On the Theory of Gravitational Field in Finsler Spaces - II

Continuing the last paper [1], more detailed considerations are made on the spatial structure of the Finslerian gravitational field: firstly, a unified field between the external (x)-field and the internal (y)-field is constructed from a vector bundle-like standpoint, where the intrinsic behavior (i.e., δy) of the internal vector variable y is taken into account; secondly, the connection structure and the metrical structure are determined by setting the base and dual base properly in the unified field; thirdly, a compactification process of the internal field (i.e., a mapping process of the (y)-field on the (x)-field) is considered in order to realize a four-dimensional Finslerian structure.     

On the Theory of Gravitational Field in Finsler Spaces

Some structural considerations are made on the Finslerian gravitational field: A Finslerian metrical structure such as g_λχ(x, y) = γ_λχ(x) + h_λχ(x, y) is proposed, where γ_λχ denotes the Riemann metric of Einstein's gravitational field, while hλχ the Finsler metric induced by the Riemann metric h_ij(y) of the internal field; The intrinsic behaviour of the internal variable y, which is expressed as ?ⁱ = K(x, y) y^j in the internal field, is grasped by the Finslerian parallelism δyⁱ (=0), which is reflected in the spatial structure of the external gravitational field by the mapping relation δy^χ = e(x) δyⁱ. The whole metrical Finsler connection D for g_λχ(i.e., Dg_λχ = 0) is determined by taking account of the intrinsic behaviour δy^χ.     

Gravitational field equations based on Finsler geometry

The analysis of a previous paper (see Ref. 1), in which the possibility of a Finslerian generalization of the equations of motion of gravitational field sources was demonstrated, is extended by developing the Finslerian generalization of the gravitational field equations on the basis of the complete contractionK = K _lj ^lj of the Finslerian curvature tensorK _l ^j _hk (x, y). The relevant Lagrangian is constructed by the replacement of the directional variabley ⁱ inK by a vector fieldy ⁱ (x), so that the notion of osculation may be regarded as the key concept on which the approach is based. The introduction of the auxiliary vector fieldy ⁱ (x) is shown to be of physical significance, for the field equations refer not only to the proper field variables but also to a special coordinate system associated withy ⁱ (x) through the Clebsch representation of the latter. The status of the conservation laws proves to be similar to that in the theory of the Yang-Mills field. By choosing a special Finslerian metric function we elucidate in detail the structure of the field equations in the static case.     

Über die Schwere des Lichtes in der Newtonschen und in der Einsteinschen Gravitationstheorie

The gravity theories of Newton and Einstein are giving opposite sentences about the velocity of light in gravitational field. According to the Newtonian theory the velocity v in gravitational field is greater than the velocity c in a field-free space: v > c. According to general relativity theory we have a smaller velocity: v < c. For a spherical symmetric gravitational field Newton's theory gives \documentclass{article}\pagestyle{empty}\begin{document}$ v \approx c\left({1 + \frac{{fM}}{{c^2 r}}} \right) $\end{document} but Einstein's theory of 1911 gives \documentclass{article}\pagestyle{empty}\begin{document}$ v \approx c\left({1 - \frac{{fM}}{{c^2 r}}} \right) $\end{document} and general relativity gives \documentclass{article}\pagestyle{empty}\begin{document}$ v \approx c\left({1 - 2\frac{{fM}}{{rc^2 }}} \right) $\end{document}. Therefore, the radarecho-measurations of Shapiro are the experimentum crucis for Einstein's against Newton's theory.     

X-ray,electron and neutron diffraction studies of polyacetylene tetrachloroferrate {CH(FeCl4)y}x

X-ray, electron and neutron diffraction studies of {CH(FeCl₄)_y}_x and {CD(FeCl₄)_y}_x are presented. The results suggest that the doping to the metallic regime results in the formation of a{CH(FeCl₄)_0.08}_x phase which corresponds to a partly intercalated, first-stage structure. For y < 0.08 significant inhomogeneity of the system can be observed as manifested by the coexistence of regions of partially isomerized polyacetylene and regions of {CH(FeCl₄)_0.08}_x.     

