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AMERICAN SCIENTIFIC LEADERSHIP,VII:

ITS ISODUAL FOR ANTIMATTER

Ruggero Maria Santilli

The Institute for Basic Research

Email: research@i-b-r.org

Tel. 727-688 3992

**Abstract**

We study some of the widely ignored Einstein's criticisms on his own gravitation, and show the implied lack of clear compatibility of general relativity with special relativity, the interior gravitational problem, electrodynamics, quantum mechanics and grand unifications. We show the impossible existence of gravitational waves under the validity of only one of the various insufficiencies of general relativity. We then show that a resolution of Einstein's historical doubts requires the abandonment of centuries old mathematics in favor of new mathematics specifically conceived for gravitation; we study the representation of gravitation via the novel isomathematics and related isogeometries, which is known under the name of *isogravitation;* we show its resolution of the historical compatibility problems with the achievement of a clear and unambiguous compatibility with all 20th century theories; and we finally show that isogravitation admits all sound experimental verifications that are popularly believed to apply solely to general relativity.

**1. The gravitational waves imbroglio**

Various colleagues have brought to my attention the recent withdrawal from publication of claims of "experimental verification" of gravitational waves, as discussed in articles at
,
*The Economist*,
*Scientific American*, and in other conduits.

To my knowledge, this is the first glimpse of scientific sanity in one hundred years of Einstein gravitation, because all preceding claims of "experimental verifications" were instantly published by biased editors without even a lilliputian image of the galactic severity used in the review of opposing claims, both editorial reviews generally being without a serious or otherwise credible scientific content.

Somewhat encouraged by a possible return of gravitation to scientific sanity, I decided to indicate that the impossibility to date of detecting gravitational waves is much deeper than what stated, since gravitational waves are prohibited by the historical insufficiencies of Einstein gravitation that have remained ignored in the mainstream literature for one century.

I have stated several times in my writings that the theory developed by Lorentz [1], Poincare' [2], Einstein [3], Minkowski [4] and others, known as **special relativity,** has a "majestic axiomatic structure and an impeccable body of experimental verifications" under the conditions clearly stated by Einstein, namely, for: A) point particles and electromagnetic waves; B) propagating in vacuum; C) when referred to an inertial reference frame.

Whenever any of Einstein's conditions A, B, C are violated, special relativity is at best approximately valid, and often it is totally inapplicable (rather than violated) for serious quantitative studies, as it is the case for the synthesis of the neutron from the hydrogen in the core of a star [5].

By contrast, I have stated various times that Einstein general relativity [6] is a scientific religion because of historical insufficiencies, some of which were identified by by Einstein himself and others, such as lack of clear compatibility of general relativity with special relativity, interior gravitational problems, electrodynamics, quantum mechanics and grand unifications, which incompatibilities have remained unaddressed by Einstein followers for one full century, let alone resolved in peer reviewed journals [7] (see also the view by the late J. V. Kadeisvili [8]).

I should emphatically stress that the current political condition of gravitation is solely due to Einstein's followers and definitely not to Einstein, because the latter is known to have expressed numerous severe criticisms of his own theory (some of which are reviewed below), and the current political condition of gravitation is entirely due to the lack of their treatment in mainstream literature for one full century.

In this informal presentation (which is intended to solicit critical comments by interested colleagues for the completion of the joint format paper with the same title) I report my half a century of research of Einstein's majestic vistas in contrast with those of his followers, and submit rudimentary ideas for possible advances in gravitation that, due to their complexity and the encompassing of all known quantitative sciences, will likely require generations of collegial studies by serious scholars.

It should be finally noted that the literature accumulated over a century in the field is very vast indeed. To avoid a prohibitive length, I listed below only the references of direct relevance to the addressed topics. A comprehensive list of references up to 2012 is available in the independent general review [37].

**2. First historical insufficiency of general relativity:
Ignore the refraction of star light passing through the Sun chromosphere, with consequential lack of evidence that space is curved.**

As it is well known, the conjecture of an actual, physical, curvature of space was inferred from the 1.75 arc-second "bending" of star light passing near the Sun. Half of this value, 0.87 arc-seconds, is known to be due to a purely Newtonian attraction of light.

To see it, I first recall that for Newton gravitation to be "universal" it must also attract light, and that the source of gravitation is the *energy* of a body since mass is a measure of our ignorance on inertia. Hence, I always wrote Newton's equation in the * identical * form in terms of the energy rather than mass

(1) F = g m_{1} m_{2} / r^{2} = G E_{1} E_{2} / r^{2}, G = g/ c^{4}.

The calculation of the 0.87 arc-seconds deviation caused by Newton gravitation of star light passing near the Sun surface is then an exercise for first year graduate students in physics.

``

The remaining 0.87 arc-seconds deviation have been known for a century to be due, *not* to the curvature of space, but to the *refraction of star light when passing through the Sun chromosphere* (see, e.g., Ref. [10] and references quoted therein). Additionally, the refraction if light passing through gaseous media is inherent in the experimental confirmations of Santilli IsoRedShift (IsoBlueShift) of light traveling within cold (hot) gases [11-5].

Despite one century of studies, the "actual" orbits of planets in our Solar systems have not been represented in an accurate, unique and time invariant way via Einstein gravitation, while they are exactly and unambiguously represented by Newton's gravitation and Kepler's laws. In fact, calculations based on the Riemannian geometry of the actual orbits of planets, besides not being unique due to the non-linearity of the theory, are different than physical orbits, and are not the same over time (see below).

Needless to say, gravitational waves crucially depend on the curvature of space represented via the Riemannian geometry. Until we dismiss in peer reviewed journals the mathematical, theoretical, experimental and visual evidence against the curvature of space, with consequential insufficiency of the Riemannian geometry to represent gravitation, all studies on gravitational waves may well result to be vacuous.

Ignore the electromagnetic origin of all masses, with consequential invalidation of Einstein's reduction of gravitation to pure curvature without sources.

As it is well known, since the contribution to gravity of the total electric and magnetic field of a body is of the order of 10

(2) G_{ij} = R_{ij} - g_{ij} R/2 = 0, i, j = 1, 2, 3, 4.

