ABSTRACT
The word ”time” has been used not only colloquially but also in the scientific literature in several crucially different meanings, such as:
◦ temporal variable - time, which is its basic, essential meaning, ◦ value of the temporal variable, ◦ instantaneous value of the temporal variable, ◦ numerical value of the temporal variable, ◦ temporal interval, ◦ duration. The multiple use of the word ”time” has created enormous and essential
confusion even in science. The existence, the nature, the sense, the meaning, the properties, and the
(non)uniqueness of time have been attracting human thoughts and efforts to explain them since the most ancient epoch. It seems that there is not an area of human activity where time has not attracted an attention. This is natural because time, by its always changing temporal value, imbues, impregnates, is over and in, passes through every cell of the body of everybody; because all processes and motions propagate in time, and all actions occur in time. We do not have a particular, separate sense to feel and/or to measure time values and their flow, possibly because we do that with every our cell. This could explain why every human has its own feeling, and consequently, understanding of time. The views on time have varied in the widest diapason, from claims that time
does not exist, that it is not a physical variable, that it is our fiction, that it is a result of our imagination, an expression of our subjective feeling, that it is just a mathematical parameter, that it is dependent on space, a kind of a spatial coordinate, that there are infinitely many different times with variable speeds, up to the claims that it has existed, either with the beginning once or have existed for ever, that it is a physical variable with well measurable value, that is
All those thoughts, all analyses, experiments and studies, our understanding
of physical reality and our knowledge, show that time possesses such unique characteristic nature, sense and properties, that they cannot be expressed or explained in terms of other variables, or in terms of nontemporal phenomena, processes and/or categories. They clarify that time is its own component with the self-contained nature, meaning, characteristics, and properties. This led herein to introduce the definition of time and to explain and describe its properties, i.e., to present its characteristics, in the form of an axiom Definition 46 and Axiom 47 are the keystones of this book. They present are
the synthetic summary that describes and explains the nature, the properties, the characteristics and the sense of time. By relying on them we will be able to reply to solve Problems 1 through 37, and to establish new physical principles and to develop fundamentals of the new physical and mathematical relativity theory called Consistent Relativity Theory. The unbounded variety of speeds of evolutions of (artistic, biological, chem-
ical, economical, physical, political, social, technical) processes and by them induced the unlimited diversity of the choices of time axes (of relative zero moments, of initial moments, of time scales, and of time units), and our everyday use of some of them, and not always the same of them, and not everywhere the same, enabled us to discover the existence of time fields. Time fields are governed by time axes so that they can be homogeneous or
heterogeneous, single-layer or multi-layer time fields, time - invariant or time - varying ones. They express a reach temporal structure of space. Among various coordinate transformations and time fields there are Galilean
- Newtonian coordinate transformations and time fields, and others that are different from them, from Lorentz transformations, and from Lorentzian time fields. It is shown herein that all of them form Poincaré group of the transformations. Consequently, we cannot reject Galilean - Newtonian transformations by claiming wrongly that they do not obey Einstein’s generalized distance preservation condition, i.e. that they do not form Poincaré group. There is a simple, but a crucial mathematical link, which has a physical
meaning, between the theory of dynamical systems with multiple time scales and the theory of relativity of time. The time scaling coefficients are this link. They will play important role in the temporal coordinate transformations studied in the sequel. They are missing in Einsteinian relativity theory. There is also an important interconnection between the general, common
properties of all physical variables, which are expressed in the various forms of Physical Continuity and Uniqueness Principle (PCUP), and the continuous flow of time values. The latter is expressed in the form of Time Continuity and Uniqueness Principle (TCUP). These principles appear crucial for an adequate mathematical modeling physical phenomena, processes and systems. Besides, they are significant for synthesis and implementation of fine controls of dynamical systems.
