_{1}

It is shown that two fundamental notions “space” and “time” can be reduced to the one notion—the “space”, where time appears to be the length of geodesic lines of the space adjacent to the three-dimensional Euclidean space. The whole space is a topological product of the three-dimensional Euclidean space and some another space, that may have the screw structure. Within the framework of this model, the Galilei’s inertia law and existence of limiting velocity of motions are consequences of the geometrical interpretation of time.

Space and time are fundamental physical notions, and they are basis for definition of the central notion of natural sciences―the notion of “motion”. It is impossible to express these notions in terms of more fundamental ones. The time is considered in special relativity as one of the four coordinates

This work is a соntinuation of author’s investigations of possibility to explain physical phenomena by geometrical properties of the space-time [

Notice, that the formal geometrical interpretation of time is known where time is considered as an imaginary fourth coordinate with the dimensionality of length in the four dimensional pseudo-Euclidean space that is often identified as a “space-time” (Minkowski space) [

Geodesic lines in the space define the shortest distances between any two points and for the space with the connectivity

where n is the dimensionality of the space,

We start with the simplest possible model of the space-time (

It follows from

So, our main suggestion (2) leads to the right law of the free body’s motion in our Euclidean space?to the Galilei’s inertia law, when the free body is moving with the constant velocity. And this proves suggested geometrical interpretation of time.

We see that within the framework of the suggested model the space of events is represented not by the fictitious four-dimensional pseudo-Euclidean Minkowski space, but it is represented by four-dimensional Euclidean space, where time has the special geometrical interpretation. Notice that suggested interpretation of time gives the opportunity for simple explanation of the existence of limiting velocity of any motion. Indeed, because

For limiting velocity of motions

We obtain this relation for the one-dimensional model of our three-dimensional Euclidean space. For real three-dimensional space relation (5) is obviously generalized as

This relation is one of the basic relations of special relativity [

In the preceding Section we considered Euclidean model of the space-time, and the space-time has here the simple symmetries of the Euclidean space. But there are some experimental facts, indicating that the space-time symmetry may be more complex. These are distortions of chiral symmetry in some native chemical compounds, non-conservation of parity in some interactions and distortion of charge symmetry (particle?antiparticle). In this Section we considered the model of the space-time with screw symmetry. To make the idea more understandable and descriptive we firstly consider the simplest possible low-dimensional analogy of the suggested model of the space-time that reflects important features (not all) of the suggested model of the space-time. Namely, we will consider an analogy where the three-dimensional Euclidean space is replaced by the one-dimensional one and where the adjacent space is a surface of the infinite cylinder with radius

The mathematics in this case is essentially simplified. First of all rewrite Equations (1), taking the length of geodesic L as a parameter

Let us firstly show that geodesic lines of this space are screw lines. Metric _{ }is defined for Riemannian space by relation for an element of length

For cylindrical surface with radius

From (7,8), we have for nonzero coordinates of

Such metric is called locally Euclidean one. Inserting (10) into (1,7), we see

that right sides in Equations (7) equal to zero. This means that equations for

The solution of these equations has the form

where

There is a known relation for the crew line with radius

From (13,14), we have

According to our geometrical interpretation of time, the time of the free body’s displacement in Euclidean space at the distance

Inserting (16) in (15), we see that our geometrical interpretation of time (16) leads to the right law of free body’s motion to the Galilei’s inertia law of motion with the constant velocity

As for Euclidean model (Section 2), the suggested interpretation explains also the existing of limiting velocity of motions that equals to light velocity. Indeed, it follows from (17)

This result has a simple geometrical explanation: the length of screw line cannot be larger than the corresponding straight line.

We consider here the general model of the space-time with the screw structure. Namely, we consider the five-dimensional space-time

For better representation of such space we showed at

Let us define now the form of geodesic lines in the space

Here

Inserting these values in (1), we obtain for nonzero components of

Inserting this in (10) and taking the length of geodesic L as a parameter

These equations describe complex screw lines in the five-dimensional space. We choose the simple solution, analogous to the one in Section 2, when the screw line has the form of the screw line with the constant radius

we take the solution

As in Section (2) constants

where

As a result, we have from (25,26)

According to our geometrical interpretation of time, the time

Inserting (28) in (27), we see that our geometrical interpretation of time (28) leads to the right law of free body’s motion―to the Galilei’s inertia law of motion with the constant velocity

As for Euclidean model (Section 2), the suggested interpretation explains also the existing of limiting velocity of motions that equals to light velocity. Indeed, it follows from (29)

This result has again a simple geometrical explanation: the length of screw line cannot be larger than the corresponding straight line. In limiting case

The last relation is the one of the basic relations of special relativity [

The geometrical interpretation of time is suggested where time of the free body’s displacement multiplied by light velocity equals to the length of the geodesic line in the space adjacent to the Euclidean space. The models of the space-time are considered where above adjacent space is the Euclidean one or the space with the screw structure. The suggested models give the geometrical explanation for the Galilei’s inertia law and for existence of the limiting velocity of any motion.

Within the framework of the suggested interpretation, time is intimately connected with the motion, and any kind of motions can be selected as the standard for the time measurements (periodical movements appeared to be the most convenient). No motion, no time. If the observer is immovable, then the time is connected with the movements in the clocks. If there are no clocks nearby, then the time is connected with the movements inside the observer.

Olkhov, O.A. (2017) Geometrical Interpretation of Time and New Models of the Space-Time. Journal of High Energy Physics, Gravitation and Cosmology, 3, 564-571. https://doi.org/10.4236/jhepgc.2017.34043