COMPLEX EUCLIDEAN \"PARA-SPACE\" MODEL FOR \'SR\' AS AN ALTERNATIVE FOR REAL MINKOWSKI PACE-TIME

{ To the Reader: This paper is for “private use” and is not intended to be a publication in the form below presented. That is why the way it is written is rather “loose” (question marks and remarks in squared parenthesis are in frequent use ) in the sense it contains some hypothetical assertions (not always proven) that often look like “contradicting” one the other (so the choice between them is postponed to the future). For example, it is not clear yet for me what is the best description of speed, both relativistic and the “Galilean” (defined in the text). Some essential computational advantages and theoretical reasons indicate all speeds, with exception of relativistic (real) speed of light c, should be considered imaginary. On the other hand, it seems to be beneficial to consider ‘Galilean speeds’ (along, say, the x-axis completed to the complex plane) as complex quantities, where their real parts are the usual relativistic speeds. Probably, it depends on the choice of the [model] complex plane (plane of speeds versus plane of positions as the adopted models ?) used for the speed description. Another hypothetical notion is the notion of “imaginary mass” (in ‘i kilograms’) (introduced in section 8) dictated by the (also hypothetical) consistency of the theory, in particular, its “Newtonian part”. I hope, that this “pre-publish” form of the work has a good reason as the number of included new ideas is too large to include it in one consistent version for publication. The reader may eventually choose versions she / he feels are more proper. Anyway, the paper in its current form may eventually be considered as starting point for a discussion. At first reading it can be started from Section 2. For a shorter version of this work see [2]. }

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COMPLEX EUCLIDEAN “PARASPACE” MODEL FOR ‘SR’ AS AN ALTERNATIVE FOR REAL MINKOWSKI SPACE-TIME Jerzy K. Filus Department of Mathematics and Computer Science, Oakton Community College, Des Plaines, IL 60016, USA, email: [email protected] or: [email protected]

Abstract In this work I consider the complex C3 Euclidean space as a model for special relativity theory (SR), considered to be an alternative to the real Minkowski M4 space-time manifold. While the basic M4 model for Einstein’s SR theory is extended to the complex spacetime C4, and then reduced to equivalent C3 “para-space”, all the well known SR theorems and properties carry over with the model transition. Remarkably, in the complex Euclidean C3 model presented here, both the theory and its interpretation are significantly simplified. Moreover, our model seems to have the potential to provide some natural extensions beyond special relativity toward more general theories that may include gravity and possibly also quantum mechanics. All of that can be expressed by means of the same kind of extremely simple ! (no tensors) mathematical description. The main result in this work is a new (possibly overlooked in the literature) simple relation between Einstein’s and Newton’s theories [for any magnitude of speeds !], both expressed within the same complex C3 model !. Remarkably, it turns out that the common relativistic behavior of “high speed physical objects” as modeled by R3 or M4 is only the result of the ‘Re( )’ projection of the classical Newtonian motion in C3 to the real R3 subspace. Probably the well-known “strange behavior” of ‘fast motion’ in relativistic mechanics, such as the Lorentz contraction, seems only to be a result of a “distortion” of the natural (but complex) classical (i.e., Newtonian) phenomena. This fact has an unbelievably elementary “mathematical explanation” within the new model ! . Another remarkable fact is the existence of a one-to-one relationship between the Lorentz transformations in M4 and some corresponding isometries (in the “usual metric sense”) in C3 thought of as Euclidean. These and other interesting facts are revealed and investigated.

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Key words: complex C3 “para-space” models; simple and exact Newton versus SR relation; universality of light speed in the complex model; on “imaginary physical quantities” 1. Introduction The subject of complex space or space-time physical models, either for the special or the general relativity, is fairly well explored in the theoretical and mathematical physics literature, dating from at least several decades ago (see, for example, [5]). Quite a long list of other references on that topic one can find in [7]. Some of those works also employ complex models for the description of quantum physics phenomena in the hope of unifying the two physical theories within the same geometrical concept of a complexified space or space-time. Up to now no “hard results” on that unification were established, see [3, 4]. However, in some works [see 3-7], the obtained partial results seem to indicate that, in general, that path of investigation is right. As for the physical model’s complexification presented here, there are three major issues specified in the following subsections which, in our opinion, are vital for this topic’s further development. 1.1 As some authors realized (see for example [3], [4], [7]), the question of the physical interpretation of the “adjunct” imaginary part of the mathematical models, that use complex numbers, is vital. The related possible conclusions may have far reaching consequences. It is perfectly well known among scientists (as well as philosophers interested in this subject) that in about all physical applications of complex numbers the powerful methods from complex analysis are used for either reinforcement of calculations, or to prove some important properties of the models and then the so obtained results are narrowed back to the real models such as real space-time. Regardless of the frequent use of complex number based math facilities, it is almost the rule that in the final “physically meaningful” solutions all the obtained imaginary “quantities” [ in the strict sense of ‘imaginary’ ( not the so called “imaginary parts”, which are real ) numbers such as ‘ (3i )’ ] , arising as parts of the underlying formulas, are simply rejected and then ignored. That mainly is so, because the complexified space-time itself is usually not regarded as having any ‘literally physical’ significance. In [3] and [4] the author, reviewing this rather common approach, wonders if such a negative point of view, regarding imaginary quantities, is really well grounded. He gives numerous examples that inclines one to revise this traditional stand point. In the introductory part of his work, Kaiser [4] gives an example of ‘tube domains’ that appear in the axiomatic quantum field theory. The existence of these domains results from the analytic continuation of certain functions associated with the theory of complex space-time. He considers as a hypothesis that “these tube domains may, in fact, have a direct physical interpretation as (extended) classical phase spaces. If so, this would give the idea of complex space-time a firm 3

physical foundation, since in quantum field theory the complexification is based on solid physical principles”. Similar ideas that opened the question on (any) physical interpretations of the so extended space-time models one can find in the paper by Newman [5] and its references. The problem of the interpretation of such “intelligible realms” may be considered as philosophical (especially ontological, but also epistemology seems to be strongly involved ) rather than a problem in physics. (As it may seem, any ontological solution of this problem may have no direct impact on outcomes of a purely physical approach. ) On the other hand, the usefulness of a proper interpretation lies not only in the obtained physical formulas, but in a better understanding, which, in turn, facilitates the mental processes in physical investigations. From my own point of view the physical or ontological nature of the arising “ imaginary quantities” (as it is supported by the complex models we have constructed) is, roughly and very generally speaking, associated with ‘dynamics’ (a ‘change’). That dynamics is, within the framework I consider, the source of ‘energy’ such as kinetic and, possibly also potential (fields of forces). An alternative to this ‘physical energy’ interpretation may possibly be (suggested, for example in [7], as the “psi reality”) the ‘psychological interpretation’ of the imaginaries as associated with the, still “mysterious”, phenomena of consciousness of any human, animal, or some other ‘beings’. As pointed out above, these interpretations are not necessarily relevant to a (pure) physical theory (or underlying computations), but might be considered as belonging to the ‘foundations of physics’ as a physics ‘meta-theory’ or philosophy. 1.2 A. Another major issue discussed in this paper is my claim on the “dependent” (not the ‘primary’) character of the notion of ‘time’. This topic’s consideration seems to involve a kind of ‘relatively new’ approach to the time notion’s handling. Even though this subject is occasionally mentioned in the form of loosely stated comments, I could not find in the physics literature any systematic treatment of that matter, neither in the form of an explicit formulation of an underlying theory, nor as a properly constructed model that would reflect that “dependence”. In the here considered framework this means that in the complex model, the “time coordinate” can be “derived”, in a natural way, from the Euclidean three dimensional (not entirely

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observable) complex “para-space” C3. This means that all information about time is implicitly included in that C3 model. When adopting “energetic” interpretation of the “imaginaries” in the complex spaces, it appears that the real or complex time (considered in, say, C3 x R1 or C3 x C1 , respectively) coordinates (R1 or C1 ) are, basically, redundant ! This is so because they bring no further information (about time) besides that already implicitly included in the six (real and imaginary) C3 coordinates alone. This is also to say that the measure of ‘time’ is built into the complex C3 space‘s structure as “derived from the dynamics”, in particular from the kinetic energy, here considered to be equivalent to motion. In turn, as it will be shown, the physical energy (that yields ‘time’) is easy derivable from the geometric properties of C3 such as ‘shape’. That shape is often represented simply by arguments (angles) of the involved complex quantities such as the physical body’s position in polar coordinates (on each complex coordinate plane). In my view, the physical dynamics are not present (modeled or described) in any geometry of the real R3 space. Therefore, once having only the real spaces one must rely on the additional fourth time-coordinate in order to be, at all, able to model motion or time. As follows from the definition given in next section, the imaginary coordinate added to each real one has a direct interpretation as being associated with a given (complex) position. The speed is then derived “geometrically” from the complex position without use of any notion of time. Consequently, time can be derived by an elementary use of speeds and corresponding distances. When the considered speeds (as in SR) are constant, time ( it turns out to be ‘absolute‘ in the related complex spacetime C4 as the invariant ‘absolute value’ of the ‘complex time’ while its real part is the, well known, relativistic time ) may be considered as a quantity directly proportional to distance, measured from the origin into the (later defined), “natural direction” (for the given speed) toward the complex interior of C3 (see next section). As the proportionality coefficient (speed), I have chosen to use the common speed of light c in the real space. At this point, Einstein’s universality of light speed is strongly involved. I would like to add to the above a short reflection on the common understanding of the notion of time as a “mysterious quantity”. That notion is really hard to understand if one considers it separately from motion [ or, more generally, separately from any kind of “dynamics” (understood as ‘any changes of anything’ including changing through chemical reactions, 5

