Arbitrary Vector Field Research Articles

On the geometrical properties of hypercomplex four-dimensional Lorentzian Lie groups

Abstract In this paper we first classify left-invariant generalized Ricci solitons on four-dimensional hypercomplex Lie groups equipped with three families of left-invariant Lorentzian metrics. Then, on these Lorentzian spaces, we explicitly calculate the energy of an arbitrary left-invariant vector field X and determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. Furthermore, we give a complete and explicit description of all homogeneous structures on these spaces in both Riemannian and Lorentzian cases and determine some of their types. The existence of Einstein four-dimensional hypercomplex Lorentzian Lie groups is proved and it is shown that although the results concerning Einstein-like metrics, conformally flatness and some equations in the Riemannian case are much richer than their Lorentzian analogues, in the Lorentzian case, there exist some new critical points of energy functionals, homogeneous structures and geodesic vectors which do not exist in the Riemannian case.

Scalar, vector and tensor harmonics on the flat compact orientable three-manifolds

Observations suggest that our universe is spatially flat on the largest observable scales. Exactly six different compact orientable three-dimensional manifolds admit flat metrics. These six manifolds are therefore the most natural choices for building cosmological models based on the present observations. This paper briefly describes these six manifolds and the harmonic basis functions previously developed for representing arbitrary scalar fields on them. The principal focus of this paper is the development of new harmonics for representing arbitrary vector and second-rank tensor fields on these manifolds. These new harmonics are designed to be useful tools for analyzing the dynamics of electromagnetic and gravitational fields on these spaces.

Classification of vector lines and fields by geometric and numerical methods

Geometric and numerical methods of classification of vector lines and vector fields are proposed. The division of vector lines of arbitrary vector fields into potential and spiral lines is carried out. The division of vortex vector fields into layered and spiral vector fields is carried out.It is shown that any vector field can be represented as a superposition of layered vector fields.Conditions of existence of so-called forceless spiral vector fields are established.The mathematical conditions to be satisfied by the magnetic fields of magnetic clouds of the solar wind are formulated.

Conservation Laws and Other Formulas for Families of Rays and Wavefronts and for the Eikonal Equation

In the previous studies, the author has obtained conservation laws for the 2D eikonal equation in an inhomogeneous isotropic medium. These laws are divergent identities of the form div F = 0. The vector field F is expressed through a solution to the eikonal equation (the time field), the refractive index (the equation parameter), and their partial derivatives. Besides that, equivalent conservation laws (divergent identities) were found for families of rays and families of wavefronts in terms of their geometric characteristics. Thus, the geometric essence (interpretation) of the conservation laws obtained for the 2D eikonal equation was found. In this paper, 3D analogs of the results obtained are presented: differential conservation laws for the 3D eikonal equation and conservation laws (divergent identities of the form div F = 0) for families of rays and families of wavefronts, the vector field F expressed through classical geometric characteristics of the ray curves: their Frenet basis (the unit tangent vector, principal normal, and binormal), the first curvature, and the second curvature, or through the classical geometric characteristics of the wavefront surfaces: their normal, principal directions, principal curvatures, Gaussian curvature, and mean curvature. All the results have been obtained on the basis of the general vector and geometric formulas (differential conservation laws and some formulas) obtained for families of arbitrary smooth curves, families of arbitrary smooth surfaces, and arbitrary smooth vector fields.

Conservation laws and the equivalence group

The paper generalizes differential conservation laws for the two-dimensional hydrodynamic Euler equations and their equivalent conservation laws for families of streamlines previously derived by the author. Three-dimensional analogues of these results, i.e., differential conservation laws for the three-dimensional hydrodynamic Euler equations, are presented. All conservation laws are divergent identities of the form div F = 0 and are found in two equivalent forms: in “mathematical-physical” form, where the vector field F is expressed in terms of quantities included in the Euler equations and their partial derivatives, and in geometric form. The second form represents conservation laws for families of streamlines where the vector field F under the divergence sign is expressed in terms of the classical geometric characteristics of curves — their Frenet unit vectors τ, ν, and β (the unit tangent, principal normal, and binormal vectors), the first curvature k, and the second curvature ϰ. Special attention is paid to the geometric interpretation of the obtained conservation laws and their constituent expressions. Issues related to the equivalence group of the eikonal equation and other equations are discussed.All results were obtained using the general vector and geometric formulas (differential conservation laws and other formulas), obtained by the author for families of arbitrary smooth curves, families of arbitrary smooth surfaces, and arbitrary smooth vector fields.

