Three-dimensional Minkowski Space Research Articles

Conoid Surfaces Family in Minkowski 3-Space

This study focuses on exploring a distinct family of conoid surfaces in the three-dimensional Minkowski space L3. Our main objective is to delve into the differential geometry of this family, analyzing its curvatures in detail. Furthermore, we establish the essential conditions for achieving minimality within this specific framework. Additionally, we calculate the Laplace−Beltrami operator for this family of surfaces and illustrate our findings through an example.

Magnetic spherical indicatricies in Minkowski 3-space

In this paper, we examine curves drawn by a timelike or spacelike curve[Formula: see text] on the pseudo-hyperbolic sphere [Formula: see text] and the unit Lorentzian sphere [Formula: see text]in three-dimensional Minkowski space [Formula: see text] We present Lorentz forces and magnetic curves produced by the Frenet frame [Formula: see text] on spherical indicatricies in a magnetic field of the curve [Formula: see text]. We calculate magnetic vector fields of magnetic curves for [Formula: see text]. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatricies. We obtain developability conditions for these surfaces. Finally, we illustrate magnetic flux surfaces with some interesting examples.

Krivulje u 3-dimenzionalnom Minkowskijevom prostoru

In this paper curves in threedimensional Minkowski space were analyzed and the main differences in local theory of curves in Euclidean and Minkowski space were emphasized. Special attention is paid to curves with no Euclidean counterpart. There are numerous examples of studied curves whose graphic representations were made by Mathematica software.

A family of non-modular covariant AQFTs

Based on the construction provided in our paper “Covariant homogeneous nets of standard subspaces”, Comm Math Phys 386:305–358, (2021), we construct non-modular covariant one-particle nets on the two-dimensional de Sitter spacetime and on the three-dimensional Minkowski space.

CONSTANT ANGLE RULED SURFACES DUE TO THE BISHOP FRAME IN MINKOWSKI 3-SPACE

In this present study, constant angle ruled surfaces are examined by using Bishop frame in three-dimensional Minkowski space. These surfaces are studied and classified based on the constant angle property of the surface in this research. Particularly, our analysis shows that although they are flat and so Weingarten surfaces, but they are not minimal ones.

Classification of the Orthogonal Separable Webs for the Hamilton-Jacobi and Klein-Gordon Equations on 3-Dimensional Minkowski Space

We review a new theory of orthogonal separation of variables on pseudo-Riemannian spaces of constant zero curvature via concircular tensors and warped products. We then apply this theory to three-dimensional Minkowski space, obtaining an invariant classification of the forty-five orthogonal separable webs modulo the action of the isometry group. The eighty-eight inequivalent coordinate charts adapted to the webs are also determined and listed. We find a number of separable webs which do not appear in previous works in the literature. Further, the method used seems to be more efficient and concise than those employed in earlier works.

Nontrivial Isometric Embeddings for Flat Spaces

Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be required to have the same symmetry as the metric. On the other hand, it is possible to require the embedding to be unfolded so that the surface in the ambient space would occupy the subspace of the maximum possible dimension. In the weak gravitational field limit, such a requirement together with a large enough dimension of the ambient space makes embedding gravity equivalent to general relativity, while at lower dimensions it guarantees the linearizability of the equations of motion. We discuss symmetric embeddings for the metrics of flat Euclidean three-dimensional space and Minkowski space. We propose the method of sequential surface deformations for the construction of unfolded embeddings. We use it to construct such embeddings of flat Euclidean three-dimensional space and Minkowski space, which can be used to analyze the equations of motion of embedding gravity.

On the Schrödinger map for regular helical polygons in the hyperbolic space

The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the Euclidean space, a nonzero torsion now implies two different helical curves. This generalizes recent works by the author with de la Hoz and Vega on helical polygons in the Euclidean space as well as planar polygons in the Minkowski space. Numerical experiments in this article show that the trajectory of the point X(0, t) exhibits new variants of Riemann’s non-differentiable function whose structure depends on the initial torsion in the problem. As a result, we observe that the smooth solutions (helices, straight line) in the Minkowski space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega.