emotional states, mental processes or other ) ]. I think that time, as a physical phenomena, is some feature of motions and not a phenomenon “parallel” (independent) to motions. Here, my view (and I am not alone in that) is, that besides of (physical) notion of ‘space’, the notion of ‘Energy’ (or just ‘change’) is more Fundamental than ‘Time’. 1.2 B. The above approach relying on the C3 model, may simplify dramatically the theories (leaving the established facts invariant) based on real or complex Minkowski manifolds. First of all, the M4 space-time’s semi-Riemannian “metric” is “turned back” to the natural ordinary (always “flat”) Euclidean metric of C3 (or an equivalent to the Euclidean metric). As it turns out, all the known results of both Einstein’s relativity theories hold in this simple model C3 !. But also it seems that the physical theory in C3 contains more notions and theorems (facts) than theories of the Minkowski space-times. The reason is that some of the useful notions (or facts) definable within C3 may have no proper meaning in the real Minkowski space-time frameworks. The complex Euclidean model yields other important simplifications. Namely, when gravitational (and, possibly other ?) forces are considered in the ‘general theory’ there is no need to impose any kind of curvature in C3, so the geometry of the considered ‘physical space’ remains “flat”. However, with regard to predicted numerical values of physical observables, one can expect that the results of computations in both M4 and C3 approaches to the SR theory will be the same. The point is that in C3 the underlying calculations “promise” to be much simpler, easier and faster. Going further through the theory it will become clear that the space curvatures commonly applied in physics modeling (which are ultimately equivalent to changes in metrics), are only the result of the projection of some, say, “energy curves” (representing gravitational fields) lying in the interior of C3 onto the (observable) flat real subspace R3 of C3. Within R3 this projection will result in changes in the (originally Euclidean) metrics. The just mentioned complex interior extending the real (boundary) physical space, may play, in a sense, the role of an “ambience room” ( [1] ) for the associated R3 , preventing it from actual “distortion”. Compare this situation with motivations of the Schwarz-Shield model. 1.2 C. In light of the above one can see that the transition from the real Minkowski’ M4 to the, presented here, C3 model, contains two basic steps. The first step of the model’s transition is just the complexification of the real M4 = R3 x R1 space-time toward its complex (still Minkowski’s) version C3 x C1.

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The complex space-time C4 is (originally) still Minkowskian as it is endowed with, in a sense “the same” as the real semi-Riemannian metrics. The “non-metric nature” of that semiRiemannian “metric” (actually the scalar product ) is a result of the (different than the space coordinates nature) time coordinate C1 which, in the Minkowskian context, may be seen as a kind of “strange factor”. The second step (based on the considerations in 1.2 A above ) relies on ‘dropping’ the redundant time coordinate C1 from the Minkowski’s complex space-time C3 x C1. In this way one obtains the paraspace or “space-energy” complex C3 = R3 x iR3 model. At this second step, dropping the ‘unnecessary time coordinate C1 ‘ from the Minkowskian (complex) space-time, results in turning the semi-Riemannian metrics into the regular Euclidean metric of the C3. Another important fact is the (accompanying to all that mentioned above) transition of the Lorentz transformation (that is only linear) in real or complex Minkowski’s models to an isometry in the complex Euclidean space. Probably this second step of the transition, as part of the process of the C3 model construction, makes this model and the associated two step procedure, different from other similar constructions of complex models of physics that are present in literature. See, for example, [3 - 7] with references. 1.3 Another novelty of our approach to the complexification lies in an associated (and probably new) reasoning for SR facts. Our complexification procedure (see section 2) results in an unexpected and surprising exhibition of the (hidden) “Newtonian nature” ! of the observed relativistic motion for arbitrary speeds. The unobserved “natural motion” (Newtonian) is assumed to “take place” in the complex interior of the “paraspace” C3, so that in this setting the R3 space merely plays the role of a topological boundary subset of the ‘full model’ C3. An important feature of the construction is that the need for complexification of the original space R3 (and M4 ) was an unexpected byproduct of my former investigations, rather than a deliberate, set in advance, goal. The starting point of our C3 model derivation was the existing Einstein’s SR theory within the real Minkowski M4 space-time manifold, and the Lorentz transformation was taken as a point of departure or as a “primary axiom” valid in M4. The ‘new (“true” isometric) transformation’ we derived from the Lorentz map by trying originally to find its “trigonometric version” upon the realization that the

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common factor (1 – u2 /c2) 1/2 present in the Lorentz formulas, together with the ratio u / c , can be considered as cosine and sine, respectively, of some initially “mysterious angle”, say . The (“possible”) meaning or interpretation of that “angle ” was to us an exciting question for a long time. Later it turned out that the variable  can simply be interpreted as arguments (in the polar representation sense) of the associated complex numbers, representing various physical quantities. It is “part” of the constructed, in the next step, complex model C3. The way it was done is briefly described in section 2. Resuming Here we need to make the point that the difference in defining the transition “from real to complex” domains, between this approach and the approaches that I found in literature, can be stated briefly (with all the involved “physical meanings”) as the difference between the algebraic concept of the x  x + iy extension and, the polar extension r  (r , ) i.e., the revolution of the real coordinate, say x = r, into the, so created, complex plane, by the angle . In the latter case we identify the pair (r, ) [or r = 0 case] with the complex number ‘r exp[i  ] ’. This subject is developed in the next section. In section 3 we develop this in more detail by illustrating the situation in the underlying complex coordinate planes by graphs. In section 4 we discuss the transition (the second step of the construction) from C4 complex spacetime to C3 time-free “paraspace”. In sections 5 – 8 we emphasis the Newtonian character of “physical phenomena” that “take place” in the complex ‘interior’ of the C3 model. In particular, in section 5 we define and investigate the (unlimited !) “Galilean speeds” and their relation to the observable relativistic speeds. Based on notions introduced in section 5, in section 6 we try to explain Einstein’s universality of speed of light which is an axiom in the existing relativistic theory (supported by the experiments). In section 7 we consider a fictitious experiment that might lead to exceeding the speed of light by some particles (in the real space ! ). The anticipated phenomenon is there analyzed with an illustration of the associated complex plane). In section 8 we consider the relativistic mass as a starting point. Then we extend the notion of mass to a complex quantity. It turns out that this quantity‘s “absolute value” (a little differently defined than it is used) is invariantly equal to the Newtonian’ rest mass, while its real part is the,

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well known, relativistic. To obtain this, not straightforward, result we introduced the imaginary part of the mass to be literarily imaginary (i.e., measured in, say, “i kilograms”). The defined ‘mass transformation’ we proposed to add to the Lorentz transform equalities. Invariance of the mass “absolute value” (the rest mass m0) is, of course, in agreement with Newtonian theory. As always, the “regular” relativistic behavior, also for the mass, can be “restored” upon the “projection” of the (complex) Newtonian phenomena’ description into R3 , i.e., into the observable part of the wider model. However, unlike complex space and time, the mass transformation is of a hyperbolic nature so the “projection”, in this case, is slightly differently defined in complex geometry. As mentioned, in the following text, we sometimes suggest to give some, not only mathematical, but possibly also physical (or a “para-physical” ?) meanings to imaginary quantities and thus to consider such “physical” ? units of measure like, for example, “i meters” or “i seconds”, “i kilograms” etc, … . Three appendixes are enclosed at the end of the paper. In Appendix 3 we give some suggestions on possible application of the C3 model for wider areas of physics especially quantum mechanics.

2. The Lorentz Transformation as Departure Point from the Real Minkowski M4 to the Complex C4 Model Transition 2.1 Let us start with some remarks on the ‘Lorentz transformations’ as defined in the real Minkowski M4 space-time: x - ut = x’ (1 – u2 / c2 )1/2 y = y’ z = z’ t – ux / c2 = t’ (1 – u2 / c2 )1/2 .

(1a)

In the above, one considers the motion along the x-axis with a constant speed u. 9

At first, we restrict our attention to the first line of (1a), and compare it with the corresponding part of the Galileo transformation: x - ut = x’.

(1b)

The two transformations only differ by the familiar factor (1 – u2 / c2 )1/2 . A simple pure mathematical observation gives the “trigonometric representation” of this factor, together with another, also persistently occurring in special relativity (SR) theory, quantity ‘ u / c ‘. Apparently, both quantities (1 – u2 / c2 )1/2 and u / c may be considered as cosine and sine, respectively, of some common “angle” , say . So we may apply the following notation: u / c = sin , and (1 – u2 / c2 )1/2 = cos  and rewrite (1a) to the form: x - ut = x’ cos  y = y’ z = z’ t – ux / c2 = t’ cos .

(2)

The above “Lorentz part” of the, introduced below, wider transformation we propose to consider jointly with the common mass transformation: m = m0 sec ,

(2*)

where m0 represents the rest mass and sec  = 1 / cos . Naturally, at first a possible ‘physical interpretation’ of that “angle”  appears to be “mysterious” if not “meaningless”. Let me then pursue a “thought experiment ”, starting with the following, “naïve” question [ that, by the way, is not so naïve within the purely mathematical framework ] : “What (if anything) “will happen” if we extend the factor ‘cos  ‘ in (2) by adding its natural “imaginary counterpart ”: 10

‘ i sin  ‘, where i 2 = -1 ”. In the beginning, the only motivation for that, purely mathematical “action”, is mathematical “intuition”. But then one observes that, when trying to test the above “wild guess”, as a result one obtains several (unobserved in reality) “physical properties” that are “felt” to be strongly desirable for a better “rationality” of the “physical motion ”. Notice that adding the imaginary term ‘i sin’ in the Lorentz formula (2) results in an extremely useful form of the complex transformation: x - ut = x’ exp[ i  ] y = y’ z = z’ t – ux / c2 = t’ exp[ i  ],

(3)

together with the real mass transformation: m = m0 sec .

(3*)

Here, the quantities x and t are complex numbers. Their real parts Re x, Re t can be identified with the former real quantities x, t as present in (1) and (2). They represent the result of measurements obtained by a rest observer located at position (0,0) , i.e., at the origin of the complex plane (see Figure 1). Unlike x, t in (3) the quantities x’, t’ measured by the “moving observer” who is situated on the “back of a rocket”, are nonnegative real and we have: x’ = | x – ut |

(4)

and t’ = | t – ux / c2 |, respectively.

