PHY3101 Modern Physics Lecture Notes Relativity 2. Relativity 2
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1 PHY30 Modern Pysis eture Notes Reltiity Reltiity Dislimer: Tese leture notes re not ment to reple te ourse textbook. Te ontent my be inomplete or een inurte. Some topis my be unler. Tese notes re only ment to be study id nd supplement to your own notes. Plese report ny inuries to te proessor. Einstein s Postultes Te bsene o ny ringe sit in te Mielson-Morely experiment or ny orienttion o te intererometer nd or ny time o te yer negted te eter ypotesis or ligt propgtion. igt wes re osilltions o te eletromgneti ield, nd no propgtion medium is neessry, unlike sound wes. Howeer, i Glilen trnsormtions re orret, ten Mxwell s equtions must be modiied or eery possible reerene rme to ount or dierent eloities or te speed o ligt. Einstein ssumed te opposite: tt Mxwell s equtions re undmentlly orret, but tt our intuitie Glilen trnsormtion is not. Tis led to te ollowing two postultes:. Te lws o pysis, inluding eletromgnetism, re te sme in ll inertil rmes.. Eery obserer mesures te sme lue or te speed o ligt (in uum) in ll inertil rmes. Te seond postulte is relly onsequene o te irst, beuse i Mxwell s equtions old in ll inertil rmes, ten te only possible lue or te speed o ligt is. Tese postultes embody Einstein s Speil Teory o Reltiity, irst publised in 905 in pper titled On te Eletrodynmis o Moing Bodies. ter e would inorporte grity nd elertion in is Generl Teory o Reltiity. As in Newtonin Reltiity, tere is no wy to detet bsolute motion. Only te reltie eloities between two inertil reerene rmes mtters. Tese seemingly simple postultes e extrordinry onsequenes. For exmple, wen you turn on te edligts o r, te ligt bem lees te r t reltie eloity o m/s. Howeer, someone stnding on te sidewlk lso mesures te speed o te ligt bem s independent o te eloity o te r! How n tis be? As we sll see, our onepts o spe nd time must be modiied. D. Aost Pge 8/5/00
2 PHY30 Modern Pysis eture Notes Reltiity Bsi Deinitions Eents re pysil penomen tt our independent o ny reerene rme. For exmple: ls, explosion, return o spesip, or disintegrtion o subtomi prtile. Obserers reord eents, bot te time nd sptil oordintes, in prtiulr reerene rme. For exmple, Mission Control in Houston mrking down te time nd lotion o te splsdown o spe psule. Te reerene rme in tis se is te Ert. Simultneous eents our wen te ligt signls rom two eents re n obserer t te sme time Reltiity o Simultneity: Two eents simultneous in one inertil rme re not simultneous in ny oter rme. Tis is onsequene o Einstein s Postultes. Proper time is te time dierene between two eents ourring t te sme position (Denoted by t 0 or τ ). Rest rme is te inertil rme were two eents re only seprted by time. Te rme in wi te proper time is mesured Proper lengt is te distne between two positions t rest, te lengt mesured in te rest rme. (Denoted by 0 ). Now tt we re rmed wit tese deinitions, let s explore te onsequenes o te onstny o te speed o ligt in ll inertil rmes. D. Aost Pge 8/5/00
3 PHY30 Modern Pysis eture Notes Reltiity Time Diltion We explore te rte o time in dierent inertil rmes by onsidering speil kind o lok ligt lok wi is just one rm o n intererometer. Consider ligt pulse bouning ertilly between two mirrors. We nlyze te time it tkes or te ligt pulse to omplete round trip bot in te rest rme o te lok (lbeled S ), nd in n inertil rme were te lok is obsered to moe orizontlly t eloity (lbeled S). In te rest rme S mirror t time up t time down t + t mirror τ Now put te ligt lok on spesip, but mesure te roundtrip time o te ligt pulse rom te Ert rme S: t t time up t t time down Te speed o ligt is still in tis rme, so + t / 4 t / 4 t / 4 4 t τ t t / t / So te time it tkes te ligt pulse to mke roundtrip in te lok wen it is moing by us is ppers longer tn wen it is t rest. We sy tt time is dilted. It lso doesn t mtter wi rme is te Ert nd wi is te lok. Any objet tt moes by wit signiint eloity ppers to e lok running slow. We summrize tis eet in te ollowing reltion: t γ τ γ / D. Aost Pge 3 8/5/00
4 PHY30 Modern Pysis eture Notes Reltiity engt Contrtion Now onsider using ligt lok to mesure te lengt o n intererometer rm. In prtiulr, let s mesure te lengt long te diretion o motion. In te rest rme S 0 τ Now put te ligt lok on spesip, but mesure te roundtrip time o te ligt pulse rom te Ert rme S: t t time out t time bk t t + t + t t t t t t + t t + t t / But, t τ / rom time diltion A A C C 0 γ γ / In oter words, te lengt o te intererometer rm ppers ontrted wen it moes by us. Tis is known s te orentz-fitzgerld ontrtion. It is losely relted to time diltion. In t, one implies te oter, sine we used time diltion to derie lengt ontrtion. Time diltion nd lengt ontrtion re onsequenes o te ssumption tt ll obserers mesure te sme lue or te speed o ligt. Tis mens tt time runs t dierent rtes or dierent inertil rmes. Tere is no bsolute time, time only s reltie mening. ikewise, lengt lso s only reltie mening. Eeryting depends on te reltie eloity between two objets. We only notie tese strnge eets wen te eloity is ner, oweer. D. Aost Pge 4 8/5/00
5 PHY30 Modern Pysis eture Notes Reltiity Te orentz Trnsormtion We re now redy to derie te orret trnsormtion equtions between two inertil rmes in Speil Reltiity, wi modiy te Glilen Trnsormtion. We onsider two inertil rmes S nd S, wi e reltie eloity between tem long te x- xis. y S y' S' z x z' x' Now suppose tt tere is single ls t te origin o S nd S t time t t 0, wen te two inertil rmes ppen to oinide. Te outgoing ligt we will be speril in spe moing outwrd wit eloity in bot S nd S by Einstein s Seond Postulte. x + y + z t x + y + z t We expet tt te ortogonl oordintes will not be eted by te orizontl eloity: y y z z But te x oordintes will be eted. We ssume it will be liner trnsormtion: x k x t x k x + t But in Reltiity te trnsormtion equtions sould e te sme orm (te lws o pysis must be te sme). Only te reltie eloity mtters. So k k. Consider te outgoing ligt we long te x-xis (y z 0). x t x t in rme S' in rme S D. Aost Pge 5 8/5/00
6 PHY30 Modern Pysis eture Notes Reltiity Now plug tese into te trnsormtion equtions: x k x t k t t kt x k x + t k t + t kt + nd Plug tese two equtions into te ligt we eqution: t x kt t x kt + t kt t kt + Plug t into te eqution or t: t k t + k k / γ So te modiied trnsormtion equtions or te sptil oordintes re: γ x x t y y z z Now wt bout time? x γ x t x γ x + t Plug one into te oter: x γ γx γt + t inerse trnsormtion D. Aost Pge 6 8/5/00
7 PHY30 Modern Pysis eture Notes Reltiity Sole or t : x γ x + γ t γt x γ + γ t γt x + γ t γt γ x + γ t γt t γ x + γ t γ t γ t x / So te orret trnsormtion (nd inerse trnsormtion) equtions re: x γ x t x γ x + t y y y y z z z z t γ t x / t γ t + x / Note te ollowing etures: We reoer te Glilen trnsormtion i (or 0) so tt γ Spe nd time oordintes re mixed (x,t) No nge in orm o equtions rom one rme to noter (Einstein s st postulte) Only reltie eloities mtter Also, note tt you n derie time diltion nd lengt ontrtion rom tese equtions. For exmple, i lok sits t rest in rme S t position x 0, ten n obserer in rme S mesures te period o te lok to be T γ τ. Moreoer, note tt two eents wi re simultneous in rme S (sy t time t 0 nd t positions x nd x ) re not simultneous in rme S (t t ). Te orentz Trnsormtion D. Aost Pge 7 8/5/00
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