On the Theory of Gravitational Field in Finsler Spaces - III

From the vector bundle-like standpoint, the Finslerian gravitational field is regarded as the total space of the vector bundle whose fibre is the internal (y)-field spanned by vectors {y} (i.e., the so-called internal space spanned by {y}) and whose base is the external (x)-field spanned by points {x} (i.e., the Einstein's gravitational field). Along this line, in this paper, different from a previous paper [1], the so-called mapping process of the (y)-field on the (x)-field is not taken into account and following Miron's method [2, 3], the Finslerian field equations will be derived from the Einstein's field equation for the total space. Some physical considerations will be made on those field equations.     

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.     

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.     

On weyl's gauge field in a non-local field

In this paper, a non-local field (i.e. the (x, ψ)-field) is constructed by regarding the spinor (ψ) as the internal freedom attached to each point (x). Since this field is likened to a unified field between the (x)- and (ψ)-fields, the metric is given bydσψ=gλ dx ^λψ. Concerning this, some conformally equivalent relations are considered. Next, Weyl's gauge field is introduced into the concept of connection in order to consider the gauge invariance. Finally, some essential features underlying our non-local field are grasped by formulating some fundamental equations of the spin curvature tensors.     

A differential geometrical consideration on a “nonlocal” field

In this paper, the concept of “nonlocality”, which is carried by the internal variable (ω) ascribed to each point (x), is first investigated generally in connection with the concept of “unified” gauge field. Next, as a typical example, a Finslerian “nonlocal” field is considered, which is obtained by taking for ω a vector (ξ), and two examples are proposed: one is a unified field with Randers metric and the other is a micro-gravitational field endowed with space-time fluctuation.     

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.     

Experimental Test of the Evans’ <Emphasis Type="Italic">B</Emphasis>(3)-Field: Measuring the Interaction with Free Electrons

During the past decade, M.W. Evans and his coworkers have been developing so-called “Evans” or “ECE theory” that intends to serve as an unified field theory. One of its predictions is an existence of a radiation magnetic field called a “B(3)-field” which should accompany a circularly polarized electromagnetic radiation. This field should affect free electrons in two ways: (1) the electrons should behave in the B(3)-field in the same way as in a classical magnetic field (i.e., Larmor precession) and moreover, (2) the electrons should undergo quantum interaction with the B(3)-field with the formation of a spin connection resonance. This paper presents an experimental test of the B(3)-field existence by observing the changes in trajectories of free electrons in special detector, when strong (up to 200 W/m²) continuous circularly polarized microwave radiation of a frequency of 2.45 GHz is applied. We have not detected any sign of B(3)-field in presented experiment. It follows that if the B(3)-field really exists, it should be at least 4 orders of magnitude smaller than the Evans’ theory predicts.     

Effects of notches on the order of flux-closure state formation in bi-rings by micromagnetic simulation

The effects of the number and the location of notches on the formation of flux-closure states in bi-rings with fields applied in the x direction (i.e., along the short axis direction of bi-rings) and y direction (i.e., along the long axis direction of bi-rings) are investigated using micromagnetic simulation. For the bi-rings with one notch and the bi-rings with two notches symmetric about y axis, the order of flux-closure state formation in each ring can be controlled. But the flux-closure state forms simultaneously in each ring for the bi-rings with two notches symmetric about x axis. For the bi-rings with two notches that are symmetric neither about x axis nor about y axis, only one ring can form a flux-closure state in the y-direction field and no flux-closure state can be found in rings in the x-direction field. Therefore, logic states can be defined by controlling the order of flux-closure state formation, which can be utilized in future logic devices.     

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.     

Anisotropic strain effects on light emission characteristics of CdZnO/ZnO quantum well structures

Anisotropic strain effects by strain relaxation on TE-polarized light emission characteristics of c-plane CdZnO/ZnO quantum well (QW) structures were theoretically investigated by using the multiband effective-mass theory. The CdZnO/ZnO QW structure with anisotropic strain has much larger emission intensity than conventional CdZnO/ZnO QW structure without the strain relaxation. In the case of the strain relaxation along x(or y)-direction, the x(or y)-polarized light emission is observed to be larger than the x(or y)-polarized light emission. In particular, in the case of the strain relaxation along both x- and y-directions, the increase in the spontaneous emission peak is significant. This can be explained by the fact that the internal field is reduced owing to the decrease in the piezoelectric field by the strain relaxation.     

