Research program: physical ideas, basic results, and the line of further investigations

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With the aim of constructing a rigorous and consistent theory taking into account the possible appearance of the physical properties of time, one should turn to dynamics. As was noted above, it is the dynamical principle that relates the evolution of a system in time to the action of force fields. As A.A. Logunov underlines, "if for some form of matter we have the laws of its motion in the form of differential equations, then these equations contain information on the structure of space and time" [32]. Obviously, dynamical equations should contain information not only about geometrical properties of space-time as a whole, but also about physical properties of space and time, taken separately [16,28,33].

Let us show schematically how to establish a link between the course of time in one inertial reference frame and the course of time in the other with an example of the motion of a point particle in a force field from the viewpoint of inertial frames and , moving relative to each other. From relativistic equations of motion of a mass point written in the inertial frame ,

(4)

where and are the vectors of momentum and force, respectively, one can express the time interval corresponding to the increments of radius-vector and momentum vector , in the reference frame . Similarly, one can express the time interval and the increments of vectors and in the other inertial frame moving relative to the -frame. Taking into account the Lorentz transformation laws linking the primed and unprimed increments of momenta and forces in the reference frames and , it is not difficult to establish a link between the primed and unprimed intervals of time. It is the relationship thus obtained which defines the course of time in the -frame in relation to the course of time and to the dynamical parameters, describing motion of a mass point, in the reference frame .

In view of fundamental importance of the problem considered, we shall discuss it in more detail.

The Cartesian coordinates connected with the inertial reference frame and are assumed for definiteness to be oriented in such a manner that the -axes of the frame are parallel to the -axes of the frame , the -axis and the -axis coincide with each other, and the reference frame moves with a velocity relative to the -frame along the -axis. Denote by the length of a path section in a vicinity of a point in the reference frame , which the particle covers for the time , and by and the corresponding quantities relating to the reference system . The quantities and are connected with each other by equality

(5)

where is the -component of the particle velocity in the - frame at the instant of time .

Quantity characterizes the change in the course of time in the vicinity of point on the particle’s trajectory in the reference frame as compared with the reference frame . As is seen from (5), if during some interval of time the -component of velocity of particle is changed (), on this time interval the relative course of time is changed as well ().

If the particle moves on a path section uniformly and rectilinearly, i.e. by inertia ( ), the relative course of time on the path section is remained constant: . Inasmuch as the change in the velocity of particle in an inertial reference frame is conditioned, according to the main postulate of classical mechanics, by the action on particle of a force connected with some physical field, hence, the force acting on particle is the reason of change in the course of time along the particle’s trajectory.Equality (5) describes the phenomenon of local dynamical inhomogeneity of time whose gist is that the quantity depends not only on , but also on the particle’s velocity in the vicinity of the point . As the change in the particle’s velocity is determined by the force influence on the particle, it follows from here that the force acting on a particle in some inertial reference frame is the reason of change in the course of time along the particle’s trajectory. In other words, the dynamical inhomogeneity of time means that the different instants of time on the time axis prove to be physically non-equivalent when the particle moves in a force field [36,37].

Let's go over from point to some other point , also lying on the particle’s trajectory, and write down for it the relationship analogous to (5):

(6)

By dividing last equality and equality (5) term by term, we arrive at relationship

(7)

Quantities and characterize the relative course of time between the points and on the particle’s trajectory in the reference frames and , respectively. In virtue of (7), if ,then , i.e. the relative course of time between points and in the - frame coincides with that in the -frame. To change the relative course of time between two points in one inertial reference frame as compared to the other, it is necessary that a force should act on the particle on the corresponding path section. According to (7), at if only the particle’s velocities at points and are not identical in magnitude. In this case, if , then, generally speaking, , i.e. to equal distances, which the particle passes in different regions of space in reference frame , there correspond different distances traversed by the particle in reference frame . This can be caused by both the different course of time at points and and the different velocities of particle at these points. In this connection the questions arise: How can these factors be separated from each other? How can the relative course of time be determined between the different points of space being considered in the same inertial reference frame?

