The Geometry of Autonomous Metrical Multi-Time Lagrange Space of Electrodynamics

The paper contains a geometrization of the autonomous multi-time Lagrangian function of electrodynamics. We point out that this multi-time Lagrangian function comes from electrodynamics and the theory of bosonic strings.

Section 1 contains physical and geometrical aspects that motivates us to study the autonomous metrical multi-time Lagrangian space of electrodynamics, denoted EDML n p . Section 2 constructs the canonical nonlinear connection Γ and the generalized Cartan canonical Γ -linear connection of EDML n p . Section 3 describes the generalized Maxwell equations which govern the electromagnetic field of this space. The generalized Einstein equations of the gravitational h-potential of the autonomous metrical multi-time Lagrange space are written in Section 4. The generalized conservation laws of these equations will also be described.

Geometrical and physical aspects.
In the last thirty years, many geometrical models in Mechanics or Physics were based on the notion of ordinary Lagrangian. Thus, the geometrical concept of Lagrange space was introduced. The differential geometry of the Lagrange spaces is now considerably developed and used in various fields to study the natural processes where the dependence on position, velocity or momentum are involved [2]. We recall that a Lagrange space L n = (M, L(x, y)) is defined as a pair which consists of a real, n-dimensional manifold M coordinated by x = (x i ) i=1,n and a regular Lagrangian L : T M → R (i.e., the fundamental metrical dtensor g ij (x, y) = (1/2)(∂ 2 L/∂y i ∂y j ) is of rank n and has a constant signature on T M\{0}). We point out that the Lagrangian L is not necessarily homogeneous with respect to the direction y = (y i ) i=1,n .
An important and well-known example of Lagrange space comes from electrodynamics. We recall that the Lagrangian L : T M → R which governs the movement law of a particle of mass m ≠ 0 and electric charge e, placed concurrently into a gravitational field and an electromagnetic one, is given by where the semi-Riemannian metric ϕ ij (x) represents the gravitational potentials of the space M, A i (x) are the components of a covector field on M representing the electromagnetic potentials, U(x) is a function on M which is called potential function, and c is the physical constant of light speed. It is obvious that L is a regular Lagrangian and, consequently, the pair L n = (M, L(x, y)) is a Lagrange space, which is called the Lagrange space of electrodynamics. At the same time, there are many problems in Mechanics that rely on the notion of time-dependent Lagrangian. A geometrization of time-dependent Lagrangians was also constructed in [2], the authors developing a classical rheonomic mechanics. From their point of view, a time-dependent Lagrangian is a function L : R × T M → R, where the product manifold R×T M is regarded as a vector bundle over the base space M. In this approach, the bundle of configurations of the classical rheonomic mechanics is Obviously, the gauge group (1.3) ignores the temporal reparametrizations, standing out by the absolute character of the temporal coordinate t.
In the classical rheonomic mechanics, a central role is played by the time-dependent Lagrangian of classical rheonomic electrodynamics, whose expression is The differential geometry induced by the classical time-dependent Lagrangian of electrodynamics (1.4) is found in [2]. In contrast, a geometrization of time-dependent Lagrangians, in a relativistic approach, or, in other words, a relativistic rheonomic mechanics, was created by Neagu [4], considering the bundle of configurations represented by the jet fibre bundle of order one where (t, x i ,y i ) are the coordinates on J 1 (R,M). It is obvious that the form of this gauge group (1.6) is more general than that used in the classical rheonomic mechanics and emphasizes the relativistic character of the temporal coordinate t.
According to Olver terminology [1], the relativistic rheonomic mechanics relies on the notion of Lagrangian ᏸ on J 1 (R,M) as a local function on the 1-jet space, which transforms by the ruleᏸ = ᏸ|dt/dt|. Like a distinct notion, the concept of Lagrangian function L : J 1 (R,M) → R is also involved in relativistic rheonomic mechanics. Remark 1.1. It is important to note the difference between the notions of the Lagrangian used in both relativistic and classical rheonomic mechanics. From this point of view, the reader is invited to compare them, following the expositions done in [2,4].
We emphasize that, in the relativistic rheonomic mechanics, a basic role is played by the following Lagrangian of relativistic rheonomic electrodynamics, where ψ 11 is a semi-Riemannian metric on R, A The differential geometry generated by this Lagrangian is exposed in [4].
To become general, consider the jet fibre bundle of order one [7] where T is a real, p-dimensional manifold coordinated by t = (t α ) α=1,p , whose physical meaning is that of "multi-time," M is a real, n-dimensional "spatial" manifold coordinated by x = (x i ) i=1,n , while the coordinates x i α have the meaning of partial directions or partial derivatives.
We should like to underline that the jet fibre bundle of order one J 1 (T , M) is a basic object in the study of classical and quantum field theories [8]. From a physical point of view, the 1-jet fibre bundle J 1 (T , M) → T ×M can be regarded as a bundle of configurations, in mechanics terms, because, considering the particular case of the temporal manifold T = R (i.e., the usual time axis represented by the set of real numbers), we recover the bundle of configurations (1.5) from relativistic rheonomic mechanics.
It is well known that a lot of problems in Physics and Variational Calculus rely on multi-time Lagrangian functions L depending on first order partial derivatives, which are viewed as functions defined on the total space of the 1-jet fibre bundle J 1 (T , M). A well-known example, which comes from Physics, is given by the "energy" Lagrangian function L used in the Polyakov model of bosonic strings, where ψ αβ (t) (resp., ϕ ij (x)) is a semi-Riemannian metric on the manifold T (resp., M). We recall that the extremals of the Lagrangian ᏸ = L |ψ| are exactly the harmonic maps between the semi-Riemannian spaces (T , ψ) and (M, ϕ).
In this context, a geometrization of a multi-time Lagrangian function L : J 1 (T , M) → R is imposed. Recently, a differential geometry, attached to certain multi-time Lagrangian functions, was created in [6]. In order to present the main concept of this geometry, fix a semi-Riemannian metric ψ = ψ αβ (t γ ) on the temporal manifold T . The fundamental geometrical concept used in the geometrization of a multi-time Lagrangian function is that of metrical multi-time Lagrange space, represented by a pair , symmetric, of rank n and having a constant signature. We point out that the differential geometry of metrical multi-time Lagrange spaces is now considerably developed in [5,6].
By a natural extension of previous examples of Lagrangian functions, we can give a very important example of metrical multi-time Lagrange space, considering the general Lagrangian function L which comes from electrodynamics and theory of bosonic strings, namely, Now, in order to unify all Lagrangian entities exposed above, we introduce the following geometrical concept.
where h αβ (t γ ) (resp., g ij (x k )) is a semi-Riemannian metric on the temporal (resp., spatial) manifold T (resp., M), U Remark 1.3. The nondynamical character (i.e., the independence with respect to the temporal coordinates) of the spatial metric g ij (x k ) determined us to use the terminology of autonomous in the previous definition.
The aim of this paper is to develop the differential geometry and the abstract field theory on EDML n p , in the sense of d-connections, d-torsions, d-curvatures, generalized Maxwell equations and generalized Einstein equations.

The geometry of autonomous metrical multi-time Lagrange space of electrodynamics EDML n p .
In this section, we will apply the general geometrical development of a metrical multi-time Lagrange space [6], to the particular space of electrodynamics EDML n p . To begin this development, consider the energy action functional associated to the multi-time Lagrangian of electrodynamics namely, where the temporal manifold T is considered compact and orientable, the local expression of the smooth map f is (t α ) → (x i (t α )) and x i α = ∂x i /∂t α . In this context, a general result from [6] implies the following result.

In other words, these extremals verify the generalized harmonic map equations attached to the multi-time dependent spray (H, G),
(2.8) Following the general exposition from [6], by a direct calculation, we can determine the adapted components of the generalized Cartan canonical connection of the autonomous metrical multi-time Lagrange space of electrodynamics, together with its torsion and curvature adapted local d-tensors.
where H γ µαβ (resp., r m ijk ) are the local curvature tensors of the semi-Riemannian metric h αβ (resp., g ij ) and " |i " represents the local spatial horizontal covariant derivative induced by the generalized Cartan connection (see [6]).
(iii) The curvature R of the generalized Cartan canonical connection of the autonomous metrical multi-time Lagrange space of electrodynamics is determined by two local adapted d-tensors, namely, H η αβγ and R l ijk = r l ijk , that is, exactly the curvature tensors of the semi-Riemannian metrics h αβ and g ij .

Generalized Maxwell equations on EDML n p .
To describe the generalized electromagnetism theory on the autonomous metrical multi-time Lagrange space, consider the canonical Liouville d-vector field C = x i α ∂/∂x i α on J 1 (T , M), and construct the metrical deflection d-tensors [5] where " /β ", " |j ", and "| (β) (j) " are the local covariant derivatives induced by the generalized Cartan canonical connection CΓ (see also [6]).
Taking into account the general expressions of the local electromagnetic d-tensors of a metrical multi-time Lagrange space [5], by a direct calculation, we deduce the following result.
Particularizing the generalized Maxwell equations of multi-time electromagnetic field, described in the general context of a metrical multi-time Lagrange space [5], we deduce the main result of the generalized electromagnetism on the autonomous metrical multi-time Lagrange space of electrodynamics.
where Ꮽ {i,j} represents an alternate sum and {i,j,k} means a cyclic sum.  Taking into account the expressions of the local curvature d-tensors of the generalized Cartan connection of EDML n p , by computations, we deduce the following theorem.

Generalized Einstein equations and conservation laws on
Theorem 4.1. The Ricci d-tensor Ric(CΓ ) of the autonomous metrical multi-time Lagrange space of electrodynamics is characterized by two effective adapted local Ricci d-tensors, namely, H αβ and R ij = r ij , where H αβ (resp., r ij ) are the local Ricci tensors associated to the semi-Riemannian metric h αβ (resp., g ij ).
stress-energy d-tensormust verify the local generalized conservation laws -B A|B = 0, for all A ∈ α, i, (α) (i) , where -B A = G BD -DA . In this context, by direct computations, we obtain the following result. where H µ β = h µν H νβ and r m j = g ms r sj .
Remark 4.6. Taking into account the componentsαβ andij of the new stressenergy d-tensor-appeared in the classical form (4.8) of the generalized Einstein equations, the generalized conservation laws modify in the classical form.