theory of gravitation is used to express the curvature tensor, up to order ${r}^{\ensuremath{-}5}$ on a flat background space-time, as a functional of the motion of the source of this curvature. The behavior, in the distant past, of the orbit of two particles weakly interacting gravitationally, with radiation reaction taken into account, is then used to compute the asymptotic behavior of the corresponding curvature tensor along past-directed null straight lines in the flat background. It is found, on the one hand, that the falloff of the curvature is fast enough to guarantee satisfaction of a condition to exclude incoming gravitational radiation. On the other hand, the falloff is slower than would have been expected if the conformally rescaled curvature tensor had been regular on the hypersurface at past null infinity of the flat background.

Semiclassical quantization of the relativistic Kepler problem

Semiclassical quantization conditions are applied to a variety of relativistic formulations of the problem of two spinless particles bound by a vector or scalar force. The WKB quantization conditions are found to yield the correct spectra whenever the equations of motion are separable. Miller's new quantum condition, making use of periodic orbits, is applied to almost circular orbits. It provides an approximate method of quantization for almost circular orbits in the action-at-a-distance formulation of relativistic charged particles interacting electromagnetically.

Nonuniversality of the g/e ratio in the relativistic Kepler problem

Nonuniversality of the g/e ratio in the relativistic Kepler problem

Bohr-Sommerfeld conditions in geometric quantization

The corrected Bohr-Sommerfeld quantum conditions, ∫ p dq− d = integer, are studied in the framework of geometric quantization. It is shown, in the representation given by a polarization F, that a half-form corresponds to a wave function only if it vanishes on all closed curves with tangent vectors in F for which the quantum condition is not satisfied. The constant d is determined, for each closed curve y, by the element of the holonomy group of a bundle of metalinear frames for F induced by y. This result is applied to a one-dimensional harmonic oscillator and a two-dimensional relativistic Kepler problem. In the case of the one-dimensional harmonic oscillator there are two possibilities of choosing a metalinear frame bundle for F. One choice leads to the original Bohr-Sommerfeld condition while the other leads to the corrected version with d = 1 2 . Similarly, choosing different metalinear frame bundles for F, we get for the relativistic Kepler problem the fine structures of the energy levels corresponding to spin 0 and spin 1 2 .

Quaternionic solutions for the relativistic Kepler problem with magnetic charges

The relativistic Kepler problem in Dirac form is solved by means of a quaternionic method for particles having both electric and magnetic charges. The variables in spherical coordinates can be separated in a simple way and the results obtained are similar to those which are valid for the usual hydrogen atom.

Nonrelativistic and Relativistic Coulomb Amplitude as the Matrix Element of a Rotation inO(4,2)

It is shown that the Coulomb amplitude is the matrix element of the group element ${e}^{\mathrm{i}(\ensuremath{\theta}+\ensuremath{\pi})\mathrm{J}2}$ evaluated in a continuous basis in $\mathrm{SO}(4,2)$. The method is applied to the relativistic Kepler problem without spins, but with the recoil corrections included. The form of the amplitude for this case is believed to be new.

On classical scalar field theories and the relativistic kepler problem

Two versions of classical relativistic field theory corresponding to massless scalar exchange are compared. It is found that they lead to identical field equations but different equations of motion. The motion of one particle bound by the field of another, infinitely massive particle, i.e., the Kepler problem, is examined. In one case the path is an ellipse precessing through an angle equal to minus one-sixth the value predicted by general relativity. In the other case the path is an ellipse which does not precess. The relativistic Kepler problem in the latter case is identical to that formulated by Fronsdal, who started from an approximation to the Bethe-Salpeter equation.

Partial Group-Theoretical Treatment for the Relativistic Hydrogen Atom

The Lie algebra of SU (1, 1) and its Hermitian representations are used together with spherical harmonics to solve the wave equations for the nonrelativistic q-dimensional oscillator and the relativistic Kepler problem.

Formal Properties of the Solution to the Radiationless Relativistic Kepler Problem

Formal properties of the solution to the classical radiationless relativistic Kepler problem are discussed with particular attention to the strong coupling limit, a case rarely considered. It is found that fall to the origin can occur for potentials less strongly attractive than required nonrelativistically. One conclusion is that the relativistic centrifugal barrier is less ``effective'' than the nonrelativistic kind. It is possible to describe these effects in terms of a dimensionless parameter resembling a classical ``coupling constant.'' The quantum-mechanical case is also briefly treated.

Quasi-Classical Theory of the Nonspinning Electron

We develop a modified Hamiltonian-Jacobi theory of classical mechanics following the early work of Van Vleck. This modified Hamiltonian-Jacobi theory, or quasi-classical theory, permits us to exhibit in classical mechanics many features that in the past have been exclusively associated with quantum mechanics. We deal with classical wave functions, classical operators, classical eigenvalue equations, a classical sum over paths formulation of classical mechanics, and with classical creation and destruction operators. Following Van Vleck, one can derive the WKB approximate solutions to the Schr\odinger equation from the solutions of the classical Hamilton-Jacobi equation. If we apply the methods of Keller to the nonrelativistic and relativistic Kepler problem, we derive eigenvalues from the requirement of single-valuedness imposed on the WKB solutions. It turns out that the energy eigenvalues are those given by the Schr\odinger equation and the Klein-Gordon equation, respectively. In the particular case of the harmonic oscillator there exists a canonical transformation which transforms the quasi-classical equation into an exact equation of quantum mechanics. We conjecture that if the WKB approximation and the Schr\odinger equation predict the same eigenvalues, then there always exists a canonical transformation which transforms the quasi-classical equation into the corresponding Schr\odinger equation. Finally we derive the quasi-classical equations in momentum space.