to debate open problems in
special relativity, general relativity and cosmology

Comments should be emailed to
Richard Anderson
with stated authorization for uploading in this page>


March 11, 2010; revised March 14,2010

The Current Historical Moment in Gravitation

comments by

Ruggero Maria Santilli

to all participants of the workshop

Cosmology, the Quantum Vacuum, and Zeta Functions
celebrating Emilio Elizalde's sixtieth birthday
Universitat Autònoma de Barcelona, SpainMarch 8-10, 2010

Dear Colleagues,

Allow me to express my appreciation for the hospitality during our beautiful workshop in magnificent Barcelona. I also appreciated the kind tolerance and grace toward my non-aligned comments during the meeting.

Additionally, I appreciated some of you asking the reasons for my lack of acceptance of the Riemannian formulation of gravity. Allow me to express these reasons in a respectful, but open way and without any expectation that you agree because, the moment we stop debating basic issue, we become academic politicians.

To anticipate the main results of the comments below, by no means I consider past research on Riemannian gravity as being useless. On the contrary, such a research has a truly historical character. However, science continuously evolves, and the solution of emerging new problems requires the search for appropriate reformations. As you will see below, the historical advances achieved by Riemannian gravitation remain in their full value. The same occurs for the conception of gravity by Einstein (rather than by his followers) that, following reformations suggested by new knowledge, remains in its full strength and historical relevance.

In the comments below I also take the opportunity of providing some of the primary literature in the reformations of gravity (out of several huindred papers and books), now available in free pdf download, which literature may be of interest for you in the event you want to acquire a technical knowledge on the comments below.

These problems are so many, so deep and so serious that I do not even know how to start and how to present them in an informal message such as this one. I can only try to do my best.

First, there are the HISTORICAL YET UNRESOLVED OBJECTIONS. When Einstein suggested the extremely revolutionary hypothesis that space is actually curved, there were numerous objections. The curvature of space was advertised (as still it is the case today) on the basis of the bending of light. However, physicists of the 1920s immediately objected that Newtonian gravitation is universal, thus including the attraction of light. In fact, today we know that the Newtonian bending iof light can be represented via the law

F = g' m E/r^2, g' = g/c^2, (1)

where m is the mass of the body and E = hv is the energy of the photons. Another historical objection (I also mentioned at the meeting as such) is the impossibility of representing with curvature the free fall along a straight radial line. Yet another historical objection was the impossibility of representing with curvature the weight of a stationary body, the argument being that Newton represents our weight but the Riemannian gravitation does not. I could go on.

The main point is that THESE HISTORICAL OBJECTIONS HAVE REMAINED UNRESOLVED IN REFEREED JOURNALS TO THIS DAY because of known manipulation by Einstein's "followers." Note that I do not say that these objections are correct. Again, I am saying that they have to be resolved in refereed journals for Riemannian gravitation to be a serious science.

Then, there are a number of objections voiced beginning from the 1960s on, which also are unresolved in refereed journals to my best knowledge. Firstly, there are objections on grounds that "gravitational conservation laws are not conserved" (you can prove this easily because of the "covariance" of Riemannian gravity). Then there is the objection that "Riemannian conservation laws are incompatible with relativistic conservation laws" (due to lack of a clear Minkowskian limit not resolved by tangent geometric feature). Then, there is the objection on Einstein's conception of gravity as pure curvature, that is, in the absence of any elm source, the gravitational field of a massive body is given by

G_\mu\nu = 0. (2)

In the event total elm features are present, their contriubtion to the gravitational field is so small (often of the order of 10^30 or so) to be ignorable, thus using Eqs. (2) for all cases.

In the physical reality, the mass of the electron is entirely of elm origin and its gravitational equations must have the related energy-momentum tensor in the r.h.s. When you study a neutral particle, you have the same occurrence. For instance, the mass of the pi^o particle is also of elm origin. Hence, the gravitational field of the pi^o requires a first order source in the r.h.s.,

G_\mu\nu = k T_\mu\nu, (3)

The same goes for all massive bodies. you can see explicit calculations in my old paper of 1974 written when I was at Boston University

[1] "Partons and gravitation: some puzzling questions,"
(MIT) Annals of Physics Vol. 13, 108-157 (1974)
available in free pdf download from the link

In this paper I essentially established that Eqs. (2) are irreconcilably incompatible with quantum electrodynamics to the extend that:

* Either one assumed Eqs. (2) as being correct, in which case quantum mechanics and quantum electrodynamics must be reformulated from their axioms in such a way NOT to have a first order elm source in the exterior of neutral composite states; or

* One must modify Einstein gravitation in such a way to avoid the representation of gravity via curvature and assume as the origin of the gravitational field the energy-momentum tensor representing the elm origin of mass.

For years I used to contact colleagues in gravitation suggesting their disproof of the above conclusion in refereed journals. With the passing of the decades, I realized that the field was essentially dominated by academic politics in the sense that the scientific content of studies in gravitation is irrelevant. Acceptance is established by a sufficient number of physicists aligned NOT with Einsteinian theories, but with THEIR interpretation and use of Einsteinian theories. At any rate, the conclusions of paper [1] remain dismissed in academic corridors via equivocal arguments, but remain unchallenged to this day in refereed journals, to my best knowledge, by therefore turning gravitation into a pure academic theology without serious scientific content.