A non-trivial connection for the metric-affine Gauss–Bonnet theory in D = 4

We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all connections constructed by acting with a projective transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss–Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss–Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.

On the Geometry of Higher Dimensional Heisenberg Groups

In this paper, we first completely determine all left-invariant generalized Ricci solitons on the Heisenberg group \(H_{2n+1}\) equipped with any left-invariant Riemannian and Lorentzian metric that this Lie group admits. Then, we explicitly calculate the energy of an arbitrary left-invariant vector field V on these spaces and in the Lorentzian cases we determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also obtain all of the descriptions of their homogeneous Riemannian and Lorentzian structures and explicitly distinguish their types. Finally, we investigate parallel hypersurfaces of these spaces and show that these spaces never admit any totally geodesic hypersurface. The existence of algebraic Ricci solitons and the non-existence of left-invariant Ricci solitons and Yamabe solitons on these spaces in both Riemannian and Lorentzian cases is proved. Also, different behaviors regarding the existence of harmonic maps, critical points for the energy functional restricted to vector fields and some equations in Riemannian and Lorentzian cases are found.

Some globally conserved currents from generalized Killing vectors and scalar test fields

In this article, I discuss the construction of some globally conserved currents that one can construct in the absence of a Killing vector. One is based on the Komar current, which is constructed from an arbitrary vector field and has an identically vanishing divergence. I obtain some expressions for Komar currents constructed from some generalizations of Killing vectors which may in principle be constructed in a generic spacetime. I then present an explicit example for an outgoing Vaidya spacetime which demonstrates that the resulting Komar currents can yield conserved quantities that behave in a manner expected for the energy contained in the outgoing radiation. Finally, I describe a method for constructing another class of (non-Komar) globally conserved currents using a scalar test field that satisfies an inhomogeneous wave equation, and discuss two examples; the first example may provide a useful framework for examining the arrow of time and its relationship to energy conditions, and the second yields (with appropriate initial conditions) a globally conserved energy- and momentumlike quantity that measures the degree to which a given spacetime deviates from symmetry.

Tensorial neutron tomography of three-dimensional magnetic vector fields in bulk materials

Knowing the distribution of a magnetic field in bulk materials is important for understanding basic phenomena and developing functional magnetic materials. Microscopic imaging techniques employing X-rays, light, electrons, or scanning probe methods have been used to quantify magnetic fields within planar thin magnetic films in 2D or magnetic vector fields within comparable thin volumes in 3D. Some years ago, neutron imaging has been demonstrated to be a unique tool to detect magnetic fields and magnetic domain structures within bulk materials. Here, we show how arbitrary magnetic vector fields within bulk materials can be visualized and quantified in 3D using a set of nine spin-polarized neutron imaging measurements and a novel tensorial multiplicative algebraic reconstruction technique (TMART). We first verify the method by measuring the known magnetic field of an electric coil and then investigate the unknown trapped magnetic flux within the type-I superconductor lead.

Directionality fields generated by a local Hilbert transform

We propose a new approach based on a local Hilbert transform to design non-Hermitian potentials generating arbitrary vector fields of directionality, p(r), with desired shapes and topologies. We derive a local Hilbert transform to systematically build such potentials, by modifying background potentials (being either regular or random, extended or localized). In particular, we explore particular directionality fields, for instance in the form of a focus to create sinks for probe fields (which could help to increase absorption at the sink), or to generate vortices in the probe fields. Physically, the proposed directionality fields provide a flexible new mechanism for dynamically shaping and precise control over probe fields leading to novel effects in wave dynamics.