Connection Considerations of Gravitational Field in Finsler Spaces

Some alternative connection structures of the Finslerian gravitational field are considered by modifying the independent variables (x,y) (x: point and y: vector) in various ways. For example, (x ^k ,y ⁱ ) (k,i = 1,2,3,4) are changed to (x ^k ,y ⁰) (y ⁰: scalar) or (x ⁰,y ⁱ ) (x ⁰: time axis); (x ^k ,y ⁱ ) are generalized to (x ^k ,y ⁱ ,p _i ) (p _i : covector dual to y ⁱ ) or (x ^k ,y ⁱ ,q _a ) (q _a : covector different from p _i ); (x ^k ,y ⁱ ) are further generalized to (x ^k ,y ^(a)i ) (a = 1,2,…,m), (y ^(a): (a)th vector), etc.     

Can quantum gravitational forces stop gravitational collapse?

We consider, in lowest order of the gravitational coupling constant G, the gravitational potential between two neutrons. As we have previously pointed out [1],the quantum (including spin) contributions to the gravitational field dominate for distances smaller than the Compton wavelength of the neutron. At such distances the gravitational force between two neutrons may be repulsive. In particular, the gravitational forces which are analogous to the familiar Darwin and Fermi forces of quantum electrodynamics are capable of stopping gravitational collapse. Our discussion is within the framework of Einstein's theory, but on a microscopic level. We conclude that gravitational collapse may be halted without the necessity of extending Einstein's theory à la Cartan or otherwise.     

Einsteins hermitesche Relativitätstheorie als Unifikation von Gravo- und Chromodynamik

Einstein's Hermitian Theory of Relativity as Unification of Gravo- and Chromodynamics Einstein's Hermitian unified field theory is the continuation of the Riemannian GRG to complexe values with a Hermitian fundamental tensor gμ_v = g_v*μ This complexe continuation of GRG implies the possibility of matter and anti matter with a sort of CPT theorem. — Einstein himself has interpreted his theory as a unification and generalization of the Einstein and Maxwell theory, th. i. of gravodynamics and of electrodynamics. However — according the EIH approximation —, from Einstein's equations no Coulomb-like forces between the charges are resulting (INFELD, 1950). But, the forces between two charges ?_A and ?_B have the form (Treder 1957) It is interesting that such forces are postulated in the classical models of the chromodynamics of the interactions between quarks (for the confinement of their motions. If we interprete the purely imaginary part gμ_v of the hermitian metrics gμ_v=gμ_v+gμ_v as the dual of the field of gluons then, all peculiarities of Einstein's theory become physically meaningful. — Einstein's own interpretation suggests that the both long-range fields, gravitation and electromagnetism, must be unified in a geometrical field theory. However, the potential α/r + ε/2 has a “longer range” than the Coulomb potential ～1, and such an asymptotical potential ～ ε/2 is resulting from Einstein's equations (TREDER 1957). In Einstein's theory there are no free charges with \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_A^n {\varepsilon A} $\end{document}. (Wyman 1950) because the field mass of a charged particle becomes infinite asymptotically: That means, in a chromodynamics we dont's have free quarks. The same divergence are resulting from one-particle systems with non-vanishing total charges: M～ε²r. However, if the total charges vanish because in a domain ～L³ the positive sources are compensated by negative sources, the field masses of the n-charge systems become finite. From the gravitational part of Einstein's equations we get field masses which are the masses measured by observers in distances r ? L. That means, the masses of quark systems with the colour condition \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_A^n {\varepsilon A} $\end{document} are proportional to the linear dimension L of the system.     

The Sugawara model in general relativity. A. The case for three current vectors (SU (2))

A general method of solving the equations of Sugawara's field theory of currents has been developed, and illustrated by applying it to the set of three currents. These are inserted into Einstein's field equations which have been solved together with the co-variant ‘gauge’ conditions for a gravitational field involving cylindrical symmetry. A further transformation exhibits the triad formed by the current vectors and exhibits clearly the deviations of the line-element from Schwarzschild's exterior solution. In a subsequent paper the case for eight vector currents corresponding toSU (3) will be treated in similar fashion.     