It should be recalled that Albert Einstein was a series scientist indeed. When referring to his field equations, he used to say that the r.h.s. is "made up of wood," to indicate his doubts. Einstein's followers have not taken into considerations his doubts and avoided the debate on this so central an issue, thus setting the difference between Einstein's serious approach to science, and his followers who suppress the scientific process for academic gains.

Then, there are serious geometric and structural problems on the very essence of the Riemannian treatment of gravity. In fact, a sad episode is the suppression of the Freud identity of the Riemannian geometry for about one century to protect the followers view of gravitation, rather than that by Einstein. I rediscovered this identity in the 1980s and, as editor in chief of ALGEBRAS, GROUPS AND GEOMETRIES, I had it examined by selected mathematicians of known ethical stand, such as the mathematician Hanno Rund and others who, confirmed that the Freud identity is indeed an independent and necessary identify of the Riemannian geometry.

This identity essentially eliminates Einstein's reduction of gravity to pure curvature, Eqs. (2), because, for geometric consistency, it demands a FIRST ORDER SOURCE in the r.h.s. even in the absence of any elm data, as in Eqs. (3). Following my rediscovery of the Freud identify, unpublished papers have appeared claiming that the identity is verified in the general field equations with elm sourcves. This type of "disproof" causes only damage to the field and produces no advance. In fact, the "disproof" is like citing a horse running in the field as an argument to disprove Fermat theorem, since the main inconsistency occurs in the ABSENCE of elm phenomenology. in any case, as inidcated earlier, the sources due to charges and magnetic moments also violate the Freud identity because they are NOT of first order in magnitude, but extremely negligible, as well known. These political "disproofs" essentially illustrate the very reason the Freud identify was suppressed in the literature of the field for about one century.

Note that my 1974 paper on the need for a first order source for all gravitational models, whether charged or not, was derived from purely physical grounds and written before I knew the Freud identity. Said identity gave me the geometric motivations for first order sources.

Then, there are structural problems caused by the absence of a well defined Minkowskian "limit" suitable for a smooth connection with special relativity; then, there are numerous additional problems. Perhaps you can inspect my paper published in as refereed journal in which I am not an editor

[2] "Nine theorems of catastrophic inconsistencies of general relativity
and their possible resolution via isogravitation"
Galilean Electrodynamics, Summer 2006, p. 43-79 (2006)

Unfortunately for scientific knowledge, THE ABOVE INCONSISTENCY THEMREMS HAVE ALSO REMAINED COMPLETELY UN-ADDRESSED IN REFEREED JOURNALS, as it has been the case for all objections, thus turning gravitation into a theology proffered by a few without final scientific content.

Yet, all the above issues are still the beginning of the problems. I can put you in touch with colleagues much more expert than me in these insufficiencies and their list of objections requires pages. We still remain with THE MOST SERIOUS OBJECTIONS: THOSE OF THE SO-CALLED "EXPERIMENTAL VERIFICATIONS" OF RIEMANNIAN GRAVITATION.

They are very serious indeed. You should know for your own personal career that the "experimental verifications" of Riemannian gravitation have never been accepted by the scientific community at large. A recent poll in the USA indicated that the majority of physicists rejects these verifications. Most of the physicists are famous but are silent because Einstein's followers are vociferous and do not want their scientific and personal lives to be disrupted, as it happened to me and other openly dissident physicists.

Let me tell you the main unresolved objections. The first is that Riemannian gravitation does not predict the same numerical values under the same conditions at different times (as special relativity does). This is a direct consequence of the "covariance" of the theory (rather than the invariance as in special relativity). Consequently, it has been shown that if you predict, e.g., 43" of precession at the time t = 0, there is no time evolution of Riemannian gravitation preserving that value over time, since with one or another time evolution you may get at t = 1 minute, e.g., 45,467". The second objection is that the expansion leading to the experimental verifications is NOT unique. In fact, you can select a different expansion via a different parameter and with a different truncation under which you get different numbers. An then, there is the final experimental problem, the lack of experimental detection of gravitational waves following the investment of large public funds that, as I openly indicated in the closing session of our meeting, prevents the use of the (metric formulation of the) Riemannian geometry for the representation of gravitation.

I have never been interested in a scientific career and never will. I am solely interested in serious scientific knowledge. Hence, I INVITE ANY INTERESTED COLLEAGUES TO PUBLISH REFEREED DISPROOFS OF THE ABOVE PROBLEMS IN ALGEBRAS, GROUPS AND GEOMETRIES. If you succeed, you will do a historical service to gravitation due to the obscurantism in the field imposed by Einstein followers for about one century.

However, bear in mind that each of the objections is potentially lethal. Hence, to establish Riemannian gravitation on true scientific grounds, you have to disprove all the objections. This is a very difficult, if not a hopeless task. because I tried myself for years in formulating these disproof's and failed.

An additional uneasiness surrounding the gravitation is that, even assuming you resolve ALL objections in refereed journals, THE RIEMANNIAN TREATMENT OF GRAVITATION HAS BEEN RESPONSIBLE FOR A NUMBER OF SCIENTIFIC FAILURES WITH RELATED WASTE OF LARGE PUBLIC MONEY OVER ONE CENTURY.