In 1974, when I was at MIT, via the full use of quantum electrodynamics, including advanced and regarded treatments, I proved that the electromagnetic origin of the mass requires the necessary presence in the r.h.s. of the field equations of source tensor of first order in magnitude of electromagnetic nature, irrespective of whether the body is charged or neutral [16],

(3) G_{ij} = R_{ij} - g_{ij} R/2 = k T_{ij, elm}

where K is a unit dependent constant, and the terms "first order magnitude" is referred to the condition of entirely representing the *gravitational mass* of the body considered [15]

(4) m_{grav} = ∫ T_{00} dv

The most political a physicist should admit for his/her own dignity that the mass of the electron is of entire electromagnetic origin. Therefore, field equations (2) are insufficient to represent the gravitational field of the electron in favor of Eqs. (3), (4).

But then, the same physicist should admit that exactly the same conclusion holds for the positronium, namely, he/she must admit that the gravitational mass of the positronium is of entire electromagnetic origin despite the total charge and magnetic moment being null. Therefore, Einstein's field equations (2) are basically insufficient for the representation of the gravitational of the positronium in favor of broader Eqs. (3), (4).

In defense of Einstein, I have to recall that, contrary to his followers, Einstein always expressed serious doubts on field equations (2), for instance, by calling their r.h.s. *A house made of wood,* compared to the l.h.s. which he called *A house made of marble.* It is unfortunate for scientific knowledge that Einstein's own doubts have been ignored by his followers.

I should also recall that, according to Ref. [16], the characterization of the *inertial mass* of a body requires the additional inclusion of all possible short range (e.g., weak and string) interactions, resulting in the need of an additional source in the r.h.s. of the equations

(5) G_{ij} = R_{ij} - g_{ij} R/2 = k_{1} T_{ij, elm} + k_{2} T_{ij, short range}

such that [14]

(6) m_{inert} = ∫ (T_{00, elm} + T_{00, short range})dv

Consequently, the inertial mass is bigger than the gravitational mass []14].

(7) m_{inert} > m_{grav}.

Besides the incontrovertible *physical* necessity for a source of first order in magnitude, the structure of Eqs. (5), (6) is mandated by the fifth identity of the Riemannian geometry, the forgotten *Freud identity* [17] (see also the recent treatment by the late mathematician H. Rund [18]) which establishes the need on purely *mathematic* grounds of a source tensor of first order in magnitude in the r.h.s of the field equations according precisely to Eqs. (5), (6).

In fact, the source term of the Freud identity can be decomposed into a tensor with null trace, (evidently, the electromagnetic tensor), and a tensor with with non-null trace (evidently, the tensor for short range interactions), thus providing a geometric confirmation of Eqs. (5), (6). I should indicate that the problem of a source has been debated at length in the literature, although under the interrelation as a stress-energy tensor whose literature is here omitted because well known.

The serious scholar should be aware of various claims in the literature hat Einstein's gravitation verifies the Freud identity. These claims are based on the admission indeed of a source tensor of electromagnetic nature, but restricted to the the *total* electromagnetic characteristics, thus violating condition (5) by a factor of 10^{-40} or so.

Remember that *gravitational waves are crucially dependent on Einstein's reduction of gravitation to pure geometry, Eqs, (2).* However, physical and geometric needs mandate their extension to Eqs. (5),(6), (7), for which gravitational waves cannot even be formulated,

Therefore, it should be evident to the most political a physicist that any consideration of gravitational waves is vacuous without dismissing in peer reviewed journals the electromagnetic origin of the gravitational mass and, independently, the validity of the Freud identify of the Riemannian geometry.

**4. Third historical insufficiency of general relativity:
Abandon the majestic Lorentz and Poincare' "invariance"of special relativity in favor of the "covariance" of general relativity with consequential lack of prediction of the same numerical values under the same conditions at different times.**

In my view, this is perhaps the biggest insufficiency of Einstein gravitation because it implies the inability by gravitation to possess a

Under the lack of invariance, general relativity could at best offer a kind of "polaroid picture" of gravitation [7,8]. However, such a static view of gravitation is dismissed by mathematical, physical, visual and experimental evidence on the lack of existence of the curvature of space.

Additional rather serious objections against published claims of "experimental verifications of Einstein gravitation" [9] stem from the fact that numerical prediction are not, by far, unique and unambiguous due to the *non-linearity of the field equations.* In fact, for any claim of "experimental verification" [9] I can assume a different PPM approximation with different expansions and prove dramatic divergences of Einstein general relativity from physical realities [7].

The lack of time invariance of Einstein's gravitation provides the final impossibility for gravitational waves to exist because any serious experimental verification should not only detect gravitational waves, which has been impossible for half a century despite the use of public funds, but said gravitational waves should change in time without any change of the source, which is a blatant physical impossibility.

In defense of Einstein I should indicate that, once the Riemannian geometry is assumed for the representation of gravitation, no symmetry of the line element is possible for technical reason similar to those of *the open Lorentz problem,* namely, Lorentz inability to achieve the invariance of the *locally varying speeds of light* of his time, that within physical media C = c/n, due to insurmountable technical difficulties in attempting to use Lie's theory for non-linear systems on which I worked for decades (see below for more details).

**5. Consequences of the historical insufficiencies of general relativity:
Incompatibility of gravitation with special relativity, interior gravitational problems, electrodynamics, quantum mechanics, and grand unifications.**

There comes a point in the life of a serious scientist in which realities have to be admitted. The Riemannian geometry does indeed admit a unique and unambiguous reduction to the Minkowskian geometry via tangent, limit and other procedures.

However, it is known for a century that general relativity does not jointly admit a limit to special relativity, of the type for which special relativity uniquely and unambiguously admits a limit into the Galilei relativity. As one among many impossibilities, there exists no consistent limit of the covariance of general relativity into the fundamental Poincare' invariance of special relativity.

Another serious incompatibility is that of the description by general relativity of the "exterior gravitational problem" in vacuum with the "interior gravitational problems" that dominated the scientific scene in gravitation until the advent of Einstein's theory. This is a serious incompatibility because its resolution prohibits the use of the Riemannian geometry due to the need of a geometry not only without curvature, but also (as indicated in Fig.1) with a metric having a dependence on coordinates x, as well as density μ, temperature τ, frequency ν, etc. g = g(x, μ, τ, ν, ...) (see Section 5 et al below).

Another aspect that should be admitted to prevent exiting from physical reality is the irreconcilable incompatibility between Einstein gravitation and electrodynamics to such an extent that [16]

5A) Either one assumes Einstein's gravitation as being valid, in which case electrodynamics must be revised from its foundations so as to eliminate the electromagnetic origin of the mass; or

5B) One assumes electrodynamics and its inherent electromagnetic origin of the mass as being valid, in which case, Einstein gravitation must be revised from its foundations.

