We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin's maximum principle we proved that the equations of the horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general relativity theory. This is a Kaluza -- Klein problem of classical and quantum physics solved by methods of sub-Lorentzian geometry. We study the geodesics sphere which appears in a constant magnetic field and its singular points. Sufficiently long geodesics are not optimal solutions of the variational problem and define the nonholonomic wavefront. This wavefront is limited by a convex elliptic cone. We also study variational principle approach to the problem. The Euler -- Lagrange equations are the same as those obtained by the Pontryagin's maximum principle if the restriction of the metric tensor on the distribution is the same.
Deep Dive into The Equations of Motion of a Charged Particle in the Five-Dimensional Model of the General Relativity Theory with the Four-Dimensional Nonholonomic Velocity Space.
We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin’s maximum principle we proved that the equations of the horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general relativity theory. This is a Kaluza – Klein problem of classical and quantum physics solved by methods of sub-Lorentzian geometry. We study the geodesics sphere which appears in a constant magnetic field and its singular points. Sufficiently long geodesics are not optimal solutions of the variational problem and define the nonholonomic wavefront. This wavefront is limited by a convex elliptic cone. We also study variational principle approach to the problem. The Euler – Lagrange equations are the sam
Distribution on a smooth manifold M is a family of subspaces A(x) ⊂ T x M (of the tangent bundle of the manifold) with the same dimension, smoothly parametrized by the points of manifold [1,2,3,4]. For each x ∈ M on A(x) the bilinear form u, u x is defined which can be equally described by the metric tensor g ij (x). The metric tensor of a distribution smoothly depends of the point of the manifold. The absolutely continous path x(t) is called horizontal, iff x (t) ∈ A(x(t)) for almost any t. In sub-Riemannian geometry the metric tensor g ij (x) of the distribution A is positively defined. In this geometry we study the shortest horizontal curves connecting two given points. If the distribution is completely nonholonomic and the Riemannian manifold is full, then any two points of the manifold can be connected by the shortest horizontal geodesic. On manifolds and also on distributions with metric tensor the geodesic is the Euler -Lagrange solution for the length functional J(x(•), u(•)) = T t 0 u(t), u(t)
x(t) dt. In sub-Lorentzian geometry the metric tensor g ij (x) of the distribution A has the signature (+, -, . . ., -). In this geometry we study the longest horizontal curves connecting two given points [5]. The equations of motion of a charged particle in the electromagnetic and gravitational fields can be obtained as the solutions of the variational problem for the distribution [6]. Consider the 5-dimensional manifold M 5 with the 4-dimensional distribution A defined by means of the differential form ω(x) = 3 i=0 A i (x)dx i + dx 4 , where the covector (A i ) i=0,…, 3 is the 4-potential of the electromagnetic field. Consider the length functional
where L(x, u) = m u, u
is the pseudonorm of the vector u, m is the mass of the particle. Consider the problem of maximizing the length functional on the set of horizontal curves connecting two given points. Solution of this problem as we proove in this paper leads to equations d dt
where F jk are defined by (5). These equations are the same as the equations of motion of a particle with the charge p 4 in the electromagnetic and gravitational fields of the general relativity theory. This wording of the problem is different from the classical problem of physics [7,8] of minimizing the action functional on a 4-dimensional manifold:
where e is the charge of the particle, c is the speed of light. This paper is devoted to the problem of unification of electromagnetic, gravitational and other fundamental interactions [9,10,11].
In this model the particles are considered on the 5-dimensional manifold M 5 with the special smooth structure. Allowed are only smooth coordinate transformations for which ∂y i ∂x 4 = 0, i = 0, . . ., 3,
Components ∂y 4 ∂x i , i = 0, . . ., 3, can be interpreted as gauge transformations (9). Since the coordinate transformations are smooth, ∂ ∂x 4
∂y i ∂x j = ∂ ∂x j ∂y i ∂x 4 = 0, i, j = 0, . . ., 4. Therefore the cylindricity condition ∂v ∂x 4 = 0 for any tensor field v is invariant. In the tangent bundle T M the subspace Lin {∂ 4 } is invariant. In the cotangent bundle T * M the subspace Lin {dx 0 , dx 1 , dx 2 , dx 3 } is invariant. Assume that the metric tensor and the 4-potential of the electromagnetic field do not depend of x 4 . This condition is invariant at the coordinate transformations with the property (4). The 4-dimensional distribution A is the velocity space, i.e. the set of all possible velocity vectors of the particle. The metric tensor of the distribution A defines 4-dimensional convex elliptic cone, completely similar to the cone of future of the general relativity theory. Causal structure in the proposed theory is similar to the causal structure of the general relativity theory [12,13]. In the presence of the electromagnetic field the manifold A is nonholonomic.
The electromagnetic field is defined in physics by the antisymmetric tensor (F jk ) j,k=0,…,3
Here E is the 3-dimensional vector of the electric field tension, H is the 3-dimensional vector of the magnetic field tension. There is the 4-potential (A i ) i=0,…,3 which defines the tensor of the electromagnetic field:
4-Potential is not defined unambiguously. It is defined with an arbitrary gauge transformation
, where f is any smooth function. Let U be some coordinate neighbourhood on the smooth manifold M 5 . The 4-dimensional distribution A on U can be defined by the differential form
This definition is correct since at the coordinate transformation with the property (4) we have
Therefore the coordinate transformations on the 5-manifold lead to transformations of the 4-potential
that include 4-dimensional coordinate transformations of the general relativity theory and gauge transformations of the 4-potential of the electromagnetic field [6] as in [7]. Locally the distribution A can be defined by the basis vector fields e i = ∂ ∂x i -A i ∂ ∂x 4 , i = 0, . . ., 3. Vector fields on a manifold can be considered as operators of differentiation. During the
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