Note on the Noether charge and holographic transports

We clarify the relation between the Noether charge associated to an arbitrary vector field and the equations of motions by revisiting Wald formalism. For a time-like Killing vector, aspects of the Noether charge suggest that it is dual to the heat current in the boundary for general holographic theories. For a space-like Killing vector, we interpret the Noether charge (at the transverse direction) as shear stress of the dual fluid so we can compute the ratio of shear viscosity to entropy density by simply using the infrared data on the black hole event horizon. We test the new method for Einstein gravity and Gauss-Bonnet gravity and find that it produces correct results for both cases even in the presence of additional matter fields.

English

In this paper we first classify left-invariant generalized Ricci solitons on some solvable extensions of the Heisenberg group in both Riemannian and Lorentzian cases. Then we obtain the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and obtain all vector fields which are critical points for the energy functional restricted to vector fields of the same length. Furthermore, we determine all homogeneous Lorentzian structures and their types on these spaces and give a complete and explicit description of all parallel and totally geodesic hypersurfaces of these spaces. The nonexistence of harmonic maps in the non-abelian case is proved and it is shown that the existence of Einstein, Einstein-like metrics and some equations in the Riemannian case can not be extended to their Lorentzian analogues.

On the geometrical properties of hypercomplex four-dimensional Lie groups

Abstract In this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.

Nabla derivatives associated with nonlinear control systems on homogeneous time scales

The backward shift and nabla derivative operators, defined by the control system on homogeneous time scale, in vector spaces of one-forms and vector fields are introduced and some of their properties are proven. In particular the formulas for components of the backward shift and nabla derivative of an arbitrary vector field are presented.

Generation of arbitrary vector fields based on a pair of orthogonal elliptically polarized base vectors.

We present an arbitrary vector field with hybrid polarization based on the combination of a pair of orthogonal elliptically polarized base vectors on the Poincaré sphere. It is shown that the created vector field is only dependent on the latitude angle 2χ but is independent on the longitude angle 2ψ on the Poincaré sphere. By adjusting the latitude angle 2χ, which is related to two identical waveplates in a common path interferometric arrangement, one could obtain arbitrary type of vector fields. Experimentally, we demonstrate the generation of such kind of vector fields and confirm the distribution of state of polarization by the measurement of Stokes parameters. Besides, we investigate the tight focusing properties of these vector fields. It is found that the additional degree of freedom 2χ provided by arbitrary vector field with hybrid polarization allows one to control the spatial structure of polarization and to engineer the focusing field.

Spinor Lie derivatives and Fermion stress–energies

Stress–energies for Fermi fields are derived from the principle of general covariance. This is done by developing a notion of Lie derivatives of spinors along arbitrary vector fields. A substantial theory of such derivatives was first introduced by Kosmann; here, I show how an apparent conflict in the literature on this is due to a difference in the definitions of spinors, and that tracking the Lie derivative of the Infeld–van der Waerden symbol, as well as the spinor fields under consideration, gives a fuller picture of the geometry and leads to the Fermion stress–energy. The differences in the definitions of spinors do not affect the results here, but could matter in certain quantum-gravity programs and for spinor transformations under discrete symmetries.

Two-dimensional imaging in hyperbolic media-the role of field components and ordinary waves.

We study full vector imaging of two dimensional source fields through finite slabs of media with extreme anisotropy, such as hyperbolic media. For this, we adapt the exact transfer matrix method for uniaxial media to calculate the two dimensional transfer functions and point spread functions for arbitrary vector fields described in Cartesian coordinates. This is more convenient for imaging simulations than the use of the natural, propagation direction-dependent TE/TM basis, and clarifies which field components contribute to sub-diffraction imaging. We study the effect of ordinary waves on image quality, which previous one-dimensional approaches could not consider. Perfect sub-diffraction imaging can be achieved if longitudinal fields are measured, but in the more common case where field intensities or transverse fields are measured, ordinary waves cause artefacts. These become more prevalent when attempting to image large objects with high resolution. We discuss implications for curved hyperbolic imaging geometries such as hyperlenses.

