Autoparallel Curves Research Articles

Quantum geodesics in quantum mechanics

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus Ω1 depending on the Hamiltonian p2/2m + V(x), and a flat quantum connection ∇ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or “geodesics”) is exactly Schrödinger’s equation. The connection ∇ preserves a non-symmetric quantum metric given by the canonical symplectic structure lifted to a rank (0, 2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to obtain a novel flow generated by the Klein–Gordon operator on Minkowski spacetime with a background electromagnetic field, by formulating quantum “geodesics” on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include quantum geodesics that look like a relativistic free particle wave packet and a hydrogen-like atom.

A novel approach to autoparallels for the theories of symmetric teleparallel gravity

Although the autoparallel curves and the geodesics coincide in the Riemannian geometry in which only the curvature is nonzero among the nonmetricity, the torsion and the curvature, they define different curves in the non-Riemannian ones. We give a novel approach to autoparallel curves and geodesics for theories of the symmetric teleparallel gravity written in the coincident gauge. Then we apply our autoparallel equation to a Schwarzschild-type metric and give remarks about dark matter and orbit equation.

Demystifying autoparallels in alternative gravity

Autoparallel curves along with geodesic curves can arise as trajectories of physical test bodies. We explicitly derive autoparallels as effective post-Riemannian geometric constructs, and at the same time we argue \emph{against} postulating autoparallels as fundamental equations of motion for test bodies in alternative gravity theories.

Minimum Time Problem Controlled by Affine Connection

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.

Focusing conditions for extended teleparallel gravity theories

In the context of extended theories of teleparallel gravity f(T) we derive the focusing conditions for a one-parameter dependent congruence of timelike auto-parallels of the Levi-Civita connection. We also consider the f(T) field equations for a general metric tensor before moving on to consider a spatially flat Robertson-Walker space-time. Following this, we study the expansion rate for a one-parameter dependent congruence of timelike auto-parallel curves of the Levi-Civita connection. Given the fact that test particles follow auto-parallels of the Levi-Civita connection, the torsion-free Raychaudhuri equation is used in order to determine the desired focusing conditions. Finally we study the obtained focusing conditions for three f(T) paradigmatic cosmological models and discuss the satisfaction or violation of these conditions. Through this, we find f(T) models that allow for the weak and strong focusing conditions to be satisfied or violated. It is mentioned that this behaviour can also be found in the so-called f(R) and f(Q) theories.

All the trajectories of an extended averaged Hebbian learning equation on the quantum state space are the e-geodesics

In this paper, two families of trajectories on the quantum state space (QSS) originating from a synaptic-neuron model and from quantum information geometry meet together. The extended averaged Hebbian learning equation (EAHLE) on the QSS developed by the author and Yuya (Far East Journal of Applied Mathematics, vol.47, pp.149-167, 2010) from a Hebbian synaptic-neuron model is studied from a quantum-information-geometric point of view. It is shown that all the trajectories of the EAHLE are the e-geodesics, the autoparallel curves with respect to the exponential-type parallel transport, on the QSS. As a secondary outcome, an explicit representation of solution of the averaged Hebbian learning equation, the origin of the EAHLE, is derived from that of the e-geodesics on the QSS.

Fiber averaged dynamics associated with the Lorentz force equation

It is shown that the Lorentz force equation is equivalent to the auto-parallel condition $\,^L\nabla_{\dot{{x}}}\dot{{x}}=0$ of a linear connection $^L\nabla$ defined on a convenient pull-back vector bundle. By using a geometric averaging method, an associated {\it averaged Lorentz connection} $\langle\,^L\nabla\rangle$ and the corresponding auto-parallel equation are obtained. After this, it is shown that in the ultra-relativistic limit and for narrow one-particle probability distribution functions, the auto-parallel curves of $\langle\,^L\nabla\rangle$ remain {\it nearby} close to the auto-parallel curves of $^L\nabla$. Applications of this result in beam dynamics and plasma physics are briefly described.

