On the intrinsic gravitational repulsion

Chaos, Solitons and Fractals 20 (2004) 683–700 www.elsevier.com/locate/chaos

On the intrinsic gravitational repulsion Jakub Czajko

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P.O. Box 700, Clayton, CA 94517-0700, USA Accepted 26 September 2003

Abstract Newton’s hesitation to publish his Principia was presumably caused by his inability to explain why volumes were irrelevant to radial action of gravity, which appeared somewhat incomplete without possible nonradial effects. His doubts about completeness of his gravitational theory are justified by results from several 20th century experiments, which suggested presence of a nonradial impact of gravity in addition to the radial attractive force that he has introduced. Existence of nonradial effects is implied by mathematical and physical preconditions, if total energy is to be conserved, and it has already been confirmed by experiments. Nonradial potentials do not affect directly the radial Newtonian potentials, which generate the radial force of gravitational attraction between bodies, but they decrease energy of the rays that run across gravity fields. The nonardial potentials also contain repulsive effect of gravity that becomes dominant at very large distances where the usual attractive radial force practically vanishes. They provide thus global mechanism for an accelerated expansion of the universe and so they explain the lack of superclusters in it. They also imply gravitational contraction, yet another new long-range effect that may explain formation of galaxies.
2003 Elsevier Ltd. All rights reserved.

1. Deep reasons for newtons doubts about gravity In times when solutions to scientific problems were routinely encrypted, and a virtual court of peers was convened to decide whether Newton or Leibniz has invented calculus, Newton postponed publication of his work for no apparent reason. His gravitational theory was correct, though not as precise as he might have wished. Yet he was not a perfectionist, for he was pushing astronomer Flamsteed to publish unfinished back then observational data. Since Newton effectively censored himself, he may have suspected that his gravitational theory was somewhat conceptually flawed, despite its theoretical success. Yet he was unable or unwilling to formulate his doubts in clear scientific terms so that others could also evaluate their merits. After hesitating for some 20 years, Newton has reluctantly published his Principia in 1687. His doubts have far deeper reasons than he realized back then, however. Since more independent variables can determine physical states of massive bodies in space than his theory of gravity utilized, he may have suspected that perhaps the radial potential that was virtually implied by his gravitational theory gives somewhat incomplete description of gravity. He might have wondered why volumes of our sun and planets were quite irrelevant to gravitational phenomena, even though gravity acts in at least 3D space [1,2]. Yet Newton’s sticking to force as the only effect of gravity, and his refraining from posting hypotheses, may have prevented him from discovering fully geometric, 3D character of mass-induced gravity. Devising and then testing hypotheses is a legal way of doing science today, however. No matter how many Newtons and Einsteins dazzle us with their ingenious theories, we must set them aside and explore new ways, if certain experiments

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Tel.: +1-925-681-2058. E-mail address: [email protected] (J. Czajko).

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2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.09.010

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defy predictions of their theories. But we should take seriously their doubts, for the doubts often open up doors to what may lie beyond their great theories. Personally, I felt deeply betrayed by Einstein when ‘‘Russian school of Einsteinbashing’’ (Logunov, Denisov & Co.) [2] exposed grave inconsistency in his approach to gravity. Theoretical inconsistencies should call for reevaluation of theory’s conceptual foundations as well as scrutiny of the mathematics deployed therein. I also relived the pain of Newton’s hesitation and just hoped that his doubts were due to a simple mistake. His hesitation proved quite reasonable, however. By AD 2000 his doubts are vindicated [1], and several formerly unexplained nonradial and mixed (i.e., partly radial) experiments [2] sustain his ‘‘worst’’ suspicions and thus justify his long hesitation. In fact, all former theories of gravity share the intrinsic incompleteness he was so afraid of, because they all were basically radial. Both Newton’s and Einstein’s theories treated gravity only in radial terms, just as radial interaction between two massive bodies. Since two nonradial directions are always present in any 3D space, then all previously developed theories of gravity need certain nonradial extensions––just as many formerly unexplained experiments suggested [2,3]. Although the usual radial impact of gravity is not affected directly by the nonradial effects, the radial effects alone could not explain nonradial and mixed gravitational phenomena [2]. When calculating the force of gravity that ties moon to earth, Newton took at first the distance between surfaces of these two planets. Only after the distance was measured between their gravity centers, he obtained correct results. Since force is a radial interaction between two bodies by definition, volume (or specific gravity for that matter) is irrelevant to it. Volume can affect only certain, unknown back then, nonradial effects of gravity (NEG), which his theory did not predict. Had it been recognized as such, the fact that radial attractive force of gravity should be taken between gravity centers could have indicated 3D character of gravity. Note that both Newton and Einstein considered the source of gravity as 1D phenomenon, whose effects then can be decomposed into 3D or 4D impact. However, the NEG implies more, namely the need for tangential and binormal gravitational potentials and so it calls for a major paradigm shift in physics. Formerly unexplained experiments suggested presence of the NEG as direct consequence of local energy conservation law [2]. Without taking the NEG into account, physics was evidently incomplete. Newton’s hesitation indicates that he was aware of that incompleteness and was presumably tormented by the fact that his theory of gravity was quite successful without being conceptually complete. Although the incompleteness may be mathematically clear today, that part of mathematics did not exist in Newton’s lifetime. Nevertheless, his very kin analytic instinct and his geometrically inclined mode of thought fueled his doubts for over 20 years, which is very cruel intellectual torture by any human standard. Yet no other scientist was really concerned about that, as far as I know, and instead of trying to solve it, or at least address the whole issue, physics made it anathema and the Newton’s doubts virtually became taboo. Nonetheless, his conceptual doubts were not a minor inconsequential quest for an improvement. He did not try to merely polish his theory, but to inquire what happens on equipotential surfaces. As one could easily see it, whatever may happen along equipotential trajectories would require certain tangential and/or binormal (hence nonradial) potentials. I would dare to say even more: Even if quite nothing would physically happen on equipotential surfaces, we would still need to prove that nonradial potentials are equal to zero, because existence of such potentials is geometrically necessary. If the radial potential is present in a 3D space, these nonradial ones must exist too, even if their physical impact could somehow disappear. Although an impact of the nonradial potentials may be optional or physically inessential in some circumstances, their abstract geometrical existence is absolutely mandatory. The tacit denial of incompleteness of classical mechanics (CM) hurts all of physics. Eventually Newton got over his doubts about gravity, but he did not really overcome them. We should be grateful for that, because the quest for NEG is difficult to address even today. Nonetheless, both the CM and quantum mechanics (QM) really need a solution to the Newton’s quandary. The incompleteness of the CM is perhaps the greatest scientific failure since Almagest, because Newton’s doubts were mishandled. Yet unsettled issues did not vanish. They are haunting physics ever since. He recognized that possibility for he suffered mental breakdown, presumably as a consequence of unresolved conflicts due to his doubts, I suppose. His hesitation was a monument to his exceptional scientific integrity. It has also confirmed his steadfast belief that every happening in nature should have a physical cause. Had he been unsuccessful, he might have fought the obstacles and probably never gave up. Seeing oneself ‘‘rewarded’’ for being wrong can make one with so great integrity desperate. It may even drive one crazy. Because he realized that he obtained quite correct results from incomplete premises, he may have suspected that by making his premises more complete he could screw up his already fine results. This conundrum was demotivating. If one cannot see any foe to fight against and does not expect victory anyway, then one tends to quit. Newton eventually left science for administration, a move that no philosopher (i.e., ‘‘lover of science’’) would do, according to Plato. He did not really quit, however. He was pursuing science till his death. I am not rewriting history, but am saying that we should pay attention to conceptual dilemmas. Even if one cannot see how incompatible was CM with tenets of differential geometry, analyzing doubts of a great mind, which sensed that his otherwise splendid theory was flawed, could have prevented some mistakes and avoided complacency, if not solved its problems in less than three centuries. Despite huge success of his Principia, he knew that it needed amends. He could

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not pinpoint any fault, while others were afraid to dissent or embarrassed to admit their doubts. Challenging a theory, without pulling it down or attacking its creators, actually means that one has learned it, or understood its deep-seated limitations and just tries to overcome them. As a matter of fact, he was not alone. When one looks at Einstein sticking out his tongue (http://www.msnbc.com/ news/435007.asp), and recalls that at about that point in his life he destroyed research papers he was working on, one may wonder whether he already got suspicious that his interpretation of gravity was faulty too. Faults do not disqualify the affected theory entirely. Both Einstein and Newton devised best theories of their times. But thinking continues and no theory can pretend to solve all aspects of the issues it deals with. One standing on ‘‘giants’ shoulders’’ could see farther than they did. General theory of relativity (GTR) generalized only the 1D radial impact of gravity––its output side. Its 10 potentials (metric coefficients) are merely (‘‘decomposed’’) components of the 4D impact caused by the mass-induced radial linear 1D potential. GTR did not address Newton’s doubts regarding the 3D source of gravity––its input side. The radial approach considers only actions between masses along the field’s gradient (or radius for spheroidal massive bodies). Former physics failed to see possible nonradial impact of gravity and tacitly dismissed the Newton’s doubts. The dismissal sparked explanations based on allegedly conservative forces, whereby mathematical physics (MP) and pure mathematics (PM) evaded their most controversial issues. Einstein discovered gravitational frequency shift when he realized that photons must get ‘‘tired’’ (redshifted) when they run radially against gravitational field. However, formerly unexplained experiments showed that photons also got tired when they run tangentially or across gravitational fields with mixed tangential and radial components on their trajectories [2]. The radial gravitational potential raises an attractive force between bodies. It taxes energy of the particles that move radially. But in order to loose an extra energy while moving along an equipotential surface, purely nonradial gravitational ‘‘drag’’ (or potential) should exist [2]. If there would be none, no matter how small, then most of differential geometry could just end up in a wastebasket. Nonradial and radial effects of gravity are entirely linearly independent (mutually exclusive) and thus complementary. This means that the NEG affect only the energy of photons, but have no direct impact on the usual radial force of gravitational attraction [2]. I could not prove this fact before, but I can do it now, because I designed the tools for that. I knew the tools already while writing the paper [2], but I could not introduce any two conjectures that mutually rely upon each other. I should introduce every one of them independently of each other and then I can use them without raising suspicions. Also in this paper I cannot say all that I think I know, but I must wait till all tools necessary for that are quite independently introduced. Unlike the radial Newtonian potential, which acts only between bodies and apparently depends on radial distance from its source’s gravity center, the nonradial (i.e., tangential and binormal) potentials vary with tangential and binormal distances, respectively. Nonradial potentials should always vary with some equipotential (nonradial) distances, or with the corresponding to them spherical angles taken along the equipotential surface. Mathematically speaking, these two requirements are self-evident. Physically, however, consequences of such statements may appear as too extensive, or perhaps even as too revolutionary. But because many experiments support that [2], we cannot just discard them. Physics should never disregard experimental evidence. I am not saying that one should wait till all uneasy questions are answered, for Newton would have to wait 313 more years, but that covering up controversies could not sweep them away. All doubts should be openly discussed, no matter how unimportant these may seem to those great minds that are not troubled by logical dissonances. Making the Newton’s doubts anathema compromised scientific research in physics. We can play down consequences of the tacit suppression of his doubts and say that we were not quite right on just one single issue, or we can admit that we were wrong for over three centuries, all way down from CM to QM. Conceptual damage caused to MP and PM by silencing his doubts is enormous. It is extremely important to change former paradigms and consequently our attitude toward physics. One cannot suppress doubts and pretend to be correct forever. Surely there should be no mistake on this: if unbiased experiments showed unaccounted-for frequency decrease, then either the energy conservation law is invalid or our physics was incomplete. It is my opinion, however, that the energy conservation law is beyond any reasonable doubt. Yet energy and potential always come as a pair. Only the total sum of kinetic and potential energy is conserved. If an extra energy was spent while traveling along an equipotential surface, in addition to the energy that is required to sustain the given motion, then a potential tangent to the surface must exist. Otherwise magic and physics could be synonyms. Nonradial potentials are necessary thus for both physical and mathematical reasons. I have already derived the nonradial potentials from experimental evidence [2]. Now I will synthesize the NEG from preconditions of spatial generalization of the radial potential. Even if the NEG would not exist, one should still try to devise a 3D action of gravity in 3D space and then perhaps prove that all the, other than radial, components of the action of such a generalized potential do not, or could not exist. Since existence of the NEG is necessary also from the standpoint of differential geometry, one should not discard it because its tiny impact was overlooked. Although Newton and Einstein can be excused for neglecting the necessity of

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existence of NEG, we must not follow their traditional ways of thinking. Many experiments remain unexplained, while mathematics is tweaked to suppress some allegedly surreal consequences of some physical and mathematical theories [1,3]. However, if the physical reality, which is given in experiments, appears somewhat surreal, then perhaps it is because former physics lost touch with the reality it was supposed to investigate. The major fault of former mathematical approach to physics is that most analyses of physical phenomena were restricted to only those aspects, which physicists already admitted as possible prior to their analyses. Mathematics was seldom used as a tool to discover laws of physics, apart from justifying one’s pre-existing intuitions. However, we need a nonpostualtive, synthetic mathematics (SM) as well as a built upon it synthetic approach to physics. The SM should operate on physically meaningful geometrical objects and utilize syntheses instead of derivations, whenever derivations would require postulates that may be unwarranted. The base for such syntheses could be operational consistency and construction of objects from preconditions of their existence, such as conservation laws and abstract transformation rules. I assume that abstract operations must not destroy geometrical objects of physics and that transformations of such objects should always comply with conservation laws. Mathematically speaking, we should avoid objects that cannot be constructed. Physically speaking, we must construct objects that can be operated on. We should synthesize geometrical objects from logical prerequisites for their construction. We must then ‘‘physicalize’’ geometrical objects (or operationally synthesize them) so that they could be operational. Physical existence is thus contingent on feasibility of abstract operations as well as on conservation laws, which have mathematical underpinnings too. The synthetic mathematical approach that I emphasize is aimed at a new mathematical way to discover either new phenomena or some new features of the already known ones. My conjecture in the present note is this: since gravity operates in 3D space, its action should be 3D too. I admit thus presence of a generalized 3D potential, not only the 3D impact of the 1D radial Newtonian potential alone, as our former physics did. Unless proven otherwise, nonradial components of the generalized 3D potential are not geometrically impossible. Quite on the contrary, they are necessary. Not only has former physics never considered seriously possible existence of the NEG, but it tacitly made them anathema by entertaining fancy idea of conservative forces. Perhaps we ought to ask ourselves the question: What features should our physical reality possess in order to comply with virtual demands of formerly unexplained experiments? The answer to this question is sine qua non condition of responsible approach to development of modern physics. We cannot afford to build mathematical and physical theories apart from experimental evidence; neither can we entrust physics to the artsy PM.

2. Spatial gravity needs also nonradial potentials Nonradial components of the generalized, 3D vector gravitational potential UðV; H; ZÞ have already been derived from experimental evidence [2]. I will show next that they are also theoretically necessary, and by the way expand them. For a star or planet of mass M and radius R, the potential U at a given height h above its surface (hence radial distance r ¼ R þ h, h P 0) comprises three distinct components in purely spherical coordinates ðr; q; pÞ: the usual radial (vertical) potential V ðrÞ ¼ GM=r and two new, hypothetical nonradial potentials, namely tangential (or horizontal) potential HðLq Þ and binormal (or zenithal) potential ZðLp Þ. Although the nonradial potential functions H ð Þ and Zð Þ must contain also the radial distance r as the constant parameter that determines the equipotential sphere they are tangent to, they should vary with some spherical distances L, or corresponding to them spherical angles q and p, which should be measured along the equipotential sphere in their respective directions [2]. From an algebraic point of view, functions can depend on any variable. From a geometric viewpoint, however, directional quantities should depend on variables that vary in the same directions as do these quantities, even when the driving variable is absent in some formulas. Even if an independent variable is absent in a formula, it may still drive this and other related formulas. An implicit variable is just as good as any explicit one. If an apparent independent variable is absent, it is neutral to the structural law represented by the formula, but it must not be treated as a nonexistent one. Implicit (i.e., concealed) variables can reappear after linear integration in the direction in which the variable varies. An implicit variable can be called Ôhidden’, if its possible impact is neglected. Implicit variable can be called Ôparameter’, if its varying cannot be directly represented in the given frame of reference. By structural law I mean here an abstract, usually purely mathematical, basically external law of composition of some physical magnitudes. Inverse square law of propagation (whether gravity or light) in terms of potentials is a structural law of physics. Structural law is a template for physical laws, while abstract operational law is a template for operations. Being purely mathematical does not make structural laws unphysical or less than physical, but it indicates presence of certain quite independent, abstract mathematical structure that should be considered during all calculations.

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Unlike the radial potential’s local directional derivative DV ðrÞ ¼ M=r2 that is often viewed as function of the radial distance r, directional derivatives of these two nonradial potentials in an infinitesimally small surrounding of the given point, should be independent of tangential and binormal distances Lq and Lp , respectively, because no separate source exists for the NEG. The very same mass causes both radial and nonradial effects, and the usual radial potential is always constant on any local equipotential sphere. Thus in other words, the local rate of change of nonradial potentials in any equipotential direction is constant, but it does not mean that the rate always equals to zero, as former physics assumed. Yet the nonradial potentials themselves should depend on their respective distances L (or on the spherical angles q and p that would correspond to these spherical distances), once their directional derivatives are integrated along equipotential paths. Since radial distance is always constant in all nonradial directions on any equipotential sphere, then it cannot really induce any varying NEG. NEG can only be driven by such a nonradial, codirectional variable that does vary in the very same direction as the NEG in question. This tiny distinction is extremely important for proper comprehension of functional relations between variables in both: geometry and physics. Although the art-like PM and MP used to ignore this nuance, we should respect it. Centuries of erring did not make any mistake right yet. Let us synthesize these requirements now. For any gravitational source of constant mass its radial potential V has radial directional derivative DV ðrÞ V ðrÞ ¼ GM=r ) DV ðrÞ ¼ GM=r2 :

ð1Þ

Because the density of matter––as determined by specific gravity Q––may be different for different massive bodies, we must take it into account [2] and therefore for a source of mass M with uniform density of both mass and matter, the tangential nonradial potential H should be determined as follows: Z q Z 0 DH ðr; Lq Þ ¼ ðKLq =QÞ DV ðrÞ dLq ; ð2Þ 0

Lq

where the nonradial (equipotential) directional derivatives should equal to DH ðrÞ ¼ KLq V ðrÞ=Q ¼ KLq GM=Qr2 :

ð3Þ

According to Eq. (2) we obtain nonradial potentials by linear integration of local directional derivatives in independent tangent directions q and p Z q Z 0 DH ðr; Lq Þ ¼ KLq ðGM=Qr2 Þ dLq ) H ðqÞ ¼ Kq GMq=Qr; ð4aÞ 0

Lq

Z

0

Lq

DZðr; Lp Þ ¼ KLp

Z

p

ðGM=Qr2 Þ dLp ) ZðpÞ ¼ Kp GMp=Qr;

ð4bÞ

0

where for the spherical angles q and p we get dLq ¼ r dq and dLp ¼ r dp. The potential functions H ð Þ and Zð Þ just crop up from the linear integration with respect to the independent variables q and p. They appear complementary to the radial potential V ð Þ, when full 3D action of gravity cannot be excluded. Existence of the radial potential implies presence of nonradial potentials, if gravity really is a 3D feature of mass and matter. The distinction between mass and matter is explained in paper [2]. Radial impact of gravity is just a radial slice of its whole 3D impact. One obtains thus the nonradial potential functions from geometrically necessary conditions for existence of the radial one. Radial and nonradial potentials always come hand in hand. While the gravitational constant G is present in all three potentials, the specific gravity Q affects only the nonradial potential functions. Note that for nonuniform distribution of mass and matter the M and Q would be some functions. Eq. (4) have already been experimentally confirmed near surfaces of earth and sun [2]. The constants K can be estimated from experiments conducted near surfaces of massive bodies [2]. Although the spherical distances dLq and dLp , and the corresponding to them spherical angles q and p are absent in the two nonradial directional derivatives DH ðLq Þ and DZðLp Þ, they drive the nonradial processes as their implicit variables. Yet one may say that nonradial potentials vary linearly, whereas the radial potential varies with inverse distance (or radius). But in differential geometry the inverse radius denotes linear radial curvature, and so all three potentials actually vary linearly with certain variables. The two nonradial potentials are nominally different, but they belong to the same conceptual class of linear potentials of any physical field. These nonradial potentials act and behave similarly to the usual radial Newtonian potential. Without the NEG physics was incomplete. As a second time derivative of a 3D function of motion, acceleration is 3D geometric object too [3]. Radial directional derivative DV ðrÞ represents only a part of the entire 3D function of acceleration [3]. One should not take just one part of the 3D acceleration and forget about all the other parts. Just as the

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radial acceleration, which by Equivalence Principle equals to the usual radial impact of gravitational field, causes radial gravitational frequency shift, so do also the nonradial ones, for they cause nonradial frequency shifts [2]. Since former physics always dealt only with radial aspects of gravity, its implicitly radial predictions disagreed with results of numerous nonradial experiments and also with the definitely mixed ones when it comes to (local) conservation of energy [1–3]. If theory has a problem with conservation of local energy, then it is either physically invalid, or it is not really a physical theory. The GTR is not a sensu stricto physical theory, because it clearly disregards local energy conservation laws even for radial interactions [2,3]. Yet it is a formidable cosmological theory of an amalgamated gravitational pseudo-field, provided that all the massive bodies can be treated as entirely dissolved within the pseudo-field. Einstein mystified acceleration by replacing it with action of an amalgamated field, but this made his field observer-centered (nonunique) and therefore also a kind of irrelevant. In infinitesimally small surrounding one has to apply a local acceleration in order to change velocity. Einstein’s inability to grasp global mathematical consequences of his special theory of relativity (STR), and his neglect of spatial flow of time in particular [1,3], prompted him to twist mathematics, presumably unintentionally. GTR is a brilliant escape from physics into cosmology at the expense of energy conservation law. The presence of nonradial potentials is thus necessary consequence of the unquestionable existence of the usual radial potential. Neglecting NEG had detrimental impact on former understanding of the ways the nature works. If experiments disagree with theoretical predictions, we should rewrite our theories, for it is the nature that sets standards of soundness. Eq. (4) suggest that Newton’s doubts were justified, indeed. Yet the nonradial (tangential and binormal) impact of gravity differs from the radial force of gravitational attraction. It is like a grazing effect tangent to the equipotential surface of the source. It affects only energy of the recipient’s body, for it is linearly independent of the radial effects that are locally perpendicular to it. Because if force is a compound linear product of potentials, then we cannot combine potentials, but we should produce those forces first, as I will show below. Perhaps because of that, the tiny impact of NEG was overlooked. The discovery of nonradial potentials [2] is not just a minor adjustment that can correct an unfortunate mistake and allow us to continue business as usual. It prompts for changes to several unspoken paradigms of physics and mathematics. In a very subtle way it calls for changes to our approach to all exact sciences, mathematics included. The new way of thinking has already showed impact on issues related to flow of time. For if the allegedly linear time can be affected by these nonradial potentials in two extra, quite linearly independent ways, then we have to recognize three independently varying linear components of time flow, and consequently thus definitely spatial 3D flow of time. Physical time reveals 3D variability in time-based dimensions. STR virtually supports this conclusion for 2D flow of time [1]––and also for 3D time flow, if extended onto artificially accelerated motions [3].

3. Impact of nonradial potentials on former theories Nonradial potentials cause frequency decrease (i.e., redshift) in rays that pass near surfaces of massive bodies [2]. Since potentials are basically static (motion-independent), they can be superimposed on the effects of arbitrary motion [3]. This does not contradict the famous Einstein’s view of gravity as curvature of spacetime. Quite on the contrary, the nonradial potentials complement that view. They provide an excess over Einstein’s prediction for deflection of light by sun [2]. The excess was observed in several quite independent experiments [4]. As a matter of fact, Einstein has recognized possible tangential effects, but he omitted them as too slight on the earth [5]. Although tiny and therefore unnoticed before, the impact of the nonradial potentials near surfaces of large and/or massive stars is overwhelming. It is measurable and makes qualitative difference thereon [2]. Taken together, the radial and nonradial potentials show that gravity is a fully geometric, directional feature of mass and matter in 3D space [2]. This does not defy the GTR, which is large-scale cosmological theory. Since in the GTR all massive bodies are dissolved into its amalgamated pseudo-field, the notion of surface is in most cases quite irrelevant to the GTR and so are the NEG, which are gravitational effects that happen near surfaces. From the point of view of the universe as a whole the tiny NEG may be negligible, although it could help us to understand the physical reasons for formation of stars and planets as well as physical reason for accelerating expansion of the universe. However, the NEG is extremely important also for precise determination of distances via spectral analysis of the rays that come from distant stars and in general, for field-related local gravitational and electromagnetic phenomena. NEG require paradigm shift. Nonradial potentials reveal entirely different kind of physical reality, which Einstein has already rejected as spooky. Yet if physical reality appears spooky, and it is only the reality that counts, then––from the standpoint of (many-valued) logic––the infrastructure of physics (i.e., mathematics) was somehow deficient. Geometry must not be reduced to just issues of curvature and shape; neither should the universe be reduced to a sandboxlike abstract topological manifold. Since 90% of the known universe’s mass seems to be missing, then either the

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universe is leaking or some beans in the amalgamated cosmic soup are unaccounted for. What is missing in all radial theories of former physics is total balance of energy. For mass and energy are just two references to basically the same thing. Since only the total energy (potential and kinetic) is conserved, then energy cannot always be balanced without accounting for all three linear potentials: radial, tangential and binormal, if a motion is considered. Neutrinos alone may not suffice to explain all the missing mass. The NEG itself is very tiny, local near surface effect, but it has an impact on energy and therefore could contribute to solution of the curious accounting problem of cosmology. The results of Pound–Rebka–Snider experiments, in one of which gamma rays were redshifted while rising in a tower [6], indicated that radio signals emitted from the top of an antenna should be blueshifted when received at the bottom of it. However, when the receiver moved away from the antenna along practically equipotential surface of the earth, the signal’s frequency decreased with increasing distance [7–10]. Such an extra redshift acquired along equipotential surface needs tangential potentials, if energy is to be conserved. Radio signals emitted on earth [10] and the rays from Taurus A that passed near sun [11] obey the energy conservation law once tangential potential is taken into account, and they are governed by the same physical law [2]. Former gravitational or electromagnetic theories could not explain these experiments [12] without the NEG. Nonradial slice of energy is quite real. It is the extraneous energy lost by the ray from Taurus A that passed near sun [2]. Its extraneous redshift was unambiguously observed [11], but it remained unexplained and was not recognized as NEG until I predicted an equipotential (nonradial) effect of gravity [1,2]. To say that these nonradial and mixed experiments were simply unexplained is too polite. They actually revealed that former gravitational theories defy the energy conservation law. Eq. (4) define physical potentials, but unlike the old radial potential which acts between two bodies, the nonradial ones imply presence of a new, passing-by kind of interaction. When radial theories dealt with gravity, they decomposed its radial impact along three or four dimensions. Both CM and GTR only dealt with 3D bodies affected by basically 1D gravity, but did not deal with 3D gravity driven by three distinct and independent (orthogonal) potentials. Since NEG cause frequency shifts on nonradial components of photon’s trajectory, we have 3D source and 3D effects on 3D motion, so that we deal with essentially nine, quite independent components [2,3], as if the CM suggested presence of a new, 9D spatial structure. In fact, the very same conclusion has been drawn in a quite different, abstract mathematical way [13]. Although relatively simple, such conclusions could explode MP and PM, by picturing gravity as a phenomenon that happens in a 9D hyperspace, which is a compound multispatial structure, not just single space [13]. Clear distinction between spatially acting gravity field and simply spatial representation of radial forces is extremely important for modern physics, especially for superstrings. CM seems to need nine distinct dimensions just to effectively deal with gravity. One can reduce dimensionality for simplicity of computations, but not for theoretical derivations. For dimensions are not objects, but localizers like symmetries. One cannot compactify geometric dimensions any more than directions like East–West or up–down. Even on the earth’s poles, for example, where the usual direction East–West becomes irrelevant, no compactification takes place. Compactification of dimensions was an ad hoc ‘‘explanation’’ of the fact that no higher than fourth dimension had been identified. In fact, physically measurable fifth and sixth dimension have been found sound and uncompactified [1,3]. Abstract reasonings can spin out of control in absence of SM, which should provide effective reality checks for PM and MP. PM did not always clearly distinguish objects from their properties or methods, which sometimes causes theoretical confusion. Descriptive notions––such as symmetry or dimension––are not geometric objects that could be concealed. These are just properties of certain objects called spaces i.e., handles for the methods that operate on some geometric objects immersed in the spaces. One can hide one’s hands, but not abstract directions such as Ôleft’ or Ôright’; neither the dimensions that correspond to the directions could be effectively concealed or disabled, even though they might be disregarded. One cannot make symmetry or dimension disappear or ineffective without having changed the objectsspaces that they describe. Former PM was quite purposely devised with blatant disregard for physics and entirely apart from experiments. Geometric objects may appear totally unphysical in topology. Separation of mathematics from physics created an art-like PM, which practically crippled the MP that depends on geometry. However, physical world reveals very subtle geometric design whose full comprehension requires both analyses of experimental data and syntheses of logical contingencies. We need to understand the world’s geometric design in order to properly interpret physical experiments and observations [14]. Previously developed mathematics oftentimes considered itself as entirely independent of physics. Nevertheless, successes of abstract mathematics have been overstated, while the guiding role of physics was underestimated. It was probably the elegance of the old Euclid’s method of derivations that caused us to believe in a linear accumulation of knowledge. However, the progress of hard sciences is neither linear nor orderly acquired. PM started generalizing mathematical concepts before mathematicians understood its abstract internal structures. We should redesign mathematics, and then try to reinvent physics on the new, redesigned set of mathematical foundations. Although existence of 4D+ world is not questioned, we oftentimes tend to assume that the number of local coordinates, which is free to choose [15], is the same as the number of geometrical dimensions. But the number of local

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dimensions may not be arbitrary. It seems that full description of electron may require 6D spacetime with two extra time-based dimensions [16]. In fact, full description of anything physical immersed in 3D space seems to require 6D hyperspace composed of 3D length-based space (LBS) and certain 3D dual time-based space (TBS) [3]. Even the action of radial gravity alone is proportional to time parameter [17], which strongly suggests that energy-momentum should be always conserved locally. Yet the density of energy–momentum tensor in the GTR is not conserved [18], which makes one wondering whether the GTR needs certain local corrections, were it not for the fact that the GTR is not really physical theory, but a cosmological one [2,3,13,14]. The mathematical method of noncovariant ‘‘psedotensors’’ that Einstein used in the GTR does nullify energy flux [19], which can make one wonder whether tensor calculus can be sensibly deployed in 4D GTR. Moreover, if the energy–momentum would have to be actually conserved in GTR-like cosmological theories, then massive bodies could not really move along the geodesics of Riemannian spacetime [20]. GTR has a big problem. Surely one can devise no-nonsense replacement for Einstein’s GTR [21], but the question remains: Have we really understood the CM completely? One may be surprised to see that even after over 200 years of thinking an entirely new logical approach can emerge. A totally new (linear) equation of constrained motion, with external constraints represented as matrices, was very ingeniously devised quite recently, wherein at each instant of time, the acceleration vector is adjusted in a manner directly proportional to the extent to which the constraints are not satisfied at that instant of time [22]. We see surprisingly new aspects of the allegedly finished CM. Einstein’s GTR has oversimplified gravity but at the same time mystified acceleration. Actions of gravity should be determined also locally by intrinsic properties of mass and matter, not only by global distribution of mass–energy in the universe. Physical theories are built in response to contemporary understanding of the status quo in physics, which could change tomorrow with unexpected results of new experiments. Einstein got several issues right, but not all of them, I am afraid. Some mathematical methods developed by PM are often questionable. Moreover, MP tends to apply otherwise valid results of PM beyond the area wherein they can make sense. I have already shown how blatant PM and MP disregards their own greatest achievements [1–3,13], and will show more examples of that. Without changing our attitudes toward abstract mathematics we are working hard to create yet another Almagest. We should also relax the arrogant ‘‘command & control’’ attitude towards doing physics and mathematics. Without changing our approach to making judgments about what is right or wrong in mathematics and physics, we are doomed to perpetuate even their minor, accidental miscarriages. One just cannot judge new development by its ability to preserve some old stuff. Continuity of accumulated knowledge should not be our goal. We should admit possibility of big surprises and tolerate constructive doubts. Newton hesitated for so long and finally gave up, but his splendid example of endurance under such enormous pressure from his own conscience, I suppose, is remarkable. The tacit suppression of his doubts that was, often inadvertently, perpetrated by so many of his illustrious successors would be a slap in his face. His doubts have been revived by some curious––formerly unexplained––experiments, whose challenge the PM and MP tacitly denied.

4. Impact of nonradial potentials on future theories Presence of tangential potential retrodicts differential rotation of the sun, and predicts its apparent anomalous rotation [2]. Besides the orbital speed of the earth around the sun, which can either produce redshift or blueshift, rotation of the sun itself appears anomalous. Redshifts in solar spectra taken from the sun’s receding limb are enlarged by the tangential redshift, whereas the rotational blueshifts from its opposite (approaching) limb are diminished by the same tangential redshift. It appears thus as if the sun rotates away faster than it rotates towards an observer on earth [2], which was sometimes blamed on poorly understood angular momentum [23]. Similarly, the NEG ‘‘causes’’ the apparent differential rotation of the sun, which seems to spin slower at higher latitudes than in equatorial plane, even though only circular speed slows down towards poles, with diminishing circle [2]. If differential rotation of the sun were real, it should have ceased long ago for the sake of its stability [24,25]. Although rudimentary differential rotation of the sun’s photosphere may exist, evaluation of solar spectra gave a ‘‘confused’’ picture [26], which I called anomalous [2]. The NEG clears the confusion. Obviously the impact of nonradial gravitational potentials on frequency makes the relationship between distances and frequency shifts that is seen in spectra from faraway galaxies less reliable than we used to believe. The cumulative effect of nonradial potentials from a large or massive star itself, when spectra are taken from its limb, and the possible nonradial impact of other stars near the ray’s path, can thus significantly enlarge their observed redshifts [2]. Also the rotational and orbital speeds of the star itself can distort its spectra, though in either direction (red or blue). Since one cannot distinguish single stars in distant galaxies, then their observed redshifts should be averaged, preferably without

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the high-redshift extremes. Only the points of convergence of their high- and low-redshift spectra––in immediate vicinity of each such point––can still be unbiased. For only spectra taken exactly from the geometric center of each given star’s disk plane can give us assurance that the static nonradial impact of the star itself is avoided. Due to the unanticipated impact of the nonradial potentials, distances to faraway galaxies are overestimated, I am afraid. Their receding speeds may therefore be slower than some redshifts seem to indicate. If the presently accepted redshift-distance relation were correct, then the galaxies with very high redshifts should be too distant for us to see them in the first place [27]. The high-redshift galaxies show differential magnification, which may be caused by greater lensing of hotter regions [28], or––at least in part––by the NEG. Hence the visible universe may be much younger and lighter than we thought [29]. The Sloan Digital Sky Survey estimated the most distant quasars at 80 billion light years now, even though the universe is up to 20 billion years old [30]. Though not impossible, such estimate would indicate very fast expansion, which seems rather incompatible with the Boomerang data that suggested purely baryonic universe with no trace of dark matter [31]. Without the NEG, the state of our universe is rather underdetermined. Even for interacting galaxies the number and luminosity evolution alone are insufficient to predict what is observed [32]. Since redshifts could also be used for calculation of diameters of possible gravitational lenses, some of which could be black holes [33], the overlooked impact of NEG might have significantly distorted the previously estimated distances to them. The apparent radial acceleration of runaway spacecrafts was a spectacular example of NEG at work [34]. Since planets still move as usual, something else must be responsible for the alleged acceleration. If an extra radial force would affect distant spacecrafts, it should disturb trajectories of planets too. Radio signals emitted by a faraway spacecraft should be blueshifted by the gravitational field of earth, sun and other planets, along the respective radial components of the signal’s trajectory. Yet along tangential parts of its path with respect to the sun (and other planets), an extra redshift is accumulated. The extra redshift diminishes the expected overall blueshift, and so it seems as if the spacecraft is somehow diverted from its prescribed trajectory [1]. When NEG is neglected, one may be tempted to assume that an extra radial force caused the alleged diversion. But lack of any impact on the planets proves such an extra force unphysical [1,34]. One can check my conjecture by measuring frequency decrease of radio signals from those spacecrafts, when the signals pass near sun [2]––as long as we can hear them [35]. The experiment designed to detect extra dimensions via deviation from the usual (radial) gravitational interactions did not find any [36]. In a sense, the NEG is radical departure from the unspoken assumption of all former radial theories that viewed gravitational field as a single radial entity. Although unanticipated effects may reveal extra dimensions, such effects may not be deviations. The two extra dimensions that have already been found [1,3] have raised the total count of physically measurable dimensions to six and there seem to be three more in 9D multispatial hyperspace [13]. El Naschie, who investigated hierarchy of spaces, has demonstrated that discrete finite dimensionality is an expectation value of that on an infinite-dimensional Cantorian manifold [37,38]. His result couples semi-structured, continuous topological manifolds with fully geometric, quasi-discrete linear vector spaces (LVSs) spanned over such continua. It also points to the possibility of various representations of the same abstract objects in either algebraic or topological or geometric terms, depending on context. Although attributed to geometric spatial structures and abstract sets, dimensions also depend on spatial representations, because geometric vector spaces are constructs over topological manifolds. Geometric LVSs are structural objects [1,3,13]. As vectors, potentials need two points to be defined. However, since the primary radial potential is associated with the mass that generates it, then it always exists at every point outside the mass. The nonradial potentials on the other hand, are accumulated (integrated) between points on trajectories. They are static in an existential sense, but they are dynamically acquired. Their emergence is thus always dynamic, not fixed. They are tied to motion between two distinct points. They can only emerge when integrated (i.e., accumulated) along the radial angles (or equipotential paths), as in Eq. (4). Hence radial experiments with (radial) dropping of neutron waves or atoms [39,40] could not really preclude any possible nonradial impact of gravity. They proved that Newton was right in all radial cases, but this does not mean that nonradial impact of gravity cannot exist. They also confirmed that NEG do not produce a force that would directly affect radial attraction, but left the Newton’s doubts unanswered. Let me show why the nonradial potentials do not and cannot directly affect the usual radial forces of gravity.

5. Mathematical meaning of nonradial potentials If masses are treated as vectors then forces emerge from foreign images (representations) of dot products of collinear vectors of the potentials [13] jFj ¼ G½ðiVM ðMÞÞ ðiVm ðmÞÞ ¼ GMm=r2 ;

ð5Þ

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where the VM ðMÞ ¼ M=r and Vm ðmÞ ¼ m=r are the respective (local) potential functions at distance r from masses M and m, and Ôi’ is the imaginary unit. This is new formulation for abstract multispatial hyperspace where the local potentials are foreign geometrical objects with respect to the force F [13]. The point is that masses can vary in an abstract 3D massbased linear vector space that belongs to a 9D multispatial hyperspace (MH). Minus sign from i2 is cancelled by dot products of coordinate versors (i.e., unit vectors j, k, l). Eq. (5) shows that the dot product of any two foreign radial potentials (i.e., represented in a foreign LVS) generates force in the primary LVS [13]. Now imagine that a photon moves along an equipotential trajectory close to the sun’s surface. In the radial direction toward the center of the sun a radial attractive force is created. But along equipotential trajectory foreign ‘‘work’’ vector results from dot product of sun’s tangential potential and the photon’s radial potential. For single point-like particle practically does not possess its own nonradial potential, for its surface is practically nonexistent. Hence the photon does carry only its radial potential, even in the nonradial directions with respect to the sun. Therefore the dot product of the tangential potential of sun and a radial potential of the point-like photon virtually gives work (or energy) instead of force. Taking Eqs. (4) and (5) with tangential potential VM ðMÞ of the sun and radial potential of the photon Vm ðmÞ we obtain work jWj ¼ G½ðiVM ðMÞÞ ðiVm ðmÞÞ ¼ GMmq=r2 ;

ð6Þ

which does affect energy of the photon [2], but has no direct impact on the sun’s radial force of gravity. We should not mix proverbial apples with oranges. Experiments confirmed the energy change [2]. From this point of view the two radial experiments [39,40] did not really disprove nonradial potentials, but confirmed their complementary nature. These and the like experiments actually showed that nonradial potentials do not directly affect any radial ones, which fact was never contested. Experiments to re-verify certain scientific beliefs are always desirable, but no radial experiment can ever prove nonexistence of noninvasive nonradial or mixed effects. Newton showed how to measure the values (quantities) of forces, but he did not define force in a structural way. Eq. (5) supplies thus structural definition of forces as geometrical objects [13] in addition to their physical representations. The fundamental component of physical field is potential whereas forces and energy are really secondary, compound magnitudes. We can talk thus about two kinds of physical fields that would correspond to the distinction of interior and exterior operational magnitudes in the very sense of Elie Cartan [1,13]. Sustainable field, like that of the basically static radial potential always exists. Though static by nature of its source, the nonradial potentials, just like electromagnetic ones, are dynamically persistent. I will compare these types of potentials elsewhere. Fields of forces, however, are vectorized scalar products of two potentials––they are dynamically created [13]. I am using the term Ôpersistence’ in its usual meaning, because it also pertains to the object-oriented approach that is used in computer sciences. I have called Ôforeign’ those vectors that cannot be represented in the given space’s primary coordinate system [13]. Eq. (5) emerged from abstract mathematical investigations. There are logical and physical reasons for its validity. It suggests existence of mass-based space (MBS) that would raise the number of geometrically identifiable dimensions to 9 within multispatial hyperspace mass–length–time (MLT) of CM [13]. The East–West neutrino asymmetry [41] may also be explained by it. By decreasing energy of particles near surfaces of large bodies [2], the NEG could cause mass-dust to conglomerate and form stars and/or planets. Energy decrease together with diverting their trajectories makes capture of slowing dust particles feasible. Apparently we got two distinct kinds of physical fields. For the potentials’ functions are actually functionals of physical field, when they emerge from linear integration of their one-directional derivatives. The radial directional derivative DV ðrÞ ¼ GM=r2 determines radial acceleration, which it is second time derivative of function of motion. It points thus to presence of certain geometric object that sits higher in the hierarchy of dimensions than the radial potential itself. Since the effective geometric dimensionality of such a higher object should be greater than three, the higher-dimensional object may appear as an exterior structure that encapsulates timeless 3D objects as its inner (interior) components. The object-oriented approach to geometry is instrumental to our comprehension of abstract hypergeometric structures. From purely algebraic point of view the combined effect of all the objects does resemble tensor. But tensor representation alone is insufficient for true representation of very complex structures of higher-dimensional geometric objects [13]. Although tensor calculus was developed with a generalization of vectors in mind, it delivered an oversimplification of geometry––or just a postulative algebraization of certain geometric issues, which have not been understood in their entirety. One can treat point-set manifolds as spaces, but such a treatment does not turn its points into physically meaningful vectors. The generalized potential function U ð Þ is an exterior potential, for it does correspond to an exterior object in the sense of Elie Cartan [1,3,13,42–46]. Its components can be seen as interior potentials. Since all three distinct directional derivatives should exist, then the exterior object should possess at least six quite distinct dimensions, where each triple must belong to one of two mutually dual 3D LVSs of a MH. Hence any physically meaningful geometric dimensionality

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is evidently quantized [1,3,13]. One may add only whole 3D LVS in order to increase the number of physically meaningful dimensions, but one must not add just one or two dimensions to a problem considered in object-oriented terms. Dimensions come in triples [13]. The interior field of force does determine only the radial linear directional rates of change of energy and momentum. Hence the nonradial potentials should be present for the sake of the geometric completeness of the whole physical picture. Having three quite independent directions in any 3D LVS, we should explore all three components of such a generalized potential, not only the radial one, which Newton’s theory came up with. This feature is a mathematical requirement, or an abstract structural principle, if you will. No exact theory can afford to ignore structural or operational mathematical principles, which are deeper than any laws of physics, and still remain exact. Nonetheless, one could say that an action requires force. As true as it may sound, the NEG act quite differently from radial forces. NEG is an entirely new and unexpected physical interaction. We never anticipated such as if ‘‘shoulderrubbing’’ though quite noninvasive kind of interaction. Consider the ray from Taurus A that passed near sun on its way to earth [11]. The ray as if paid a toll fee (payable in physical currency convertible to energy) just for passing by. I could say that sun squeezed energy out of the ray. Yet sun was just a spectator of the ray’s passing by, so to say. But when the ray has been intercepted, its frequency was lower than it should be if the sun would not affect the ray [2,11]. Even if one would decompose the ray’s path into two subbranches, one being Taurus A fi earth and the other (with respect to the sun) Taurus A fi sun fi earth, then any radial impact of the sun on the subbranch Taurus A fi sun must practically cancel its radial impact on the subbranch sun fi earth [2]. Since the ray’s frequency determines its energy, then some–– other than radial––potential must have acted in order for the ray to lose some energy. Its energy on the two subbranches can be superposed [2]. NEG impose fundamental shift of physical and geometric paradigms. Newtonian theory of gravitation has implicitly required double abstract spatial structure LBS–MBS, even though this virtual requirement has never been acknowledged before [13]. The STR virtually required presence of a dual spatial structure LBS–TBS [1,3]. When merged together, these two requirements suggest an existence of a triple abstract multispatial structure MBS–LBS–TBS whose acronyms seem to extend the former meaning of the MLT system of CM onto MH [13]. Whether distance between stars is measured in meters or seconds or in light years, it is still the same distance. Einstein understood only one side of the codependency. For Einstein this was just an issue of physics shaping geometry, as if these two could exist in a master–slave kind of relationship. Starting tentatively from the Einstein’s point of view, however, one could see that even local geometry involves much more than just the shape of its objects, and its relationship with physics is actually very multifacted one. Geometry should be concerned with preservation of geometric objects under any abstract operations, while physics should control all conservation laws. Neither Newton nor Einstein fulfilled these requirements completely. Eqs. 5 and 6 contain structural operational template and its implementation in physics. Without abstract structure for scalar operations on vectors physical forces were just lucky guesses. Mathematics is not just a crutch for physics, but a guide in determining exact nature and scope of physical interactions. If a theory merely incorporates a subtheory, then it just combines the two without any actual generalization. The GTR generalized Newtonian theory of gravitation, but it only included the STR without a generalization thereof. Gravity maintains primary field of potentials and compound field of forces. GTR dealt only with the interior radial force field. It is criticized for lack of energy conservation [47–50], and for an incompatibility of momentum and forces [51]. The double structure of gravity field upholds GTR-like theories as great cosmological tools that were not designed for nonradial or mixed effects. On the other hand, however, entirely physical, theory of gravity that would respect the energy conservation law seems also necessary. If 4D spacetime is to be a derived concept, it must be based on something more fundamental [52]. This statement is true not only in the Kaluza–Klein theory (KKT). Even though the concept of spacetime is often associated with the GTR, it is local concept [1]. GTR was unable to predict results of many local experiments, including those with flying atomic clocks. These experiments need accelerations and tangential potentials on order to become reconciled [14,53,54]. Tangential potentials expand the spacetime concept onto hyperspace [1,3], however. They are mathematical objects and as such are not limited to gravity. Surface electric fields are also known [55]. In the hyperspace that comprises multiple 3D LVSs, one can see separate abstract symmetries. This is essential for the KKT, where gauge symmetries acting on internal coordinates are distinct and independent of the 4D spacetime symmetries that act on external coordinates––for they act on separate spaces [56]. When treated as single space, spacetime does not make sense, because time flow is not independent of length-based coordinates and corresponding to them dimensions [13,14]. These issues shall be discussed elsewhere. Actually Eq. (4) support the hidden message implicitly conveyed by the GTR that there is no physical gravity field and so quantization of gravity (understood as physical manifestation of spacetime) means quantization of the spacetime itself [57]. This statement could be taken almost literally, for multispatial hyperspace may comprise many single 3D spaces. Although bosons can be described in terms of classical field, this is quite impossible for fermions, for

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indeterminacy relation between amplitude and phase would violate Pauli principle [58]. Nevertheless, since density matrix (or wave function) represent observer’s knowledge of the quantum system in question [59], dual spatial representation in hyperspace gives us a chance to increase the very knowledge by possibly introducing additional relationships. The ultimate geometric objects are thus state vectors. In the QM no reciprocal action of particle on the wave exists [60]. Moreover, probability amplitude is a projection of one state vector onto another [61]. Although the wave function completely determines behavior of QM system [62], there are some degrees of the completeness [63]. As a matter of fact, bra hAj and ket jAi vectors are actually dual [64] and the duality complies with all the rules that govern geometric dual LVSs [65]. Hence the geometric duality that erects abstract multispatial hyperspaces [13] makes it feasible to beef-up the QM without going back to the concept of classical field, at least in principle. Although the concept of object apparently breaks down in the former QM, it does not have to be so [66]. Perhaps only single-space images of objects break down. We should come up with such an abstract notion of geometric object that would not break down. Although such objects may not always be directly observable, their actions should. Perhaps the QM’s dictum that definite states emerge when an observation is made is just an evasion [67]. Copenhagen interpretation (CI) of QM sounds like old ‘‘good’’ metaphysics. Perhaps the QM should not use purely syntactic systems created by the PM [68]. When a new theory creates apparent paradox or absurd without being inconsistent, it could mean that the assumed mathematical infrastructure of former phusics might have been somewhat oversimplified. Hence instead of attacking it we should ask what would the reality really look like, wherein the alleged absurd would make perfect sense. The reality, in which many apparent absurds of relativity, QM and CM disappear, reveals multispatial structure of hyperspace. Conceptual problems do not really disqualify those theories that raise them, as long as experiments back their predictions. But they indicate inadequate theoretical infrastructure, which usually means somehow deficient mathematics. We need SM to overcome that.

6. NEG may cause accelerating expansion of universe Recent observations hinted at accelerated expansion of the universe, but there was no universal mechanism for that. Any expansion would require repulsion, but repulsion alone would not suffice for logical explanation of what is observed, namely that each galaxy apparently recedes from each other at an increasing speed. One needs both an attraction and a repulsion that would not be cancelled by each other, in order to account for such an accelerated expansion. Moreover, the ‘‘engine’’ that causes the universal repulsion should somehow increase its impact with increasing distance and be of intrinsic, global rather than local nature. The latter three requirements effectively disqualify thus any local repulsion that is fueled by pressure. Former physics entertained negative pressure as the source of gravitational repulsion [69–71]. However, it does not fulfill the aforesaid requirements. An aircraft can take off from an airport, but it cannot cause other airplanes to recede from each other as well. We need a global repulsion that is also as universal as the usual attractive force of gravity is. No interaction known to former physics fits the description we are looking for. Nevertheless, a very long-range repulsion that would not disturb long-range attraction seemed impossible to devise in former 1D, radial world of gravity. Directional derivative DV ðrÞ ¼ GM=r2 of global radial potential V ðrÞ ¼ GM=r of constant source mass M determined the radial, M-centered source force field in Eq. (1). Eq. (4) have been defined near surfaces of massive bodies, so that both mass and specific gravity were assumed as practically constant in that setting. Eq. (4) have been experimentally confirmed only for near surface phenomena [2]. In general, however, both mass and specific gravity can be functions of distance too. Even in apparently empty space there is some mass scattered. So if volume of ‘‘greater sun’’ would extend to the earth, for example, its total mass would increase whereas its specific gravity would decrease. At great distance from massive body M, varying densities of mass and matter would change the whole impact of its potentials, especially the long-range effects of its nonradial potentials. In general terms, the total radial directional derivative is now defined as DV ðr; MðrÞÞ ¼ oV =or þ oV =oM ¼ GðM=r2  1=rÞ:

ð7Þ

If the total directional derivative for the tangential potential H is defined as DH ðq; r; MðrÞ; QðrÞÞ ¼ oH =or þ oH =oQ þ oH =oq þ H =oM;

ð8Þ

then total nonradial directional derivatives from Eqs. (4) and (8) are as follows: DH ðq; r; M; QÞ ¼ Kq GðMq=Qr2 þ Mq=rQ2  M=rQ  q=rQÞ;

ð9aÞ

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DZðp; r; M; QÞ ¼ Kp GðMp=Qr2 þ Mp=rQ2  M=rQ  p=rQÞ:

695

ð9bÞ

Derivatives of nonradial potentials supply thus two extra Q-driven factors in them, which I will call nonradial contraction rate derivative henceforth DCðQÞ ¼ GMðqKq þ pKp Þ=rQ2

ð10Þ

and also two clearly repulsive components of the total nonradial derivatives, which I will call repulsive expansion rate derivative henceforth DEðq; pÞ ¼ GMðKq þ Kp Þ=rQ:

ð11Þ

The repulsive expansion rate and the nonradial contraction rate are new, long-range effects. They would emerge––as significant factors––at distances where the attractive force has practically vanished. I have silently assumed that the mass increase due to dust is insignificant so that Eqs. (10) and (11) are only approximations. The MðrÞ and QðrÞ are actually certain functions now. Although the repulsive expansion rate derivative acts radially, it is an effect caused by the nonradial potentials just as the nonradial contraction is. As one can see the combined (two-directional) repulsive expansion rate function Eð Þ and the combined contraction rate function Cð Þ have built-in break (1=r) that slows them down and a built-in accelerator (1=Q) that slowly increases over distance. The radial gravitational attraction diminishes very rapidly with distance as (1=r2 ). At some point, the gravitational repulsion overcomes gravitational attraction and the tiny repulsion and contraction effects become the dominant effects of gravity at very large, presumably intergalactic distances. This point is so remote from the gravity center, that repulsion can be observed only for agglomerations of stars, such as galaxies. Note that galaxies do not really have definite boundaries, but the specific gravity Q can always be computed for larger and larger volumes as long as it does not infringe onto an inner volume of another neighboring galaxy. For most practical purposes, the inner volume of every galaxy can be delimited thus by the surface on which gravitational repulsion overcomes the radial attraction of given galaxy. Beyond the inner volume, the repulsion fuels an accelerating expansion. The contraction Cð Þ may explain clustering of stars into galaxies while the repulsive expansion Eð Þ could explain an accelerating expansion of the universe as well as the lack of superclusters in the universe. The repulsive effect of the nonradial potentials as shown in the expansion rate function Eð Þ is locally very tiny [2]. However, it grows with increasing distance. Since galaxy is made of billions of stars and a lot of practically empty space, the intrinsic gravitational repulsion grows with diminishing specific gravity Qð Þ. The balancing role of specific gravity Q has been experimentally confirmed [2]. There is nothing postulative about it. The specific gravity effectively normalizes gravity by virtually introducing as if volumes into the equations. Eq. (11) gives us very simplified picture of repulsion from a spheroidal, single-star ‘‘galaxy’’ with uniform distribution of mass and matter, of course. Yet this means that gravity not only attracts local matter, but it also forms its local agglomerations by compressing them, and then protects them from destruction by repulsing other agglomerations.

7. Summary and conclusions The presence of nonradial effects of gravity is required by existence of the usual radial potential as well as by rules of differential geometry. It does not exclude an existence of dark matter and energy [72]. An impact of nonradial effects was experimentally confirmed for local near surface phenomena [2]. The most important effects implied by nonradial potentials are as follows: • frequency decrease in the rays that go near surfaces of large massive bodies––a tiny, very short-range effect; • long-range gravitational contraction; • very long-range gravitational repulsion. Although the two latter effects are hypothetical, they have been derived from experimentally confirmed and mathematically necessary equations. The frequency decrease does not pertain to the source of gravity, as in [73], for instance, but strictly to rays running nonradially, across a gravitational field. The new effects do affect energy and momentum and as such are not comparable to anything in the GTR, which has no classical limit because of its very essence [74]. These effects are intrinsic to the nonradial potentials and therefore their impact is global even though they are defined in terms of differentials in infinitesimally small surrounding of given center of mass.

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I was really perplexed by the fact that Newton hesitated for so long and effectively censored himself before deciding to publish his great Principia. Without being overly suspicious, and with no doubts whatsoever that his splendid work was the best approach to gravity made in terms of forces, I was nevertheless a little bit suspicious that there may be something very troublesome behind his hesitation. Although I never cracked the cold case directly, when I realized that my inability to reconcile the discrepancies between predicted and observed time rates of the flying atomic clocks by any motion-related means [53] could imply an existence of some basically static, but other than radial potentials, I was almost sure that I suffered from the same kind of theoretical ‘‘influenza’’ as Newton presumably did. I would have to start almost from scratch just as he did, while always staying aware of implications of the GTR, which was the best approach to gravity in terms of curved spacetime, back then. The task was daunting and the atmosphere very tense and discouraging. We should not be afraid of being wrong while thinking or perhaps embarrassed by having some ‘‘dark’’ theoretical doubts, however. He, who has no doubts, has ceased to learn. We should tolerate doubts if we want to make progress. Otherwise we will continue to repeat old, tentative truths along with unrecognized mistakes. Even some allegedly eternal mathematical truths are merely tentatively valid propositions [13]. Newton’s suspicion that his theory of gravity may be incomplete proved true. This incompleteness does affect all former theories of gravity, because they all were radial and the incompleteness was in regard to some nonradial effects. Yet at the root of his dilemma was deep mathematical issue. He has sensed it, for he rebelled against the static picture of reality provided by the Cartesian analytic geometry [75]. It does not mean that Cartesian analytic approach to geometry is wrong, for it is not. It is oversimplified vision of geometrical issues that poses problems. The oversimplification makes some derivations from its assumptions conceptually unreliable. Des Cartes rightly deserves credit for ingenious algebraization of geometry. But algebraization should not degrade the fine granularity of geometrical thinking, for it would result in a reductionism that may eventually lead to oversimplifications. As elegant as it is, algebraic geometry is only admissible as a complementary approach to geometry––not as its substitute––and they both must agree [13]. Newton understood, I think, that he should restrict his splendid theory in order for it to stay afloat. His first definition of Quantity of Matter––as a measure of matter that arises from density and volume jointly [76]––makes it clear that he realized that one must neutralize volume. At that time, there was no unambiguous distinction between matter and mass––the confusion still persists [2]. This definition and the whole Scholium of his Principia, is a fantastic scientific performance, given the evidence he had at hand. But he was obviously troubled by the outstanding question: if I am successful and yet incomplete, then what did I leave out? He might have hoped that this unanswered issue will be resolved some day, and I am not contesting his tranquilizing thoughts. As with all tranquilizers, they create a problem when overused and thus cause stupefying effects. I do not want to sound like Karl Popper, but one may safely assume that when a theory fits too well today’s perception of reality, which is always incompletely understood, then it may be simply wrong. One can derive any local predictions about concrete facts from it, but to prove or disprove theorems about the (surrounding us) reality from it is dangerous. Only resolution of logical conflicts can give insights about the reality. Experimental data should be used to confirm the predicted consequences of those conjectures that were built upon these insights. What seems obvious today must not be used against any mathematically necessary conclusion that results from a tentative resolution of logical conflicts. Newton’s great physical instinct and his mathematical intuition still inspire us today. His tacit doubt about the source of gravity was correct and his fear of quite possible incompleteness of his radial approach has eventually been vindicated. Had his theory been in error, he would probably have solved it. But the riddle that took 20 years of his life was far greater, for it touched the structure of physical reality. Contemporary mathematics tried to avoid such questions by any means possible, including obstruction of truth and a code of silence that virtually censored its own greatest achievements [13]. But unsolved problems do not really go away when we just stop pursuing them. Sooner or later they will resurface again. It is important to follow up on the old unsolved problems, for they may hold keys to future theories. I am not advocating cautious hesitation that could unwisely postpone publication of otherwise elaborate ideas. Quite on the contrary––it is extremely important to push forward theories without waiting for their completeness. It is the arrogant suppression of opposite and/or alternative views that often derailed progress in development of new physical theories. Newton’s hesitation is an example of an honest pursuit of mathematical truth for most of his adult life. We became slaves of our limited comprehension and the use of language. The way Newton has defined absolute (‘‘true’’) motion advanced him over ancients [77], but at the same time crippled his understanding of motion and consequently gravity, I suppose. The concept of force was seen as cause of motion in ancient and medieval mechanics, whereas Newton changed it to mean a cause of change of motion [78]. Yet for orbiting astronaut who feels no force, gravity in terms of curvature rather than forces makes more sense [79]. Although astronomer Halley finally persuaded Newton to publish his Principia [80], he was looking for a second astronomical opinion [81] before publishing its second edition. It seems that Newton was reluctant to leave his doubts unanswered. He was ahead of his time when it comes to

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physics and mathematics. For just like myself, he also believed that a well-founded geometrical proof should possess a synthetic rather than an analytic form [82]. I was too lazy to get that from him, so I have discovered it on my own. Newton might have struggled with various kinds of forces, because he had distinguished innate (inserted or implanted) force [83]. But unlike Berkeley and others who could not imagine infinitesimals or idealized notions such as instantaneous velocity and ridiculed them as ‘‘ghosts of departed quantities’’ [84], he was definitely not afraid of using physically explainable (i.e. one that can be interpreted in physical terms, such as fluxions) mathematics to its fullest. Therefore I tend to think that Newton was certainly aware and somewhat disturbed by what he might have perceived as incompleteness of his concept of force, and consequently, as perhaps just a little bit flawed his geometric treatment of gravity. However, his contemporary astronomical data was rather imprecise by today’s standards and experiments that would allow him to seriously reconsider his theoretical methods would certainly require atomic clocks that could measure discrepancies in nanoseconds. Although I do not really see any way out of the Newton’s dilemma before modern times, I think that truly open, uncensored talks about doubts like his could have accelerated at least the development of theoretical physics and could have prevented alienation of mathematical methods from its physical roots, and perhaps the degeneration of mathematics could also be avoided. Failure of Almagest could have been avoided by switching just one single answer from No to Yes. Ptolemy ridiculed the logical suggestion that earth may revolve about the sun, for indeed all objects not actually standing on the earth would move in opposite direction just as stars do [85], without the Newton’s force of gravity. Ptolemy’s single overconfident though logically incomplete answer wasted thus almost two millennia of doing sciences. Existence of NEG implies very strange world, in which nonlocality seems quite natural. In fact, twin particles separated in one space may still appear close together in a space dual to the primary one [3,13]. Hence if their state vectors would span both the primary and its dual space, then there would be no need to send any superluminal (or even a subluminal) signal between the twins, for these twins would virtually always stay connected in either the primary or in its dual space (channel). Such twins are perhaps not objects, but their representations in each of the dual spaces. For a representation (or an image, if you will), being in two places at once seems not impossible––at least in Colorado. A new way to imagine a physical object as being in two different places at once has been investigated by El Naschie [86]. Having made appealing case for the possibility that single point in 4D space may be perceived as yet another 4D space [87] that is embedded within itself, he upheld thus his idea of conjugate complex time, which can become the new backbone of future geometrical physics that could be founded upon abstract generalization of the principle of projective duality [3,99,100]. On the micro scale, the imaginary time can mean (physically) that spin of particle corresponds to rotation of its temporal trajectory about an observer’s time axis [88]. From an abstract mathematical point of view, however, it can be seen as geometric decomposition of time flow in a dual space to the primary space of motion [1,3]. This result was based on El Naschie’s approach to complex time and Cantorian spacetime [89–93]. Time behaves spatially just like length-based quantities even though it is not declared as a parameter in geometry. Elie Cartan successfully deployed the method of moving reper as an independent frame [44,94], which shows time as a counterpart of the usual length-based 3D space [3]. Any physical theory must solve the issues of time flow and the structure of physical fields just in order not to waste time, if for no other reason. Also any physical theory must comply with differential geometry. None of former theories of (radial) gravity solved issues of time flow to my satisfaction. All the former radial theories were also in blatant contempt of differential geometry, as far as I can tell. We need an overhaul of physics. Newton’s gravity was defined in terms of attractive forces. The Einstein’s GTR was conceived in terms of curvature of spacetime and defined in terms of pseudopotentials of an amalgamated gravitational field. Among all other ingenious theories of gravitational interactions, the Relativistic Theory of Gravitation (RTG) and Yilmaz Theory of Gravity (YTG) should at least be mentioned, if not discussed. Some flaws of the GTR are discussed in [95] as well as in publications endorsing other theories of the GTR-class. By the GTR-class I mean tensor-based theories with either more than 3D geometric spaces or with time assumed as a complementing dimension within a single space. The RTG is very meaningful cosmological theory [96,97] and so is probably the YTG if one is afraid of black holes [98]. However, by late 1930s the era of unbridled, postulative mathematics abruptly ended, even though many mathematicians and physicists did not notice that it is over. Although many nimble works of Banach, Elie Cartan, G€ odel, Sierpinski, Slebodzinski and several other humble mathematicians have not been quite understood, sometimes not even by their creators themselves, postulative mathematics collapsed. Therefore also the physics that was founded upon the faulty mathematics must be reinvented. My papers gave examples of deficiencies in both mathematics and physics along with solutions to their accumulated problems. Relatively simple ideas of G€ odel and Sierpinski (fractals, continua) have been accepted and developed. But many advanced results still wait to be reinterpreted. Although physical world was designed and created for us to live in it, apparently its proper comprehension was not set as the prerequisite for its inhabitation, or we would likely end up in hell.

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