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We discuss the generalized Lagrange structure of a deformed Minkowski space (DMS), , namely a (four-dimensional) generalization of the (local) space-time based on an energy-dependent “deformation” of the usual Minkowski geometry. In , local Lorentz invariance is naturally violated, due to the energy dependence of the deformed metric. Moreover, the generalized Lagrange structure of allows one to endow the deformed space-time with both curvature and torsion.

It is well known that symmetries play a basic role in all fields of physics. In particular, in relativity the most fundamental symmetry is local Lorentz invariance (LLI). Over the last two decades there have been tremendous interest and progress in testing LLI [

A formalism of this second kind is Deformed Special Relativity (DSR), namely a (four-dimensional) generalization of the (local) space-time structure based on an energy-dependent “deformation” of the usual Minkowski geometry [

The paper is organized as follows. In Section 2, we review the basic features of DSR that are relevant to our purposes. Lorentz violation in DSR is discussed in Subsect. 2.2. Section 3 deals with the generalized Lagrange structure of

The geometrical structure of the physical world-both at a large and a small scale—has been debated since a long. After Einstein, the generally accepted view considers the arena of physical phenomena as a four-dimensional space-time, endowed with a global, curved, Riemannian structure and a local, flat, Minkowskian geometry.

However, an analysis of some experimental data concerning physical phenomena ruled by different fundamental interactions have provided evidence for a local departure from Minkowski metric [

All the above facts suggested to introduce a (four-dimensional) generalization of the (local) space-time structure based on an energy-dependent “deformation” of the usual Minkowski geometry of M, whereby the corresponding deformed metrics ensuing from the fit to the experimental data seem to provide an effective dynamical description of the relevant interactions (at the energy scale and in the energy range considered).

An analogous energy-dependent metric seems to hold for the gravitational field (at least locally, i.e. in a neighborhood of Earth) when analyzing some classical experimental data concerning the slowing down of clocks.

Let us shortly review the main ideas and results concerning the (four-dimensional) deformed Minkowski spacetime

The four-dimensional “ deformed” metric scheme is based on the assumption that spacetime, in a preferred frame which is fixed by the scale of energy

with

Metric (1), (2) is supposed to hold locally, i.e. in the spacetime region where the process occurs. It is supposed moreover to play a dynamical role, and to provide a geometric description of the interaction considered. In this sense, DSR realizes the so called “Finzi Principle of Solidarity” between space-time and phenomena occurring in it^{1} (see [^{2}.

We notice explicitly that the spacetime

As far as phenomenology is concerned, it is important to recall that a local breakdown of Lorentz invariance may be envisaged for all the four fundamental interactions (electromagnetic, weak, strong and gravitational) whereby one gets evidence for a departure of the spacetime metric from the Minkowskian one (in the energy range examined). The explicit functional form of the metric (2) for all the four interactions can be found in [

1) Both the electromagnetic and the weak metric show the same functional behavior, namely

with the only difference between them being the threshold energy

2) for strong and gravitational interactions, the metrics read:

with

Let us stress that, in this case, contrarily to the electromagnetic and the weak ones, a deformation of the time coordinate occurs; moreover, the three-space is anisotropic^{3}, with two spatial parameters constant (but different in value) and the third one variable with energy in an “over-Minkowskian” way (namely it reaches the limit of Minkowskian metric for decreasing values of

As a final remark, we stress that actually the four-dimensional energy-dependent spacetime

Let us remark the mathematically self-evident, but physically basic, point that the generalized metric (2) (and the corresponding interval (1)) is clearly not preserved by the usual Lorentz transformations. If

(where

However, in DSR it is possible to introduce generalized Lorentz transformations which are the isometries of the deformed Minkowski space

and leaving the deformed interval (1) invariant, namely

Therefore, unlike the case of a standard LT, a deformed Lorentz transformation generates a similarity transformation which preserves the deformed metric tensor. Let us also notice the explicit dependence of

The explicit form of the deformed Lorentz transformations can be found in [

It is clear from the discussion of the phenomenological metrics describing the four fundamental interactions in DSR that the Minkowski space

where

However, let us notice that DSR can be considered as a metric gauge theory from another point of view, on account of the dependence of the metric coefficients on the energy. Actually, once the MGP has been applied, by selecting the suitable gauge (namely, the suitable functional form of the metric) according to the interaction considered (thus implementing the Finzi principle), the metric dependence on the energy implies another different gauge process. Namely, the metric is gauged according to the process under study, thus selecting the given metric, with the given values of the coefficients, suitable for the given phenomenon.

