The Special Theory of Relativity

Transcription

1 The Speial Theor of Relaii

2 The Speial Theor of Relaii Chaper I. Conradiions in phsis?. Galilean Transformaions of lassial mehanis 3. The effe on Mawell s equaions ligh 4. Mihelson-Morle eperimen 5. insein s posulaes of relaii 6. Coneps of absolue ime and simulanei los

3 Galilean Newonian Relaii Galileo Galilei Isaa Newon Definiion of an inerial referene frame: One in whih Newon s firs law is alid. onsan if F arh is roaing and herefore no an inerial referene frame, bu we an rea i as one for man purposes. A frame moing wih a onsan eloi wih respe o an inerial referene frame is iself inerial. Relaii priniple: Laws of phsis are he same in all inerial frames of referene

4 Inuiions of Galilean Newonian Relaii Wha quaniies are he same, whih ones hange? Lenghs of objes are inarian as he moe. Time is absolue. Mass of an obje in inarian in for inerial ssem Fores aing on a mass equal for all inerial frames Veloiies are (of ourse) differen in inerial frames (Galileo ransformaions) Posiions of objes are differen in oher inerial ssems (Galileo oordinae ransformaion)

5 Galilean Transformaions A lassial (Galilean) ransformaion beween inerial referene frames: View oordinaes of poin P in ssem S Noe; Inerse ransformaion?

6 Galilean Transformaions In mari form

7 Relaii priniple: The basi laws of phsis are he same in all inerial referene frames s a Laws are he same, bu pahs ma be differen in referene frames

8 The domain of eleromagneism Mawell s equaions Gauss Farada Ampere/ Mawell Inegral form da Q /ε B da d B d Φ B µ I µ ε Φ B Differenial form ρ /ε B B B µ J µ ε

9 Differenial eor analsis for reaing Mawell s equaions in Caresian oordinaes (an be done in spherial): F F F F Diergene Curl F F F F F F F ˆ ˆ ˆ Gradien V V V V ˆ ˆ ˆ Laplaian V V V V Proof heorems on seond deriaies ( ) F ( ) ( ) F F F ( ) f Produ (hain) rules ( ) ( ) ( ) ( ) ( ) A B B A B A A B B A

10 Deriaion of he wae equaions in auum (no harge, no urren) B Calulae: B B µ ε B ( ) ( ) ( B) µ ε µ ε Similarl derie: B µ ε B

11 leromagnei wae equaions µ ε B µ ε B Noe ha: f f ν f is in general a wae equaion 855; eleri and magnei measuremens 8 µ ε 3.7 m / s Measuremen of he speed of ligh m / s Fieau m / s Fouaul 858 Hisor of he speed of ligh: hp:// Mawell: B B

12 Mawell s equaions µ ε wih µ ε James Clerk Mawell Ligh is a wae wih ranserse polariaion and speed Problems: In wha inerial ssem has ligh he ea eloi Wha abou he oher inerial ssems Waes are known o propagae in a medium; where is his eher How an ligh propagae in auum? Laws of elerodnamis do no fi he relaii priniple?

13 Mawell s equaions do no obe Galilei ransform Consider ligh pulse emied a ime ; a ime > in frame {,,,} In he moing frame So: {,,, } Appl Galilei ransform ( ) ( ) Simple approah:

14 Mawell s wae equaion ransformed Appl i o he wae equaion in (,) dimensions alulae differenials (diffiul?): Calulae field deriaies using he hain rule : Then also seond Spaial par Temporal par

15 Mawell s wae equaion ransformed II Inser in Mawell wae equaion This is no an eleromagnei wae equaion

16 The Mihelson Morle perimen Nobel 97 Alber Mihelson dward Williams Morle "for his opial preision insrumens and he sperosopi and merologial inesigaions arried ou wih heir aid" Quesions: Wha is he absolue referene poin of he her? In whih direion does i moe? How fas? Alber Abraham Mihelson her onneed o sun (ener of he unierse)? 4 arh ~ 3 m / 8 ~ 3 m / s s } ~ 4 Moion of he arh Should produe an Obserable effe

17 The Mihelson Morle perimen ais ( / ) Noe: we adop he lassial perspeie

18 The Mihelson Morle perimen ais /

19 The Mihelson Morle perimen / / / ( ) / Inerferomeer: If, hen no effe on inerferomeer If, hen a phase-shif inrodued Bu his is no obsered (auall diffiul o obsere)

20 The Mihelson Morle perimen Roae he inerferomeer T ( ) Approimae: / << / T Numbers: ~3 4 m/s /~ -4 l ~l ~ m ( ) 3 T 7 6 Visible ligh: λ~55 nm f~5 4 H s Then: / Phase hange (in fringes) f T / Should be obserable! Deeabili:. fringe

21 Conlusion: The Mihelson Morle perimen This inerferomeer was able o measure inerferene shifs as small as. fringe, while he epeed shif was.4 fringe. Howeer, no shif was eer obsered, no maer how he apparaus was roaed or wha ime of da or nigh he measuremens were made. The possibili ha he arms of he apparaus beame slighl shorened when moing agains he eher was onsidered b Loren. Hendrik A Loren Nobel 9 "in reogniion of he eraordinar serie rendered b heir researhes ino he influene of magneism upon radiaion phenomena" Loren onraion

22 Possible soluions for he eher problem. The eher is rigidl aahed o arh. Rigid bodies onra and loks slow down when moing hrough he eher 3. There is no eher

23 Alber insein A new perspeie

24 Alber insein On relaii

25 Posulaes of he Speial Theor of Relaii. The laws of phsis hae he same form in all inerial referene frames. Ligh propagaes hrough emp spae wih speed independen of he speed of soure or obserer This soles he eher problem (here is no eher) The speed of ligh is he same in all inerial referene frames

26 Simulanei One of he impliaions of relaii heor is ha ime is no absolue. Disan obserers do no neessaril agree on ime inerals beween eens, or on wheher he are simulaneous or no. Wh no? In relaii, an een is defined as ourring a a speifi plae and ime. Le s see how differen obserers would desribe a speifi een.

27 Simulanei Though eperimen: lighning srikes a wo separae plaes. One obserer beliees he eens are simulaneous he ligh has aken he same ime o reah her bu anoher, moing wih respe o he firs, does no.

28 Simulanei From he perspeie of boh O and O he hemseles see boh ligh flashes a he same ime From he perspeie of O he obserer O sees he ligh flashes from he righ (B) firs. Who is righ?

29 Simulanei Here, i is lear ha if one obserer sees he eens as simulaneous, he oher anno, gien ha he speed of ligh is he same for eah. Conlusions: Simulanei is no an absolue onep Time is no an absolue onep

30 Time Dilaion a) Obserer in spae ship D proper ime b) Obserer on arh speed is he same apparen disane longer Ligh along diagonal D D / 4 D / / γ This shows ha moing obserers mus disagree on he passage of ime. Cloks moing relaie o an obserer run more slowl