Das Korrespondenzprinzip in der Kosmologie I. Der isotrope Kosmos als NEWTONscher Grenzfall der relativistischen Gravitationstheorien

According to the principle of correspondence (in HEISENBERG 's formulation) each general relativistic theory of gravitation must give a NEWTON ian representation for an isotropic cosmos with the ROBERTSON -WALKER metric. Indeed, the FRIEDMANN equations can be interpreted as the expression for the HAMILTON ian H of a closed NEWTON ian system of the cosmic fundamental particles, written in the rest-system of the center of gravity. In this HAMILTON ian H only the relative-coordinates and the relative-velocities of the particles are present and one can write H without absolute quantities but only with MILNE 's relative-quantities. The time-independence of the HAMILTON ian H = 0 is the FRIEDMANN equation. – This NEWTON ian deduction of the FRIEDMANN equation is more general than the relativistic deduction and than MILNE 's deduction for a NEWTON ian fluid, too. In the general NEWTON ian form H the parameter f M of the active mass can be an arbitrary function of the cosmic time t. The choice f = f(t), M = M(t) defines the divers modifications of relativistic cosmology. – In general relativity fM = const and M = const are resulting from EINSTEIN 's equations and from EINSTEIN 's principle of equivalence.     

Solution of a generalized equation of transfer in a two-region slab

The general equation of transfer in a two-region slab of unequal thickness with general boundary conditions has been solved by an analytical method developed by Menninget al. (1980). The scattering is regarded as isotropic and the source function is taken as a general one to accomodate different types of problems.     

General vacuum solution for Brans-Dicke-Bianchi type-II

In this paper we have obtained the general vacuum solution for Bianchi type-II in the Brans-Dicke theory for total anistropyR ₁R ₂R ₃. It is known that by use of our method, we can find the general solution for Bianchi type-II vacuum case in the general relativity theory, first given by Taub (1951). Some physical properties of this model are also discussed.     

Recent progress in the theory and application of symplectic integrators

In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the formH=T(p)+V(q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets.     

Expanding universe: thermodynamical aspects from different models

The pivotal point of the paper is to discuss the behavior of temperature, pressure, energy density as a function of volume along with determination of caloric EoS from following two model: w(z)=w ₀+w ₁ln(1+z) & . The time scale of instability for this two models is discussed. In the paper we then generalize our result and arrive at general expression for energy density irrespective of the model. The thermodynamical stability for both of the model and the general case is discussed from this viewpoint. We also arrive at a condition on the limiting behavior of thermodynamic parameter to validate the third law of thermodynamics and interpret the general mathematical expression of integration constant U ₀ (what we get while integrating energy conservation equation) physically relating it to number of micro states. The constraint on the allowed values of the parameters of the models is discussed which ascertains stability of universe. The validity of thermodynamical laws within apparent and event horizon is discussed.     

Adiabatic indices in a convecting anisotropic plasma

Adiabatic indices for a non-dissipative anisotropic convecting plasma are analyzed, and general expressions for the effective adiabatic index and the partial adiabatic indices parallel (γ_∥) and perpendicular (γ_⟂) to the magnetic field are obtained. It is shown that, in the general case, the value of the effective adiabatic index is not an universal constant and depends on the plasma temperature anisotropy and on the properties of the plasma motion. The values of γ_⟂ and γ_∥ are shown to be independent of the plasma parameters being completely determined by the characteristics of the plasma flow.     

Relative equilibrium solutions in the four body problem

Beyond the casen=3 little was known about relative equilibrium solutions of then-body problem up to recent years. Palmore's work provides in the general case much useful information. In the casen=4 he gives the totality of solutions when the four masses are equal and studies some degeneracies. We present here a survey of solutions for arbitrary masses, discussing the manifolds of degeneracy. The ordering of restricted potentials allows a counting of the number of bifurcation sets and different invariant manifolds. An analysis of linear stability is done in the restricted and general cases. As a result, values of the masses ensuring linear stability are given.     

