SUPERLUMINAL INTERACTION, OR THE SAME, DE BROGLIE RELATIONSHIP, AS IMPOSED BY THE LAW OF ENERGY CONSERVATION PART II: GRAVITATIONALLY BOUND PARTICLES

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1 SUPERLUMINAL INTERACTION, OR THE SAME, DE BROGLIE RELATIONSHIP, AS IMPOSED BY THE LAW OF ENERGY CONSERVATION PART II: GRAVITATIONALLY BOUND PARTICLES Tolga Yarman Okan Uniersity, Akfirat, Istanbul, Turkey ABSTRACT Preiously, based on just the law of energy onseration, we figured out that, the graitational motion, depits a rest mass ariation, throughout. Consider for instane, the ase of a planet in an ellipti motion around the sun; aording to our approah, an infinitesimal portion of the rest mass of the planet, is transformed into extra kineti energy, as the planet approahes the sun, and an infinitesimal portion of the kineti energy of it, is transformed into extra rest mass, as it slows down away from the sun, while the total relatiisti energy remains onstant throughout. The same applies to a motion drien by eletrial harges; this onstituted the topi of the preeding artile (Part I of this work). One way to oneie the mass exhange phenomenon we dislosed, is to onsider a jet effet. Aordingly, an objet on a gien orbit, through its journey, must ejet mass to aelerate, or must pile up mass, to deelerate. The speed U of the jet, strikingly points to the de Broglie waelength λ B, oupled with the period of time T, inerse of the frequeny ν, delineated by the eletromagneti energy ontent hν, of the objet; is hν originally set by de Broglie, equal to the total mass m of the objet (were the speed of light taken to be unity). This makes that, jet speed beomes a superluminal speed, U = λ / T = /. This result seems to be important in many ways. Amongst other things, it may mean that, either graitationally interating marosopi bodies, or eletrially interating mirosopi objets, sense eah other, with a speed muh greater than that of light, and this, in exatly the same way. In whih ase though, the interation oming into play, exludes any energy exhange. Thus energy annot of ourse go faster than light, but information an be arried without any basis of energy. Furthermore, our approah, indues immediately the quantization of the graitational field, in exatly the same manner, the eletri field is quantized. Our onjeture may be heked with isolated eletri harges, suffiiently far away from eah other; as unusual as it may seem, they should interat with eah other, right away, eletrostatially, or magnetially. Whereas it remains diffiult to apply the present idea to ommuniation faster than the speed of light (essentially for, the fore of onern, dereases as the square of the distane), it appears nonetheless plausible to implement it, to omputation faster than the speed of light. B 1

2 1. INTRODUCTION In the first part of this work, along our approah, we hae elaborated on the motion of two eletri harges is-à-is eah other. More speifially, we hae onsidered the losed system of an eletron rotating around a proton. 1 The proton being muh more massie than the eletron, we supposed that, as they unite, to form the hydrogen atom, the proton remains untouhed. Hene, the eletron s internal dynamis weakens as muh as the binding energy oming into play. The bound eletron should then weigh less than the free eletron, and this, as muh as the binding energy. Reall that this onsideration led us to a new equation of motion, followed by a new relatiisti quantum mehanial set up.,3 The weakening of the internal energy of the eletron would be hard to hek; nonetheless one an well erify that the bound muon deay rate retards as ompared to the free muon deay rate, and this, as muh as the binding energy oming into play. 4,5,6,7,8,9 Our predition about this ourrene, in effet, remains better than any other aailable preditions. 1,11 Note that our approah, is in full ompatibility with all existing quantum eletrodynamial data. In Part I, based on the framework we hae set up regarding the eletri interation, we were able to derie the de Broglie relationship, thus through essentially the law of energy onseration. The tools we used on the way, points to a striking result. It is that, the eletri interation an take plae with speeds muh higher than the speed of light (without howeer any exhange of energy, between interating harges). For stati harges, the interation seems to be spontaneous. In this part, we are going to handle, the graitational interation. In fat, at first, we had deeloped our approah, mainly, to desribe the graitational motion, more speifially, to desribe the end results to the General Theory of Relatiity (GTR). 1,13 Thus, as a result of our approah, not only that, we are able to derie the de Broglie relationship, is-à-is the eletri interation, taking plae within the frame of an atom, but as unusual as this may be, we an well land at it, within the frame of the graitational interation, too. In a way though, this should be onsidered quite natural, owing to the exat similarity between Coulomb s law (written for eletri harges), and Newton s law (written for masses); then again, reall that, we onsider these fundamental laws for stati objets, exlusiely. We ould add this Part II, to the preeding Part I. Yet, first of all it is already oluminous. More importantly, it does not seem easy to modify, well established beliees, swiftly.