Invariant conserved currents in generalized gravity

We study conservation laws for gravity theories invariant under general coordinate transformations. The class of models under consideration includes Einstein's general relativity theory as a special case as well as its generalizations to non-Riemannian spacetime geometry and nonminimal coupling. We demonstrate that an arbitrary vector field on the spacetime manifold generates a current density that is conserved under certain conditions, and find the expression of the corresponding superpotential. For a family of models including nonminimal coupling between geometry and matter, we discuss in detail the differential conservation laws and the conserved quantities defined in terms of covariant multipole moments. We show that the equations of motion for the multipole moments of extended microstructured test bodies lead to conserved quantities that are closely related to the conserved currents derived in the field-theoretic framework.

On the geometrical properties of solvable Lie groups

Abstract In [3] Bozek introduced a class of solvable Lie groups M2n+1. Calvaruso, Kowalski and Marinosci in [9] have studied homogeneous geodesics on these homogeneous spaces with an arbitrary odd dimension. In [1] we have studied some other geometrical properties on these spaces with dimension five. Our aim in this paper is to extend those geometrical properties for an arbitrary odd dimension in both Riemannian and Lorentzian cases. In fact we first obtain all of the descriptions of their homogeneous Lorentzian and Riemannian structures and their types. Then we calculate the energy of an arbitrary left-invariant vector field X on these spaces and in the Lorentzian case we prove that no left-invariant unit time-like vector fields on these spaces are critical points for the space-like energy. There is also a proof of non-existence of invariant contact structures and left-invariant Ricci solitons on these homogeneous spaces.

Metric Relativity and the Dynamical Bridge: Highlights of Riemannian Geometry in Physics

We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge. We describe the notion of equivalent (dragged) metric $ \widehat {g}_{\mu \nu }$ which is responsible to map the path of any accelerated body in Minkowski space-time onto a geodesic motion in such associated $\widehat {g}$ geometry. Only recently, the method introduced by Einstein in general relativity was used beyond the domain of gravitational forces to map arbitrary accelerated bodies submitted to non-Newtonian attractions onto geodesics of a modified geometry. This process has its roots in the very ancient idea to treat any dynamical problem in Classical Mechanics as nothing but a problem of static where all forces acting on a body annihilates themselves including the inertial ones. This general procedure, that concerns arbitrary forces—beyond the uses of General Relativity that is limited only to gravitational processes—is nothing but the relativistic version of the d’Alembert method in classical mechanics and consists in the principle of Metric Relativity. The main difference between gravitational interaction and all other forces concerns the universality of gravity which added to the interpretation of the equivalence principle allows all associated geometries—one for each different body in the case of non-gravitational forces—to be unified into a unique Riemannian space-time structure. The same geometrical description appears for electromagnetic waves in the optical limit within the context of nonlinear theories or material medium. Once it is largely discussed in the literature, the so-called analogue models of gravity, we will dedicate few sections on this emphasizing their relation with the new concepts introduced here. Then, we pass to the description of the Dynamical Bridge formalism which states the dynamic equivalence of nonlinear theories (driven by arbitrary scalar, spinor or vector fields) that occur in Minkowski background to theories described in associated curved geometries generated by each one of these fields. We shall see that it is possible to map the dynamical properties of a theory, say Maxwell electrodynamics in Minkowski space-time, into Born-Infeld electrodynamics described in a curved space-time the metric of which is defined in terms of the electromagnetic field itself in such way that it yields the same dynamics. It is clear that when considered in whatever unique geometrical structure, these two theories are not the same; they do not describe the same phenomenon. However, we shall see that by a convenient modification of the metric of space-time, an equivalence appears that establishes a bridge between these two theories making they represent the same phenomenon. This method was recently used to achieve a successful geometric scalar theory of gravity. At the end, we briefly review the proposal of geometrization of quantum mechanics in the de Broglie-Bohm formulation using an enlarged non-Riemannian (Weyl) structure.