1 st axiom: PRINCIPLE OF RELATIVITY The laws of physics are the same in every inertial frame of reference.

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1 SPECIAL ELATIVITY Alber EINSTEIN inrodued his SPECIAL THEOY OF ELATIVITY in 905. To undersand he heory, we will firs reiew he bakground, he heoreial and experimenal deelopmens sine Newon. SPECIAL means jus, no general. The heory is abou inerial frames only. eall ha, for Galileo and Newon, suh frames had a enral imporane in mehanis and osmology, as, by he PINCIPLE OF ELATIVITY, no laboraory-based measuremen ould esablish one raher han anoher as a res, or in moion. s axiom: PINCIPLE OF ELATIVITY The laws of physis are he same in eery inerial frame of referene. Galileo and Newon were familiar wih he relaionships among inerial frames. If in he frame S he oordinaes of an een are ( x, ), hen, in a frame S moing a relaie eloiy in he x-direion is oordinaes are ( x, ). If we le he origins oinide a x x 0, 0, hen he oordinaes are relaed by x x These equaions are he Galilean Transform. (Adding equaions for y and z is riial). All of mehanis is inarian under he Galilean ransform. This means ha he same formulae for, e.g., he ollisions of billiard balls or for he Coriolis fore, are reoered if we subsiue x for x, for, and hen simplify. The eloiy simply drops ou. Mehanis is he same in any inerial laboraory, whaeer eloiy i may be raelling a. James Clerk MAXWELL ook he experimenal findings of Ampère, Faraday, and Coulomb on eleriiy and magneism, and synhesised hem ino a beauiful sysem of four iner-linked equaions (see EMF, Semeser B). His Treaise on Eleriiy and Magneism was published in 873. Ligh urned ou o be an eleromagnei wae, wih a eloiy of m s -. I soon beame lear ha Maxwell s equaions are no inarian under he Galilean Transform. One inerpreaion was ha here is a saionary res frame, in whih ligh has he eloiy. This would be he luminiferous eher, a posulaed medium for ligh o wae in. Howeer, his inerpreaion did no agree wih experimen (sellar aberraion, he Mihelson-Morley aemp o measure he speed of he earh hrough he eher, e). Oher people sough ways o reain he Priniple of elaiiy. Consider he more general linear ransform, x ax b αx β

2 wih he oeffiiens a, b, α and β being funions of. I urns ou ha he simple requiremen ha he bak ransform (x and expressed in erms of x and ) resris he ransform o he form x ax a a x a a wih he single parameer a appearing in eah of he four oeffiiens. Noe ha he Galilean Transform is obained by puing a. The Lorenz Transform: I was Lorenz who noied ha Maxwell s Equaions are inarian under his ransform if a has he alue a The ransform wih his alue of a is herefore alled he Lorenz Transform. Noe ha a ordinary speeds, a is ery lose o uniy. E.g. for speeds ypial of saellies, say 0 miles per seond, a A firs, no-one knew wha his ould mean. Fizgerald noied ha his ransform would shoren moing bodies, and his has led o he erm, Fizgerald onraion, being used in relaiiy. The Lorenz ransform applied o eleri and magnei fields does oner hem ino eah oher, so ha an eleri harge saionary in he frame S produes only eleri field. Viewed from he frame S moing relaie o S, magnei field is seen oo. This is why Einsein s key 905 paper was iled, On he Elerodynamis of Moing Bodies (in German). Wha Einsein showed was ha all of his required a new heory of spae and ime. We shall deelop ha heory, ogeher wih a rigorous definiion of wha i is ha is being hanged. There are wo major misonepions in he lieraure: One is ha Newon s Laws of Moion were shown o be unrue, or only approximaely rue a small eloiies. Tha is no so. As oulined aboe, i is assuming ha he Priniple of elaiiy is rue, ha Newon s Laws are rue, and ha Maxwell s equaions are rue, ha fores a new heory of spaeime. The oher is ha elaiiy deried from wo axioms, o The Priniple of elaiiy o The Priniple of he Consany of he Speed of Ligh In fa, Einsein ofen presened elaiiy in his way in laer books, and we will do so here.