(5)

The rest observer does not see, by his senses, objects expressed by complex numbers x, t but we will assume he can “see” these ‘intelligible realms’ (together with all the interior of underlying complex planes) “mentally”, i.e., in the mathematical model. In the literature the ‘imaginary (part of) time t often is formally associated with the “motion” at an ‘imaginary (space) distance’ (see [3]).

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The notion of ‘complex time’ is relatively widely applied in the literature, especially in quantum field theory. It is often considered in association with the constructions of four-dimensional complex manifolds. These manifolds often are considered to be complex extensions of the real M4 –Minkowski spacetime, so the corresponding “metrics” (actually, as given by the semiRiemannian scalar product “metrics” are not metrics in the regular topological sense) remains non-Euclidean (see, for example, [3], [6] ), and for this reason such complex spacetimes are essentially different (in topological structure) from the common complex Euclidean space C4 that we intend to apply in association with (3). Before going to next section, first recall that for the speed u we have: u = c sin.

(6)

Second, let us make the following general assumptions: x = ct, x’ = ct’.

(7)

The natural assumptions (7) get rid of the semiRiemannian metrics on C4. Now, the common ‘spacetime interval’ loses its original meaning as the ‘norm’, “becoming” always equal to zero. The new metrics we postulate on the set C4 is the Euclidean, and transformation (3) is understood to be the regular isometry on this Euclidean C4. From now on, it is vital to distinct between two different models: (C4, m), (C4, d). For the same model’ universe C4, m is the semiRiemannian “metrics” [the Minkowski case] and d is the Euclidean, or Euclidean equivalent, metrics [the Euclidean case]. Remark 1. Equalities (7) strictly relate any measurement of time within the same free-float frame (i.e., the same ) to any corresponding measurement of distance travelled by light during this time and vice versa. Creating this (different than the Lorentzian) framework we, for example, avoid the situation of “pure time” measurement as done by a watch alone, i.e., “time distance” between two epochs both happening at the same place in space where the watch is situated. According to our approach, to every such time period always corresponds the distance travelled by light. Since every possible, well determined watch gives the same, based on its inner dynamics, measurement, we choose the one based on the measurement of the distance travelled by light (eventually reflected by a mirror back to a ‘staying in one place’ observer). 12

With such assumed ‘distance - time’ relation, the points of C4 cease to have the usual “events” interpretation, so now they are simply “points” (Euclidean).

3. Simplified Illustrations We analyze the (complex versus real) motion as illustrated in Figure 1:

Figure 1. In Figure 1, both the ‘rest observer’ and the ‘moving observer’ (placed in the back of rocket) are assumed to be situated at the origin O of the x’ + ix’* complex plane (of complex positions) at the moment t = 0. [ Unlike in the real space (line), directions of the x-axis (as “seen” by the rest observer) and the x’-axis (for the observer in the rocket) differ by the angle . ] As it holds in SR, the rocket that the rest observer really sees (by his “physical senses”) moves in the real positive direction of the x’-axis. (Recall that at t = 0, the back of the rocket, for both observers, is situated at the origin O.)

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However, its front, for the rest observer, is at point A, while for the rocket observer the front is at point B past A (due to the usual Lorentz contraction for the rest observer’s measurement ). Suppose that the rest observer is able to “mentally see” the whole complex space (one plane, in this case) as illustrated in Figure 1. Then he may come to the conclusion that, as a matter of fact, the formerly observed Lorentz contraction of the rocket was an “illusion of senses (and physical instruments ?)”, and it is only due to the rotation of the rocket’s path x from the path x’ by the angle . What he is really able to see by his senses is only a “shadow” (OA) of the “true rocket” (OB’) that moves along the line x toward the interior of the complex space. The “true length” of the rocket remains invariant since for the lengths we have |OB’| = |OB|. In this context, the observed (by the rest observer) front A of the rocket is simply the projection of the “naturally positioned” complex front B’. Notice also that, as the speed u grows, the corresponding angle  grows too and, according to (6), u  c as  /2. In the mathematical model we will consider a “natural body’s (complex) position” x as that being spread out between the rear point O and the front point B’, where point B’ (the complex “position” coordinate) will be identified with a single complex number x . Its absolute value r = |x| can be thought of as the (invariant in , so in u) “natural length” |OB’| of the moving rocket. The adjective “natural” for the word “direction” will be given regardless the fact that this direction is out of direct physical (and sensual) observation. As for the body’s real direction OA, we would propose to call it the “observable direction”. Needless to say, knowledge of the ‘rocket’s’ observable parts OA and OB (each given only to one of the two observers) uniquely determines the natural position OB’ of the rocket, where B’ is a complex number (on the rotated x-axis). For that, notice that cos  = |OA| / |OB|. Therefore, in this sense, complex positions are at least “mentally observable” (understood). Remark 2. A difficulty that arises is the “destiny” of the rocket. Namely, as it goes in its natural direction x it becomes arbitrarily “far” from the real space because of the constant increase of the imaginary part of its position. This may be seen as unnatural even if the rocket’s projection (“shadow”) on the x’-axis still obeys the rules of SR. In order to find some way out of this difficulty we propose the following “rest observer’s interpretation” as is illustrated in the following Figure 2.

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Figure 2. In Figure 2 we placed two different rest observers. The “old” observer is placed, as before (Figure 1) at the origin O, while additional observer is placed further at some (variable) point O’ on the real x’-axis. Suppose, after a period of time , the back of the rocket (or rather its projection on the x’-axis) is passing the point O’. For the “old” at-rest observer the rocket (its back and front) is now spread out between the points O’’ and D’’ while, as we assume, for the O’ at-rest observer, it is spread between O’ and D’.

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In other words, the situation for O’- observer, just after the considered period of time  passed, is exactly the same as it was for the O - observer when that period of time started. We will then assume that the (space) intervals O’’D’’ and O’D’ are ‘equivalent’ (as parallel and having the same real projection O’C for both rest “sensual observers” ). Thus we propose to equate them. What is essential for both observers is that the natural direction as determined by the common angle , is the same. Now, consider the following 1-dimensional figure: t, t’

.

.

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----------------- ----------------------- --------------- ---------------------------------------------------- 0 α β Figure 3. Figure 3 geometrically illustrates Lorentz’ real time transformation t’  t [now, t is real] , whose analytic version is given as the fourth line of transformation (1a) or equivalently (2). So, at the moment, we restrict the analysis to the real line only (the t-axis and t’-axis are distinct but parallel with the same origin, but different units). Referring also to Figure 1, suppose both observers are located at the origin x = x’ = 0 , which, at time t = t’ = 0 corresponds to the position of the back of the rocket. Now realize that the “information” (observation) about the rocket’s front will be available after the times  = tA = |OA| / c for the rest observer and  = t’B = |OB| / c for the rocket observer. More generally, these relations are satisfied by any x and any corresponding x’ , since, according to (7), we have the relations t = x / c and t’ = x’ / c . The fact that  /  = tA / t’B = cos is due to the well-known time dilation phenomenon. This phenomenon has an explanation if we consider the complex version of time as modeled by the complex plane illustrated by Figure 4 below. At this point we will again consider the complex version (3) of the Lorentz transformation.

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Figure 4. For clarity we now consider the basic “ rotation of the (complex) time t ” to be performed about the origin O instead of the arbitrary point t0 = ux/c2. Analytically this corresponds to the assumption that x = 0 in (3). Now we transform the formulas (1) and (2) into (3), and set the originally real (parallel) lines x and x’ as well as t and t’ into the complex plane (Figures 1 and 4 respectively). Doing so, one recovers the (“absolute”) time invariance, since the line t (initially parallel to t’) is just rotated by the angle . As the result of that rotation the real time point  (corresponding to the real space position of point B) turns into complex time ’ (corresponding to the complex position B’). The Newtonian ‘time invariance’ here is recovered upon taking the absolute value of the “complex time” ’ (considered as “absolute time”).This can be expressed by (in the same way as for the space) the relation: |O| = |O’|. The dilated [relativistic] time |O| (the real part of the complex time), as observed by senses and physical instruments of the rest observer, turns out to be the projection of the invariant (absolute) time |O’| onto the real line of time. Actually, it is the projection of the complex time O’ whose absolute value | O’| is the invariant “absolute time” (possibly, in the Newtonian sense ?).

