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THE R.M. SANTILLI FOUNDATION PROMOTING BASIC SCIENTIFIC ADVANCES AND SCIENTIFIC ETHICS |
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Palm Harbor, Florida, November 9, 2011 Unsubscribe link at bottom This announcement is also uploaded in the link http://www.santilli-foundation.org/teaching-grants.php
PART i:
Dear Colleagues,
We have all seen comments on the CERN/GRAN SASSO announcement of neutrinos traveling faster than the speed of light in vacuum (superluminal speeds), which comments range from great enthusiasm for expected advances, to personal criticisms of Einstein as well as of special relativity (SR), with statements such as "Einstein was wrong," "Violation of special relativity," and the like. To honor the memory of Albert Einstein, following my fifty years of research in the field (see the recent general review [1]), I would like to indicate that, under the proper physical conditions and mathematical treatment, superluminal and subluminal speeds are indeed compatible with SR axioms and no violation can be scientifically claimed. I present below the argument in a language accessible to the general physics audience.
This impossibility is due to the fact that, under the indicated assumptions, we can only have action-at-a-distance interactions derivable from a potential (technically known as variationally selfadjoint interactions - SA - see monographs [2] that I wrote when at MIT in 1974-1977 and finalized following a seminar course when at Harvard University in 1977-1981). In this case, it takes infinite energy to accelerate any particle at the speed of light, and the achievement of any superluminal speed can be proved to violate causality, energy conservation, and other physical laws, in full agreement with the 20th century formulation of SR. The use of directly unverifiable conjectures is then dismissed by these violations.
The only possibility known to me for achieving superluminal speeds in a way verifying causality, energy conservation and other physical laws, is that particles and electromagnetic waves propagate within physical media (conditions historicalle referred to as interior dynamical problems). As I indicated since 1982 [3], due to the absence of "point-like wavepackets," the latter conditions imply the necessary existence of contact, zero-range interactions due to wave overlapping. These interactions are non-linear (in the wavefunction), non-local (of integral type not reducible to points) and non-potential, thus not representable with a Hamiltonian (variationally nonselfadjoint interactions - NSA [2]). In turn, the non-Hamiltonian character implies that the time evolution is necessarily non-unitary. The verification of causality and any other physical law for superluminal speeds is then reduced to the use of the appropriate mathematics whose achievement required decades of solitary research (see below).
I would appreciate the indication of superluminal treatments without NSA interactions, with the understanding that I can prove their violation of causality, energy conservation and other physical laws.
(1) x = (xμ) = (x1, x2, x3, t), m = Diag. (1, 1, 1, - c2), I = Diag. (1, 1, 1, 1),
historical line element and light cone in (3, 4) space
(2a) x2 = ( xμ mμν xν ) I
= ( x12 + x22 + x32 - t2 c2) I
the latter cone implying that c, the speed of light in vacuum, is the maximal causal speed, and I = Diag. (1, 1, 1, 1) being the basic unit of the Lorentz-Poincare' (LP) symmetry.
Consider now the characterization of interior dynamical problems via the most general possible, symmetric, and non-singular space M*(x, T*m, I*) with line element in (3+1)-dimension in the x-coordinates of the experimenter (for the general treatment, see monographs [4])
(3) x2 = [ xμgμν(x, v, a, E, d, ψ, ∂ψ, ...) xν ] I*(x, v, a, E, d, ψ, ∂ψ, ...) =
where: the n's are called the characteristic quantities of the medium considered with values (derived from experimental data normalized to the value n = 1 for the vacuum) characterizing the inhomogeneity and anisotropy of the medium; I* is a 4x4 positive-definite matrix assumed as being the inverse of T* for reasons indicated below, with diagonal realization
(4a) I*(x, v, a, E, d, ψ, ∂ψ, ...) = Diag. ( n12, n22, n32, n42) = 1 / T > 0,
the metric g = T*m, and therefore the characteristic n-quantities, have the most general known nonlinear, nonlocal and nonpotential dependence on coordinates x, velocities v, acceleration a, energy E, density d, wavefunctions ψ, their derivatives ∂ψ, and any other needed quantity; and the speed of light c of invariant (2a) is replaced by the local speed
(5) C = c / n4(x, v, a, E, d, ψ, ∂ψ, ...)
thus being bigger, equal or smaller than c, depending on local physical conditions; light cone (2b) is lifted into the form in the space s-direction
(6) xs2/ns2 - t2 c2/n42= 0
with resulting new maximal causal speed
(7) Vmax, s = c (ns/n4)
modified shift law in first approximation (see [4] for the general form)
(8) ν' ≈ (1 ± vsn4/c ns + ...) ν
and related equivalence principle
(9) E* = ms c2 (ns2/n42) = ms V2max, s
A few comments are in order. Firstly, it should be indicated that laws (4)-(9) are uniquely and unambiguously derivable from the universal symmetry of general line element (3) under condition (4), as indicated below. The same expressions coincide with conventional expressions only when spacetime is homogeneous and isotropic, e.g., for ns/n4 = 1. Note the abandonment of the speed of light as the maximal causal speed in favor of the covering geometric law (7) that, of course, recovers c as the maximal causal speed for the vacuum. The abandonment of c as the maximal causal speed is necessary for interior dynamical problems since the media here considered are generally opaque to light. Note also that, in empty space, the maximal causal speed, inertial masses, energy equivalence, etc. are the same in all space directions. By contrast, within physical media, the maximal causal speed, inertial masses, etc., cannot be the same in all directions due to the indicated inhomogeneity and anisotropy (as it is the case, e.g., for our atmosphere), thus being dependent on the desired direction in view. This implies the prediction that the energy equivalence of spinning (thus anisotropic) particle is a constant, but its inertial mass in the equatorial plane is different than the mass of the same particle along the spin axis.
It is easy to see that, despite the above seemingly major divergences, the most general known symmetric and nonsingular spacetime M*(x,T*m, I*) in realization (3)-(9) is isomorphic to the abstract spacetime M(x,m,I) in conventions realization (1),(2) and, consequently, superluminal speeds are admitted by SR axioms under the appropriate physical conditions and mathematical treatment. This result can be proved intuitively in a variety of ways. An effective argument is given by the observation that. imn the transition from invariant (2a) to (3), conventional spacetime coordinates are mutated for a given amount while the corresponding unit is mutated by its inverse,
(10) xs → xs/ns, Is → ns
thus preserving the original geometric features.