We here have another endless series of problems. Let me tell you a couple. Curvature has prohibited any consistent operator formulation of gravity over one century with thousands of papers under public financial support, all easily proved to verify the THEOREMS OF CATASTROPHIC MATHEMATICAL AND PHYSICAL INCONSISTENCIES OF NONCANONICAL AND NONUNITARY THEORIES. For an outline of the vast literature, you may inspect Richard Anderson et al. Chapter 3, Section 3.9 of the link

The argument is elementary: unlike special relativity, Riemannian gravity is a noncanonical theory. Hence, its operator image must be nonunitary for consistency (otherwise you DO NOT have the operator image of Riemannian gravity). It has been known for one century that nonunitary theories violate causality, probability and other fundamental laws. For instance, half-odd-integer values of angular momenta have been rejected in quantum mechanics because they violate unitarity and no paper has been ever published showing a quantum treatment of half-odd-integer angular mopmenta. Yet, organized Einstein followers have managed to allow the publication of thousands of papers in "quantum gravity" most of which are catastrophically inconsistent because with a nonunitary time evolution!

Additionally, it is know to serious scholars in the field that the Riemannian treatment of gravity has prohibited a consistent grand unification of gravitation and electroweak interactions, with additional thousands of papers published under public financial suppoort that can bve proved as being catastrophically inconsistent because:

1) It is easy to prove that a grand unification of Riemannian gravitation and electroweak interactions is catastrophically inconsistent due to the CURVATURE of gravitation and the flatness of electroweak interactions, e.g., because of major inconsistencies occurring for electroweak interactions under Lorentz and other transformations, essentially carrying the problems of gravitation to electroweak interactions thus crashing their beauty.

2) It is also easy to prove that grand unifications of Riemannian gravity and electroweak interactions are catastrophically inconsistent due to the COVARIANCE of gravitation. When treated alone, electroweak theories are simply magnificent with a full mathematical and physical consistency due to their INVARIANCE under spacetime and gauge symmetries. But, when coupled to a covariant gravitation, they become inconsistent because all the inconsistencies caused by covariance are carried over to the electroweak interactions.

3) It is easy to prove that grand unification of Riemannian gravitation and electroweak interactions are catastrophically inconsistent because of the failure by the Riemannian geometry to provide any consistent representation of ANTIMATTER. I mentioned in the introduction of my talk that Riemannian geometry is structurally unable to represent neutral antimatter, and when antimatter is charged there are separate inconsistencies. By comparison, electroweak interactions have a beautiful democracy between matter and aninatter. The imbalance in joining such structurally inequivalent theories is then obvious to all scholars in good faith, and so are the ensuing inconsistencies.

I could go on and on, but I do not want to abuse of your time. You can see a detailed treatment of the above aspects 1).2), 3) in my monograph

[3] Isodual Theory of Atimatter
With Applications to Antigravity, Grand Unifications and Cosmology

Springer, 2001

Again, as indicated at the meeting, you can ask for a complimentary copy until they last by sending your complete mailing address to Richard Anderson

In closing, I should stress that the great majority of colleagues have worked in the Riemannian treatment of gravitation in full honesty due to the lack of knowledge of the above problems, an occurrence that has motivated these comments. The few colleagues who knew the above problems, but elected to ignore them for personal gain,s are rapidly decreasing in number under increasing international denunciations, particularly when operating under public financial support. In any case, equivocal obstructions of due scientific process are myopic, and often self-destructive, because physical truth always emerges.\

I beg you, most sincerely, NOT to consider the above comments as implying the waste of all the past research done in Riemannian gravitation, because such a position would be completely unethical and utterly ascientific. As you will see below, ALL future advances in gravitation are and will be based on the historical discoveries permitted by the Riemannian geometry.

However, scientific knowledge advances inexorably and, unless we renew our scientific vistas, we risk to leave the field of serious research in favor of academic theologies. As illustrated by the important talk by Christian Corda on the current absence of gravitational waves, and as i indicated in comments following that talk, we are indeed living a historical moment in gravitation that can be indicated with the following main lines:L

1) Research on Riemannian gravitation should continue, because supported by all true scientists, including dissident scientist in the field, such as myself.

2) It is now necessary to conduct systematic studies of basically "new" theories of gravitation as the sole condition to avoid a severe judgment by posterity.

3) It is necessary to conduct comparative studies of Riemannian and new theories of gravitation as the sole basis for a serious condition of the field.

They are many, studied by various groups and, to avoid an evident scientific discriminations, they should be ALL consider and compared to Riemannian gravity. One of them is the new line of research indicated to me by colleagues at the meeting under study in Italy by Mauro Francaviglia and his group (of which i was unaware), given by the affine treatment of gravity without metric. I can ask colleagues in the "new" gravitation to indicate others.

One clear understanding is that TO BE TRULY "NEW," THEORIES SHOULD BE BASED ON ANY GEOMETRY EXCEPT THE METRIC FORMULATION OF THE RIEMANNIAN GEOMETRY. For instance, even though appealing, the so-called "modified theories" of gravitation are not expected to pass the test of time because still based on the Riemannian geometry.

I have spent years and years in searching for new theories of gravitation. Allow me to outline the main difficulties i stumbled into as well as the main results of the research, because they could be useful to colleagues seeking new theories of gravitation, as well as, more general, an evolutionary renewal in physics for the third millennium, thus honoring and maintaining preceding results (rather than "revolutionary" departure from prior history).

In essence, after countless trials and errors, I had to address the very hearth of the problem: the need to achieve a formulation of gravity admitting a universal symmetry as it is the case for the majestic consistency of special relativity in vacuum. After working for years, I discovered that this objective could not be achieved with the mathematics of the Riemannian geometry, that is, with conventional numbers, fields, spaces, differential calculus, algebras, geometries, symmetries, topologies, etc.

