RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS

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1 RELTIVISTIC GRVITY ND THE ORIGIN OF INERTI ND INERTIL MSS K Tsarouhas To ite this version: K Tsarouhas. RELTIVISTIC GRVITY ND THE ORIGIN OF INERTI ND INERTIL MSS. 07. <hal > HL Id: hal Subitted on 3 Feb 07 HL is a ulti-disiplinary open aess arhive for the deposit and disseination of sientifi researh douents, whether they are published or not. The douents ay oe fro teahin and researh institutions in Frane or abroad, or fro publi or private researh enters. L arhive ouverte pluridisiplinaire HL, est destinée au dépôt et à la diffusion de douents sientifiques de niveau reherhe, publiés ou non, éanant des établisseents d enseineent et de reherhe français ou étraners, des laboratoires publis ou privés. Distributed under a Creative Coons ttribution 4.0 International Liense

2 RELTIVISTIC GRVITY ND THE ORIGIN OF INERTI ND INERTIL MSS K. I. TSROUCHS Shool of Mehanial Enineerin National Tehnial University of thens, Greee bstrat. In physis there are two asses, the ravitational and inertial, and an open question: what is the relation between these two asses? The answer to that ust pass throuh the explanation of inertia. In the experients that have been ade, it has been shown that the ratio of inertial ass to ravitational ass of an objet is onstant and equal for eah aterial. This has led throuh the General Theory of Relativity in responses whih led to draati preditions onfired experientally. In this paper we will follow a opletely different path and will ive different answers but our preditions will inlude earlier preditions that have been onfired experientally. Creatin and usin a theory in a radially different way of thinkin, whih affets the whole of physis, we will explain how the inertial fores are enerated. In this proess we will define the physial quantity of inertial ass. The nature of inertial ass, the way it hanes and the effet on the easureent of tie, will show that it is an extreely useful tool for the explanation of dark atter and dark enery.. INTRODUCTION Inertial fores appear in a non-inertial referene syste. But what deterines an inertial referene syste? ordin to Newton, an inertial referene syste is a syste that oves at a onstant veloity with respet to the absolute spae. ordin to Mah's priniple, inertial referene syste is a syste that oves with onstant veloity with respet to the distant stars. If the rest of the Universe deterines the inertial frae, it follows that inertia is not an intrinsi property of the atter, but arises as a result of the interation of atter with the rest of the atter in the Universe. lfred Einstein was inspired by Mah's priniple. The General Theory of Relativity attepted to interpret inertia, onsiderin that it is the ravitational effet of the whole Universe, but as pointed out by Einstein, it failed to do so []. Einstein showed that the field equation of General Relativity iplies that an objet in an epty Universe has inertial properties. Followin Mah's priniple, Dennis Siaa in a paper in 95 attepted to interpret inertia usin a odel where ravitational fores are desribed by equations siilar to Maxwell equations []. In this paper, the inertia of an aeleratin objet is an indutive effet of all other objets of the Universe and the priniple of equivalene is a result of the theory and need not be introdued axioatially. But no answer was iven to the question of how inertial fores appear exatly at the tie when a body is aeleratin. In Eletrodynais inertia has the nae radiation reation. In an attept to explain the radiation reation in 945 Wheeler and Feynan reated a new theory, the bsorber theory of radiation [3][4]. ordin to this theory, the radiation reation in an aelerated hare oes fro all other hares in the Universe and travels with advaned waves, waves that propaate fro the future to the past, to reah the hare exatly the oent when it starts aeleratin. Paul Dira also used advaned waves in a paper in 938 [5], in order to interpret the radiation reation. The Wheeler-Feynan theory revived one ation at a distane forulation of Eletrodynais as derived by Shwarzshild, Tetrode

3 and Fokker [6][7][8][9]. It is known as the ation at a distane theory beause hares an interat with one another only over finite distane without a ediatin entity (a field) that arries this ation fro one partile to the other. It is possible to forulate lassial eletrodynais as an ation at a distane theory and dispense with the field onept altoether [0][][]. Here we will follow Mah's priniple in order to find a satisfatory explanation of the oriin of inertia and inertial ass. If the inertia of an objet is derived fro the interation of the objet with all the other objets in the Universe, then, the fores with infinite rane that an enerate inertial fores are the ravitational and eletrial fores. The fore between two ravitational asses at rest is desribed by Newton's law of ravitation. The eletri fore between two point hares at rest in vauu is desribed by Coulob's law. Both these laws are of the sae forat. But while the two fores bean with reat siilarity, they ended up desribed by two opletely different theories, the General Theory of Relativity and Eletrodynais. But as ephasized by Poinare [3][4], if equilibriu is to be a frae-independent ondition, it is neessary for all fores of noneletroaneti oriin to have preisely the sae transforation law as of the Lorentz-fore. GRVITY Let's suppose that we have two objets with ravitational asses and positive eletri hares, at a point where the external ravitational and eletri fields are zero. The two objets are plaed at a distane fro eah other so that the attrative ravitational fore is equal in anitude and opposite in diretion to the repulsive eletri fore exerted between the. There the objets are plaed at rest, in respet to us, and they will reain at rest as they are in equilibriu. But what is observed by another observer ovin in relation to the two objets? Let's iaine that if they ollide, they will explode. It is ipossible for one observer to see an explosion and for another not. The Speial Theory of Relativity requires that the two objets ust be in equilibriu also for an observer ovin with a onstant veloity in relation to the. If we want the syste of two objets to be in equilibriu for eah inertial observer, the ravitational fore should be transfored in exatly the sae way as the eletri fore is transfored in different inertial referene systes. In order for this to happen, both fores ust be desribed by siilar equations. Sine the equations of eletroanetis are relativisti invariants and verified by nuerous experients, it is loial to think that ravity should also be desribed by siilar equations. Moreover, as we will prove in setion 6, the ravitational and eletri fores atin on an objet, affet the spaetie etri pereived by this objet. So, the idea is that the ravitational and eletri fores are desribed by siilar equations and both fores urve the spaetie with siilar way. It is known and well douented that we an produe all the equations of eletroaneti theory in Minkowski spaetie usin the Speial Theory of Relativity, on ondition that [5]:. Coulob's law desribes the fore exerted between the two eletri point hares at rest in vauu.. The eletri hare is invariant. It has the sae value in all inertial referene systes. 3. n eletri hare produes a salar potential and a vetor potential, whih toether for a four-vetor. The potentials produed by a hare ovin in any way depend only upon the