3 SETTING UP A SPACE TIME COODINATE SYSTEM. Eens are fundamenal. They happen in spae and ime hey hae a where and a when. They are independen of a referene frame used o refer o hem, bu we hae o se up a referene frame o gie alues o he where and when o desribe he eens. Galileo, Newon and ohers inrodued he onep of an Absolue Frame of eferene whih was defined o be a res Absolue es. Then all frames moing a a onsan eloiy relaie o his frame are alled Inerial Frames of eferene. In hese frames, Newon s Laws of Moion hold, wihou fiiious fores. Frames whih are roaing or aeleraing are alled non-inerial and are deal wih in General elaiiy. Lengh and Time Measuremen Le frame S be moing a onsan eloiy relaie o S. Le all axes be parallel, and be along he x- axis of S. Take 0 when he origins oinide, when O O. Eens are obsered in boh frames, and labelled by oordinaes in spae and ime, ( x, y, z, ) or ( x, y, z, ). The spaial oordinaes are measured by mere rules or yardsiks and he emporal oordinae by loks. Comparison of obseraions in S and S requires omparison of he yardsiks and loks used by he obserers in S and S. I is easy o ompare wo yardsiks when hey are saionary jus lay hem alongside eah oher and equally easy o synhronise wo saionary loks and o hek ha hey run a he same rae. Then we make wo assumpions based on he Priniple of elaiiy: One yardsiks are ompared wih eah oher a res, heir lenghs remain unhanged when in moion. One loks are synhronised and ompared, hey run a he same rae when in moion. Noe ha hese assumpions hold in Newonian spae-ime; we shall see laer ha hey boh remain rue in Speial elaiiy. (Tha is, here is no onraion of a moing yardsik of he sor Fizgerald enisaged.) Le een P our a he posiion r ( x, y, z) a ime in S, and a he posiion r ( x, y, z ) a ime in S. Then r r is he Galilean ransformaion ha gies oordinaes of P in S if oordinaes in S are measured. And

4 r r is he Galilean ransformaion ha gies oordinaes of P in S if oordinaes in S are measured. Transform of Lengh Le a rod of lengh L be a res in S. Measuremen of is lengh onsiss of finding he oordinaes of is ends. This requires wo eens, whih need no be simulaneous, yielding, e.g., x and x and a lengh of L x x. An obserer in S, howeer, will wan he wo measuremens eens o be simulaneous. The lengh ransforms as L x x x x ( x x ) ( ) whih gies he righ answer, L x x, only if (or if he orreion for lak of simulaneiy is made). Measuremens of he same physial quaniy, made in S and S, are made under differen physial ondiions. Transform of Time Inerals Le a lok be a res in S and wo eens P and P ake plae nex o i (i.e. a he same plae.) The obserer in S need only look a he lok o obsere he posiions of is hands a he wo eens. The obserer in S sees he wo eens ourring a wo differen plaes, and so she has o look a wo separae loks in her own frame (or a one moing lok in he S frame). Measuremens of he same physial quaniy, made in S and S, are made under differen physial ondiions. Transforms of Veloiy and Aeleraion We merely need o differeniae wih respe o or x x dx dx d dx u u d d d d d x d x d a a d d d In general, u u a a

5 Transforms of Mass and Fore Obserers in S and S agree on aeleraion. They ahiee onsiseny if, haing wrien, F ma F m a hey agree ha mass and fore are unhanged and ha Newon s nd Law is unhanged. (Tha is wha inarian under a ransformaion means.) Transforms of Momenum and Energy I follows ha obserers in S and S agree ha momenum and mehanial energy are onsered, een hough hey will disagree on he alues (as hey depend on ). Consider a ball hrown wih iniial eloiy omponens upwards, 0 y and horizonal, 0 in S. The obserer in S will wrie, x 0 y ½ ETo mgh ½m x ½m The obserer in S moing a 0 will wrie x onsan in S E To m g h ½m Sine, as we hae seen, m m g g h h y y i follows ha E To differs only by a onsan from E To and is herefore also onsered. The analysis for momenum is ery similar, and leads in he same way o he onlusion ha p differs from p only by a onsan and is herefore also onsered. y

