Is the Kinematics of Special Relativity incomplete?

An analysis of composite inertial motion (relativistic sum) within the framework of special relativity leads to the conclusion that every translational motion must be the symmetrically composite relativistic sum of a finite number of quanta of velocity. It is shown that the resulting space-time geometry is Gaussian and the four-vector calculus has its roots in the complex-number algebra, furthermore, that Einstein's"relativity of simultaneity"is based on a misinterpretation of the principle of relativity. Among others predictions of the experimentally verified rise of the interaction-radii of hadrons in high energetic collisions are derived. From the theory also follows the equivalence of relativistically dilated time and relativistic mass as well as the existence of a quantum of time (fundamental length) and its quantitative value, to be found in good accord with experiment.


Introduction
Modern physics developed experimental methods the results of which in principle confirm special relativity as proposed by Albert Einstein as well as its further mathematical shaping mainly by Hermann Minkowski. At the same time new physical phenomena were discovered in high energy (collider) physics which usually are not brought in connection with the kinematics and mechanics of special relativity. These phenomena, for which a convincing physical explanation has not been found yet, can be grouped as follows: 1) The rise of the interaction-radius and total cross-section of elementary particles (hadrons) with increasing energy (of the beam); 2) The shrinkage of mean-free-paths of ultra relativistic particles (nucleii) in material media; 3) The obvious existence of shortest life-times of particle resonances of the order 10 s, -24 where "s" means second. As widely known, special relativity rests on two premises: The invariance of the physical laws for all observers, independent of the state of inertial motion (principle of relativity); The constancy of the velocity of light in a frame of rest independent of the velocity of the source, Ernst Karl Kunst 2 (1) wherefrom the Lorentz transformation results automatically.
A reconsideration of the kinematics of special relativity results in a novel definition of the concept of velocity between any two inertial frames of reference and a modification of the Lorentz transformation. In some aspects the predictions of this new kinematic view deviate markedly from special relativity especially at velocities near that of light and, together with a reviewal of the "relativity of simultaneity", explain the previously mentioned experimentally verified physical phenomena both qualitatively and quantitatively as being of relativistic origin.
Imagine a system S (x , y , z , t ) moving inertially at constant uniform speed "w" 2 2 2 2 2 parallel to a system (x, y, z, t) and the latter moving at the velocity "v" relative to an observer resting in the coordinate source of a sytem S (x , y , z , t ) "at rest", 1) It is demonstrated that the resulting relativistically composite velocity u = (v + w)/ (1 + vw/c ) of S -as observed from S -is variable, dependent on the respective 2 2 1 value of v and w, where but v + w is always constant. Einstein considered in order to prove that the relativistic sum of two velocities which are slower than light always results in a velocity slower than light [1]. We posit v = c -, w = cand = + , whereby always > ( -) > 0. From (1) out -as well as any other composite velocity with no point . Therefore, 2) must be 0 false and 3) true so that (2b) attains the only possible form which guarantees the light signal to propagate via 0. 5) Hence between any two inertial frames of reference S and S , moving relative 1 2 to each other with constant uniform speed, an inherent preferred reference point 0 always exists at rest -for the time of the translational motion -relative to S and S 1 2 implying their velocity relative to to be symmetrically equal and oppositely 0 directed and relative to another permanently a relativistic sum (2).
The chain of argumentation and the result that any velocity u be symmetrically composite also is valid for the velocities "v ", "v " etc. Thus, a point also must 1 2 1 exist between and S , a point between and , and S , or S and so forth The fact that every velocity is quantized in the order of (3) and all elements of (3) Thus, velocity is always composite according to (4), which usually falsely is taken to be non-composite. It is clear that even in the ultra relativistic region, where v 0 c, direct measurements would unveil no difference to the classical apparently non composite velocity "v". But this cannot be true for measurements of momentum or energy, which are based on the electron Volt (eV). A protron (antiprotron), accelerated in an electrical field with the potential difference of one eV, would reach the subrelativistic velocity v v 3 × 10 cm/s. Thus, with rising energy or 0 4 momentum a systematic deviation of the correct composite value according to (4) from the special relativistic one of the order of magnitude must be taken into consideration, where "n" means any number (multiplicity) of eV, p momentum on the strength of naturally composite velocity according to (4), p 0 special relativistic momentum, m rest mass, v conventional velocity, and 0 Lorentz factor on the strength of the conventional and the composite concept of velocity, respectively. According to (4)