4 3 veloity and position at the retarded tie. Chanes in these potentials are propaatin in vauu at the speed of liht. The ethodoloy for doin this is desribed in any of the lassi books on eletroanetis [6][7][8]. To apply the sae ethodoloy in Gravity and to obtain equations siilar to those of eletroaneti theory, the sae prerequisites ust be et. But all we have is Newton's law of ravitation that is siilar to Coulob's law for eletri point hares. If we want to apply the sae ethodoloy, we need the other two as well. We will aept the postulated: Priniple The ravitational ass is invariant. It has the sae value in all inertial referene systes. Priniple ravitational ass produes a ravitational salar potential and a ravitational vetor potential that toether for a four-vetor. The potentials produed by a ravitational ass ovin in any way depend only upon the veloity and position at the retarded tie. Chanes in these potentials are propaatin in vauu at the speed of liht. So we arrive at the followin equations, siilar to Maxwell s equations, whih we will all the equations of the ravitoaneti theory in Minkowski spaetie and in SI units. E (.) B 0 (.) 0 4 G 0 B E (.3) B 0 j F ( E B ) (.5) : ravitational ass density E :the ravitational field t t 4 G (.4) 0 j : ravitational ass urrent density B : the ravitational aneti field G : Newton s universal ravitational onstant : the speed of liht in vauu. : ravitational ass The potentials produed by a ravitational ass ovin with veloity at a distane r, is as the Lienard-Wiehert potential: 4 r r / 0 (.6) 4 r r / 0 In the above equations, what is in square brakets refers to the retarded position of the ravitational ass. Startin fro the potentials, in order to find fields, we have the equations: E (.8) and B (.9) t In this way, we easily obtain a relativisti invariant theory for Gravity, the Gravitoaneti theory. (.7)

5 4 3. PRINCIPLE OF GENERL RELTIVITY The inertia appears in objets when they are aelerated. So the next step on the road to interpret inertia is to aept the extension of the Priniple of Relativity for all referene systes, not only for inertial ones. There an be no privileed referene systes. We will do this with the next priniple. Priniple 3 The laws of physis are the sae in all fraes of referene. The priniple of relativity is extended to all oveents. In order to use Lorentz transforations to any referene syste we should use the instantaneous speed. There is no absolute aeleration just as there is no absolute speed. With any experients arried out in a losed laboratory, an observer aeleratin annot establish whether:. He is ovin with aeleration and all other objets in the Universe are at rest.. He is at rest, while all the other objets in the Universe are ovin with aeleration ( ). n eletri hare eits eletroaneti radiation when aeleratin. This follows fro the equations of eletroanetis. Sine the sae equations apply to ravitoanetis, a ravitational ass when aeleratin will eit ravitoaneti radiation whih will propaate in vauu at the speed of liht and is desribed by the sae equation as that of eletroaneti radiation. ordin to the priniple of eneral relativity we aepted, aeleration is relative, so the radiation of a ravitational ass or an eletri hare when aeleratin is also relative. This eans that when a ass or hare is aelerated in relation to an observer, it eits radiation pereived by the observer. If, for another observer B, there is no relative aeleration, there will be no radiation. That is, the radiation eitted by an objet is always taken fro another objet. No radiation is eitted just in epty spae. The radiation has neessarily a sender and a reipient. Beause aeleration is relative, the inertial fore annot be derived fro the sae objet whih is aeleratin. It ust be due to those objets in relation to whih it is aelerated. Suppose there are only two objets in the Universe, and B. If B sees objet aelerate, then B reeives radiation eitted by. Beause aeleration is relative, sees B aelerate. So reeives radiation eitted by B. Both objets should reeive radiation if aeleration is relative. Otherwise, we an disern whih objet is aelerated. But at the sae tie an inertial fore is exerted on both objets if aeleration is relative. In order for all this to happen, one way is to aept the followin proedure: Objet loated at a tie t at point, with position vetor r, is aeleratin for B. So eits radiation at tie t, whih will reah B at tie t, when B is loated at position, with position vetor r. Radiation fro will exert a fore F on objet B. Objet B at a tie t at point, with position vetor r, is aeleratin for. So B eits radiation at tie t, whih will reah at tie t when is loated at position, with position vetor r. Radiation fro B will exert a fore ( F ) on objet. Radiation fro B to is propaated opposite in spae and tie, in relation to the radiation fro to B. The tie t related to tie t with the equation:

6 5 t r r t (3.) The sin in the equation (3.) is deterined by the priniple of ausality. The ause ust preede the effet. If aelerates first, pereives as inertial fore the radiation of B. If B aelerates first, B pereives as inertial fore the radiation of. If is aeleratin first, the radiation is propaated to the objet B with retarded waves. Objet pereives as inertial fore the radiation of B whih is propaated fro objet B on the objet with advaned waves. Both waves, retarded and advaned, are solutions of the Maxwell equations. 4. CTION-RECTION n objet exerts a fore on another objet with radiation. But there is no reason to treat the fore of radiation differently fro the other fores. So we aept that the proedure desribed above applies to all fores. The Priniple of General Relativity leads us to the relativisti eneralization of the third law of Newton: Priniple 4 When an objet oves in any way and is loated at tie t at a point with position vetor r, it exerts a fore-ation F at tie t, on an objet B, loated at a point with position vetor r, ovin in any way, then B exerts a fore-reation ( F ) on Α at tie t with t r r t. Coents on Priniple 4. What is transitted and has the effet of exertin fore on one objet is the potential of another objet. The ation is the ause and the reation is the effet. Without ation there is no reation. Thus, the priniple of ausality is not violated. The above proedure always happens as a whole, i.e. ation is always aopanied by reation. So the onept of the field is auxiliary and not essential. It is only useful for alulations. t this point, we ust ake it lear that if we deal with phenoena ourrin on our planet, the reation oes fro soe thousandths or illionths of a seond in the future. In lassial physis we aept that the fores are transitted instantaneously, and prove the priniple of onservation of oentu fro the third law of Newton. Now, with the relativisti eneralization of the third law of Newton, we an prove the priniple of onservation of oentu also for fores that are transitted with finite speed. Strane as it ay see, if we want to interpret phenoena suh as quantu entanleent, the possibility for a partile to reeive the reation fro all other partiles at exatly the oent when soe fore exerted on it, appears neessary [9].

7 6 4. INERTI Havin reated all the tools we need, we an now ove on and onsider an aelerated objet. 5. Study of the otion of an objet with only ravitational ass Suppose we have a Universe where all objets ontained are at rest. Suppose we have a point K where 0, is the ravitational salar potential at point K fro all the Universe. We plae an objet S, with ravitational ass, at rest at point K. The resultant ravitational fore exerted on the objet S is zero. The ravitational vetor potential is zero, at the rest frae of S, sine all objets are at rest in relation to S. fore F beins to be exerted on objet S and it starts to ove. Suppose at tie t objet S is ovin with veloity and aeleration in relation to the other objets in the Universe. Chanes of the potentials of objet S are propaated at the speed of liht to the other objets, where they arrive at various later ties t i, different for every objet, with: ri r t ti where r i is the distane traveled to reah every objet. Sine veloity and aeleration are relative, at the rest frae of S, the other objets ove with veloity ( ) and aeleration ( ) at the speifi ties t i. The hanes of the potentials of the other objets propaate bakward in spae and tie and arrive at the tie t at objet S. These hanes ause at tie t, a fore ( F ) to objet S that we all ravitational inertial fore. Now the advaned ravitational vetor potential it will be fro equation (.7): ( ) at the rest frae of S at tie t is not zero but advaned The ravitational salar potential hanes but ontinues to be: [ ] adv 0, beause if an objet is in equilibriu to point K beause 0, it is in equilibriu for any observer. What is in square brakets refers to a later tie than tie t. With this in ind, we an then reove the square brakets: (5.) The ravitational fore exerted on an objet with ravitational ass ovin at speed v, when it is in a ravitational field E and a ravitational aneti field B is fro eq. (.5): F ( E v B ) (5.) Where E and t B

8 7 In the rest frae of S, beause v =0 the ravitational aneti fore is: v B 0. Moreover, as entioned earlier, 0 ( [ ] adv 0 with the square brakets). So at t, the fore exerted on S is: Equation (5.3) toether with the (5.) beoes: F E F t (5.3) F t (5.4) If we define as ravitational inertial ass in the inertial ass due to ravity: in (5.5) The equation (5.4) beoes: F in t (5.6) We define as ravitational oentu p : p (5.7) in Beause in is positive, speed and oentu have the sae diretion as required. So if you replae (5.7) to (5.6) we have: F p (5.8) t F t in in t (5.8a) This is known to us as inertia fore of an objet that has only ravitational ass. We end up to the onlusion that the inertial fore reeived by an objet S with ravitational ass when aeleratin is a ravitational fore and is due to the aeleration of all the other objets in the Universe in relation to the objet S. So we an express the seond law of Newton as follows: When an objet is aelerated, the fore exerted on the objet to aelerate it, needs to be equal in anitude and opposite in diretion to the inertial fore. It is obvious that the seond law of Newton is now a onsequene of the theory that we follow, and there is no need to introdue it axioatially.

9 8 We aept that the ravitational inertial ass of an objet S depends on the advaned ravitational salar potential of all other objets in relation to whih it is aelerated. The advaned ravitational salar potential depends only upon the veloity and position at an advaned tie. Usin (5.5) and (5.) the ravitational vetor potentional So the ravitational oentu beoes: in in p of an objet S and the advaned ravitational vetor potential fro all the other objets at the rest frae of S, are onneted with the equation: p (5.9) It is worth repeatin that to et here we have aepted that the ravitational inertial ass of an objet S is: in The total inertial ass hare, the ravitational inertial ass in will siply be alled inertial ass. For an objet that has no eletri in is equal to the inertial ass in. in in If we take the ratio of the inertial ass to ravitational ass of an objet, we have: in (5.0) We see that the ratio of the inertial ass of an unhared objet to the ravitational ass is independent of the objet. It depends only on the advaned ravitational salar potentials due to all other ravitational asses in the Universe in respet to whih it aelerates. So this ratio is equal for eah objet when the inertial ass is easured at the sae point in spae. This explains why all objets fall with the sae aeleration in a ravitational field. The above results are onfired experientally very preisely. t another point in spae, while the ravitational ass reains the sae, the inertial ass hanes so the ratio has another value. Consider now an objet that akes free fall in the ravitational field of another uh bier objet K. The aeleration the objet obtains is suh, that the inertial fore to be exerted on it is equal in anitude and opposite in diretion to the ravitational fore it aepts fro K. So the resultant ravitational fore on it is zero. Therefore: The rest syste of an objet with ravitational ass whih akes free fall in a ravitational field is equivalent to an inertial referene syste. n objet S with ravitational ass whih is aelerated in relation to the other objets reeives an inertial fore that is nothin else than a ravitational fore. So:

10 9 referene syste onsistin only of ravitational asses whih akes an aelerated otion in a spae where there is no ravitational field is equivalent to an inertial referene syste at rest in a ravitational field. In the General Theory of Relativity the priniple of equivalene for the ravitational fores is aepted as an initial axio. Here we have proved that the priniple of equivalene for the ravitational fores is a onsequene of the theory. In the General Theory of Relativity we aept that the inertial ass is equivalent to the ravitational ass. Here we have proved that the two asses an be opletely different physial quantities but their ratio is equal for eah objet at the sae point, but not the sae everywhere. Let us now exaine how the ravitational inertial ass of an objet S is transfored when S oves with veloity in relation to the other objets. In equation (5.5): in the ravitational ass and the speed of liht, are invariant quantities. So the ravitational inertial ass will be transfored in the sae way as the ravitational salar potential. We define ravitational salar potential restin as 0. The 0 is the ravitational salar potential fro all other objets, in the rest frae of S, when all other objets are at rest in relation to S. If the Universe onsists of N disrete ravitational asses and eah is loated at a different distane fro the objet S, then: 0 N i 4 r (5.) 0 i i The distane r i is easured in the referene syste where all objets are at rest. When S oves with veloity in relation to the other objets, the ravitational salar potential of all other objets, in the rest frae of S is transfored in the sae way as the eletri salar potential [5],[8][0]: 0 (5.) Equation (5.5) shows that when S oves with veloity in relation to the other objets the ravitational inertial ass will vary in the sae way: in in 0 (5.3) where in 0 is the ravitational rest ass. It is the ravitational inertial ass in the rest frae of objet S when all other objets are at rest in respet to S.

11 0 We now see fro equation (5.3) that the equation for the relativisti ass in the Speial Theory of Relativity is proved diretly and not indiretly, but with two iportant differenes. The rest ass of an objet S, is the inertial ass in the referene syste where all the objets are at rest, inludin S. In Speial Theory of Relativity the rest ass of an objet S, is the inertial ass in the referene syste where only the objet S is at rest. The ravitational inertial ass of an objet S varies only when S oves in relation to the other objets and not when S oves in relation to an observer. If the Universe onsists of N disrete ravitational asses and eah ass is loated at a different distane fro an objet, the ravitational rest ass of that objet is: N i (5.4) 4 r in0 0 i i The distane r i is easured in the referene syste where all the objets are at rest. 5.. Dark atter Fro equation (5.4) we observe that the ravitational rest ass of an objet S depends on the distane of the other objets in respet to whih it is aelerated. If we have an objet only with ravitational ass, the total rest ass in is the ravitational rest ass 0 in 0. So then (5.4) beoes: N i 4 r (5.5) in0 The distribution of atter in the Universe is not ontinuous. If it was ontinuous, the equation (5.5) would be doinated by the very distant ravitational asses, so uh that the ravitational rest ass would be onsidered pratially onstant and independent of the objet position in relation to other objets with very lare ravitational ass suh as the Milky Way, our alaxy. Let us think that our alaxy with a diaeter of about one hundred thousand liht years is far fro the losest lare alaxy, the ndroeda alaxy, whih is alost idential to our own, one and a half illion liht years. The inertial ass of a star loated at an averae distane fro the enter of the Milky Way is affeted alost fifty ties ore fro the Milky Way than the ndroeda alaxy. In order to find fifty alaxies like ndroeda, we have to inlude alaxies that are alost twenty illion liht years far. So it sees that the position where a star is situated in a alaxy, affets sinifiantly the inertial ass of the star. In plaes with hiher density of stars the inertial ass of a star will be reater. Two stars with the sae ravitational ass have different inertial asses if they are in a different position in a alaxy. The star that is far fro the enter has a lower inertial ass than the star that is loated near the enter. To balane the ravitational attration the star that is far away fro the enter will ove with reater speed than the speed that it would have if the inertial ass was onstant. This hiher speed is observed. The inability to explain why speed is hiher has driven to the onept of dark atter. This onlusion arises fro a very siple approah we have taken. Certainly we need to do ore preise alulations. 0 i i