6 Transforms in Speial elaiiy I is mos elegan o proeed direly from he Lorenz inariane of Maxwell s equaions of eleromagneism. Howeer, his is mahemaially raher diffiul. Mos books, and indeed Einsein s own laer books, proeed from a deduion from Maxwell s equaions ha he speed of ligh is ~ m s -. Inariane of physial laws o he eloiy of he laboraory requires ha he speed of ligh is hen in any inerial frame. If wo obserers, in S and in S, measure he eloiy of he same ligh beam, hey mus boh obain he alue. This onradis he Galilean ransform, aording o whih if one obserer measures, he oher mus obain. There was some experimenal eidene for a onsan speed of ligh. Firs, from obseraions of he moons of Saurn, i was lear ha he speed of ligh does no depend on he eloiy of he soure unlike a rifle bulle. Sine ligh was known o be a wae, his was no surprising, sine waes rael a a speed defined relaie o he medium, no he soure. A medium for ligh waes alled he Aeher had been posulaed. Mihelson and Morley (887) proposed o measure he speed of he Earh hrough he aeher, by omparing he speed of ligh parallel o he Earh s moion and perpendiular. They failed. This was he seond piee of experimenal eidene for a onsan speed of ligh. The Mihelson-Morley Experimen. The experimen uses a Mihelson Inerferomeer: Mirror M (moable) Monohromai ligh beam Beam-splier (halfsilered mirror) Mirror M (fixed) Deeor

7 The wo arms are of lenghs and. If, he wo pahs inerfere onsruiely and he deeor dees ligh. This is alled a brigh fringe. If he pah lenghs are differen, hen if λ n inerferene is sill onsruie and a brigh fringe is obsered. If λ ½ n hen inerferene is desruie and no ligh falls on he deeor (a dark fringe). If M is moed, a suession of ligh and dark fringes is obsered. Use as a Speedomeer for he Earh The idea is o pu he insrumen wih one arm (say, arm ) parallel o he Earh s moion, and he oher arm,, perpendiular o he moion. The eloiy of he Earh in is orbi around he sun is ~ 0-4, so his is he kind of resul expeed. If he ligh akes ime ½ o go from he beam-splier o mirror M, he apparaus moes a disane s ½ in ha ime. The disane he ligh raels is no, bu l So 4 4 l In beam, he ime aken o go from he beam-splier o he mirror and bak is So he differene in ime in arms and is Leing, puing β / and using he binomial heorem ( / is small), β This ranslaes ino an apparen pahlengh differene of 8 ~ 0 β.

8 This pahlengh differene may be expressed as a number of waelenghs, β n λ λ whih for m (using a muliple pah insrumen), β 0-4, λ 550nm, gies n 0.4. The apparaus is now roaed hrough 90, inerhanging he wo arms. The fringe shif is n n Mihelson and Morley rekoned hey ould see a fringe shif as small as n 0.0. Their measured resul was null, i.e. n 0 ± 0.0. (Laer improed ersions of he experimen ahieed n 0 ± 0.00.) Conlusion: The apparaus does no work as a speedomeer for he Earh. One inerpreaion is ha he speed of ligh is measured o be whaeer he speed of he laboraory. Tha Eleromagneism has no absolue frame of referene. So his resul has ery ofen been used (by Einsein himself and by mos exbook wriers) o jusify saing wo Posulaes and deriing Speial elaiiy from hem: The Priniple of elaiiy The Priniple of he Consany of he Speed of Ligh. We shall follow ha pah, despie Einsein s own laim ha he hadn heard of he Mihelson-Morley experimen by 905 when he published Speial elaiiy!