Yet another reality that has to be faced following one century of failed attempts is that Einstein's gravitation is incompatible with quantum mechanics for several reasons. The reason most important in my view is that a gravitational theory formulated on a Riemannian space is necessarily non-canonical (variationally non-self-adjoint [19]) at the classical level. Therefore, any consistent "quantization" must be "non-unitary" with the consequential of the Theorems of Catastrophic Inconsistencies [18] and ensuing loss of physical value, e.g., due to the violation of causality laws.

The moment of truth for serious scientists also implies the admission that Einstein gravitation is incompatible with grand unified theories, if nothing else, because of failed attempts over one full century, beginning with the failed attempt of unifying gravitation and electromagnetism by Einstein himself.

**6. Problems to be solved for grand unifications**

Following studies on grand unifications for decades, the incompatibility of a grand unification of Einstein gravitation with electroweak interactions are the following (see, later on monograph [37]):

6.I. The physical consistency of electroweak interactions on a the flat Minkowski space cannot be salvaged when joined to theory on the curved Riemannian space because the inconsistencies of the latter carry over to the former;

6.II. Within a grand unification, the covariance of Einstein's gravitation carries over to electroweak interactions, by therefore destroying their gauge invariance and, therefore, the very notion of electroweak interactions;

6.III. Electroweak interactions contain both particles and antiparticles, while Einstein gravitation solely represent matter, thus rendering any grand unification technically impossible and catastrophically inconsistent if attempted.

I should mention a recent trend of extending the applicability of special and general relativities to the classical representation of antimatter. Serious scholars should be alerted that this is a mere political manipulation because it prevents any representation of matter-antimatter annihilation, and violates other physical laws.

**7. Rudiments of IsoMathematics**

The most important lesson I learned in my fifty years of research is that * the protracted lack of solution of physical problems is generally due to insufficient mathematics, rather than to physical issues.*

I believe that this is precisely the case for gravitation, namely, all problems treated above are caused by the use of an excessively insufficient mathematics, that essentially dates back to the Newton-Leibniz differential calculus. Only after the achievement of a more adequate mathematics, open physical problems can be quantitatively and effectively addressed.

To see the case, note that *no theory of gravitation will resist the test of time unless it possesses an invariance similar to that of the Poincare' symmetry in special relativity so as to predict the same numerical values under the same conditions at different times.* The best known way to achieve an invariant theory of gravitation is via its formulation in a Minkowski spacetime. But then, we have the known insufficiencies of formulating gravitation via special relativity. The use of new mathematics is then the only promising solution.

This occurrence forced me to construct the isotopic (intended as axiom-preserving) lifting of 20th century applied mathematics known as *isomathematics* [19], that I initiated when I was at the Department of Mathematics of Harvard University in the late 1970s under DOE support.

Isomathematics is based on the isotopic lifting of the conventional associative product AB between generic quantities A, B (such as numbers, functions, matrices, etc. into *isoproduct*) [19b]

(8) A×B = AT*B, ,

where the quantity T*, called the *isotopic element,* is positive definite but otherwise posses an arbitrary functional dependence on all needed local quantities, such as time t, coordinates r, velocities v, accelerations a, density μ, temperature τ, frequency ν wavefunction ψ, etc. T* = T*(t, r, v, a, μ,τ ,ν, ψ, ...) > 0.

I introduced product (8) for the primary intent of achieving a representation of extended, non-spherical and deformable particles, which is notoriously impossible via 20th century mathematics, but elementary via the isotopic element (see below for examples).

I also suggested isomathematics to achieve a generalization of the Lie's theory info a form applicable for the first time to non-linear, non-local and non-Hamiltonian systems (that is, variationally non-self-adjoint systems not entirely representable with a Hamiltonian [19a]).

Following initial tentative papers, I presented a systematic isotopic lifting of the various branches of Lie's theory in monograph [19b], the resulting theory is today known as the *Lie-Santilli IsoTheory,* and it is based on the *isocommutation rules* (Isotopes of Lie's second theorem)

(9) [X_{i}, X_{i} × X_{j} - X_{j} × X_{i} = X_{i} T* X_{j} - X_{j} T* X_{i} = i C_{ij}^{k} X_{k}

and their integration into a finite *isogroup* here illustrated for simplicity via one dimensional time evolution

(10) A(t) = e^{X T* t i } A(0) e^{- i t T* X}

with evident non-linear, non-local and non-Hamiltonian characters.

In Vol. [19b] I then presented a concrete realization of the Lie-Santilli isotheory given by the Birkhoffian generalization of classical Hamiltonian mechanics and its universality for the representation of all infinitely possible, well behaved, non-Hamiltonian systems.

However, the new isosymmetries activated the *Theorems of Catastrophic Inconsistencies of Non-Canonical and Non-Unitary Theories* when formulated via the mathematics of canonical and unitary theories, respectively [18].

Therefore, while being at the JINR in Dubna, Russia, I was forced in 1993 to reformulate isomathematics on new numbers, today known as *isonumbers* n* = nI*, where n represents ordinary real, complex or quaternionic numbers, and I* is the new left and right generalized unit, known as *isounit* [20]

(11) I* (t, r, v, a, μ,τ ,ν, ψ, ...) = 1/T*, I* x n* = n* x I* = n*

The entire mathematics was then reformulated on isofields. Yet, the all fundamental invariance under the time evolution remained elusive. This forced me to lift the Newton-Leibniz differential calculus into * Santilli IsoDifferential Calculus* first presented in mathematical memoir [21] of 1996, with basic isodifferential

(12) d*r* = I* d [(r T*) = dr + r I*dT*

and related *isoderivative*

(13) ∂*F*(r*,...)/∂*r* = T* ∂ F*(r*,...)/∂ r*.

where F* = FI*, r* = rI* etc. as a necessary condition for their values to be isonumbers.

The isodifferential calculus permitted the achievement of preliminary maturity for isomathematics, with ensuing numerous scientific as well as industrial applications. A comprehensive presentation of isomathematics for physicists is available in monographs [22] and a comprehensive presentation for mathematicians is available in monographs by S. Georgiev [23].

