### THE RELATIVE THEORY OF TIME APPLIED TO THE CINEMA (by VLADIMIR V. ARISTOV, Moscow)

In the basis of the given model, connected with the consideration of a succession of visual images of the world, lies quite a simple analogy with the cinema. Such comparison is able to clear out a lot in the suggested relative constructions. From the other hand, the reverse «influence» is also inevitable, since along with it the time model of the cinematograph is interpreted in a certain way (at least, some of its important aspects).

Before I start expounding in this way the position of the model I need to make closer definitions, where historical excursus to the basic notions will right away give the idea of the problem’s principles. We will be particularly interested in the questions of the external reproduction of the plurality of events given through the motion (let it be the most obvious motion) of planets, for example. But the relative model (of observing the outside world) stands next to the inside contemplation, where time is perceived as a series of interior visual images or a succession of emotional experiences. The terms «relative» and «relation» themselves need an explanation. Relative (the word «relativist», as a rule, is connected to the theory of relativity) means connected with something, not having an independent nature. In this sense the relative time withstands substantial ideas (realizing a certain schematic character of this opposition), where with the notion of time is always connected some substance, with the course of which is associated the time flow. Various books tell about these notions, their evolution (check out, for example, [1-3]). Both conceptions have long standing traditions.

The primitive thought, the protophilosophical ideas are very interesting, a lot of works are dedicated to that, but we have no time to discuss it. We can only quote from [3]: «Man in the philogenetic and ontogenetic childhood perceives the world in a relative way, and this, of course, requires to pay special attention to the specific character of the primitive and the child’s perception of time and space... During two thousand years man projected on the world the space of his kinesthesia (the Euclidean space) and thought it to be a priori, universal». And really, they had to depart from the Kantian apriorism with its predetermined euclidity, although some prototypes in their original form are probably essential. So not only time and space, recorded from outside on tabula rasa (clear board) should be studied, but the structure of this tabula rasa itself can be the subject of an in-depth research. I will note by the way that in general the work with projections, worked out by man for time and space, is extremely important because it’s the clash of the interior visuality taken outside and «the objectively depicted» visuality in the cinematograph that plays an essential role.

Beginning from the ancient authors, the relative idea is quite clearly presented. In the dialogue «Thimei» Plato says [4]: «Therefore it came to him to create some moving resemblance of the eternity: he set up the sky, together with him it creates for the eternity, being one whole, the eternal image, moving from number to number that we have called time...and for time to be born from God’s reason and thought, there appeared the Sun, the Moon and five other celestial bodies called planets to determine and keep the numbers of time.» A persistent mentioning of calculating time and its connection to the image of number is very characteristic here, it will be later developed in the theory of Aristotle, where time is defined as number and the measure of motion. This quite developed philosophical theory contains important questions, and up to the present moment the answers to some of them are still not found. For us the most important is its quite definite relativity and a precise formulation of the problem, expressed, for example, by the following phrase [5]: «Time is the number of motion, and «now», just like the being moved, is like a digit of the number.» In the works of some stoic philosophers we can find important mentions of the distance of time [6].

On the philosophical level we can already see the solution of the problem which in a certain «negative» sense was clear to Plotin (he was against the relative definitions of time and space which were external concerning the inner world): «It is impossible for time to be a motion... if you take all [possible] motions and turn them all into one» [7]. At the same time the metrical properties of time are reproduced which we usually associate with a moving clock hand. The views on time of Augustin the Blessed are well known and described. He indicates the connection between time and motion which is very important for us, he «almost» reaches such description but doesn’t make the last step in this direction.. Here are his words from [8]: «I heard one wise man say that the motion of the Sun, the Moon and the stars is time, but I don’t agree with it. Why not consider time the motion of all bodies? If the heavenly bodies stopped, and a potter’s wheel went on moving, then there wouldn’t be time to measure its rotations?» And then he exclaims: «That’s why it seems to me that time is nothing but an extension, but of what? I don’t know, may be, of soul itself.» Those elusive definitions of time through external subjects, that negation of such possibility which we often meet in Augustin’s works, contains, nevertheless, the image of time as a moving plurality. It brings us closer to the perception of something general where none of the things in particular can serve the image of the unstoppable time. But may be all together they can make up this image? It is interesting to compare such relative ideas with the often quoted saying of Leibnitz that the Universe is a clock and of his opponent Newton who supposed that the Universe contains a clock.

Metrication, the modeling of elementary intervals of duration, the shifting of moments of time is essential for understanding the most important aspects of reproduction of the desirable properties of time. Yet Aristotle discussed the problem of «now» which must not have extension. The relative theory points out the possibility of description of elementarily small intervals of time, and thus it differs from other approaches. Bergson with his basic notion of «duration» («duree») said that «now» can not be divided in principle, which made the problem of the system of chronology hardly realized in the exact science (although incredibly attractive in the existential philosophical, poetic, plastic sense). Bergson wrote [9]: «Changeability is self-sufficient, it is a thing in itself... Nowhere the substantiality of changeability can be seen and felt as much as in the domain of the inner life.» The statement of Pouancare, directly answering Bergson, is characteristic [10]: «This duration could not be an instrument of the man of science; it could become one only having been totally changed, having become «spatial», as says Bergson. Really, it is necessary for it to become measurable; what can not be measured, can not be a subject of science... the properties of time - are only the properties of clocks, just like the properties of space are only the properties of measuring instruments».

From the theoretical point of view the suggested relative model is an attempt to develop also the ideas about time and space, denoted in the theory of relativity. I would distinguish especially the significance of the ideas of Ernst Mach, his ideas about time and space can be called the principle of Mach (this is a wider philosophical notion compared to that in the theory of relativity). He supposes that time and space don’t exist out of the moving sum of particles; what’s more, there is a possible mutual influence of micro- and macrocosms. I imagine that Einstein felt unsatisfied by the state of the contemporary theory (there was more than just the impossibility of building the unified field theory), when he spoke about the desirability of creating the theory of clocks and rulers. The ideas of creative significance of mathematics in all the physical and natural philosophical knowledge were important for Plank, Gilbert, Einstein, Pouancare, Eddington, Dirac. The two elements of divisibility and indistinguishable singularity - the physical atomicity and the numerical mathematical atomicity are now able to be combined in the new understanding of time in a neopythagorean attempt to create a physical theory that can be called «the physical mathematics» - by analogy and contrary to the well known discipline of mathematical physics.

Now I will speak on the main principles of the model [10-12], comparing it with the shooting technique. When a film is being rewound backward, you get an impression that time (for the events depicted on the film) has turned backwards. Although there is no doubt that the global time continues to flow forward. We are talking here about one of the most important properties of time - its inability to turn back. If we could depict on some miraculous film all events in the world, this film being played back would show the reverse motion of the world time. The first essential supposition concerning the suggested model is connected with the mentioned idea that with the exhaustive description of the world (or some limited system that we regard as a «world») the main properties of time can be conveyed. If in one photograph, in one frame there is information about all events of the system, and in the exhaustive description you can go as far as the atomic level, then it may be considered that the provided information is enough to describe a certain moment of time correlated with a certain frame of the film. Naturally, the question arises: how do we have to describe events, how to formalize and depict the changes?

When the frames change, with a visible difference of the picture, we notice that the way the objects were arranged about each other has changed. The second supposition concerning the model is that the desirable formalization can be made on the language of spatial relations. That is a moment of time is characterized by a set of mutual distances between all the subjects of the given frame. Strictly speaking: each particle which can be fixed in a frame is characterized by its three coordinates in the given three-dimensional coordinate system. When the frames change these distances change also. We don’t doubt it being convinced by our empirical experience of perception and «positivistic» experience of photographing (it will be said later whether we should define what is motion initially). And by the change of mutual distances (or the coordinates of all particles) we can tell how the picture of the world has changed, and thus to tell what interval of time has passed between two neighboring frames. I will note that frames (here we should appeal to our sensual experience, to the sensations given by our sensory organs) should change quite quickly so that the changes in the position of objects wouldn’t be significant, that is there wouldn’t be objects returning to their former position in the second frame.

Such idea that a lot - if not everything - can be portrayed, conveyed with the help of light, is actually a certain philosophical (or metaphysical) conception. The world as light. It is not for nothing that in the Russian language these two words are synonyms. Doesn’t this idea lie in the basis of the world relations for some film directors? On the theoretical plane, what I am talking about is that the knowledge of the world (material or ideal, it doesn’t matter here) can be conveyed with the help of the light depicted on hypothetical all-receptive photographs. With the help of these cinema frames we can get an idea about time (here we consider the space to be given, although the positions of particles near the marks of the system of coordinates we find out anyway with the help of the photographs). Incidentally, the reading of the clock of the traditional physical theory can be compared to this average spatial change, that’s where we get the basic mathematical equations of this theory from. Based on the built time and space description we can get all the physical theoretical knowledge. The equations of the suggested model lead to the conclusion that elementary particles, identical in the contemporary understanding (for example, two electrons) can be distinguished. Although the level of difference is extremely low and lies far beyond the limits of the possibilities of contemporary measuring instruments. This most subtle hypothetical distinction in fact means an illumination with some soft dim light whose properties we are yet to find out. But it really will correspond the predictions, that would mean the possibility of recognition of the individuality and the history of every particle in the world.

They often ask: what is the basic idea of the model? Is motion the initial notion? Doesn’t there happen some confusion, some mixing up of essences, don’t we take time for an a priori notion and only make up the measure of such motion-time? But in our constructions we reproduce the mode of measuring by the physical clock. Thus there is no point in talking about the initial presence of some «time» as motion if there is an indicated algorithm of building the model which we call time. Therefore we can consider the constructive building of time to be ousting the time-substance. That is in this model there is no time as the initial notion, but we can say that we reproduce (ultimately) all the attributes of real time. The same way a physical cinematographic «building» being depicted on film (I won’t mention yet other methods) doesn’t simply reproduce something given but builds up its own model of time.

We should stop at the procedure that lets us actually correlate the time and space values. We need to correlate the measured flow of the universal time (expressed in the working stroke of a clock) not only with a selected period of time but with all sum of motions in the world that you can possibly observe. You can not confine yourself to just one object when regarding time, for the object can stop in its motion or its behavior intuitively can be uneven. But finding the average, summing up all its movements can fulfill all the needed requirements. It conveys the uniform motion and the continuity. It is finding the mathematical function providing the correspondence with the certain physical correlation that solves the given problem of modeling. In the philosophical respect we can see the solution of a certain problem which was mentioned before when I was quoting the works of Plotin and Augustin. Time as a measurable value, as some extension, as a duration which is fixable by the consciousness and which is reproduced by the clock, can be modeled on the level of motion. What’s more, if quantity of particles in the system is small, time built in such a way will not be even, measured because the static mistake will be great.

The essential in the given model is its «speculativity», here the ultimate objects that can be conceived with the human mind become significant in a concrete model which can be physically realized. Incidentally, the inner and the outer vision strive for each other. The general objection that we can not describe the position of all the particles of the world when building the universal time is canceled by the observation that one can not as well describe the position of all particles, for example, in a cubic centimeter of air but the statistical physics, when calculating the average, operates such values.

In the suggested construction we will consider it possible for a time interval to be expressed through the motion (spatial motion) of all the particles of the system, on a larger (ultimate) scale - all particles of the world. This connection is expressed mathematically as an equality of a time interval to some sum of spatial changes. The factor of different measures (time and space) standing before the sum (it would be more precise to say the average quadratic) turns out to be the inverse value of the speed of light in vacuum. Thus one measure - time - turns out to be expressed through another physical measure (which is considered to be independent in the standard theory) - space. A moment of time which is understood as a frame on our film, is characterized only spatially. Time is being brought out, removed here, in this sense there is «no time here», and it is the movie camera that carries out this eliminating role leading to the diminishing of essences, being a mediator between time and space.

Here the kinematic correlation determined by the equation is written down practically without any concrete measures, and the connection with the physical interpretation of the values is carried out with the help of one measured fundamental constant - the speed of light. In a most simple case dynamic equations are derived from this basic kinematic equation. One succeeds in getting the equations of motion of the special theory of relativity, and within the limits of low speeds, compared to the speed of light - usual equations of Newtonian mechanics. Correspondingly we get the expressions determining the correlation of the laws of conservation. It is important to emphasize that all the given equations, owing to the absence of concrete measures, are actually based on mathematical correlations which, in their turn, are based on mathematical axioms and their consequences. Thus, here is indicated the connection of the existing physical laws (postulates) with the postulates of the mathematical character. The mathematical correlations are «projected» into the physical correlations with the help of the corresponding instruments, in this case - the clock. Therefore out of a large number or opportunities that can give mathematical correlations, the instruments having quite certain mode of description choose quite a limited number of physical correlations.

Acquiring the speed of light, in the motion of a particle there must appear differences from the standard description of the theory of relativity (and this effect can not be checked now because in the modern rocket boosters the speed needed for detection is many times higher the accomplishable now).

Sure, not all properties of time are reflected on this level where building the model, reproducing the equations of the classical reversible mechanics, played the leading role. Although it is obvious that the question of the impossibility to turn back time, of its connection with thermodynamics should be posed here (I, actually, talked about it before). One «photograph», one frame give complete information on the condition of the system, it is a spatial image of a moment of time. It is important to say that this is not just one point of the time axis determined by 3N parameters (N is the number of particles in the system). Any changes happening in the world will cause changes in the mutual position on particles which will be characterized by changes in their position on «photographs» and changes of the corresponding radius-vectors. Thus we can characterize any changes happening in the system while time continues to flow. But if any two «photographs» fully coincide, we could say that two moments of time have coincided (strictly speaking, for a dynamic coincidence not only radius-vectors should fully coincide but also their increments). In that case we could say that time has returned back. Since there is a very large number of particles in the system, such