4. Transition from the Complex C4 to the Complex C3 Euclidean Time-Free Model 4.1 Looking at Figures 1 and 4, one can see that two geometric figures on the complex planes (one depicted by the points O, A, B, B’ and the other by the points O, , , ’) are similar, and, under the assumption c = 1, even congruent. So, in a good sense, they are “identical”. Analytically, this is reflected by recognizing that, upon assumptions (7), the first and fourth rows in (3) are simply proportional, and, for c = 1, “identical”. If there is no hidden error in the above considerations (?), then one, probably, should delete the time transformation in (3), since it always can be restored without any additional information. Realize then the following feature of transformations (3). First, upon assumptions (7), by multiplying both sides of the time transformation: ‘ t – ux / c2 = t’ exp[ i  ] ’

by c,

18

one obtains ‘back’ the space transformation: x - ut = x’ exp[ i  ] . To see this, note that ux / c = ut. Consequently, the time transformation ‘ t – ux / c2 = t’ exp[ i  ] ’, is just proportional to the space transformation, and is redundant as a kind of repetition of the space transformation, and therefore it can simply be deleted from the Lorentz transformation (3) without losing any significant information. Recall that upon assumptions (7) we have no longer the Lorentz and Minkowski framework, and no longer ‘events’. (?) If correct, all of the above is in accordance with the idea of deleting the time coordinate from the original complex space-time Minkowski model C4 in order to adopt the Euclidean C3 “paraspace”, or, in other words, the “space-energy model”. Going further with this thought, one can, eventually, rewrite the complex Lorentz transformation (3) into its equivalent reduced form: x - ut = x’ exp[ i  ] y = y’ z = z’

(8)

where we dropped the equation t – ux / c2 = t’ exp[ i  ]. This latter equation might be considered merely as derived (i.e., ‘logically dependent’ ) from the space transformations (8). Nonetheless, time t is still present in (8) explicitly, even though the variable t does not “directly belong” to C3. To give (8) a more fundamental “logical form”, independent from the foreign variable t, one can replace the first equation in (8) by the following “time-free” equivalent one: x (1 – sin ) = x’ exp[ i  ] , which is now expressed in pure geometric terms. Time t , obviously, can be recovered as x / c and t’ as x’ / c. The proposed full version of the “time-free complex paraspace’ Lorentz transformation” for the motion observable along the real x’ -axis, one can write as follows: 19

x (1 – sin ) = x’ exp[ i  ] y = y’ z = z’

(8*)

One also may try to give a geometric sense to the “non-geometrically looking” quantity ‘c’ (i.e., the observable speed of light) by identifying it with the right angle . This (geometric) “orthogonality” of the ‘speed of light’ would make sense upon extending the ‘one-to-one relationship’ (6), between the “ordinary” speed u and the acute angle , to the “limit case”:  = 90 degrees. That “orthogonality (or singularity) of c” may also be associated with, say, a “pure imaginarity” notion, when associating the mathematical notion of “imaginary” with a physical “dynamics” (or, more specifically, kinetic energy) (?). That is, however, a task for future. Below we still will be using the defined time symbol ‘t’ since it cannot be removed from the real Galilean transformation valid in real space, and also for clarity. 4.2 Let us compare the Galileo transformation x’ = (x – ut) with the complex Lorentz equation: x’ = (x – ut) exp[ - i  ] ( which is just another version of the first row in (8) ). Realize, that the latter equation is a simple composition of the ‘ Galilean shift ‘ (x – ut), with, say, the “Lorentz-Einstein rotation” exp[ - i  ], about the new origin ut in the same complex plane. Now [ upon regarding the considered transformations as defined for all three basic complex planes (i.e., the complex coordinates in the C3 ) independently of each other ] it is evident that any composition of the (Galilean) translation with the (Lorentz-Einstein) rotation within each basic (complex) plane forms an isometry of C3 which always remains in a one-to-one relationship with the corresponding space part of the real Lorentz transformation (1a) in M4 . Recall that these original Lorentz mappings are (only) linear and not isometries in the ‘regular metrics’ sense. At this point the following comment should be made. 20

Suppose that the isometries introduced above again take place in one (complex) plane that, topologically, is two dimensional. Within the classical Lorentz space-time basic framework we too have the corresponding transformation of the (real) planes, (x, t)  (x’, t’) , where time is explicitly and essentially involved. To compare this with our case, it seems the time coordinate was replaced by the ‘imaginary part of space’, which turns out to have an ‘energy interpretation’. As for the (“Galilean”) shifts, say, (x0, t0) = (ut , ux / c2 ), they are simply related by x0 = c t0. The strict correspondence between the ‘physical’ speed u and the ‘geometric’ quantity  suggests the geometric character (or genesis) of motion in C3. Evidently, in this ‘time free’ model, the motion (or, more generally, ”dynamics”), is reduced to the geometric notion of ‘shape’ (here equivalent to the angle  ) without referring to time. 5. On Speed Transformations We now continue our consideration of how far the mechanical “motion within the complex interior” of C3 is Newtonian (Galilean). Relations (4) and (5) establish the invariance of (the absolute value of) time and length in C3 with respect to changes of the speed u . The next question that arises is about the (possibly, Galilean ?) behavior of the speeds in the interior of our complex model. (These speeds are unlimited, but always smaller than the corresponding “semiGalilean” speeds of light, see below). In order to be consistent with the (hypothetically Newtonian in C3) theory, I propose to consider [ besides the relativistic, i.e., constrained, speeds u ( < c ) ] the hypothetical ‘natural unbounded speeds’ : U = u sec .

(9)

The Reasoning: Referring to Figure 1, the reason that relation (9) holds, is that, during the ‘rocket front’ motion (in the model), its natural and observable positions B’ and A, respectively, always (in the geometric sense) share the same real part |OA| of the two distances. Recall that, according to the complex Lorentz transformation (3), for any (complex, having the argument ) time moment t, its absolute value | t | along the ray OB’, as calculated for x = 0, is the same as the value t’ along the real axis. [ Obviously, its real part Re (t), i.e., the ‘relativistic time’ is, also in accordance with SR, shorter. ]

21

Thus, at any time  = |t| = |t’| , the variable A can be obtained from variable B’ by a geometric projection. As the two points move together, at any common time epoch , the ratio of the distance made by B’ to the distance made by A is always equal to the same trigonometric quantity sec = 1/cos. Thus, since the (absolute) times of both motions are the same (= ), the ratio of “speed” U of B’ to the speed u of A must also be equal to sec. Consequently, the speed U of the point B’ moving in the complex (natural) direction  can be analytically expressed by (9).  From now on formula (9) is considered to be a definition of (unlimited) “Galilean speeds “ U (which also deserves the name “natural speeds”) in the complex interior directions . We will also call them “r-speeds” (or “radial speeds”), where r denotes the ray OB’ (see Figure 1), as well as its length (i.e., the absolute value of the complex number B’). The composition of any two natural speeds U, V reduces to the ordinary arithmetic addition and subtraction, while their projections’ u, v composition must satisfy the common Einstein’ formula for “adding” speeds. (The, mainly geometric, proof of this assertion is given in Appendix 2.) Realize that every non-zero Galilean r-speed U is always greater than the corresponding observed “Einsteinian speed” u. Since, in the relativistic theory case, we have u = c sin, one obtains for the corresponding Galilean speed U: U = u sec= (c sin) sec= c tan shortly: U = c tan 

















The consequence of relation (10) is that the Galilean speed U is finite but unbounded ! For any non-zero speed u and the corresponding Galilean U, one obtains that if / 2 then u  c and U   ! . Similarly we may define (more natural than the usual common light speed ‘c’ which is only in the real direction, i.e., = 0 ) the “semi-Galilean speed of light”, say C, when a ray of light is “shed ” into a complex interior direction determined by the argument . Physically this means that the light ray is sent from the rocket which moves into the same “direction” (with natural speed U = c tanand relativistic speed u = c sin ).  22

This “unobservable speed of light” C is given by:

C = c sec,

(11)

so that c < C, whenever   0. The reasoning for (11) is the same as the reasoning directly following (9), this time applied to the light motion. Note that C = c, only when the light is shed (from rest) in the observable (real) direction x’, ( = 0). Recall, that the latter ‘real direction’ is the only direction “we” (and our instruments) can sense. Here realize that in the complex model all the physical bodies (not just their observed “shadows” in the real subspace R3 ), but light, can only “fully move” in the real direction with speed zero (since tan 0 = sin 0 = 0, but for light: sec 0 = 1), see next: [ Remark 3. This would indicate the, not so clear for me yet, fact that the value of any speed (but speed c of light in real direction), should, probably, be declared ‘imaginary’. One might risk the assertion that any relativistic speed < c is the same as an “imaginary length” (“absorbed” from the actual real length by the Lorentz contraction ?). Possibly, to be at all an imaginary quantity, is the same as that (primary, originally real) quantity when in motion. In this sense perhaps one can tell that “imaginarity” is the source of dynamics (possibly also of consciousness, compare with [7] ). Notice that at the moment it’s hard to find for these hypothetic assertions any firmer grounds than an intuition or computing advantage. One possible understanding of speeds’ ( c ) “imaginary nature” is to consider them on separate complex plane (of speeds, see Figure 7 in Appendix 2) and not just on the plane of positions depicted in Figure 1 and 6. ] The reason for the name “semi-Galilean” for speeds C is that the ‘truly natural’ direction for the (Galilean) light speed is given by the argument  = /2, i.e., parallel to the imaginary axis. This, at least suggests the “fact”, that the (true) ”Galilean speed of light itself” is infinite (at least in the mathematical sense of the limit and within the mathematical model). The latter is a direct consequence of the (purely) “mathematical fact” that sec(/2) “takes on” an ‘infinite value’, in the sense of the limit of C = C() for  /2 - . Another interesting fact regarding speeds in the complex model is that the ratio U / C of any Galilean speed U and the corresponding semi-Galilean speed C of light is invariant. Namely, the proportion U / C = u / c holds for every 0 ≤ /2. This property follows the elementary ‘arithmetic fact’ that U / C is defined as ‘ u sec  / c sec  = u / c’. 23

Remark 4. The case /2 is physically not entirely clear where it concerns the Galilean speeds. This is so because the limit of any (semi)Galilean speed, as /2 , is infinite. This is a simple fact in the mathematical model, but the corresponding physical interpretation remains unclear. On the other hand the limit of the projections U  u (for / 2 ) of Galilean speeds equals c. The latter (by a “continuity argument ”) might be understood in that the “projection” of the infinite “speed of light”  = Cmax , into the real line, is c. To avoid this unsecure matter we, eventually, may declare [ as another, separate, version of the created theory ] all the Galilean speeds U, including semi-Galilean C, as undefined for /2 , and the “infinite speed” considered as only the speeds’ (unbounded) behavior in an arbitrary small left-neighborhood of the angle /2. In that case, the semi-Galilean speeds of light C <  could be the only speeds of light and therefore the more proper name for them would be just “Galilean”. Nevertheless, in the following we will keep the notions of “semi-Galilean” and “infinite speeds” of (full ?) light for consistency reasons, and to provide a better understanding of the phenomenon of ‘universality of light speed’ discussed in next section. One should, however, keep in mind that both theories (with and without infinite speed of light ) are optional and, possibly, describe the same physical phenomena. Moreover, realize that for the set of relativistic speeds the case /2 is very well defined (as speed c), see formula (6).

