Newton s E = mc 2 Two Hundred Years Before Einstein? Newton = Einstein at the Quantum Scale

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1 Newton s E = Two Hunded Yeas Befoe Einstein? Newton = Einstein at the Quantu Scale Espen Gaade Haug Nowegian Uniesity of Life Sciences August 7, 07 Abstact The ost faous Einstein foula is E =, while Newton s ost faous foula is F = G. Hee we will show the existence of a siple elationship between Einstein s and Newton s foulas. They ae closely connected in tes of fundaentaaticles. Without knowing so, Newton indiectly conceptualized E = two hunded yeas befoe Einstein. As we will see, the speed of light (which is eual to the speed of gaity) was hidden within Newton s foula. Key wods: E =, enegy, kinetic enegy, ass, gaity, elatiity, Newton and Einstein. Did Newton Discoe E = Two Hunded Yeas Befoe Einstein? The Italian geologist and industialist, Olinto de Petto, speculated thee yeas befoe Einstein that the old, well-known foula E = had to be eual to E = when soething oed at the speed of light, whee is the object s speed, c is the speed of light, is the ass, and E is the enegy. In his 904 book Electicity and Matte, Thoson [] pesented what he called a kinetic enegy foula fo light, which he descibed as E = c. Einstein [] is still likely the fist to atheatically poe the E = elationship between enegy and ass. Today, oeoe, we know that the kinetic enegy foula and est ass foula ae not the sae. Hee, howee, we will show that Newton basically conceied the sae atheatical elationship between enegy and atte, as hidden in his gaity foula (see [3]) two hunded yeas befoe Einstein; if only he had siply known the adius and ass of a fundaentaaticle. The est ass of any fundaentaaticle can be witten as = h c, () whee is the educed Copton waelength. Fo exaple, the est ass of an electon is gien by e = he c kg. () On this basis, we obtain the following elationship between enegy ass: E =, (3) whee p is the Planck ass (see [4]) and is the educed Copton waelength of the fundaental paticle (this is actually the paticle s extended adius, see [5, 7]). That is, the est enegy ebedded in any fundaentaaticle is eual to the gaity (enegy) between two Planck asses sepaated by the educed Copton waelength of the fundaentaaticle of inteest. Moeoe, the kinetic enegy of a fundaentaaticle is gien by e-ail espenhaug@ac.co. Thanks to Richad Whitehead fo assisting with anuscipt editing. E = was suggested by Gottfied Leibniz in the peiod In the oiginal foula he used notation E = MV,butThosonclealystatedwheeV is elocity with which light taels though the ediu....

2 E k = c c G p p. (4) And when <<c, we can use the fist te of a seies expansion and closely appoxiate the kinetic enegy by Futheoe, we ust hae E k = G pp. (5) c. (6) And what we can call the est foce of a fundaentaaticle is gien by In othe wods, F = c. (7) Newton = Einstein, whee is the extended adius of the fundaentaaticle in uestion (the educed Copton waelength of that paticle). Moing on, the elatiistic foce ust be F = = G p p. (8) Based on Haug s ecent eseach, the axiu elocity of a Planck ass paticle (ass gap paticle) supisingly is zeo. The Planck ass paticle is at est in any efeence fae and theefoe the only inaiant paticle. This can only happen if the Planck ass paticle only lasts an instant (see [0, 8]). The Planck ass paticle is the collision point between two photons (the building blocks of photons), and it only lasts fo one Planck second as easued with Einstein-Poincae synchonized clocks. This eans that the foce fo any paticle oing at its axiu elocity ust be F = pa p = p = pc. (9) lp A Planck ass paticle should not be confused with two Planck asses: we can hae a heap of potons aking up a Planck ass, yet this is not a Planck ass paticle. A Planck ass paticle has a educed Copton waelength of. Fo non-planck asses, it sees that one pehaps needs to pefo a elatiistic adjustent fo gaity, but this is likely not the case fo gaity between light paticles (photons). Futheoe, we ust hae the following elationship fo elatiistic oentu p = p p = G Based on this, the elatiistic enegyoentu elation can then be witten as. (0)

3 E = p +( ) E = p p +( ) 0 u E = t@g pp A c u E = tg p p s E E E = s + + G pp c + p G p +. () Table suaizes the atheatical elationship between special elatiity and Newtonian gaity (Newton-inspied foulas). Mass Relatiistic ass Enegy Relatiistic enegy E = c Kinetic enegy E k = c Kinetic enegy ( <<c) Einstein = Newton c = G pp E = E k = G pp Foce F = c Relatiistic foce F = c Relatiistic enegy oentu elation p p = G E = p p +( ) G pp Table : The table shows soe siple atheatical elationships between the Einstein special elatiity foulas and Newton-inspied foulas. Table illustates the elatiistic liit fo the Einstein and Newtonian foulas. This is based on the axiu elocity fo anything with a est ass ax = c l p. 3

4 Einstein = Newton Relatiistic ass ax = ax ax c Relatiistic enegy E ax = = G pp ax ax c Kinetic enegy E k,ax = = G pp ax ax Relatiistic oentu p = ax ax c Relatiistic foce F ax = = G pp ax ax = p = p G pp = p l p = pc Table : The table shows the elatiistic axiu liit fo enegy, kinetic enegy and foce based on Haug s axiu elocity foula. The Speed of Gaity and Light Hidden Within the Newton Gaitational Foula? It is often assued that the speed of gaity in Newton gaitational theoy is instantaneous o altenatiely that the Newton foulas contains no infoation about the speed of gaity. Hee we will uestion this iew and clai that the speed of gaity in the Newton foula ust be the speed of light. The speed of light sees to be hidden within Newton s gaitational constant. Haug [9, 0] has ecently suggested that the Newton gaitational constant is a coposite constant gien by G = c 3 h, () whee is the Planck length, h is the educed Planck constant and c is the speed of light. One could easily conclude that knowing big G is necessay fo calculating the Planck length, and that big G theefoe ust be oe fundaental, and that the Planck length ust be a deied constant. Howee, it is fully possible to find the Planck length while haing no knowledge of big G (see []). Still, Newton could not hae discoeed this, since he was unawae of the Planck constant and the Planck length at that tie. This eans the Newton gaitational foula can be e-witten as F = G M = c 3 h M p h c = Mc p This foula is soewhat di eent than the foula (euation 7) we hae descibed fo fundaental paticles. The eason fo this is that we now hae genealized the foula fo objects of any size. Fo this foula, we can see that the speed of light is ebedded in the Newton foula. Newton actually knew the speed of light uite accuately. In his 704 book Opticks [], Newton, based on Røe s findings, calculated that it would take seen to eight inutes fo light to tael fo the Sun to the Eath. Howee, Newton did not link the speed of light to the speed of gaity; on the contay, he sees to hae belieed that gaity was an instantaneous foce. Still, the speed of light (gaity) is hidden in his gaitational foula. We in no way clai that Newton knew that his gaitational foula ebedded in the gaitational constant contained the speed of light. But Newton was not able to easue the gaitational constant in his foula, and he did not know that such easueents ae actually indiectly dependent on the speed of light as well as the Planck length and the Planck constant. The Newton gaitational constant was fist indiectly easued uite accuately in 798 by Caendish [3], who was also unawae that it was a coposite constant. If Newton had known the Planck length and the Planck constant, he could hypothetically hae found the speed of light fo gaitational obseations, such as the oon s obital elocity. If he had done so, he also could hae deteined that the speed of gaity was likely identical to the speed of light. (3) 4

5 3 Altenatie Way of Witing the Gaitational Foulas fo Any Object The analysis aboe leads us to an altenatie way to wite the any well-known gaitational foulas. Table 3 shows the standad gaitational foulas of Newton (and geneal elatiity fo bending of light). We see that all altenatie foulas contain the ass atio ultiplied by the adius atio. This is in ou iew uch oe intuitie than the standad foulas containing big G. In atheatical atois the Planck length is an indiisible paticle s diaete and the Planck ass paticle s adius. Futheoe, as entioned aboe, the Planck length can be easued independent of any knowledge of big G. The oe intuitie e-witten foulas suppot ou iew that big G is a coposite constant. And because it is a coposite constant, it actually eoes intuition when used without knowledge of what it actually is. The discoey of the Newton gaitational constant was natually a geat achieeent. In no way do we clai that the gaitational constant is unipotant. Howee, we clai that it is a coposite constant that consists of een oe fundaental entities, naely the Planck length, the educed Planck constant and the speed of light. Standad way Gaitational foce F = G M Obital elocity o = Escape elocity e = Bending of light Red shift Acceleation field GM GM = 4GM z() GM g = GM Altenatie way F = Mc p M l o = c p p e = c M p =4 M p z() M p g = c M p Table 3: The table shows the standad and altenatie ethods of witing gaitational foulas without using big G. All altenatie foulas contain the ass atio ultiplied by the adius atio. Again, we beliee these foulas ae oe intuitie than the taditional foulas containing big G. 4 Conclusion We hae pesented soe inteesting atheatical elationships between Einstein s special elatiity foulas and Newton s foulas. It sees like the Newtonian gaity foulas ae likely non-elatiistic. Still, they ae oe than that, as they also see to be the elatiistic liit based on Haug s axiu elocity foula. We do not clai that Newton knew about E = diectly. But indiectly, Newton essentially had an enegyass elationship foula that was alid fo any fundaentaaticle. He eely lacked the concept of educed Copton waelength and the adius fo fundaentaaticles. Futheoe, we hae seen that all the gaitational foulas contain the ass atio ultiplied by the adius atio. By witing the gaitational foulas in this way, we aoid big G and obtain uch oe intuitie foulas linked to soething that is conceptually easie to undestand. People can gasp the concepts of ass atio, adius atio and to soe extent pue enegy, but what does the gaitational constant epesent? This is had to gasp because, as we ague, it is a coposite constant. Refeences [] J. J. Thoson. Electicity and Matte. New Yok Chales Scibne s Sons, 904. [] A. Einstein. On the electodynaics of oing bodies. Annalen de Physik, English tanslation by Geoge Bake Je ey 93, (7),905. [3] I. Newton. Philosophiae Natualis Pincipia Matheatics. London,686. [4] M. Planck. The Theoy of Radiation. Doe 959 tanslation, 906. [5] E. G. Haug. Unified Reolution, New Fundaental Physics. Oslo, E.G.H. Publishing, 04. [6] E. G. Haug. The gaitational constant and the Planck units. A siplification of the uantu eal. Physics Essays Vol 9, No 4, 06. 5

6 [7] E. G. Haug. The Planck ass paticle finally discoeed! The tue God paticle! Good bye to the point paticle hypothesis! [8] E. G. Haug. The ultiate liits of the elatiistic ocket euation. The Planck photon ocket. Acta Astonautica, 36,07. [9] E. G. Haug. Planck uantization of Newton and Einstein gaitation. Intenational Jounal of Astonoy and Astophysics, 6(),06. [0] E. G. Haug. The gaitational constant and the Planck units. A siplification of the uantu eal. Physics Essays Vol 9, No 4, 06. [] E. G. Haug. Can the Planck length be found independent of big G? [] I. Newton. Opticks: O, a Teatise of the Reflexions, Refactions, Inflexions and Colous of Light. London, 704. [3] H. Caendish. Expeients to deteine the density of the eath. Philosophical Tansactions of the Royal Society of London, (pat II), 88,798. 6