Metric gravitation with a two parameter family of static spherically symmetric space-times

Among the variety of all conceivable metric theories of gravitation, Lorentz curvature dynamics is the most geometric extension of Einstein's field equations to fit the solar system data. In this framework two parameters determine the asymptotic form of a static spherically symmetric space-time (without imposing Einstein's conditions); these two parameters are the active gravitational mass of the source and the PPN parameter γ. The Lorentz connection is shown to satisfy covariant evolution equations which preserve either of these two parameters; furthermore, right and left oriented space-times differ in their Lorentz connection. Deviations from the Schwarzschild character find an interpretation in terms of a new object, the Lorentz curvature energy-momentum tensor, which always vanishes identically under the restriction of Einstein's conditions. These deviations contribute strongly to the gravitational force only in the neighbourhood of the Schwarzschild sphere.     

Some structurological remarks on a nonlocal field

In this paper, continued from the last paper (Ikeda, 1974), two kinds of structurological generalizations of our nonlocal field (i.e., the (x, ψ) field) are considered physicogeometrically. One is a Finslerian generalization, where the base field [i.e., the (x) field] is extended to a Finslerian field and Weyl's gauge field (i.e., the electromagnetic potential) is physically identified with the directional vector adopted as the internal variable in the ordinary nonlocal field theory. Another is a generalization by which the spinor (ψ) itself is taken as an independent variable, where some inherent characteristics ofψ are fused into the spatial structure. The latter is regarded as a “nonlocalization” of the (x) field accomplished by attachingψ to each point, in the true sense of the word. Particularly, the spatial structures of these generalized nonlocal fields are described in detail.     

Theory of Gravitational-Inertial Field of Universe

In this paper the basic proposition is a generalization of the metric tensor by introduction of an inertial field tensor satisfying ?_ig_lm ? g_lm;i ≠ 0. On the basis of variational equations a system of more general covariant equations of gravitational-inertial field is obtained. In Einstein's approximation these equations reduce to the field equations of Einstein. The solution of fundamental problems of generl taheory of relativity by means of the new equations give the same results as Einstein's equations. However application of these equations to the cosmologic problem leads to following results: 1. All Galaxies in the Universe (actually all bodies if gravitational attraction is not considered) “disperse” from each other according to Hubble's law. Thus contrary to Friedmann's theory (according to which the “expansion of Universe” began from the singular state with an infinite velocity) the velocity of “dispersion” of bodies begins from the zero value and in the limit tends to the velocity of light. 2. The “dispertion” of bodies represents a free motion in the inertial field and Hubble's law represents a law of motion of free bodies in the inertial field - the law of inertia. All critical systems (with Schwarzschild radius) are specific because they exist in maximal inertial and gravitational potentials. The Universe represents a critical system, it exists under the Schwarzschild radius. In the high-potential inertial and gravitational fields the material mass in a static state or in the process of motion with decelleration is subject to an inertial and gravitational “annihilation”. Under the maximal value of inertial and gravitational potentials (= c²) the material mass is completely “evaporated” transforming into a radiation mass. The latter is concentrated in the “horizon” of the critical system. All critical systems –“black holes”- represent geon systems, i.e., the local formations of gravitational-electromagnetic radiations, held together by their own gravitational and inertial fields. The Universe, being a critical system, is “wrapped” in a geon crown. The Universe is in a state of dynamical equilibrium. Near the external part of its boundary surface a transformation of matter into electromagnetic-gravitational-neutrineal energy (geon mass) takes place. Inside the Universe, in the galaxies takes place the synthesis of matter from geon mass, penetrating from the external part of the world (from geon crown) by means of a tunneling mechanism. The geon system may be considered as a natural entire cybernetic system.     

Über mögliche Grenzen einer Gravitationstheorie

On Possible Limits of a Gravitation Theory This work intends to answer the question why gravitational phenomena are described just by a theory which make use of a Riemannian metric tensor that has to be a solution of the known Einstein equations. The answer is thought of as a link from Maupertuis's principle to Einstein's equations, which appear as a transformed integrability condition. The metric tensor is first formally introduced and direction dependent (a Finslerian one) and then reduced to a Riemannian one by means of the condition of local Lorentz invariance. The construction seems to show that the gravity concept could not have a sense at the atomic level as a consequence of the central role which plays the particle-motion in the whole.     