6. An Attempt to Explain Einstein’s Universality of Speed of Light The (adopted) infinity of the Galilean speed of light, if “true”, should give some more understanding for Einstein’s ‘universality of speed of light principle’, that in SR is considered as an (empirically established) axiom with no specific theoretical explanation. [ Below, we apply the rule (see, Appendix 2) that, the composition of the Galilean and of semiGalilean speeds (only) reduces to the ordinary arithmetic sum or difference. ]. 6.1 We adopt the ‘Galilean relation’ (i.e., arithmetic difference) between the speeds of any (fast) body, which moves in the complex]direction, and the Galilean speed of light sent forward from that body. The difference between these two Galilean (so none of them is semi-Galilean) speeds will be ‘infinite’, regardless of the arbitrarily high but, by necessity, finite speed of all the other physical bodies. At this point notice that only the Galilean (and not semi-Galilean) speed of light is infinite. This universality of Galilean speed of light as an infinite quantity among all other being finite, in a way, carries over to the corresponding SR speeds.

24

The universality of the (observed, considered in SR) light speed ‘c’ is then supported by the “fact” (of the “mathematical nature” and, perhaps of a physical too ?) that the observed value ‘c’ is the result of the “projection” of the, orthogonal to the real direction, infinite speed of light. Such a projection can be defined mathematically (with no guarantee of existence of a physical sense), where the mathematical notion of limit ( as   /2) is essentially employed. Anyway the ‘observed’ speed c may be seen as being ‘topologically equivalent’ to the (point at) “infinity” Cmax = , while every other speed u < c should be thought of as being the “finite” (projection of the finite vector). Recall that u  c whenever U  , for   /2. 6.2 Once again, consider the natural directions as determined by a specific value of the argument  with - /2 < < /2 . Recall that at every such direction,  determines the corresponding (unique) relativistic speed u = c sin The relation between the relativistic and non-relativistic speeds is given by U = u sec  In particular, the ‘magnitude’ of the light semiGalilean speed C in the direction  is, similarly, given by C = c sec where c is the usual speed of light (the “minimal” speed of light). Combining the above with the relation ‘u = c sin ’, we obtain the corresponding Galilean speed Uas defined by relation (10). The above listed formulas imply that if  = 0, than U = u = 0. However, (if to assume the imaginarity of all speeds  c, see Remark 3) in the “rest direction” = 0, the relativistic speed of light “at rest”, still takes the common non-zero real value c and thus all the speeds u, U, and C take their minimum values (with the only non-zero minimal speed Cmin = c). These and other curious properties of the speeds will be better understood once their nature as complex (physical) quantities will be exhibited in a more explicit way in what follows. 6.3 Assume, as above, that all the Galilean and the Einsteinian speeds, such as U and u , are purely imaginary (so they may as well, and more correctly, be denoted as U = iU and u = iu, respectively). Then the complex semi-Galilean speed of light in the  direction, say Ɔ , is given by the formula: Ɔ = c (1 + i tan  ).

(12)

25

Without loss of generality we may restrict ourselves to the case 0 ≤ ≤ /2. The absolute value of the complex velocity Ɔ is the positive real number | Ɔ | = C, by virtue of the formula: C = c | (1 + i tan  ) | = c sec  . According to the adopted assumptions we then have: U = i c tan ,

(13)

and u = i c sin  

















Now, without the assumption of infinity of the [ Galilean ] speed of light, we may establish two other arguments for the universality of the minimal speed of light c . Argument 1. For that universality realize that: C2 - U2 = c2,

(15)

which holds by virtue of the trigonometric rule: sec2 - tan2 = 1,

(15*)

valid for every  whenever defined. The universality then relies on the independence of the difference C2 – U2 of the speed’s squares from the angle so, equivalently, on the “independence of c” from the ‘natural’ speed U of the light’s source. Of course, in the context of the complex (C3) space, c is considered as a measure of “how much light is faster” than every U and not how fast is the light, which is the same value in the relativistic (R3) case. Argument 2. Even more explicitly than in (15), the ‘speed of light’s universality’ follows from the complex representation of the speeds (or rather, in this case, the ‘velocities’) as given by formulas (12) and (13). Now realize that for the difference of the two “-velocities” we have a remarkably simple and natural intuitive relationship: Ɔ - U = c !

(16)

26

The left hand side of (16) is independent of the two velocities as both are represented by the same ‘natural direction’  in the complex plane. Recall that, according to formulas (12) and (13), we have for the velocity of the regular body (the light source) U = i c tan  and for the corresponding velocity of light sent from that source Ɔ = c + i c tan . 6.4 As mentioned, formula (16) seems to express Einstein’s Universality Postulate in accordance with the simple common physical intuition as the constancy of the speeds’ difference [ recall our assertion (in Appendix 2) that (unlike for relativistic) for the Galilean and semiGalilean speeds composition is defined by arithmetic addition and subtraction ]. In this setting realize that the light’s semi-Galilean velocity Ɔ in its (full) complex form (12), is the only ‘complex speed’ that has a non-zero real part, always equal to the same value c. As for the two angles given by | = / 2, formula (16) still holds if it is interpreted as the limit, for |  / 2 . Obviously, that limit as the limit of a constant function, always exists and equals c . The foregoing results, in particular the speed of light universality, expressed by the very intuitive formula (16), strongly suggest the imaginary nature of all speeds except the (real) speed c . This (hypothetical) interpretation resembles the close similarity of the complex C3 model with the real R6 configuration space (3 space coordinates plus 3 velocity coordinates) for a (classical) particle.

7. On a Possibility of ‘Exceeding the Speed of Light’ in Real Directions

First note that, even if the phenomenon expressed in the above title, would occur, this does not, in any way, violate Einstein’s SR. The main reason for this consistency with SR is that the hypothetical phenomena are quite apart from the SR ‘domain of applicability’. Recall that special relativity does not consider accelerations (positive nor negative). Meanwhile, the considered phenomenon of exceeding the (observable) light speed ‘c’ requires very strong decelerations provided by proper media. Some of the phenomena like that are well known in the literature ( see, for example the “Cherenkov radiation” phenomenon), and some experiments may, possibly, support the claim formulated below. Here we present some (possibly new) theoretical insight, and, hopefully, better understanding of these phenomena. 27

Consider the following Figure 5.

28

29

Figure 5 7.1 Now consider a point-like body A moving “very fast” in the observable positive direction x’. Its relativistic speed u is assumed to be very close to the observed speed of light c. Recall that on the real axis x’ all the observed physical behavior of the body and that of the light entirely obeys Einstein’s SR laws. The same body’s behavior along the natural direction OB’ (see Figure 5) is different and (hypothetically) Newtonian. Assume that in this case the angle  = BOB’ is very close to, but less than, / 2. Let a (classic) “point-particle” A on the x’ –axis be the full body’s (or particle’s) (B’)’s shadow (i.e., projection). According to the previous considerations, if particle A ‘approaches’ the speed value c , the speed (along the complex x-axis) of B’ arbitrarily grows to infinity. Suppose that after 1 unit of time, as measured by the rest observer along x’-axis, the relativistic body A made the distance |OA|. In parallel to this motion, the whole (complex) body B’ “traveled” (within the same unit of [complex] time’ absolute value) the corresponding distance |OB’|. Here, we assume that the distance |OB’| in the interior of the complex plane is, at least, several times longer than that observed distance |OA| along the real line x’. With a little dose of imagination one can take the point of view that the particle’s “true position” (on the real x’-axis !) “is” B rather than A , as the ‘actual distance’ should be considered invariant. That statement, as being a product of imagination, is, at the moment, only hypothetical and, eventually, might be false. It, however, enforces “scientific intuition” enabling it to risk the reasoning in the following ‘thought experiment’. 7.2 Now let us describe the, mentioned above, thought experiment using as an illustration Figure 5. Suppose that one suddenly stops body A ( therefore, at the same time, also stops “its full extend” B’ ) , without destroying its essential inner structure, so that it quickly (after making the long “quick arc” about the origin, together with the whole ray OB’ toward OB ) “finds itself” at rest, i.e., on the x’-axis, but (approximately) at the point B, instead of being at (or close to) point A. ( Notice that if no such ‘stopping action’ ever took place then the body would only be placed “not far” from A , say at a point A + Δ A, for a small distance Δ A .)

