Signals, Relativity and Optimality in Nature and Technology
Механика
Математическое моделирование
УДК 531
Signals, Relativity and Optimality
in Nature and Technology
E.A. Galperin
Departement de mathematiquesUniversite du Quebec a Montreal
C.P. 8888, Succ.Centre Ville,Montreal, Quebec H3C 3P8, Canada
All processes in Nature and technology are realized by transmittal of forces and actions (information) with certain signals which takes time and is oriented concurrently to the flow of time. This includes the propagation of fields at finite (possibly variable) velocities.The process evolution (motion) follows certain path or propagation route which is always optimal with respect to some criteria (known or unknown) within natural or technological bounds. This provides for an orderly deterministic or stochastic (under disturbances or in probabilistic description) evolution of a process. Transmittal of forces (information, actions) at finite velocities implies the relativistic effects considered in [A. Einstein,Zur Elektrodynamik der bewegte Körper. Ann. der Physik, 17 (1905)891–921] with respect to the rays of light as the carrier of relativity in observation.Naturalsynchronization oftime in different reference systems at rest or in motion is conditioned on thephysical processes (signals) that transmit the information in process evolution, and it is achievable only within some margin of accuracy. Natural time delays in transmission of action by physical processes are intertwined with relativistic phenomena in a structure of mutual interdependence. This requires aunified study of process evolution, with the information transmittal, time uncertainty, optimality and relativityas the basic elementsin their intimate interrelation at finite velocities, in both deterministic and stochastic environments. Analysis of relations between these basic elements in process evolution is presented in this paper which opens new perspectives for research and development in physics, engineering and technology.
Keywords: Signals;Relativity; Optimality; Abstract and Real Time.
1
Signals, Relativity and Optimality in Nature and Technology
1.Introduction
Back in1924, the first volume of Methods of Mathematical Physics by Richard Courant and David Hilbert was published by the firm of Julius Springer, and in the preface Courant says: "Since the seventeenth century, physical intuition has served as a vital source for mathematical problems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from the roots of mathematics in intuition, have concentrated on refinementand emphasized thepostulational sideof mathematics, and at times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific development may split into smaller rivulets and dry out…" The drive for innovation at all costsgained so much popularity and prominence that certain natural laws and properties were not noticed in some surrealistic considerations promoting new theories and notions,like the absolute time,the infinite speed, the instantaneous actions. As an example, we reproduce the announcement in the Noticesof AmericanMathematical Society, p. 453,of March2012:
"*2-4 Superluminal Physics & Instantaneous Physics – as new trends in research (electronic conference), University of New Mexico, 200 College Road, New Mexico.
Description: In a similar way as passing from Euclidean Geometry to Non-Euclidean Geometry, we can pass from Subluminal Physics to Superluminal Physics, and further to Instantaneous Physics (instantaneous traveling). In the lights of two consecutive successful CERN experiments with superluminal particles in the Fall of 2011, we believe these two new fields of research should begin developing. A physical law has a form in Newtonian physics, another form in the Relativity Theory, and different forms at Superluminal theory and at Instantaneous (infinite) speeds – according to the S-Denying Theory spectrum. First one extends physical laws, formulas and theories to superluminal traveling and to instantaneous traveling. Afterwards one founds a general theory that unites all theories at low speeds, relativistic speeds, superluminal speeds, and instantaneous speeds – as in the S-Multispace Theory.
Deadline: Papers should be sent by July 1, 2012, to Professor (name omitted).
Information:
The consideration of Newtonian absolute time, instantaneous transmission of actions and instantaneous propagation of light and certain fields may serve as an approximation to reality, bypassing relativity. However, the"Instantaneous Physics…at Instantaneous (infinite) speeds…" as a general approach to sciences is product offantasy, and often just wishful thinking according to the following citation:
"Experimental results can be clouded by wishful thinking. Back in 1953, Nobel Prize-winning chemist Irving Langmuir coined the expression "pathological science" to describe a process in which a scientist seems to follow the scientific method but unconsciously strays in favor of wishful thinking. Pathological science is distinct from fraud; it is essentially faulty science promoted by people who are somehow blind to the evidence against their own ideas" (Montreal Gazette of September 15, 2012,page B5.)
Indeed, the above expressions "Instantaneous Physics,Traveling,Speeds", and the like are self-contradictory. If, say, a mass is instantaneously transmitted from one place A into another place B, then, in fact, that mass at the same moment of being at A is also at B, as prescribed by the word instantaneously which means "at the same moment of time".If applied to the transmission of time, it would mean that all points of the Universe would have the same time, and if the bodies of the Universe are allowed to change in their own ways, there would be uncountable sets of same time values meaning something (what ??) at all spots of the Universe. For this reason, the words instantaneous speed, change, etc., have no sense, being absurd. In Nature, all velocities are finite, and thetime-values at different points may be equal, or different due to transmission of time at finite velocities.
