Announcements. Lecture 6 Chapter. 2 Special Relativity. Relativistic Dynamics. Relativistic Kinetic Energy. Relativistic Momentum

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1 Announemens HW: Ch.-70, 75, 76, 87, 9, 97, 99, 104, 111 HW1 due: now, HW due: /08 (by lass hour) No lab his week; New TA (Ganga) Physis Colloquium (Thursday a 3:40m) Quiz 1: Feb 8 h. (30 min) -- Cha. *** Course Web Page ** h://highenergy.hys.u.edu/~slee/40/ Leure Noes, HW Assignmens, Shedule for hephysis Colloquium, e.. Ouline: Leure 6 Chaer. Seial Relaiiy Relaiisi Dynamis Relaiisi Momenum Relaiisi Kinei Energy Toal Energy Momenum and Energy in Relaiisi Mehanis Thursday: General Theory of Relaiiy Ne Week Quanum Physis Newon s nd Law: Relaiisi Momenum he momenum of a arile, m is inarian (does no deend on he eloiy) eressed in erms of 3-eors, inarian under G.Tr. (bu no L.Tr.) Relaiisi Kinei Energy In relaiisi mehanis, he one of energy is more useful han he fore : f Relaiisi form of he nd Law (inrodued by Einsein): where definiion of he momenum in relaiisi mehanis 1 0 Eamle: Calulae he momenum of an eleron moing wih a seed of By ignoring relaiisi effes, one would ge kinei energy of a arile of he mass mmoing wih seed

Relaiisi Kinei Energy In relaiisi mehanis, he one of energy is more useful han he fore : f Toal and Res Energies We ee his resul o be redued o he lassial KE a

(raher han Cl. M.) when K and E 0 beome of he same order of magniude. The energy and momenum are onsered (he onsequene of uniform and isoroi sae).

2 Relaiisi Kinei Energy In relaiisi mehanis, he one of energy is more useful han he fore : f Toal and Res Energies We ee his resul o be redued o he lassial KE a low seed: 1 0 Le s rewrie he eression for K in he form: he oal energy he res energy Limi of small seed: We mus use Rel. M. (raher han Cl. M.) when K and E 0 beome of he same order of magniude. The energy and momenum are onsered (he onsequene of uniform and isoroi sae). For an isolaed sysem of ariles: kinei energy of a arile of he mass mmoing wih seed Relaiisi Kinei Energy Show ha E + m 4 follows from u mu and E u m for momenum and energy in erms of m and u Anoher Eamle:

3 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 0) E INTERNAL m ) (m E + E TOTAL ENERGY E INTERNAL ENERGY

4 Eressions for (oal) Energy and Momenum of a arile of mass m, moing a eloiy u 1 Classial Limi mu <<1 mu E m + Classial Limi Kinei Energy KE 1 mu <<1 NEW mu E m + FAMILIAR kinei energ

5 Energy Maer Energy Maer Aomi Bomb (Chaer 10): Energy is CREATED From he Mass of Nulei (Inernal energy is ransformed ino kinei energy) Is here Absolue Causaliy? Le s assume ha he order of eens is hanged in some 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? > 0 < 0 Is ha Possible? and are he ime inerals beween he same wo eens obsered in S and S, reseiely

6 $ % & + $ % & + 1 Using Lorenz ransformaions. 0 0 < > if hen $ % & + $ % & + 1 Using Lorenz ransformaions. 0 0 < > if hen > < $ % & $ % & + $ % & + 1 Using Lorenz ransformaions. 0 0 < > if hen > < $ % & Imossible Is here Absolue Causaliy? 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? YES NO NO

7 Problems Relaiisi Dynamis Some Eamles Q: Wha is he momenum of an eleron wih K m? Q: How fas is a roon raeling if is kinei energy is /3 of is oal energy? Problems 1. Wha is he momenum of an eleron wih K m?. How fas is a roon raeling if is kinei energy is /3 of is oal energy? Problem An eleron whose seed relaie o an obserer in a lab RF is 0.8 is also being sudied by an obserer moing in he same direion as he eleron a a seed of 0.5 relaie o he lab RF. Wha is he kinei energy (in MeV) of he eleron o eah obserer? K K

8 Problem An eleron whose seed relaie o an obserer in a lab RF is 0.8 is also being sudied by an obserer moing in he same direion as he eleron a a seed of 0.5 relaie o he lab RF. Wha is he kinei energy (in MeV) of he eleron o eah obserer? K K The eleron seed as seen by he moing obserer K : 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? In he lab IRF K: In he moing IRF K : 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) m: m/: Problem 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. Thus, he rearding oenial differene hoon hoon 1 (b) before afer