12 5.. Study of the otion of an objet with an eletri hare Let us now study the ase that in the Universe there are eletri hares at rest. Suppose we have a point K where 0, is the eletri salar potential fro all the eletri hares. We plae an objet S with eletri hare q at rest in point K. The resultant eletri fore exerted on the objet S is zero. Sine all the other objets with eletri hare are at rest, in relation to S, the eletri vetor potential due to the in the rest frae of S is zero. The proess followed is idential to that whih we followed in the previous setion 5.. fore beins to be exerted on the objet S and it starts to ove. Suppose at tie t the objet S is ovin with veloity and aeleration in relation to the other eletri hares. Chanes of the eletri potentials of objet S are propaated at the speed of liht to the other eletri hares, where they arrive at various later ties t i, different for every eletri hare, with: t t i r r i where r i is the distane traveled to reah every eletri hare. Sine veloity and aeleration are relative, at the rest frae of S, the other eletri hares ove with veloity ( ) and aeleration ( ) at different tie oents in the future. The hanes of the eletri potentials of the other eletri hares propaate bakward in spae and tie and arrive at tie t at objet S. These hanes ause at tie t, a fore ( F ) to objet S that we all eletri inertial fore. be: Now the advaned vetor eletri potential in the rest frae of S at tie t is not zero but it will ( ) advaned The salar eletri potential, in the rest frae of S, hanes but will ontinue to be [ ] 0 beause if a hare is in equilibriu to point K, it is in equilibriu for any observer. What is in square brakets refers to a later tie than t. With this in ind, we an then reove the square brakets: adv (5.6) The eletri fore exerted on an objet with eletri hare q ovin at speed v, when it is in an eletri field E and a aneti field B is the Lorentz fore: F q( E v B) (5.7) e E and B t In the rest frae of S, beause v =0, no aneti fore is exerted on S. Moreover, we have said that 0 ( [ ] 0 with the square brakets). adv

13 So at t, the eletri fore exerted on S is: If we now define as eletri inertia ass (5.8) Fe qe Fe q t in the inertial ass aused by eletri fores: e in e q (5.9) If we define as eletri oentu p the oentu due to eletri fores: e Then (5.6) beoes: pe in e pe q p e q (5.0) (q) pe Fe Fe t t (5.) Fe t ine ine t (5.a) t this point we ust ephasize that the eletri inertia ass aordin to equation (5.9) is not always positive, as is the ravitational inertia ass. If all hares are either positive or neative, the eletri inertial ass is neative. We onlude fro equation (5.9) that the ratio of the eletri inertial ass of a hared objet to the hare is independent of the objet: in e q This ratio depends only on the advaned salar potentials due to all other eletri hares in the Universe in relation to whih the objet aelerates. So this ratio is equal for eah hared objet when the eletri inertial ass is easured at the sae point in spae. Let us now exaine how the eletri inertial ass of an objet S is transfored when S oves with veloity in relation to the other hares. In equation (5.9): ine q the hare q and the speed of liht are invariant quantities. So the eletri inertial ass will be transfored in the sae way as the eletri salar potential. We define eletri salar potential restin as e0.the e0 is the eletri salar potential fro all other hares, in the rest frae of S, when all other hares are at rest in relation to S. If the Universe onsists of N disrete eletri hares and eah is loated at a different distane fro the objet S, then: N qi e0 4 r (5.) 0 i i The distane r i is easured in the referene syste where all hares are at rest.

14 3 When S oves with veloity in relation to the other hares, the eletri salar potential of all other hares, in the rest frae of S is known that is [5],[8][0]: e0 (5.3) Equation (5.9) shows that when S oves with veloity in relation to the other hares the eletri inertial ass will vary in the sae way: in e0 ine (5.4) where in is the eletri rest ass. It is the eletri inertial ass in the rest frae of objet S when e0 all other hares are at rest in respet to S. If the Universe onsists of N disrete eletri hares and eah hare is loated at a different distane fro an objet, the eletri rest ass of that objet is: N qqi 4 r (5.5) ine0 0 i i The distane r i is easured in the referene syste where all the objets are at rest. 5.. Universe only with eletri hares If there were only eletri hares in the Universe, the total inertial ass eletri inertial ass. Then fro equation (5.9) we et: in e in q would be equal to the in Thus, in a Universe with only eletri hares all hares in an eletri field would have the sae aeleration. hared objet S whih is aelerated in relation to the rest of the Universe reeives an inertial fore that is nothin else than an eletri fore. So in a Universe with only eletri hares apply a priniple of equivalene: referene syste onsistin only of eletri hares whih akes an aelerated otion in a spae where there is no eletri field is equivalent to an inertial referene syste at rest in an eletri field. The rest syste of an objet with only eletri hares whih akes a free fall in an eletri field is equivalent to an inertial referene syste.

15 4 5.3 Study of the otion of an objet with both ravitational ass and eletri hare In order to estiate the total inertia fore exerted on the objet S, we add the equations (5.8) and (5.): e F p p p inertia F inertia (5.6) t t t in Finertia in t t The total oentu p is the su of ravitational oentu p and eletri oentu p e : (5.6a) p p p p ( ) (5.7) e in ine The total inertial ass is the su of ravitational ass and eletrial inertial ass: in in in in e in q (5.8) The transforation of total inertial ass when the objet S oves with veloity in relation to the other objets is: in0 ine0 in0 in in (5.9) where in is the total rest ass. It is the inertial ass in the rest frae of objet S when all other 0 objets are at rest in respet to S. In ase we have N objets in the Universe with ravitational ass and M objets with eletri hare, the total rest ass in is: 0 N M i qqi ( ) 4 r 4 r in0 (5.30) 0 i i 0 i i The oentu p of an objet S and the advaned ravitational and eletri vetor potential fro all the other objets at the rest frae of S, are onneted with the equation: p q (5.3) In ase we onsider the otion of an eletron in a seiondutor rystal, we have to take into aount the hane in the inertial ass of the eletron beause of the existene of other eletrial hares in the spae that it oves.