On multi-region problems in radiative transfer

TheF _N method is used to solve radiative transfer problems, based on the general anisotropically scattering model, in multi-layer atmospheres.     

Exact solutions: classical electron model

Rahaman et al. (Astrophys. Space. Sci. 331:191–197, 2010) discussed some classical electron models (CEM) in general relativity. Bijalwan (Astrophys. Space. Sci. 334:139–143, 2011) present a general exact solution of the Einstein-Maxwell equations in terms of pressure. We showed that charged fluid solutions in terms of pressure are not reducible to a well behaved neutral counter part for a spatial component of metrice ^λ . Hence, these solutions represent an electron model in general relativity. We illustrated solutions in terms of pressure briefly with de-Sitter equation of state and charged analogues of Kohler Chao interior solution as a special cases.     

Global Solution of Dynamical Systems a Report to Z. Kopal, on what Followed Since

Two basic problems of dynamics, one of which was tackled in the extensive work of Z. Kopal (see e.g. Kopal, 1978, Dynamics of Close Binary Systems, D. Reidel Publication, Dordrecht, Holland.), are presented with their approximate general solutions. The ‘penetration’ into the space of solution of these non-integrable autonomous and conservative systems is achieved by application of ‘The Last Geometric Theorem of Poincaré’ (Birkhoff, 1913, Am. Math. Soc. (rev. edn. 1966)) and the calculation of sub-sets of ‘solutions précieuses’ that are covering densely the spaces of all solutions (non-periodic and periodic) of these problems. The treated problems are: 1. The two-dimensional Duffing problem, 2. The restricted problem around the Roche limit. The approximate general solutions are developed by applying known techniques by means of which all solutions re-entering after one, two, three, etc, revolutions are, first, located and then calculated with precision. The properties of these general solutions, such as the morphology of their constituent periodic solutions and their stability for both problems are discussed. Calculations of Poincaré sections verify the presence of chaos, but this does not bear on the computability of the general solutions of the problems treated. The procedure applied seems efficient and sufficient for developing approximate general solutions of conservative and autonomous dynamical systems that fulfil the PoincaréBirkhoff theorems. The same procedure does not apply to the sub-set of unbounded solutions of these problems.     

New photoelectric observations of VW Cephei

The aim of the present paper is to present the newB andV light curves of the eclipsing binary VW Cep, obtained with the 48-in. reflector of the National Observatory of Athens, Greece.In the introduction general information for the system is given. In Section 2 some observational and reductional details are given and the obtained light curves are represented. Section 3 deals with the period of the system which was found to continue its shortening. Finally, in Section 4, a general discussion concerning our light curves is given.     

On a uniformly rotating magnetic polytrope

The equilibrium general magnetic field inside a magneto-rotating star, assumed to be a polytrope, has been determined more accurately, for large general magnetic field. Furthermore the effect of such field on the structure and oscillations of a slowly rotating polytrope has been studied forn=1,0, 1.5, 2.0, and 3.0.     

Comparing selfinteracting scalar fields and R + R3 cosmological models

We generalize the well-known analogies between m²φ² and R + R² theories to include the selfinteraction λφ⁴-term for the scalar field. It turns out to be the R + R³ Lagrangian which gives an appropiate model for it. Considering a spatially flat Friedman cosmological model, common and different properties of these models are discussed, e. g., by linearizing around a ground state the masses of the resp. spin 0-parts concide. Finally, we prove a general conformal equivalence theorem between a Lagrangian L = L(R), L′L″ ≠ 0, and a minimally coupled scalar field in a general potential.     

A unified theory of polynomial expansions and their applications involving Clebsch-Gordan type linearization relations and Neumann series

Motivated by their potential for applications in several diverse fields of physical, astrophysical, and engineering sciences, this paper aims at presenting a unified study of various classes of polynomial expansions and multiplication theorems associated with the general multivariable hypergeometric function (studied recently by A. W. Niukkanen and H. M. Srivastava), which provides an interesting and useful unifiation of numerous families of special functions in one and more variables, encoutered naturally (and rather frequently) in many physical, quantum chemical, and quantum mechanical situations. Several interesting applications of these general polynomial expansions are considered, not only in the derivations of various Clebsch-Gordan type linearization relations involving products of several Jacobi or Laguerre polynomials, but also to associated Neumann expansions in series of the Bessel functionsJ _v (z) andI _v (z) (and of their suitable products).     

Wave propagation in a magnetically structured atmosphere

The propagation of waves in a magnetic slab embedded in a magnetic environment is investigated. The possible modes of propagation are examined from the general dispersion relation, both analytically and numerically, for disturbances which are evanescent in the environment. Approximate dispersion relations governing propagation in a slender slab of field are derived both from the general dispersion relation and from an application of the slender flux tube approximation.Several different situations, representative of both photospheric and coronal conditions, are considered. In general, the structures are found to support both fast and slow, body and surface, waves. Under coronal conditions, for two dimensional propagation, disturbances propagate as fast and slow body waves. The fast body waves are analogous to the ducted shear waves of seismology (Love waves).     

Stationary Configurations for Co-orbital Satellites with Small Arbitrary Masses

We derive general results on the existence of stationary configurations for N co-orbital satellites with small but otherwise arbitrary masses m _i , revolving on circular and planar orbits around a massive primary. The existence of stationary configurations depends on the parity of N. If N is odd, then for any arbitrary angular separation between the satellites, there always exists a set of masses (positive or negative) which achieves stationarity. However, physically acceptable solutions (m _i > 0 for all i) restrict this existence to sub-domains of angular separations. If N is even, then for given angular separations of the satellites, there is in general no set of masses which achieves stationarity. The case N=3 is treated completely for small arbitrary satellite masses, giving all the possible solutions and their stability, to within our approximations.     

The general solution of the HENON–HEILES problem

The general solution of the Henon–Heiles system is approximated inside a domain of the (x, C) of initial conditions (C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution space of the problem. In the case of the Henon–Heiles potential we calculated the families of periodic solutions that re-enter after 1–108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ε (Poincaré parameter), which ascertains that at least one periodic solution is computed and available within a distance ε from any point of the domain (x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ε = 0.07. The same solution is further improved by “zooming” into four square sub-domain of (x, C), i.e. by computing sufficient number of families that reduce the density parameter to ε = 0.003. Further zooming to reduce the density parameter, say to ε = 10⁻⁶, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary. The stability of all members of each and all families computed was calculated and presented in this paper for both the large solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections. All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received, consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon–Heiles Problem,” including their stability and is available at request. It is concluded that approximation of the general solution of this system is straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties.     

Cosmological Creation of Matter in Conformal Cosmology

The effect of rapid cosmological creation of vector W and Z bosons is studied in the framework of conformal cosmology, which unifies the general theory of relativity and the standard model for strong and electroweak interactions.     

Ohm’s law for plasma in general relativity and Cowling’s theorem

The general-relativistic Ohm’s law for a two-component plasma which includes the gravitomagnetic force terms even in the case of quasi-neutrality has been derived. The equations that describe the electromagnetic processes in a plasma surrounding a neutron star are obtained by using the general relativistic form of Maxwell equations in a geometry of slow rotating gravitational object. In addition to the general-relativistic effect first discussed by Khanna and Camenzind (Astron. Astrophys. 307:665, 1996) we predict a mechanism of the generation of azimuthal current under the general relativistic effect of dragging of inertial frames on radial current in a plasma around neutron star. The azimuthal current being proportional to the angular velocity ω of the dragging of inertial frames can give valuable contribution on the evolution of the stellar magnetic field if ω exceeds 2.7×10¹⁷(n/σ) s⁻¹ (n is the number density of the charged particles, σ is the conductivity of plasma). Thus in general relativity a rotating neutron star, embedded in plasma, can in principle generate axial-symmetric magnetic fields even in axisymmetry. However, classical Cowling’s antidynamo theorem, according to which a stationary axial-symmetric magnetic field can not be sustained against ohmic diffusion, has to be hold in the general-relativistic case for the typical plasma being responsible for the rotating neutron star.     

Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.