A Special Hour with Relativity

Transcription

1 A Special Hour wih Relaiviy Kenneh Chu The Graduae Colloquium Deparmen of Mahemaics Universiy of Uah Oc 29, 2002

2 Absrac Wha promped Einsen: Incompaibiliies beween Newonian Mechanics and Maxwell s Elecromagneism. We will sae he spaceime hypoheses and he posulaes Special Relaiviy and use hem o derive he ransformaion law beween inerial frames in Special Relaiviy, which implies ha he Minkowski meric on spaceime can be defined. We will discuss four phenomenon prediced by Special Relaiviy: ime dilaion, lengh conracion, breakdown of universal simulaneiy and he win paradox.

3 Newonian mechanics is relaivisic oo! Newon s Second Law: F = mẍ Galilean Transformaion beween inerial observers in Newonian Mechanics: { = x = x v, where v is he consan relaive speed beween he wo observers. Since he Second Law involves only ẍ, i holds for boh observers. Roughly speaking, inerial observers are non-acceleraing observers. Their observaions are no conaminaed by acceleraion of heir own reference frames.

4 Relaiviy is a Good Thing. The invariance of Newon s Second Law under Galilean Transformaions is called Newonian Relaiviy. Newon, Einsein, and basically physiciss of all ime consider ha naural laws should ake he same form for any inerial observers i.e. all physical laws should exhibi relaiviy. Mahemaically, relaiviy means ha he equaions describing any physical law should be invarian under he group of ransformaions beween inerial observers.

5 The crux is: How do inerial frames ransform? The ruh is: Newon didn know! He adoped Galilean ransformaions basically because he waned (Newonian) relaiviy and he realized ha his Second Law is invarian under Galilean ransformaions. And, of course, Galilean ransformaions were he absoluely reasonable hing o adop based on everyday experience all ha was available o Newon a he ime.

6 Newon s misake : He assumed universal ime! Recall Galilean Transformaion: { = x = x v, The adopion of he firs equaion in effec acily (and wrong in ligh of Special Relaiviy) assumes he exisence of universal ime: I is possible o define a universal ime agreed by all inerial observers in he enire universe as follows: Equip all inerial observers wih idenical and synchronized clocks. Then hese clocks say synchronized everywhere in he universe, for all ime, for all inerial observers! Phrased as such, i should be obvious ha = is in fac a huge assumpion and Newon knew i!

7 Signs of Weakness of Newonian Mechanics A he urn of he las cenury, here arose a new rising sar among he physical heories: Maxwell s Elecromagneism Lorenz derived he ransformaions under which Maxwell s equaions are invarian. They are called he Lorenz ransformaions. The VERY Bad News Is: Lorenz ransformaions are differen from Galilean ransformaions!

8 Signs of Weakness of Newonian Mechanics (con d) Also, Maxwell s equaions predic ha he speed of ligh in a vacuum is a consan ( m/s) for EVERY observer for whom Maxwell s equaions hold! Bu Newonian mechanics predics ha he speed of a ligh signal (or any moving objec) as observed by wo inerial observers mus differ by he relaive speed of he wo observers. In a nushell, Newonian Mechanics and Maxwell s Elecromagneism are incompaible unless relaiviy is o fail for one of hem.

9 Three Obvious Opions 1. There is an eher, he medium in which ligh ravels, which provides a preferred inerial frame for elecromagneism. Boh Newon and Maxwell were righ, bu relaiviy fails for elecromagneism. Maxwell s equaions hold only in he Eher frame. 2. Maxwell was wrong. Relaiviy holds for boh Mechanics and Elecromagneism, bu Maxwell s elecromagneism needs o be modified. 3. Newon was wrong. Relaiviy holds for boh Mechanics and Elecromagneism, bu Newon s mechanics needs o be modified.

10 The Michelson-Morley and Similar Experimens Assume he Eher exiss. Newonian relaiviy implies ha inerial observers ravelling a a non-zero speed wih respec o he Eher frame mus measure a differen speed of ligh. A vas amoun of effor was spen o devise ever more sophisicaed experimens o measure hese discrepancies of ligh speed. Bu, every single one failed: The measured ligh speed was invariably ha same consan prediced by Maxwell!

11 The Emission Theories: Aemps o modify Maxwell s Elecromagneism I suffices o say: All hese aemps were conradiced by experimens.

12 In comes Einsein!

13 Special Relaiviy = Minkowski Geomery The mahemaics of Special Relaiviy is Minkowski geomery: (Graviy-free) spaceime is modelled by he following disance funcion (meric): 4 equipped wih Le P 1 and P 2 be wo poins in spaceime (will be called evens). Le P 1 = ( 1, x 1, y 1, z 1 ) and P 2 = ( 2, x 2, y 2, z 2 ) be he respecive coordinaes of hese wo poins as measured by any fixed inerial observer. Then he (possibly negaive!!) disance beween P 1 and P 2 is given by: P 1 P 2 2 = ( 1 2 ) 2 (x 1 x 2 ) 2 (y 1 y 2 ) 2 (z 1 z 2 ) 2.

14 Special Relaiviy = Minkowski Geomery (con d) The definiion P 1 P 2 2 = ( 1 2 ) 2 (x 1 x 2 ) 2 (y 1 y 2 ) 2 (z 1 z 2 ) 2, of course, acily assumes ha he RHS is independen of he inerial observer whose coordinaes for P 1 and P 2 are being used.

15 Our Plan: To prove he well-definiion of he Minkowski meric 1. Sae he definiion of inerial observers. 2. Sae he assumpions on Spaceime. 3. Sae he wo posulaes of Special Relaiviy. 4. Derive, using he above assumpions, he ransformaion law beween inerial observers. This will urn ou o be he Lorenz ransformaions. The Lorenz ransformaion law in urn implies he well-definiion of he Minkowski meric on Spaceime.

16 Definiion of an Inerial Observer Inerial Observer := Un-accelaraed self-es Newon s Firs Law holds for hem Able o measure ime inervals and spaial disances Assumpions on Spaceime 1. Homogeneiy: No experimen performed by any inerial observer can deermine a preferred even (poin) in Spaceime. 2. Isoropy: No experimen performed by any inerial observer can deermine a preferred direcion in Spaceime.

17 The Two Posulaes of Special Relaiviy 1. Principle of Relaiviy: All physical laws (no jus mechanical ones, as wihin he framework of Newonian mechanics) are idenical in all inerial frames. i.e. he oucome of any physical experimen is he same when performed wih idenical iniial condiions relaive o any inerial frame. 2. Consancy of he Speed of Ligh: Ligh signals in vacuum propagae recilinearly wih he same speed, a all imes, in all direcions, in all inerial frames. wih respec o he assumpions on Spaceime

18 Coordinaizaion of Spaceime by an Inerial Observer 1. She picks an arbirary spaceime poin as origin. 2. Her abiliy o measure ime inervals allows her o se up a ime axis (her world line) in spaceime. 3. She defines simulaneiy in her own reference frame using he consancy of he speed of ligh.

19 Coordinaizaion (con d) O P 0 = = Q

20 Le O and O be wo inerial observers, ravelling a fixed relaive speed v, who coincide in spaceime a one (and only one) spaceime poin. For simpliciy, we work in only one spaial dimension. By spaceime homogeneiy, we may assume he wo observers ake heir even of coincidence as heir common origin. By spaceime isoropy, we may assume, by re-aligning he spaial axes of he wo observers if necessarily, ha O ravels in he posiive x-direcion of O and he wo posiive x-axes are aligned.

21 Le P be an arbirary spaceime poin and le P = and P = ( x ) and O respecively. ( x be he coordinaes of P according o O ) We wan o derive how he wo ses of coordinaes are relaed. Wrie ( ) ( ) x = f. x

22 Claim: Spaceime homogeneiy implies ha f : 2 2 mus be linear. Since O and O have a common spaceime origin, we mus have ( x ) = ( a b c d ) ( x ) = ( a + bx c + dx ). Then, ( ) 2 (x ) 2 = = (a 2 c 2 ) 2 + 2(ab cd)x + (b 2 d d )x 2? = 2 x 2.

23 Recall ( ) 2 (x ) 2 = = (a 2 c 2 ) 2 + 2(ab cd)x + (b 2 d d )x 2. Now, Consancy of Speed of Ligh ( ) 2 (x ) 2 = 0 2 x 2 = 0 ( ) ( 0 ) So for P = x 0 wih ( ) 2 (x ) 2 = 0, we have or equivalenly, 0 = [(a 2 + b 2 c 2 d 2 ) ± 2(ab cd)] 2, { a 2 + b 2 c 2 d 2 = 0 ab cd = 0 (1)

24 The worldline of O is described by x = 0 x = v according o O and O respecively. By spaceime isoropy, he worldline of O mus hen be described by Noe ha minus sign! x = 0 x = v.

25 Hence, along he worldline of O: x = 0 x = v, ( ) ( ) ( a + bx x = becomes ) ( a c + dx v = c which immediaely implies c = av. ), A similar consideraion of he worldline of O : x = 0 x = v implies c = dv. Thus av = c = dv = a = d, and ab cd = 0 = b c = 0 = b = av.

26 We now have ( Solving for ( ( x x x ) ) : ) = = a(v) 1 a(v)(1 v 2 ) ( 1 v v 1 ( 1 v v 1 ) ( x ) ) ( Bu, repeaing he preceding argumen wih he roles of O and O inerchanged, i.e. v v, we ge ( ) ( ) ( ) 1 v = a( v) x v 1 x. x. ). I follows ha 1 a(v)(1 v 2 ) = a( v) = a(v) a( v) = 1 1 v 2.

27 Claim: a(v) = a( v). Q: Wha is a( v)? A: I is he ime dilaion facor of O s clock as observed by O. Indeed, he worldline of O is given by x = v and x = 0 according o O and O respecively. ( ) ( ) ( ) 1 v = a( v) = v v 1 0 = a( v) = a( v), (along he worldline of O ) which says precisely ha according o O, he ime of O flows a he consan rae of 1/a( v) imes ha of O. This is ime dilaion! Spaceime isoropy now implies a(v) = a( v). Claim 2 is proved.

28 So, a(v) a( v) = 1 1 v 2 a(v) = a( v) = a(v) = a( v) = 1 1 v 2. And, we finally have he ransformaion law we are afer: ( ) ( ) ( ) 1 1 v x =, 1 v 2 v 1 x which is precisely he Lorenz ransformaion, under which Maxwell s equaions are invarian!

29 The Minkowski meric on spaceime is indeed well-defined! A sraighforward calculaion now shows ha he equaliy ( ) 2 (x ) 2 = vx 1 v 2 2 v + x 1 v 2 = 2 x 2 always holds if he ransformaion is Lorenz. 2 =

30 1 v 2 = O O P : x = 0, Time Dilaion x = v / 1 v 2 0 < = 1 v 2 <,

31 O O : x = 0, = x/v Breakdown of Universal Simulaneiy 0 = 0, x = vx = 0 = = v 1 v 2 x, x = 1 1 v 2 x = = vx.

32 O O Lengh Conracion P : = 0, x Q : = 0, x = 1 v 2 x 0Q O = 1 v 2 x < x = 0P O

33 Renzo 2 x 2 = cons Emina Emina The Twin Paradox

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