3 Thus, it seemed more appropriate to allow the interested reader to think about the approah we presented, and try it, on the basis of the deriation of something he knows and aepts well, i.e. the usual de Broglie relationship, arising within the frame of the miro atomi world, and only then, take him to onfront with a similar finding, now deried within the frame of the maro elestial world. This is how, we hoped to oerome, the great surprise, and onseratie reations, many of us will most likely exhibit, gien that, ia the approah we present herein, muh of our atual oneption, would somehow be stunned, and perhaps modified, had our results turned out to be loser to nature s behaior. Nonetheless, already at this stage, it turns out quite omforting that, our approah appears to restore the broken or non-existing links, or annoying inompatibilities, between different disiplines of atomi physis, quantum mehanis, relatiity, and elestial mehanis, whereas, it is the desire of all of us to establish just a whole omplete pakage of oneption for the unique nature existing out there, whether it is question of miro aspets, or maro aspets, of it. Thus, our approah seems to reinstate the broken link between the end results of the GTR, and Newton Mehanis. Though, it indiates that Newton s law of graitational attration is alid for stati masses stritly, just like the Coulomb s law is only alid for stati harges. In other terms, the graitational mass entering Newton s law of graitational attration, is not the inertial mass, entering the Newton s desription of fore, more preisely, [fore] = [inertial mass] x [aeleration], i.e. the Newton s Seond Law of Motion. These two masses turn out to be different;,3 it is that [graitational mass] = [inertial mass] / [Lorentz dilation fator]. This result is eidently, against the phrasing of the priniple of equialene (PE) of Einstein (whih states that the two masses of onern, should be equal). Before we present our stand point, we feel the need of elaborating on this point. Experiment Ahieed at the General Physis Institute, in Mosow, Supporting the Derease of the Graitational Mass with eloity At any rate, the aboe relationship suggests that a moing partile in a graitational field would weigh less than the same partile at rest, in the same loation in that field. V. Andree effetiely reported at the PIRT Conferene held in July 5, in Mosow that, a pendant load irradiated at the General Physis Institute of the Russian Aademy of Sienes, by high energy eletrons, omes to weigh less than its untouhed twin ounterpart. 14 The author of the present atile, right after Andree s presentation, suggested that, the effet must be due to energizing, the alene eletrons of the atoms of the load in onsideration (whih happened to be duraliminium); these eletrons (based on our finding is-a-is the graitational mass and the inertial mass), beome pratially weightless. A quik alulation indeed proes this point of iew, whih shall be elaborated on, in a subsequent artile. The onlusion is that, heated eletrons weigh less than normally bound eletrons, and this, points to a lear iolation of the ommon interpretation of the PE. 3

4 It is not that we Question, or we Do Not Aept the Priniple of Equialene (PE), It is that we are in No Way Bound to Utilize it! Anyhow, through the approah that will be presented herein, we do not at all need to make use of the PE. It is not that, at this stage, we question the PE. It is that for the present theory, we are in no way bound to utilize it. Whether it is alid or not, we do not really bother, with suh a question. After all, to us, it represents a prodigious and useful analogy, but this is not really the point. It happens that we still attrat seere reations, when we say we do not need the PE. Some people think we must need it (and, as we will soon see, we really do not need it). Aording to these people, it is further silly to question whether it is right or wrong; it is surely right and no doubt we need it, they laim. Here is a joke eluidating this situation, the author has made with a ery fine sholar and a ery lose friend of his, who firmly adoated this priniple, and ould not aept the fat that we do not need it, whereas eery one else needs it: - All right, Dear Camarade, but you did not need any other woman than your wife, for the hildren, she gae you. It is not that you ould not hae hildren with another women; it is simply that, you did not need any other woman for the hildren, you had with her. This is exatly what we mean, by stating We do not need the PE to establish our approah. The fat that we do not use it, does not, on the other hand, onstitute any ontradition with the theory of relatiity, more preisely with the Speial Theory of Relatiity (STR), whih onstitutes our main framework. Quite on the ontrary, our approah, as will be elaborated below, remedies the inonsistenies, otherwise oming into play. In any eent, in order to dissole the upset we unwillingly reate regarding the PE, this priniple ought be analyzed and larified. A Disussion About the Ambiguity of the PE, as Defined by Most Authors: Tuning of the Definition of the PE, for the Present Purpose As we hae just stated; aording to Newton, the graitational mass, is the mass to be plugged in the expression of Newton s law of graitational attration, whereas the inertial mass is the mass to be plugged in the expression of Newton s Seond Law of Motion. Newton himself, before anyone else, doubted about the equality of these two masses (and we find indeed that they are not equal). As many ery many authors openly state, the PE appears to state that these two masses are equal. 4

5 But the GTR rejets the onept of fore whih, in Newton s desription made of mass and aeleration, inoles the inertial mass ; the GTR also rejets the onept of Newton s law of graitational attration, whih inoles the graitational mass. Thus the original ground onsidered by Newton, to appraise the two masses in question, is totally wiped out. On the other hand, inertia, by definition, is a harateristi, an objet deelops as a resistane to any ariation, in its motion. Thus the onept of inertia is assoiated with the hange, the objet undergoes, through a gien motion. Regarding the PE, the motion of onern is that of the aelerated eleator, as assessed by an outside obserer, fixed in regards to distant stars. Thus, Einstein onsiders the rest mass of an objet, lying on the floor of the eleator, aelerating (as assessed by the outside fixed obserer) upward (along the diretion drawn from the floor to the eiling of it). This is what onstitutes the basis of the PE. Then, the effet of aeleration on the objet of onern, is onsidered to be equialent to the effet of graitation (yielding the aeleration, in question). This indeed seems to be a striking, but as we show elsewhere, an inaurate analogy. 15,16,17,18 Anyway, by analogy, so far we only desribe a rest mass sitting in a graitational field. We do not desribe a mass in motion, in a graitational field (suh as a planet around the sun, as originally isualized by Newton, in regards to the equialene of inertial mass and graitational mass). Through Einstein s analogy, in order to desribe a mass in motion in a graitational field, we hae to go bak to the inside of the aelerated eleator, and see what happens to the objet originally at rest on the floor of this, say, if it is thrown, in a gien diretion. Only the omparison of the rest mass of the objet originally lying on the floor of the eleator, and the relatiisti mass of this objet brought to a gien motion inside the eleator, in regards to a gien obserer (to be speified), an (ia Einstein s analogy) bring an answer to the question introdued by Newton, regarding the graitational mass, and the inertial mass ; it is obious that these are not either equal within the frame of the GTR. Thus anyone, aring about the PE should not really be frustrated right away, when we say that the graitational mass and the inertial mass are not really equal. It is that they are not equal in the way Newton first questioned. Unfortunately there are quite a number of books, whih are ineitably misleading when they introdue the PE, through Newton s original question, and they aim to proide an answer to it in a domain where Newton s original tools are demolished by Einstein. We would like to add two major points. 5