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. before hoon afer hoon 1 Problem momenum onseraion 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. before afer energy onseraion momenum onseraion energy onseraion (a) (b) In he ener-of-mass RF: before afer relaie seed From before Sae & Time / Energy & Momenum Relaiiy mies u sae &ime - also energy &momenum When onering from one inerial frame o anoher In he ime dilaion and lengh onraion formulas, ime is in he lengh formula and lengh is in he ime olume hrough he eloiy (i.e. eloiy lengh/ime) The oal energy of a arile is deenden on i s kinei energy and is mass. Een when he arile is no moing, i.e. no KE, i has (inernal) energy. Mass is anoher form of energy Moreoer, energy an show u as mass. 35 In he oal energy formula, momenum(or kinei energy) and mass energy are relaed There are ombinaions of sae/ime and energy/momenum ha obserers in any inerial frame will measure he as he same For energy and momenum his inarian says ha all obserers an agree on mass an obje has when i s a res 36

10 General Relaiiy General Relaiiy General relaiiy is he geomeriheory of graiaion ublished by Alber Einsein in Many rediions of general relaiiy differ signifianly from hose of lassial hysis. I is he urren desriion of graiaion in modern hysis. Eamles of suh differenes inlude graiaional ime dilaion, he graiaional red-shifof ligh, and he graiaional ime delay. I unifies seial relaiiy and Newons law of uniersal graiaion, and desribes graiy as a geomeri roery of sae and ime. General relaiiys rediions hae been onfirmed in all obseraions and eerimens o dae. In ariular, he uraure of sae-ime is direly relaed o he four-momenum (mass-energy and momenum). Howeer, unanswered quesions remain, soluion is he quanum graiy sounds quie omliae.. The relaion is seified by he Einsein s field equaions, a sysem of arial differenial equaions. (Graduae leel ourse) 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

11 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 Equialene Prinile Try some eerimens Aeleraing referene frames are indisinguishable from a graiaional fore Consan eloiy? Consan ael.? See wha his means 0 o o 0 o o Floor aeleraes uward o mee ball Canno do any eerimen o disinguish aeleraing frame from graiaional field 43 44

12 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 0 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 Whih of hese is a sraigh line? Whih of hese is a sraigh line? A A B B A. A C B. B C. C D. All of hem A. A C B. B C. C D. All of hem 47 48

a ured surfae. Mass and Curaure General relaiiy says ha any mass will gie saeime a uraure Moion of objes in saeime is deermined by ha uraure 49 50 Idea behind geomeri heory Maer bends sae and ime.

13 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 saeime a uraure Moion of objes in saeime is deermined by ha uraure 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 1/r is roorional o he loal energy and momenum densiy The roorionaliy onsan is A es of General Relaiiy Can es o see if he ah of ligh aears ured o us Loal massie obje is he sun Can obsere aaren osiion of sars wih and wihou he sun where G is Newons onsan 51 5

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

14 Sae is Cured? Einsein said o iure graiy as a war in sae Keler s Laws an all be elained by moemen around hese ukers 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 Eanding Unierse (alhough Einsein missed he hane o redi i He didn belieed) Graiaional Time Dilaion Graiy wars boh sae and ime A 10,000 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 1976 o wihin 0.01%.. Graiaional Radiaion When a mass is moed, he uraure of saeime hanges If a mass is osillaed, riles of sae-ime uraure arry he signal Graiaional radiaion arries energy andmomenum and wiggles mass in is ah Correions are now sandard in he synhronizing saellies This orreion needed in addiion o he seial relaiiy orreion for GPS 55 56

Eidene for Graiy Waes In 1974, 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

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.

15 Eidene for Graiy Waes In 1974, 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 57 Phy107 Fall The big bang In 199 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 14 billion years ago. In 1964 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 Nobel Prize in 006 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 006 Nobel rize was gien o he eole who designed he COBE eerimen whih was sensiie enough o see hose luming in he CMB Phy107 Fall Phy107 Fall

16 / Mawell s Equaions of eleromagneism (1873) Relaiisi mehanis, El.-Mag. (1905) Classial hysis Conlusion Relaiisi quanum mehanis (197-) Quanum mehanis (190 s-) Newonian Mehanis, Thermodynamis Saisial Mehanis h/s Quesion: Should we use relaiisi or lassial aroah o desribe he moion of an eleron in H aom?

Relativistic Dynamics

Relativistic Dynamics 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