Deceptive Physics

Since the school years (or maybe even earlier), we are used to associating physical objects with mathematical constructs; however, only the most naive will believe that such abstraction are what physical objects really are. On the elementary level, some prudent professors still try to explain the conventionality of physical notions to the students: there always exist certain characteristic thresholds determining the safe range for the measurable physical quantities; any theory at all would no longer be meaningful as we approach the limits of its applicability, so that to drive us to a different qualitative picture implying some other quantitative estimates. With all that, entering the course of theoretical physics, a student is bound to experience a kind of psychological shock: there is no matter of fact any more, as the "exact" mathematical methods are put on the top of the list; that is, physical intuition is to give place to mathematical sense. A smart formalism is then believed to necessarily correspond to something in nature, so that the esthetical pleasure and the admiration of the prestidigitator skills become the primary criterion of truth.

The general theory of relativity serves as a classical example. The famous Einstein's equation has been introduced on a purely formal reason as the most general covariant structure containing the derivatives of the metric tensor up to the second order. The beauty has absolutely fascinated physicists so that almost nobody has tried ever since to discover a meaningful physics behind the formalism; all we can meet is but various excuses, bringing nature under the ready-made scheme. Quite similarly, modern suggestions of grand unification are most attractive due to the universality of the basic strategy of deducing all the observables from a common symmetry, so that the many ways of its violation would produce numerous physical landscapes. The method of theoretical physics mimics mathematician's mentality: there is nothing but a formal theory, and all the rest can only be treated as it "models" littered with the implementation details.

From the scientific viewpoint, it is these minor peculiarities that are of real interest; we go into particulars seeking for significant deviations from the formal scheme. When an experimentalist doggedly tracks down any instrumental errors, an increase in accuracy is never an end in itself, or an attempt to approach a allegedly self-existent mathematical abstraction; the real motive is to come upon the limits of our current conceptualizations, to guess the moment when it's hard time to stop it and switch to a different occupation. This is much like summing up an asymptotical series, or iterationally solving the equations for an essentially ill-posed problem: there is no sense in going on if any attempt to proceed would only worsen the result.

All the physical theories (with no exceptions) are only valid in this "asymptotical" sense, far away from any boundaries. When a physical parameter becomes infinitesimally small (or infinite), this is an indication of violating the limits of applicability: the formal model in use does not correspond to the real behavior of the physical system any longer. Truly physical theory is indeed a hierarchy of special models adjusted each to its own scale, and satisfying certain usage conditions. No formula, however elegant, can pretend to provide a perfect picture of the world; such impressive abstractions are mere metaphors, or witticisms. Nature can allow for some level of subjective dictates, for us to be able to arrange our affairs within an arbitrary pattern; still, one day, one is to get rid of the deceptive stereotypes, which is the more painful the deeper they have rooted in our minds.

The common-life prejudice of space and time seems to be one the most ingrown kind. We all perfectly know that everything happens somewhere and some time. The place we call a spatial point; a accomplished event is a label for a moment (or instant) of time. Later one could discover that the episode also involves many adjacent points; well, let us refer to it as a spatial area, with the reservation of the right to distinguish individual points whenever necessary. Similarly, lasting (spread in time) events take the place of simple instantaneity, though the (at least principal) possibility of detecting individual instants still dwells on us as a picture of durations built of infinitesimal ticks of the clock. Now, just impose the standards, denoting the points and instants with numbers; the rest will follow from sheer math… Mathematicians will gracefully lend us the appropriate formal constructs, such as the sets of the continuum cardinality. Since the abstraction of a number does not imply any physics, space and time appear to exist on themselves, regardless of any moving bodies or wave propagation; in this way we just "embed" any physics in a mathematical space, so that it is only the modes of the transition from one abstract point to another that can to be meaningfully discussed.

