Discussions. The answers to questions and remarks

Session of the scientific seminar on temporology, November 13, 2001 (below are made references to the papers given in References to Section II 'Research Program').

1. What do you mean by the dynamical inhomogeneity of time? When does this phenomenon appear?

Answer. Let us consider the motion of a classical point particle under the action of a force field in the inertial reference frames and , moving relative to each other. The Cartesian coordinates connected with the reference frames 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 [33,34]:

(1)

where

(2)

is the -component of the particle velocity in the -frame at the instant of time . As is seen from (1), if , the quantity depends not only on , but also on the instant of time , that is the different instants of time on the time axis prove to be physically non-equivalent. Therein lies the essence of the phenomenon of local dynamical inhomogeneity of time. The dynamical inhomogeneity of time means that the course of time in a vicinity of point in the reference frame changes as compared to the course of time in the frame . As a measure of the change in the course of time at a point one may use the ratio (see (1))

(3)

Obviously, the dynamical inhomogeneity of time arises provided that , i.e. at .

2. Why can the force be considered as a reason of the dynamical inhomogeneity of time?

Answer. As can be seen from (2) and (3), if during some interval of time the particle moves with a constant velocity (), i.e. by inertia, then on this time interval

that is the different instances of time are physically equivalent. Thus, the dynamical inhomogeneity of time is conditioned by the change in particle's velocity, i.e. it is caused by the action of a force on particle.

3. Let us assume that the force is such that . In this case and thus by virtue of (2) and (3) the course of time is constant, contrary to the statement made above that force is the reason for changing the course of time.

Answer. The reasoning above is erroneous for the following reason. According to the relativistic equations of motion, if , then , but . From equality


it follows that .

4. What is meant by the change of the course of time along the trajectory of motion of particle in the same inertial reference frame? What quantity may serve as a measure of change of the course of time in this case?

Answer. Let us explain, first of all, what the constancy of the course of time along the particle's trajectory means. Let a dot particle move with constant velocity () relative to an inertial reference frame . Obviously, by virtue of the constancy of particle's velocity, the particle passes the path sections and of equal length () , situated in a vicinity of points and lying on the trajectory, for equal intervals of time:

(4)

where

The condition (4), which is carried out for any points and lying on the trajectory, expresses itself the constancy of the course of time along the trajectory of motion of free particle. Evidently, in the case of free particle the condition of the (4) type is carried out in any inertial reference frame. This means that for free particle the course of time along the trajectory does not vary with time in any inertial reference frame.
If the particle moves relative to an inertial reference frame under the action of an external force, the situation essentially changes because of changing the velocity of the particle with time (). Even if in the reference frame

(5)

and, hence, in this reference frame the condition (4) is carried out (i.e. in the -frame the course of time is constant), in any other inertial reference frame , moving relative to , the course of time will change along the trajectory as it follows from (1) and (2). An example of the force field, in which the flow of time along the trajectory is constant, is homogeneous magnetic field: a charged particle moves in this field with the velocity submitting to condition (5) (see [33,34]).
Before introducing a measure of change of the course of time along trajectory in the same inertial reference frame, we shall notice, that the course of time along trajectory may change only as a result of the action on particle of a force field, when the velocity of the particle varies from point to point. If the force field is absent (the particle moves with constant velocity), there is no reason for changing the course of time.
Let us pass from the inertial reference frame , in which an external force operates on the particle, in such a reference frame , in which the action of external force is completely compensated for by force of inertia at that point of space at which the particle is placed. In the reference frame , the particle is in the state of imponderability, that is it is free. The reference frame is quasi-inertial: in some vicinity of the point, at which the particle is in the state of imponderability, the reference frame behaves as inertial. As in the state of imponderability there is no force influence on the particle and, thus, there is no reason for changing the course of time, the course of the proper time of particle should be constant. Choosing identical intervals of proper time and , corresponding to points and lying on particle's trajectory, and taking into account the relationship

(6)

from equality

(7)

one can derive [37]:

(8)

where is the time interval in the reference frame , corresponding to proper time of the particle in the reference frame when the particle moves along the

trajectory in a vicinity of point

. It is the quantity

(8) that characterizes the

change in the course of time in a vicinity of point in comparison with the course of time in a vicinity of point . The course of time along the trajectory of motion of particle in some inertial reference frame is constant, if for any two points and lying on the trajectory. As is seen from (8), the course of time along trajectory varies with varying the particle's velocity, i.e. in the presence of an external field acting on the particle (the force influence is the reason for changing the course of time in the same inertial reference frame).

For the time intervals and the particle covers the path sections of length

and

(9)

respectively. By virtue of (8) and (9) one can obtain:

(10)

The ratio

(10) takes account of the change in the course of time at points

and

.

The condition

yields (see.(8)):

,

5. The dynamical inhomogeneity of time seems to be an artefact resulting from an alternative and confusing definition of time in the systems of reference moving relative to each other, which does not change the equations, but allows one to dream of controlling time.

Answer. In my papers, there is no new definition of time different from the definition given by the special theory of relativity (STR). The crux of the problem is that the force in relativistic mechanics is not only the reason of acceleration of particle in an inertial reference frame, but also the reason of change of the course of time along particle's trajectory. This is the main physical content of the STR.

6. What is called in the article the change in the course of time represents a trivial dependence of time intervals, during which the particle covers the path sections of identical length, upon velocity of particle on these path sections: with changing the particle's velocity the lengths of these intervals change.

Answer. This statement is wrong. The mistake is caused by the wrong understanding of the sense of quantities and in (8). The quantities above have no sense of time intervals during which the particle covers identical distances in the vicinities of points and (see equalities (10)). These quantities have the following meaning: they are those time intervals which correspond to the identical intervals of proper time of the particle being in the state of imponderability. To establish, whether the course of time along the trajectory of particle in a force field changes, it is necessary to compare the course of time when the particle moves freely (i.e. under conditions when there is obviously no reason for changing the course of time) with that in the presence of a force influence on the particle. It is from such a comparison that the formula (8) for the relative course of time between points and is derived.

It should be emphasized that, when the particle moves freely, quantity (8) is always equal to 1, whereas when moving under the action of a force field, generally speaking, . Last inequality points to the dependence of the flow of time on the state of motion of the particle in force field, i.e. it testifies to the occurrence of the physical properties of time. This result is of fundamental importance as it is indicative of the feasibility of controlling the course of time.

7. What is the magnitude of the change in the course of time?

Answer. If the particle moves with the velocity small in comparison with that of light ( ), the change of the course of time is relativistically small (it is of the order of

).

8. What is meant physically by the change of the course of time in various areas of space in inertial reference frame? Is it possible to observe this effect?

Answer. From the physical point of view, the change of the course of time in one area of space in comparison with the other means that the time scale in one area differs from that in the other. The content of the paper consists in the proof that the time scale along the trajectory of motion of particle depends upon the force influence on the particle. This phenomenon is quite observable, and it should be taken into account because, without regard for it, wrong expressions can be received both for the intervals of time between physical events and for the distances between points at which these events occur (see formula (10)).

9. One would think that the equality

(11)

which follows from the Lorentz transformations, points to the existence of the dynamical inhomogeneity of spatial coordinates similar to the inhomogeneity of time. Is such an interpretation correct?

Answer. As is seen from (11), the distance , traversed by a particle in the -frame, depends not only on the distance , which the particle traverses in the -frame, but also on the instant of time . This dependence cannot be treated, however, as a manifestation of the dynamical inhomogeneity of spatial coordinates. Formula (11) is a trivial consequence of kinematics: it expresses the simple fact that the distance traversed by particle depends on its velocity, which may change with time (as a result of the action of a force). Really, in classical mechanics, on the basis of the Galilean transformations, we have successively:

,

From this it follows that

Up to the factor last formula coincides with (11). In the case of relativistic mechanics we have analogously:

.

From here, using the velocity addition law and the Lorentz transformations for time, we arrive at formula (11).

Thus, as distinct from spatial coordinates, time in relativistic mechanics may be deformed (i.e. time scale may change) under the action of an external force. It should be emphasized that the active role of time in dynamics is due to the dynamical principle: the latter is formulated in terms of time, not of spatial coordinates.

10. It would be desirable to elucidate the physical content of the notions 'the physical properties of time' and 'the physical properties of space'. What are the distinction and similarity between them?

Answer. I shall begin with the answer to the second part of the question. Common to the properties mentioned above is that they are due to the dependence of time and spatial characteristics of physical systems upon material processes occurring in them, while distinctions are associated with the fact that time coordinate stands out in relation to spatial coordinates from both geometrical and physical points of view (see [37]).

The physical properties of time signify the dependence of the course of time in some area of space on the character of the physical processes occurring in this area. The reason of the change of the course of time is the force field acting on a system.

By the physical properties of space is meant the fact that the spatial area, in which a force field acts, turns to a physical medium having the capability to interact directly with other particles.

It should be emphasized that because of the indissoluble connection between time and space the occurrence of physical properties of space inevitably leads to the occurrence of physical properties of time.

11. It is well known that the existence of the dependence of the course of time on gravitational field is substantiated in the general theory of relativity (GТR) . What is new in your work?

Answer. First of all, I shall make a quotation from [43] (see p. 303): 'Already in the special theory of relativity the flow of true time is different for the clocks moving relative to each other. In the general theory of relativity the true time flows in different ways also at different points of space in the same reference frame. This means that the interval of proper time between two events occurring at some point of space and the interval of time between the events taking place simultaneously with them at the other point of space, generally speaking, are different from each other'. As 'the gravitational field is nothing but the change of the space-time metrics'([43], p. 313), one can assert, apparently, that the change of the course of time is due, from the point of view of the GTR, to the change of the 4-dimensional space metrics.
In the approach being developed, the gravitational field is considered as an ordinary force field and the motion of particle is assumed to occur in the pseudo-Euclidean space-time. The formulas received (see [37], and also formulas (8) and (9) from Section II) describe the change of the course of time in an arbitrary force field at different points of space in the same inertial reference frame. According to the results received, the change in the course of time in a force field is by no means connected with the change of the space-time metrics. It is caused by the action of the force field on a particle in inertial reference frame and follows directly from the dynamical principle underlying relativistic mechanics.

12. The general theory of relativity predicts the red shift of spectral lines in gravitational field. Is there anything new about the shift of spectral lines in your theory?

Answer. As is seen from our results, the shift of spectral lines in gravitational field is connected not to the change of the space-time metrics, but to that the gravitational field is a force field. According to our approach, force is the reason of the change in the course of time and, hence, the reason of the shift of spectral lines of atoms (the case in point is merely that part of the shift which has a bearing on the change in the course of time).

It should be emphasized that, from the point of view of quantum electrodynamics, the shift of spectral lines of atoms is caused by interaction of atoms with force fields. This interaction results in the occurrence of the energy level shifts of atoms and, hence, is the reason of the shift of spectral lines. Evidently, the energy level shifts contain a component connected with the change of the course of time, and its magnitude can be calculated by our formulas.

If the force field is gravitational, the picture should remain the same: the shift of a spectral line of atom, caused by the gravitational field, should consist of two components - one of them is connected with the change of the course of time in gravitational field and the other has no bearing on this change.

13. What is the basic result of your research on the problem of time?

Answer. The basic result is that the strict proof is given of the Kozyrev hypothesis about the existence of physical properties of time, and a general relationship is received connecting the course of time on one path section of a particle moving in a force field with the course of time on the other in the same inertial reference frame. Briefly, basic conclusion may be formulated as follows [37]: in relativistic mechanics, the force acting on a particle in an inertial reference frame is the reason of change of the course of time along the particle's trajectory.