The basic problem I failed to solve for years is the impossibility of achieving any invariance of a line element on a curved manifold via the use of the 20th century formulation of Lie's theory, as established by numerous technical problems that historically motivated the replacement of the magnificent invariance of special relativity with the problematic covariance of general relativity.

The problem was similar to that for the solution of the Lorentz problem underlying my mathematical, theoretical and experimental studies on the isoredshift

[4] "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 1-43.

I am here referring to the historical inability by Lorentz to achieve, via the use of Lie's theory, the invariance of the speed of light universally known at his pre-Einsteinian time, the locally varying speed of light within physical media with an arbitrary functional dependence of the index of refraction on local; variables, $C = c/n(t, r, v, ...). This difficulty forced Lorentz to restrict his studies to the constant speed of light c in vacuum, by writing in any case a very important page of scientific history.

Additional evident insufficiencies of the 20th century formulation of Lie's theory were due to its sole capability of representing linear, local-differential, and Hamiltonian systems, compared to the general nonlinearity, nonlocality as well as non-Hamiltonian character of the physical reality. Hence, I was forced to build a new mathematics specifically conceived for the above indicated problems.

When I was at the Department of Mathematics of Harvard University under DOE support, I proposed in 1978 the isotopies of Lie's theory based on the isotopic (that is, axiom-preserving) lifting of Lie's theory in its enveloping associative algebra (Poincare'-Birkhoff-Witt Theorem), and Lie groups, and Li algebras here symbolically indicated as follows

AB = assoc. => AxB = ATB, T fixed, (4a)

A(t) = eHtiA(0) e-itH => A(t) = eHTtiA(0) e-itTH, (4b)

idA/dt = AH - HA => idA/dt = AxH - HxA = ATH - HTA, I = t/T>0, (4c)

where: T is positive-definite but otherwise possesses an arbitrary nonlinear, nonlocal integral as well as non-Hamiltonian functional dependence on all needed local variables, T = T(t, r, v, |>, ...) > 0; AxB = ATB is called isoassociative product; and I = 1/T is the new unit of the theory called isounit. The resulting theory is today called the Lie-Santilli isotheory.

Note that, since T is positive-definite, isotopic formulation (4) and the conventional Lie formulation coincide at the abstract, realization-free level by conception and construction, to such an extent that they can be both formulated with the same abstract symbols only subjected to different realizations with dramatically different physical implications indicated below., This illustrates the "evolutionary," rather than "revolutionary character of my studies.

The above lifting of Lie's theory quickly produced all desired new invariances, as well nas the desired joint treatment of Hamiltonian interactions represented with H, and nonlinear, nonlocal and non-Hamiltonian interactions represented with isounit, However, the original invariance soon emerges as verifying the Theorems of Catastrophic Mathematical and Physical Inconsistencies of Noncanonical and Nonunitary Theories recalled earlier. In fact, Lie-Santilli isotheories are manifestly noncanonical at the classical ;level and nonunitary at the operator level.

The resolution of the above inconsistency theorems required about two decades of studies. In fact, I was forced to build, for evident consistency, the isotopic lifting of the entire applied mathematics I know into a form admitting I(t, r, v, |>, ...) as the correct left and right unit at all levels,. This effort including the liftings of fields, spaces, differential calculus, geometries, algebras, symmetries, topology, etc. because all dependent on the basic unit.

The first major difficulty in the resolution of the inconsistency theorems was caused by the rather general beliefs in mathematics that "all numbers, as solution of the axioms of a numerical field (for the case of characteristic zero) have been classified ands are known." I am a physicist and, without a base field verifying numerical axioms, I cannot do physics because I cannot predict numbers as needed for experiments. But the achievement of the needed symmetries had forced be to generalize the basic unit of all fields, thus hitting a basic stumbling block.

In 1993, while visiting the Joint Institute for Nuclear Research in Dubna, Italy, I realized that the historical classification of numbers was based on the tacit assumption of the trivial unit 1 dating back to pre-biblical times. I then discovered that the conventional axioms of a numerical field do not restrict the basic unit to have the trivial value 1 > 0, because they remain fully valid for an arbitrary positive-definite unit I(t, r,v, |>, ... ) = 1/T > 0, provided that the field is reformulated accordingly. This lead to new numbers today known as Santilli isonumbers, first published in the paper

[5] "Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals
and "hidden numbers" of dimension 3, 5, 6, 7."
Algebras, Groups and Geometries Vol. 10, 273-322 (1993)

In particular, the above paper demonstrates that isofields are fields that merely escaped the historical classification of all numbers. The first detailed mathematical treatment of the new numbers was provided by the Chinese mathematicians C.-X. Jiang and is available in free pdf download

[6] Jiang CX. Foundations of Santilli Isonumber Theory. International Academic Press 2001, available as free download from

Despite the resolution of the problem on the base field, the consistency of the invariance of arbitrary line elements on a curved manifold remained elusive. I spent years in inspecting all isotopies including fields, spaces, algebras, geometries, symmetries, etc., but I could not find the origin of the mathematical and physical inconsistencies of the early formulations of the Lie-Santilli isotheory. The absence of a consistent invariance delayed experimental and industrial applications for years. Finally, while working in complete isolation at the Institute for Basic Research in Florida, I discovered in 1995 that the origin of this additional stumbling block was another, centuries old, widespread belief in mathematics, namely, that "the ordinary differential calculus is independent from the assumed basic unit." Once I discovered that the differential calculus is indeed dependent on the basic unit, particularly when the latter depends on the differentiation variable, its lifting was simple and immediate, resulting in the new isodifferential calculus that I first published in the mathematical memoir

[7] "Nonlocal-integral isotopies of differential calculus, mechanics and geometries,"
Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 7-82 (1996)

Actually, the above quoted entire issue of the Journal was dedicated to the new mathematics under the editorship of P. Vetro, and contains a review paper by the Georgian J. V. Kadeisvili on the Lie-Santilli isotheory.