16 Zero inertial ass Suppose in our Universe, doinated by the ravitational asses, we have two partiles and with ravitational asses, and eletri hares q, q at a distane r. ll other hares are far away. Equation (5.30) where we have two eletri hares is applied. N i qq For partile in0 ( ) (5.3) 4 r 4 r 0 i i 0 Suppose the two partiles have like hares. In this ase, the eletri inertial asses of the partiles are neative. We see fro equation (5.3) that when they approah eah other, the absolute value of the eletri inertial ass inreases. When the two partiles approah eah other the total inertial ass of partiles dereases. When they approah eah other at a ertain distane, the absolute value of the eletri inertial ass is equal to the ravitational inertial ass and so the total inertial ass is zero. t a shorter distane the inertial ass of partiles beoes neative. Let us now alulate this distane. Partile with ravitational ass has ravitational rest ass: N i in0 (5.33) 4 0 i ri The equation (5.33) ives us the partile's rest ass of the ontribution of ravitational fores. When the hared partile approahin at a distane r fro partile, the rest ass dereases and beoes zero when: If we obine (5.33) with (5.34) we et: q q (5.34) r r N i 4 0 i i 4 0 qq in o 40 r (5.35) We onsider that the ravitational rest ass reains onstant as the two partiles approah. Fro (5.35) we an alulate the distane where the rest ass beoes zero: r qq (5.36) 40 in o In a weak eletri field the rest ass is approxiately equal to the ravitational rest ass: ino ino When easurin the rest ass of a partile in a weak eletri field, ravitational rest ass is basially easured. So equation (5.36) beoes: qq r (5.37) 4 0 ino

17 6 Equation (5.37) for two protons ives us: Equation (5.37) for two eletrons ives us: r,53 0 r, With values: qp 9 7,60 0 C, p,67 0 K, qe 9,60 0 C, e 9, K When the partiles arrive at a distane less than the iven value, the inertial asses of partiles are neative. What neative inertial ass eans is an open question. 6. THE SPCE TIME METRIC PERCEIVED BY N OBJECT We want to exaine if spaetie etri pereived by an objet with ravitational ass and eletri hare depends on its position in a ravitational and eletrial field. Here we will show that ravitational and eletri fores hane spaetie etri pereived by an objet ovin in it and not the etri of spaetie itself. The etri of spaetie itself is flat. When two different objets with different ravitational ass and eletri hare are at the sae point of a ravitational and eletri field, eah one will pereive a spaetie with different etri. n objet oves freely when no fore is exerted on it or the resultant fore exerted on it is zero. We restate Newton s First Law of Motion for urved spaetie as follows: Priniple 5 Every objet that oves freely is an inertial observer and oves in a straiht line in the spaetie that it pereives. In an inertial observer tie passes at a onstant rate and the lenth of a rod is onstant. Three ruial lassi experients verify the General Theory of Relativity []: the red shift of spetral lines eitted by atos in a reion of stron ravitational potentional, the defletion of liht rays that pass lose to the Sun and the preession of the perihelion of the planet Merury. ll the three above derive fro Shwarzshild etri. Many efforts have been ade in the past to produe Shwarzshild etri by usin only the priniple of equivalene, Newton s law of ravitation and the Speial Theory of Relativity [][][3][4][5][6][7]. These efforts divided the sientifi ounity in the past. Today it is aepted that there is one way to produe the equations that deterine the hane of the rate of tie and the hane of lenth in a ravitational field usin only Newton s law of ravitation, the Speial Theory of Relativity and the priniple of equivalene [8][9]. Sine the priniple of equivalene is a onsequene of the new theory and Newton's law of ravitation is valid in this theory, this partiular way will be followed here. Havin a new theory of ravity different fro the General Theory of Relativity we should et the sae results in the three ruial experients that we entioned. For this reason we will ove on to derive the spaetie etri that pereives an objet with ravitational ass and eletri hare. This spaetie etri will inlude Shwarzshild etri

18 7 6.. Free fall in a ravitostati field We have an objet S of ravitational ass M. The S is stati and has spherially syetri ass distribution. We use as oordinate syste a spherial oordinate syste with zero position at the enter of the spherial objet S.. Free fall of a lok We have three idential loks:, B and C. Eah lok has ravitational ass. We let lok at rest at a distane r fro the enter of S in position. We let lok B at rest at infinity in position B and in the sae radial diretion as lok. We let lok C ake a free fall in the ravitational field of objet S with zero initial speed startin fro the position B. Clok C will ove in a straiht line that onnets and B (fiure ). When we say infinite, we ean far enouh to assue that the ravitational field is zero. We want to opare the rate of lok with the rate of lok B at infinity. Beause lok C akes a free fall it is an inertial observer, so tie at lok C passes at a onstant rate throuhout the whole free fall. So, if let lok C ake a free fall startin fro position B with zero veloity, the lok C will work at the sae rate with lok B that is at rest at infinity. When lok C is at position B, if dt is the tie interval that lok C easures between two events ourrin at lok B and dt B is the proper tie interval that lok B easures between the sae two events, then: dt dt B Therefore when lok C passes fro point, with veloity Fiure and opares its own rate with the rate of lok, it is like oparin the proper rate of lok with the proper rate of lok B. When lok C passes fro point, lok C and lok reeive the sae fore of the ravitational field beause they have the sae ravitational ass. Thus we do not need the effet of ravity to opare the rates of loks and C. To opare the rate of lok C with the rate of lok we only need the Lorentz transforations beause of the relative speed they have. When lok C passes fro point, it easures the tie interval dt between two events ourrin at lok. Clok easures the tie interval dt for the sae two events that is the proper tie. ordin to Lorentz transforations: C υα S B dt dt (6.) Where: (6.) ordin to what we said before dt is also the proper tie for the lok B, so equation (6.) relates the proper tie of lok B at infinity with the proper tie of lok at a distane r. So dt is the oordinate tie and dt is the loal tie. In order for lok C to easure an infinitesial tie interval on lok, lok C needs two tie instants, so we aept that the speed of lok C is onstant at infinitesial distanes fro position.