Geometric kinematics of persistent rigid motions in three-dimensional Minkowski space

This paper investigates whether Lorentz geometry can be systematically implemented to study the geometric kinematics of persistent rigid motions in three-dimensional Minkowski space. The notion of persistence of a one-parameter rigid motion is identified by the property that the instantaneous twist of the motion has a constant pitch. The main difference between three-dimensional Euclidean and Minkowski spaces is that the pitch of twists will take three different values in Minkowski space depending on the causal character of the curve on which the motions are based. Furthermore, based on the fundamentals of screw theory, the paper establishes necessary and sufficient criteria for modeling the persistence of some significant frame motions, including Frenet–Serret, adapted frame, and Bishop motions. Then, the axode surfaces of these special motions and their geometric concepts are defined. Finally, for a thorough treatment of special frame motions in three-dimensional Minkowski space, this paper reveals some illustrative examples of persistent rigid motions and axode surfaces.

3d conformal fields with manifest sl(2, \u2102)

In the present paper we construct all short representation of so(3, 2) with the sl(2, ℂ) symmetry made manifest due to the use of sl(2, ℂ) spinors. This construction has a natural connection to the spinor-helicity formalism for massless fields in AdS4 suggested earlier. We then study unitarity of the resulting representations, identify them as the lowest-weight modules and as conformal fields in the three-dimensional Minkowski space. Finally, we compare these results with the existing literature and discuss the properties of these representations under contraction of so(3, 2) to the Poincare algebra.

Directional magnetic and electric vortex lines and their geometries in Minkowski space

In this study, we firstly introduce a different type of directional Fermi-Walker transportations along with vortex lines of a non-vanishing vector field in three-dimensional Minkowski space. Then we consider some geometric quantities, which are used to characterize vortex lines, in order to express angular velocity vector (Darboux vector) of the system in terms of these quantities. Later we present timelike directional magnetic vortex lines by computing the Lorentz force. Hence, we reach a remarkable relation between timelike directional magnetic vortex lines and angular velocity vector of vortex lines with a nonrotating frame in Minkowski space. We also determine the timelike directional electric vortex lines by considering the electromagnetic force equation. We finally investigate the conditions of being uniform for magnetic fields of timelike directional magnetic vortex lines and we improve such a remarkable approach to find the electromagnetic curvature which contains many geometrical features belonging to timelike directional magnetic and electric vortex line.

On cylindricity of submanifolds of nonnegative Ricci curvature in a Minkowski space

We consider Finsler submanifolds of nonnegative Ricci curvature in a Minkowski space which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag curvature in a three-dimensional Minkowski space which is not locally convex.

Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations

Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a Frenet null frame by a gauge transformation that belongs to the little group contained in the Lorentz group SO(2, 1) and provides an analog of the Hasimoto transformation (relating a parallel frame to a Frenet frame for curves in Euclidean space). The Cartan structure equations of the transformed frame are shown to encode a hereditary recursion operator giving a two-component generalization of the recursion operator of Burgers equation, as well as a generalization of the Cole-Hopf transformation. Three different hierarchies of integrable systems are obtained from the various symmetries of this recursion operator. The first hierarchy contains two-component Burgers-type and nonlinear Airy-type systems; the second hierarchy contains novel quasilinear Schrödinger-type (NLS) systems; and the third hierarchy contains semilinear wave equations (in two-component system form). Each of these integrable systems is shown to correspond to a geometrical flow of a family of elastic null curves in three-dimensional Minkowski space.

Entire surfaces of constant curvature in Minkowski 3-space

This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a consequence of our classification of surfaces with bounded prescribed Gaussian curvature, sometimes called the Minkowski problem, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Some applications to minimal Lagrangian self-maps of the hyperbolic plane are obtained.