Transport properties of a and c axis oriented (Y1?xCax)Ba2Cu3Oy thin films

Anisotropy and Hall effect measurements have been performed in calcium-doped, i.e., overdoped YBa₂Cu₃O_y ((Y_1−xCa_x)Ba₂Cu₃O_y) thin films witha andc axis orientations. In highly overdoped films (x=0.4), the anisotropy of the normal resistivity decreases and a drastic change in Hall conductivity in the mixed state is observed. The change in Hall conductivity in the overdoped region is consistent with recent experimental results for La_2−xSr_xCuO₄ films and seems to be common in highT _c superconductors.     

High-temperature impedance and modulus spectroscopy characterization of La/Mn modified PbTiO3 nanoceramics

Nanocrystalline samples of Pb_1−yLa_y(Ti_1−xMn_x)_(1−y/4)O₃ (PLMT) (y=0.06, x=0, 0.04, 0.07 and 0.10) were prepared by mechanical activation process (i.e., ball milling) followed by some annealing. The formation of single phase tetragonal crystal structure is confirmed by high-resolution X-ray diffraction study and by High resolution transmission electron micrographs (HRTEM), nano-scale compounds. The electrical behavior (i.e., impedance (Z) and electrical modulus (M)) of PLMT ceramics was studied by impedance spectroscopy technique in high temperature range. This study was carried out by means of the simultaneous analysis of the complex impedance (Z^?) and electrical modulus (M^*) functions in a wide frequency range (1 kHz-1 MHz). Impedance analysis has shown the grain and grain boundary contributions by an equivalent circuit model. Modulus analysis has provided vast information on charge transport processes. The simultaneous representation of the imaginary part of impedance and electric modulus (Z″, M″) vs. frequency revealed the localization of relaxation. The activation energy obtained from relaxation data may be attributed to oxygen ion vacancies.     

Noncentrosymmetric cubic helical ferromagnets Mn1 ? yFeySi and Fe1 ? xCoxSi

Two systems of noncentrosymmetric cubic helical magnets Mn_{1 − y} Fe _y Si (y = 0.06, 0.08, 0.10) and Fe_{1 − x} Co _x Si (x = 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.50) have been compared. The concentration dependences of the critical temperature and magnetic field have been obtained using small-angle polarized-neutron scattering and analyzed in the framework of the Bak-Jensen model. It has been established that, among the two interactions that play the main role in these systems, i.e., the isotropic symmetric ferromagnetic exchange and the Dzyaloshinskii-Moriya isotropic antisymmetric interaction, the former interaction determines the critical temperature in the Mn_{1 − y} Fe _y Si system and the latter interaction determines this temperature in the Fe_{1 − x} Co _x Si system.     

Electron–hole symmetry and spin–orbit splitting in IV–VI asymmetric quantum wells

It is shown that, due to the electron–hole symmetry of the fundamental gap of the lead–salts (PbTe, PbSe and PbS), the Rashba spin splitting in their flat band asymmetric quantum wells is much reduced with the usual equal conduction and valence band-offsets. Different from the III–V case, we find that the important structure inversion asymmetry for the Rashba splitting in IV–VI quantum wells with different left and right barriers is not a material property (i.e., barrier height, effective mass or band gap) but results from the band alignment. This is shown by specific envelope function calculations of the spin-dependent subband structure of Pb_1−xEu_xTe/PbTe/Pb_1−yEu_yTe asymmetric quantum wells (x≠y), based on a simple but accurate four-band kp model for the bulk band structure near the gap, which takes into account band anisotropy, nonparabolicity and multi-valley effects.     

The emergence of a Kaluza-Klein microgenometry from the invariants of optimally Euclidean Lorentzian spaces

It is shown that relativistic spacetimes can be viewed as Finslerian spaces endowed with a positive definite distance (ω⁰, mod ωⁱ) rather than as pariah, pseudo-Riemannian spaces. Since the pursuit of better implementations of “Euclidicity in the small” advocates absolute parallelism, teleparallel nonlinear Euclidean (i.e., Finslerian) connections are scrutinized. The fact that (ω^μ, ω₀ ⁱ) is the set of horizontal fundamental 1-forms in the Finslerian fibration implies that it can be used in principle for obtainingcompatible new structures. If the connection is teleparallel, a Kaluza-Klein space (KKS) indeed emerges from (ω^μ, ω₀ ⁱ), endowed ab initio with intertwined tangent and cotangent Clifford algebras. A deeper level of Kähler calculus, i.e., the language of Dirac equations, thus emerges. This makes the existance of an intimate relationship between classical differential geometry and quantum theory become ever more plausible. The issue of a geometric canonical Dirac equation is also raised.     