Following the discovery of the isodifferential calculus, it was finally easy to prove the achievement of consistent t symmetries characterized by the Lie-Santilli isotheory, with the resolution of all inconsistency problems of the earlier formulations known to be. The resulting new mathematics is today known as Santilli isomathematics, or isomathematics for short. It should be stressed that, by the very conception of the isotopies, isomathematics is new in the sense that it consists of new realizations of existing mathematical axioms, thus illustrating again the "evolutionary," rather than "revolutionary" character of the studies (the broader Lie-admissible genomathematics indicated below introduces structurally new axioms).

The first independent mathematical treatment of the isotopies was provided by the late mathematician Grigorios T. Tsagas, Chairman of the Department of Mathematics of Aristotle University in Thessaloniki, Greece and by the applied mathematician D. S. Sourlas also from Greece, in the monograph that appeared before the discovery of the isofields and the isodifferential calculus

[8] Sourlas DS. and Tsagas GT. Mathematical Foundation of the Lie-Santilli Theory. Ukraine Academy of Sciences 1993, available as free download from

The best currently available, independent mathematical presentation of the new isomathematics is that provided by two Spanish mathematicians in the monograph

[9] Falcon Ganfornina RM. and Nunez Valdes JN.
Fundamentos de la Isoteoria de Lie-Santilli.
International Academic Press 2001,

Besides a rigorous formulations of the new mathematics, the above mathematicians have achieved an advance of truly fundamental character, today known from their initials as the FGNV isotopology, Recall that the topology of the Riemannian geometry is local-differential, thus ideally suited to represent the dynamics of point-like particles, historically known as the exterior dynamical problem. The FGNV isotopology has achieved, for the first time in history, to my knowledge, the characterization of the dynamics of extended, nonspherical and deformable particles moving within a physical medium, thus being the correct topology for all future studies on interior dynamical problems. Only following decades of research on the needed new mathematics, the formulation of the new gravity was elementary, according to the following rules for any given (nonsingular) Riemannian metric g(r):

1) Decompose g(r) into the product of the Minkowski metric m = Diag. (1, 1,1, -1) ands a 4x4 metric T(r)

g(r) = T(r) m; (5)

2) Assume as the fundamental isounit of the theory the inverse of T(r)

I(r) = 1/T(r); (6)

3) Reconstruct the entire mathematics of the Riemannian geometry, including the machinery of Christoffel's symbols, etc., in such a way to admit I(r) as the correct left and right unit at all levels.

The above SIMPLE REFORMULATION of Riemannian gravitation has the following important consequences:

1) The first desired feature is the disappearance of curvature because the resulting geometry is locally isomorphic to the Minkowskian, rather than the Riemannian geometry. This can be seen from the fact that, when the Riemannian deformation of the Minkowski metric m => g(r) = T(r)m is computed with respect to a unit which is the inverse of the gravitational content T(r), curvature is gone.

2) The second desired consequence is that the absence of curvature, combined with the Lie-Santilli isotheory, allows the construction of a universal symmetry for all possible Riemannian line elements, today called the Lorentz-Poincare'-Santilli (LPS)isosymmetry which results in being locally isomorphic to the conventional Lorentz-Poincare' (LP) symmetry (due to the positive-definiteness of all isounits under the conditions here assumed), as expected from the Minkowskian character of the geometry.

3) The third desired consequence is a fully consistent operator formulation of gravity given by the mere embedding of gravity in the unit of relativistic quantum mechanics. The consistency is guaranteed by the fact that the emerging new mechanics verifies all axioms of the conventional mechanics by the very nature of the isotopies, otherwise the construction is incorrect,

4) Another desired feature is an axiomatically consistent grand unification based on the general formulation on a flat manifold, governed by the universal LPS isosymmetry, with full democracy between matter and antimatter permitted by the isodual conjugation of gravity.

5) Yet another desired feature is the preservation of all the physical discoveries achieved via the Riemannian geometry, such as the preservation of black holes that are now merely re-interpreted as the zeros of the (space component of the) isounit

I_k^k(r) = 0, I_4^4(r) => infinity; (7)

as you can see by representing the Schwartzchild metric via the isounit (merely given by factorizing out the Minkowski metric and computing the inverse).

6) Still another important feature is the full preservation of Einstein's field equations, although in thire form (3) dictated by physical and geoemetrical aspects discussed in Section !, and their mere reformulation in a geometry whose basic unit is I(r)

G_)]mu\nu = k T_\mu\nu, (8a)

g(r) = T(r)m, I)r_ = e/T(r_. (8b)

7) In this way, subject to independent scrutiny, all historical, physical and geometrical objections are eliminated. For instance, the generators of the LP and LPS symmetries are the same, thus guaranteeing not only the true conservation of total gravitational quantities but also their compatibility with the relativistic ones; there is a clearly defined Minkowski limit given by I(r) => Diag. (1, 1, 1, 1); the historical objections on bending, free fall, et al. are resolved due to absence of curvature; etc.

