Relativistic Dynamics

Transcription

1 Announemen Course webage h://highenergy.hys.u.edu/~slee/4/ Tebook PHYS-4 Leure Se., 5 Leure Noes, HW Assignmens, Physis Colloquium, e.. Relaiisi Dynamis Chaer Seial Relaiiy. Basi Ideas. Consequenes of Einsein s Posulaes 3. The Lorenz Transformaion Equaions 4. The Twin Parado 5. The Doler Effes 6. Veloiy Transformaion 7. Momenum & Energy 8. General Relaiiy & a s Look a Cosmology 9. The Ligh Barrier. The 4 h Dimension Ouline: Relaiisi Momenum Relaiisi Kinei Energy Toal Energy Momenum and Energy in Relaiisi Mehanis General Theory of Relaiiy Ne Week Quanum Physis

2 Relaiisi Kinei Energy Show ha E + m 4 follows from γ u mu and E γ u m for momenum and energy in erms of m and u 4 E E m INVARIANT E z y z y z y ( ( ( m E m E m E + + E TOTAL ENERGY Reoluionary Cone Wha abou E m? (m E + E m? (m E + E TOTAL ENERGY E INTERNAL ENERGY (when

3 E INTERNAL ENERGY E INTERNAL m Eressions for (oal Energy and Momenum of a arile of mass m, moing a eloiy u E TOTAL ENERGY E + (m Classial Limi Classial Limi mu << mu << mu E m + NEW mu E m + FAMILIAR kinei energ

4 Kinei Energy KE Energy Maer u Energy Maer Aomi Bomb (Chaer : Energy is CREATED From he Mass of Nulei (Inernal energy is ransformed ino kinei energy

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

6 Le s assume ha he order of eens is hanged in some referene frame S < > Is ha Possible? and are he ime inerals beween he same wo eens obsered in S and S, reseiely Le s assume ha he order of eens is hanged in some referene frame S < > Is ha Possible? and are he ime inerals beween he same wo eens obsered in S and S, reseiely $ % & + $ % & + Using Lorenz ransformaions. < > if hen $ % & + $ % & + Using Lorenz ransformaions. < > if hen > < $ % & +

7 Using Lorenz ransformaions. & $ % & $ % + + >?? Is here Absolue Causaliy? YES Migh ause reede effe in one referene frame bu effe reede ause in differen referene frame(s? NO if > < hen & $ % + < > Imossible e.g. an someone see you firs die, and hen see you ge born? NO Relaiisi Dynamis Problems. Wha is he momenum of an eleron wih K m? Some Eamles. How fas is a roon raeling if is kinei energy is /3 of is oal energy?

8 Problems. Wha is he momenum of an eleron wih K m? 4 E + m + E m K $ % m $ % m 4m m 3m & & Problem An eleron iniially moing wih momenum m is assed hrough a rearding oenial differene of V ols whih slows i down; i ends u 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?. How fas is a roon 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 m is assed hrough a rearding oenial differene of V ols whih slows i down; i ends u 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 m m m m m: ( ( ( Thus, he rearding oenial differene 5 m/: E m $ + ( m m % & Δ E E E m.3m.3(.5 ev.5 ev ( * 5 V.5 V An unsable arile of mass m moing wih eloiy relaie o an inerial lab RF disinegraes ino wo gamma-ray hoons. The firs hoon has energy 8 MeV in he lab RF and raels in he same direion as he iniial arile; he seond hoon has energy 4 MeV and raels in he direion oosie 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 arile. before hoon afer hoon Problem E m E m Δ E m. 5 ev V. 5 V (b (

9 An unsable arile of mass m moing wih eloiy relaie o an inerial lab RF disinegraes ino wo gamma-ray hoons. The firs hoon has energy 8 MeV in he lab RF and raels in he same direion as he iniial arile; he seond hoon has energy 4 MeV and raels in he direion oosie 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 arile. (a (b before m ( / m ( / hoon afer hoon h h m ( / m ( / E E h h E + E h h E E 8MeV 4MeV 4MeV E + E 8MeV + 4MeV MeV h h Problem ( h h ( ( momenum onseraion energy onseraion ( a 4 ( b 3 m E + E MeV MeV / /9.3 Problem A moing eleron ollides wih a saionary eleron and an eleron-osiron air omes ino being as a resul. When all four ariles hae he same eloiy afer he ollision, he kinei energy required for his roess is a minimum. Use a relaiisi alulaion o show ha K min 6m, where m is he eleron mass. 4 E + m 4E energy onseraion before ( ( ( ( E m + E m + E + m 4E 4 In he ener-of-mass RF: before afer 4 momenum onseraion % 6 $ & ( ( E + Em + m 6( E 6 ( m + ( afer ( m ( ( ( E ( + 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 seed 4 + ( 48/49 V m K m m 6 General Relaiiy General Relaiiy General relaiiy is he geomeri heory of graiaion ublished by Alber Einsein in 96. I is he urren desriion of graiaion in modern hysis. I unifies seial relaiiy and Newons law of uniersal graiaion, and desribes graiy as a geomeri roery of sae and ime. In ariular, he uraure of sae-ime is direly relaed o he four-momenum (mass-energy and momenum. The relaion is seified by he Einsein s field equaions, a sysem of arial differenial equaions. (graduae leel ourse Many rediions of general relaiiy differ signifianly from hose of lassial hysis. Eamles of suh differenes inlude graiaional ime dilaion, he graiaional red-shif of ligh, and he graiaional ime delay. General relaiiys rediions hae been onfirmed in all obseraions and eerimens o dae. Howeer, unanswered quesions remain, soluion is he quanum graiy sounds quie omliae..

