Reductio ad Absurdum Modern Physics Incomplete Absurd Relativistic Mass Interpretation And the Simple Solution that Saves Einstein s Formula.

Transcription

1 Reductio ad Absudum Moden Physics Incomplete Absud Relativistic Mass Intepetation And the Simple Solution that Saves Einstein s Fomula. Espen Gaade Haug Nowegian Univesity of Life Sciences Januay, 208 Abstact This note discusses an absudity that is ooted in the moden physics intepetation of Einstein s elativistic mass fomula when v is vey close to c. Moden physics (and Einstein himself) claimed that the speed of a mass can neve each the speed of light. Yet at the same time they claim that it can appoach the speed of light without any uppe limit on how close it could get to that special speed. As we will see, this leads to some absud pedictions. If we asset that a mateial system cannot each the speed of light, an impotant question is then, How close can it get to the speed of light? Is thee a clea-cut bounday on the exact speed limit fo an electon, as an example? O must we settle fo a mee appoximation? Key wods: Relativistic mass, maximum velocity of subatomic paticles, bounday condition, Haug maximum velocity. Intoduction Einstein s elativistic enegy mass fomula [, 2] is given by Futhe, Einstein commented on his own fomula mc 2 q. () v 2 c 2 This expession appoaches infinity as the velocity v appoaches the velocity of light c. The velocity must theefoe always emain less than c, howevegeatmaybetheenegiesusedto poduce the acceleation Camichael (93) [3] came up with a simila statement in elation to Einstein s theoy: The velocity of light is a maximum which the velocity of a mateial system may appoach but neve each. We cetainly agee with Einstein s fomula. 2 Ou question is, How close can v be to c? Moden physics says nothing about this, except that it can appoach c, but neve each c. Does this mean that one can make it as close to c as one wants? This is what we will look into hee, and we will show that without a moe specific bounday condition on v this can lead to tuly absud pedictions. Einstein s elativistic mass equation pedicts that a mass will keep inceasing as the velocity of the mass appoaches the velocity of the speed of light. If v = c, thenthemasswouldbecomeinfinite. Einstein and othes have given an ad hoc solution to the poblem, namely in claiming that indeed the elativistic espenhaug@mac.com. Thanks to Victoia Teces fo helping me edit this manuscipt. Also thanks to Alan Lewis, Daniel Du y, ppaupe, and AvT fo useful tips on how to do high pecision calculations. This quote is taken fom page 53 in the 93 edition of Einstein s book Relativity: The Special and Geneal Theoy. English tanslation vesion of Einstein s book by Robet W. Lawson. 2 As a matte of fact, I have poven that Einstein s fomula is consistent with atomism, a belief system that I have eason to believe contains the ultimate depth of eality.

2 2 mass neve can become infinite, as this would equie an infinite amount of enegy fo the acceleation. Still, they also seem to claim that the speed of subatomic paticles can get as close to c as one would want. The discussion above is also fully elevant at today s univesity campus. Fo example in the excellent text book Univesity Physics by Young and Feedman (206) 3 states that When the paticle s speed v is much less than c, this is appoximately equal to the Newtonian expession...in fact as v appoaches c, the momentum appoaches infinity. Hee I have maked pat of the sentence in bold. univesity text book by Walke [5] we can ead 4 Similaly, in anothe well-known and excellent As v appoaches the speed of light, the elativistic momentum becomes significantly lage than the classical momentum, eventually diveging to infinity as v! c. Simila in the univesity physics text book by Cutnell and Johnson [6] we can ead 5 As v appoaches the speed of light c, the p v 2 /c 2 tem in the denominato appoaches zeo. Hence, the kinetic enegy becomes infinitely lage. Howeve, the wok-enegy theoem tells us that an infinite amount of wok would have to be done to give the object an infinite kinetic enegy. Since an infinite amount of wok is not available, we ae left with the conclusion that the objects with mass cannot attain the speed of light c. I do not diectly disagee; mathematically this is coect. My point is that moden physics does not give an exact limit on how close v can get to c, and we will soon see how this leads to absud elativistic masses and kinetic enegies. In the othewise excellent book on special elativity by Satoi [7] we can ead 6 Accoding to equation (7.2), the kinetic enegy of a body appoaches infinity when its speed appoaches c. Thisimpotantpedictionisconfimedbytheexpeimentaldata. I will claim that these statements ae patly wong, o at least they ae not pecise. No expeiment has shown that the kinetic enegy appoaches infinity. What has been shown is that the kinetic enegy inceases apidly as a paticle is acceleated towads a velocity significantly close to the speed of light. In 965, Max Bon [8] statedthat 7 Aglanceatfomula(78) 8 fo the mass tells us that the values of the elativistic mass m become geate as the velocity v of the moving body appoaches the speed of light. Fo v = c the mass becomes infinitely geat. Fom this it follows that it is impossible to make a body move with a velocity geate than that of light by applying foces: Its inetial esistance gows to an infinite extent and pevents the velocity of light fom being eached. Actually long ago, in 893 Thomson [9] wote 9 When in the limit v = c the incease in mass is infinite, thus the chaged sphee moving with velocity of light behaves as if its mass wee infinite... Natually, Thomson did not know about Einstein s theoy of special elativity, as it was published 2 yeas late. Still his equations pointed to a simila esult concening mass when v appoaches c. Fo futhe exploation, see a list of efeences stating simila pespectives in the Appendix. 2 The Absudity of the Electon Following Moden Physics Incomplete Relativistic Mass Intepetation An electon is a vey small so-called fundamental paticle with a est-mass of appoximately m e kg. Next let s look at the elativistic mass of the electon as v appoaches, but neve eaches, the speed of light. 3 4th edition page 238, fo full efeence see [4]. 4 Fouth edition, page Ninth edition page Page Page Bon is hee efeing to the Einstein elativistic mass fomula. 9 Page 2. Actually Thomson used V as symbol fo the speed of light and w fo the velocity of the object, we have eplaced these with c and v in the citation to make it easie to follow.

3 3 Absud One Kg Mass Electon Assume an electon is acceleated (by a giant exploding sta, o by the coe of a galaxy, fo example) to the following velocity v = c That is 70 nines behind the decimal point followed by the numbe 586, o we could say it is with nines instead of zeos afte the decimal point. It gives a elativistic mass fo a single electon of appoximately kg. Absud Moon Mass Electon Assume an electon is acceleated (by fo example a giant exploding sta, o by the coe of a galaxy) to the following velocity v = c That is 6 nines behind the decimal point followed by the numbe 23, o we could say it is with nines instead of zeos afte the decimal point. It gives a elativistic mass fo a single electon of appoximately kg, that is basically equal to the est-mass of the Moon. That is quite amazing, a tiny electon that suddenly has a elativistic mass equal to the est-mass of the moon! Whee can we find such electons? Absud Eath Mass Electon Assume an electon is acceleated to the following velocity v = c That is 9 nines behind the decimal point followed by the numbe 884, o we could say it is with nines instead of zeos afte the decimal point. It gives a elativistic mass fo a single electon of kg, that is basically equivalent to the est-mass of the Eath. Absud Sun Mass Electon Assume an electon acceleated to the following velocity v = c That is 30 nines behind the decimal point followed by the numbe 895. It gives a elativistic mass fo a single electon equal to the est-mass of the Sun, that is about kg. Absud Milky Way Mass Electon Assume an electon is acceleated to the following velocity v = c The elativistic mass of the electon at this velocity is equal to the est-mass of the Milky Way, that is about 0 2 sola masses. Still, the electon is taveling below the speed of light, so this does not go against mainsteam moden physics. Insane Obsevable Univese Electon Assume an electon is acceleated to the velocity of v = c

4 That is 74 nines behind the decimal point followed by the numbe 6, o we could say it is with nines instead of zeos afte the decimal point. It gives a elativistic mass fo a single electon of appoximately kg, that is basically equal to the est-mass of what main fame physics claims is the appoximately mass of the obsevable univese, see [0,, 2, 3]. That is quite amazing, a tiny electon that suddenly has a elativistic mass equal to the est-mass of the whole obsevable univese. Moden physics leads to absud kinetic enegies fo subatomic paticles The table below lists the elativistic kinetic enegy of an electon taveling at vaious velocities, all below the speed of light. All of these velocities ae valid inside the famewok of moden physics, as it stipulates no pecise speed limit on the velocity of an electon as long as it falls below the speed of light. Velocity of electon Relativistic electon mass Kinetic enegy: a Ton TNT equivalent: b % of light: = est-mass of (9 ns in font) Moon J (9 ns in font) Eath J (9 ns in font) Sun J (9 ns in font) Milky Way J Table : The table shows the kinetic enegy fo an electon taveling at vaious velocities below the speed of light. a The Kinetic enegy is calculated as E k = mc2 mc 2. v2 c 2 b One ton TNT equivalent is about 4.84 giga joules. Why don?t we see a single electon (o othe subatomic paticle) with a elativistic mass equal to (even at the most modeate level) the est-mass of the Moon? Such an electon would have enomous kinetic enegy, causing a gigantic impact with collision with the Eath, o othe planets in ou sola system. We suspect that mainsteam physics does not have a good answe to this question. Maybe such fast-taveling electons exist, but they ae ae and theefoe have a vey low pobability of occuing? What if, as a countepoint, a single electon wiped the dinosaus out? Ae we doomed? And why have we not head physicists discussing such velocities fo electons? Pehaps they simply do not like to talk about such things, as they have no good explanations fo why such vey fast electons have neve been obseved. 3 A Simple Solution to the Absudity that Saves Einstein s Relativistic Mass Fomula Einstein s special elativity fomula is pefectly coect, but it lacks an exact bounday condition on the velocity fo mass. Such a bounday condition has ecently been deived by Haug [4, 5, 6, 7, 8]. The maximum velocity any subatomic paticle can take as measued by Einstein-Poincaé synchonized clocks 0 is given by v max = c whee is the educed Compton wavelength of the mass in question, and l p is the Planck length [20, 2]. When inseted into Einstein s elativistic mass equation, this show that the maximum elativistic mass that any fundamental paticle can take actually is the Planck mass. The Planck mass is appoximately kg. It is enomous compaed to the electon, but still it is miniscule compaed to the mass of the Moon, Eath o the Sun. Futhe, the Planck mass only can last fo an instant, as pointed out by Haug. In othe wods, this seems to make pefect sense. Futhe, an electon can tavel at a velocity vey close to that of the speed of light, but its maximum velocity will still be significantly below what is descibed above. The maximum velocity fo an electon is s 2 (2) v max = c 2 e = c (3) 0 This also holds tue if measued with clocks synchonized with vey slow clock tanspotation method, see [9].

5 5 Because thee is some uncetainty in both the exact Planck length and the educed Compton wavelength, thee is some uncetainty aound this velocity, but it must be vey close to this numbe. We can est assued that the electon (o any othe mass) can neve each a elativistic mass close to even one kg, so thee is no chance that a single electon will cause much ham, no matte how fast it is acceleated. This is because thee is a maximum velocity that limits both its kinetic enegy and its elativistic mass. Will moden physics accept the existence of a maximum speed limit fo subatomic masses based on atomism o will they keep holding on to thei absud beliefs? If they do not accept the maximum velocity fo subatomic paticles given by atomism, then they must accept the following absudities: That thee is a wavelength shote than the Planck length. Something that is highly unlikely and impossible unde atomism. That thee is a maximum fequency highe than the Planck fequency. Something that is highly unlikely and impossible unde atomism. That an electon can take a elativistic mass simila to that of the Moon, the Eath, the Sun, and even the Milky Way, o even lage masses. This is, at best, tuly absud! Ou theoy shows that no subatomic paticle can take a elativistic mass highe than the Planck mass. That thee is no limit on the elativistic Dopple shift. This is also highly unlikely. Haug [5] has shown that the limit hee is the Planck fequency Dopple shift. Fo a subatomic paticle, thee is a momentum close to infinity. This is absud. The maximum momentum of a subatomic paticle is actually just below the Planck momentum. Fo a subatomic paticle, thee is a kinetic enegy close to infinity. This is, again, absud. The newly intoduced maximum velocity puts a seies of limits on subatomic fundamental paticles : The maximum fequency is the Planck fequency: f max =2 c l p. The maximum elativistic Dopple shift is equal to the Planck fequency. The maximum elativistic mass a subatomic paticle can take is the Planck mass. The maximum elativistic momentum a subatomic paticle can take is just below the Planck momentum. The maximum kinetic enegy a subatomic paticle can take is close to h l p c. The maximum elativistic length contaction of a subatomic paticle is 2l p,whichisthelengthof the Planck mass. 4 Ways to Wite the Maximum Velocity Fomula Thee ae seveal ways to wite the maximum velocity fo subatomic paticles that will all give the same answe; hee we pesent some of them. In tems of educed Compton wavelength v max = c 2 (4) In tems of paticle mass v max = c s m 2 m 2 p whee m is the est-mass of the paticle and m p is the Planck mass. As a function of Newton s gavitational constant Gm v max = c 2 hc All of these fomulas ae basically the same, but each one equies somewhat di eent input: s lp v max = c 2 = c m 2 Gm = c 2 2 m 2 p hc (5) (6) (7)

6 6 Electon the maximum velocity Fo an electon, the maximum velocity can be witten as function of the dimensionless gavitational coupling constant v max = c p G (8) This is no supise, since the dimensionless gavitational coupling constant is given by G = m2 e = l2 p m 2 2e p 5 Beakdown of Loentz Invaiance at the Planck Scale? The maximum velocity fomula fo anything with est-mass would mean Loentz invaiance beaks down at the Planck scale. Based on this view, the Planck paticle, the Planck length, and the Planck time, unlike any othe aticle, length o time, seem to be the same no matte what fame they ae obseved fom. The view that Loentz invaiance could be boken at the Planck scale appeas to be consistent with what is pedicted by seveal quantum gavity theoies, see fo example [26]. Loentz symmety is suppoted by a long seies of tests, but it has neve been tested at anything even close to the Planck scale (at distances close to the Planck length, o Planck enegies) so one should be caeful to use expeimental evidence as an agument against this idea. 6 Conclusion We conclude that in stating that a mass must tavel moe slowly than the speed of light, while at the same time asseting that it can appoach the speed of light, we get absud pedictions. Examples include the idea that an electon could attain a elativistic mass equal to the est-mass of the Moon, the Eath, the Sun, the Milky Way, o even entie galaxy clustes. Haug has ecently addessed this absudity by showing q that thee must be a pecise maximum velocity fo anything with mass given by v max = c Refeences [] A. Einstein. Ist die tägheit eines köpes von seinem enegieinhalt abhängig? Annalen de Physik, 323(3):639 64, 905. [2] A. Einstein. Relativity: The Special and the Geneal Theoy. Tanslation by Robet Lawson (93), Cown Publishes, 96. [3] R. D. Camichael. The Theoy of Relativity. JohnWiley&Sons,93. [4] R. A. Feedman and H. D. Young. Univesity Physics with Moden Physics, 4th Edition. Peason, 206. [5] J. S. Walke. Physics, Fouth Edition. Addison-Wesley,200. [6] J. D. Cutnell and K. W. Johnson. Physics, 9th Edition. JohnWiley&Sons,Inc.,202. [7] L. Satoi. Undestanding Relativity. Univesity of Califonia Pess, 996. [8] M. Bon. Einstein s Theoy of Relativity. Dove, 965. [9] J. J. Thomson. Recent Reseaches in Electicity and Magnetism. Oxfod Univesity Pess, 893. [0] G. J. Whitow. The mass of the univese. Natue, 85:65 66,946. [] P. Diac. The Pinciples of Quantum Mechanics. Oxfod Univesity Pess, 947. [2] J. C. Cavalho. Deivation of the mass of the obsevable univese. Intenational Jounal of Theoetical Physics, 34: ,995. [3] M. A. Pesinge. A simple estimate fo the mass of the univese. Jounal of Physics, Astophysics and Cosmology, 2,2009. Fo infomation about the dimensionless gavitational coupling constant see [22, 23, 24, 25]. 2.

7 7 [4] E. G. Haug. The Planck mass paticle finally discoveed! Good bye to the point paticle hypothesis! [5] E. G. Haug. A new solution to Einstein s elativistic mass challenge based on maximum fequency [6] E. G. Haug. The gavitational constant and the Planck units. A simplification of the quantum ealm. Physics Essays Vol 29., No 4, 206. [7] E. G. Haug. The ultimate limits of the elativistic ocket equation. The Planck photon ocket. Acta Astonautica, 36,207. [8] E. G. Haug. Can the Planck length be found independent of big G? Applied Physics Reseach, 9(6), 207. [9] E. G. Haug. Unified Revolution: New Fundamental Physics. Oslo, E.G.H. Publishing, 204. [20] M. Planck. Natulische Masseinheiten. De Königlich Peussischen Akademie De Wissenschaften, Belin, p. 479., 899. [2] M. Planck. Volesungen übe die Theoie de Wämestahlung. Leipzig: J.A. Bath, p. 63, see also the English tanslation The Theoy of Radiation (959) Dove, 906. [22] J. Silk. Cosmogony and the magnitude of the dimensionless gavitational coupling constant. Natue, 265:70 7, 977. [23] I. L. Rozental. On the numeical values of the fine-stuctue constant and the gavitational constant. Soviet Jounal of Expeimental and Theoetical Physics Lettes, 3(9):9 27,980. [24] M. D. O. Neto. Using the dimensionless Newton gavity constant g to estimate planetay obits. Chaos, Solitons and Factals, 24():9 27,2005. [25] A. S. Buows and J. P. Ostike. Astonomical each of fundamental physics. Poceedings of the National Academy of Sciences of the United States of Ameica, (7):3 36,203. [26] F. Kislat and H. Kawczynski. Planck-scale constaints on anisotopic Loentz and CPT invaiance violations fom optical polaization measuements. Physical Review D, 95, 207.