8) Intriguingly, all the above features are only the beginning. It is easy to see that THE FUNCTIONAL DEPENDENCE OF THE ISOUNIT IS UNRESTRICTED, ONLY ITS POSITIVE-DEFINITNESS IS REQUESTED. Consequently, we can use isounits with the additional velocity and other dependence

I(r, v, ...) = 1/T(r, v, ... (9a)

Interval x^\mu T_\mu^\rho(r, v, ...)m_\rho\nu x^\nu (9b)

9) Then, all preceding features refer to EXTERIOR GRAVITATIONAL PROBLEMS. The new theory permits for the first time known to me a geometrically consistent treatment of INTERIOR GRAVITATIONAL PROBLEMS in which the speed of light is a local variable, resulting in the isoredshift I presented at the meeting, with the mere identification

I_4^4 = c^2/n^2(r, v, v, ...). (10)

see paper [4] for detils.

10) Additionally, there is a change of geometry in going from the exterior to the interior case., In fact, the metric of the interior problem is essentially Finslerian although the geometry is not because it is flat.

11) Inspection of the new geometry, now called the Minkowski-Santilli isogeometry, has shown that it includes as particular case all infinitely possible metrics in (3+1)-dimensions, thus resulting in a unification that appears of difficult understanding by mathematicians due to its novelty, called Isotopic unification of the Minkowskian, Riemannian, Finslerian and all other possible geometries in (3+1)-dimensions. In fact, the axioms of the geometry are unique, while the identification of the particular realization, such as Minkowskian, Riemannian, Finslerian, etc., is merely given by the selection of the isounit.

A flurry of research is now going on. For instance, conventional studies on black holes emerge as geometrically correct, but as an exterior abstraction of gravitational collapse. The physically correct treatment is the interior tretment of gravitational collapse, e.g.,

I_k^k(r, v, ...) = 0, I_4^4(r, v, ...) => infinity; (11)

that apparently preserves all features of black holes, except its singularity which is removed, resulting in the so-called brown holes. I should stop at this point to avoid excessive length.

The above theory is today known as Santilli Einsteinian gravitation, or isogravitation for short, where I suggested the prefix "iso" to honor Albert Einstein due to the preservation of his main physical conceptions and their mere reformulated in a way to avoid inconsistencies.

I presented isogravitation for the first timer at the 1992 M. Grossmann Meeting at SLAC (following the initiation of the studies back in 1978.....), as you can see the proceedings

[10] "Isotopic quantization of gravity and its universal Poincare' symmetry"
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),

I dedicated the]is paper to Remo Ruffini in recognition of his serious scientific commitment for allowing qualified evolutionary views at the M. Grossmann Meetings, as well as for his efforts in bringing Italy at the forefront of studies in gravitation.

The universal LPS isoinvariance of all infinitely possible Riemannian, Finslerian and other line elements was treated in detail in a 1993 paper I wrote when visiting the JINR in Dubna, Russia,

[11] "Nonlinear, nonlocal and noncanonical isotopies of the Poincare' symmetry,"
Moscow Phys. Soc. Vol. 3, 255-280 (1993)

My most important geometric contribution in the new gravitation is that on the isotopies and isodualities of the Minkowskian geometry presented in the 1995 memoir

]12] "Isominkowskian geometry for the gravitational treatment of matter
and its isodual for antimatter,"
Intern. J. Modern Phys. D Vol. 7, 351-407 (1998)

I first presented the axiomatically consistent grand unification on a flat space with a universal invariance inclusive of antimatter at the 1998 M. Grossmann Meeting in Israel

[13] "Unification of gravitation and electroweak interactions"
Proceedings of the Eight Marcel Grossmann Meeting in Gravitation,
Israel 1997, T. Piran and R. Ruffini, Editors,
World Scientific, pages 473-475 (1999),

A readable general review of isomathematics and isogravitation for interior problems in various quantitative sciences is the monograph by R. Anderson et al. currently under completion with preliminary version in html language and all references in free pdf download you can inspect from the link

My most detailed mathematical and physical presentation following the discovery of the isofields and isodifferential calculus is that available in the second (1995) edition of the two volumes published by Naukova Dumka, Kiev,. Ukraine,

[14] Santilli RM. Elements of Hadronic Mechanics. Ukraine Academy of Sciences, Kiev, 1995, Vol. I [14a], Vol. II [14b], available in free pdf downloads from

My own comprehensive presentation is under way ibn the five volumes of the series

[15] Santilli RM. Hadronic Mathematics, Mechanics and Chemistry. International Academic Press 2008, Volumes I [15a], II [15b], III [15c], IV [15d] and V [15e], available as free downloads from

The first independent review of the physicval part of my studies was provided by colleagues following a series of lectures I deliuvered in 1991 at the in Trieste, Italy, thaks to an invitation by my dear supporter Abdus Salam (that he attended despite his very advanced impedfiments, showing his stature as True Scientist dedicated to novel human knowledge until his death),

[16] Aringazin AK., Jannussis A., Lopez DF., Nishioka M. and Veljanosky B. Santilli's Lie-Isotopic Generalization of Galilei and Einstein Relativities. Kostakaris Publishers, Athens, Greece 1991, available as free download from

An important independent perspective from the viewpoint of nonassociative algebras, was provided by the mathematician Jaak Lohmus and his associates from Estonia in the onograph