19 8 Β. Free fall of a rod We have three idential infinitesial rods for easurin lenth:, B and C. Eah rod has a ravitational ass. We let rod at rest at a distane r fro the enter of the objet S in position. We let rod B at rest to infinity in position B and in the sae radial diretion as rod. We let rod C ake a free fall in the ravitational field of objet S with zero initial speed startin fro the position B. ll the rods are parallel to the straiht line that onnets with B (fiure ). We want to opare the lenth of rod with the lenth of rod B. Beause rod C akes a free fall it is an inertial observer, so the lenth of rod C is onstant throuhout the whole free fall. So if let rod C akes a free fall startin fro infinity with zero veloity, the rod C will have the sae lenth with rod B that is at rest at infinity. When rod C is at position B, if dr is the lenth of rod B as easured by rod C and dr B is the proper lenth of rod B, then: dr dr B Therefore when rod C passes fro point, with veloity and it opares its own lenth with the lenth of rod, it is like oparin the Fiure proper lenth of rod with the proper lenth of rod B. When rod C passes fro point, rod C and rod reeive the sae fore of the ravitational field beause they have the sae ravitational ass. Thus we do not need the effet of ravity to opare the lenths of rods and C. To opare the lenth of rod C with the lenth of rod we only need the Lorentz transforations beause of the relative speed they have. When rod C passes fro point, it easures that the lenth of rod is dr. Rod easures its own lenth, whih is dr, that is the proper radial distane. ordin to Lorentz transforations: dr dr (6.3) ordin to what we said before dr is also the proper radial distane for the rod B, so equation (6.3) relates the proper radial distane of rod B at infinity with the proper radial distane of rod at a distane r. So dr is the radial oordinate distane and dr is the loal proper radial distane. Sine lenth easureent needs two points at a ertain distane and between the two points spae tie etri hane for rod, we an approxiately aept that in infinitesial distanes fro point spaetie is flat. We an alulate the anitude of veloity by applyin the onservation of enery when observer C akes a free fall for non-relativisti veloities. t the beinnin observer C is at infinity, where the potential enery is zero. Thus the enery of the syste is initially zero and the equation for onservation of enery is: M M G 0 G (6.4) r r in in C υα Σ B Therefore: GM (6.5) r in

20 9 By replain (6.5) at (6.) we have: dt GM dt (6.6) rin By replain (6.5) at (6.3) we have: dr dr GM rin Fro the equations (6.6) and (6.7) we draw onlusions that apply to the ravitational field. These are:. The lok at position oes slower than lok B at infinity. Therefore, we onlude that when the loks are in a ravitational field, they are slowed.. The rod in position is loner than the parallel rod B at infinity. Therefore, we onlude that a rod loated in the ravitational field outside a stati and spherially syetri distribution of ass expands in the radial diretion. It is known and well douented that outside a stati and spherially syetri distribution of ass, the spaetie etri for a point that is at a distane r fro the enter of the objet an be written in spherial oordinates [8]: rr (6.7) ds 00 r dt r dr r d sin d (6.8) This etri should ive us to infinity the Minkowski etri in spherial oordinates: So we ust have the boundary onditions: 00 ds dt dr r d sin d (6.9) li r. t r και r t li. (6.0) The spaetie distane between two events happenin at the sae point in spae, dr d d 0, is: But ds dt is the proper tie at position r. r 00 rr ds r dt (6.) So the proper tie at position r is: dt dt (6.) 00 Fro the equations (6.) and (6.6) and fro the boundary onditions (6.0) we have:

21 0 r GM r in 00 (6.3) The spaetie distane ds between two events happenin at the sae tie at different points in infinitesial distane fro the position but on the sae radial diretion, dt d d 0 is the proper radial distane dr. Fro the equation (6.8) when dt d d 0 we et: ds dr (6.4) rr But the proper radial distane at position r is: dr dr (6.5) rr Fro the equations (6.5) and (6.7) and fro the boundary onditions (6.0) we have: rr GM rin (6.6) If we replae the equations (6.3) and (6.6) in the (6.8) we et: GM dr ds ( ) dt r ( d sin d ) r GM in rin (6.7) Equation (6.7) desribes the spaetie etri pereived by an objet with ravitational ass and inertial ass in, who is at a distane r outside a stati objet S with ravitational ass M and spherially syetri distribution ass. We refer to spaetie etri pereived by an objet and not to the spaetie etri that exists independent of the objet. Certainly, as we have said, the ratio of ravitational ass to inertial ass at the sae point is the sae for every objet so the spaetie etri pereived by a ravitational ass is the sae for every ravitational ass. So it ould be onsidered that this etri is the etri of spaetie itself, but this is wron, and this will be apparent when we have both eletri hares and ravitational asses. The ratio of ravitational ass to inertial ass is the sae for every objet at the sae point. In our position in the alaxy we have aepted that this ratio is equal to one. (6.8) Thus, when the ratio is equal to one, equation (6.7) ives us the etri: in