10 Seial Theory of Relaiiy The wo osulaes: Seial Theory of Relaiiy: General Theory of Relaiiy: BUT: earh Aelleraing frames loally? Seial Theory of Relaiiy: General Theory of Relaiiy: Deals also wih Aeleraing - LOCALLY INERTIAL FRAMES Deals elusiely wih globally INERTIAL FRAMES - onsan loally Aeleraion Profound Link Graiaional Fore

11 Equialene Prinile Try some eerimens Aeleraing referene frames are indisinguishable from a graiaional fore??? Consan eloiy? Consan ael.? See wha his means o o o o Floor aeleraes uward o mee ball Canno do any eerimen o disinguish aeleraing frame from graiaional field Ligh follows he same ah Pah of ligh beam in our frame Veloiy Veloiy +a o Veloiy +a o Is ligh ben by graiy? If we an disinguish an aeleraing referene frame from graiy and ligh bends in an aeleraing referene frame hen ligh mus bend in a graiaional field Ligh o Pah of ligh beam in aeleraing frame o Bu ligh doesn hae any mass. How an graiy affe ligh?? Maybe we are onfused abou wha a sraigh line is 44

12 Whih of hese is a sraigh line? Whih of hese is a sraigh line? A A B B A. A B. B C. C C D. All of hem 45 A. A B. B C. C C D. All of hem 46 Sraigh is shores disane They are he shores disanes deermined by wraing sring around a globe. On a globe, hey are alled grea irles. 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 ah of all objes in ured sae-ime The onfusion omes in when you don know you are on a ured surfae. Mass and Curaure General relaiiy says ha any mass will gie sae-ime a uraure Moion of objes in sae-ime is deermined by ha uraure 47 48

13 Idea behind geomeri heory Maer bends sae and ime. Bending on a wo-dimensional surfae is haraerized by he radius of uraure: r Einsein dedued ha /r is roorional o he he loal energy and momenum densiy The roorionaliy onsan is 8πG where G is Newons onsan 5 A es of General Relaiiy Can es o see if he ah of ligh aears ured o us Eddingon and he Toal Elise of 99 Q:: Can we es o see if he ah of ligh aears ured o us? Aaren osiion of sar Aual osiion of sar Measure his angle o be abou.75 arseonds Loal massie obje is he sun Can obsere aaren osiion of sars wih and wihou he sun 5 5

14 Sae is Cured? Einsein said o iure graiy as a war in sae Cured Sae-ime: (Ineroninenal flighs Keler s Laws an all be elained by moemen around hese ukers Eeryhing moing is affeed, regardless of mass 54 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 Eanding Unierse (alhough Einsein missed he hane o redi i He didn belieed 55 Graiaional ime dilaion Graiy wars boh sae and ime A, km aboe he Earh s surfae, a lok should run 4.5 ars in faser han one on he Earh Comaring iming ulses from aomi osillaor loks onfirms he graiaional ime dilaion in 976 o wihin.%. Correions are now sandard in he synhronizing saellies This orreion needed in addiion o he seial relaiiy orreion for GPS 56

15 Graiaional Radiaion When a mass is moed, he uraure of sae-ime hanges Graiaional radiaion arries energy and momenum and wiggles mass in is ah Eidene for Graiy Waes In 974, Joseh Taylor and his suden Russell Hulse disoered a binary neuron sar sysem losing energy as eeed from graiaional radiaion Dire Deeion of Graiy Waes LIGO is a olleion of large laser inerferomeers searhing for graiy waes generaed by eloding sars or olliding blak holes Phy7 Fall 6 59 The Big Bang In 99 Obseraion of nearby and far away galaies indiae ha eeryhing is reeding from us. Key hysis needed o undersand his is he simle Doler shif of ligh waes. Waes from soures moing away from us are srehed ou or lower frequeny. Eraolaing bakwards indiaes ha all he galaies originaed from he same soure 4 billion years ago. In 964 radiaion from he early sages of ha elosion was deeed. Again he Doler shif was he key sine he waes were shifed o low frequeny - mirowae Phy7 Fall 6 6

16 Nobel Prize in 6 For he unierse o sar small and eand sae and ime mus be hing ha an eand(or onra General relaiiy was key hysis needed o undersand ha roess Howeer, a simle model of ha would redi suh a unierse would no hae lums of maer(sars, galaies Unless hose luming were resen ery early on 6 Nobel rize was gien o he eole who designed he COBE eerimen whih was sensiie enough o see hose luming in he CMB