Multi-affinity and multi-fractality in systems of chaotic elements with long-wave forcing

Multi-scaling properties in quasi-continuous arrays of chaotic maps driven by long-wave random force are studied. The spatial pattern of the amplitude X(x,t) is characterized by multi-affinity, while the field defined by its coarse-grained spatial derivative exhibits multi-fractality. The strong behavioral similarity of the X- and Y-fields respectively to the velocity and energy dissipation fields in fully-developed fluid turbulence is remarkable, still our system is unique in that the scaling exponents are parameter-dependent and exhibit nontrivial q-phase transitions. A theory based on a random multiplicative process is developed to explain the multi-affinity of the X-field, and some attempts are made towards the understanding of the multi-fractality of the Y-field. Received 16 November 1998     

Friedman-like Robertson–Walker model in generalized metric space-time with weak anisotropy

A generalized FRW model of space-time is studied, taking into consideration the anisotropic structure of fields which are depended on the position and the direction (velocity). The Raychaudhouri and Friedman-like equations are investigated assuming the Finslerian character of space-time. A long range vector field of cosmological origin is considered in relation to a physical geometry where the Cartan connection has a fundamental role. The Friedman equations are produced including extra anisotropic terms. The variation of anisotropy z _t is expressed in terms of the Cartan torsion tensor of the Finslerian manifold. A physical generalization of the Hubble and other cosmological parameters arises as a direct consequence of the equations of motion.     

通过掺杂V和F提高碳包覆的磷酸铁锂锂离子电池的电化学性能

在原位聚合合成方法的基础上,结合两步烧结过程制得LiFe_1-xV_x(PO₄)_(3-y)/₃F_y/C.V和F掺杂对碳包覆的磷酸铁锂材料的结构、形貌和电化学性能有影响.通过XRD、FTIR、SEM、充/放电测试和电化学阻抗谱对材料的结构、形貌和电化学性能进行了表征.结果表明,V和F的掺杂并没有破坏橄榄石结构中的LiFePO₄/C,但可以提高晶体结构的稳定,降低电荷的转移阻抗,提高锂离子扩散速度,改善了LiFePO₄/C材料的循环性能和高倍率性能.     

Analyses of intrinsic inhomogeneity and metal segregation in samples of Ag–Ge–Se glasses

Ge_ySe_(1−y) glasses are semiconductors but when Ag is added above certain threshold concentration, Ag_x[Ge_ySe_(1−y)]_(100−x) glasses behave as fast ionic conductors [Ureña et al., Solid State Ionics 176 (2005) 505]. This peculiar behavior may be attributed to the intrinsically inhomogeneous nature of these glasses where zones rich in metals coexist with zones of the host material. The conductivity transformation may be ascribed to the percolation of the Ag rich phase [Pradel et al., J. Phys.: Condens. Matter 15 (2003) S1561].Ag_x[Ge_0.25Se_0.75]_(100−x) glasses either massive or as films were obtained by melt quenching and pulsed laser deposition (PLD), respectively, in compositions belonging to the Se rich corner of the ternary phase diagram. Their amorphous nature and intermediate range order was checked employing X-ray diffractometry (XRD), the short range order was characterized by extended X-ray absorption fine structure (EXAFS) (Ge and Se K absorption edge) and their microstructure was characterized by scanning electron microscopy (SEM) and small angle X-ray scattering (SAXS).     

Anomalous dielectric nonlinearity and dielectric relaxation in <Emphasis Type="Italic">x</Emphasis>BST-(1−<Emphasis Type="Italic">x</Emphasis>) (LMT-LNT) ceramics

xwt%Ba_0.6Sr_0.4TiO₃-(1−x)wt%[0.4La (Mg_0.5Ti_0.5)O₃-0.6(La_0.5Na_0.5)TiO₃] (x=0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 0.95) ceramics were prepared via a traditional solid-state reaction route. Interesting anomalous dielectric nonlinearity (ADN)—permittivity increased with dc bias electric field (E-field), and low-temperature dielectric relaxation (LTDR) behaviors—were observed within a x range of 0.30∼0.70 for the first time. Based on our experimental facts, it was suggested that the LTDR was originated from a charge-associated process between electron-oxygen vacancy pairs during a thermal stimulation, while the ADN was related with a metastable state of polarized nano-regions (PNRs).