6 Few Words About the Highly Aurate Measurements Worshiped With Regards to the PE From Our Stand Point The first one onsists in the reasonably preise measurements endorsing the equialene of the, so to say, inertial mass and graitational mass. (We say so, beause as explained aboe, the wording of suh an equialene, in many books, if not unaeptable, is misleading.) Thus, we are, by reiewers and friends, seerely asked the following question: - How an you (if not deny), but still dare to bypass suh a trustable result? As useful as it may be, for us, the PE is of ourse well thought-of, but still like eletroni tubes, that is bound to be forgotten, when replaed by transistors. It is that we do not need it anymore, within the frame of the present approah. One again, it is of ourse, monumental, it sered a lot, yet please we would like the reader understand it, we do not needed anymore We are truly sorry for those who are fond of it, but we hae happened to show that the law of onseration of energy does all the job. That is the way it is, and we will a little bit more elaborate on this, below. It is true that our findings puts at stake the PE, but this priniple is already, srupulously, questioned. 19, The equality of the graitational mass and the inertial mass, based on the approah presented herein, is an approximation whih is aeptable, only if the eloity of the objet in motion is small, as ompared to the eloity of light in empty spae. It is anyway interesting to note that, all the highly preise measurements regarding the relatie diergene of these two masses (how eer they are defined), are performed on Earth (where the obserer is moing with Earth), so that the preision they produe, no matter how fine this may be, aording to our approah, should be onsidered, as misleading. In effet, sine along our finding, [graitational mass] = [inertial mass] / [Lorentz dilation fator] (the aboe relationship), it seems that one should not rely on the experiments in question, any more then he should ount on the null result of the Mihelson Morley experiment 1 (whih, being performed on Earth, fails to detet the motion of Earth around the sun, or else). In other terms, the Galilean Priniple of Relatiity, whih is the main ingredient of the STR, forbids that we an on Earth, detet any suh differene, based on the eloity of motion in question (sine otherwise we should be able to tell, how fast we are ruising in spae, and we annot). Not knowing that, the equality of graitational mass and inertial mass, is only approximate, one may still insist (just the way it is done regarding the experiments in question) that, suh an equality an well be established on Earth. But the rotational eloity of Earth around itself is 1667 km/hour. Hene one should attain a preision of, i.e. better than /.6 x 1-1, whereas the highest preision reahed so far, is barely, this muh. 6

7 We an yet well rely, as Newton himself predited, on measurements based on a possible polarization of Earth and the Moon, through their motion around the sun. The polarization aording to our approah, must result from the fat that the speed of the Moon with respet to the sun, as the Moon rotates around Earth, aries slightly. The speed of Earth around the sun is about 3 km/s. The speed of the Moon around Earth is about.6 km/s. Thus the speed of the Moon with respet to the sun is about 3 km ±.6 km. The detetion of a differene in the falls of Earth and that of the Moon toward the sun, thus requires a measurement of the distane between the Earth and the Moon, with a preision better than (.6 km /s )/, i.e. ~ 4x1-1 3,4 whereas the preision atually reahed is still below this. Note further that, een through the fastest obserable elestial motions, suh as that of binary stars, around eah other (where the objets moe with speeds around 1 km/s), the differene between the graitational mass and the inertial mass, remains still undetetable. The Classial Approah, Deeloped on the Basis of the PE, is Somewhat Equialent to the Present Approah - Taking into Aount the Mass Derease due to Stati Binding - in Desribing an Objet Sitting in a Graitational Field The next major point we would like to bring to the attention of the reader, at this stage, is the following: The lassial approah deeloped on the basis of the PE, is somewhat equialent to the present approah, in desribing, the hanges an objet sitting (at rest) in a graitational field, would display. Indeed, following our approah, as will be summarized, below, an objet brought quasistatially into a graitational field, suffers a mass defiieny, and this, as muh as the graitational binding energy oming into play. The onept of binding energy is worked out in Part I, of this work, with regards to the losed system hiefly made of the pair of proton and eletron. The idea is exatly the same, with regards to the losed system made of the pair of a elestial body and an objet bound to it. Suh a derease of rest mass is, owing to the law of onseration of energy, broadened to embody the equialene of mass & energy drawn by the STR, nothing else, but a derease of the oerall rest relatiisti energy of the objet in hand. This yields, a weakening of the internal energy of the objet, at hand; thus the lassial red shift, whih is substantially, the same result yeld by the GTR. More generally, along with our approah, we onsider the stationary quantum mehanial desription of the objet in onsideration. Now, if we injet in this desription an arbitrary derease of different masses taking plae in there, and this, by the same amount, to represent the mass derease the objet would display in the graitational field, then quantum mehanially, we find that, the total energy of the objet, or the same, the eigenalue of the quantum mehanial desription of it, dereases just as muh. Thus, the red shift. Conomitantly, the size of the objet strethes, as muh. What we would like to stress here, is that, in fat suh a proedure, irtually ahiees, what the PE does, with regards to the objet sitting on the floor of the aelerated eleator, assumed to represent the graitational field. This is exatly why, we affirm that we do not need the PE, sine we are well able to get the end results of the GTR (up to a preision, atual measurements delineate), ia essentially, the law of onseration of energy. 7

8 There are though harateristi differenes between our approah and the GTR (although, as mentioned, the end results, are ery similar). The major differene is that the PE an be applied to an objet, subjet to a graitational attration, exlusiely, whereas our approah an be applied to any objet, subjet to any fore. This makes that, whereas the stik meter ontrats in a graitational field, only if it is lied, parallel to the attration diretion, in our approah, it is strethed just as muh, and this uniformly, in all diretions. (Reall that in the GTR, originally, only if the stik meter is lied along a diretion perpendiular to the graitational attration, as assessed by a distant obserer, it remains untouhed.) Furthermore, the mass of the objet in the GTR inreases in the graitational field, whereas in our approah it dereases. But then, all that with regards to the GTR, yields not only inompatibilities, and possibly a iolation of the laws of energy onseration and momentum onseration, but also bloks the assoiation of the graitational attration, with any other fore field, for one is unable to oneie a PE with regards to fores, other than the graitational attration (sine, there is no suh thing as eletri mass, or nulear mass, that an be isualized like the graitational mass ). It is furthermore important to reall that the GTR does not allow the quantization of the graitational effet. These problems do not really onstitute the topi of this artile. How eer, being targeted with onseratie reations, we are somewhat obliged to mention them. We are aordingly bound to present a lue as to what, differenes between the GTR and our approah our, although both theories, end up with pratially the same results. Clue to the Question of How Come that our Results are the Same as the End Results of the GTR, But Still Display Charateristi Differenes To make things easy, let us onsider a rotating dis. The PE, tells us that a lok nailed to the dis and rotating with it, is equialent to its twin sitting on a graitational ground, supposing that the entrifugal aeleration of the dis and the graitational aeleration, affeting respetiely both loks, are equal. The entrifugal aeleration of the dis and the graitational aeleration, separately, delineate respetiely the aelerational world whih we all AW, and the graitational world whih we all GW. Elsewhere 15,16 we hae proen that the lassial PE is based on a non-onform analogy, i.e. it does not embody a one to one orrespondene between the AW and the GW. Although it furnishes good results, it appears to be totally erroneous. For one thing, the PE oerlooks the rest mass defiieny the objet brought to the fore field should undergo, and this onstitutes a lear iolation of the law of energy onseration. 8

9 Reall that, Kündig, almost half a entury ago, measured the time dilation of an objet in rotation, and published results that were laimed to be firmly in agreement with the lassial predition. 17 Howeer, deidedly he has misproessed the data, 18 and the measured time dilation effet, turns out to be muh greater than that predited by the lassial theory. The remedial of the lassial PE, leaes it unneessary. We do not need it. On the ontrary we an straight, derie it, based on our approah. Thus, we state it like this: - Both graitation and aeleration alters the rest mass in the same manner; the rest mass dereases as muh as the energy neessary to furnish to the objet in order to remoe it from the fore field. Thus graitation is not a priileged field, at all. Any field, een that aused by a entripetal fore due to a rotational motion, does the same. Thene the graitational theory the author of the present artile, has deeloped, well oers, the end results of the GTR, and beyond, sine it turns out to be alid (not only for graitation, but) for all fields. It exludes any singularity, and treats the photon like any ordinary objet. The derease of the mass of the bound partile, ia our quantum mehanial theorem (we will soon reall), applied to the internal dynamis of the partile in hand, hanges both the period of time, and the size of spae to be assoiated with the internal dynamis in question, in exatly the same manner, at either an atomisti sale, or a elestial sale. Our approah yet, leads to a different metri than that of the GTR. Thus, briefly, the mass of an objet embedded in a graitational field (in fat, any field, it an interat with), is dereased as muh as the binding energy oming into play. (Aording to the GTR, it inreases.) The length of it, aording to our approah strethes, just as muh, and this uniformly. (Aording to the GTR, it ontrats, and this, originally, along the diretion of attration, only.) The period of time assoiated with the objet, dilates along our approah. (Aording to the GTR, it dilates just as muh, but also delineates a singularity.) How eer, our approah remedies the splitting between Newton s approah and Einstein s approah It restores the inompatibilities arising between the STR, and the GTR, suh as the breaking of the fundamental relatiisti result, [energy] = [mass] x [speed of light]. Along the same line, it does not allow the law of energy onseration, nor the law of momentum onseration to break down (ontrary to, as pointed out, what happens within the frame of the GTR). Furthermore, beause it leaes unneessary the usage of the PE, i.e. the basis of the GTR, it proides us with a whole different horizon. We ome to be able to establish a natural link between our approah and quantum mehanis, otherwise badly hindered by the GTR. In effet, the frame we draw, desribes in an extreme simpliity, both the atomi sale, and the elestial sale, on the basis of respetiely, Coulomb fore (written for stati eletri harges only), and Newton s fore (written for stati masses only). Thus the frame we draw, yields exatly the same metri hange and quantization, at both atomi and elestial sales. 9

10 In partiular, graitationally bound loks shall, aording to our approah, retard as implied by the amount of the graitational binding energy oming into play; thus not only this, but, loks anyway bound to any field, they interat with, should also retard. For instane, an eletrially harged lok, bound to an eletri field, must ome to slow down, just like loks bound to graitational field, slow down. More speifially, a muon deay s rate, when bound to an atomi nuleus, as pointed out aboe, retards as muh as the binding energy, oming into play. This shows that, the metri hange indued by protons nearby a nuleus is exatly the same as the metri hange indued by the mass nearby a elestial body. These happen to be the heartening harests of our approah. Most important of all, our approah (based on just the law energy onseration, broadened to embody the mass & energy equialene of the STR), allows, as we will elaborate throughout, the deriation of de Broglie relationship, just based on the law of energy onseration, with regards to the graitational field (just like we did it, with regards to the eletri field), thus indeed, the quantization of the graitational field (not any different than the quantization of the eletri field). This is what we will undertake in this artile. About the de Broglie Relationship In the preious Part 1, we hae formulated the de Broglie relationship as λ B = λ [Eq.(6) of Part I] ; (1) λ B is the de Broglie waelength; λ is gien by h ν = m ; () m is the rest mass of the objet in onsideration; is the speed of light in empty spae; ν is the frequeny of the eletromagneti radiation, were the objet in hand (totally) annihilated. By definition, we hae λ = = λν ; (3) T thus, T is the period of time of the wae oming into play. 1

