Hello, my favorite forum members :-)

I have just posted new manuscript to the journal, so I would like to know your feedback ;-)

Before it will be reviewed and published, it is still ATM, so I post it here.

Please, find below the body of the article

Title:Fundamental interactions as a property of space-time

Abstract:

In the article it is shown, that tiny change in the definition of four-position drives to derivation of electromagnetic field tensor as natural property of the space-time. In presented picture, fundamental interactions are not something additional to the space-time - they become part of it and the derivation comes directly from the foundings of Special Relativity and Relativistic Quantum Mechanics.

1. Section. Introduction

Present days we describe field phenomena in many ways e.g. [1], [2], [3], [4] however we still did not repeat the success of A.Einstein. As it is known, in General Relativity, gravity is not an interaction but some property of the space-time geometry. In most field theories [5], field is still something additional to the space-time (instead of natural result of its presence) or authors modify metric tensor following idea from General Relativity.

The motivation standing behind this article was to find such description of the space-time, which in natural way deliver equations of electromagnetism, mass, and maybe other phenomenas. It looks like it has succeeded much easier, than anyone would expect.

In the article at first we will come back to the root of Special Relativity to state some simple hypothesis about the nature of space-time. Next we will discuss Lagrangian Mechanics and consider consequence of this hypothesis to relativistic mechanics. In effect we will obtain Lorentz force and electromagnetism resulting as natural consequence of discussed space-time definition.

Author believes, that this article may be important step on the path to unify interactions. It also opens new areas for scientific research.

We will use Einstein summation convention and metric signature . We denote as test body proper-time where ; we denote for rest mass and as reduced Planck constant [6].

2. Section. Hypothesis

As it is known in flat Minkowski space-time [7] we define four-position as with proper-time given as . Thanks to above, from historical point of view, we use Minkowski metric with four-velocity giving .

Let us notice, that without affecting metric tensor, we may assume that

(1)

where is some constant chosen such way, that

(2)

This way four-position product gives now

(3)

what gives

(4)

Above redefinition does not affect metric tensor, either four velocity value, however - it will affect action and Lagrangian Mechanics, what we will show below.

3. Section. Results

3.1 Chapter. Preparation

Let us recall classical Lagrangian Mechanics [8] where relation between action (Hamilton's principal function), generalized momentum and Lagrangian we define as follows

(5)

where is for generalized momentum and H for Hamiltonian. Here we will narrow to elementary particles and fundamental interactions only, and following [8] we will define action as we do it in Relativistic Quantum Mechanics

(6)

where we have introduced four-momentum

(7)

With redefined norm of four-position (3), the action in (6) is equal to

(8)

We should here discuss some important issue. In classical mechanics, action is the function of [tex](t, \vec{r})[\tex], thus Lagrangian is function of and Hamiltonian is function of . We therefore have to assume, that action defined this way depends only on .

We will then follow de Broglie hypothesis [9] and assume, that for elementary particles and fundamental interactions, four-momentum may be expressed as

(9)

where is some wave four-vector dependent only on four-position. Thanks to above assumption wave-particle dualism becomes important founding of this derivation.

Thanks to this assumption we get Lagrangian

(10)

Let us introduce four-force

(11)

and four-acceleration

(12)

where . Since

(13)

then we get from (10)

(14)

thus

(15)

Comparing (15) to (4) we get relation which will be important soon

(16)

We may also easy calculate

(17)

thus we may describe dynamics of the system by equation

(18)

where seems to play important role here, determining behavior of velocity and position. Let us therefore introduce four-vector (with the dimension of four-momentum), defined as

(19)

According to (17), it is orthogonal to four-velocity, giving . From (10), using Lagrangian Mechanics property we may calculate generalized momentum as

(20)

We have to stop at this moment to consider, how is dependent on .

T.B.C...