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Science is a collection of alternative ideas, and as such must not be based on voting in which there is only one winner. Alternative subjective views and subjective personal ideas must not be considered as a temporary preparatory situation for the nal showdown that will eliminate `incorrect' opinions (by some omnipotent judge refutation-falsication-disprove experiment) and lead to universal agreement and consensus on the only-one true idea. In this understanding, science must never be objective, and must persist forever to be a bundle of subjective ideas in which everybody has a right to be right. Objective science is dead science. However, the current academic world embrace only one right vision of science and does not allow for alternatives theories. This parochial attachment to a singular and monolithic vision of science is harmful to science and must be jettisoned.

The central concept of the theory of relativity is relative velocity. The velocity of a material body is not an intrinsic property of this body; it depends on the free choice of reference system. Relative velocity is thus reference-dependent; it is not an absolute concept. We stress that even zero-velocity must be relative. Every reference system possesses its own zero-velocity relative to exactly that one system. The theory of relativity formulated in terms of relative velocities, with many zero-velocities, does not imply the Lorentz isometry group. Moreover, we discuss a conceptual dichotomy: two different rival concepts of reference system: the Minkowski space-time observer-monad as time-like vector field, versus the Einstein space-time coordinate tetrad.

7th Biennial Conf. on Classical and Quantum Relativistic Dynamics of Particles and Fields.

Electric and magnetic elds are relative. They depend not only on a choice of electromagnetic sources via Maxwell equations, but also on a choice of observer, a choice of material reference-system. In 1908 Minkowski defined electric and magnetic fields on a four-dimensional spacetime, as tensorial concomitants of observer. Minkowski dened Lorentzgroup-covariance of concomitant tensor eld as group-action that commute with contractions. Present-day textbooks interpret Lorentz-group-covariance of concomitant tensor dierently than Minkowski in 1908. In 2003-2005 Tomislav Ivezic re-invented Minkowski's group-covariance. Different interpretations of group-covariance, lead to different relativity transformations of electric and magnetic fields.

An objective of present article is to explore third possibility, implicit in [Minkowski 1908, x11.6], where a set of all relativity transformations of all material observers forms a groupoid category, which is not a group.

1. Dear Professor Selleri, 2
2. Inertial transformation and weak relativity 3
3. The inertial transformations is a groupoid, and not quasi-group 6
4. Special relativity do NOT negate privileged reference 6
5. Non-uniqueness of the Lorentz boost 6
6. Lorentz group in terms of rotation subgroups 8

The central concept of the relativity theory is a relative velocity. Relativity of the velocity means that the velocity is not absolute concept. The velocity of positive-mass body is not an intrinsic property of this body; it depends on the free choice of the reference system, it is reference-dependent. We stress that the zero velocity must be relative. Every reference system possess the own zero-velocity relative to exactly this system. Many zeroth-velocities contradict to group structure, and therefore the relativity theory in terms of relative velocities must be formulated within the groupoid structure, which is not a group. There is a dichotomy: two di
erent concepts of a positive mass body, a tetrad versus a monad.

The central concept of the (special) relativity theory is a concept of a relative velocity.
Relativity of the velocity means that velocity is not absolute concept. The velocity
of a massive body is not an intrinsic property of this body, but depends on the free choice of
the reference system, it is reference-dependent. The definition of the relative velocity does
not depends on existence or absence of the privileged reference system aether, also the extra
assumption of constant relative velocity is not so important, inertial reference system is also of
secondary relevancy. There is no big difference with this respect among Galilean and Lorentz-
group relativity theories. The concept of a relative velocity depends on the axiom of what it the reference system, the Einsteinian coordinate system versus the Minkowskian fluid vector field. 

Presented at International Conference on Applied Analysis, Quer?etaro, 2007. Segundo Congreso Cientifico Tecnologico, Cuautitlan 2007. A solution of the Maxwell differential linear equations, the electric and magnetic fields, is said to be the electromagnetic radiation, if and only if there is a transport of energy, i.e. if the Poynting vector does not vanishes in no-one reference system. It is known that this is the case if and only if holds the two non-linear algebraic conditions, E.B = 0, and, E² = B². It is proposed to solve first the non-linear algebraic equations, and after look for solutions of the linear differential Maxwell's equations. In this it is shown that each electromagnetic radiation needs no more than two scalar fields introduced by Robert Yamaleev in 2005. These scalar fields are conceptually different from introduced by Edward Whittaker in 1904, and are distinct from Peter Debye potentials