The Riemann tensor and the Bianchi identity in 5D space–time

The initial assumption of theories with extra dimension is based on the efforts to yield a geometrical interpretation of the gravitation field. In this paper, using an infinitesimal parallel transportation of a vector, we generalize the obtained results in four dimensions to five-dimensional space–time. For this purpose, we first consider the effect of the geometrical structure of 4D space–time on a vector in a round trip of a closed path, which is basically quoted from chapter three of Ref. [5]. If the vector field is a gravitational field, then the required round trip will lead us to an equation which is dynamically governed by the Riemann tensor. We extend this idea to five-dimensional space–time and derive an improved version of Bianchi's identity. By doing tensor contraction on this identity, we obtain field equations in 5D space–time that are compatible with Einstein's field equations in 4D space–time. As an interesting result, we find that when one generalizes the results to 5D space–time, the new field equations imply a constraint on Ricci scalar equations, which might be containing a new physical insight.     

A background-dependent approach to the theory of gravitation II. Relativistic gravitational equations

On the basis of the results of Paper I and guided by a Machian view of nature, we find new gravitational equations which are background dependent. Such equations describe a purely geometrical theory of gravitation, and their dependence on the background structure is through the total energy-momentum tensor on the past sheet of the light cone of each space-time pointx [θ^μν x, say], i.e., through the integral on the past sheet of the light cone ofx of the parallel transport of the energy-momentum tensor from the space-time point in which it is defined tox along the geodesic connecting the two space-time points. Following Gürsey, we assume that the source of the De Sitter metric is not the cosmological term, but, rather, the energy-momentum tensor of a “uniform distribution of mass scintillations” [T ^μν x, say].T ^μν x, indeed, turns out to be equal to the metric tensor times a constant factor. As a consequence, in any local inhomogeneity A of a space-time whose background structure is determined by the Perfect Cosmological Principle,θ ^μν turns out to be approximately equal to the metric tensor times a constant factor, providedT=g _αβ T ^αβ is sufficiently small and the structure of the past sheet of the light cones of the space-time points belonging to Λ is not too much perturbed by the local gravitational field. As a consequence, in Λ the new equations approximately reduce to Einstein's equations. If one considers a “superuniverse model” in which our universe is considered as a local inhomogeneity in a De Sitter background, then from the above result there follows a fortiori the agreement of the new gravitational equations with the classical tests of gravitation. Furthermore, the dependence on the background structure is such that the new equations (i) incorporate the idea that the frame has to be fixeddirectly in connection with cosmological observations, and (ii) are singular in the absence of matter in the whole space-time. Moreover, (iii) the coupling constant turns out to be dimensionless in natural units (c=1=?), and (iv) a local inertial frame in a De Sitter background is determined by the condition that with respect to it the background structure is homogeneous in space and in time and is Lorentz invariant.     

Electro-geometrodynamics

By distinguishing between the metric of a Riemannian geometry and the interval defining function it is demonstrated that both Einstein's gravitational field equations and Maxwell's electromagnetic field equations can be generated from a single geometry.     

The geometric nature of Lie and Noether symmetries

It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the form [(x)\ddot]ⁱ+G_jkⁱ[(x)\dot]^j[(x)\dot] ^k+f(xⁱ)=0{\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0} where f(x ⁱ ) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.     

Gravitation and Riemannian space

The field equations of the quadratic action principle of relativity are solved, assuming a weak perturbation of the basic structure, which is a highly agitated Riemannian lattice field of a very small lattice constant. A field emerges which can be interpreted as the weak gravitational field of an apparently Minkowskian space. This field does not coincide with Einstein's theory of weak gravitational fields. Whereas the redshift remains unchanged, the light deflection becomes reduced by11.1% of the value predicted by Einstein.     

Molecular orbital study of germanium,germanium-gallium and germanium-arsenic clusters

The quasirelativistic INDO/1 method has been used to generate molecular orbitals for some {Ge _x }, doped {Ge _x Ga _y } and {Ge _x As _y } clusters. These one-electron energy levels predefine the density of states (DOS) and/or hole functions. The effect of the cluster size (x=24, 56, 92) and that of dopants on the DOS profiles are discussed. The calculations are compared with those generated by periodic crystal orbitals of the EHT quality.