22 GM dr ds ( ) dt r ( d sin d ) r GM r (6.9) Equation (6.9) is the well known Shwarzshild etri. Thus, all the phenoena that eere fro Shwarzshild etri also eere fro equation (6.7). These are: The red shift of spetral lines eitted by atos in a reion of stron ravitational potentional. The defletion of liht rays that pass lose to the Sun if we aept that liht oves in a straiht line in spaetie pereived by the objet that reeives the liht. The preession of the perihelion of the orbit of the planet Merury. t another point in our Galaxy, the ratio of ravitational to inertial ass is not equal to one. The phenoena are the sae qualitatively but not quantitavely. 6. Dark Enery Fro equation (6.7) arises that the equation relatin the proper tie at a distane r, fro the enter of S in position (fiure ), with proper tie at infinity is: dt dt GM rin (6.0) The equation (6.0) desribes the red shift of spetral lines whih is eitted by an ato in a ravitational field and is reeived by an objet whih is out of the ravitational field. Fro the Shwarzshild etri (6.9) arises that the equation relatin the proper tie at a distane r, fro the enter of S in position (fiure ), with proper tie at infinity is: dt dt GM r (6.) s it eeres fro equation (6.), red shift of spetral lines eitted by an ato in a star is the sae for every star with the sae ravitational ass. However as it eeres fro equation (6.0), the red shift of spetral lines eitted by an ato in a star depends on the radio of the ravitational ass to the inertial ass of the atos eittin the liht. s the Universe expands, inertial ass of all objets dereases as tie oes by as it eeres fro equations (5.5) and (5.4). In this way, the inertial ass of an ato that eits liht dereases as tie oes by. s it eeres fro equation (6.0) the liht eitted by two idential supernovas Ia at different oents in the history of Universe, they will have different red shifts. When we see a supernova Ia at a distane l it appears as it was at t. When we see a supernova Ia at a distane l it appears as it was in t. When l l then t t.

23 Thus the inertial ass of the atos of a loser supernova Ia is saller than the inertial ass of the atos of a further idential supernova Ia. So, as it eeres fro equation (6.0) loser supernovas have reater red shift. This phenoenon has been observed but the inability to explain why the red shift of spetral lines is reater has led to the theory that the Universe expands in an aeleratin way beause of dark enery. Certainly we need to do ore preise alulations but the hanes of ravitational inertial ass due to the expansion of the Universe are very useful for the explanation of the dark enery theory. 6.3 Free fall in a Universe that ontains only eletri hares s we said in setion 5.., in a Universe with only eletri hares all hares in an eletri field would have the sae aeleration. So the appropriate priniple of equivalene is valid here. We repeat exatly the sae thouht experient we ade in setion 6. with the only differene that the loks and the infinitesial rods have eletri hare q and the objet S that is stati and has spherially syetri distribution of eletri hare, has eletri hare Q. For this reason it is not neessary to repeat all the proedure but to refer only to the points in whih they are different. We want to opare the rate of the lok and the lenth of the infinitesial rod loated in position with a lok and a parallel infinitesial rod at infinity in position B where the eletri field is zero (fiure, ). We hoose the hares q and Q to be unlike in order to be attrated. When lok-rod C that ake free fall, with zero initial speed startin fro the position B, passes fro point, lok-rod C and lok-rod reeive the sae fore of the eletri field beause they have the sae eletri hare. Thus we do not need the effet of eletri fore to opare the rates of loks and C and the lenth of rods and C. To opare the physial quantities we only need the Lorentz transforations beause of the relative speed they have. Fro the onservation of enery for non-relativisti veloities we have: Qq KQq (6.) 0 in K e r r ine Thus: KQq (6.3) r ine nd we end up to the equations: dt KQq dt (6.4) and rin e dr dr KQq rin e (6.5) Followin the sae proedure as the one we followed in setion 6. we oe to the equation for the spaetie etri pereived by an objet with eletri hare q and inertial ass in e, who is at a distane r fro a spherially syetri and stationary objet with eletri hare Q.

24 3 KQq dr ds ( ) dt r ( d sin d ) r KQq ine rin e (6.6) In setion 6. and setion 6.3 we found the followin phenoena: When we have loks and rods whih onsist only of ravitational ass, the rate of the loks and the lenth of the rods in the diretion of the field, hane when they are in a ravitational field When we have loks and rods whih onsist only of eletri hare, the rate of the loks and the lenth of the rods in the diretion of the field, hane when they are in an eletri field. If we have an objet C with both a ravitational ass and an eletri hare q in the field of an objet S, with ravitational ass M and eletri hare Q, both phenoena happen at the sae tie. 6.4 Free fall in a ravitostati and an eletrostati field We repeat exatly the sae thouht experient we ade in setion 6. with the only differene: the loks and the infinitesial rods have ravitational ass and eletri hare q the objet S has ravitational ass M eletri hare Q and is stati and has spherially syetri distribution of ravitational ass and eletri hare. Beause we repeat exatly the sae thouht experient, it is not neessary to repeat all the proedure but to refer only to the points in whih they are different. We hoose the hares q and Q to be unlike in order to be attrated. We want to opare the rate of the lok and the lenth of the infinitesial rod loated in position with a lok and a parallel infinitesial rod at infinity in position B where the ravitational and eletri fields are zero (fiure, ). ordin to priniple 5, an infinitesial objet C (lok or rod) with ravitational ass and eletri hare, whih oves freely in the ravitational and eletri field of the objet S, is an inertial observer. When objet C passes fro point, objet C and objet reeive the sae fore of the ravitational and eletri field beause they have the sae ravitational ass and eletri hare. Thus we do not need the effet of ravitational and eletri fore to opare the rates of loks and C and the lenth of rods and C. To opare the physial quantities we only need the Lorentz transforations beause of the relative speed they have. We an estiate the anitude of veloity by applyin the onservation of enery when objet C akes a free fall fro infinity for non-relativisti veloities. t the beinnin objet C is at rest at infinity, where the potential enery is zero. Thus the enery of the syste is initially zero and the equation for onservation of enery is: M Qq GM KQq G K 0 (6.7) in r r rin rin Thus: GM KQq (6.8) r r in in nd we end up to the equations: