The special theory of relativity

Transcription

1 The special thery f relatiity The preliminaries f special thery f relatiity The Galilean thery f relatiity states that it is impssible t find the abslute reference system with mechanical eperiments. In ther wrds: The laws f mechanics are the same in eery inertial frame f reference. By the end f the XIX century, the physicists still beliee that there is an abslute frame f reference, and abslute time, but s far had been impssible t find it by mechanical eperiments. By this time hweer a lt f new and ery sensitie eperimental methds hae been deelped, using electrmagnetic waes, and electrdynamics. Physicists think that the unierse is fill ut with hypthetical aether r ether (it was assumed that the ether must be luminiferus, r ery tenuus, in rder t allw the planets t me freely thrugh it), and this is the abslute frame f reference, and the light waes prpagate in this hypthetical medium ether, just as the sund wae in the air. The duble rle f ether is: it is the abslute frame f reference (the abslute rest), and the medium in which the light prpagates. If ether eists, it is pssible t measure the speed f the arth with respect t ether. During the late nineteenth and early twentieth centuries, intensie effrts were made t find eperimental eidence fr the eistence f the ether. The Michelsn-Mrley eperiment was an effrt t detect mtin f the earth relatie t ether. This ery imprtant eperiment was taken in 1887 by Michelsn and Mrley, and was a significant rle in the birth f special thery f relatiity. The eperimental deice is called Michelsn-Mrley interfermeter. The eperiment was repeated seeral times and despite the rbital elcity the earth appeared t be at rest relatie t the ether. This and all similar eperiments yielded cnsistently negatie results and the ether cncept has been discarded. This negatie result baffled physicists f the time, and t this day the Michelsn-Mrley eperiment is the mst significant negatie result eperiment eer perfrmed. instein was wh discarded the cncept f ether and the abslute frame f reference and he stated the pstulates f the special thery f relatiity. 1. There is n abslute r preferred inertial system. The laws f physics are the same in all inertial frames. It is impssible t find difference between them by any physical eperiment.. ther des nt eist; the light waes prpagate istrpic way withut medium. The speed f light is the same in all inertial system c= km/s.

2 The Special Thery f Relatiity was wrked ut in 1905 by instein. The wrd special means that it deals nly with inertial reference frames. There is a cntradictin between the special relatiity and the Galilean relatiity. Cnsider the well knwn Galilean crdinate transfrmatin equatins between tw inertial frames, K and K. Let the -aes f the tw frames lie alng the same line, but let the rigin O f K me relatie t the rigin O f K with cnstant elcity. The rigins cincide at time t=0. = cnstant, and t = 0, when O = O K K y y O O ach bserer recrds the same eent, which might be the detnatin f tiny flashbulb, and assigns space and time crdinates t the eent, namely,y,z,t and,y,z,t. What are the relatins between these tw sets f numbers? = + t = t y = y y y = the Galilean transfrmatin equatins z = z z = z t = t t = t d d d = = = dt dt dt = y = y z = z elcity transfrmatin This equatin is in cntradictin with principle f special thery f relatiity, that the speed f light is cnstant. The cause f this is that the same time scale is applied t = t. The time is nt the same fr each bserer. The Galilean transfrmatin equatins are crrect in the imprtant regin c, but fail as c. The crrespnding equatins are the s called Lrentz transfrmatin equatins, withut pring: ( ) γ ( ) = γ t = + t y = y y = y z = z z = z t = γ t t γ t = + c c γ = 1 1 c Lrentz-factr

3 The transfrmatin equatins hae sense until < c (γ -must be real number). In the Special thery f Relatiity the speed f light is an upper limit fr the speeds f material bjects. If << c r c, the Lrentz equatins reduce t Galilean equatins. S fr slw bject the classical r Newtnian mechanics is crrect. As the transfrmatins are linear in the ariables, they can be changed t differentials r finite differentia. It means instead f we can use d r Δ. Δ = γ ( Δ Δt) Δ = γ ( Δ + Δt ) Δ y = Δy Δ y = Δy Δ z = Δz Δ z = Δz Δ t = γ t t γ t Δ Δ Δ = Δ + Δ c c ( ) γ ( ) d = γ d dt d = d + dt dy = dy dy = dy dz = dz dz = dz dt = γ dt d dt γ dt d = + c c The Lrentz equatins hae seeral interesting cnsequence, elcity transfrmatin, time dilatatin, and length cntradictin. elcity transfrmatin: At gien calculate. d γ ( d + dt ) + = = = dt γ dt + d 1+ c c If = c, then c+ c+ = = = c. c+ 1+ c c c Time dilatatin: Let measure a time interal in K Δ t, fr the bserer in K this interal is Δ t, and Δ t > Δ t, which is called time dilatatin, we ften erbalize it as the ming clcks run slw. Length r Lrentz cntradictin: Let measure the length f a rd in K Δ, fr the bserer in K this length is measured Δ, and Δ <Δ, and it is called length cntradictin. Relatiistic dynamics

4 Due t the pstulates f the special thery f relatiity, the laws f physics are the same in all inertial frames. 1. The Mawell equatins are eact equatins, because they are inariant under the Lrentz crdinate transfrmatin.. The Newtn s laws f mtin are inariant under the Galilean crdinate transfrmatin, but in case f relatiistic mechanics we hae t apply the Lrentz transfrmatins, and they are nt inariant under Lrentz crdinate transfrmatin. This requires generalizatin in the laws f mtin, and the definitins f mmentum and energy. In the special thery f relatiity we can apply the equatin: dp F dt =, the frce is the time rate f change f mmentum, where p = m, but the mass is nt cnstant any mre. At arund 1901 Kaufmann measured the dependence f the inertial mass (r mmentum) f an bject n its elcity. actly he measured the electrn charge t mass rati fr different elcities f the electrn. In 1901 he was able t measure a decrease f the charge-t-mass rati, thus demnstrating that mass r mmentum increases with elcity. The ariatin f mass with elcity is gien by the relatiistic mass frmula: m m= mγ =, where m is the rest mass, that is, the mass measured when the particle 1 c is at rest with respect t the bserer, and γ is the Lrentz factr. m m 0 O 0,5 1 c m is the relatiistic mass, m 0 is the rest mass. If, c, then m. The relatiistic equatin f mtin d dm d F = ( m) = + m dt dt dt Since we generalized the Newtn s II law t bring it int accrd with the principle f relatiity, it is nt surprising that the energy cncept als requires generalizatin. In classical r Newtnian mechanics the definitin f the mmentary pwer f frce is dt 1 P = F, and the pwer law P =, where T = m, the kinetic energy. dt This equatin states that the time deriatie f the kinetic energy is equal t the pwer deliered by the frce. This is the s called pwer law.

5 In case f relatiistic mechanics the frm f the pwer law: d P= ( mc ). dt It means that we can keep the riginal frm f the pwer law, if we intrduce the relatiistic kinetic energy as: T = mc + K, where K is cnstant. It is a triial agreement, that if =0, then T=0. 0 = mc + K, K = m c. 0 Thus the relatiistic kinetic energy: 0 ( γ 1) T = mc mc = mγc mc = mc. Of curse if is much smaller than c, we can 1 btain the classical kinetic energy epressin T = m. mc = T + m c = mc, ttal r relatiistic energy = mc, rest energy T, kinetic energy = T + 0 The rest energy is a new cncept. If =0, then T=0, and = The best knwn result f the special thery f relatiity is the s called mass energy equialence. = mc. That is the cnseratin f ttal energy is equialent t the cnseratin f relatiistic mass. Mass and energy are equialent they are cnnected by a uniersal cnstant the speed f light. In classical physics we had tw separate cnseratin principles: cnseratin f mass, and cnseratin f energy. In relatiity these merge int ne cnseratin principle, that f the cnseratin f mass-energy. The mst imprtant eperimental erificatin f the mass energy equialence is the Cckrft Waltn eperiment (193). After discering the particle acceleratrs, resting Li atms were bmbarded by accelerated prtns. This was the first artificial splitting f a nucleus. It was als the first transmutatin using artificially accelerated particles. The fllwing reactin tk place: H + 3Li He+ He Li H He In the reactin it was fund fr the rest masses: m + m > m, s a mass defect appeared He Li H Δ m = m m m < 0. But at the same time the kinetic energy f the tw He nuclei is greater than the kinetic energy f the prtn and Li nucleus: T + T > T + T, that is kinetic energy grwth appeared He1 He H Li Δ T = THe + T 0 1 He > T Li + TH > Due t eperience in the reactin mass cnerted int kinetic energy Δ T +Δ mc = 0. This reactin was the first eperimental prf f instein's part f rest energy was cnerted int kinetic energy. = mc. We can say that sme