The rigorous proof of the above intuitive argument required decades of solitary research due to the necessary construction of a new mathematics today known as Santilli isomathematics, where the prefix "iso" stands precisely to honor Albert Einstein (as well as other founders of 20th century physics) because intended in the Greek meaning of being "axiom preserving." In essence, the proof of the isomorphism of M*(x,T*m,I*) and M(x,m,I) requires lifting the unit I = Diag. (1, 1, 1, 1) of the conventional Minkowski space into a positive-definite, thus diagonal, generalized unit I* with a totally unrestricted functional dependence (4). For consistency, such a lifting required the reconstruction of the entire mathematics underlying SR into a form admitting I* as the basic right and left unit at all levels, including numbers, functional analysis, differential calculus, metric spaces, Lie theory, topology, etc.
In the hope of minimizing major misrepresentations by colleagues and editors alike not expert in the field, I should stress that the elaboration of NSA interior dynamical problems via the conventional mathematics of SR activates the Theorems of Catastrophic Mathematical and Physical Inconsistencies I recalled in my preceding message (reproduced on the website http://www.santilli-foundation.org/Announcement-Super.php). As indicated earlier, the new unit I* represents the physical characteristics of the medium considered. All NSA theories, when treated with conventional mathematics, are known as being necessarily non-canonical at the classical level and non-unitary at the operator level [2].
However, non-canonical or non-unitary transformations on conventional spaces do NOT preserve the basic unit by central assumption. Consequently, conventional treatments of structures (3)-(9) imply the lack of conservation of I* over time, that is, the transition from the original interior problem at time t = 0 to a physically different problem (medium) at a later time. Isomathematics was built precisely to leave a given isounit I* that is, the interior problem under consideration, invariant at all times. After decades of study and the search for any possible alternative, the selection of a generalized unit for the representation of NSA interactions resulted as being the only one assuring the same invariance as that of Hamiltonian-SA interactions. The construction of the new isomathematics was then unavoidable.
As a guide for colleagues and editors seriously interested in honoring (rather than abusing) Einstein's name, let me recall that the first mandatory step was the lifting of Lie's theory into (axiom-preserving) Lie-isotopic form based on the isoassociate product AxB = AT*B of generic quantities A, B (numbers, vector fields, operators, etc.) that I proposed when at the Department of Mathematics of Harvard University in memoir [5] of 1978 and then formulated more rigorously in monographs [2]. The resulting theory is today known as the Lie-Santilli isotheory (see independent studies [6,7,8]).
The original isotopies of Lie's theory turned out as beion g inconsistent because they were based on the unit I of Lie's theory, and consequently they did not conserve I*. The only possible solution was the generalization of conventional numerical fields into the new isofields, namely, rings of elements a, b, etc. called isonumbers with the new multiplication a#b = aT*b admitting I* = 1/t* (called Santilli isounit, T* being called the isotopic element) as the left and right multiplicative unit, under which assumptions all axioms of a numeric field are verified. Isofields were first achieved in Ref. [9] of 1993 (see also monograph [9] by the Chinese mathematician C-X. Jiang of 2000). The discovery of the isofields requested a compatible reformulation of functional analysis, metric spaces, Lie-isotopic theory, etc. since all these methods are notoriously formulated over a numeric field
This second generation of the needed mathematics also resulted in being inconsistent because of the lack of preservation of the same numerical predictions under the same conditions at different times, whose resolution required additional laborious (and solitary) research. The problem was finally identified where I expected the least, in the ordinary differential calculus that, after remaining essentially unchanged since Newton's time, turned indeed out as being dependent on the assumed basic unit of the underlying carrier space when dependent on the differentiation variable. This discovery permitted the construction of the novel isodifferential calculus first achieved in memoir [10] of 1996 (see also the monograph by the Spanish mathematicians Ganformina and Valdes [11]).
As an illustration, consider the conventional differential dr of the variable r and assume that the isounit depends on r, I* = I*(r) = 1/T*(r) > 0. Then, for consistency, the variable r must be replaced by an isovariable r* = rI*, and dr must be lifted into the isodifferential form d*r* = T*(r) d[rI*(r)] that recovers dr when the isounit does not depend on r, but is otherwise structurally different. The lifting of the differential calculus into the isotopic form finally permitted the achievement of the necessary invariance, but only after the third and final reconstruction of the entire preceding formulations (for a general formulation one can study monographs [12] and independent treatments [1,13]).
Interested colleagues, editors or students can find a presentation of this long scientifiuc journey initiated at MIIT, continued at Harvard and completed in various institutions. in the World Lecture Series (WLS) [22], Level II, Lectures IIA and IIB.
Isomathematics was subsequently used for the systematic, step-by-step lifting of all aspects of the LP symmetry, including (see Refs. [4] for a general treatment, and the second announcement of our Foundation of 1q1/25/09 www.santilli-foundation.org/Announcment-2.php) the isotopies of: the rotational symmetry [14]; the Lorentz symmetry in classical formulation [15]; the Lorentz symmetry in operator formulation [16]; the SU(2)-spin symmetry [17]; the Poincare' symmetry [18]; the spinorial covering of the Poincare' symmetry [19]; and the Minkowskian geometry [20].
The general treatment of the LPS isosymmetry is available in monograpghs [12]. An introductory lecture on the LPS isosymmetry is available in free view or download modes in Level III, Lecture IIIA of WLS [22], with lectures on a number of experimental verifications available in Level V. As an example. the universal invariance of line element (3) when projected in conventional (3,4) dimensions, first achieved in Ref. [15], is given by:
(11a) x' 3 = γ* [ x3 - β* x4 (n3 / n4) ],
The proof that new laws (5) to (9) are uniquely and unambiguously derivable from the LPS isosymmetry is rather instructive for colleagues unaware of these advances.
However, LPS transformations (11) can be written in the way formally identical to the conventional transformations
(12a) x*' 3 = γ* ( x*3 - β* x*4 ),
where
(13) x*3 = x3 / n3, x*4 = x4 / n*3
(14) γ* = 1 / ( 1 - β* )1/2, β* = V / C,
V = v / n3, C = c / n4, x4 = t c.
This completes the proof that the LPS isosymmetry is isomorphic to the conventional LP symmetry and, consequently the oriifd that SR axioms do indeed admit arbitrary superluminal or subluminal speeds under proper physical conditions and mathematical treatment.
The resulting isotopies of SR were initially called by the author "Isospecial Relativity" [4]. Subsequently, the term "special" resulted in being redundant for the reason indicated below, and the name predominantly used today by experts in the field is that of Santilli IsoRelativity (IR) [1].
It should be recalled that isomathematics, the Lie-Santilli isotheory and the LPS isosymmetry were developed for the primary purpose of constituting the foundation of the non-unitary, axiom-preserving covering of (nonrelativistic and relativistic) quantum mechanics known as (nonrelativistic and relativistic) hadronic mechanics, where the term "hadronic" indicates its primary applicability to the structure of hadrons, nuclei and stars with the most general known SA and NSA interactions caused by the necessary mutual penetration of the wavepackets and/or charge distribution of the constituents (see the general presentation [12]).
It should be also recalled that the above Lie-isotopic formulations constitute a particular case of the broader Lie-admissible formulations for irreversible processes that, in turn, constitute a particular case of the yet broader multi-valued hyperformulations for biological structure, multiuniverses and otehr structures. All this is solely applicable to matter, thus requiring an anti-Hermitean map called isoduality for antimatter at the various levels of study (see the general treatment [23] and Proceedings [24] of the January 2011 Lie-admissible Conference in Nepal)).
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Palm Harbor, Florida, November xxx, 2011
PART II:
Dear Colleagues,
Thank you for the large number of emails you sent me for questions and comments on my preceding message of November 9, 2011 entitled
With sincere apologies, I simply cannot answer all these emails and, regrettably, I have been forced to collect the most representative questions and comments and outline then in this follow up message via the use of the bibliography of the preceding message.
The preceding message was solely intended to outline my way of honoring the memory of Albert Einstein, not via the widespread abuse of his name by pretending the applicability of SR for conditions the theory was not intended for and never verified directly, but by showing that, under the appropriate physical conditions and mathematical treatment, superluminal and subluminal speeds do verify SR axioms. Therefore, the recent confirmation at CERN/GRAN SASSO of neutrinos faster than the speed of light in vacuum ]31\ do verify SR axioms and no criticism of Einstein or of SR is scientifically valid.
From an inspection of the emails I have received, it appears that the above point has been conveyed and accepted by serious scholars, particularly in view of the identical reformulation of the Lorentz-Santilli transformations (11) in the traditional Lorentz form (12), the latter admitting arbitrary causal speeds bigger or smaller than the speed of light in vacuum.
Hence, in this message I address the majority of the questions and comments I have received that deal with the separate issue of the proper formulation and interpretation of the isotopies of SR known as isorelativity (IR) that, following a laborious and solitary research, I first presented in paper [15] of 1983, extended in monographs [4] of 1991, and finalized in monographs [12] of 1995, with a general update in the five volumes of Refs. [23] of 2008 that include experimental verifications in diverse fields. Please understand that, in attempting to clarify the issues, I must go up in mathematical treatment so as to prevent misunderstandings as much as possible.
The main difficulty I have seen in the various questions and comments, of course in full good faith is the inspection and interpretation of IR via the mathematics and physics of SR. This is equivalent to inspecting and appraising quantum mechanics via the formalism of Newtonian mechanics. Alternatively, the inspection of IR via the formalism of SR is as inappropriate as the inspection and appraisal of SR via the isomathematics of IR.
In any case, I believe that there cannot be a truly new physical theory without a truly new mathematics, and there cannot be a truly new mathematics without new numbers. This is the reason that, to study - as a physicist - interior dynamical problems necessary for serious research on new radiation free fusion and new clean energies, I had to spend the great majority of the fifty years of studies herein reported in search of new numbers, since everything else can be easily derived via simple compatibility arguments at both mathematical and physical levels.
1) The coordinates of M* must be isoscalar,, i.e., have the form x* = xI* (where x is the usual four-coordinate);
2) The metric K* must be an isometric, i.e., have isoscalar as elements
3) The isometric K* must be the most general conceivable, symmetric and nonsingular (thus diagonalizable) metric in (3_1)-dimension with elements given by isoscalars and, therefore, always admitting the factorization K* = T*mI* (see my contribution at the 1994 Marcel Grosmann meeting [55]), where T* is a positive-definite (ordinary) 4x4 matrix called the isotopic element, and m is the conventional metric of the ordinary Minkowski space in the selected version (1), i.e. m = Diag. (e, e, e, -c^2).
Note that, as a central condition for consistency, the isounit I* of the isospace M* must coincide with the isounit of the base isofield F* to avoid catastrophes in the physical content. In conventional 20th century treatments, these units are not unified, because the unit of the Minkowski space is given by I= Diag. (1, 1, 1, 1) while that of the base field is the trivial number 1. In the studies herein reported this 20th century fashion is abandoned in favor of a simple, yet important unification of the units of both M and F for evident compatibility with the isotopies.
Under the above conditions, the isoline element of isospace M* over F* is given by
(15) x*2* = (x*)T # K* # x* ,
where
(16) x* = xI*
where the upper index T stands for transpose, the symbol # represents isoproduct and equation numbers are a continuation of those in Part I. To achieve a better understanding, isoline element (16) should be compared to the corresponding version of the conventional Minkowski line element
(17) x2 = (x)T m x.
One can see that, under isotopies, coordinates x are replaced by coordinates x* = xI*, the Minkowski metric m is replaced by the isometric K* , and the conventional associative product ab is replaced by the isoproduct a#b = aT*b.
The technical understanding of isorelativity requires the awareness that, when formulated on the Minkowski-Santilli isospace M* over the isofield F*, isorelativity coincides with special relativity to such an extend that the maximal causal speed in M* is the speed of light c in vacuum as a necessary condition for M* being a consistent isotopy of the conventional Minkowski space M. The same holds for the Lorentz-Poincare'-Santilli (LPS) isosymmetry and the Lorentz-Poincare' (LP) symmetry and the physical laws.
The mechanism at the foundation of the isotopies consists in mutating the speed of light by the amount characterized by the index of refraction, c => C = c/n_4, while the corresponding unit is mutated by the inverse amount, 1 => 1* = n_4, thus leading to the preservation of the numerical value of the speed of light c in vacuum also on Minkowski-Santilli isospace (due to the multiplication of a quantity and its unit in the isotopic line element). The same happens for other quantities and physical laws. Note that this mechanisms is solely possible in a consistent and invariant way via the use of Santilli's isomathematics (see Refs. [12], Vol. II for brevity).
The above preservation of SR axioms and physical laws under isotopies is the property I used to honor the memory of Lorentz, Poincare', Einstein, Minkowski, and the other founders of 20th century knowledge, as presented in my preceding message, and now reformulated in a mathematically more rigorous way.
(18a) x*2* = (x*)T # K* # x* = [xμ (T*μρ mρν xν] I* = (18b) x = (xμ) = (x1, x2, x3, t), m = Diag. (1, 1, 1, - c2),
(18c) I*(x, v, a, E, d, ψ, ∂ψ, ...) = Diag. ( n12, n22, n32, n42) = 1 / T > 0,
where the n's are the characteristic quantities of the medium considered as indicated in Part I. Universal invariant (18) is primarily intended to geometrize the physical medium considered with particular reference to the local speed of "electromagnetic waves," C = c/n_4.where, n4 is the familiar index of refraction normalized to the value n4 = 1 for the vacuum (see point I-3.2 on the impossibility of a consistent reduction of all; elm waves to photons). This implies that elm waves are propagated by the ether as a universal substratum, as necessary for light to propagate in a straight line within a transparent medium such as water. Note that, contrary to popular 20th century "fears" (because technically unsubstantiated) the ether as a universal substratum does not violate SR in the vacuum because we can never identify the presumed privileged inertial system at rest with the ether.
Once the need for the index of refraction in universal invariant (18) is understood, the introduction of the space characteristic quantities is a mere consequence, e.g., due to symmetrization or the mere application of conventional Lorentz transforms. In any case, a primary objective of the isotopies of SR is that of representing extended, therefore non spherical and deformable particles (see the tile of ref. [15]), for which scope the space characteristic quantities are mandatory, as shown by the experimental verifications of Vol. IV, refs. [23].
ISOAXIOMS FOR LINEAR INTERIOR MOTIONS
(19a) x*' 3 = γ* ( x*3 - β* x*4 ),
The physical interpretation of isorelativity is then provided by the following isoaxioms (first introduced in Ref. [4] of 1991, then in monographs Ref. [12] of 1995, specialized in Paper II of Refs. [46] of 2011 for the interuor of the scattering region, finalized in Ref. [47]):
ISOAXIOM I: The maximal causal linear speed within a physical medium is given by expression (7), i.e.,
(20) Vmax,s = c (ns/n4) = C (ns;
ISOAXIOM II: The additional of speeds within physical media follows the
law
(21) Vtot = (v1 + v2) / (1 + v1v2 / V2max,s);
ISOAXIOM III: Within physical media, the behavior with speed of mass, time and length follows the isotopic laws
(22a) m* = m / √(1 - (v2 / V2max,s),
ISOAXIOM IV: The frequency isoshift within physical media follows the isotopic
law
(23) ν' = γ* ( 1 - β* cos* α* ) ν ≈ (1 ± vsn4/c ns + ...) ν ;
(where we are forced to use isofunctional analysis for consistency);
ISOAXIOM V: The mass-energy isoequivalence within physical media follows law (9), i.e.,
(24) E* = ms c2 (ns2/n42) = ms V2max,s.
We have then proved the following
PROPOSITION II-2.1: Isorelativity coincides with special relativity under the replacement of the speed of light in vacuum c with expression (20) as the maximal causal speed within physical media, special relativity being recovered uniquely and unambiguously when motion return as being in vacuum.
The above isoaxioopms have been experimentally verified in all quantitative sciences, including classical physics, particle physics, nuclear physics, superconductivigty, chemistry, astrophysics and cosmology (see Vol. IV of Refs. [23] for a comprehensive rpesentation and Part III ofg this message for a recent summary available in the site
A simple but effective illustration of the above isoaxiomns is that for interior dynamical problems in water. In this case, the attempted use of SR leads to known inconsistencies [4,12], e.g., because of electrons traveling in water faster than the local speed of light (this is the origin of the Cerenkov light). Therefore, the use of the speed of light in vacuum as the maximal causal speed in water (evidently in trying to salvage causality) causes the violation of the relativistic law of addition of speeds, because the sum of two speeds of light in water does not yield the speed of light in water. If one assumes for maximal causal speeds the speed of light in water, we have the violation of causality due to faster physical electrons. These occurrences are also the best illustration of the impossibility for the local speed of light being the maximal causal speed within physical media.
The use of IR resolves the above inconsistencies. In fact, water can be considered as homogeneous and isotropic to a considerable extent, in which case n_s = n_4 = 2/3, the maximal causal speed in water is indeed the speed of light c in vacuum, and the local speed of light is C = 2c/3 as per experimental evdience. The main advantage over SR is the representation lof all experimental data with the verification of the isoaxioms and without the violation of the addition of velocities, of causality and other laws.
ISOAXIOMNS FOR ROTATIONAL INTERIOR MOTIONS
In this case SR is inapplicable for a number of reasons, such as the presence of centrifugal accelerations, the absence of inertial reference systems, etc. By contrast, IR holds due to the universality of isoinvariant (18) for all possible symmetric spacetimes. As an example, the considered disc within a physical medium can be represented with the line element
(25a) x2 = R-1 ( xs2/ns2 - t2 c2/n42) I*
where s represents a generic radial orientation in the plane of the disc, n_s is a geometrization of the centripetal force and n_4 its symmetrization in the forth direction. In this case, isosymmetry (19) become
(26a) x*' s = γ* ( x*3 - β* x*4 ),
where v_t is the tangenmtial velocitty of the disc at the radius R and V_max, t is the maximal causal tangential velocity (20) also at the radius R.
Therefore, we have the followingisoacioms for interior brotational motions [47]
ISOAXIOM I': The maximal causal angular velocity, centrifugal acceleration and centrifugal force for a disc of radius R and mass m rotating within a physical medium are the quantities at which the tangential speed at R is the maximal causal speed of Isoaxiom I, amd are given by
(27a) Ωmax = ωc (ns/n4);
where ωc is the angular velocity for which the tangential speed at R is the speed of light in vacuum.
ISOAXIOM II': The additional of angular velocities within physical media follows the
law
(28) Ωtot = (ω1 + ω2) / (1 + ω1ω2 / Ω2max,s);
ISOAXIOM III': Within physical media, the behavior with the rotational velocities of mass, time and length follows the isotopic laws
(29a) m* = m / √(1 - (ω2 / Ω2max) = m / √(1 - (a / Amax),
where ω is the measured angular velocity and a is the centrifugal acceleration at R for the angular velocoty ω.
ISOAXIOM IV': The frequency isoshift for rotational motions within physical media follows the law
(30) ν' = γ* ( 1 - β* cos* α* ) ν ≈ (1 ± vt / Vmax,t + ...) ν = (1 ± ω n4/ωc ns + ...) ν = ω / Ωmax + ...) ν
ISOAXIOM V': The mass-energy isoequivalence within physical media follows the law (where c is now the tangential speed at the radius considered)
(31) E* = m Ωmax2 R2 = m Amax R .
The best iexperimental verificaiton of the above rotational axioms is that provided via the recent Mossbauer experiment by A. Kholmetskii and his associates (see Ref. ]28] and Lecture 9 of WLS [25]) for IsoAxioms IV' that can be written
(32) ν' ≈ [ 1 ± (n4 / ns) (ω / ωc) + ...) ν.
resulting in the numerical value calculated as an average of first and second order contributions
(33) n4 / ns = 1.2
namely, for experiment [28], we have the property (34) Ωmax = ).80 ωc
In examining the above measurement, one should keep in mind that the rotational isoaxioms have been formulated for the general case of a disc rotating within a physical medium. However, for the case of experiment [28], the presence of air can be ignored for the hard gamma of the Mossbauer effect. This point illustrate the importance of Kholmetskii's experiment because it has measured deviations from Einsteinian axioms specifically caused by accelerations and other non inertial effects. Kholmetskii's experiment also identify the primary representation by the characteristic n-quantities of these non inertial effects, besides the additional contributions from the physical medium when appreciable or applicable.
For experimentalists (or theoreticians who feel uneasy learning new mathematics), I have developed a very simple procedure to map any given formulation for exterior conditions of point particles in vacuum solely under SA interactions into the corresponding interior conditions with the addition of NSA interactions (while leaning hamiltonian interactions unchanged). It consists in:
1) Identifying the characteristics of the desired interior conditions or of the desired NSA/non-Hamiltonian interactions due to motion of extended objects within physical media (see below for examples), thus identifying the isounit I*;
2) Introducing a noncanonical transform at the classical level (nonunitary transform at the operator level) verifying the identity
(35) U U† = I* = 1/T*; and
3) Submitting the totality of the formalism of the original exterior conditions to the above noncanonical (nonunitary) transform, including all quantities and all their operations with the warning that, in the event "any" quantity or operation is not mapped, there are hidden catastrophic inconsistencies.
In fact, under the above rules, we have the map of the unit I = Diag. (1, 1, 1, 1) of the original LP symmetry into the isounit I* of the covering LPS isosymmetry, the map of numbers n into isonumbers n*, the map of the conventional associative product AB between generic quantities A, B. into their isoproduct, the map of the exponentiation of a quantity into the isoexponentiation, etc.
(36a) I → I' = U I U† = I*,
It is then evident that the above noncanonical (or nonunitary) map of the Lorentz-Poincare' symmetry leads uniquely and unambiguously produces the covering Lorentz-Poincare'-Santilli isosymmetry in all its aspects (enveloping algebra, Lie algebra lie group, representation theory, etc.). Note the non triviality of the map since the isotopic element T*(x, v, a, E, d, ψ, ∂ψ, ...) appears in the exponent of the group structure/ This most general possible non-linearity, non-locality and non-canonical is also seen in the LPS isotransforms (11) by remembering that for all characteristic quantities we have the functional dependence n - n(x, v, a, E, d, ψ, ∂ψ, ...).
Once the covering isotopic formulation is achieved via the above truly elementary procedure, we remain with the crucial invariance over time, e.g., the prediction of the same numerical values under the same conditions at different times. It is at this point where the use of isomathematics becomes mandatory. Recall that Santilli's isounit I* represents the characteristics of the considered interior system. Hence, the use of any additional noncanonical (or nonunitary) transform will alter the numerical value of the isounit,
(37) I* → I*' = U I* I U† ≠ I*,
thus performing the transition from the original interior system to a different one, e.g., we may have the transition from the synthesis of the neutron in a star to the chromosphere surrounding a given quasar.
However, noncanonical (nonunitary) transforms treated with 20th century mathematics can be identically reformulated via isomathematics by reconstructing canonicity (unitarity) at the isotopic level. This is done very simply with the identical reformulation
(38) U = U*T*1/2
under which
(39) UU† = U* # U*† =
U*† # U* = I*.
It is then easy to see that, under the above isocanonical (isounitarty) transforms, we do indeed achieve the needed invariance over time in which absence noncanonical (nonunitary)theories verify the Theorems of Catastrophic Inconsistencies outlined in the announcement of September 4, 2011 (available in the website http://www.santilli-foundation.org/Announcement-Super.php). In fact, it is easy to see that, in the Minkowski-Santilli isospace over isofields, the isounit I* and the isotopic element T* are numerically preserved,
(40a) I* → I*; = U* # I* # U*† = I*,
and so is the entire theory. In conclusion, when treated with the appropriate mathematics, nonlinear, nonlocal, non-Hamiltonian interactions do indeed preserve all the axiomatic properties and consistencies of classical Hamiltonian mechanics (quantum mechanics). As a matter of fact, this occurrence is so strong that, in the event "any"feature of the original theory is not preserved, the isotopies have not been correctly constructed.
In closing, I should indicate that I constructed isomathematics, isorelativity and the LPS isosymmetry for the primary purpose of being the foundations of the covering of Relativistic Quantum Mechanics (RQM) known as Relativistic Hadronic Mechanics (RHM), since the latter is needed for truly "new" and "clean" energies and fuels so much needed by society. Needless to say, RHM can also be constructed from RQM via the systematic application of a nonunitary-isounitary transform to the totality of the quantities and their operations of RQM that we cannot possibly review in an email (see volumes [12] for details).
It is hoped that the preceding outline has indicated that, again under the proper mathematical treatment, the applicability of SR can be dramatically broadened to the point of achieving "direct universality" for all possible reversible exterior and interior dynamical problems. We here refer to the representation of all systems of the class considered (universality), directly in the coordinate of the experimenter (direct universality) while irreversible processes require the more advanced Lie-admissible treatment (see Refs. [23]).
However, the mathematical, theoretical and experimental implications of this broadening are far from trivial, as outlined below. In considering them, colleagues, editors and students are suggested to abandon the familiar 20th century features for exterior problems and understand that we are dealing with new 21st century vistas with far reaching possibilities for scientific technological and industrial advances [1].
3.1. Experimental verifications via superluminal speeds
3.2. Experimental verifications via subluminal speeds.
A) the impossibility for photons to reach a numerical representation of experimental data, such as the angle of refraction (since photons will scatter in all directions);
B) the impossibility to represent the large reduction of the speed of light, e.g., in water (photon scattering can only represent a small percentage of speed reduction as serious scholars are requested to verify with calculations rather than words);
C) the impossibility of representing the propagation of light along a straight line (that would require a large number of photons traversing a large number of atoms and nuclei in a straight line);
D) the evident impossibility of reducing to photons elm waves of large wavelength that have exactly the same phenomenology within physical media as visible light; E) the impossibility, e.g., by radio antenna, of emitting elm waves with large wavelength as a sequence of photons; and other inconsistencies.
The above occurrences mandated the general use of Maxwell's electromagnetic waves with the resulting emergence of the historical Lorentz problem, that is, the invariance of the local speed of light (5). The LPS isosymmetry then applies due to its universality established by various authors (see, e.g., Ref. [1]) thus requiring its independent verification not only for superluminal but also for subluminal speeds. An important prediction of the LPS isosymmetry for the case C = c/n < c, first identified in Refs. [4] of 1991, is a shift of the frequency of light propagating within transparent physical media without any relative motion between the source, the medium and the detector, today known as Santilli IsoShift (IS). The prediction is due to the general functional dependence of the n-characteristic quantities, thus including that on the speed, under which isoshift law (8) can be written
(41) ν'v = 0 ≈ (1 ± vsn4/c ns + ...) ν = (1 ± N + ...) ν
where N is a constant for a given medium and frequency. The anomalous shift is then called an IsoRedShift (IRS) when the shift is toward the red (light loses energy E = hv to cold media), IsoBlueShift (IBS) when light acquires energy from hot media, and NoIsoShift (NIS) when the energy lost by light is equal to that gained from the medium. In Ref. [4] I then predicted that the tendency toward the red of Sunlight at Sunset is an IRS and proposed the verification or disproof of the hypothesis by following the Sun spectrum from the Zenith to the horizon. Following about twenty years of rejections of my proposal by astrophysics laboratories around the world, I conducted myself the measurement in 2009 of IRS at our laboratory in Florida and confirmed the prediction [32-35]. The results were then independently confirmed via extensive and repeated measurements presented in Refs. [36]. Note that all tests refer to direct Sunlight thus excluding scattering due to its impossibility of propagation along a straight line. Presentations of the above aspects can be found in WLS [22], Level V and WLS [25], Lecture 3.
3.3. Application to the absence of universe expansion, big bang and all that.
a) the conjecture of the expansion of the universe to maintain the validity of Doppler's law from the sole experimental evdience, which is the cosmological redshift pf light from fart away galaxies;
b) plus the conjecture of the acceleration of the expansion due to the dependence of the redshift on the distance;
c) conjectures a) and b) requiring the additional conjecture that 95% of the universe is constituted by the invisible "dark energy" whose existence has been conjectured in the hope of representing the conjecture of the expansion of the universe and the conjecture of its acceleration;
d) plus the additional unreassuring conjecture that billions of galaxies at the edge of the known universe travel at speeds bigger than that of light.superluminal speeds that are necessary for the numerical representation of the redshift of light from said edge of the known universe (a condition oddly accepted for entire galaxies by cosmologist who dismiss superluminal speeds in our environment);
e) plus the further conjecture of the big bang requiring the universe being void of galaxies in a radius of about 14 billion light years, plus the prediction of the the decrease, rather than the increase of the expansion, and other known inconsistencies;
f) plus the yet additional conjecture that the background radiation originated at the time of the big bang some 14 billions years ago, when known calculations established that, in that case, the background radiations would have been absorbed by galaxies and intergalactic dust billions of years ago;
g) the above plethora of interconnected unverifiable conjecture, each voiced in support of a preceding unverifiable conjecture, implying a return to the Middle Ages with Earth at the center of the universe, since the cosmological redshift is the same for the same distance in all directions from Earth, in which case supporters of the above plethora of conjecture come forward with yet another. perhaps the most audacious hyperbolic conjecture of them all, namely, that spacetime itself behaves like rubber (a conjecture necessary for uniform expansion throughout the universe without a central point of expansion as inherent in the big bang), all conjecture being individually studied so as to avoid their hyperbolic character when joined, and all conjectures being clearly intended to maintain the validity of 20th century SR formulations under conditions they have been experimentally disproved.
Note that the evidence on Santilli intergalactic IRS not only eliminate the entire plethora of conjectures, but also provided a continuous source for the background radiation. The same evidence on Santilli IRS eliminates dark matter since innergalactic spaces, as visible in telescopes, are physical media (see also de M. A. Souza spiral galaxies without any need for dark matter in Proceedings [25]). Dark energy is eliminated not only from the absence of the expansion and its acceleration as well as equivalence principle (9) yielding values of the energy equivalence for superdense masses much bigger than those derived via the conventional equivalence law. In fact, dark energy is eliminated by the rather moderate average value 8c for the maximal causal speed in the interior of black holes, quasars, stars, and other hyperdense masses, which is a rather moderate amount for the extreme interior conditions here considered [32,37]. To defend against his name from widespread abuse, Albert Einstein clearly stated that the equivalence principle E = mc^2 is valid for point-like wavepackets moving in empty space because, under these conditions, c is indeed the maximal causal speed. The extension of the same equivalence principle to the extreme conditions in the inferior of black holes and other astrophysical bodies without due scrutiny is indeed an abuse of Einstein's name and must be denounced as such. Presentations of the above aspects can be found in WLS [25], Lecture 6.
3.4. Application to grand unification and operator gravity
3.5. Application to the synthesis of neutrons from hydrogen atoms
(42a) p+ + e- => n + v
for the primary intent of salvaging the conservation of the angular momentum. However, quantum mechanics was not salvaged in this process since the mass of the neutron is 0.872 MeV bigger than the sum of the rest energies of the proton and the electron, in which case the Schroedinger equations no longer admits physically meaningful solutions. This occurrence forced the submission in 1978 of a nonunitary axiom-preserving covering of quantum mechanics under the name of hadronic mechanics [5] as the only known way of admitting "positive binding energies" for extended particles in conditions of total mutual penetration. An exact and time invariant representation of "all" characteristics of the neutron in synthesis (15) was then achieved in 1990 at the nonrelativistic level [41] and in 1996 at the relativistic level [19] (see Kadeisvili general review [42]). A main point of these results is that, during his time, Fermi had no other option than that of representing the proton as a point, because that is the sole possibility for quantum mechanics. Hadronic mechanics has permitted the representation of the proton as an extended and hyperdense charge distribution, in which case, there is no possibility of formulating the hypothesis of the neutrino because of the emergence of an angular momentum (absent at Fermi's time) originating from the compression of the electron inside the hyperdense and spinning proton. This produces constrained orbital angular moment of the electron equal yo the proton spin (an occurrence prohibited in quantum mechanics but quite natural for the nonunitary structure of hadronic mechanics). It then follows that the total angular momentum of the electron compressed inside the proton is null, the spin of the neutron coincides with that of the proton, and there is no possibility of achieving the spin 1/2 needed for the neutrino hypothesis. Additionally, there is no possibility of identifying the energy needed for the formulation of the neutrino hypothesis. It should be noted that a necessary condition for the representation of all characteristic of the neutron in its synthesis (15) crucially depends on the tangential speed C = 1,56 c for the electron constrained within the hyperdense proton, a numerical value confirmed by the Bose-Einstein correlation and other fits of particle data 9see Volume IV of Refs. [23]).
Despite these advances, the need for a mechanism delivering missing for synthesis (15) persists. This need suggested the formulation of the etherino hypothesis [43], not as a particle to replace the neutrino, but as a longitudinal impulse propagated by the ether delivering the missing quantities [43]. Rather than voiding the results of the neutrino experiments, these studies indicate that said experiments may well be the tip of a scientific iceberg with implications beyond our imagination at this time. In fact, it is possible that, in lieu of detecting hypothetical neutrinos (that cannot be directly detected by conception), the experiments may well detect longitudinal impulses propagating in the ether, so as to render plausible its passage through large bodies with minimal scattering (something rather implausible for contemporary massive neutrinos). In turn, the CERN/GRAN SASSO announcement [31] may well be yet another form of superluminal communications besides Nimtz vacuum tunneling [26,27]. Intriguingly, as it is the case for the slow-down of elm waves within transparent media, it may well be that the superluminal speed detected by CERN/GRAN SASSO is decreased in its propagation under ground with the possibility of bigger superluminal speeds for the propagation of the same impulse in vacuum. A presentation of the (rather large number of) historical but unsolved problematic aspects of the various neutrino conjectures can be found in WLS [22], Level I, Lecture IE.
3.6. Application for gthe absence of symmetry breaking
3.7. Mutation of spacetine
All in all, the above remarks confirm that, rather than having achieved terminal stage as preferred by a few, our scientific knowledge is just at its infancy and so much remains yet to be discovered by young minds of a new age.
REFERENCES
[1] I. Gandzha and J. Kadeisvil, New Sciences for a New Era
Sankata Printing Press, Nepal (2011), free pdf download from
[2] R. M. Santilli, Foundation of Theoretical Mechanics,
Volume I (1978), and Volume II (1982),
Springer-Verlag, Heidelberg, Germany,
[3] R. M. Santilli, "Can strong interactions accelerate particles faster than the speed of light?"
Lettere Nuovo Cimento Vol. 33, pages 145-153 (1982)
[4] R. M. Santilli, Isotopic Generalizations of Galilei and Einstein Relativities,
Vols. I and II (1991), Ukraine Academy of Sciences
[5] R. M. Santilli, Hadronic J. Vol. 1, 223-423 and 574-901 (1978),
http://www.santilli-foundation.org/docs/Santilli-58.pdf
[6] D. S. Sourlas and G. T. Tsagas, Mathematical Foundation of the Lie-Santilli Theory,
Ukraine Academy of Sciences 91993),
[7] J. V. Kadeisvili, Santilli's Isotopies of Contemporary Algebras, Geometries and Relativities,
Ukraine Academy of Sciences, Second edition (1997),
[8] R. M. Santilli, "Isonumbers and Genonumbers of Dimensions 1, 2, 4, 8,
their Isoduals and Pseudoduals, and "Hidden Numbers" of Dimension 3, 5, 6, 7,"
Algebras, Groups and Geometries Vol. 10}, 273 (1993),
[9] Chun-Xuan Jiang, Foundations of Santilli Isonumber Theory,
International Academic Press (2001),
[10] R. M. Santilli, "Nonlocal-Integral Isotopies of Differential Calculus, Mechanics
and Geometries," in Isotopies of Contemporary Mathematical Structures,
P. Vetro Editor, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 7-82 (1996),
[11] R. M. Falcon Ganfornina and J. Nunez Valdes,
Fundamentos de la Isoteoria de Lie-Santilli,
International Academic Press (2001),
[12] R. M. Santilli, Elements of Hadronic Mechanics.Volumes I and II
Ukraine Academy of Sciences, Kiev, second edition 1995,
[13] J. V. Kadeisvili, "Foundations of the Lie-Santilli Isotheory,"
in "Isotopies of Contemporary Mathematical Structures,"
P. Vetro Editor, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 7-82 (1996),
[14] R. M. Santilli, ''Isotopies of Lie Symmetries, I and II,"
Hadronic J. Vol. 8, 36 and 85 (1985),
[15] R. M. Santilli, "Lie-isotopic Lifting of Special Relativity
for Extended Deformable Particles,"
Lettere Nuovo Cimento Vol. 37, 545 (1983),
[16] R. M. Santilli, "Lis-isotopic Lifting of Unitary Symmetries
and of Wigner's Theorem for Extended and Deformable Particles,"
Lettere Nuovo Cimento Vol. 38, 509 (1983),
[17] R. M. Santilli, "Isotopic Lifting of the SU(2) Symmetry with
Applications to Nuclear Physics,"
JINR rapid Comm. Vol. 6. 24-38 (1993),
[18] R. M. Santilli, "Nonlinear, Nonlocal and Noncanonical
Isotopeis of the Poincare' Symmetry,"
Moscow Phys. Soc. Vol. 3, 255 (1993),
[19] R. M. Santilli, "Recent theoretical and experimental evidence
on the synthesis of the neutron,"
Communication of the JINR, Dubna, Russia, No. E4-93-252 (1993),
published in the Chinese J. System Eng. and Electr. Vol. 6, 177 (1995,
[20] R. M. Santilli, "Isominkowskian Geometry for the Gravitaitonal
Treatment of Matter and its Isodual for Antimatter,"
Intern. J. Modern Phys. D {\bf 7}, 351 (1998),
[21] R. M. Santilli, "Isorepresentation of the Lie-isotopic SU(2)
Algebra with Application to Nuclear Physics and Local Realism,"
Acta Applicandae Mathematicae Vol. 50, 177 (1998),
[22] J. Pace, Chairman, World Lecture series
[23] R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry.
Volumes I, II, III, IV and V
International Academic Press 2008,
[24] Third International Conference on the Lie-admissible
Treatment of Irreversible Processes, held at
Kathmandu University, Nepal, January 4 to 8, 2011
http://www.ku.edu.np/iclatip/ proceedings available from the linp
[25] San Marino Workshop on Astrophysics and Cosmology for Matter
and Antimatter, September 5 to 9, 2011
http://www.workshops-hadronic-mechanics.org
main lectures available in free view or download from the link
[26] G. Nimtz, "Do evanescent modes violate relativistic causality?"
Lectures Notes in Physics, Springer-Verlag, Berlin-Heidelberg (2006).
[27] G. Nimtz, "Experimental confirmation of superluminal communications,"
to appear in the Proceedings of the 2011 San Marino Workshop in
Astrophysics and Cosmology for Matter and Antimatter,
For a DVD in the lecture, please visit
[28] A. L. Kholmetskii, T Yarman, O.V. Missevitch, "Detection of
extra energy shift between emission and absorption lines in
Mossbauer experiments in rotating systems," to appear in the
Proceedings of the 2011 San Marino Workshop in
Astrophysics and Cosmology for Matter and Antimatter,
For a DVD in the lecture, please visit
[29] A.L. Kholmetskii, O.V. Missevitch, and R. Smirnov-Rueda,
"Experimentally Observed Anomalously Small Retardation of
Bound Electromagnetic Fields in Near Zone and Possible Physical
Implications," to appear in the Proceedings of the
Proceedings of the 2011 San Marino Workshop in
Astrophysics and Cosmology for Matter and Antimatter,
For a DVD in the lecture, please visit
[30] A. Bunthaler, H. Falcke, G. C. Bower et al., III Zw2,
The first superluminal jet in a Sayfert galaxy, A&A 2000; 357, L45.
[31] CERN/GRAN SASSO press release on superluminal neutrinos
http://public.web.cern.ch/press/pressreleases/Releases2011/PR19.11E.html
[32] R. M. Santilli, "Experimental Verifications of Isoredshift with Possible
Absence of Universe Expansion, Big Bang, Dark Matter, and Dark Energy,"
The Open Astronomy Journal, 2010, Vol. 3, page 1-43m
[33] R. M. Santilli, Contributed paper in the
Proceedings of the International Conference on Numerical Analysis
and Applied Mathematics, Rhodes, Greece,
September 19-25, 2010, T. E. Simos, Editor,
AIP Conference Proceedings Vol. 1281, pp. 882-885 (2010)
[34] R. M. Santilli, "Iso-Minkowskian Geometry For Interior
Dynamical Problems," Contributed paper in
Cosmology, Quantum Vacuum, and Zeta Functions,
Diego Saez-Gomez, Sergei Odintsov Sebastia Xambo
Editors, Springer, 2011.
[35] R. Anderson, Confirmation of Santilli IsoRedShift and IsoBlueShift
[36] G. West and G. Amato, "Experimental Confirmation of
Santilli's IsoRedShift and IsoBlueShift," to appear in the
Proceedings of the 2011 San Marino Workshop in
Astrophysics and Cosmology for Matter and Antimatter,
For a DVD in the lecture, please visit
[37] R. M. Santilli, "Absence of Universe Expansion, Expansion
Acceleration, Big Bang, Dark Matter and Dark Energy
from the Experimental Confirmation of IsoRedShifts
and IsoBlueShifts, to appear in the
Proceedings of the 2011 San Marino Workshop in
Astrophysics and Cosmology for Matter and Antimatter,
For a DVD in the lecture, please visit
[38] R. M. Santilli, "Partons and Gravitation: some Puzzling Questions,"
(MIT) Annals of Physics, Vol. 83, 108-157 (1974),
[39] R. M. Santilli, "Nine Theorems of Inconsistency in GRT
with Resolutions via Isogravitation,"
Galilean Electrodynamics, Summer 2006, Vol. 17, 43 (2006),
[40] R. M. Santilli, Isodual Theory of Antimatter with Applications to
Antigravity, Grand Unifications and Cosmology, Springer (2006),
[41] R. <. Santilli, "Apparent consistency of Rutherford's hypothesis on the
neutron as a compressed hydrogen atom,"
Hadronic J. Vol. 13, 513-531 (1990)
[42] J. V. Kadeisvilki, ''The Rutherford-Santilli Neutron," Hadronic J. {\bf 31}, 1 (2008),
available in free pdf download from the link
[43] R. M. Santilli, "The etherino and/or the Neutrino Hypothesis?"
Found. Phys. Vol. 37, p. 670 (2007)
[44] R. M. Santilli, "Isotopic quantization of gravity and its universal isopoincare' symmetry."
in the Proceedings of "The Seventh Marcel Grossmann Meeting in Gravitation,, R. T. Jantzen, G. M. Keiser and R. Ruffini, Editors, World Scientific Publishers pages 500-505(1994).
[45] R. M. Santilli, "Unification of gravitation and elkectroweak iunteractions," in the Proceedings of the "Eight Marcel Grossmann Meeting in Gravitation,, Israel 1997, T. Piran and R. Ruffini, Editors, World Scientific, pages 473-475 (1999)
[46] "Nonunitary Lie-isotopic and Lie-admissible scattering theories of hadronic mechanics,"
[47] "Formulation, interopretation and verification of isorelativity,"
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