**8. Rudiments of IsoMechanics**

The primary reason I introduced isomathematics is the lifting of Newton's equations into the form, first presented in Ref. [21]

(14) m*×d*v*/d*t* = F^{sa
}

today known as *Newton-Santilli IsoEquations,* that allows the first known representation of the actual shape of bodies via isounits for the velocities of the type

(15) I*_{v}(t, r, v, a, μ, τ, ν, ...) = Diag. (n_{1}^{2},
n_{2}^{2},
n_{3}^{2}) e^{Γ(t, r, v, a, μ, τ, ν, ...)}, n_{k}(t, r, v, a, μ, τ, ν, ...) > 0, k = 1, 2, 3,

as well as the embedding in the isodifferential of of all non-Hamiltonian (variationally non-self-adjoint [19]) forces the exponent are embedded in the isodifferential. and only Hamiltonian (variationally self-adjoint [19]) here represented by the exponent Γ, while in the r.h.s. of Eqs. (14) we solely have potential (variationally self-adjoint [19] forces.

In turn, the Newton-Santilli isoequations admit the first known representation via a variational principle for non-Hamiltonian systems, such as a body moving within a resistive medium [21]

(16) δ* A* = δ* ∫ (p*×d*r* - H*×d*t*) = 0

thus permitting the first known use of the optimal control theory for the shape, e.g., of a wing moving within a fluid.

In turn, the availability of an isoaction principle for non-Hamilton systems has allowed the isotopic lifting of classical Hamiltonian mechanics into its covering *Hamilton-Santilli isomechanics* with basic *Hamilton-Jacobi-Santilli isoequations* [21] (see the details study in Refs. [22])

(17) ∂*A*/∂*t* + H* = 0, ∂*A*/∂*r* - p* = 0, ∂*A*/∂*p* = 0,

Still in turn, the availability of the latter isoequations has permitted the first known, axiomatically consistent, unique and unambiguous, operator map of non-Hamiltonian systems into a covering of quantum mechanics introduced in 1978 under the name of *hadronic mechanics* [19,22], with *Schroedinger-Santilli Isoequations* [21]

(18) i*×∂* ψ*(t*, r*)/∂*t* = H*×ψ*(t*, r*) = H* T* ψ*(t*, r*) = E*×ψ*(t*, r*) = Eψ*(t*, r*)

with related *isolinear momentum*

(19) p*×ψ*(t*, r*) = - i*×∂*_{r*}ψ*(t*, r*) = - i I*∂ψ*(t*, r*)

and their isounitarily equivalent *Heisenberg-Santilli isoequations* [21] for the isotime evolution of an operator A* in the infinitesimal form

(20) i*×d*A*/d*t* = [A*, H*]* = A*×H* - H*×A* = A*T*H* - H*T*A*

and integrated finite form (10), where the asterisk denotes formulation on an iso-Hilbert space over the isofield of isocomplex numbers [22].

For readers not familiar with the field, I should recall that *isomechanics is a non-unitary "completion" of quantum mechanics much along the celebrated argument by Einstein-Podolsky and Rosen.* However, non-unitary theories formulated on a conventional Hilbert space over a conventional field violate causality. Hence, the identical reformulation of non-unitary theories via isomathematics is crucial for mathematical and physical consistency of isomechanics (see monographs [22] for comprehensive presentation).

I should also mention that *isomechanics eliminates the divergencies of quantum mechanics* because the value of the isounit (15) is generally very large. Consequently, the value of the isotopic element T* is very small, thus permitting the conversion of divergent or weakly convergent quantum series into strongly convergent forms via the systematic use of isoproduct (8). This feature is particularly important for approximate solutions of interior particle problems, as well as of non-linear gravitational equations.

Finally, the non-initiated reader should be aware that quantum mechanics and isomechanics coincide at the abstract level by conception and construction to such an extent that they can be expressed via the same equations, merely subjected to different realizations of the associative product. Out of my decades of research in the field, I believe that this feature is crucial to assure consistency and causality of isomechanics

**9. Rudiments of IsoGravitation for Matter**

The main result of my studies in gravitation is that he conjecture of the curvature of space is the dominant origin of the incompatibilities of Einstein's gravitation with other 20th century sciences, besides being disproved by visual and experimental evidence (Figure 1).

Therefore, the main objectives of isogravitation are the reformulation of field equations via a basically new geometry without curvature, show the compatibility of the reformulation with 20th century sciences, and prove the applicability to said reformulation of all sound experimental verifications popularly believed until now to apply solely to general relativity.

Following decades of preparatory research on the new isomathematics and isomechanics, I presented isogravitation for matter at the 1992 Marcel Grossmann Meeting in Gravitation [24] via the following elementary rules:

*RULE 9-I: Decompose any non-singular Riemannian metric g(x) in (3+1)-dimensions into the product of the the Minkowski metric η and the 4×4-dimensional gravitational isotopic element *T*

(21) g(x) = T*_{gr}(x) η,

*where the positive-definite character of T* ^{gr}(x) is assured by the topology of the Riemannian space;*

*RULE 9-II: Assume the inverse of the isotopic element as the gravitational isounit*

(22) I*_{gr}(x) = 1/T*_{gr}(x)

*RULE 9-III: Reformulate the entire general relativity into such a form admitting I* _{gr}(x) as the correct left and right unit at all level, including numbers, functional analysis, differential calculus, geometries, etc.*

The fundamental spacetime of isogravitation verifying the above conditions is given by the infinite family of isotopies of the Minkowski space M(x, η, I) with spacetime coordinates x, metric η = Diag. 91, 1, 1, - c^{2}) and unit I = Diag. (1, 1, 1, 1), which was first introduced in Ref. [25] of 1983 for the classical profile and Ref. [26] of the same year for the operator counterpart.

The new spacetime is nowadays called the *Minkowski-Santilli isospacetime,* it is denoted M*(x*, η*, I*), and it is characterized by the infinite family of isotopies of coordinates into isocoordinates (a necessary condition to be isonumbers),

(23) x → x* = xI*,

metric into isometric

(24) η → η* = T*_{grη
}

unit into the isounit with related isotopic element

(25) I*(x, μ, τ, ν, ψ, ...) = Diag. (n _{1}^{2},
n_{2}^{2},
n_{3}^{2},
n_{4}^{2}) = 1/T* > 0, n_{k} > 0 , k = 1, 2, 3, 4,

(26) T*_{gr} (x, μ, τ, ν, ψ, ...) = Diag. (1/n_{1}^{2},
1/n_{2}^{2},
1/n_{3}^{2},
1/n_{4}^{2}),

and line element into the isoline element

(27) x*^{2*} = x*^{i}xη*_{ij}x*^{j} =
(x_{1}^{2}/n_{1}^{2} +
x_{2}^{2}/n_{2}^{2} +
x_{3}^{2}/n_{3}^{2} -
t^{2} c^{2}/n_{4}^{2})I*,

where one should note the multiplication by the isounit as a necessary condition for the isoline element to be an isonumber, and we have ignored for simplicity the exponential factor in the isounit and isotopic element representing non-Hamiltonian interactions as in Eqs. (15) (see Refs. [22] for the full treatment).