[17] L\^{o}hmus J., Paal E., and Sorgsepp L. Nonassociative Algebras in Physics. Hadronic Press 1994, available as free download from

A very nice geometrical and physical review is that provided by J. V. Kadeisvili in the second (1997) edition of his Naukova Dumka monograph

[18] Kadeisvili JV. Santilli's Isotopies of Contemporary Algebras, Geometries and Relativities. Ukraine Academy of Sciences, Kiev 1997. Second edition, available as free download from

A readable comprehensive review with most original literature available in free pdf download is under way by members of the Santilli Founbdation and now partially available in html format from the link

[19], Anderson R., Cloonan M. and Gandzha I.
New Sciences for the new Millennium:
Mathematical, Physic al and Chemical Discoveries of Ruggero Maria Santilli,

International Academic Press, to appear, partially available in html format at

In closing, I would like to stress that studies on new gravitational theories are at their infancy and so mucky remains to be done, because the complexity of gravitation is beyond our capabilities at this writing. In particular, isogravitation needs to be confronted with experimental data and the results compared with those of Riemannian gravitation.

Note that all the above comments refer to exterior and interior formulations of gravity that are reversible over time. A much more complex and intriguing problem is the construction of irreversible Lie-admissible covering of the above Lie-Santilli;i isotheories, that allow the construction of basically new geometries with a non-symmetric metric representing motion forward in time, and its transposed representing motion backward in time. These more general theories will be studied at our forthcoming meeting in Nepal from December 20, 2010 to January 7, 211, see the announcement

IrTo illustrate that Einstein has mere initiated, rather than ended, geometric studies in gravitation, and that, despite all the above exterior and interior, reversible and irreversible advances for matter and antimatter, our understanding of gravitation remains at its infancy, It is sufficient to note that all these studies merely deal with the "description" of gravitation. A much more fundamental and intriguing problem is the identification of the "origin" of gravitation (see its definition in Ref. [1]). That, and and other fascinating problems are left to young minds of any age who can conduct innovative research in complete oblivion of pre-existing theories, including mine.

Needlessd to say, any critical comments would be gratefully appreciated.

Best wishes to all

Ruggero Maria Santilli

Hotel H 1898, Barcelona, Spain
March 11, 2010


Geometrically appealing and politically rewarding, but physically implausible characters of the expansion of the universe, big bang, dark matter and dark energy.

Caracas, Venezuela, March 14, 2010

Dear Prof. Santilli,

Thank you for your unique and penetrating outline of the unreassuring, on one side, and exciting, on the other size, situation of gravitation, as well as for sending it to me for review. My only suggestion is that your outline of unreassuring stands should be completed with your own outline of the shortcomings of the various "hyperbolic conjectures" you presented at the 2007 Bolu Conference in Turkey, and published in the proceedings

Proceedings of the 2007 International Conference on Dynamical Systems, Bolu, Turkey.
Ozel C.l and Kay V., editors. International Academic Press, 2007,

I outline below your views in my open language for the benefit of colleagues, because the debate of the insufficiencies has been suppressed in the literature by organized interests on preferential lines.

Expansion of the universe. You taught me that the weakness of this conjecture rests on its very basic law, the adulterated Hubble's law: the cosmological redshift of galaxies due to their (claimed) motion away from us is proportional to their distance from Earth. In fact, this law implies a progressively increasing acceleration of galaxies also proportional to the distance from Earth. Besides an evident return to the Middle Age conception of Earth at the center of the universe, the conjecture requires an astronomical amount of energy for the continuous acceleration of billions and billions of galaxies over billions and billions of years, which acceleration is positively not explained by the big bang (see below). Such a view is geometrically appealing on pure mathematical grounds, as shown by the widespread interest among geometers, as well as politically rewarding, evidently because the conjecture is aimed at maintaining the validity of Einstein special relativity throughout the universe, including, in your words, "conditions for which the relativity was not conceived for or ever tested." However, the expansion of the universe and its acceleration are physically implausible to such an extent that, until the origin of the needed astronomical amount of energy is clearly and plausibly identified, the conjecture of the expansion of the universe has no credible foundation on serious scientific grounds, e.g., those used in refereeing of contemporary papers. The situation of astrophysics and cosmology becomes "unreassuring" in your words if one compares the extremely farfetched need of immense energy of totally unknown origin with the very plausible, historical alternative hypotheses, such as that of the "tired light" you properly review in the above quoted paper. Additionally, we should not forget that Hubble died without ever accepting the expansion of the universe for much the same reasons as yours, as it has been also the case for renouned scientists such as Zwicky, de Broglie, Vigier, and many others, all silenced by organized interests on Einstein. It is hoped, but I am not sure due to known resiliency of organized academic interests and their impunity, that your momentous experiment on the isoredshift has eliminated for ever this "hyperbolic conjecture" via the experimental evidence that light loses energy E = hv to the intergalactic medium primarily composed of light energy, thus eliminating not only the expansion of the universe, but also its acceleration, since the cosmological redshift becomes proportional to the travel of light within the intergalactic medium in full verification of Hubble law, but without the arrogant conception of Earth at the center of the universe. Additionally, your isoredshift experiment provides an excellent quantitative representation of the background radiation, since the energy loss by light to the intergalactic medium cannot disappear and it is converted precisely to the cosmological background radiation. Hence, the isorfedshift establishes the basic: unadulterated Hubble's law: the cosmological redshift of galaxies is proportional to their distance from Earth. see the paper:
The Open Astronomy Journal, 2010, Vol. 3, page 1-43.

