The Einstein-Maxwell field equations,II

In order to arrive at more general results solving Einstein-Maxwell's equations our investigation is centered around an electromagnetic spin tensor, which must be chosen in such a way that conservation laws still hold. This notion of the combined tensor is of course closely linked with the unified field equations. We shall avoid in this way the problem of the form of the matter tensor and neglect non-linear gravitational terms in the Ricci tensor. Then, the field equations have as solutionsh _ij=h _ij ^(P) +h _ij ^(h) , whereh _ij ^(P) are particular solutions, which are obtained by direct calculations andh _ij ^(h) are solutions of h _ij ^(h) =0. The quantitiesh _ij ^(P) are purely electromagnetic in nature, whileh _ij ^(h) may represent purely gravitational terms. The results obtained complete the ones which have been published already in the preceeding paper (Dionysiou, 1980a; which will hereafter be referred to as Paper I).     

Multi-dimensional cosmological models in effective string gravitation. I

We consider multi-dimensional cosmological models in the low-energy field theory of strings with a boson gravitational sector containing a metric, dilaton field, and antisymmetric Kalb-Ramon field. We study the conformal properties of the action and show that in the general conformal representation the theory is equivalent to a generalized scalar-tensor theory with a Lagrangian of nongravitating matter dependent on the dilaton. We find exact solutions of the flat homogeneous anisotropic model with structure R×M¹×...×Mⁿ and with equation of state p_i=a_i in the space Mⁱ. We discuss the picture of cosmological evolution in different conformal representations.Translated fromAstrofizika, Vol. 38, No. 2, 1995.     

Unrestricted second-order tensor virial equations for linear oscillations of magnetic configurations

This paper deals with the second-order tensor virial equations for the linear oscillations of a gaseous mass in the presence of a magnetic field. It is shown that the commonly used linearized versions of the tensor virial equations are restricted integral equations that incorporate the linearized equation of motion but not the boundary condition. These restricted equations only allow trial functions that fulfil the boundary condition and are of limited practical value.The unrestricted variational principle for the linear oscillations of a magnetic configuration is used to derive a more general formulation of the second-order tensor virial equations so that the linear trial function _i =X _ij x _j can be used to study the oscillations of a configuration with a magnetic field that extends in the exterior vacuum. The unrestricted virial equations have been applied to Ferraro's model and approximate results for the eigenfrequencies and eigenfunctions have been obtained for nine oscillation modes.     

Exact solution of perfect fluid massive string cosmology in Bianchi type?III space-time with decaying vacuum energy density?Λ

Exact solution of Einstein’s field equations is obtained for massive string cosmological model of Bianchi III space-time using the technique given by Letelier (Phys. Rev. D 20:2414, 1983) in presence of perfect fluid and decaying vacuum energy density Λ. To get the deterministic solution of the field equations the expansion θ in the model is considered as proportional to the eigen value s² ₂\sigma^{2}_{~2} of the shear tensor s^j _i\sigma^{j}_{~i} and also the fluid obeys the barotropic equation of state. The vacuum energy density Λ is found to be positive and a decreasing function of time which is supported by the results from recent supernovae Ia observations. It is also observed that in early stage of the evolution of the universe string dominates over the particle whereas the universe is dominated by massive string at the late time. Some physical and geometric properties of the model are also discussed.     

Cylindrically symmetric inhomogeneous universe with electromagnetic field in string cosmology

Cylindrically symmetric inhomogeneous string cosmological model in presence of electromagnetic field is investigated. We have assumed that F ₂₃ is the only non-vanishing component of F _ij . To get the deterministic solution, it has been assumed that the expansion (θ) in the model is proportional to the eigen value σ ¹ ₁ of the shear tensor σ ⁱ _j . The physical and geometric aspects of the model are also discussed.      

A recurrence relation for inclination functions

When the terms of the series expansion for the gravitational potential of the Earth are expressed in terms of the orbital elements of an arbitrary Earth satellite, the orbital inclination,i, appears in each, term as the argument of a function of inclination only. For the special case when the field is axi-symmetric, studied in an earlier paper, a recurrence relation was given for a normalized inclination function,A ^k l(i), with two parameters. The present paper gives a recurrence relation for a general normalized function,K ^k lm(i), with three parameters.     

Gravitation und weitreichende schwache Wechselwirkungen bei Neutrino-Feldern Gedanken zu einer Theorie der solaren Neutrinos

The strange non-evidence of the solar-neutrino current by the experiments of DAVIS et al. postulates a fundamental revision of the theory of weak interactions and of its relations to gravitation theory. (We assume that the astrophysical stellar models are not completely wrong.) – Our paper is based on PAULI 's grand hypothesis about the connection between weak and gravitational interactions. According to PAULI and BLACKETT the (dimensionless) gravitation constant is the square of the (dimensionless) FERMI -interaction constant and according to the hypotheses of PAULI, DE BROGLIE , and JORDAN the RIEMANN -EINSTEIN gravitational metric g_ik is fusioned by the four independent WEYL ian neutrino fields (β-neutrinos and β-antineutrinos, μ-neutrinos and μ-antineutrinos). This fusion gives four reference tetrads h_i^A(x^l) as neutrino-current vectors, firstly. Then, the metric g_ik is defined by the equation g_ik = η_AB h_i^Ah_η^B according to EINSTEIN 's theory of tele-parallelism in RIEMANN ian space-times. The relation of the gravitation field theory to FERMI 's theory of weak interactions becomes evident in our reference-tetrads theory of gravitation (TREDER 1967, 1971). – According to this theory the coupling of the gravitation potential h_i^A with the matter T_ιⁱ is given by a potential-like (FERMI -like) interaction term. In this interaction term two WEYL spinor-fields are operating on the matter-tensor, simultanously. Therefore, the gravitation coupling constant is PAULI 's square of the FERMI -constant. Besides of the fusion of the RIEMANN -EINSTEIN metric g_ik by four WEYL spinors we are able to construct a conformal flat metric ĝ_ik = ϕ²η_ik by fusion from each two WEYL spinors. (This hypothesis is in connection with our interpretation of EINSTEIN 's hermitian field theory as a unified field-theory of the gravitational metric g_ik and a WEYL spinor field [TREDER 1972].) Moreover, from the reference-tetrads theory is resulting that the WEYL spinors in the “new metric” ĝ_ik are interacting with the DIRAC matter current by a FERMI -like interaction term and that these WEYL spinors fulfil a wave equation in the vacuum. Therefore, we have a long-range interaction with the radiced gravitational constant \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {\frac{{tm^2 }}{{hc}}} $\end{document} as a coupling constant. That means, we have a long-range interaction which is 10¹⁸ times stronger than the gravitation interaction. – However, according to the algebraic structure of the conform-flat this long-range interaction is effective for the neutrino currents, only. And for these neutrinos the interaction is giving an EINSTEIN -like redshift of its frequences. The characteristic quantity of this “EINSTEIN shift” is a second gravitation radius â of each body: N = number of baryons, m = mass of a baryon.) This radius â is 10¹⁸ times larger than the EINSTEIN -SCHWARZSCHILD gravitation radius a = fM/c²: But, this big “weak radius” â has a meaning for the neutrinos, only.–The determination of the exterior and of the interior “metrics” ĝ_ik is given by an “ansatz” which is analogous to the ansatz for determination of strong gravitational fields in our tetrads theory. That is by an ansatz which includes the “self-absorption” of the field by the matter. For all celestial bodies the “weak radius” â is much greater than its geometrical dimension. Therefore, a total EINSTEIN redshift of the neutrino frequences v is resulting according to the geometrical meaning of our long-range weak interaction potential ĝ_ik = ϕ²η_ik. That means, the cosmic neutrino radiation becomes very weak and unable for nuclear reactions. Theoretically, our hypothesis means an ansatz for unitary theory of gravitation and of weak interaction. This unitary field theory is firstly based on EINSTEIN 's hermitian field theory and secondly based on our reference-tetrads theory of gravitation.     

On the Tetrad Formulation of the Equations of Motion in General Relativity

It is shown that the equation of motion Du _j/Ds = (e/mc ²)F _ji u ⁱ , a natural generalization to the curved spacetime of the Heaviside-Lorentz law of ponderomotive force, is equivalent to the metric independent and covariant Van Dantzig's equations of motion dx ^j _[jpi] = 0 or L _v p _i = 0, where p _i is the conjugate momentum 4-vector and v a vector determined by the condition p _i v ⁱ, only with respect to holonomic coordinates. With respect to an anholonomic system, the Heaviside-Lorentz equation is a particular case of the VD equations valid for a privileged class of anholonomic frames, those consisting of orthogonal unit vectors.     

Vibrational stability of rotating stars

The aim of the present paper will be to detail the explicit form of the equations which govern first-order oscillations of fast-rotating globes of self-gravitating fluids; with due account taken of the effects arising from the centrifugal as well as Coriolis force. As such configurations oscillate in general about distorted figures of equilibrium, the equations governing them can be conveniently expressed in terms of the Clairaut coordinates, associated with distorted spheroidal figures, and introduced in our previous paper (Kopal, 1980) for this purpose.In Section 2 which follows a brief outline of our problem, the equilibrium properties of fast-rotating configurations or arbitrary structure will be formulated. In Section 3 we shall carry out a separation of the variables in the equations of motion, and reduce the partial differential equations of the problem to an equivalent system of ordinary differential equations, by an expansion of expressions for the velocity componentsU, V, W in terms of tesseral harmonicsY _n ^m (, ). The explicit form of such a system, including the effects of all tesseral harmonics of orders up tom=n=4, will be specified in Section 3 for configurations whose equilibrium form is a sphere; while in Section 4 this latter condition will be relaxed to allow for the equilibrium configuration to become a rotational spheroid.In the concluding Section 5 we shall convert the complex form of our equations of motion into real terms, amenable to a solution-analytical or numerical-in terms of real variables; and shall establish the boundary conditions necessary for a specification of the characteristic frequencies of oscillation.     

Well behaved parametric class of relativistic charged fluid ball in?general relativity

The paper presents a class of interior solutions of Einstein–Maxwell field equations of general relativity for a static, spherically symmetric distribution of the charged fluid. This class of solutions describes well behaved charged fluid balls. The class of solutions gives us wide range of parameter K (0≤K≤42) for which the solution is well behaved hence, suitable for modeling of super dense star. For this solution the mass of a star is maximized with all degree of suitability and by assuming the surface density ρ _b =2×10¹⁴ g/cm³. Corresponding to K=2 and X=0.30, the maximum mass of the star comes out to be 4.96 M _Θ with linear dimension 34.16 km and central redshift and surface redshift 2.1033 and 0.683 respectively. In absence of the charge we are left behind with the well behaved fourth model of Durgapal (J. Phys., A, Math. Gen. 15:2637, 1982).     

Note on aTheorem of Relativistic Hydrodynamics

A well known theorem of relativistic hydrodynamics states that the streamlines of an isentropic perfect fluid are the future-pointing timelike (FPT) curves extremizing the integral J = ∫ _S1 ^S2 fds, where f is the so-called index function and s the proper time on the world line of the fluid particle. The integral is taken over all possible FPT curves with regular representations xⁱ = xⁱ (s) joining the fixed end events E₁, E₂. The purpose of this note is to show that the streamlines of an adiabatic perfect fluid can likewise be regarded as extremizing curves of the functional J provided the class of admissible curves consists of those FPT curves satisfying the side condition uⁱ∂_iS = 0, uⁱ unit 4-velocity and S the specific proper entropy of the fluid, with the first end point fixed and the second being the end point variable. __________ Published in Astrofizika, Vol. 48, No. 4, pp. 641–647 (October–December, 2005).     

‘Quantized’ dimensions of sunspots and penumbra phenomenon

Several authors have studied solutions of Einstein's field equations for a conformally invariant scalar field with trace-free energy-momentum tensor for the Robertson-Walker models forK = 0, ± 1. The relationship of these solutions to a previously existing one by Som (1985) is discussed. TheK = 0 model derived by Innaiah and Reddy (1985) is shown to be a special case of the Bianchi type-I models due to Accioly, Vaidya and Som (1983a).     

On most general exact solution for Vaidya-Tikekar isentropicsuperdense star

The energy density of Vaidya-Tikekar isentropic superdense star is found to be decreasing away from the center, only if the parameter K is negative. The most general exact solution for the star is derived for all negative values of K in terms of circular and inverse circular functions. Which can further be expressed in terms of algebraic functions for K = 2-(n/δ)² < 0 (n being integer andδ = 1,2,3 4). The energy conditions 0 ≤ p ≤ αρc ², (α = 1 or 1/3) and adiabatic sound speed conditiondp dρ ≤ c ², when applied at the center and at the boundary, restricted the parameters K and α such that .18 < −K −2287 and.004 ≤ α ≤ .86. The maximum mass of the star satisfying the strong energy condition (SEC), (α = 1/3) is found to be3.82 Mq_· at K=−2/3, while the same for the weak energy condition (WEC), (α =1) is 4.57 M_ _⊙ atK=−>5/2. In each case the surface density is assumed to be 2 × 10¹⁴ gm cm^-3. The solutions corresponding to K>0 (in fact K>1) are also made meaningful by considering the hypersurfaces t= constant as 3-hyperboloid by replacing the parameter R ² by −R² in Vaidya-Tikekar formalism. The solutions for the later case are also expressible in terms of algebraic functions for K=2-(n/δ² > 1 (n being integer or zero and δ =1,2,3 4). The cases for which 0 < K < 1 do not possess negative energy density gradient and therefore are incapable of representing any physically plausible star model. In totality the article provides all the physically plausible exact solutions for the Buchdahl static perfect fluid spheres. This revised version was published online in July 2006 with corrections to the Cover Date.     

Satellite motion in a non-singular gravitational potential

We study the effects of a non-singular gravitational potential on satellite orbits by deriving the corresponding time rates of change of its orbital elements. This is achieved by expanding the non-singular potential into power series up to second order. This series contains three terms, the first been the Newtonian potential and the other two, here R ₁ (first order term) and R ₂ (second order term), express deviations of the singular potential from the Newtonian. These deviations from the Newtonian potential are taken as disturbing potential terms in the Lagrange planetary equations that provide the time rates of change of the orbital elements of a satellite in a non-singular gravitational field. We split these effects into secular, low and high frequency components and we evaluate them numerically using the low Earth orbiting mission Gravity Recovery and Climate Experiment (GRACE). We show that the secular effect of the second-order disturbing term R ₂ on the perigee and the mean anomaly are 4″.307×10⁻⁹/a, and −2″.533×10⁻¹⁵/a, respectively. These effects are far too small and most likely cannot easily be observed with today’s technology. Numerical evaluation of the low and high frequency effects of the disturbing term R ₂ on low Earth orbiters like GRACE are very small and undetectable by current observational means.     

A photometric study of BO Canum Venaticorum

Photometric BV light curves of BO CVn obtained in 1992 and new times of minima are presented. The primary minimum shows a transit, whereas the secondary minimum, shows an occultation. The system may be classified as an A‐type W UMa system. A complete study of minima allows one to detect a possibly increasing period by about 0.037 s/yr. This indicates that the conservative mass transfer rate from the less massive component to the more massive one is 1.57 10^—10M_⊙ /yr. Because of the variable period, the new ephemeris is determined for future observations. Using the Wilson‐Devinney code a simultaneous solution of the B and V light curves is also performed. The analysis shows that the system is in a contact configuration with q = 0.205 ± 0.001 and fillout factor (f) = 0.18, T₁ = 7240 K (fixed), T₂ = 7150± 10 K. The high orbital inclination i = 87°.54 ± 0.26 was con firmed by photometric observations of the secondary minimum.     

Scalar field as dark energy accelerating expansion of the Universe

The features of a homogeneous scalar field ϕ with classical Lagrangian L = ϕ_;i ϕ^;i /2 − V(ϕ) and tachyon field Lagrangian L = −V(ϕ)√1 − ϕ_;i ϕ^;i causing the observable accelerated expansion of the Universe are analyzed. The models with constant equation-of-state parameter w _de = p _de/ρ_de < −1/3 are studied. For both cases the fields ϕ(a) and potentials V(a) are reconstructed for the parameters of cosmological model of the Universe derived from the observations. The effect of rolling down of the potential V(ϕ) to minimum is shown. Published in Ukrainian in Kinematika i Fizika Nebesnykh Tel, 2008, Vol. 24, No. 5, pp. 345–359. The article was translated by the authors.     

Solution of Riemann–Hilbert problems for determination of new decoupled expressions of Chandrasekhar’s <Emphasis Type="Italic">X</Emphasis>- and <Emphasis Type="Italic">Y</Emphasis>-functions for slab geometry in radiative transfer

In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness can be expressed in terms of Chandrasekhar’s X- and Y-functions. The nonlinear nonhomogeneous coupled integral equations which the X- and Y-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann–Hilbert problems using Plemelj’s formulae. Solutions of those Riemann–Hilbert problems are obtained using the theory of linear singular integral equations. New forms of linear nonhomogeneous decoupled expressions are derived for X- and Y-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known N-function and two new unknown functions N ₁- and N ₂- in the complex plane for both nonconservative and conservative cases. The N ₁- and N ₂-functions are expressed in terms of the known N-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the H-function for a semi-infinite atmosphere are obtained as a limiting case.     

A proposed correction to the solar abundances of carbon and oxygen utilizing new and accurate theoretical forbidden transition probabilities

Previously published solar abundances of oxygen and carbon can be corrected to be logN(O) = 8.93 and logN(C) = 8.60 on the hydrogen log-scale when new accurate forbidden electric quadrupole transition probabilities A _Q(s^–1) are used. Such A _Q's, based on the new atomic structure and electron correlation theory, developed recently by Sinanolu and coworkers, are reported for the (¹ S ₀-¹ D ₂) lines of [C i], [N ii, [O i] and [O iii] and the (² P-² D) lines of [N i] and [O ii]. The available experimental values are also given for comparison.Work supported by Grant No. GP-29471 from the U.S. National Science Foundation.     

Bianchi Type I Magnetized String Cosmological Model in General Relativity

Bianchi Type I magnetized string cosmological model following the techniques used by Letelier and Stachel, is investigated. To get a determinate model, we assume a condition ∊ = λ (geometric string) where ∊ is the rest energy density, λ the string tension density and expansion (θ) is proportional to eigen value σ₁ ¹ of shear tensor (σ _j ⁱ ), which leads to A = ℓ (BC) ⁿ where A, B, C are metric potentials and ℓ and n are constants. The behaviour of the model in presence and absence of magnetic field is discussed. The physical and geometrical aspects of the model are also discussed.     

Third and fourth order resonances in Hamiltonian systems

The stability of the origin of an autonomous Hamiltonian system is investigated when the system possesses a third or fourth-order resonance.H ₂, the quadratic part ofH isH ₂=ⁿ _{i=1 i} J _i and the resonance condition is ⁿ _i=1 k _{i i} where thek 0,i = 1, 2, ...,n are the natural or fundamental frequencies. It is shown that the only case in which the origin can be unstable is ifk _i0,i=1,2,...,n. The condition for instability is then given in terms of the coefficients of the higher order terms in the Hamiltonian. The transfer of energy between modes is also investigated when a near-resonant condition exists.