Relying on the reasoning given above, it is natural to suppose that the change in the course of time atthe points lying on the particle’s trajectory can be caused only by the action of a force field on the particle. Indeed, in the absence of force field, when the particle moves freely, there are no reasons for changing the course of time. Let's go over from the inertial reference frame , in which the motion of a particle takes place under the action of a force field, to such a noninertial reference frame , in which the inertial force is completely compensated for by the force field action at that point of space where the particle is at rest. Apparently, in the -frame the particle moves by inertia, i.e. it is in a free state (imponderability state) [41,42]. Since in this state the force effect on the particle is absent and, thus, the reason for changing the course of time is lacking, in this reference frame the course of time should be uniform at the point where the particle is placed. By choosing in the -frame at the point where the particle is placed two equal in magnitude intervals of time corresponding to two different points and lying on the particle’s trajectory in the inertial reference frame , and by performing then the inverse transition from the reference frame to the -frame, we may define the magnitude of relative course of time between points and in the reference frame .

In [37] is received the following general expression for the relative course of time between the points lying on the trajectory of motion of a particle in a force field in an inertial reference frame:

(8)

where is the velocity of particle at the instant of time relative to the inertial reference frame . Expression (8) can be written in the following equivalent forms:

(9)

where and are the kinetic and potential energies of particle at the instant of time , provided that equality is fulfilled. As is seen from (8), if the particle’s velocity is small as contrasted to the speed of light, the change of the course of time is a relativistically small quantity of the order of .

Note that equality (5) is one of the relationships entering into the Lorentz transformations for coordinates and time, and consequently it is not new. A new point is that an analysis of this relationship is given, as applied to the motion of point particle in a path under the action of a force field, and on its basis a number of physical consequences is derived concerning the course of time, which were not discussed in literature till now and were formulated for the first time in [28,33]. These consequences are important not only for elucidating the physical nature of time, but also for a deep insight into the physical content of relativistic mechanics, and thus they deserve consideration in more detail.

According to A. Logunov, the main content of the special theory of relativity (STR) consists in that “all physical processes proceed in space-time possessing pseudo-Euclidean geometry” ([32], p.26). Undeniably, from the mathematical (geometrical) point of view, in the formulation presented above the essence of the STR is expressed correctly, still the physical content of relativistic mechanics, the physical essence of dynamical principle underlying it, is that the force is not only the reason of particle’s acceleration in an inertial reference frame, but also the reason of the change in the course of time along particle’s trajectory. It should be emphasized that existence of a link between the force and the course of time along particle’s trajectory follows directly from the fact that space and time are connected with each other to form a single 4-dimensional space.

Thus, the fundamental difference between relativistic mechanics and Newtonian one lies not only in the fact that in Newtonian mechanics time is of an absolute character and does not change in going over from one inertial frame of reference to another, and in the STR it ceases to be identical in all inertial reference frames. In relativistic mechanics time with necessity acquires physical properties, which are conditioned by the action on particle of a force connected with a physical field. As a result of the force action, the course of time is continuously changed along the particle’s trajectory.

As is known [43,44], the existence of dependence of the course of time upon the gravitational field potential is predicted by the general theory of relativity (GTR). According to the GTR ([43], p.303), time flows differently at the different points of space in one and the same reference frame. Since “gravitational field is nothing more nor less than a change of the space-time metric” ([43], p.313), one can assert, apparently, that the change in the course of time is due, in the view of the GTR, to the change of the 4-space metric. It should be emphasized that in the present paper gravitational field is considered as an ordinary force field, and the particle motion is supposed to occur in pseudo-Euclidean space-time. The main formulas of the article [37], (8) and (9), describe the change in the course of time in an arbitrary force field at different spatial points in one and the same inertial reference frame. As is seen from the results received, the change in the course of time in a force field is in no way connected with the change of space-time metric. It is conditioned by the force field action on particle in inertial reference frame and is a direct consequence of the dynamical principle underlying relativistic mechanics.

It should be emphasized that the existence of dependence of the course of time on the state of motion of particle in a force field points to the feasibility of controlling the course of time using force fields.