30

This phenomenon is quite natural in the complex mathematical model, but cannot be fully observed in the real part of the complex “paraspace” C3. As for the “promised” hypothetical conclusion on the possibility of exceeding the ‘observable speed of light’ c, we would anticipate what follows. The phenomenon that might be noticed by an observer at rest (possibly also situated at point A), may eventually rely on the “mysterious disappearing” of some “suddenly stopped” particles, and then “finding them”, or their traces, somewhere much further ahead. This anticipated “observation”, if it happens, could be interpreted in the following way. Within a very short period Δt of the rest observer’s (real) time, the (classical) particle “made” the distance |AB| , instead of, say, the much smaller distance ΔA (so, ΔA / Δt = u) ahead of A , where ΔA is considered to be small. If this would happen then the “speed” of the particle, calculated as the ratio |AB| / Δt , could be (as it follows from the visual proportions on Figure 5) several times bigger than c . Realize that Δt was applied, above, twice as the rest observer’s period of time, since it was twice applied along the same horizontal axis of motion (and horizontal axis of the complex time). [ Moreover, the following possibility may also be anticipated. Suppose that the cause of the sudden ‘stopping’ of the considered fast particle (in a ‘very small’ neighborhood of point A on the real axis x’) was a solid, but relatively thin, ‘screen’, [or a very “fat” liquid ? ] entirely contained in the real subspace R3 of C3 and perpendicular to the real x’-axis. In such a situation, we would (hypothetically) expect that a physical observation, as limited to R3 , might yield the “conclusion” that such particle(s), or just a fraction of their total amount, did not break the screen, and yet still were found “on the other side of that screen”, in some, relatively remote, distance behind the screen, say at the point (or a little “behind” it) B. If this really would happen, then one possibly could ‘explain’ it using the (just mathematical at the moment) fact that there was “enough room” to “omit” (instead of “crossing”) the screen, in the (wider than the containing the screen ‘physical’ R3) C3 “space of the whole action”. A “path” the particle(s) might follow could be the, mentioned above, arc B’B. Now the particle going through the complex interior in C3 just “omits” the screen as it is situated in the real subspace only. Notice that R3 as a subset of C3 represents (topologically) only the boundary of C3. Let us once again recall that this “behavior” of (“classical”) particles, as well as its explanation is so far hypothetical. ]

31

If all the above would be true, then the considered object could “move” with a theoretically arbitrary high speed, much faster than c, even in the real physical space (that is, in the case we considered, spread out along the x’-axis). As for the corresponding physical experiment, one should, if possible, prepare the laboratory conditions for being able to “produce” such a fast motion. Thus, first of all, one needs to determine a proper method for “stopping” that fast particles quickly enough, but possibly without “destroying” all of them. So that at least some of the particles could survive, i.e., each particle with a positive probability. Notice, finally, that the (higher than c) “motion” of a particle from A to B along the x’-axis may not be detected if the process of measuring by physical instruments is based on any kind of electromagnetic waves. The particle might be “too fast” to be “caught” by a “slow” radiation used by the physical instruments ?. The only facts that would be registered could be an event of hitting, by the superfast particle, screen B at an earlier time than if it traveled not faster than light. 8. Mass Invariance in the Complex Space 8.1 As for mass transformation (3*), it seems that it also “must” have a natural complex extension. Since the real part of mass (2*) transforms differently than space and time (2), our guess is that, unlike the circular space and time revolution (3) ( by  ), the complex extension of mass transformation (3*) could, possibly, be generated by a hyperbolic revolution. Our proposition is to consider the complex extension of (3*) as: m = m0 sec  + [ i m0 ] tan ,

(3**)

where, to get more sense out of it, the additional (created by the motion) “mass” [ i m0 ] tan should rather be considered (literarily) imaginary ! and measured in, say “ i-kg” units. Under this “physical imaginarity” assumption, one obtains that the (hyperbolic) “absolute value” of the complex mass also is speed invariant: | m | = m0. The foregoing equality is due to the trigonometric rule: sec2  - tan2  = 1, and so | m |2 = m02 (sec2  - tan2  ) = m02.

(3***)

32

Invariance of the above mass “hyperbolic absolute value” with respect to (3**) is then equivalent to its independence from any speed related to the corresponding . The main reason for the choice of ‘imaginary mass’, together with the above “hyperbolic framework”, is to obtain consistency with Newtonian’s mechanics. Use of the square parenthesis [ . ] in (3**) is the, here proposed, convention. Consequently, it will be important to distinguish between say ‘5 [i kg]’ , which represents the ‘imaginary mass’, and (‘5 [kg] i ’ , which involves the usual ‘5 kg’, as the ‘imaginary part’ of a complex number, i.e., the real valued physical quantity. Here (when physical factors are involved ) the math rule of associativity of arithmetic multiplication is deliberately violated. Another interesting fact is the similarity (possibly duality) between the (hyperbolic) transformation of mass (3**), (3***) and the (circular) transformation of the “semi-Galilean” speed of light C given by (15) and (15*). Notice the similar roles of the “minimal” quantities m0 and c (the observable speed of light) in the two formulas. The difference relies on the fact that the minimal light speed c has the direction of the real line while the minimal mass m0 “spreads” along the “hyperbolic radius” corresponding to the angle  in the “mass’ hyperbolic complex plane” (where the hyperbola plays the role of a “circle”). [ Anticipating, by analogy, we may presume that, moving along the real x’-axis, electrical charge has an increased (by the coefficient sec) [absolute] value along the unobserved direction  , corresponding to speed, while its value in the real direction is invariant (“minimal”). An underlying, hypothetical, additional “imaginary electric charge” (probably) might somehow be identified with a “magnetic charge” which produces a corresponding magnetic field. Consequently the magnetic field B could be considered as an “imaginary electric field”, say, B = i E*. So an imaginary charge would produce an imaginary electric field (i.e., the magnetic) ??. Hypothetically, there might be a kind of “skew symmetry” between laws of mechanics and electrodynamics. ] Anyway, it seems that the rule of light speed universality, as expressed by (15) and (16), and the rule of the rest mass preservation given by (3**) and (3***) may have the same roots (energy ?). Remark 5. The mass transformation (3**) should probably be added to the restricted complex Lorentz transformation (8*) to obtain the, possibly full, “Lorentz space-mass transformation” as follows:

33

x (1 – sin ) = x’ exp[ i  ] y = y’ z = z’ m = m0 sec  + [ i m0 ] tan.

(17)

To keep things simpler we, deliberately, omitted considering the “hyperbolic complex plane of mass” in the theory’s model. This would lead back to C4, possibly, again, with no Euclidean metric. Besides, notion of mass (likewise ‘electric charge’) seems, at first, not to be geometric. Eventual consideration on this set of problems we leave for the future. 8.2 To give some more arguments for the mass invariance and the associated notion of ‘imaginary mass’, let us take, as another assumption, the invariance of momentum with respect to the transition from relativistic to classical phenomena and eventually “back”. This latter assumption is, of course, much more natural than the, assumed before, existence of imaginary mass, which, in turn, was a sufficient condition for mass-invariance. Thus, possibly, momentum invariance, as formulated below, can stand as an argument for rationality of our considerations in subsection 8.1. According to the assumption then, the relativistic momentum px’ (along the x’-axis) defined by px’ = mu , (where m = m0 sec  ) equals the natural (Newtonian) momentum px in its complex direction of x. We then have, px’ = (m0 sec )u = m0 U = px where m0 is the invariant rest mass and U is the natural Gallilean unlimited speed of the considered body. This hypothetical preservation rule (i.e., px’ = px ) mathematically corresponds to the ordinary associativity of arithmetic multiplication: px’ = mu = (m0 sec )u = m0 (sec  u) = m0 U = px .

(18)

The simplicity and some beauty of equalities (18) suggest their truthfulness and indirectly stand as an argument for existence and rationality of the imaginary mass as introduced in 8.1.

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FINAL REMARKS ON UNIFYING FEATURES OF THE EUCLIDEAN C3 AS THE POSSIBLE MODEL FOR MORE GENERAL PHYSICS For sake of simplicity, in the following considerations we will deliberately use the Euclidean C4 model with the (redundant) time coordinate, instead of (sufficient for the mentioned below physical theories) model C3. Consider first the complex C4  C4 Lorentz transformation as defined by (3) on the Euclidean version of C4 . Recall that in this “complex space-time” case, the fourth row (time) can be obtained from the first, merely by multiplying the first (space) row by the coefficient 1 / c , which represents the reciprocal of the minimal speed of light. Observe the following three features of this complex transformation: 1) As a result of applying the real part, the ‘Re( )’ operator to both sides of (3), one recovers the usual real Lorentz transformation given by (1) or (2), using the notation (1 – u2 / c2 )1/2 = cos . Now, however, the so obtained real linear (Lorentz) transformation (x, t)  (x’, t’) (again in the Minkowski real space-time), is not an isometry (in “true sense”) anymore. Also, the resulting metric in the obtained “(x, t) plane” is not Euclidean. So now all “returns” to the previous M4 space-time with its common Riemannian semi-metric. Recall that in the latter situation, the Maxwell equations are well transformed by that real part of (3), and thus the classical electrodynamics as well as the SR theory hold. Obviously, this is not a good model for Newtonian mechanics. 2) Suppose that instead of the previous ‘real part’ operator Re( ), we apply the ‘absolute value’ operator [ that may be denoted by ‘Abs( )’ ] to both sides of the first three rows of the Lorentz transformation (3). Now, if to neglect the shift of time by the term ux / c2 , (which is close to zero for small u, x), then we obtain “back” the familiar Galileo transformation !, as given by (1b) together with the (approximate) t = t’ equality. So this “absolute value form” of transformation (3) well transforms Newton equations but is not useful as model for the Maxwell’s nor for Einstein’s theories. 3) The full complex transformation (3) or (8) can be considered as the ‘extension’ of both classes [ in the above cases 1) and 2) ] of the transformations that corresponded to Einstein’s or Newton’s theories respectively. Such an extension of the above two models may, probably, be parallel to some extension of both Einstein’s and Newton’s theories to some wider theory.

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Such a general theory would contain the Einstein’s 1) and the Newton’s 2) mechanics as special cases! . That wider theory might contain all or some of the theories of, generally speaking, ‘quantum physics ’ ?. This topic, however, is out of scope of this paper ( but still, { as this work is not intended to be in a form of a publication } see the Appendix 3. ).

References 1. Burns, K. and Gidea, M. “Differential Geometry and Topology With a View to Dynamical Systems.” Taylor & Francis Group, LLC, 2005. 2. Filus, J. “A Complex “Time-Free” Model for Both Classical and Relativistic Motion”. a shorter version of the current paper (to be submitted), 2015. 3. Kaiser, G. “Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis.” North-Holland, Amsterdam, 1990. 4. Kaiser, G. “Physical wavelets and their sources: Real physics in complex spacetime” Center for Signals and Waves www.wavelets.com. 2003. 5. Newman, E. T. “Maxwell’s Equations in Complex Minkowski Space”, J. Math. Phys., 14, pp 202-203, 1973. 6. Rauscher, E. and Targ, R. “Complex Space-Time Metrics.” Journal of Scientific Exploration 15, pp 331-354, 2001. 7. Rauscher, E. and Targ, R. “Investigation of a Complex Space-Time Metric to Describe Precognition of the Future.” AIP Conference Proceedings, Vol 283, pp 121-146, 2006. 8. Schutz, B. F. “A First Course in General Relativity.” Cambridge University Press, Cambridge, U.K., 1985.

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APPENDIX 1 Recall formula (9) from section 5, which defines the absolute value | U | = U = u sec  of the ‘(complex) speeds’ U along the natural direction determined by  in the underlying complex space. Here we give second (alternative) reasoning for that formula. First realize that, according to the considerations in section 4, it is good for benefit of the theory that any speed is defined (or postulate) only geometrically without using, as primary, the notion of time. Now, our basic notion will be the speed of light c (only), which may be identified with the geometric notion of the right angle (as the specific value of the argument ), see notice in subsection 4.1 and relation (6). As it is consistent with SR, the real part of any ‘complex speed of light’ is always c. The Alternative Reasoning for (9): To prove formula (9) [in section 5] in alternative way, consider the following Figure 6.

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Figure 6.

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Suppose a photon of light was sent from “back of the rocket”, placed at the complex plane’s origin O, in a complex direction determined by the angle defined by the rocket’s relativistic speed u). The photon’s “speed” along the radius OB’ is assumed to be constant as the real part c of this speed is well known to be the constant. The latter conclusion on the speed constancy can be derived from more basic geometric fact that for any, randomly chosen, photon’s complex position P on the radius OB’ (O ≤ P ≤ B’), the corresponding real parts of the position of the photon satisfy: Re(P) = Re(a) = a.

(Q)

Recall that only the value ‘a’ is physically observed as the photon’s position, while P is given as the “intelligible value” which can only be “seen” by one’s mind in the mathematical model. In the same model the imaginary part Im(P) = b, as corresponding to the complex P, is uniquely determined too. It is very natural to assume that the real and imaginary part of the photon’s complex speed C at any point P are proportional to the corresponding distances a and b, respectively. Realize that ratio of these speeds must be the same as the lengths ratio: b/a = tan independently of the choice of point P (all P’s produce similar right triangles). So the constancy of the photon’s speed follows from the constancy of speed of a (= c) and constancy of the angle  when u is constant. Since the real component of the speed is known (from SR) to be c, the imaginary component of the photon’s speed must be ‘ c tan ’ (or, possibly, ic tan ). Concluding, the complex speed C of light, when sent from the rocket moving with the real speed u (equivalent to the angle ), satisfies: C = c + ic tan .

(10)

Taking the absolute value from both sides of the latter equality on obtains: | C | = C = c sec .

(11)

As can be seen from the above derivation of C , the speed of the rocket in the complex direction  affects the (“original”) speed c by “increasing” it by the imaginary (“hidden” for observation) value ‘ ic tan ’. It seems then reasonably to ‘guess’ the absolute value U of speed U of the rocket in the  direction to satisfy U = c tan . In order to check this hypothetical value of U, realize that its projection (i.e., multiplication by cosresults with the speed value csinalong the real axis which according to (6), is the known relativistic speed u. Realization that U = c tan c sin ) sec  = u sec  terminates the proof of equality (9).  The complex speed U of the rocket is obtained as: U = u + iu tan 

(12)

This must be true for (9) to hold as (9) is a direct consequence of (12), so (9) is a necessary

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condition for (12). Moreover, given Re(U) = u, formulas (9) and (12) are just equivalent. Also note the mutual similarity of formulas (10) and (12). In parallel to the latter we have the similarity of the two formulas (9) and (11), which I propose to call the speed’s “Newtonization”.

APPENDIX 2 Here I will prove the following Proposition: Proposition. Suppose V1, V2 are Galilean speeds corresponding to the relativistic speeds v1, v2, respectively, therefore satisfying the relations Vi = vi sec i (for i = 1, 2), while each angle i corresponds to both the speeds vi = c sini and Vi = c tani . Then the composition V of the speeds V1, V2 reduces to the ordinary arithmetic sum V = V1 + V2 if and only if the composition v of the relativistic speeds (with c = 1 for the real speed of light) v1, v2 obeys Einstein’s formula v = v1  v2 = (v1 + v2) / (1 + v1v2). Moreover, the values of the compositions v, V are related as V = v sec, where  is the angle defining both speeds v, V.  Speeds’ Descriptions in Two Distinct Complex Models: For a, basically geometric, proof (below) of the above Proposition see both Figure 6 in Appendix 1 and the following Figure 7.

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Figure 7.

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Realize that the behavior of complex speeds is different than the “behavior” of position and that of (complex) time. On the other hand, there are significant similarities as well, so we will, in parallel (depending on the considered speeds’ properties), represent the considered speeds in two slightly different models as illustrated by Figure 6 and Figure 7. In Figure 6 (the complex plane of positions) we place speeds’ directions in accordance with the changing position of a moving physical body. Thus, generally speaking, if  is the speed defining angle then, according to Figure 6, the Galilean, complex, speed V = v + i vtan, spreads out along the slant direction OP, while its relativistic (real) part v = csin spreads along the horizontal direction Oa. When considering the speeds alone, without referring to the changing position of the moving body, their deeper nature [compare this with Remark 3 in section 5 and the considerations in subsection 6.4,] turns out to be better described as either purely imaginary or, at least, their complex plane direction is vertical (possibly, they may be considered real, just ‘physical speeds’ multiplied by the ‘math quantity’ i ). The only exception is the relativistic speed of light c (here c = 1), which, as the real quantity, has a horizontal direction. Thus, unlike all other graphs in this paper, Figure 7 represents the complex plane of the speeds so that complex points (vectors) w + iw* on the plane (but not all of them) correspond to complex speeds, either relativistic or Galilean. In Figure 7, the Galilean speeds are represented by vertical chords (“vectors”) which for convenience I placed on the vertical line crossing the real point R = c = 1. Admit any vertical chord on the plane, if one represents a positive speed, has the (speed’s) direction “from down to up”, and the opposite direction for negative speeds. The real positive component c (here represented by OR) of any semiGalilean (and relativistic) speed of light is assumed to be directed “from left to right”. Consequently, directions of semiGalilean speeds of light are slant like the directions of OQ or OP. One has also to admit that the same (as in the model illustrated by Figure 7) speeds, when modeled by the complex plane of the ‘physical positions’ (Figure 6), are differently placed as the same objects but differently described. So that all the relativistic speeds are placed in the horizontal direction and both Galilean and semiGalilean light’ (complex) speeds have the slant direction according to their “real and imaginary parts”, which are closely related to the changing (complex) positions. In Figure 7, the two Galilean speeds V1 = tanA and V2 = tanB (here we may neglect the geometric meaning of an angle B = arctan V2) are placed on the chords RQ and QP, respectively. Consequently, the corresponding semiGalilean speeds of light are placed on the chords OQ and OP. Their (invariant) “relativistic contributions” are placed on the chords ON and OZ*, respectively. Their lengths satisfy |ON| = |OZ*| = c = 1 as the arc of the circle in Figure 7 has the radius c = 1. To simplify the considerations, from all the possible relativistic speeds in Figure 7, we choose to consider only positive, i.e., those which are placed (Figure 6) along the x’-axis for bodies moving in the right direction. This restriction does not decrease any generality. Returning to the speed’ model (Figure 7), notice that all positive relativistic speeds are placed on vertical chords, whose bottom ends (the “initials”) are points on the horizontal chord OR. Their upper ends 42

(defined by the ‘original definition’ before eventual later shifting to a next to [also vertical] equivalent position, such as M*W is equivalent to MN), such as point N, should lie on the circle centered at point O and having radius 1. Once this choice is being made, realize that the relativistic speed v1 (as, originally, modeled by the chord MN) has the magnitude |MN| = sinA, while, at the same time, the corresponding Galilean speed is V1 = tanA, so that we have the usual speeds relation V1 = v1 secA. Also, we have V2 = v2 secB, for some proper angle B. Proof of the Proposition: Consider Figure 7. We chose (as an “educated guess”) the composition V (= |RP| ) of the Galilean speeds V1, V2 (as both lying, adjacently, on the same vertical line and not overlap each other) as satisfying the rule of arithmetic addition V = V1 + V2. This rule, geometrically, is in accordance with the graph (Figure 7). From this assumption , I seek to derive the corresponding, well known, relativistic rule v = v1  v2 = (v1 + v2) / (1 + v1v2), [c = 1], which will stand as ‘necessary condition’ for V = V1 + V2 . For this goal, realize that the relativistic counterpart v of the composition V is (or can be) represented by the chord M*Z*. [ Also, by the argument of similarity of the triangles ORP and OM*Z*, speed v2 is the “relativistic partner” of the speed V2 (as V2 / v2 = secB, while V2 = tanB ). ] The composition v (as any single speed modeled by the complex plane of speeds, Figure 7) must be represented by one chord (here M*Z*) on one vertical line. This chord contains as its “parts” speed v2 (ZZ*, that corresponds to QP = V2) and speed v1 (in order not to exceed the line of the circle and to be in one vertical line with v2 [the speeds composition into a one speed]). The v1 representation M*W (so |M*W| = |MN| = v1) is obtained from its original definition MN by the (insignificant in this model) parallel shift to the left. Now, it is clear that, for both speeds not being zero, their composition v1  v2 is always smaller than the arithmetic sum v1 + v2 . The reason for that is that their representations M*W = v1 and ZZ* = v2 must overlap on the chord ZW. Assume, |ZW| = . It is clear that v = v1  v2 equals v1 + v2 - . What then remains is to find the value . As can be seen in Figure 7, the length of the chord M*Z** (which has no physical meaning but a geometric) equals v1 + v2 since |Z*Z**| was chosen to be equal . Since, (for both i = 1, 2) 0 ≤ vi ≤ 1 (we also admit speed of light vi = c = 1) we can identify each vi with a “random event” and its length pi = |vi| with the geometric probability of that event defined on the “probabilistic space”, which is the closed interval [0, 1] the subset of the real numbers line R1. As it is well known, the corresponding “random events” are Lebesgue measurable subsets of [0, 1] , so that the geometric probability of each subinterval of [0, 1] is identified with its length. It’s natural to assume that the “events” v1, v2 are (stochastically) independent. In such a framework, by probabilistic arguments, including the stochastic independence, one obtains  as the geometric probability:  = Pr [(v1  v2)  v)] = |v1| |v2| |v|. Identifying the intervals (random events) v1, v2, v with their lengths (“probabilities”) one obtains  = v1v2v with the unknown value v.

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Since, (by Figure 7 ) we have also  = (v1 + v2) – v, one obtains the value v as the only solution of the following simple equation: (v1 + v2) – v = v1v2v.

(E)

Solving (E) with respect to v one obtains: v = (v1 + v2) / (1 + v1v2).

(E*)

So, from the fact that V = V1 + V2 we derived that, correspondingly, v = v1  v2 = (v1 + v2) / (1 + v1v2), which is a very well-known fact in SR. Here, this fact turned out to be the necessary condition for the Galilean formula V = V1 + V2. To prove the converse that the ‘Einsteinian addition’ v1  v2 implies the Galilean V1 + V2 , one should again analyze Figure 7. A simple geometric interpretation of the graph brings the answer. Knowing (E*) from SR as well as applying the information v = |M*Z*|, one can see that “transforming the story” from the vertical line M*Z* to the line RP one obtains that the adjacent images V1, V2 of the “chords” v1, v2 do not overlap (with an inessential exception of the point Q). So, in geometric language, the length |RP| (= V) is the regular sum of lengths |RQ| (= V1) and |QP| (= V2). By this, purely geometric, argument we can prove that v = (v1 + v2) / (1 + v1v2) implies V = V1 + V2. The latter, together with the Einsteinian speeds addition, shows the Galilean arithmetic rule for the (Galilean) speeds composition. Moreover, by similarity of the triangles ORP and OM*Z* and from the relation |OP| / |OZ*| = sec, one directly obtains V = v sec, which is in accordance with the definition of the Galilean speed V, given relativistic v. Recall  is the (common for both speeds v, V) defining angle  ROP. This recognition terminates proof of the Proposition.  One also can say, that the Galilean law ‘ V = V1 + V2 ’ (as proven above) follows the geometric fact that all the Galilean speeds (not semiGalilean speeds of light) can be placed (originally) on one vertical line, given by the equation Re(z) = c, for z belonging to the complex plane of speeds. The additivity ‘+’ of those speeds follows usual linearity of any straight line. Unlike that, any two distinct relativistic speeds, [as originally defined by chords with bottom point on OR (as v1 is defined by the chord MN) ] “lie” on different vertical lines. When taken together to one line, to form the composition, they always, by nature, overlap. This difference of behavior of the two types of speeds representations follows from the fact that one of them is unbounded and the other (relativistic) bounded. At the end of this Appendix note a nice algebraic fact. Let (W, + ) be the group of all (including negative and zero) the Galilean speeds with the arithmetic addition ‘+’. Realize that the set of all relativistic speeds E also forms a group with the Einsteinian associative and commutative addition ‘’. Let h: W  E be the transformation that to each speed VW assigns the 44

corresponding (defined by the same argument ) speed vE, so that v = V cos is the projection (Figure 6). It follows from the above Proposition that, for any pair of speeds V1, V2 in W and for v1 = h(V1) and v2 = h(V2) in E, we have: h(V1 + V2) = h(V1)  h(V2) = v1  v2. Thus the projection h( ) ( in the ‘position model’ described by Figure 6 ) is the algebraic homomorphism of the groups (W, + ) and (E, ). h( ) is also an injection of the set W onto the set E, so it is an isomorphism. We thus have that the groups (and, in parallel, the theories) of all the Galilean speeds and of all the relativistic speeds are algebraically identical. They are also topologically identical (homeomorphic) as (upon the assumption c = 1) the intervals [-1, 1], [- , +] of relativistic and Galilean speeds respectively, are homeomorphic in the natural way (through multiplication of any v by the corresponding “coefficients” sec ). Also it can be checked that the operations ‘+’ and ‘’ are continuous in the corresponding topologies. Notice at this point too that the isomorphisms h( ) and h-1( ) are continuous since the projection is continuous. So the groups (W, + ) and (E, ) are equivalent also as topological groups. This, together with the comments in [2] (section 4.2), suggests (some) equivalence of the Einstein and Newton theories as both modeled by the same complex C3.

APPENDIX 3. ‘QM-LIKE theories’ and the Complex “Rotating Stick” Let us make a note on the possibility of applying the para-space C3 model for a description within it, some most basic facts or phenomena of QM theory. At first let us introduce a proposition for the following mathematical description of a single ‘free elementary particle’ (such as an electron) as a small, possibly 1-dimensional, “stick” rotating with an angular speed  in the complex plane { x + ix* }. [ AN ADJUSTMENT (separately added later): Preserving the validity of all other considerations about the “stick”, one can reinterpret its physical meaning that this entity rather represents a quark of the particle instead of the particle itself. The 3-dimensional, by nature, particle may be considered as a composition of (for example) three interconnected, one dimensional, ‘rotating sticks’, each rotation placed in a different complex plane (included as coordinates in C3) so that none of the two (out of three) planes are parallel. ] In parallel, let us observe the “shadow” (projection) of this stick on the real x- axis, and recall that these observations are the only possible ‘direct experimental measurements’ at the disposal of QM. 45

According to my modest knowledge in this area, it may be a fact that these (strongly distorted) R3 data represent that reality which QM theory analyzes and interprets ??? Let the (hypothetical) “stick” [ for simplicity sake, its motion is reduced only to one complex plane { x + ix* } ] be modeled by the following harmonic oscillator ( which does not represent the wave function ): ‘r exp[ i (t - ) ]’, where t denotes time, and, with “absence” of linear speed (u = 0) of the particle, the phase shift  (which has the same interpretation as “ = arcsin (u / c)” considered throughout this paper) is zero. The observed (real) part of the hypothetical stick’s (position) is given by x(t) = ‘r cos ( (t - ) )’ and may represent the ‘real length of the particle’. This (“position”), when observed, “is” a “wave” (with length 1/ and progressing time t) while the stick as a whole is a “particle”. [ Here, I suppose that the above defined x(t) does not represent a “string” as a whole entity. This is only a (real) part of an entity. ] As for the ‘free particle’ momentum (or, in this setting, rather “speed”) it is “measured” by the imaginary part of the stick ‘r exp[i (t - ) ]’ , i.e., by the function: v(t) = ‘r sin (t - )’ . In light of the above it may seem that the Heisenberg Rule of uncertainty is only valid in the real part (R3) of the complex reality, so it holds in a different model than the C3 mathematical model. My impression is that in the para-space model C3, the position of a free particle, ‘r cos (t - ) = x(t) uniquely determines ! its speed: v(t) = ‘r sin (t - )’ [or, possibly, just:

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v(t) = dx / dt = - r  sin(t - ) ? ; so if  would be equivalent to the mass (?) the latter might be the momentum rather than speed only. ]. It probably means that Heisenberg’s rule is ‘unnecessary’ ? when in C3 . On the other hand, it certainly holds in a QM model like algebra(s) of Hermit operators defined on a proper Hilbert space (of the complex functions defined on R3 , so on real space) of states. As for the phase shift  , it is intended to indicate the linear speed along the x- axis, of the so modeled “particle”. In general, this  is random (or “random” could be ‘the very moment’ when a physical measurement is taken ) since many unpredictable impacts from the environment (like temperature) affects it. Going back to the macro-scale considered above in this work, the speed u , as related to the ‘expected value’ of the random variable  , is a statistical average of billions of particles (contained in the considered physical body) for which a single particle’s phase shift  is random. Thus, such a macro phenomenon like the Lorentz contraction is, again, the statistical average of all the particles’ contribution to it, with a very small (practically undetectable) standard deviation. Consequently, the “net-value” of  exhibits itself as deterministic quantity. [ With the, mentioned above, quark interpretation the angular speed ‘’ of the ‘stick’ should be replaced by three, possibly distinct, angular speeds 1, 2 , 3 (with possible interpretation as “colors”?). ] As the last note (without proofs) I would like to add that (according to our hypothesis) the following statements may hold: 1. Potential energy of a physical body is equivalent to the average value of the random (?) angular speeds  of all its particles (as well as to a number of the particles in the given body), while the kinetic energy may depend on average value of (random)  . [ For the quark interpretation it seems to be reasonable to set  = a1 + b2 + c3 , with a, b, c being real, possibly variable, coefficients ? ] 2. The necessary minimum value 0 of  is, probably, responsible for the rest mass m0.

It seems to be possible that in a constant (gravitational or other ?) field presence, the quantity (or maximum value ?) ‘(t - )’ remains constant (or ‘stable’ in time t), while, separately,

47

 and  may vary ?. The resulting by the field, motion might be equivalent to transforming the quantity  to the quantity . [ By the way, the (rest) mass resistance against an acting force may, possibly, be of the same origin as forces exerted by an angular momentum against changes. So, change of axis of rotation of the “rotating stick” (by an angle, say Δ ) may, according to the ‘rule of the angular momentum preservation’ exert a force to counteract the change. The total inertia of macro physical body may be the sum (or statistical average) of all its particles contributions to the resistance. Every such a particle contribution may, in turn, be random. ?? ]

3. I presume that time t (in seconds) and mass ‘m’ (as the average of angular speeds  ? measured in ‘1 / second’ ) are related as (physical) reciprocals [ time as reciprocal of energy ? ] to each other ! This relation would result with the following ‘product invariance’ (a duality ?): m0 t0 = mt = constant. (see transformations (2) and (2*), in this paper). ?… (November 4, 2015 ).

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