It is easy to see that the projection of the isoline element (27) in conventional spacetime is the most general possible symmetric (thus diagonalized) and non-singular line element in (3+1)-dimensions, thus including as particular cases all possible Minkowskian, Riemannian, Fynslerian and other line elements (it should be noted that non-symmetric line elements for the geometric representation of irreversibility require the broader Lie-admissible genomathematics [22])

The most important feature of the Minkowski-Santilli isospacetime is that of being *isoflat,* that is, its curvature is identically null when defined on isofields and elaborated via isomathematics. One can see this feature by noting that, under isotopies, we have the lifting of the Minkowskian coordinates while the corresponding unit is lifted by the inverse amount, thus preserving the original flatness

(28) x_{k} → x_{k}* = x_{k}/n_{k}^{2}

(29) I_{k} → I*_{k} = n_{k}^{2}

In any case, isotopies must preserve the original axioms by central condition and technical realization. This means that, when properly treated, the isotopies of the Minkowski spacetime must preserve the original flatness *despite the dependence of the isometric on local coordinates*.

Thanks to the prior construction of the Lie-Santilli isotheory [19], the universal isosymmetry of isoline element (27) was constructed in only one page of Ref. [25]; it is today called nowadays the * Lorentz-Santilli isosymmetry;* it is characterized by the original symmetry plus the isotopic element (26); and can be written for the (3, 4)-plane (see Refs. [22] for the general case)

(30) x'^{ 3} = γ* [ x^{3} - β* (n_{3} / n_{4})x^{4} ],

(31) x' ^{4} = γ* [ x^{4} - β* ( n_{4} / n_{3})x^{3} ].

where

(32) γ* = 1 / ( 1 - β*^{2} )^{1/2}, β* = (v / n_{3}) / (c / n_{4}),

As one can see, it is evident that the Lorentz-Santilli isosymmetry is locally isomorphic to the original symmetry by conception and realization. It is also evident that this local isomorphism is crucial for attempting a grand unification of gravitation and electroweak interactions without structural incompatibilities (Section 11).

Following the original isotopies [25,26], systematic studies were done on the isotopies of all most significant spacetime symmetries. Ref. [27] was devoted to the isotopies O*(3) of the rotational symmetry O(3) to achieve the invariance of all topology preserving deformations of the sphere; Refs. [27,28] were devoted to the isotopies SU*(2) of the SU(2) spin symmetry; Ref. [29] presented for the first time the isotopies P*(3.1) of the Poincare' symmetry P(3.1) with the first proof of providing the universal invariance of all possible non-singular, Riemannian line elements; and Ref. [30] was devoted to the isotopies of the spinorial covering of the Poincare' symmetry. Independent papers [31,32] confirmed the universal character of the Lorentz-Santilli isosymmetry for all infinitely possible symbiotic line elements in (3+1)-dimensions.

Another central feature of the isospacetime is that its geometry, first introduced in paper [33] of 1998 and today known as the *Minkowski-Santilli isogeometry,* admits the entire machinery of the Riemannian geometry, such as covariant derivative, Christoffel symbols, etc. merely reformulated in terms of the isodifferential calculus, Eqs. (12), (13). This is evidently due to the fact that, unlike the Minkowski metric η, its isotopic covering admits the most general possible functional dependence, under the sole condition of positive-definiteness of the isotopic element, Eq. (26).

Therefore, in line with Eqs. (3)-(6), we have the *isoequations for the exterior gravitational problem*

(33) G*_{ij} = R*_{ij} - η*_{ij}(x)× R*/2* = k* × T*_{ij, elm}(x)

under the condition

(34) m_{grav} = ∫ T*_{00}(x) × d*v*

and the broader *isoequations for the interior isogravitational problem*

(35) G*_{ij} = R*_{ij} - η*_{ij}(x, μ, τ, ν ψ, ...) × R*/2* = k*_{1} × T*_{ij, elm}(x) + k*_{2}× T*_{ij, short range}(x, μ, τ, ν ψ, ...)

under the conditions

(36) m_{inert} = ∫ [T*_{00, elm}(x) + T*_{00, short range}(x, μ, τ, ν ψ, ...)] × d*v*

(37) m_{inert} > m_{grav}.

one should note to avoid erroneous perceptions that the elements of the isometric are isoscalars.

Yet another important feature at the basis of the original proposal [24] is that * isogravitation admits an axiomatically consistent, unique and unambiguous operator map into the relativistic isomechanical branch of hadronic mechanics merely characterized by the embedding of gravitation in the basic unit of relativistic quantum mechanics.*

The point important for axiomatic consistency, as well as the elimination of solutions violating causality, is that *the axioms of operator isogravitation are the conventional axioms of relativistic quantum mechanics merely submitted to a broader realization.*

We can also say that *gravitation was already existing in the "hidden variables" of quantum mechanics, only realized as "hidden operators."* To see this point, consider the right modular associative action of a Hamiltonian on a Hilbert state, H ψ = E ψ. The Schroedinger-Santilli isoequations (18) in their gravitational realization are given by the axiom-preserving modular isotopic action

(38) H*× ψ* = H* λ ψ* = Eψ*, λ = T*_{gr}(x, μ, τ, ν, ψ, ...)

The important point is that the conventional modular action H ψ = E ψ and its isotopic covering H*× ψ* coincide at the abstract, realization-free level.

To illustrate isomechanical isogravitation, Ref. [24] presented the following explicit example. The familiar Schwartzchild line element

(39) ds^{2} = r^{2}(dθ^{2} + sin^{2}θ^{2}) + 1, (1 - 2M/r)^{-1} dr^{2} - (1 - 2M/r)dt^{2}

can be represented isotopically in a fully equivalent way in the Minkowski-Santilli isospacetime with the isotopic element

(40) T*_{gr} = Diag. [1, 1, (1 - 2M/r)^{-1}, (1 - 3M/r)]

where one should note the suggesting reformulation of bravitational singuilarities in terms of the zeros of the basic isounit of the gtheory.

Then, the isotopies of the Dirac equation, first introduced in Ref. [30],

(41) (η*^{ij}×γ*_{i}×∂*_{j} + i*×m*×c*^{2})ψ* = 0

(42) γ*_{k} = γ_{k}/n_{k}, k = 1, 2, 3, γ*_{4} = i γ_{4}/n_{4}.

(43) {γ*_{i}, γ*_{j}}* = γ*_{i}xγ*_{j} + γ*_{j}xγ*_{i} = γ*_{i}T*_{gr}γ*_{i}+ γ*_{j} T*_{gr}γ*_{i} = η*_{ij}

characterize the *Dirac-Schwartzchild isoequation* because the Schwartzchild metric appears directly in the value of the anti-isocommutators rules of the Dirac-Santilli isogamma matrices.

It should be stressed that **The compatibility of isogravitation with 20th century sciences is direct and immediate.** Compatibility with special relativity is immediately established by the fact that its universal isosymmetry is locally isomorphic to the conventional Poincare' symmetry. Compatibility of the physical laws of isogravitation with those of special relativity is then an immediate consequence.

The compatibility of isogravitation with the interior gravitational problem is established by the completely arbitrary functional dependence of the isometric that allows, apparently for the first time, quantitative studies of interior gravitational problems, such as the propagation of star light within the Sun chromosphere (Fig. 1).

Compatibility of isogravitation with electromagnetism is established by the electromagnetic origin of the gravitational mass appearing in Eqs. (33). Compatibility with quantum mechanics is inherent in the very notion of isotopes and it is used at the foundation of the very proposal of isogravitation [24[. Compatibility with grand unifications will be discussed in Section 11.

The verification by isogravitation of the sound experimental tests of general relativity is discussed in the joint formal paper to be released following comments on this informal presentation. Subject to verifications, it appears that the Lorentz-Santilli isosymmetry uniquely and unambiguously identifies the isotopies of the various axioms of special relativity [22] that, in turn, appear to admit a verifications via the indicated gravitational experiments [9].

I limit myself to the indication that, for a given Riemannian metric g(x) with elements g_{ij}, the Lorentz-Santilli isosymmetry uniquely and unambiguously identifies the *isotime isodilatation in the (3, 4)-plane*

(44) t' = t / √(1 - (v^{2} g_{44}^{2} / c^{2}g_{33}^{2}),

that appears indeed to verify current experimental data on gravitational time dilations, this time, in a time invariant way. Similar isoaxioms [22] specialized to Riemannian metrics appear to verify the remaining experimental data.

An insidious aspect open to possible misrepresentation is that the exterior metric used in the experimental verification is the metric of the solutions of the isofield isoequations * with the electromagnetic origin of the mass,* Eqs. (33), and *not* the metric for the original Einstein equations (2).

**10. Rudiments of the Isodual IsoGravitation for AntiMatter**

Despite all the above advances, attempts at an axiomatically consistent grand unification of electroweak and gravitational interaction continued to be inconsistent and not worth their presentation in a scientific paper, because *Einstein gravitation as well as isogravitation solely apply for matter-bodies,* thus preventing any consistent unification with electroweak theories that are bona-fide theories of particles and antiparticles.

A solution of the latter problem required the construction of yet another new mathematics, specifically conceived for the *classical* representation of *neutral* (or charged) antimatter-
bodies. The transition from matter to antimatter required the new mathematics to be anti-isomorphic to isomathematics, so as to be consistent with experimental data, including matter-antimatter annihilation.

Following numerous failed attempts, when I was at he Department of Mathematics of Harvard University, in the early 1980s I finally succeeded in identifying the needed map, that I called *isodual* for the transition from matter to antimatter

(45) I*^{d} = - I*^{†}(-x^{†}, -v^{†}, -ψ^{†}, ...) < 0

that must be applied to the totality of quantities and their operations of isomathematics without any exclusion. The resulting new mathematics is today known as *Santilli isodual isomathematics* and includes isodual isonumbers, isodual isofunctions, isodual isodifferential calculus, isodual isoalgebras, isodual isogeometries, etc. (see monograph [36] for a comprehensive study and Ref. [37] for an independent general review).

Following the construction of the isodual isomathematics it was necessary to construct the isodual image of classical and quantum theories, with particular reference to the *isodual Lorentz-Santilli isosymmetry* and the axiomatically consistent classical representation of the gravitational field of neutral (or charged) antimatter-bodies. The compatibility of the emerging*isodual theory of antimatter* was assured by the equivalence of the isodual map with charge conjugation (for brevity, one may inspect monograph [36]).

**11. Rudiments of IsoGrandUnification**

in my view, a most important implication of the search for axiomatically consistent grand unification is the shift from the "description" of gravitation to a study of its "origin." In fact, Ref. [16] is crucially dependent on the abandonment of the "unification" of gravitation and electromagnetic interactions in their "identification" under appropriate field equations. Ref. [16] also submitted experiments for the possible laboratory creation of a measurable gravitational field that appear feasible nowadays thanks to the availability of highly sensitive detectors, such as those based on neutron interferometry.

Only following the above scientific journey I was finally in a position to present at the 1997 Marcel Grossmann Meeting in Gravitation a grand unification of electroweak and gravitational interactions with the inclusion of matter and antimatter at all classical and operator levels [34] (see also Ref. [36]). the grand unification essentially consistent in the embedding of gravitation in the gravitational isounit of electrostatic interactions. (see monograph [36] for brevity).

Of course, I do not know whether this type of grand unification is verified in nature since, by following Albert Einstein, I pride myself to have a self-criticism generally stronger then those by reviewers, but I believe that the studies have provided at least much needed new vistas in gravitation for further advances by colleagues.

**Comments**

Qualified critical comments for the completion of the joint format paper would be received with appreciation either via my email research@i-b-r.org or via my phone 727-688 3992. Comments by anonymous or alias authors are welcome since I am solely interested in physical content.

Interested colleagues should know that our Foundation has research grants for the continuation of the studies on isogravitation with priority on studies on its experimental verification.

**Acknowledgments**

I would like to thank the organizers of the Eight Marcel Grossmann Meeting in Gravitation, held at the Hebrew University, Jerusalem, June 22 to 27, 1997, for having my talk presented by the session chairman and publishing my paper [34] in the proceedings despite my being incapacitated to attend.

**References**

[1] H.A.Lorentz, Amst. Proc. Vol. 6, 809 (1904).

[2] H.Poincar\'{e}, Compte Rendues, Paris Vol. 140, 1504 (1905).

[3] A.Einstein, Ann. Phys. Vol. 17, 891 (1905).

[4] H.Minkowski, Nachr. Ges. Wiss. Gottingen Vol. 43, (1908).

[5] R. M. Santilli, 2014 Keynote Lectures "IsoLinear, IsoInvariant IsoRelativity"

http://www.world-lecture-series.org/isorelativity-2014-i

http://www.world-lecture-series.org/isorelativity-2014-ii

http://www.world-lecture-series.org/isorelativity-2014-iii

[6] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation,
W. H. Freeman and Company (1973)

[7] R. M. Santilli, "Historical Insufficiencies of General Relativity
and their possible resolution via Isogravitation,
Galilean Electrodynamics, Vol 17 No. 3, page 43 (2006)

www.santilli-foundation.org/docs/Incons.GravFinalGED-I.pdf

[8] J. V. Kadeisvili, "Obscurantism on Einstein's gravitation,"

http://www.santilli-foundation.org/inconsistencies-gravitation.php

[9] C. M. Will, "The Confrontation between General Relativity
and Experiment," Living Rev. Relativity, Vol. 9, p. 3 (2006).

[10] S. Beeson, Steven and J. W. Mayer, *Patterns of Light
Chasing the Spectrum from Aristotle to LEDs,* Springer (2008)

[11] R. M. Santilli, "Experimental Verifications of IsoRedShift with Possible Absence of Universe Expansion, Big Bang, Dark Matter, and Dark Energy," The Open Astronomy Journal 3, 124 (2010),
http://www.santilli-foundation.org/docs/Santilli-isoredshift.pdf

[12] R. M. Santilli, "Experimental Verification of IsoRedShift and its Cosmological Implications," AIP Proceedings Vol. 1281, pp. 882-885 (2010)

http://www.santilli-foundation.org/docs/Isoredshift-Letter.pdf

[13] G. West and G. Amato, "Experimental Confirmation of Santilli's IsoRedShift and IsoBlueShift," Journal of Computational Methods in Sciences and Engineering, 12, 169 (2012),

http://www.santilli-foundation.org/docs/Confirmation-IRS-IBS.pdf

[14] R. M. Santilli, G. West and G. Amato. "Experimental Confirmation of the IsoRedShift at Sun at Sunset and Sunrise with Consequential Absence of Universe Expansion and Related Conjectures, " Journal of Computational Methods in Sciences and Engineering, 12, 165 (2012).

http://www.santilli-foundation.org/docs/Confirmation-sun-IRS.pdf

[15] H. Ahmar, G. Amato, J. V. Kadeisvili, J. Manuel, G. West, and O. Zogorodnia, "Additional experimental confirmations of Santilli's IsoRedShift and the consequential expected absence of universe expansion," Journal of Computational Methods in Sciences and Engineering, 13, 321 (2013),

http://www.santilli-foundation.org/docs/IRS-confirmations-212.pdf

[16] R. M. Santilli, "Partons and Gravitation: some Puzzling Questions,"
(MIT) Annals of Physics, Vol. 83, 108-157 (1974),

http://www.santilli-foundation.org/docs/Santilli-14.pdf.

[16] P. Freud, Ann. Math. 40 (2), 417 (1939)

[17] H. Rund, Algebras, Groups and Geometries Vol. 18, 267 (1991).

[18] R. M. Santilli, ''Lie-admissible invariant representation of irreversibility for matter and antimatter at the classical and operator levels," Nuovo Cimento B {bf 121}, 443 (2006),

http://www.santilli-foundation.org/docs/Lie-admiss-NCB-I.pdf

[19] M. Santilli, * Foundation of Theoretical Mechanics,*Volume I (1978) [19a], and Volume II (1982) [19b], Springer-Verlag,

http://www.santilli-foundation.org/docs/Santilli-209.pdf

http://www.santilli-foundation.org/docs/santilli-69.pdf

[20] 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),

http://www.santilli-foundation.org/docs/Santilli-34.pdf

[21] 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),

http://www.santilli-foundation.org/docs/Santilli-37.pdf

[22] R. M. Santilli, * Elements of Hadronic Mechanics,*
Vol. I (1995) [22a], Vol. II 91995) [22b],
Academy of Sciences, Kiev, available in free pdf downloads from

http://www.santilli-foundation.org/docs/Santilli-300.pdf

http://www.santilli-foundation.org/docs/Santilli-301.pdf

[23] S. Georgiev, *Foundations of the IsoDifferential Calcuilus,* Volumes I, II, III, Nova Scientific Publishers (2014 and 2015).

[24] R. M. Santilli, "Isotopic quantization of gravity and its universal isopoincare' symmetry" in the {\it Proceedings of "The Seventh Marcel Grossmann Meeting in Gravitation,} SLAC 1992, R. T. Jantzen, G. M. Keiser and R. Ruffini, Editors, World Scientific Publishers pages 500-505(1994),

http://www.santilli-foundation.org/docs/Santilli-120.pdf

[25] R. M. Santilli, "Lie-isotopic Lifting of Special Relativity for Extended Deformable Particles," Lettere Nuovo Cimento Vol. 37, 545 (1983),

http://www.santilli-foundation.org/docs/Santilli-50.pdf

[26] R. M. Santilli, "Lie-isotopic Lifting of Unitary Symmetries and of Wigner's Theorem for Extended and Deformable Particles," Lettere Nuovo Cimento Vol. 38, 509 (1983),

http://www.santilli-foundation.org/docs/Santilli-51.pdf

[27] R. M. Santilli, "Isotopic Lifting of the SU(2) Symmetry with Applications to Nuclear Physics," JINR rapid Comm. Vol. 6. 24-38 (1993),

http://www.santilli-foundation.org/docs/Santilli-19.pdf

[28] 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),

http://www.santilli-foundation.org/docs/Santilli-27.pdf

[29] R. M. Santilli, "Nonlinear, Nonlocal and Noncanonical Isotopirs of the Poincare' Symmetry," Moscow Phys. Soc. Vol. 3, 255 (1993),

http://www.santilli-foundation.org/docs/Santilli-40.pdf

[30] 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),

http://www.santilli-foundation.org/docs/Santilli-18.pdf

[31] J. V. Kadeisvili, "Direct universality of the Lorentz-Poincare'-Santilli
isosymmetry for extended-deformable particles,
arbitrary speeds of light and all possible spacetimes" in {\it Photons: Old problems in Light of New Ideas,} V. V. Dvoeglazov Editor
Nova Science (2000, available as free download from

http://www.santilli-foundation.org/docs/Santilli-25.pdf

[32] A. K. Aringazin and K. M. Aringazin, "Universality of Santilli's iso-Minkowskian geometry" in {\it Frontiers of Fundamental Physics,} M. Barone and F. Selleri, Editors
Plenum 91995), available as free download from

http://www.santilli-foundation.org/docs/Santilli-29.pdf

[33] R. M. Santilli, "Isominkowskian Geometry for the Gravitational Treatment of Matter and its Isodual for Antimatter," Intern. J. Modern Phys. D Vol. 7, 351 (1998),

http://www.santilli-foundation.org/docs/Santilli-35.pdf

[34] R. M. Santilli, "Unification of gravitation and electroweak interactions" 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),

http://www.santilli-foundation.org/docs/Santilli-137.pdf

[35] R. M. Santilli, "Isotopic grand unification with the inclusion of gravity," Found. Phys. Letters {\bf 10}, 307 (1997),

http://www.santilli-foundation.org/docs/Santilli-04.pdf

[36] R. M. Santilli, *Isodual Theory of Antimatter with Application to Antigravity,
Grand Unification and the Spacetime Machine,* Springer 2001,

http://www.santilli-foundation.org/docs/santilli-79.pdf

[37] I. Gandzha and J. Kadeisvili, *New Sciences for a New Era:
Mathematical, Physical and Chemical Discoveries of
Ruggero Maria Santilli,* Sankata Printing Press, Nepal (2011),

http://www.santilli-foundation.org/docs/RMS.pdf

============================

Subject: Request for comments on isogravitation

Date: Thurs, 25 Feb 2015 22:33:07 +0100

From: Aaaaaa Bbbbbbb

Reply-To: Aaaaaaa Bbbbbbb

To: Ruggero Santilli research@i-b-r.org

Dear Prof. Santilli,

Congratulations for your historical contributions in gravitation. Your visual, mathematical and experimental evidence on the lack of curvature of space are shocking and made me feel a bit brain wasted by my academic environment. There are a web of questions in my mind, but I have to confess that they may be due to my lack of technical knowledge of the field. To avoid abusing your time, we can perhaps discuss them after I have studied at least the most important references. I definitely recommend publication of your paper because, as you state in the closing remarks, it will take generations to understand and resolve open issues. I have moved one of my graduate students to have his Ph. D. Thesis on "Santilli isogravitation" ands shall recommend you as one of the reviewers.

Regards

Aaaaaa Bbbbbb

Professor of Physics

ZZZZZ University, Wwwwww

**--------------------------------------------**

Subject: Request for comments on isogravitation

Date: Fri, 26 Feb 2015 10:05>06 +0110

From: Bbbbbbbb Ccccccccccc

Reply-To: Bbbbbbbbb Cccccccccc

To: Ruggero Santilli research@i-b-r.org

Dear Ruggero,

Gravitation will never be the same following the paper on gravitation you sent me. Any continued use of the curvature of space following your smashing proof of its lack of existence would raise lots of eyebrows, especially if made under taxpayers money without disproving your work. Following your paper on isogravitation, I decided to stop being an "Einstein follower" and become a "Santilli follower." Please let me know whether you can deliver a one week seminar course on your isogravitation at our department.

With great excitement, I remain

Yours, Truly

Ccccccccc Dddddddd

Professor of Physics

University of QQQQQ, Kkkkk

**--------------------------------------------**

Subject: Request for comments on isogravitation

Date: Sat, 27 Feb 15:11:21 +0100

From: Rrrrr Tttttt

Reply-To: Rrrrrr Ttttttttt

To: Ruggero Santilli research@i-b-r.org

Dear Dr. Ruggero Maria Santilli,

Thank you for your extremely exciting paper that numerous basic novelties kept me aware for days. We are experimentalists and are primarily interested in the experimental verifications of your isogravitation, including the all important "laboratory creation" of gravitation you propose in Ref. [16]. We are also interested in other experimental verifications of basic axioms similar to those of special relativity, only specialized to solutions of Eqs. (3-4). Since these gravitational axioms are not identified in the draft we received, please send me their derivation from the Lorentz-Santilli isosymmetry so that we can identify the gravitational deviations from special relativity that should be confronted with meaningful available tests and/or subjected to new tests.

Ddddd Eeeeee

Department of Physics and SAstrophysics

University of Ssssssm Fffff

**--------------------------------------------**

Prof. Santilli,

I have a lot of questions that I can perhaps formulate after understanding a bit deeper your studies. At this moment I am puzzled by your statement in the comparison of Eqs. (39) and (40) that the Riemannian singularities are turned into zeros of the isounit, since that is not the case for the space component. Could you please elaborate? By the way, I support your studies on the "limitations" of special relativities and the "insufficiencies" on general relativity because we may lose big inn\ not doing them.

David Ssssssss

Israel

\
Hello Dave,

Thanks for caring to ask questions. You have a valid point. My informal email is insufficiently technical also due to great limitations in writing formulae in HTML language. The correct statement should have been that the Riemannian space singularities are turned into infinities of the "space component" and zeros of the "time component" of the isotopic element.

You may be interested in knowing that when we pass from Eqs. (2) to (33) I have been unable to find any divergency as conventionally understood, thus implying the apparent turning of "black holes" into
"brown holes" while essentially maintaining everything we know in the field since merely prevent what I believe is impossible in nature, to have local infinities.

Thanks for supporting our line of research. The most important supporters for the continuation of the studies in the field, of course without necessarily accepting them and jointly supporting studies along other lines, have from Israel, including Prof. Larry Horwitz from Tel Aviv I consider one of the biggest living physicist.
Regards

Ruggero

**--------------------------------------------**

Mr. Santilli

Do you have any shame in solely listing your own papers?

James Aaaaaa

James

I do not have shame, but I am embarrassed indeed. The documented fact is that, through the years, I have proposed studies on isogravitation to numerous physicists, including some of my collaborators, and our Foundation has even offered grants for research on isogravitation, to no avail whatsoever, because all contacted colleagues freaked out at the idea of conducting an in depth scientific process on one of the biggest sacred cows of 20th century physics: the belief that space is curved. Consequently, I understand you are disturbed by this, but you have to face the reality that, besides various reviews, I am the only embarrassed contributor in isogravitation to date, by praying that this situation will change soon.

RMS

============================

Sherri Johnson info@hadronicpress.com

============================

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