Doppler Effect. PHYS-2402 Lecture 6. Chapter 2. Announcement. Feb. 3, Special Relativity. Quiz 1: Thursday 30 min; Chap.2

Transcription

1 Announemen Course webpage hp://highenergy.phys.u.edu/~slee/40/ Tebook PHYS-40 Leure 6 Quiz : Thursday 30 min; Chap. Feb. 3, 05 HW (due /0) 0, 6, 36, 4, 46, 5, 55, 70, 76, 87, 9, Doppler Effe Chaper Speial Relaiiy. Basi Ideas. Consequenes of Einsein s Posulaes 3. The Lorenz Transformaion Equaions 4. The Twin Parado 5. The Doppler Effes 6. Veloiy Transformaion 7. Momenum & Energy 8. General Relaiiy & a s Look a Cosmology 9. The Ligh Barrier 0. The 4 h Dimension

2 Ouline: Relaiisi Dynamis E = INTERNAL ENERGY E INTERNAL = m Relaiisi Momenum Relaiisi Kinei Energy Toal Energy Momenum and Energy in Relaiisi Mehanis General Theory of Relaiiy E = TOTAL ENERGY E = p + (m ) Epressions for (oal) Energy and Momenum of a parile of mass m, moing a eloiy u Kinei Energy = KE

3 u u u Is here Absolue Causaliy? Migh ause preede effe in one referene frame bu effe preede ause in differen referene frame(s)? e.g. an someone see you firs die, and hen see you ge born?

4 Is here Absolue Causaliy? Le s assume ha he order of eens is hanged in some referene frame S Migh ause preede effe in one referene frame bu effe preede ause in differen referene frame(s)? e.g. an someone see you firs die, and hen see you ge born? > 0 ' < 0 Is ha Possible? and are he ime inerals beween he same wo eens obsered in S and S, respeiely Le s assume ha he order of eens is hanged in some referene frame S > 0 ' < 0 Is ha Possible? and are he ime inerals beween he same wo eens obsered in S and S, respeiely Using Lorenz ransformaions. & ' = $ % & ' = $ % > 0 if hen ' < 0 + +

5 $ % & + = $ % & + = ' ' Using Lorenz ransformaions. 0 ' 0 < > if hen > < $ % & + 0 $ % & + = $ % & + = ' ' Using Lorenz ransformaions. 0 ' 0 < > if hen > < $ % & + 0 Impossible >?? Is here Absolue Causaliy? Migh ause preede effe in one referene frame bu effe preede ause in differen referene frame(s)? e.g. an someone see you firs die, and hen see you ge born? YES NO NO Some Eamples Relaiisi Dynamis

6 Problems. Wha is he momenum of an eleron wih K =m? Problems. Wha is he momenum of an eleron wih K =m? 4 E = p + m + E m K p= $ % m = $ % m = 4m m = 3m & ' & '. How fas is a proon raeling if is kinei energy is /3 of is oal energy?. How fas is a proon raeling if is kinei energy is /3 of is oal energy? ( 3 3 K = E = m + K) E = 3m E = m V / ( ) V 8 = 3 V $ % = = / & ' 9 3 ( V ) Problem An eleron iniially moing wih momenum p=m is passed hrough a rearding poenial differene of V ols whih slows i down; i ends up wih is final momenum being m/. (a) Calulae V in ols. (b) Wha would V hae o be in order o bring he eleron o res? Problem An eleron iniially moing wih momenum p=m is passed hrough a rearding poenial differene of V ols whih slows i down; i ends up wih is final momenum being m/. (a) Calulae V in ols. (b) Wha would V hae o be in order o bring he eleron o res? (a) = + = + = E p m m m m p=m: ( ) ( ) ( ) Thus, he rearding poenial differene 5 p=m/: E = m $ + ( m ) = m % & Δ E = E E = ' m 0.3m 0.3( ev) =.5 0 ev ( ) * (b) ( ) 5 V =.5 0 V E = m E = m Δ E = m. 0 5 ev V =. 0 5 V

7 An unsable parile of mass m moing wih eloiy relaie o an inerial lab RF disinegraes ino wo gamma-ray phoons. The firs phoon has energy 8 MeV in he lab RF and raels in he same direion as he iniial parile; he seond phoon has energy 4 MeV and raels in he direion opposie o ha of he firs. Wrie he relaiisi equaions for onseraion of momenum and energy and use he daa gien o find he eloiy and res energy, in MeV, of he unsable parile. before phoon afer phoon Problem An unsable parile of mass m moing wih eloiy relaie o an inerial lab RF disinegraes ino wo gamma-ray phoons. The firs phoon has energy 8 MeV in he lab RF and raels in he same direion as he iniial parile; he seond phoon has energy 4 MeV and raels in he direion opposie o ha of he firs. Wrie he relaiisi equaions for onseraion of momenum and energy and use he daa gien o find he eloiy and res energy, in MeV, of he unsable parile. (a) (b) before m ( ) / m ( ) / phoon afer phoon ph ph m ( ) / m ( ) / E E = ph ph = E + E ph ph = E E = 8MeV 4MeV = 4MeV = E + E = 8MeV + 4MeV = MeV ph ph Problem momenum onseraion energy onseraion ( a) 4 = = = () b 3 ( ph ph ) ( ) ( ) m = E + E = MeV MeV / /9.3 Problem A moing eleron ollides wih a saionary eleron and an eleron-posiron pair omes ino being as a resul. When all four pariles hae he same eloiy afer he ollision, he kinei energy required for his proess is a minimum. Use a relaiisi alulaion o show ha K min =6m, where m is he eleron mass. p 4 p E + m = 4E energy onseraion before ( ) ( ) ( ) ( ) E = m + p E = m + p E + m = 4E p = 4p In he ener-of-mass RF: p ' before afer p = 4p momenum onseraion % 6 $ & ( ) ( ) E + Em + m = 6( E ) = 6 ( m ) + ( p) afer ( m ) ( ) p ' ( ) ( E) ( p ) + Em + ( m ) = 6( m ) E = 4 m / m = 7m E ' 4m = E γ m ( m ) ( ) ' + V = = = = '/ + 3/ K E m m = = 6 ' = ' = γ' = ' 3 = ' = relaie speed 4 + ( 48/49) ' V = m K m m = = 6 General Relaiiy General relaiiy is he geomeri heory of graiaion published by Alber Einsein in 96. I is he urren desripion of graiaion in modern physis. I unifies speial relaiiy and Newon's law of uniersal graiaion, and desribes graiy as a geomeri propery of spae and ime. In pariular, he uraure of spae-ime is direly relaed o he four-momenum (mass-energy and momenum). The relaion is speified by he Einsein s field equaions, a sysem of parial differenial equaions. (graduae leel ourse)

8 General Relaiiy Many prediions of general relaiiy differ signifianly from hose of lassial physis. Speial Theory of Relaiiy The wo posulaes: Eamples of suh differenes inlude graiaional ime dilaion, he graiaional red-shif of ligh, and he graiaional ime delay. General relaiiy's prediions hae been onfirmed in all obseraions and eperimens o dae. Howeer, unanswered quesions remain, soluion is he quanum graiy sounds quie ompliae.. BUT: earh Aelleraing frames? Speial Theory of Relaiiy: General Theory of Relaiiy: loally Speial Theory of Relaiiy: Deals elusiely wih globally INERTIAL FRAMES - = onsan General Theory of Relaiiy: loally Deals also wih Aeleraing - LOCALLY INERTIAL FRAMES

9 Aeleraion Profound Link Equialene Priniple Aeleraing referene frames are indisinguishable from a graiaional fore??? See wha his means Graiaional Fore Try some eperimens Ligh follows he same pah Veloiy = +a o Consan eloiy? Consan ael.? Pah of ligh beam in our frame Veloiy = Veloiy = +a o =0 = o = o =0 = o = o Floor aeleraes upward o mee ball Canno do any eperimen o disinguish aeleraing frame from graiaional field Ligh =0 = o Pah of ligh beam in aeleraing frame = o