11 Eqs. () and (3) lead h λ =. (4) m Diiding the two sides of Eq.(1), by, yields T λ T B λ = T = [Eq.(7-a) of Part I]. (5) Let us define for oneniene, the LHS, as U B : U B λ T B =. (6) Thus, ia Eq.(), beomes U B U B = [Eq.(7-) of Part I]. (7) Eqs. (5) and Eq.(6), tell us that, through a motion, suh as the Bohr rotational motion of, say the eletron around the proton, in the hydrogen atom, eah time the inherent periodi phenomenon of the eletron (assumed by de Broglie, and) desribed by λ, beats; something else too beats all the way through the stationary orbit of perimeter λ B. In other words the de Broglie waelength, beomes the waelength of the pulse of frequeny ν = ν / 1 /, propagating with the speed / * [f. Eq.(6-b) of Part I], where ν is the inerse of T. * Below, we first summarize our preious work, whih led to a noel equation of graitational motion (Setion ). Then, we desribe our jet model, simulating the graitational interation (Setion 3). Next, based on our approah, and essentially the law of energy onseration, we derie the de Broglie relationship, for graitational interation (Setion 4). A general onlusion, regarding both the eletri interation and the graitational interation, is drawn, afterwards (Setion 5). * = λ B ν = λ B ν 1. [Eq.(6-b) of Part I] 11

12 . PREVIOUS WORK: A NOVEL APPROACH TO THE EQUATION OF GRAVITATIONAL MOTION In a preious work,,3,1,13 a whole new approah to the deriation of the Celestial Equation of Motion was ahieed; this, well led to all ruial end results of the GTR(in a few lines only, and already in an integrated form). Thus, we had started with the following postulate, essentially, in perfet math with the relatiisti law of onseration of energy, thus embodying in the broader sense the onept of mass, though we will hae to preise, aordingly, the notion of field. Postulate: The rest mass of an objet bound to a elestial body amounts to less than its rest mass measured in empty spae, the differene being, as muh as the mass, equialent, to the binding energy oming into play. A mass defiieny onersely, ia quantum mehanis yields the strething of the size of the objet at hand, as well as the weakening of its internal energy, ia quantum mehanial theorems proen elsewhere, 5 still in full onformity with the STR. An easy way to grasp this is to onsider Eq.(). If the mass is dereased due to binding, so will be the frequeny. Thus the red shift. Eq.(4) makes that the size is aordingly strethed. In order to alulate the binding energy of onern, we make use of the lassial Newtonian graitational attration law, yet with the restrition that, it an only be onsidered for stati masses. Lukily we are able to derie the 1/distane dependeny of the graitational fore between two stati masses, still just based on the STR. 6 Thus, the framework in onsideration fundamentally lies on the STR. Heneforth, one does not require the priniple of equialene assumed by the GTR, as a preept, in order to predit the end results of this theory. m Let then be the mass of the objet in onsideration, at infinity. When this objet is bound at rest, to a elestial body of mass M, assumed for simpliity infinitely more massie as ompared to m, this latter mass will be diminished as muh as the binding energy oming into play, to beome m(r), so that 1 m(r) α(r) = m e, (8) (mass of the bound objet at rest) where α(r) is GM α (r) = ; (9) r 1

13 G is the uniersal graitational onstant; r is the distane of m(r) to the enter of M, as assessed by the distant obserer; reall that, after all, G is not as uniersal as one may think it is, sine we hae shown that it depends on the loation. 13 Note that, as Eq.(8) delineates, the present theory exludes singularities, thus blak holes. 1 Now suppose that the objet of onern is in a gien motion around M ; the motion in question, an be oneied as made of two steps: i) Bring the objet quasistatially, from infinity to a gien loation r, on its orbit, but keep it still at rest. ii) Delier to the objet at the gien loation, its motion on the gien orbit. The first step yields a derease in the mass of as delineated by Eq.(8). The seond step yields the Lorentz dilation of the rest mass m(r) at r, so that the oerall mass m γ (r), or the same, the total relatiisti energy of the objet in orbit beomes m m (r) γ = m(r) = m e α(r) ; (1) (oerall mass, or the same, total energy of the bound objet, on the gien orbit) is the loal tangential eloity of the objet at r. Here, the reason for whih we refer to the distane r, as measured by the distant obserer, is that, we onsider the Newton s law of graitational attration, speifially in that frame. 6 The total energy of the objet in orbit [i.e. m γ (r) ] must remain onstant, so that for the motion of the objet in a gien orbit, one finally has m e α γ (r) = m = Constant = m. (11) γ (total energy written by the author, for the objet in motion around the sun) 13

14 This is in fat, the integrated form of the general equation of motion, we will furnish right below; it is interesting to note that we hae arried at it, in just three lines [i.e. Eqs. (8), (1) and (11)]. The differentiation of the aboe equation leads to GM 1 dr = r d. (1-a) This equation an be put into the form [differential form of Eq.(11), leading to the equation of motion] or in the form G α r dr (t ) M m e = m γ, (1-b) r r dt α GM e r dr(t ) 1 r = 1, (1-) + α r dt [etorial equation written based on Eq.(1-a), or the same, equation of motion written by the author, ia the law of energy onseration, extended to oer the relatiisti mass & energy equialene ] written in terms of the proper quantities only, ia the relationship α r = r e α re, (1-d) (length strethed in the graitational field) as indued by Eq.(8), and a theorem proen elsewhere 1 [whih an be quikly ross heked ia Eq.(4)]; r is the etor bearing the magnitude, and direted outward. It should be realled that, though onsisting in a totally different set up, than that of the GTR, Eq.(11), thus Eq.(1-a) and (1-), amazingly yields results idential to those of this theory, within the frame of the seond order of the orresponding Taylor expansions, 13 yet in an inomparably easier manner. Eq.(1-b) is the same relationship as that proposed by Newton, exept that the graitational fore intensity is now dereased by the fator e α r. Note that our approah, just like the lassial Newtonian approah, leads to the law of linear momentum onseration, as well as the law of angular momentum onseration (ontrary to what is drawn within the frame of the GTR, where these lassial laws are broken). / 14

15 It is peuliar to show how our approah leads in general to the law of linear momentum onseration. Nonetheless it does not take long to show how it leads to the law of angular momentum onseration. One an see this readily, by multiplying the etor Eq.(1-), by the etor r. Thus the related ross produt beomes: d (t ) m γ(r )r x =. (1-e) dt [the ross produt of Eq.(1-b) by r ] The integration of this equation yields r = x (t ) Constant. (1-f) (Angular Momentum Conseration law deried from the author s set up) This result is important, sine it tells us that, our approah is quite ompatible with Kepler s laws. 7,8 Furthermore, we find it remarkable that within the frame of our approah, we are able to treat a light photon just like any other partile. Thus Eqs. (11), (1-a), (1-b) and (1-) are perfetly alid for a photon, just like any objet affeted by the graitational field. This has two interesting onsequenes: 1) The first one is that the photon as small as this may be, bears a kernel; de Broglie speulates on this point in his dotorate thesis, and alulates the rest mass of the 44 photon to be 1 grams, if an eletromagneti wae of waelength of 1 kilometer, moes with a speed, relatiely 1 times faster than an eletromagneti wae of 9 waelength of 3 kilometers. ) Seondly, yes, there is a eiling to the propagation of the eloity of light in empty spae, yet this is, the speed of a photon of pratially infinite energy. This is the Lorentz inariant eloity of light, whih enters our equations. At this stage it seems useful to draw the following table displaying the differenes between our approah and the standard approah. 15

16 Table Differenes Between the Standard Approah and Present Approah, Based on the Pair of Sun and Planet Standard Approah Present Approah Fore Between the Sun and the Planet, Altogether at Rest r m Mm G M α G r (as assessed, by the distant obserer) e Total Energy of the Statially Bound Planet GM m(r ) = m r (Newtonian Approah) m(r ) = m α (Einsteinian Approah) m(r ) = m e α Total Dynami Energy Rest Energy + Potential Energy + Kineti Energy (Newtonian Approah) The onept of potential energy, as onsidered lassially, is misleading. Total Dynami Energy of the Sun and the Planet in Motion m (r ) γ = m α m (r ) γ = m e α (The Einsteinian total energy) Fore Between the Sun and the Moing Planet G Mm r (Newtonian Approah) G M α m e 1 r 16

17 In order to irument onseratie reations, we better spell few words regarding the disrepany (as small as this may be), displayed by the raw before the last one, in the aboe table, between our result and the orresponding result yeld by the GTR, 3 i.e. α α e m γ ( r) = m = Constant, ersus m γ(r) = m =Constant. (1-k) (total relatiisti energy of a gien objet, predited by the GTR, ersus that predited by our approah) Our result, oinides up to the seond order of the orresponding Taylor expansion, with the result furnished by the GTR (i.e. the first relationship, written aboe). There is no easy way to interpret the singularity appearing in the numerator of the Einsteinian equation appearing aboe, whereas not only that it is possible to asertain what the numerator of our expression is all about, but also the set up of it, is almost eident. We would like to reall that, in fat, the singularity arises beause of the non-onformal analogy between the aelerational world and the graitational world, leading to the set up of the GTR. 15 Thus we are not a bit disappointed not to hae obtained idential same results in omparison with the GTR. The two approahes are based on ompletely different setups. And as pointed out, although the GTR furnishes good results, it seems to be totally erroneous. For one thing, it oerlooks the rest mass defiieny the objet brought to the fore field should undergo, and this is a lear iolation of the law of energy onseration. Strikingly, it athes up with this, through a serie of mistakes, deilishly ompensating eah other. Thene the disrepany oming into play does not a bit disfaor our approah, up against the one worked out within the frame of the GTR. What about the singularity, yielding to the blak hole onept, and so ery many aompanying works. Too bad! There is no singularity underlined by the present approah. Reall that our result is yet fully onsistent with what Yilmaz would hae written, 31,3 in the same way as that presented by Landau and Lifshitz leading to the GTR s relationship, then with the exponential orretion, Yilmaz proposed to Einstein s metri, i.e. α e m γ ( r) = m =Constant, (1-l) (relationship that would hae been written by Yilmaz, had he followed the same way as that presented by Landau and Lifshitz, with regards to the GTR, thus along with the orretion he proposed to Einstien s metri) though the set up would still be ery different. Further, we would like to mention that, likewise, Loguno landed at a similar result, with no singularity, whatsoeer

18 3. MASS SUBLIMES INTO KINETIC ENERGY, AND KINETIC ENERGY CONDENSES INTO MASS, THROUGHOUT THE MOTION: A JET MODEL Aording to our approah, m γ (r ) of Eq.(11) (i.e. the total relatiisti energy of the planet) ought to be onstant, all along the planet s journey around the sun. As the planet speeds up nearby the sun, it is that, an infinitesimal part of its rest mass, somehow sublimes into extra kineti energy (the planet aquires, as it aelerates). In other words, the extra kineti energy oming into play, is fueled by an equialent rest mass. As the planet slows down away from the sun, through its orbital motion, it is that, a portion of its kineti energy, somehow ondenses into extra rest mass, on the orbit. In other words, the rest mass spent preiously to fuel the extra kineti energy, is now brought bak. One way of oneiing this phenomenon, is to think in terms of a jet effet. Thus, within the frame of suh a modeling, in order to aelerate, the planet would throw out an infinitesimal net mass from the bak, just like an aelerating roket. Conersely, in order to deelerate, it would absorb an infinitesimal net mass, from the front. Whether in reality, the whole thing works out this way or not, we do not know it. For the present purpose, we do not need to know it, either. Though, we ame to be able to refer to a mehanism whih an proide us, with the end result we hae dislosed (i.e. the rest mass hange throughout), that we require, in order to take are of the ariation of the kineti energy, in relation to the ariation of the stati graitational binding energy, thus in relation to the ariation of the rest mass of the bound objet (imposed by the law of energy onseration). Hene, we an well base ourseles on it, to make useful preditions. Now refer to Eq.(1-a). Thus suppose that around the loation r, the planet of rest mass m(r), and eloity, aelerates as muh as d, through an infinitely small period of time dt, around the time t. This ourrene, aording to our approah, is insured by an amount of jet of mass -dm(r) thrown by the planet, through the period of time dt, with a tangential jet eloity U, with respet to the fixed sun. The law of momentum onseration requires that (f. the deriation proided in Part I, regarding an eletri motion, whose frame turns out to be stritly idential to that of the graitational motion), m γ ( r) d = Udm( r) ; (13-a) (kik reeied by the planet due to the jet effet, on a retilinear motion) note that the quantity dm(r), by definition, is negatie [so that dm(r) is a positie quantity]. dm(r) = m(r+dr)-m(r) <, when the planet aelerates ia throwing out mass; m(r) is the planet s mass at r. 18

19 Reall that aboe, we happened to hae assoiated the jet speed U with the rest mass ariation dm(r ), the planet displays on the way. We did it on purpose, just the way we did on the leel of Eq.(16-a) of Part I. The reason is that, as we will see, it is the rest mass dm(r ) that an be determined diretly, here, from the related stati graitational binding energy; moreoer, dm(r ) may well be zero, while U still remains defined. At any rate, here again, not to yield misinterpretations, the jet momentum U dm(r ) better be written, as, should dm(r ) dm(r ) U = V = γ V dm(r ) V = ( γ V dm(r ) ) V = dm(r ) ( γ VV), (13-b) V (momentum of the jet expressed in different terms) where γ V is 1 γ V =, (13-) V and V is the jet speed of the relatiisti mass γ V dm(r ), so that γ V V = U. (13-d) Eq.(13-b) is remarkable, gien that the RHS of it, i.e. γ V dm(r ) V an be read either as ( γ V dm(r ) ) V, or as ( γ V V ) dm(r ). In the former ase the relatiisti mass ( γ V dm(r ) ) multiplies the speed V to yield the relatiisti momentum of the jet in onsideration. The latter ase beomes interesting, as we will see, for the ase where dm(r ) is zero, pointing to an interation with no net mass ariation ; in this ase, the produt γ V V = U is to be onsidered as a whole; in any ase, the produt γ V V, is to be onsidered as a whole, sine it turns out that we will end up with U, as just one speifi quantity, and not separately γ V and V. This outome is remarkable, beause, as simple as it may look, along with our approah, it underlines the partile-wae duality (not only for the atomisti world, as disussed in the preious Part I, but) for the elestial world, as well: The relatiisti momentum ( γ V dm(r ) ) V, eidently points to a partile harater of the eletron, whereas γ V V = U as a whole, taking plae in the produt dm(r ) ( γ VV) (as we will soon disoer) indeed works as the key of the wae-like harater of the objet in hand; this beomes partiularly eident when dm(r ) anishes. 19

20 For the graitational interation too, we will disoer that / U = (, and / ) / V = ; for this reason we propose to all U the wae-like jet speed, or the superluminal jet speed, and V the relatiisti jet speed ; whereer we write just jet speed, we will mean the tangential wae-like speed U. Based on our jet model, heneforth, there seems reason to beliee that, een when the jet mass is null, whih happens to be the ase for a irular motion [f. Eqs. (16), (17) and () of Part I], whateer is the information we would expet to be arried by the jet speed, this information is still transferred. It is that, when there is no hange in the speed of the moing objet, the jet mass is aordingly zero. But as we will one again derie (here, is-à-is the graitational motion), U ultimately depends only on the eloity of the moing objet. Thus een if the jet mass is null, U is finite. In other words, whateer is the information arried by the jet speed U, this information an still be transferred, along with no mass, thus no energy transfer, whatsoeer. This is interesting sine we ame to say that: - The graitational information, just like the eletri information, an well be transferred with no need of energy exhange. 4. DERIVATION OF THE DE BROGLIE RELATIONSHIP, AND SUPERLUMINAL SPEEDS Let us now multiply Eq.(13-a) by : m (r)d = Udm(r). (14) γ The law of energy onseration requires that, the quantity - dm(r), appearing at the RHS of this equation, must ome to be equal, to the hange in the orresponding kineti energy, whih in return, must be equal to the hange in the orresponding graitational stati binding energy [f. Eq.(8)]. Thus M m(r) dm(r) = G dr. (15) r (ariation of the rest mass, in terms of the stati, graitational binding energy) Note, on the other hand that, when the planet aelerates, it gets loser to the sun; in this ase dr [just like, dm(r)], turns out to be a negatie quantity. Equating the LHS of Eq.(14), with the produt of the RHS of Eq.(15) and U [ia Eq.(1)], leads to α α e M m e m d = UG dr. (16) r