SYMMETRIC TELEPARALLEL GRAVITY: SOME EXACT SOLUTIONS AND SPINOR COUPLINGS

In this paper, we elaborate on the symmetric teleparallel gravity (STPG) written in a non-Riemannian space–time with nonzero nonmetricity, but zero torsion and zero curvature. First, we give a prescription for obtaining the nonmetricity from the metric in a peculiar gauge. Then, we state that under a novel prescription of parallel transportation of a tangent vector in this non-Riemannian geometry, the autoparallel curves coincide with those of the Riemannian space–times. Subsequently, we represent the symmetric teleparallel theory of gravity by the most general quadratic and parity conserving Lagrangian with lagrange multipliers for vanishing torsion and curvature. We show that our Lagrangian is equivalent to the Einstein–Hilbert Lagrangian for certain values of coupling coefficients. Thus, we arrive at calculating the field equations via independent variations. Then, we obtain in turn conformal, spherically symmetric static, cosmological and pp-wave solutions exactly. Finally, we discuss a minimal coupling of a spin-1/2 field to STPG.

Autoparallel curves and Riemannian geodesics for materially uniform but inhomogeneous bodies

According to the theory of materially uniform but inhomogeneous bodies two geometric structures can be defined on the body manifold when a uniform reference is known. The first is given by the material connection and the second by the intrinsic Riemannian metric. Two important classes of curves related with these structures are the geodesic and the autoparallel curves. The goal of the present contribution is to define and characterize mechanically these kind of curves for materially uniform but inhomogeneous bodies. We propose the use of these curves for constructing the stress-free non-Euclidean material manifold that plays the role of the reference configuration for the dislocated problem. A generic scheme for the construction of this manifold based on geodesics/autoparallel curves is given as well as a discussion related with the field of internal stresses. Attention is then focused on a continuous distribution of edge dislocations. We solve numerically the geodesic equation that corresponds to a solid body with a continuous symmetry group. By using the norm for the dislocation density tensor we conclude that the higher the dislocation density the greater the deviation from the straight line is. For the same distribution of dislocations, but for a solid body with a discrete symmetry group, we solve analytically the autoparallel curves.

On Almost φ-Lagrange Spaces

We initiate a study on the geometry of an almost φ-Lagrange space (APL-space in short). We obtain the expressions for the symmetric metric tensor, its inverse, semispray coefficients, solution curves of Euler-Lagrange equations, nonlinear connection, differential equation of autoparallel curves, coefficients of canonical metrical d-connection, and h- and v-deflection tensors in an APL-space. Corresponding expressions in a φ-Lagrange space and an almost Finsler Lagrange space (AFL-space in short) have also been deduced.

On the structure of infinitesimal almost projective transformations in the tangent bundle of a general space of paths

We establish necessary conditions for a vector field on the tangent bundle of a general space of paths to be an infinitesimal almost projective transformation in the case when the tensor fields determining the complexes of autoparallel curves are Yano-Okubo-Kagan complete lifts of tensor fields from the base manifold.

Infinite-dimensional second order ordinary differential equations via [formula omitted

The vector bundle structure obtained on the second order (acceleration) tangent bundle T 2 M of a smooth manifold M by means of a linear connection on the base provides an alternative way for the study of second order ordinary differential equations on manifolds of finite and infinite dimension. Second order vector fields and their integral curves could provide a new way of representing and solving a wide class of evolutionary equations for states on Fréchet manifolds of sections that arise naturally as inequivalent configurations of a physical field. The technique is illustrated by examples in the framework of Banach and Fréchet spaces, and on Lie groups, in particular discussing the case of autoparallel curves, which include geodesics if the connection is induced by a Riemannian structure.

On the linear connection and curvature in Newtonian mechanics

The trajectories of a scleronomic, holonomic particle motion in an otherwise general force field are autoparallel curves in a linear connected, symmetric, ’’almost’’ semimetric space. The Riemann–Christoffel curvature tensor and its concomitants belonging to the dynamical affinity are defined, and the physical meaning is discussed.

On the Geodesic and Autoparallel Curves in a Generalized Metric Space

On the Geodesic and Autoparallel Curves in a Generalized Metric Space