Big bang. You taught me that this additional conjecture is also appealing on pure geometric grounds, and politically rewarding, because it consists of an unverifiable conjecture formulated to support the original unverifiable conjecture of the expansion of the universe, both conjectures being voiced for the support of Einstein';s theories. However, you also taught me that the conjecture is physically implausible because: 1) The big bang conjecture is based on a return to the Middle Ages of our Earth at the center of the universe; 2) Following a primordial explosion some 15 billions years ago, our galaxy, let alone our solar system, could not possibly exist; 3) All visible galaxies should be at the ultimate edge of the visible universe. as established by the very principle of an explosion, thus violating Hubble's law; 4) The big bang is in additional disagreement with gravitational attraction because, whatever expansion could exist at such a distant edge of the universe, it should decrease rather than increase; 5) The background radiation cannot possibly be a "proof" of the big bang because a graduate student can show the complete absorption of the radiation by galaxies over 15 billions years; etc. The physical implausibility of the big bang is also established by the fact that, to represent the current distribution of galaxies in the universe, one must assume some form of absurd "resistance" in empty space (!) that has somehow slowed down the expansion of selected chunks of primordial energy; the existence of our galaxy at the center of the explosion should be explained as some "chunk of matter" that was somewhat mysteriously left over by the primordial explosion that, as such, is no longer an explosion; etc. It is astonishing how "hyperbolic hyperbolas" can acquire a large number of followers just because the are aligned with "Einsteinian interests". Also, your "irreconcilable disagreements" with Steven Weinberg on related matters when both of you were at Harvard University are known to the scientific community and should not be forgotten, because serious science is not done via a "capillary international organization of academic power," but solely via physical truth.

Dark matter The physical implausibility of this additional "hyperbolic hyperbola" is also beyond imagination because, as clearly stated in your papers: 1) In the event uniformly distributed, dark matter cannot have any measurable effect; 2) Dark matter has to be capriciously placed, say, in front of a given star to get the desired dynamical anomaly; but them 3) The dynamics of a nearby star is way off. You have reached the top of academia in the U. S. with a position at Harvard supported by the DOE and, in this capacity, you have eyewitnessed the birth of a number of hyperbolas. In this case, the evidence facing organized academic interests was serious because, prima facie, violating Newton and inevitably Einstein's laws. hence, high level organized interests concocted a farfetched conjecture to deviate attention from the real meaning of the evidence. The dark matter hyperbola is clealy appealing to mathematicians, and politically rewarding for its transparent aim at salvaging Newton's and Einstein's theories. Nevertheless, it is astonishing to see so many physicists embracing such a farfetched conjecture just because voiced by high ranking academic interests without any verification whatsoever of the physical plausibility! I understand that experiments to look for dark matter are either under way or under funding. Their outcome is easily predictable, as it was the case for the neutrinos. In fgact, I expect the experimenters will simply voice a new conjecture of hypothetical unverifiable particles to "prove" the unverifiable conjecture of the big bang that, in turn, was voiced to support the unverifiable conjecture of the expansion of the universe, How low ethics in physics has collapsed! It is hoped, but I am no sure, that this misuse of public funds is halted until the huge implausibilities ofd dark matter are resolved in one way or another in refereed journals. It is hoped, but i am not sure, that serious scholar from now on will admit your experimental evidence that the radially decreasing gas density in a galaxy causes a radially decreasing isoredshift, thus representing in thgeir entirety current experimental data on the redshift of stars, from which the galactic dynaucs are derived. In the process, Newton's gravitation is recovered completely but, contrary to the desire of organized academic interests, Einstein special relativity is gone because not conceived for or valid within physical media according to vast evidence.

Dark Energy. As you showed me, this conjecture too has been formulated for the intent of imposing the validity of special relativity everywhere throughout the universe, including in the interior of stars, quasars and lack holes, as evidently needed to have the universal validity of E = mc^2. However, as you correctly indicate, in this case the originator of the conjecture had indeed serious technical motivations, because the elimination of dark energy would require superluminal speeds that, prior to your isorelativity, violate causality. The advent of your isorelativity, and its numerous experimental verifications for interior problems in various sciences have established the causal character of arbitrary speeds within the interior of stars, quasars and black holes, thus characterizing an energy equivalence of the universe as needed to eliminate dark energy. For instance, you indicate that a rather modest average maximal causal speed in the interior of black holes and other astrophysical objects C = 10 c eliminates dark energy because E = mC^2 is close to 100 times E = mc^2 (I assume colleagues remember , as your students know, that the source of the gravitational field is characterized by energy and not by mass).

Implication of your isoredshift. I would like to recall, as you indicates in various works, that there are no empty spaces in the universe. Consequently, the entire universe is an "interior dynamical problem" that, as such,. is governed by the Lie-Santilli isotheory, your isorelativity, and related isoredshift for low densities or isoblueshift for very high densities. Consequently, new astrophysical models should locally admit your isorelativity, rather then special relativity, to be in line with the mathematical, theoretical experimental and other evidence you have accumulated in about 50 years of research.

Thanks for allowing me a free use of your writings, and for you teaching, above all, for showing me the dignity of opposing political manipulations of scientific knowledge.

Yours, Truly

Jerdsay V. Kadeisvili
temporary email: