The Minkowski's space–time is consistent with differential geometry of fractional order

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Abstract

The recent discovery of fractional Taylor's series for nondifferentiable functions, f(x+h)=Eα(hαDxα)f(x), where Eα() denotes the Mittag-Leffler function, and Dxα is the so-called modified Riemann–Liouville fractional derivative, opens the possibility of a differential geometry of fractional order. It is shown that the Lorentz pseudo-metric is quite consistent with this fractional geometry, in other words that the fractional velocity of light does not affect the matter. The fractal effect is pictured by change of scale in space and time.

Introduction

Since the pioneering work of Nelson [1], [2] which “proved” the Schrödinger equation by using some concepts drawn from stochastic mechanics, there have been a huge number of papers on stochastics and fractals in physics, and more especially in quantum mechanics [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. As a matter of fact, fractals and stochastics are very much inter-related, as far as a function which is continuous everywhere but nowhere differentiable cannot be replicated, and thus, necessarily exhibits random-like features. On the other hand, as far as we have understood, fractals would be of special interest to construct a unified framework for physics, since then by using its self-similarity properties, one could switch from microphysics to macrophysics merely on zooming.

Recently has been discovered a fractional Taylor's series for nondifferentiable (but fractional differentiable) functions, which opens the possibility of a differentiable geometry of fractional order, and the question of course, is how we can duplicate the Minkowski pseudo-metric in this framework? Does this duplication require new assumptions of physical nature on light velocity?

The purpose of the present Letter is exactly to examine this question and it is organized as follows. Firstly, we shall bear in mind the definition of the so-called “modified Riemann–Liouville derivative of fractional order” which we introduced recently to cope with some drawbacks involved in the classical definition, and then we shall give the expression of the fractional Taylor's series of nondifferentiable functions as we obtained it recently. Given this prerequisite, we shall outline how differential geometry of fractional order could be constructed in quite a natural way in this framework. Then one examines how one can generalize the Minkowski pseudo-metric in this framework, by introducing the fractional velocity of light.

The present work is only a formal development based on fractional calculus and differential geometry, and there remains to get more insight in the physics behind this approach.

In the next sections two and three we shall summarize some results which are detailed in [19], [20], [21], and to this end, we shall use at will and for convenience the notationsDαf(x)=f(α)(x)=dαf(x)dxα for the fractional derivative of order α.

Section snippets

Fractional derivative via fractional difference

Definition 2.1

Let f:RR, xf(x), be a continuous (but not necessarily differentiable) function, and let h>0 be a constant discretization span. Define the forward operator FW(h), i.e. (the symbol:=means that the left side is defined by the right one)FW(h).f(x):=f(x+h); then the fractional difference of order α, 0<α<1, of f(x) is defined by the expressionΔα.f(x):=(FW1)α.f(x)=k=0(1)k(αk)f[x+(αk)h], and its derivative of fractional order is defined by the expressionf(α)(x)=limh0Δαf(x)hα,0<α1.

This

Main definition

A generalized Taylor expansion of fractional order (F-Taylor series in the following) reads as follows:

Proposition 3.1

Assume that the continuous function f:RR, xf(x) has fractional derivative of order kα, for any positive integer k and any α, 0<α<1, then the following equality holds, which is [20]f(x+h)=k=0hαkΓ(1+αk)f(αk)(x),0<α1, where f(αk) is the derivative of order αk of f(x).

With the notationΓ(1+αk)=:(αk)!, one has the formulaf(x+h)=k=0hαk(αk)!f(αk)(x),0<α1, which looks like the classical one.

Corollary 3.1

Introduction to differential geometry of fractional order

In the present section, all the notations and parameters are local ones (i.e. for this section only) and mainly we shall use the notation of tensor calculus regarding the indexes of co-variant and contra-variant co-ordinates.

Given this prerequisite, it is easy to see that the fractional calculus of the preceding section provides an easy way to extend some results of elementary differential geometry.

As it is customary, we consider a generic point M defined by the rectilinear co-ordinates xj, j=1,

Invariance principle of fractional order

On assuming that there exists a finite resolution in space and time co-ordinates due to some coarse graining phenomenon, in quite a natural way, we are led to introduce a mechanics of fractional order which involves the fractal velocity να=dαx/dtα, 0<α1. In this framework, the basic principle of the special relativity will become:

Principle

Given two Galilean systems S(0,xyz) and S(0,xyz), the expressions(dsα)2:=cα2(dt)2α(dαx)2(dαy)2(dαz)2 and(dsα)2:=cα2(dt)2α(dαx)2(dαy)2(dαz)2, where cα

Application to the strip modeling

We assume that the space co-ordinate is not x but rather is the complex-valued variablez:=x+iξ, where ξ is a real valued state parameter which, in a first approach, can be considered as picturing some lack of definition. This complex modeling pictures the fact that we do not deal with a line, but rather a strip: this is a strip modeling (see for instance [8]). We assume further that ξ is a nondifferentiable function in such a manner that we can write the αth increment dαξ in the formdαξ=να(dt)α,

Concluding remarks

We now have at hand a fractional calculus based on differential of fractional order, which could serve as a point of departure to construct a differential geometry of fractional order. This could be meaningful for instance in a framework in which the trajectory of particles would be affected by some coarse grained phenomenon, which would cause that the state parameter would not be differentiable, but only fractional differentiable.

If one introduces the fractional velocity of light, then one

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