21 Here, we an replae, by the same quantity, furnished by Eq.(1-a). d Thene, the jet eloity U, as assessed by the distant obserer, turns out to be U =. (17) (the jet speed as referred to the outside fixed obserer) This equation is amazingly the same as Eq.(7), if the jet eloity U is taken to be same as U B, of this latter equation. It only depends on the eloity of the objet of onern. We an, anyway, write Eq.(17) as U λ =, (18) T ia the usual definition of the eloity of light, i.e. Eq.(3). Now, if we propose to write the jet eloity U, in question, in terms of the period of time, of the eletromagneti wae, we assoiate with the mass expression of a waelength, in terms of λ λ, i.e. m T, along Eq.(); we ome to the λ T λ = T, (19) whih is nothing else, but the de Broglie waelength [f. Eq.(1)]: λ = λ B = λ ; () (de Broglie waelength obtained from the wae-like jet speed, deried in here) reall that we did not hae to allow any restrition, while landing at this relationship. Tough it remains interesting to takle with it, primarily for a onstant eloity, thus for a irular motion. 1

22 For suh a motion, for the nth leel, one an write (f. the onluding footnote in the Introdution of Part I): n nh λ Bn = nλ = = πr n, (1) m r n, being the radius of onern. n (de Broglie waelength on the n th orbit) This relationship, numerially would sere, to alulate n, if other quantities are gien: n πrn m n n = ; (-a) h n, for earth rotating around the sun, for instane, approximately beomes 7 n 1x1. (-b) This number appears to be ery large; yet this affets in no way the alidity of the foregoing approah. The fat that a relationship similar to Eq.(), an be obtained, on the basis of an eletri interation (the way it was ahieed in Part I), already onstitutes a diret deriation of the usual de Broglie relationship. We an, on the other hand, alulate the eloity u of the jet eloity, with respet to the planet. As a rough approximation, one an write, u U +, (3) where is the speed of the eletron, with respet to the proton (assumed at rest); we will all u the superluminal jet speed, sine, as larified right below, it is always greater than the speed of light. (Note that in our approah the eiling annot be reahed, unless the photon bears an infinite amount of energy. 13 ) Obiously we do not know the rule regarding the addition of superluminal eloities, with eloities lying under the speed of light. Nonetheless, the examination of Eq.(17), makes our task easy. The two interesting ases indeed our for = and =. For =, U = ; thus one an, right away guess that, in this ase, u must be infinite. For =, U = ; thus one an guess that, in this ase u must be.

23 Hene, in onformity with the usual oneption regarding the tahyon postulation, 34,35 we an well establish that, the superluminal interation speed u (with respet to the objet in question), aries between (for the objet of onern, at rest), and (for the objet moing with the speed of light). As tautologial as it may seem, this yields the fat that, light annot interat with anything, ia a speed aboe the speed of light (sine its superluminal jet speed is, at best, ). One final remark should onsist in the following: Our approah seems to bring an answer to the quest of graitational interation ahieed with a speed muh faster than the speed of light. The fat that the sun and the planets must interat with eah other with a speed at least a hundred and nine times larger than the speed of light, is to be known, sine Laplae. 36 The Frenh Sientist, more than two enturies ago, had disoered that if the graitational field propagates with the mere speed of light, then the sun and the planets would get torn apart. This would yield the doubling of the Earth s distane from the sun, in 1 years. Amazingly this disoery of Laplae, no doubt, beause it is onsidered to ontrary theory of relatiity, does not take plae in ery many text books. It is unfortunate that many physiists, are aordingly unaware of Laplae s findings. Via the same onsiderations, one with reent data, an onlude that the speed of propagation of graity omes to amount een muh greater eloities. 37,38 5. GENERAL CONCLUSION Herein, based on just energy onseration, we figured out that, the graitational motion, just like the eletri motion, depits some sort of rest mass exhange, throughout. One way to oneie this phenomenon is to onsider a jet effet. Aordingly, an objet on a gien orbit, through its journey, must ejet mass to aelerate, or must pile up mass, to deelerate. The eloity U of the jet (as referred, not to the objet, but to the fixed outside obserer), strikingly delineates the de Broglie waelength, oupled with the period of time T, displayed by the orresponding eletromagneti energy ontent of the objet [as required by Eq.()]. This result seems to be important, in many ways. o We were able to derie the de Broglie waelength, in regards to both the eletri motion (Part I) and to the graitational motion (Part II); this makes that, the objets interating ia graitational harges, do behae in exatly the same way, as that displayed by quantum mehanial objets, interating ia eletri harges. It is interesting to note that our approah an be generalized to any fore field existing in nature. o Our approah appears to be apable to remoe, at least partly, the assumption about the unpreditability, or indeterminism to be made, to oer the quantum weirdness, suh as that displayed by the EPR experiment. 39 The immediate ation at a distane indued by the feature we dislosed, is-à-is de Broglie waelength, indeed seems to take are of it. It is that, Einstein seemingly rejeted to beliee in quantum mehanis, beause the gedanken (then erified) EPR experiment, pointed to an ation at a distane, ourring with a speed muh faster than the speed of light. Howeer, we hae dislosed throughout that, suh an ation is possible. Note that reent measurements seem 4, 41 to bak up our arresting dedution. 3