Lorentz, Poincaré, Einstein and other founding fathers of the modern physics are said to initiate a revolutionary denial of such prehistoric visions, for the theory of relativity to gain an entirely new physical sense. Does that correspond to what we finally get? Frankly speaking, the idea of the real existence of mathematical abstractions has in no way been affected: the only difference of relativistic mechanics from that of Galileo and Newton is in the abandoning the time-independent three-dimensional space in favor of a combined space-time with the signature (1, 3); it is into this abstract container that we embed all the physical processes, which, in this picture, are far from any processual character, rather resembling purely geometrical objects, trajectories and flows. One step in advance, two steps backwards. The considerations of general covariance bring no bright spots, as the material fields are only summoned to distort the same ideal space, which crookedness we identify with gravity, just for formal convenience.

To be honest, one has to indicate that working physicists were never entirely comfortable with all those conceptual strains. In the early days of the relativity theory, they made spontaneous attempts to explain things to the general public (and thus get more assurance for themselves). All in vain. By the voluntary decision, the junk has been swept under the cover, to get rid of the moral inconveniences once and for ever; let the fundamental structure of the theory be an a priori postulate. For a diffident pretext, take the absence of any experimental evidence for the contrary, in the span of many decades.

As a lucky turn, quantum theoreticians came up with a pack of tricky technologies that the public was even less capable to grasp. Once again, in its tender years, quantum mechanics seemed to entirely destroy the classical space-time reasoning, eliminating the very necessity of raising any clamor. Unfortunately, the relief was not but ephemeral, since the only difference from the classical case was in putting the same space-time inside the physical system, so that we can no longer measure it in any direct manner, satisfying ourselves with plausible conjectures. That is, yet another step off the meaningful physics towards the mathematical thickets. Well, there is nothing really new in principle; still, the lack of technical background leaves no choice to a layman beyond the blind faith in the competence of the sage ones. The scientific revolution did not dispel superstition, but rather fed it up to bring science to the edge of religion.

Well, I certainly lay it on somewhat thick. Nothing is yet lost. Practical demands are to introduce the necessary corrections, and the ends will meet anyway. Physicists are not supposed to know why. They've got a hell of their professional concerns, with no time left for a hardcore philosophy.

With all that, a few explanations may be quite appropriate in the popular literature (especially for children). Thus, David Bohm, in his book about the special theory of relativity [W. A. Benjamin, Inc., 1965], righteously indicates that the spatial coordinates and time have no sense on themselves; they are always related to some common (socially established) measurement schemes, appearing as an outcome of interaction between the physical system and the instrument. For such entities, that are not imposable a priori, but must be revealed in the course of a thorough analysis of experimental data, the thought about a nontrivial interdependencies of space and time is quite natural; in this context, relativistic theory should no longer bring about any psychological tension. If so, why not move a little bit further? To determine velocities, even more indirect procedures are to be involved, eventually expressible in terms of coordinate and time measurements. Why, then, not fancy the speed of light as an artefact based on the traditional instrumental setup? As long as we bind the technologies of determining length and duration to the process of light propagation, the constancy of that speed is a sheer tautology rather than a physical principle. So far, physicist have just nothing to compare. By the way, the dimensionality of space-time can be immediately related to the number of parameters required to describe electromagnetic phenomena.

No popular writers are that bold. They do not develop their oblique hints and quickly drop the introductory part, to proceed to computation; playing with numbers is much easier than search for the foundations of physics. Here, one could witness utterly anecdotic situations. For instance, the relativity of simultaneity and distances is often illustrated by a number of mental experiments in a geometry entirely based on the assumption of velocities much smaller than the speed of light; the results thus obtained are then triumphantly fed to the reader as the convincing evidence of the impossibility of faster-than-light motion; that is, absolutely, no other options! Logically, it is perfectly clear that light exchange between the bodies with their relative velocity higher than the speed of light will produce a very peculiar pattern which is to be separately discussed. In the same deceptive line, the usual derivation of the Lorentz transform on the basis of common symmetries and the constancy of the speed of light also involves implicit under-light geometry, in the critical points; that is why it can hardly ever be considered as convincing enough.

However, there are much deeper reasons for the instinctive rejection of the rude relativism. What seems the most unacceptable is the abstract identification of the physical space and time with mathematical constructs that could as well apply to almost anything at all. Still, in nature, there are physical processes, and human activities are essentially dependent on that. One physical process (or an activity) can be compared with another. With all that, treating real things and events as point-like is only possible in a very special context, with a specific choice of the scale. When something tiny moves slowly enough, we can trace its trajectory as a sequence of momentary positions, points. On the contrary, the motion of extended bodies with very high velocities may present an entirely different picture. The usual coordinate representation is a risky enterprise for the physical speeds comparable with the speed of the probing signal; the more so for much faster velocities.

Virtually, one can easily guess what needs to be adjusted. The already mentioned Bohm's book contains a brightly indicative phrase [Ch. XIII]: if the observer in the embankment measured the position of the rear of the train at one moment and the front an hour later, he could ascribe a length of 60 miles or more to the train, evidently an absurd result.

What exactly comes out to be absurd? As a matter of fact, we need to determine the length of an extensive object; there are no preliminary considerations to decide which values are physically acceptable and which are not. Admit that the resident observer's clock is graduated in millennia; one hour is practically nothing on this scale, and the train will virtually look "spread" over many miles of the track. For a train-based observer, the length of the stock will be much less than that, in the same time scale; this leads us to the idea of length contraction in moving frames of reference. In this context, simultaneity becomes largely relative as well; however this primarily depends on the typical dimensions of the objects and the observer rather than on relative motion. By the way, such relativity is entirely compatible with the common sense: when we say that a train is (for instance) in Paris, we are not only ignoring the lengthiness of the train (taking it as a point, in its entirety), but also Paris is understood as something integral and indivisible (though, on another scale, one could think of specifying the particular position of the train within Paris). Compared to the Solar system, Earth can readily be represented with a massive point; on the contrary, for a hypothetical observer from a nearby star, the variations of the Sun's luminosity due to Earth passing the solar disk would be a regular method for determining the relative size of the planet. Similarly, one day might mark an instant in the life's span, while being felt as eternity in agonizing suspense.

In the example of Bohm's train, the observed form of the train may manifest intricate variations depending on the railway timetable. The very notion of the train's length has no meaning in this case; in the best, one could introduce a quantitative estimate of the minimal and maximal diameter of the observable structure. Effectively, we thus observe the topology of a certain segment of the railway system; other trains (serving as physical probes) would extend the picture and make it more specific. In the general case, we need to account for the modes of motion, thus switching to a kind of phase trajectory incorporating speed variations and halts. Space and time will naturally intertwined in this picture, so that our observations, beside the structure of the railway network, also provide information on the actual timetable; within the range of its permanency, there is a physics of space-time invariants.

On the other hand, thinking of a train as an extended object, what do we mean speaking of the train's front and rear? For those who measure distances in Angstroms, the question is too far from trivial! Under certain conditions, the very idea of a spatial (or temporal) boundary may need revision, or should even be suppressed. Admit that, for a standing train, some spatial distributions will realistically represent its boundaries; still, one can never be sure that the type of distribution is to remain the same in the state of a rapid motion, or on a different scale.

That is, the illustrative absurdity primarily refers to the idea of the absolute character of spatial points, their infinitesimality, regardless of the chosen scale. A physical point on itself can be of any complexity; as long as this inner structure does not pertain to the current task, we treat this system as a point. In real life, any "point" can be unfolded in large area as soon as our practical needs demand accounting for the underlying structural features. In other words, geometry and topology are not the absolute attributes of the world, but rather a characteristic of the particular mode of motion, an effective definition of a physical system. As long as our practical manipulations obey certain rules, the physics of the matter will show up as a set of invariants; approaching the limits of the model's applicability, we are bound to introduce significant corrections, or entirely revise the theoretical framework. For example, the notorious relativistic contraction of lengths and durations with the growth of a body's speed is sure to eventually violate the structure of the working space-time scale, so that the too short segments collapse to zero within the existing uncertainty; this demands a different experimental setup, with the former boundaries either becoming essentially distributed or entirely irrelevant and meaningless. Of course, along with coordinates, the same holds for any other physical values, such as mass, charge, potential, temperature, the number of particles etc.

The choice of the basic and derived physical quantities is no less relative. Thus, admit that velocity is defined by the ratio of a distance to a duration. Then, with both lengths and durations only determinable up to the characteristic parameters of the scale, speeds too must logically be treated as conditionally measurable, so that the constancy of a speed value in any transitions from one frame of reference to another is only meaningful in a macroscopic sense, far from the critical durations and lengths. For very short distances, one has to account for boundary effects; with times below certain values, the dispersion of the measurable speeds may grow beyond any limit, and the notion of an exact speed is to be replaced with some speed distribution. If, however, it were practically important to fix the exact values of a speed, this would mean that either distances or durations should be treated as secondary quantities, related to the motion of something with that a priori constant speed. The scales of time and length would then be interdependent, for any overall scale change to be effectuated in a consistent manner. Speaking of a constant speed, we actually demand that the chosen velocity scale admitted its precise representation with a number, while the physical system remained far from too small velocities, and consequently there should be a higher limit for any speed value. No need to remind that, like any physical value at all, velocities can be determined in respect to all kinds of scales, and any constancy cannot be but an abstraction.

Mathematics is justifiably usable in physics, provided there is a clear understanding of what exactly we are going to study (or employ). The structure of science is to reproduce the actual conditions of observation (or exploitation), so as to obtain a qualitatively correct picture of what really happens. Scientific formalization is never entirely abstract: it corresponds to the organization of a specific activity. For instance, when a small body (a physical point) moves too fast, a human observer cannot visually track the individual displacements and, instead of a point, sees a track, a segment of a line gradually drifting in the field of vision. The speed of the signal that we use to determine the position of the point (here, this is visible light) is high enough, and we locate all the intermediate positions; still, as the rate of image processing is much lower, the geometry of the observable motion will be determined by this strongest restriction. For some modes of motion, the transversal sixe of the moving body will be much smaller that longitudinal; one could thence conjecture that the thickness of the trail gives the true size of the body, while the observed lengthwise extension is mere artefact, instrumental effect. Still, like any hypothesis, this one is yet to be carefully validated. Say, if we get an instrument with smaller processing delays and find that the longitudinal dimensions significantly decrease, this supports the point particle assumption; on the contrary, if there is no drastic contraction, we have to conclude that the physical system is a finite-length filament. Any indirect observation schemes are, of course, possible as well; this is a usual (albeit somewhat risky) practice, to compare data from different sources. Ideally, to confirm the point-like nature of a physical object, we need a measurement device with the inner delays much less than the time necessary for the object to shift by a distance of the order of its transversal size. This is the obligatory condition for us to be able to speak of the position of a material point, and its spatial coordinates. With no means of increasing the temporal resolution of observation, we will have to account for the spatial spread of the object as a physical fact.

Such an approach may seem to reduce physics to mere optical illusion, while in reality… Still, what do you mean by "reality"? Thus, having established that a truck is to be observed within a certain block of space during some finite time, we should not get in there just for time said; otherwise, the physical fact may be aggravated by a medical incident (probably lethal). For practical decisions, the "true" physical size of the truck is of no importance; all we need is the mode of motion. Provided a theory predicts the point-like behavior, one might try to cross the track in a pre-calculated lucky moment; in fact this means becoming yet another physical instrument operating within a different scale and therefore capable of elaborating (albeit with the experimentalist's life at stake) our scientific model. Nothing subjective about this reasoning: substituting a human observer for any physical body would change nothing.

However, the temporal resolution of an instrument is not a mere numerical estimate. Any human activity develops on many levels, each with its own scale, characteristic lengths and times. Sometimes, time delays between the consecutive measurements are much shorter than typical times of the system's motion; that is how we register the system's "momentary" states. In other cases, the system may go quite a way from one act of observation to another, and its motion will then appear as a series of leaps; under certain conditions, there is a kind of "stroboscopic effect", a seemingly stationary structure. Quite plausibly, the observable space-time is but a kind of such an illusion! Indeed, to be able to measure anything, we get in resonance with some physical process and all the rest refer to this global formation, a frame of reference. Yet another observer would build his frame of reference in his individual pace; still, if we do it on the basis of the same referent process (for instance, light propagation), our frames of reference somehow correspond to each other, with the optional possibility of formally translating any measurement results from one language to the next (of course, with the due precautions about the compatibility of the scales and staying in the inner area, far from the limits of the model's applicability).

Following on, we conclude that, for each physical model, there is a number of dedicated scales supporting our capability to produce any meaningful (physically verifiable) hypotheses. Nature knows no absolute scale equally applicable to any physics at all. Physical interactions do not merely influence the structure of space-time (its metric, connectives, curvature); they rather give birth to space-time as such. Thus, the Coulomb law may well be interpreted as a definition of space in electrostatics: the squared distance is inversely proportional to the force of interaction. Similarly, the length of a road is determined by the wayfarer's fatigue (the work spent); the time on the way is related to the level of the traveler's involvement in the motive effort. Even the banal application of a rule is a kind of motion, from one mark to another.

A simple example: the swing of a pendulum determines the space of the possible positions (the angles of deflection). Add here an accurate enough clock to get an impression of a continuous space. However, due to the variations of the (angular) velocity as an effect of the pendulum's interference with the clock, the points of this oscillation space are not equally spaced , which result in a dynamical metric ("gravitation field"). Measuring time with a significant temporal discontinuity could bring about an apparent lattice structure, making the motion of the other system resemble chaotic wandering in that grid space.

Just to get more food for thought, consider the formal reduction of the angular speed to zero on the borders of the possible deflection range. As we have already seen, in nature, there is no such thing as a perfect zero; so, what could we see in a different scale? Judging by superficial logic, there is no other choice: one has to stop first, to start moving in the opposite direction… Well, just recall that it is only in mathematics that a harmonic oscillator is (by definition) what obeys the equation of the harmonic oscillator. In physics, we only use a mathematical abstraction within the range of its applicability, with no far-reaching allegations. In a different scale, the model of harmonic oscillator may become less realistic, and we'll have to fairly consider a physical pendulum, working out the boundary details, up to binding mechanics to thermodynamic considerations. Eventually, with a yet deeper insight, the very notion of a momentary deflection (a spatial point) becomes physically meaningless, as the pendulum may take several positions at once (like a quantum object, or anything like that).

Now, let us get back to Bohm's trains for a while. As expected, the physical simultaneity of detecting spatial positions and the momentary geometry of a train are essentially dependent on the scales applied, and observable motion can, in some cases, appear as a spatial structure. A universal space-time common for everybody can only be an illusion, an artefact, as commonality in the construction of a frame of reference is yet another abstraction. However, the motion of physical systems does not stick to this (formal) diversity. Our mental experiments pose the hardest problem ever: what do we actually mean by the notion of a train? How a piece of matter observed in many places in a number of time moments by different observers can figure in theory as the same and only object? In everyday life, things do not just move from one place to another; they tend to drastically change! Can a human being be treated as the same person entering a different age? Well, admit a few additional cars get coupled to a train at some intermediate station; will it be the same or another train for an outer observer? To judge by the train number in the timetable, there is no difference; physically, we have to cope with a different geometry, and it is not obvious what should be taken for the train's length in the general case. Now, the same train number in the timetable may refer to quite different stock, as determined by the car numbers indicated in the waybill. Does that correspond to the same physics? It is here that the true relativity enters the game: the definition of a physical system depends on the mode of its inclusion in some environment that need not be directly related to any physics at all. For humans, their conscious activity is to provide such a reference. In inanimate nature, one is to consider a hierarchy of physical processes that can be unfolded in any particular direction, for the higher layers to determine lower-level physics outlining the essential traits of their dynamics. The notion of a point is thus becoming fuzzier.

Take a trivial illustration: a small physical body moves along a straight line. The body is to disappear in one place and reappear in another. Since, by assumption, there is some motion, the body will certainly occupy these spatial positions in different time moments. Conversely, the moments of departure and arrival are bound to be registered in different spatial points. Nevertheless, we somehow have moral right to measure the distance thus covered (as if the dimensions of a stationary object were concerned) and divide it by the travel time (as if read off from the same stationary clock) to obtain the speed of the body as a measure of motion. For the time being, let us put aside the mass and inner complexity. One way or another, when the theory of relativity is expected to offer purely local picture of space-time, this is no more than an illusion; as a practical idea, any motion is essentially nonlocal since it takes the deliberately different as the same. The problems of quantum theory are even harder: the configuration space is infinitely dimensioned, and there are many way to pass from one configuration to another; by the basic premise, there is no way to observe that displacement.

Still, the quantum analogy allows to, at least metaphorically, visualize motion as occupying several positions at once. Let the state of a physical body's motion be represented by a weighted combination of the initial and final states; an appropriate normalization will ensure the integrity of the system (that is, we are discussing the displacements of the same thing); the way of switching between the coefficients from the pair (1, 0) to the pair (0, 1) depends on the overall character of motion (the chosen scale).

Alternatively, a classical metaphor might come as handy: considering the final size of any physical point, the process of displacement cam be pictured as a smooth flow from one container to another, with some overlap of the old and the new; the separation of the initial and final states is also dependent on the scale selection.

In either approach, the system shows up to be hierarchical: on the lower level, there are the initial and final state on themselves, abstracted from motion; the unity of the two (the state of motion) is achieved on a higher level. There is yet higher level of physical scales (frames of reference) serving to link the former two levels to each other (which cannot be done from within). Once again: the admissible scales are never arbitrary, and they constitute an indispensable layer in the structure of any physical system; this is how nature is made. The selection of a scale is in no way an observer's caprice; we can only choose from the objectively possible in accordance with the character of physical interactions, including the interaction of the physical system with the observer. As it comes to motion in nature, it may be hard to say who observes what: there is interaction of qualitatively different physical systems, which defines a higher-level system, a physical bond. It may sometimes be impossible to split the integral picture of interaction into the interacting systems as such and the process of interaction (represented by a physical system of a certain kind, the carrier of interaction); the structured theory is accurate enough in the inner domain, far from the critical points, physical boundaries (mathematically modeled by singularities, infinitely small or infinitely great values).

On the practical side, the singular regions are not generally accessible. Just because our instrumentation is to eventually be correlated with the size of our biological bodies. Technological progress necessarily expands the range of the achievable scale; however, switching a scale may be quite a challenge. For instance, to discern the structure of the light barrier (thus confirming the impossibility of the exact determination of the speed of light), one may need to reach the speeds within some trifling part of percent (dozens of decimal orders). It is highly likely that the effect could be revealed in much slower movements; but it is not clear where to seek for it. No doubt, indirect measurements are the most powerful method in science; however the risk of stretching the results to a theoretical tradition is much higher than in direct measurement.

One way or another, there is a fundamental fact: the very possibility of motion is related to the nonlocal nature of physical systems, their presence at several places and times at once. Moreover, these particular times and places are merely an expression of coexistence, interdependence, participation in the same process. This implies hierarchy: the discernible on one level is closely coupled on the next. The train as a whole is formed of a number of components (like the cars, the engine, the platforms etc.) considered, in some approximation, as an integrity; the character of the junction determines the inner space of the train; on a higher level, the train will look like a point (or a region) of a different, outer space structured in accordance with the mode of its inclusion in a yet higher-level integrity. To keep this hierarchical construct together, one needs to somehow relate one level to another: they do not exist in separation, as one level is reflected in another. That is, we have all the reasons to describe the structure of the inner space in terms of the outer space, and the other way round, to "embed" the inner space in the outer. The ways of such interconnection may largely vary, but they will never be arbitrary, since they can be represented by the elements of yet another level of the same hierarchy. The constraint of the same value of the speed of light for the inner and outer observers is one of the options; alternatively the inner and outer pictures may be interconnected in a quantum-mechanical manner: macroscopic (higher-level) quantities are thought of as the asymptotic states of inner motion (related to the choice of the basis). In thermodynamics we don't have any outer space at all, and everything physically happens "in the same point"; still, this point is not infinitely small: it has a definite volume, so that (on some intermediate level) the whole thermodynamic system can be split into subsystems represented by a collection of connected volumes. The transition from one mode of description to another depends on the overall character of motion; this is yet another indication of the limited usability of traditional formalism in mechanics, with the premise of precisely measurable coordinates.

Mathematics is indifferent to the physical structure of space. However, the very choice of model makes relativistic physics intrinsically nonlocal. Indeed, the covariance requirement is based on the assumption of the feasibility of global coordinate systems, with as global coordinate transforms. This means virtually implies the treatment of such coordinate systems as pertaining to the inner space of some higher-level physical system (a frame of reference, a global observer, or a god if you wish). On the contrary, for a material point, this coordinate description refers to the outer space, while the inside of the moving body is represented by a flat (1, 3)-dimensional space describing the available choices to move on (the direction and pace of transition to the neighboring point of the outer space). Effectively, this is an inner observer ordering the knowledge of the outer world by the degree of remoteness. Since the incoming data reflect the relativistic restriction on the signal speed, the picture of the world for the inner observer will certainly have a kind of horizon, with know information of the outer world beyond that limit. However, it does not mean that the outer world is also limited in the same manner: logic only forbids discussing too remote events within the current observation scheme. The structure of the inner space as such does not depend on any outer circumstances; this (formally infinite) space exists all at once, in its entirety, with no restrictions on the comparison of spatial regions separated by a distance of any scale. In the same way, the outer space is immediately given to the outer observer, with no need to gradually construct it depending on the succession of events. The reflexive character of this hierarchy is normally introduced by the correspondence principle: the geometry of space-time becomes nearly flat in the vicinity of every point. One level thus gets projected onto another. Still, the nature of the bond is entirely determined by using electromagnetic field as common scale for any other processes. In general, the correspondence may be less trivial, as the inner space of a material point can contain degrees of freedom that are absent in the outer world, and conversely, some outer coordinates can be statistically degenerate in the inner space.

With any choice of a scale, its structure is different from that of formal mathematical spaces by the presence of finite discrimination thresholds. A physical point is not mere number; it is not infinitely small, but rather spans over some spatial zone within which the observer (the other components of the same physical system) could not distinguish one thing from another. This has nothing to do with the discreteness of the container space; all we can assert is that very close points will merge on the chosen scale, behaving like a single material point in respect to physical interactions essential for the system as a whole. In principle, system dynamics can lead to the spatial separation of the formerly mingled material points (with their transition from one zone to another), with the effect of the birth of several equally elementary "particles". The specific features of the organization of the physical space (including zone sizes and the degree of their separation) are related to global symmetries. Thus, the presence of a limitation on the admissible values of a physical quantity (a boundary) is to produce lattice-like zone structures (similar to musical scales). That is, the finite speed range and the existence of the relativistic barrier will produce a lattice space on the global scale; the same holds for spaces containing all kinds of geometrical singularities (similar to an event horizon). No need to say that any physical symmetries cannot be but approximate, so that apparent boundaries will only show up for a very remote observer.

An analogy with atoms (or any other compound quantum systems) would be quite appropriate here. For the motion of free particles, we observe continuous spectra; with an increase of the coupling strength, specific non-monotonicity will be superimposed on that background; finally, the formation of bound states will manifest a number of narrow spectral lines, which, however, get broadened due to the virtual exchange with the spectral continuum. Similarly, for each physical system, the geometry of space-time arises in the interactions of some distinct parts: for weak interdependence, the physical space can be satisfactorily represented by mathematical spaces; closer coupling leads to geometrical and topological peculiarities, though no borders can be absolutely sharp.

Physical bodies are not like mathematical points; their inner organization will influence the outer motion. To move a material body from one place to another is not so easy, and it can never be pictured as a momentary jump. In the hierarchical model, the outer position of a physical body implies unfolding its inner hierarchy starting from some initial node. To displace the such a structured point, one needs, first, to fold the inner hierarchy, and then unfold it in a different manner, from another vertex: kinematically, any motion is a special case of hierarchy conversion. The more developed the inner structure, the higher the inertia of motion; in mechanics the overall measure of this inertance is known as mass. The interaction of massive bodies means their inclusion in a higher-order integrity; since this implies additional effort of folding and unfolding, interaction will contribute to the total mass. However, such coupling may require a transition to a different physical scale and a rearrangement of the inner and outer spaces; that is why the mass of a compound system need not be always greater than the formal mathematical sum of the incident masses: the disjoint and compound systems exist on different levels of the physical hierarchy.

Joining separate physical system in an encompassing system normally means introduction of new scales, both in space and time. Thus, when we are to determine the distance between the instruments of a human size separated by hundreds, thousand, or even millions of miles, the size of the resulting system is far beyond the original scales, and a special procedure is required to switch to the new dimensions. As long as we cannot surpass the common-life magnitudes, the structure of the higher-level scale will remain merely a working hypothesis developed upon some reasonable (in our opinion) conceptual grounds. In particular, we cannot observe the outer space in its entirety, as an outer observer would; we cannot go beyond the horizon imposed by the finite interaction propagation speed. Similarly, the necessity of coordinating events separated by times much longer that their local span, we have to base our conclusions on some hypothetical considerations, which may be far from the real structure of the compound system. The problems of that kind are inevitable; but they are not the hardest challenge to face. There is a principal moment: how do we paste two independent instances of space-time together, in the common space-time of the compound? Thus to combine the mechanical motion of two material points, we are to somehow distribute the original eight (outer) coordinates (six spatial and two time-like) between the inner and outer degrees of freedom of the new system. In non-relativistic mechanics, such coherent motion is normally described by the coordinates of the center of mass, while the reduced spatial coordinates and time refer to the inner motion (provided that the inner and outer scales are qualitatively different). In relativistic mechanics, the notion of the center of mass is ill-definable, which is basically related to the conceptual confusion: the speed of light is said to be the same for both the inner and outer observers, which is a logical fallacy, since each level of hierarchy requires a scale of its own, and one cannot identify the quantities of different nature. Physically, inner movements should be much faster than the characteristic speeds of outer motion; that is, from far away, why we only see an averaged picture, a massive point. To keep the notion of a material point for faster inner motion, comparable to outer displacements, the observer must be placed beyond the scope of the both; that is, we need a scale much rougher than the inner scales. Otherwise, instead of a point, we will observe something distributed in the outer space (and in outer time); relativistic kinematics does not pertain to the systems like that.

As a mere formality, we are free to link anything to anything at all. This invokes yet another mathematical abstraction, the notion of a set. Still, physics is basically a science about interactions. Physical system do not result from a formal comparison of different movements; on the contrary, the feasibility of adding things up is related to the real hierarchy of the world, which contains both the level of the union and the level of its components. The occurrence of various "emergent" properties in abstract system is sheer illusion. Yes, such combination are theoretically possible; but they have nothing to do with physics: at best, this is a different science, with its own spatial and temporal notions.

It's high time for a brief summary. There is no "true" geometry of space-time, and no way to meaningfully introduce it in physical theory. There are physical system of many kinds, and their geometry (and topology) is determined by the nature of inner interactions and the mode of inclusion to some production environment (a frame of reference), Inner and outer motion develop in different scales; there are objectively formed (natural) scales. Any formal derivation is only valid on the condition of clear separation of these levels, far from the transitional areas. In particular, the mechanics of a relativistic point (as well as its superstructure, the general theory of relativity) is only applicable in smooth (non-singular) motion, when the system objectively behaves as a number of points placed and evolving in a common space-time. The universality of such a picture is deceptive, and future discoveries are bound to significantly extend the range of available paradigms.