WHAT CAN BE OBTAINED FROM THE SUBSTANTIAL CONCEPTION OF TIME ? L. S. Shikhobalov

In memory of Nikolay Alexandrovich Kozyrev

who saw in time the vital basis of the Universe

...

Time is one of the most fundamental ideas of physics. It, or, more precisely, the

variable describing it (usually denoted by the letter t after the word “time”), enters into the

equations of motion of Newton’s classical mechanics, Schrödinger equation of quantum

mechanics, the equations describing system evolution in thermodynamics and statistical

physics and many other equations of practically all divisions of physics. Besides, time remains

one of the greatest mysteries of nature. Such questions of principle as: “What is the

stream of time?”, “Does the direction of time exist or not?” and a number of others have

not yet been solved conclusively and rigorously.

Modern scientific world outlook knows two essentially different conceptions of

time, the relational one and the substantial one (Chernin 1987; Molchanov 1977, 1990;

Space... 1983). According to the first one, there exists no time “per se” in nature and time

is no more than a relation (or a set of relations) between physical events. In other words,

time is a specific manifestation of the properties of physical bodies and changes occurring

in them. The second conception, the substantial one, assumes, vice versa, that time is an

independent phenomenon of nature, a specific kind of substance, coexisting with space,

matter and physical fields. The relational conception of time is conventionally associated

with the names of Aristotle, G.W.Leibnitz and A.Einstein. The most ardent adherents of

the substantial conception of time are Democritus, I.Newton and N.A.Kozyrev.

Nowadays physics is based exclusively on the relational conception of time. This

manifests itself in the fact that only matter and physical fields are regarded in all physical

theories as material objects, without any time substance of a “specific kind” involved. With

such an approach it is impossible to determine purely logically, whether a time substance

exists or not in reality, since it is impossible to prove the presence or absence of something

which is not defined.

The aim of the present paper is to formulate the fundamentals of a physical theory

based on the alternative, substantial conception of time. N.A.Kozyrev’s ideas about an active

role of time in the phenomena of our World (Kozyrev 1991) impelled the author to do

this work.

...

All physical events occurring in nature are ordered in a certain way. This is apparent

from the fact that space and time localizations of events obey a strictly fixed law: they

form a manifold possessing completely determined properties. It is usually called the

space-time manifold or simply space-time. Within the problems solved by special relativity

one may consider this manifold to possess the geometry of Minkowski space. Let us recall

the corresponding definition.

The four-dimensional real pseudo-Euclidean space of signature (1, 3) is called

Minkowski space. (Sometimes the signature (3, 1) is used.) Like any Euclidean space,

Minkowski space comprises three elements: a basis set, a vector space with scalar multiplication

of vectors, called the associated space to the Minkowski one, and a mapping assigning

a vector of the associated space to each ordered pair of points from the basis set.

With regard for such a construction Minkowski space is sometimes called a point-vector

space. The vectors of the associated space and the points of the basis set are conventionally

called vectors and points of Minkowski space itself, while the metric form defined on

the direct product of the associated space with itself, is called the metric form of Minkowski

space.

In special relativity it is conventional to denote by the term “Minkowski space” just

the manifold formed by space-time locations of physical events, i.e. the space-time. The

points of this manifold are called world points or events. (The latter reflects the fact that a

“physical event” is understood here in an idealized sense, namely, as a position of a point

object at a given place of space at a given time instant.)

It should be emphasized that in special relativity Minkowski space formed by

points-events is treated as a physical reality but not just as a mathematical abstraction. It is

of importance that Minkowski space is a unified manifold, unseparated into space and

time, in which it fundamentally differs from our intuitive image of the Universe. The fact

that we perceive time and space separately is related apparently to the specific character of

our organs of sense (to which we adjust our physical instruments), lying in our ability to

perceive only those characteristics of physical systems which correspond not to Minkowski

space vectors themselves but separately to their spatial and temporal components. Note

that the components of the same vector, calculated in different frames of reference, may

take different values. It is due to this fact that the spatial size of a body or the time interval

between two events may take different values being measured in different frames of reference,

which is a well-known effect of relativity.

The substantial conception of time, underlying the subsequent constructions, has a

long history. Along with the substantial conception of space, it dates back to Democritus’

ideas ascribing a special kind of being a empty space. This conception has been most fully

embodied in the Newtonian notion of absolute time. According to I.Newton, absolute time

and absolute space are self-sufficient entities, independent both of each other and of material

objects contained and processes occurring in them. It could be said that the Newtonian

ideas of time have completed the formation stage of the substantial conception of time.

The further important step in the development of the substantial conception of time

was made by N.A.Kozyrev (Kozyrev 1991). In his book “Causal or asymmetric mechanics

in linear approximation”, published in 1958, N.A.Kozyrev formulated a number of axioms

endowing time with properties in addition to duration, due to which time interacts with

different physical objects and processes. He called these properties of time physical or active.

...

To clarify the difference between Newton’s absolute time, independent of anything at all, and Kozyrev’s

changeable time interacting with the objects of nature, the following example could be given. In

mechanics, while describing solids, the notions of perfectly rigid and deformable bodies are used. Postulating

that a solid is a perfectly rigid body, we restrict its kinematical properties to the capability of moving

as a whole. Abandoning the idea of perfect rigidity and assuming that the body may be deformed, we obtain

an object with a variety of kinematic properties. Such a body can both move as a whole and be deformed

reversibly or irreversibly. It can contain fixed or moving internal sources of stress, various waves

propagating, etc. Similarly, N.A.Kozyrev’s abandoning the idea of absolute time and endowing time with

properties besides duration can far enrich this notion, one of the most fundamental in physics.

Unfortunately, N.A.Kozyrev did not provide a rigorous mathematical formulation

of the notion of time substance in his papers. It should be noted that he did not use the

term “substance” with respect to time at all and spoke less certainly about time as a

“phenomenon of nature” which through its “active properties” may affect the course of

events. The absence of a clear definition of time substance is a feature of other publications

dedicated to the substantial concept of time as well. Besides, these publications neglect the

fundamental difference between the time substance and any other physical field and matter.

Namely, the time substance, if it exists, is necessarily an object of the fourth dimension,

orthogonal to the three-dimensional space embracing matter and fields. Just this conclusion

concerning the properties of the time substance undoubtedly follows from relativity.

Allowing for the aforesaid, we shall construct the theory on the basis of the following

approach. Let us combine the substantial conception of time and the fundamental

premise of modern physics that space and time form a single manifold. For simplicity we

restrict ourselves to the case studied by special relativity when the above manifold is the

four-dimensional real pseudo-Euclidean space of signature (1, 3), i.e., Minkowski space

(see Section 2). Thus we adopt the following postulate.

Postulate I. Space and time form a unified four-dimensional substance; it is

endowed with Minkowski space geometry and possesses certain physical properties due to

which it interacts with matter, physical fields and processes occurring in it.

We call the postulated object space-time substance and denote it by S.

In this paper we shall not specify the physical properties of the substance S but just discuss the

consequences following from this postulate and a few postulates formulated later.

Since physics is a science of three-dimensional bodies, it is reasonable to introduce

a notion unifying all the three-dimensional material objects, i.e. matter and physical fields.

This unification is conventionally called physical space. For short, we shall call it our

World.

Matter and physical fields as structures of the space-time substance

The substantial model of space-time under consideration admits different versions

of relations between our World (i.e., matter and physical fields) and the time substance S.

By one of the versions, our World and the substance S are mutually independent

physical realities. At first sight such an approach appears to be plausible; however, it is unsatisfactory

because it leaves unresolved the problem of metric transfer from the substance

S to the matter and fields. The situation is further aggravated by the fact that if the matter

and the fields are independent of the substance S, it is admissible to consider a limiting

case when there is no substance S at all. What happens in such a case? Are the matter and

the fields left without a metric, or, maybe, they possess a specific metric of their own,

which, according to the contents of Section 10, can be uncoordinated in different spacetime

points? No apparent answer to these questions is seen. Meanwhile, as shown by practice,

if the foundations of a theory leave unanswered any questions of this kind, concerning

the most fundamental features of the phenomena to be described, there is little hope that

such theory would answer them after a deep elaboration.

However, another version of the relations between the World and the time substance

is possible. We will take this version as a basis. It is established by the following

postulate.

Postulate IV. The matter and all the physical fields which form our World are

not independent physical entities but are specific structures of the space-time substance.

Our World as a whole is a solitary wave (like a soliton) propagating in the space-time

substance.

The adoption of this postulate is justified by a primary nature of the notions of

space and time as compared with those of matter and field; this nature manifests itself in

the fact that the former can, at least in principle, exist without the latter, while the reverse

is not true. Indeed, the idea of Minkowski space, unfilled with matter or fields, is quite

meaningful as long as it can be given a rigorous mathematical description; unlike that, the

idea of a material body having no spatial characteristics, in particular, occupying no (even

zero) spatial volume, as well as the idea of a material process having no temporal characteristics,

are deprived of any physical content.

This subordinate type of relation may be exemplified by the relation between a crystal and crystal

lattice defects contained in it, such as vacancies, dislocations and others. The example of dislocations is

the closest to our topic. For this defect, being an elementary carrier of a crystal’s plastic deformation, an

equation of motion has been derived, the notion of mass has been introduced, forces acting on it due to

other defects have been calculated, etc. All that shows that a dislocation behaves in the corresponding theory

as an independent material object (Hirth and Lothe 1967; Shikhobalov 1978, 1982, 1990, etc.) However,

actually a dislocation is not an individual material body. One cannot take it away from a crystal and

study separately by a microscope. It is just a specific state of the crystal itself, a specific structure in it, so

that a dislocation cannot exist without a crystal. Just this subordination relation between objects, such that

one of them is only a structure of the other, although it behaves in some respects as an independent material

body, is the one adopted for the model being described.

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If the time substance S is endowed with pseudo-Euclidean geometry, the wave of

our World, mentioned in Postulate IV, is, in general, different in different coordinate systems

(although its propagation directions are coordinated, see Section 2). If, on the contrary,

it is endowed with proper Euclidean geometry, then the wave of our World is unique

for all the coordinate frames. Note that in both these cases the wave of our World has a

flat shape. If the model is extended to the case studied by general relativity, the substance

S should possess the geometry of a pseudo-Riemannian space. Thereby, due to specific

effects predicted by this theory, both the substance S and our World wave will be appreciably

curved near structures of high energy.

The effect of matter and fields on the time substance geometry can be illustrated on the example

of disclinations, crystal lattice defects related to dislocations. A crystal lattice in a defect-free state has flat

atomic layers, while a disclination created in it causes a deformation described by a nonzero flexuretorsion

tensor (de Wit 1970, 1973a,b,c). In a certain meaning, there is a similar situation with the substance

S. Having the geometry of flat Euclidean space if the matter and fields are absent, it acquires the

geometry of a curved Riemannian space when these are present, so that the curvature value near a certain

structure is the greater, the higher the energy (mass) of that structure. However, as the structures commonly

dealt with cause very small curvatures of the substance, these curvatures can be neglected in the

first approximation.

Thus by Postulate IV the matter and fields are certain structures of the time substance

(like condensations, vortices, dislocations, etc.). In such a version of the relations

between the World and the time substance the problem of metric transfer from the substance

to the matter and fields, posed in Section 10, is resolved at once. As mat ter and

fields are specific states of the substance it self, no special met r ic t ransfer

is required since these objects have a common met r ic with the subst

ance from the out set .

It is easily verified that Postulate IV leaves unchanged all the constructions of the

previous sections.

Apparently just one of the propositions of the previous sections could cause doubt as regards the

possibility of its extension to the case considered. It concerns the use of inversions and rotations which

transform the World M and the time substance S s epa r a t ely fr om each other. (Such transformations

were used when the model symmetry was analysed.) The plausibility of using these transformations can be

easily illustrated again on the example of dislocations in a crystal. The fact that a dislocation is a structure

over a crystal lattice, as it is well known, does not exclude the possibility of different positions of a dislocation

with respect to the lattice. Similarly, in the case of the World M the fact that it is a structure of the

time substance is not by itself an obstacle for realizing its different positions with respect to the substance

S. Therefore in the present version of the model it is admissible to use the inversions WM and WS and the

rotations YM, YS, FM and FS which transform the World M and the substance S independently of each

other. Therefore all the conclusions of Sections 8 and 9 concerning the model symmetry, remain valid for

the model version incorporating Postulate IV as well.

We would like to restrict the discussion of the present version of the model to a

few brief comments.

Evidently, the idea of a time substance satisfying Postulate IV, is in some respects

close to the quantum-field-theoretical concept of physical vacuum from which particles of

matter are created. Meanwhile, our model is free of a certain ambiguity inherent in the

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physical vacuum concept. The latter consists in the fact that the term “vacuum” in its very

sense denotes emptiness, i.e., absence of anything at all, and at the same time in quantum

field theory vacuum is endowed with certain physical properties, i.e., is actually treated as

a material object. Such an ambiguity certainly cannot favour further development of the

theory.

The suggested model version observes the famous Occam principle (Okun’ 1988,

p.187) claiming that essences should not be multiplied without necessity. Here, instead of

numerous sorts of matter and physical fields there is only one essence, the time substance,

while all the rest is just its structures.

The fact that modern physical theories are successful in describing the properties of

matter and fields without addressing to a time substance forming them, does not mean that

such a substance is absent. Recall that lately in the 19th century it was also believed

(Physicists Joke 1966, p.32) that the then available physical theories were quite sufficient

for describing the properties of matter, although nothing was known about the elementary

particles forming it. By the way, modern physics is successful in doing without the notions

of life, man, consciousness (such notions are just absent in either “Physical Encyclopaedic

Dictionary”, or in subject indices to the ten volumes of “Theoretical Physics” by

L.D.Landau and Ye.M.Lifshitz), which nevertheless does not mean that those phenomena

do not exist.

A difference between the presently introduced time substance and the known ether

models is as follows. The time substance S is four-dimensional, while ether is threedimensional.

The substance S flows across our World normally to it, while ether is at rest

with respect to the World as a whole (in this connection it is often considered as an absolute

frame of reference). The substance S possesses pseudo-Euclidean geometry and

therefore satisfies all the statements of special relativity while ether is commonly endowed

with proper Euclidean geometry, leading it to contradictions with relativity.

That the time substance has not yet been discovered by experiment, can be explained

by the fact that the physical instruments available and our organs of sense are able

to interact only with matter and fields but not directly with the time substance forming

them.

Here again it is pertinent to draw a parallel with a crystal containing a dislocation. As known

(Hirth and Lothe 1967), in an infinite crystal a rectilinear dislocation at rest is not subject to forces from

the crystal lattice. Only as a dislocation moves, the crystal lattice exerts an influence on it, hindering its

motion (the so-called Peierls resistance). However, even this influence is small as compared with the hindering

action of other defects in many crystals. Therefore it could be said that a dislocation does not “feel”

the surrounding crystal; in other words, “from the viewpoint” of a dislocation no crystal exists at all, there

is only itself and other defects of the same kind. In exactly similar fashion our sensual and instrumental

feelings might deceive us saying that there is no time substance, although maybe it is the one we are consisting

of.

Thus the suggested substantial model of space-time in its version incorporating

Postulate IV, easily resolves the question of why the metric is coordinated in different

space-time points, the question having no answer in modern physical theories. This version

of the model reduces the properties of all the physical objects of our World to those of the

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time substance. A further development of the model should consist in concretizing physical

properties of the substance which would satisfy Postulates I - IV.

Conclusion

Space-time as a four-dimensional substance and the three-dimensional World moving

through it are the basic features of the suggested model. It gives a clear meaning to the

notions of time flow and time direction and easily proves a proposition on the World symmetry

similar to the CPT theorem of quantum field theory, while the method of specifying

space-time coordinates is brought into correspondence with that adopted in mechanics. It

is shown that the observed mirror asymmetry of the World, along with its asymmetry with

respect to particles and antiparticles, can be consequences of the action on the World exerted

by the space-time substance. A version of the model has been suggested in which our

World is a specific structure of the space-time substance. It has been possible to obtain all

these results without knowing the physical properties of the substance. Their specification

is a subject of further investigations.