We have therefore a double metric gaugement, according, on one side, to the interaction ruling the physical phenomenon examined, and on the other side to its energy, in which the metric coefficients are the analogous of the gauge functions^{4}.

We want now to show that the deformed Minkowski space

Let us give the definition of generalized Lagrange space [

Consider a N-dimensional, differentiable manifold

has a fibre bundle structure. Let us denote by

Let

The natural basis of the tangent space

A local coordinate transformation in the differentiable manifold

Here,

On account of Equation (14), the natural basis of

Second Equation (15) shows therefore that the vector basis

If

then

Here,

The dual basis to the adapted basis is

A distinguished tensor (or d-tensor) field of (r,s)-type is a quantity whose components transform like a tensor under the first coordinate transformation (19) on

In particular, both

A generalized Lagrange space is a pair^{5} and of constant signature.

A function

differentiable on

A generalized Lagrange space

on^{6}, with

Of course,

Since, in general, a generalized Lagrange space is not reducible to a Lagrange one, it cannot be studied by means of the methods of symplectic geometry, on which—as is well known—analytical mechanics is based.

A linear

In terms of

The two derivatives

The coefficients of

By means of the connection

Here, the d-tensor

and is explicitly given by^{7}

The tensor

are the d-tensors of torsion of the metrical connection

From the curvature tensors one can get the corresponding Ricci tensors of

and the scalar curvatures

Finally, the deflection d-tensors associated to the connection

namely the hand v-covariant derivatives of the Liouville vector fields.

In the generalized Lagrange space

where

On the basis of the previous considerations, let us analyze the geometrical structure of the deformed Minkowski space of DSR^{8}

The derivatives

Then, it is possible to prove the following theorem [

The pair

Notice that such a result is strictly related to the fact that the deformed metric tensor of DSR is diagonal.

If an external electromagnetic field

where

namely, the connection coincides with the deformed field.

The adapted basis of the distribution

The local covector field of the dual basis (cfr. Equation (18)) is given by

The derivation operators applied to the deformed metric tensor of the space

Then, the coefficients of the canonical metric connection

The vanishing of the electromagnetic field tensor,

One can define the deflection tensors associated to the metric connection

The covariant components of these tensors read

It is important to stress explicitly that, on the basis of the results of 3.2.1, the deformed Minkowski space

Following ref. [

Consider the following metric

where

where the prime denotes derivative with respect to

According to the formalism of generalized Lagrange spaces, we can write the Einstein equations in vacuum corresponding to the metrical connection of the deformed Minkowski space (see Equations (31)). It is easy to see that the independent equations are given by

The first equation has the solution

where

This solution represents the time coefficient of an over-Minkowskian metric. For

In other words, considering

It is also worth noticing that this result shows that a spacetime deformation (of over-Minkowskian type) exists even in absence of an external electromagnetic field (remember that Equations (45) and (46) have been derived by assuming

As we have seen, the deformed Minkowski space

(horizontal electromagnetic internal tensor) and

(vertical electromagnetic internal tensor).

The internal electromagnetic hand v-fields

Let us stress explicitly the different nature of the two internal electromagnetic fields. In fact, the horizontal field

A few remarks are in order. First, the main results obtained for the (abelian) electromagnetic field can be probably generalized (with suitable changes) to non-abelian gauge fields. Second, the presence of the internal electromagnetic hand v-fields

The important point worth emphasizing is that such an intrinsic dynamics springs from gauge fields. Indeed, the two internal fields

Such a fundamental result can be schematized as follows:

(with self-explanatory meaning of the notation).

In Deformed Special Relativity, two kinds of breakdown of Lorentz invariance occur. One is straightforward, and is due to the very dependence on energy of the metric coefficients. The second is more subtle, and is related to the mathematical structure of Generalized Lagrange space, which allows one to endow deformed Minkowski space-time with both curvature and torsion.

This is a basic result, not only from the theoretical, but also from the experimental side. Indeed, a number of experiments carried out in the last two decades have shown that a variety of new physical phenomena do occur in deformed space-time [