Chapter 1 Relativity
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1 Chaper Relaii - Posulaes of Speial Relaii and Loren Transformaion The s posulae: The laws of phsis ma be epressed in equaions haing he same form in all frames of referene moing a onsan eloi wih respe o one anoher. The nd posulae: The speed of ligh in free spaes has he same alue for all obserers, regardless of heir sae of moion. Eg. (a) Phenomenon of relaii. (b) Classial phenomenon. Galilean ransformaion in lassial phsis: and.
2 Loren ransformaion in relaii: and Proof of Loren ransformaion: Suppose ha a flare is se off a a ommon origin of S and S a, and he obserers in eah ssem proeed o measure he speed ligh a and, respeiel. We hae and. k( ) k( ) Le and k[k(-) ]k (-)k f (,, ) f (,, ) ( k ) ( k ) k k(-) [k ] [ ] k k ( k ) k. Loren ransformaion holds. ( k ) Loren onraion: An obserer in S measures he lengh of a moing rod, of whih ends o and in S. The lengh in S is L L -, where L - in S L L.( )<L. The lengh of an obje in moion wih respe o an obserer is shorer han i is a res wih respe o him. Eg. A meer sik appear onl 5m long o an obserer. Wha is is relaie eloi? How long does i ake o pass he obserer?. (Sol,).5.( ) 3, se Time dilaion: A lok is a in S. When an obserer in S find he ime is, afer a ime ineral, and he ime is up o (all in S ). The obserer in S find he ime ineral is - >. A saionar lok measure a longer ime ineral beween eens ourring in a moing frame of referene han a lok does in he moing frame. Eg. (Twin parado) A woman leaes he earh in a spaeraf ha makes a round rip o a sar, 4 ligh-ear disan, a a speed of.9. How man ears ounger is she upon her reurn han his win siser who remained behind? (Sol.) 4.9-4( ).95.ears
3 Anoher simple formulaion of ime dilaion: (a): (b): (a) A ligh-pulse lok a res as seen b obserer in S, he ime ineral is he ligh pulse rael beween mirrors. Therefore we hae L. (b) A ligh-pulse lok in S as seen b obserer in S, and hene we hae () L (), L Simulanei: Consider ha a pair of ime-bombs eplodes a he same ime a and in S, respeiel. Bu in S,, ( ) - if, The eplosions do no simulaneousl our in S frame.
4 Relaiisi eloiies: Define dd, dd, dd in S, and d d, d d, d d in S. d d d, dd, dd, d d d, we hae d d d d d d and. Eg. Spaeraf A has a eloi wih respe o he earh of.9. If spaeraf B is o pass spaeraf A a a relaie eloi.5, wha eloi mus spaeraf B hae wih respe o he earh? (Sol.) Eg. Show ha is a uniersal onsan. (Proof)
5 - Relaiisi Mehanis Suppose A and B ollide a.5y, and A B. The round-rip ime T for A measured in S is T Y A, whih is idenial T Y B in S. m A A m B B m B YT Le m A m, m B m m Momenum: pm m B Y T m m Y Fore: Fdpdmdddmd d( m) Fds ds d A mby Kinei energ: T d( m) s s m A m B. d( m ) m -m d m m.( )m -m. Eg. An eleron has a kinei energ of.me. Find is eloi aording o lassial and relaiisi mehanis. (Sol.) Classial mehanis:.me J mse Relaiisi mehanis: T.6-4 m -m [ -].64 8 mse Toal energ: Em Tm Eg. Dnamie liberaes JKg when i eplodes. Wha fraion of is oal energ onens in his? (Sol.) Suppose ha he dnamie is Kg. Em 9 6,
6 -3 Doppler Effe w Classial Doppler effe: ff.( ), where w is he eloi of wae, is he w eloi of obserer, and is he eloi of soure.. > if he obserer moes oward he soure; < if he obserer moes awa from he soure.. > if he soure moes oward he obserer; < if he soure moes awa from he obserer. Eg. The eloi of sound is 34mse. A rain generaes some noise as high as 3H when is eloi is 5mse awa from an obserer who is sanding a a fied posiion. Wha frequen does he hear? w (Sol.) f 3, w34,, -5, ff.( )65.3H w Doppler effe in relaii: ff.( ), where θ is he angle beween he osθ moing direion of he ligh soure and he line from he soure o an obserer. And is he relaie eloi of he soure (> if he soure and he obserer approahing eah oher; < if he moe awa from eah oher). Speial ase : Longiudinal Doppler effe, θ ff. Speial ase : Transerse Doppler effe, θπ ff.( ) Eg. A spaeraf reeding from he earh a.97 ransmis daa a he rae of 4 pulsese. A wha rae are he reeied? (Sol.) -.97, f 4, ff pulsese Eg. (Red shif) A disan gala in he onsellaion Hdra is reeding from he earh a 6. 7 mse. B how muh is a green speral line of waelengh 5nm emied b his gala shifed oward he red end of he sperum? (Sol.) -6. 7, λ 5, ff. λλ. 65nm
7 -4 Some Phenomena of General Relaii The priniple of equialene: An obserer in a losed laboraor an no disinguish beween he effes produed b a graiaional field and hose produed b an aeleraion of he laboraor. Sarligh passing near he sun is defleed b sun s srong graiaional field: This phenomenon has been proed b obsering and measuring he preession of he perihelion of Merur s orbi.
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