Present status of projective unified field theory and its physical predictions

In this article we shortly present the basic equations of our 5–dimensional Projective Unified Field Theory and their physical interpretation. Our main intention is to confront this theory with physical experience. For this reason we treat the Einstein effects (including the exact solutions for a central body) in detail. Further we once more discuss the first order effects on the quasi–Newtonian level with respect to the equivalence principle.     

Treatment of a static spherically symmetric matter distribution on the basis of the projective unified field theory

In order to confront the Projective Unified Field Theory with physical experience it is appropriate to apply this theory also to astrophysical objects, for mathematical reasons in particular to a static spherically symmetric matter distribution as a rough model for a star (e.g. neutron star or another exotic compact object). The problem mentioned is treated here as far as possible analytically in order to prepare the basis for the numerical approach.     

Particle motion in the field of a five-dimensional charged black hole

In this paper, we have investigated the geodesics of neutral particles near a five-dimensional charged black hole using a comparative approach. The effective potential method is used to determine the location of the horizons and to study radial and circular trajectories. This also helps us to analyze the stability of radial and circular orbits. The radius of the innermost stable circular orbits have also been determined. Contrary to the case of massive particles for which, the circular orbits may have up to eight possible values of specific radius, we find that the photons will only have two distinct values for the specific radii of circular trajectories. Finally we have used the dynamical systems analysis to determine the critical points and the nature of the trajectories for the timelike and null geodesics.     

Qualitative behaviour of the trajectories of test bodies in five-dimensional spherically-symmetric gravitational field

Qualitative behaviour of the trajectories of test bodies is investigated in the five-dimensional gravitation theory for the spherically-symmetric solution. The Binnet formula is obtained for the trajectory. The region of the existence of circular trajectories as well as the regions of their stability are specially investigated using the method of numerical calculations. The values of the free parameters in the five-dimensional gravitation theory are estimated for the solar system.     

Bianchi type-III cosmological model in four- and five-dimensional bimetric theory of relativity

Four- and five-dimensional Bianchi type-III cosmological model in Rosen (1980) bimetric theory of gravitation is considered. Restricting to a particular type of background metric, it is observed that the Bianchi type-III cosmological model does not exist in case of both meson field and mesonic perfect fluid. Hence only vacuum model can be obtained.     

The missing mass problem and the cosmological constant problem in the Brans-Dicke theory

The missing mass problem and the cosmological constant problem, which arised in the theory of general relativity, are re-examined in the Brans-Dicke theory. It is shown that the two problems can be solved in simple ways in the Brans-Dicke theory.     

First integrals for the Kepler problem with linear drag

In this work we consider the Kepler problem with linear drag, and prove the existence of a continuous vector-valued first integral, obtained taking the limit as \(t\rightarrow +\infty \) of the Runge–Lenz vector. The norm of this first integral can be interpreted as an asymptotic eccentricity \(e_{\infty }\) with \(0\le e_{\infty } \le 1\). The orbits satisfying \(e_{\infty } <1\) approach the singularity by an elliptic spiral and the corresponding solutions \(x(t)=r(t)e^{i\theta (t)}\) have a norm r(t) that goes to zero like a negative exponential and an argument \(\theta (t)\) that goes to infinity like a positive exponential. In particular, the difference between consecutive times of passage through the pericenter, say \(T_{n+1} -T_n\), goes to zero as \(\frac{1}{n}\).     

The gravitational perturbation spectrum in linear satellite theory

A new method for calculating the perturbation spectrum in the framework of Kaula's linear satellite theory (LST) is introduced. The novelty of this approach consists in using recent results on the spectral decomposition of the perturbation frequencies in LST to provide a closed formulation for the amplitude and the phase of each line in the perturbation spectrum. The theory presented here can be applied to perturbations in the elements or in the radial and transverse directions due to the geopotential or to the tides. Separate algorithms are developed for application to orbits with circulating or frozen perigee.     

The time-dependent problem in the Jordan-Brans-Dicke tensor-scalar theory of gravitation

We state the time-dependent problem in the Jordan-Brans-Dicke tensor-scalar theory of gravitation. We show that in the conformally corresponding space the equations of the problem have the same form as the Einstein equations with an extra source in the form of the components of the energy-momentum tensor of the scalar field. We discuss the question whether Birkhoff's theorem can hold in the Jordan-Brans-Dicke theory. We simultaneously obtain a formula that expresses the mass of an isolated static object as a function of the distribution of pressure of matter. We also discuss the integrability of the vacuum time-dependent equations of the Einstein theory.Translated fromAstrofizika, Vol. 37, No. 3, 1994.The authors are grateful to the participants in the seminar of the Department of Theoretical Physics of Erevan State University for helpful discussions. Two of the authors (VVP and GGA) worked with partial support from grant RY 6000 of the International Science Fund (Soros Foundation).     

The linear stability of the post-Newtonian triangular equilibrium in the three-body problem

Continuing a work initiated in an earlier publication (Yamada et al. in Phys Rev D 91:124016, 2015), we reexamine the linear stability of the triangular solution in the relativistic three-body problem for general masses by the standard linear algebraic analysis. In this paper, we start with the Einstein–Infeld–Hoffmann form of equations of motion for N-body systems in the uniformly rotating frame. As an extension of the previous work, we consider general perturbations to the equilibrium, i.e., we take account of perturbations orthogonal to the orbital plane, as well as perturbations lying on it. It is found that the orthogonal perturbations depend on each other by the first post-Newtonian (1PN) three-body interactions, though these are independent of the lying ones likewise the Newtonian case. We also show that the orthogonal perturbations do not affect the condition of stability. This is because these do not grow with time, but always precess with two frequency modes, namely, the same with the orbital frequency and the slightly different one due to the 1PN effect. The condition of stability, which is identical to that obtained by the previous work (Yamada et al. 2015) and is valid for the general perturbations, is obtained from the lying perturbations.     

Kolmogorov and Nekhoroshev theory for the problem of three bodies

We investigate the long time stability in Nekhoroshev’s sense for the Sun– Jupiter–Saturn problem in the framework of the problem of three bodies. Using computer algebra in order to perform huge perturbation expansions we show that the stability for a time comparable with the age of the universe is actually reached, but with some strong truncations on the perturbation expansion of the Hamiltonian at some stage. An improvement of such results is currently under investigation.     

The main problem of artificial satellite theory for small and moderate eccentricities

Perturbation techniques based on Lie transforms as suggested by Deprit were used as the theoretical foundation for programming the analytical solution of the Main Problem in Satellite Theory (all gravitational harmonics being zero exceptJ ₂). The collection of formulas necessary and sufficient to construct an ephemeris is given in the exposition. Short and long period displacements, as well as the secular terms, have been obtained up to the third order inJ ₂ as power series of the eccentricity. They result from two successive completely canonical transformations which it has been found convenient not to compose into a unique transformation. Division by the eccentricity appears nowhere in the developments-neither explicitly nor implicitly. The determination of the constants of motion from the initial conditions has been given an elementary solution that is both complete and explicit without being iterative. The program was developed by Rom from MAO's package of subroutines forMechanizedAlgebraicOperations. Reliability tests have been run in two instances: in-track errors for ANNA 1B are only 20 cm after 210 days in orbit, while for RELAY II, they are 2.4 m, even after 350 days in orbit.     

Integration theory for the restricted problem of three axisymmetric bodies

In this paper the circular planar restricted problem of three axisymmetric ellipsoids S _i (i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S ₃ be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries).     

The symmetric tensor field in the relativistic theory of gravitation

The system of a self-gravitating scalar field is frequently used in inflationary cosmological models. In the present paper we study a more complicated system containing an extra linear tensor field _ik=_ki with minimal coupling. We determine five of the six free parameters that occur in the most general expression for the actionS of this field. In doing so we assume that in flat space-time the field _ik must be invariant under gauge transformations. In a special case theS found becomes a known expression for the action of a massless tensor field _ik. We compute the metric energy-momentum tensor that determines the contribution of _ik to the Einstein equations. We also exhibit the equations of motion of _ik in curved space-time.Translated fromAstrofizika, Vol. 38, No. 3, 1995.     

Lee yang theory for cosmological many body problem

We study the phase transition in a gravitating system by analyzing grand canonical partition function as a function of complex fugacity. We extend the Yang-Lee theory to study phase transitions in the gravitational galaxy clustering of galaxies having a variety of masses. This generalizes our previous work based on the same theory for the single-component system to a multicomponent system. We find that galaxy clustering is sensitive to masses and number densities of individual galaxies at early stages while at later stages collective behavior of the particles is more pronounced. This validates our earlier work obtained from different considerations.     

Analytical linear perturbation theory for highly eccentric satellite orbits

Kaula's satellite linear perturbation theory has been extended for the case of highly eccentric orbits by using elliptic function expansions.On leave from Institute of Applied Astronomy, St.-Petersburg     

Affinely invariant field theory

If we require the effective field equations for a local system to be second order partial differential equations, an affinely invariant theory (not presuming the existence of a metric tensor beforehand) has to be non-local, the action being a multiple integral over the manifold considered.     

The main problem of lunar theory solved by the method of Brown

Brown's method for solving the main problem of lunar theory has been adapted for the computation by machine with the help of an algebraic processor. Brown's results are first recovered and refined. The solution is then expanded to include most terms of order nine. The terms in the series for the longitude and latitude are listed with an accuracy of 0.000 01 and of 0.000 001 for the parallax.This research was supported in parts by the National Science Foundation grant MCS 78-01425.