It is proved that the Lorentz boost entails the relative velocity to be ternary: the ternary relative velocity is a velocity of a body with respect to an interior observer, as seen by a preferred exterior-observer. The Lorentz-boost imply non-associative addition of ternary relative velocities. Within Einstein's special relativity theory, each preferred observer (fixed stars, aether, etc), determine the unique relative velocity among each pair of massive bodies. Therefore, the special relativity founded on axiom, that each pair of reference systems must be related by Lorentz isometry, needs a preferred reference system in order to have the unique Einstein's relative velocity among each pair of massive bodies. This choice-dependence of relative velocity violate the Relativity Principle that all reference systems must be equivalent. This astonishing conflict of the Lorentz relativity group, with the Relativity Principle, can be resolved in two alternative ways. Either, abandon the Relativity Principle in favor of a preferred reference system. Or, within the Relativity Principle, replace the Lorentz relativity group by the relativity groupoid, with the choice-free binary relative velocities (not parametrizing isometry). The axiomatic definition of the kinematical unique binary relative velocity as the choice-free Minkowski space-like vector, leads to the groupoid structure of the set of all deduced relativity transformations (instead of the Lorentz relativity group), with the associative addition of binary relative velocities. Observer-independence and the Lorentz-group-invariance are distinct concepts. This suggest the possibility of formulating many-body relativistic dynamics without Lorentz/Poincare invariance.

International Journal of Geometric Methods in Modern Physics, V4, N5 (2007) 739-749. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents.

Presented at the Fifth Workshop Applied Category Theory, Graph-Operad-Logic, Merida, May 2006. The isometry-link problem is to determine all isometry transformations among given pair of vectors with the condition that if these initial and final vectors coincide, the transformation-link must be identity on entire vector space. Such transformations-links are said to be pure transformations, or the boost transformations. In the first part of this essay we provide the complete solution for the link problem for arbitrary isometry. We prove that a solution of the link problem is not given uniquely by the initial and final vectors. Each solution needs the third vector called the privileged or preferred vector. The triple of vectors determine the unique pure isometry-link. We apply these considerations for the Lorentz-boost, parameterized by a relative velocity, and we show that the Lorentz boost needs a choice of the preferred time-like observer, an {ae}ther. Non-uniqueness of the isometric relative velocity, apparently was not the Einstein's intention.

The categorical relativity is a groupoid category of massive bodies in mutual motions. The relative velocity is defined to be the basis-free and coordinatefree binary morphism. We are showing that coordinate-free unique definition of relative velocity in Galilean relativity becomes two different coordinate-free possibilities for relative simultaneity: binary velocity-morphism in categorical relativity, and ternary reciprocal-velocity in isometric special relativity. We are proving that the isometric Lorentz transformation needs at least threebody system. Observer-dependence and the Lorentz-covariance are different concepts.

The Poincare-Lorentz versus the Einstein-Minkowski interpretations of a formal structure of relativity are not the unique dichotomy. We propose to consider the concept of relative velocity as the primary concept with two possibilities: Voigt 1887 & Heaviside 1888, versus Einstein 1905. In categorical relativity the inverse relative-velocity-morphism v⁻¹ is interior-observer-dependent, and not absolute as in the isometric exterior formulation where v⁻¹ = −v.

In the framework of categorical relativity we consider coordinate-free transformations of adopted mathematical co-frames, and transformation of proper-times (clocks), including the transformation of the Einstein-Minkowski simultaneity. The categorical relativity does not predicts the lenght/rod material contraction, because this concept is not basis-free. The concept of simultaneity is basis-free and coordinate-free, and simultaneity in categorical relativity must be relative exactly in the same way as in the isometric special relativity.

As another example we consider the electric field registered by a moving observer: electrodynamics of moving bodies is different from the isometric special relativity with Lorentz transformations.

The kinematics of categorical relativity is ruled by Frobenius algebra, whereas the dynamics of categorical relativity needs the Fr?licher-Richardson algebra.

This work also review the mathematical and theoretical aspects of biological time-dilation and material length-contraction, with comparison with Langevin's interpretation in 1911.

Presented at XXV International Colloquium on Group Theoretical Methods in Physics, Mexico, August 2004. Following Minkowski in 1908, we consider the relative velocity to be the Minkowski space-like vector. We show that the Lorentz boost entails the relative velocity to be ternary: ternary relative velocity is a velocity of a body with respect to interior observer as seen by a preferred exterior observer. The Lorentz boost imply non-associative addition of ternary relative Einsteinian velocities. Within Einstein's special relativity theory, each preferred observer (aether, fixed stars, etc), determine the unique relative velocity among each pair of massive bodies. The special relativity founded on axiom that each pair of reference systems must be related by the Lorentz isometry, needs the preferred reference system in order to have the unique Einstenian relative velocity among each pair of massive bodies. This choice-dependence of relative velocity violate the Relativity Principle that all reference systems must be equivalent.