10 Is ligh ben by graiy? Whih of hese is a sraigh line? If we an disinguish an aeleraing referene frame from graiy A and ligh bends in an aeleraing referene frame B hen ligh mus bend in a graiaional field Bu ligh doesn hae any mass. How an graiy affe ligh? A. A B. B C? 38 Maybe we are onfused abou wha a sraigh line is 37 C. C D. All of hem Whih of hese is a sraigh line? Sraigh is shores disane B A They are he shores disanes deermined by wrapping sring around a globe. On a globe, hey are alled grea irles. A. A B. B C This an be a general definiion of sraigh, and is in fa an inuiie one on ured surfaes I is he one Einsein used for he pah of all objes in ured spae-ime C. C D. All of hem The onfusion omes in when you don know you are on a ured surfae

11 Mass and Curaure General relaiiy says ha any mass will gie spae-ime a uraure Moion of objes in spae-ime is deermined by ha uraure 4 Idea behind geomeri heory Maer bends spae and ime. Bending on a wo-dimensional surfae is haraerized by he radius of uraure: r Einsein dedued ha /r is proporional o he he loal energy and momenum densiy The proporionaliy onsan is 8πG where G is Newon's onsan 43 A es of General Relaiiy Can es o see if he pah of ligh appears ured o us Loal massie obje is he sun Can obsere apparen posiion of sars wih and wihou he sun 44

12 Eddingon and he Toal Elipse of 99 Q:: Can we es o see if he pah of ligh appears ured o us? Apparen posiion of sar Measure his angle o be abou.75 arseonds Aual posiion of sar Cured Spae-ime: (Ineroninenal flighs) 45 Spae is Cured? Einsein said o piure graiy as a warp in spae Kepler s Laws an all be eplained by moemen around hese pukers Eeryhing moing is affeed, regardless of mass Oher Consequenes of GR Time dilaion from graiy effes Graiaional Radiaion Creaed when big graiy soures are moed around quikly Similar o he eleromagnei waes ha were aused by moing eleron harges quikly Blak Holes Epanding Unierse (alhough Einsein missed he hane o predi i He didn belieed) 47 48

13 Graiaional ime dilaion Graiy warps boh spae and ime A 0,000 km aboe he Earh s surfae, a lok should run 4.5 pars in 0 0 faser han one on he Earh Comparing iming pulses from aomi osillaor loks onfirms he graiaional ime dilaion in 976 o wihin 0.0%. Graiaional Radiaion When a mass is moed, he uraure of spae-ime hanges Graiaional radiaion arries energy and momenum and wiggles mass in is pah Correions are now sandard in he synhronizing saellies This orreion needed in addiion o he speial relaiiy orreion for GPS Eidene for Graiy Waes In 974, Joseph Taylor and his suden Russell Hulse disoered a binary neuron sar sysem losing energy as epeed from graiaional radiaion Dire Deeion of Graiy Waes LIGO is a olleion of large laser inerferomeers searhing for graiy waes generaed by eploding sars or olliding blak holes 5 Phy07 Fall 006 5

14 The Big Bang In 99 Obseraion of nearby and far away galaies indiae ha eeryhing is reeding from us. Key physis needed o undersand his is he simple Doppler shif of ligh waes. Waes from soures moing away from us are srehed ou or lower frequeny. Erapolaing bakwards indiaes ha all he galaies originaed from he same soure 4 billion years ago. In 964 radiaion from he early sages of ha eplosion was deeed. Again he Doppler shif was he key sine he waes were shifed o low frequeny - mirowae Phy07 Fall Nobel Prize in 006 For he unierse o sar small and epand spae and ime mus be hing ha an epand(or onra) General relaiiy was key physis needed o undersand ha proess Howeer, a simple model of ha would predi suh a unierse would no hae lumps of maer(sars, galaies) Unless hose lumping were presen ery early on 006 Nobel prize was gien o he people who designed he COBE eperimen whih was sensiie enough o see hose lumping in he CMB