Usually, the current time is measured by clocks, with different clocks in different spots showing different current times (some of which wrong because of bad clocks). Here we consider the Time as a physical parameter which existed always, even in the epoch of dinosaurs when there were no clocks. This time-parameter is present and changing in all processes. It is convenient to consider this unique time-parameter by its uniformly increasing value which presents the positive orientation of natural time.We do not consider speeding or lagging clocks or time-functions, sometimes used to denote time with respect to which some processes may be described in a simpler way.
The time-values are transmitted by signals, usually by the rays of light [1–2], or radio waves, or sound waves, – all having finite velocities. The forces and actions are also transmitted by signals, not by the rays of light which do not propagate in metals and some other media. Transmittal of forces and actions is directional and follows one, several, or all (spherical waves) directions which are optimal with respect to some criteria (known or unknown) that assure the orderly transmission of actions. These optimality criteria hold for any small interval of time, thus presenting total optimality, in contrast to the terminal optimality imposed by technical or economic considerations. For example, Fermat’s principle of minimum time for passage of the rays of light, or the least action principles in mechanics are total optimality criteria that determine the path for rays of light or actual motion in mechanics. The optimality criteria may be not fixed, but changing in time which implies the changing directions or velocities, leading to a process corresponding to variable optimality which prescribes directions of signals. All processes evolve optimally with respect to the optimality prescribed by Nature or by technological requirements in process control which modify the optimality over some intervals of time in the way desired by people.If the processes P1, P2 evolving in different frames K1, K2 are dependent in their evolution, such dependence is realized by certainsignals transmitting the action at finite speed which implies relativity present in such interacting processes, and not only in their observation by the rays of light.A measured (identified, occurred) point-value z(t) of time, z(t) t, or some other quantity, z(t) t, depending on time, when transmitted by a physical process relates to an instant which, at the moment of reception, is already in the past. If transmission is carried over a short length with the speed of light, its time > 0 is very small, so transmitted z(t) is considered at reception as current value despite that, in fact, it is already past, the current value being z(t+), where > 0 is unknown and depends on a finite speed of information transmittal.
In this paper, some general notions about the real time synchronization, information transmittal, relativity and optimalityin their interrelation are discussed, which are important for process evolution and control in nature and technology.
The paper is organized as follows. In Sec. 2,Einstein’s definition of simultaneity is presented with further discussion related to the real time of synchronized clocks affected by the natural time delays due to information transmittal. In Sec. 3, Einstein’s relativistic model [1, 2] is considered with derivation of the calibration factor for abstract time, verification by spherical wave propagation, and observed relativistic contraction along the X-axis. In Sec. 4, the generalized relativistic transformations in real time are obtained for observations in experimental physics and astronomy. In Section 5, the relativity in transmission of energy and action is considered. Section6 presents discussion of multiple relativities as the origin of entanglement and non-locality (the EPR paradox [3]) in quantum mechanics. In Sec. 7, the relativistic effects acting on the mass in transmission of forces are discussed, and in Sec. 8 the general results and special points of interest are summarized, followed by the references immediately relative to the problems considered.
2. Definition of Simultaneity:abstract time and real time
First, we reproduce the original Einstein’s description of time and simultaneity [1, § 1]in the English translation from the Russian edition [2, pp. 8–10]. For a coordinate system "in which are valid the equations of mechanics of Newton", called "still system", or system at rest, the following is written.
"When desired to describe a motion of a material point, we specify the values of its coordinates as functions of time.Thereby it should be noted that such mathematical description has physical sense only if it is first understood what is meant by "time". We should pay attention to the fact that all our considerations in which time plays a role are always the considerations about simultaneous events". Then we read on page 9 of [2]:
"If at point A of a space there is a clock, then an observer at A can establish the time of events in immediate proximity of A by observing the simultaneous with those events positions of hands of the clock. If at another point B of the space there is also a clock (we add "identical as the one at A"), then in immediate proximity of B it is also possible to make time estimate of events by an observer at B. However, it is impossible without further hypotheses to compare timing of an event at A with an event at B; we have yet defined only "A-time" and "B-time" but not the common for A and B "time". The latter can be established by introducing a definitionthat "time" necessary for passing of a ray of light from A to B is equal to "time" necessary for passing of a ray of light from B to A. Consider that at a moment tA of "A-time" a ray of light leaves from A to B and is reflected at a moment tB of "B-time" from B to A returning back at A at a moment t’A of "A-time". The clocks at A and B will be, by definition, synchronized, if
tB tA = t’A tB . (1)
We assume that this definition of synchronization can be made in a non-contradictory manner, and furthermore, for as many points as desired, thus, the following statements are valid:
1) if the clock at B is synchronized with the clock at A, then the clock at A is synchronized with the clock at B;
2) if the clock at A is synchronized with the clock at B and with the clock at C, then the clocks at B and C are also synchronized with respect to each other.
Thus, using certain (thoughtful) physical experiments, we have established what should be understood as synchronized located in different places still clocks, and thereby we evidently achieved definitions of the concepts: "simultaneity" and "time". "Time" of an event means simultaneous with the event indication of a still clock which is located at the place of the event and which is synchronized with certain still clock, thereby with one and the same clock under all definitions of time.
According to experiments, we also assume that the value
2AB /(t’A tA) = V
(AB is the length of a segment) (2)
is a universal constant (the speed of light in vacuum).
It is essential that we have defined time with the help of still clocks in a system at rest;we shall call this time that belongs to a system at rest, "the time of still system".
"§ 2. About relativity of lengths and of segments of time(Sec. 2 of [1]).
Further considerations are based on the principle of relativity and on the principle of constancy of the speed of light. We formulate both principles as follows.
1. Laws which govern the changes of state of physical systems do not depend on which of the two coordinate systems, moving with respect to each other with a constant speed along a right line, these changes relate.
2. Every ray of light propagates in a "still" system of coordinates with certain speed V irrespective of whether the ray oflight is issued by a resting or moving source.
Thereby, formula (2) applies, and the "segment of time" should be understood in the sense ofthe above definition".In these and following citations, the quotes, notations and italicsare by Einstein, but formula numbers are ours (no numbering in the cited papers of Einstein). It is worth noting the following remarks of Einstein from his answer to Paul Ehrenfest [4]",see also [2, pp. 51–52]:
"… Principle of relativity, or more accurately, the principle of relativity together with the principle of constancy of the speed of light, should be understood not as a "closed system" and not as a system in general, but only as a certain heuristic principle containing in itself only statements about solid bodies, clocks and light signals. All other results the theory of relativity renders only because it requires the existence of links between events that were perceived before as independent.
… In the theory of relativity, we are still far from the final goal. We know only kinematics of rectilinear motion and the expression for kinetic energy of a body in translational motion if it is not interfering with other bodies (footnote: "That this is essential, we shall soon show in a separate paper",see [2, pp.60–62, § 3,"Remarks on dynamics of a solid"]).
Remark 2.1.The two principles of Einstein and his answer above are based on the results of well known physical experiments that were done using sources of light moving with velocities much less than the speed of light itself. For this reason, we interpret both principles under the restriction that a source of light moves with velocity strictly less than the speed of light.
Observers at A and B clearly do not physically coincide with the points A and B, thus, to be observed (received, registered), the time estimates of the moments of arrival at A and B in (1) must be transmitted to the observers near A and B visually or otherwise, by a physical process which takes some time 0. Thus, if we want to consider in (1)the time estimates of the moments registered by a sensor (observer), we have to agree that those estimates ofthe moments of arrival of the ray of light at A and B will not be received by the observers, or registered by the sensors, at the very same instants as the light arrives at those points, but a little later. It means that reception, orregistration,of time estimates of arrivals isnot simultaneous with the actual arrival time of the ray at A and B but relates, in fact, topast moments, due to a finite speed of information transmittal to the sensors (observers). Hence, if we want to consider the real time estimate registered by a sensor, not some arrival that actually occurred but is not yet detected(received), we have to replace the estimates in (1) by the instants of actual reception of past arrivals, and add totBcertain time interval 0 of reflection in the mirror at B which time interval is contained in time differences of (1) if reflection in a mirror is not instantaneous. This renders the equation for experimentally observed time estimates that correspond to the genuine moments of arrival already past:
(tB+ +B) (tA +A)= (t’A +A)
– (tB+ + B ), A , B (0, ] . (3)
The time estimates in parentheses we shall call real time, which is the instants registered in the sensor as times of arrival,with delays due to information transmittal. The moments indicated in (1) we shall call abstract time.
Abstract time in not a fictitious moment, – it has really occurred but cannot be known exactly. It can only be estimated up to some precision and with a delay equal to duration of information transmittal by an available physical process. Classical relativity theory operates with abstract time, thus, ignoring delays due to information transmittal. Of course, this simplifies the analysis, but makes its verification and results subject to added inaccuracies of information transmittal which in some cases may be quite large and comparable with purely relativistic effects. For this reason, it is interesting and important to consider a parallel representation of relativity theory in real time, to compare it with classical representations and results presented in abstract time.
If information transmittal were instant-taneous, or if itis ignored, then abstract and real time coincide. Real time is the time of actual reception of a signal, being it in observation or in action transmitted by the signal. Abstract time t is the time considered in thought experiments which is time past and uncertain, being in a leftneighborhood of the exact real time t + of the reception of the signal. It means that exact synchronization of clocks postulated in (1) is conditioned on duration of information transmittal and on the time of mirror reflection that may be positive of the order 10-10sec, which awaits experimental confirmation, see [5, Sec.3.4].However,synchronization in(3) can be achieved within some margin > 0 of time uncertainty.
Remark 2.2.As concerns relation (2), the time delays A of information transmittal cancel out, but the time of reflection in mirror B, if positive, is contained at left, though it does not interfere with the principle of constancy of the speed of light V which is just a little less if computed by (2) with included: 2AB / (t’A tA + ) = V.