The Symmetric Lorentz Transformation
Consider the inertial systems S and S moving uniformly parallel and oppositely 1 2 Ernst Karl Kunst 6 (7) directed relative to the natural frame at rest relative to them. The symmetric 0 transformation equations are derived by assuming the validity of: 1) The Lorentz transformation (principle of relativity), 2) The inherent rest frame of nature at rest in any translational 0 movement implying the absolute equality of the inertial systems under consideration (principle of symmetry).
It is understood that the bodies resting in the coordinate sources of S and S are 1 2 geometrically identical if they are compared with each other at rest, according to the Einsteinian definition [2]. According to postulate 2) must the transformation be absolutely symmetric in respect to the systems under consideration. Furthermore, according to both postulates observers resting in S and S must consider 1 2 themselves at rest and at the same time to move relative to and the other 0 system. Thus, the observer in S will besides the Lorentz transformation according 1 to postulate 1) -first line of (7) -, where he considers himself at rest, deduce a second transformation from the moving frame S -according to the principle of 2 relativity now considered at rest -back to his own system -now considered moving relative to and S (see Fig. 1): The dashes designate the moving system S and the open circles the reference 2 rest frame S , now considered moving. Likewise the observer resting in S will 1 2 deduce: Ernst Karl Kunst 7 According to presupposition 2) for the observer resting in either system is valid: If the upper lines of (7) and (8)  Equs. (7) to (9) have been deduced by strictly considering transformations from a system considered to be at rest to the one considered to move. The different states of motion have intentionally been made distinguishable by the use of different symbols. The proper inverse Lorentz transformation in (7) and (8) is given by the respective second line, where the former moving system now must be considered to be at rest according to the principle of relativity. The invariance of the scalar follows from (7) to (10).

The Symmetric Minkowski-Diagram; Equivalence of Four-Vector and Complex Number Calculus
Consider the preferred frame of reference of nature , relative to which S and S o 1 2 are moving at oppositely directed velocity v , and the inertial frame S' propagating n 0 ( cosi sin-) .
relative to S at the velocity v , and their paths in space-time (world-lines). Fig. 1 1 0 shows a diagram of space-time on the grounds of (7) to (10). It is evident that due to the absolute symmetry of S (S' ) and S (S ' ) relative to the triangles 1 1 2 2 0 (0, S , S' ) and (0, S , S' ) must be Pythagorean ones. (14a) As Fig. 1b shows will the special-relativistic space-like world-line S¯S through 1 multiplication by the Lorentz-factor be stretched to the (symmetric) world-line S¯S' -apart from a factor N according to (4) -and, therewith, the Minkowskian  Fig. 1 shows.
The empirical principle of relativity basically implies that any observer in whatever state of inertial motion relative to another system has to consider himself at rest in his frame of rest, with the consequence (among others) that light travels isotropically in all systems of reference alike, which fact is expressed by (11). Thus, it is clear that the hypothesis of FitzGerald and Lorentz that moving bodies are contracted by the factor in the direction of motion is not needed in special 1) Nevertheless, as widely known, introduced Einstein in special relativity the "relativity of simultaneity" to receive a "measurement rule", which allows for the "FitzGerald-Lorentz contraction" as a result of the theory. For this he maintained [2]: i) that the length l' of a "moving" rod r , as measured in the "moving" system, AB equals "according to the principle of relativity" the length l of a like rod (as compared at rest), resting relative to the former one in a system "at rest". Thus, the "moving" observer (r ) would find clocks (A and B), which are at rest relative to AB and synchronized in the system "at rest", not to be synchronous ii) so that must be valid The difference t -t means the time a light signal emitted in the system "at rest" B A needs from A to B, to be there reflected and travelling back to and reach A at the time t , where t -t = t -t .
2) It has been overlooked ever since that this proceeding is not admissible in the framework of special relativity and violates the very basis of that theory: the principle of relativity. If the observer changes from a "resting" system to a "moving" one, then the latter becomes according to the principle of relativity the observers reference rest frame at rest. This implies that the former resting system must now be considered moving relative to the new rest frame, as also follows from (4), where v results from 0 successive Lorentz transformations in respect to the observer at rest. Thus, the time intervals t -t = t -t now belong to a moving system. If the rod resting in the system now considered at rest is designated r (to distinguish it from the now ab moving rod r ), it must be valid r /c = t -t = t -t , as measured by an observer Ernst Karl Kunst 11 3) A correct transformation of the space-time coordinates of a light signal moving relative to the coordinate source of any inertial system into the respective coordinates of another system, being the rest frame of the observer, requires in any case the application of the addition theorem. This implies that conclusion ii) of 1) must be wrong. Instead the only possible and correct comparison of the lenghts of the rods r and r by the observer resting at r can be made if -at the time velocity (coinciding with the x-direction), consequently isotropically, too, to reach either end of L after the time L/(2c) = ût/2. Therefore, owing to the fact that ût' = ût , as observed from the system L at rest, an absolute symmetry of the 0 propagation of the light signals from the respective center is found: As observed from the resting system, the oppositely directed signals in the moving system also arrive simultaneously at both ends to trigger the laser signal, though dilated by the factor ût( -1)/2, as compared with the simultaneous arrival of the 0 light at either end of the resting rod after the time ût/2. Multiplication by c delivers Thus, it follows conclusively that the length L' of the moving rod must be expanded by the factor so that L' = L , as observed from the system considered at rest. 0 0 But it already is from the symmetric setup of (7) to (10) clear that any interpretation of the latter transformation equations of the x-coordinate other than to take them at face value -which results in an expansion of the x-dimension -is ruled out.
5) The reasoning 2) -4) also is in full accord with the well-known experiment of Fizeau in 1851, which in principle corresponds to Einstein's thought experiment 1), with the deviation that the velocity of light c' in the moving system (running water) is slower than in vacuo: c' = c/n, where "n" means refractive index. It has been overlooked by Einstein and ever since that Fizeau's result directly contradicts 1), because it is readily explained by the relativistic addition theorem [4] -in accord with 2) -4). From the latter directly follows Fizeau's empirical formula and, therewith, the velocity of light in the moving medium to be c' = c/n -in vacuo n = 1 and c' = c -, whereas Einstein's argumentation 1) and the derived relations ii) would lead to the pre-relativistic result c' g c/n.
Hence for an observer resting in the coordinate source of S evidently the spatial 1 dimensions of a body resting in the the moving system S' are given by 2 Thus, results where V means volume. Of course, an observer who happens to rest in the system S will deduce the corresponding result: 2 Hence (12) also is valid for the volume of a moving body: n = = V'/V. Now let a 0 system S° , propagating "within" a "real resting" system S , come to a halt within 2 1 and relative to S , e. g. a material particle within some solid material. From (11) in 1 connection with (12) we receive wherefrom is deduced and in the inverse case All other known special relativistic (optical and electrodynamical) effects also result from the first lines of (7) and (8) -owing to the coincidence with the Einsteinian Lorentz transformation.

Interaction-Radii and Geometrical Cross-Sections of Material Bodies in Ultra Relativistic Collision Events
If this theory is correct the relativistic expansion of length or volume should be noticable in ultra relativistic collisions of material particles. Because nearly all collision events in high energy physics more or less are of a grazing kind the mean geometrical dimensions of the colliding particles must be averages over all three axes x, y, and z of the according to (16) and (17) relativistically enlarged volumina. Especially the mean of the x-dimension must be Imagine two real material bodies (m g 0) being spherically symmetrical and identical in all aspects, their centers resting in the coordinate sources of S and S' , 1 2 to collide at at the time t' = t = t = 0. At this moment (7) and (16) Thus, as deduced from either system, considered to be at rest, in ultra relativistic collision events the body resting in the system considered moving must in the mean seem enlarged by the factor . Hence its mean relative geometrical interaction-0 1/3 radius averaged over the three spatial dimensions must be wherefrom the mean geometrical cross-sections follow: In either system the colliding bodies rest, the same enhancement of the interactionradius (24) or of the geometrical cross-section (25) with growing velocity will be noticed. Therefore, relative to the kinematic center , which at at the time t' = t = t 0 2 1 0 = 0 coincides with S and S , the mean total geometrical cross-section is given by 1 2 whereby from the foregoing it is clear that 1' = 1' . As is shown below, (24) and (26) 2 1 are directly related to the interaction radii as derived from high energetic collisions on the strength of the optical theorem and the total cross-sections, respectively.
Consider a particle, based at a system S° , moving through a dense material medium 2 at ultra relativistic velocity and coming to a halt within the medium. Relative to S° the 2 atoms, constituting the medium and resting relative to another, obviously represent the "real resting" system S according to (7). According to (18) and (19) a moving 1 particle S° would relative to the resting atoms seem enhanced by the factor - This result predicts a shrinkage of the interaction mean-free-paths of particles plunging through some material before coming to a halt: where means mean-free-path of a slowly moving particle and 1 the mean 1 1 ûr geo 1 × geometrical cross-section if v « c. 0

Experimental Interaction-Radii, Total Cross-Sections and Mean-Free-Paths Compared with Theory
In the following is investigated, whether the experimentally found rise of the interaction-radius of the protron on the strength of the optical theorem is in accord with (24) and, furthermore, the total cross-section 1 = 1 + 1 of hadrons at ultra tot el inel relativistic collisions in colliders possibly depends solely on the total geometrical cross-section according to (26). We restrict to protrons and antiprotrons, which we assume to be (only) geometrically alike. The geometrical cross-section of the "resting" or "slowly moved" protron (antiprotron) is measured with 1( fm ) = 10 cm = 10 mb (millibarn). Consequently, according 2 -26 2 to (24) the mean interaction-radius of the protron (antiprotron) amounts to (in Fermis) and the mean total geometrical cross-section according to (26) rises to (in millibarn) irrespective of quantum-mechanical effects. Effects of spin are considered to average out over a wide range of collision events.
But before computing the geometrical cross-sections and interaction-radii, we have yet to conceive fair approximations of N and, therewith, of as a function of the i 0 center-of-mass energy to render those computations . According to (14)  1. Therefore, to compute N -N , or N -N , respectively, it will be only a minor 1/9 1/9 2/9 2/9 1 3 1 3 error to deduce -from the approximations 1 3 if the respective E* successively is reached, where E > E > E . Therefrom is given:    It is predicted that the enhancement of the geometrical cross-section or interactionradius also delivers an explanation of the "EMC-effect".
According to (28) the interaction mean-free-paths of nucleii or particles coming to a The EMU 08 experiment at CERN studied the interactions of oxygen beams at E* = 200 and 60 GeV/nucleon in nuclear emulsion and found for inelastic events the interaction mean-free-paths for higher and lower energy beams to be 10.89 cm and 12.84 cm [13]. Extrapolation according to (28) results in 10.89 cm, too. For secondary particles this effect is experimentally well-known and controversially discussed under the term "anomalons" (secondary nuclei with abnormal short mean-free-paths after collision of primaries within some material).

Further Physical Implications
Suppose a body of mass m' = V'!', resting in the coordinate source of the frame of reference S' -where V' means volume as defined by (16) and !' density of mass -, to move inertially relative to the frame of reference S, considered to be at rest. 60 years ago [14], also explains Heisenberg's uncertainty principle: an uncertainty smaller than ûE × ût = h = cannot exist. Furthermore, it is clear that Lorentz whereT means life time and + full width, in by far the most cases nearly integers and in the others integers plus a half result, e. g. 0.98 for the top quark (computed + 1.55 GeV) implying its life-time be exactly one quantum of time, and 3.95 for the 1370 MeV "exotic" meson (+ 385 MeV), recently found at Brookhaven's AGS [15].
Let be smallest part of an one-dimensional manifold R . Then, necessarily, the It is clear that from the foregoing result further implications yet, especially for high energy physics, but which could not dealt with in this first sketchy concept of naturally composite velocity. I thank my wife Ingrid as well as my friend Dr Paul Yule for their assistance and Benjamin Kunst for helpful discussions, which contributed much to clarify the basic idea of this study.