Hamiltonian flows of Lorentzian polyhedra: Kapovich-Millson phase space and SU(1, 1) intertwiners

We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with N faces. Starting with the Schwinger representation of the su(1,1) Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for space-like vectors in the three-dimensional Minkowski space R1,2. Considering N copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with N faces whose normal vectors are space-like, up to Lorentz transformations. We identify a generating set of SU(1, 1)-invariant observables, whose Hamiltonian flows generate geometrical deformations of polyhedra. We distinguish area-preserving and area-changing deformations. We then show that the area-preserving observables form a glN(R) Lie algebra and that they generate a GLN(R) action on Lorentzian polyhedra at fixed total area. This action is cyclic and all Lorentzian polyhedra can be obtained from a totally squashed polyhedron (with only two non-trivial faces) by a GLN(R) transformation. All those features carry on to the quantum level, where quantum Lorentzian polyhedra are defined as SU(1, 1) intertwiners between unitary SU(1, 1)-representations from the principal continuous series. Those SU(1, 1)-intertwiners are the building blocks of spin network states in loop quantum gravity in 3 + 1 dimensions for time-like slicing, and the present analysis applies to deformations of the quantum geometry of time-like boundaries in quantum gravity, which is especially relevant to the study of quasi-local observables and holographic duality.

A Formulation of L-Isothermic Surfaces in Three-Dimensional Minkowski Space

The Cartan structure equations are used to study space-like and time-like isothermic surfaces in three-dimensional Minkowski space in a unified framework. When the lines of curvature of a surface constitute an isothermal system, the surface is called isothermic. This condition serves to define a system of one-forms such that, by means of the structure equations, the Gauss-Codazzi equations for the surface are determined explicitly. A Lax pair can also be obtained from these one-forms for both cases, and, moreover, a nonhomogeneous Schrödinger equation can be associated with the set of space-like surfaces.

Gauss Map and Its Applications on Ruled Submanifolds in Minkowski Space

We study ruled submanifolds in Minkowski space in regard to the Gauss map satisfying some partial differential equation. As a generalization of usual cylinders, cones and null scrolls in a three-dimensional Minkowski space, a cylinder over a space curve, a product manifold of a right cone and a k-plane, a product manifold of a hyperbolic cone and a k-plane which look like kinds of cylinders over cones in 3-space, and the generalized B-scroll kind in Minkowski space are characterized with the partial differential equation regarding the Gauss map, where k is a positive integer.

Timelike surfaces in Minkowski space with a canonical null direction

Given a constant vector field Z in Minkowski space, a timelike surface is said to have a canonical null direction with respect to Z if the projection of Z on the tangent space of the surface gives a lightlike vector field. For example in the three-dimensional Minkowski space: a surface has a canonical null direction if and only if it is minimal and flat. When the ambient has arbitrary dimension, if a surface has a canonical null direction and has parallel mean curvature vector then it is minimal. We give different ways for building these surfaces in the four-dimensional Minkowski space. On the other hand, we describe several properties in the non ruled general case in four-dimensional Minkowski space. One property is that the tangent part of Z is an asymptotic direction of the surface. We describe these surfaces in the ruled case in arbitrary dimension. Finally use the Gauss map for describe another properties of these surfaces in dimension four.

Computation of Smarandache curves according to Darboux frame in Minkowski 3-space

Abstract In this paper, we study Smarandache curves according to Darboux frame in the three-dimensional Minkowski space E 1 3 . Using the usual transformation between Frenet and Darboux frames, we investigate some special Smarandache curves for a given timelike curve lying fully on a timelike surface. Finally, we defray a computational example to confirm our main results.

Thomas precession for dressed particles

We consider a particle dressed with boundary gravitons in three-dimensional Minkowski space. The existence of BMS transformations implies that the particle’s wavefunction picks up a Berry phase when subjected to changes of reference frames that trace a closed path in the asymptotic symmetry group. We evaluate this phase and show that, for BMS superrotations, it provides a gravitational generalization of Thomas precession. In principle, such phases are observable signatures of asymptotic symmetries.