Modern Physics. Two major parts: modern relativity, first 4-6 lectures

Transcription

1 Modern Physis Fall/Winer 900, Max Plank s paper Ueber das Gesez der Energieereilung im Normalsperum, Annalen der Physik IV, 553 (90 peak in 90s/30s Two major pars: modern relaiiy, firs 4-6 leures Quanum mehanis and is appliaions, res of he ourse also main onen of Phys 3 o follow nex quarer Wha is Physis all abou? oneps and heir onneion, i.e. mahemaially formulaed equaions/laws, oneps and laws are deried from inerplay beween heory and experimen, his makes sure only good heories surie and heories ge beer oer ime some fundamenal oneps suh as spae and ime are muh older han Physis and are sor of ommon sense knowledge (Kan s a priori oneps and may een be inheried geneially

2 bu: Werner Heisenberg (in Physis and Philosophy Any oneps or words whih we hae formed in he pas hrough he inerplay beween he world and ourseles are no really sharply defined wih respe o heir meaning; ha is o say, we do no know exaly how far hey will help us in finding our way in he world.... This is rue een of he simples and mos general oneps like exisene and spae and ime The oneps may, howeer, be sharply defined wih regard o heir onneions. This is aually he fa when he oneps beome a par of a sysem of axioms and definiion whih an be expressed onsisenly by a mahemaial sheme. Suh a group of onneed oneps may be appliable o a wide field of experiene and will help us o find our way in his field. So modern physis will be in large pars onrary o our inuiion beause i deals wih he ery fas, i.e. speeds omparable o he speed of ligh and he ery small, aoms, moleules, elemenal pariles. Our (possibly inheried lak of appreiaion ha he world of he ery fas and he world of he ery small may well be ery differen from he world we are used o makes modern physis diffiul o omprehend, bu Heisenberg showed he way, see aboe, we hae o sik o he mahemaial shemes ha onne oneps and hae o redefine well known oneps, suh as spae, ime, ausaliy, o fi hese shemes

3 Relaiisi Mehanis - Speial Relaiiy 3 Galileo, Newon: Inerial referene frame: Newon s (firs law: a body oninues o be a res or oninues moing wih onsan eloiy if here is no ne fore aing on i 0 or onsan if Σa ΣF 0 beause Σ F ma and ds d m ineria, ha is where he referene frame ges is name from Galileo, Newon: all laws of mehanis are he same in inerial referene frames, all referene frames are equally alid, here is no preferred inerial referene frame lassial onep of relaiiy Classi relaions beween an een as obsered in wo differen ( 0, > 0 inerial referene frames are relaed by Galilean ransformaion

4 Galilean ransformaions 4 wo differen ses of oordinaes: spae oordinaes (x,y,z, ime oordinae a res; (x,y,z ; in moion x x + y y z z x x y y z z, een has one se of oordinaes in one sysem and anoher se of oordinaes in anoher sysem (bak ransformaion are he same exep for sign of leads o eor addiion law of eloiies, if een moes in unprimed frame wih eloiy u, and u add up u x x x ( x + ( x + ( x x + ( x + u x u x + u y u y u z u z Noa Bene: spae and ime oordinaes do no mix,

5 imporane of hese equaion is ha hey ensure he physial laws ha are inarian wih respe o hese equaions are alid eerywhere and a all imes (if we use our ommon sense ideas of spae and ime 5 Resul of Galileian relaiiy: here is no mehanial experimen ha an dee absolue moion, you an ea your dinner in an air plane (when i is no aeleraing whih is moing raher fas wih respe o he earh jus as well as on your dinner able a home whih is moing een faser wih respe o he sun In 870s -904 some new idea of how o measure absolue moion µ ε π 0 TmA C N m prediion of Maxwell s 860 se of equaions, µ 0 permeabiliy, e 0 permiiiy of free spae (i.e. auum m onsan (and now exa per definiion s aording o wha/whom has his alue??? Maxwell s own answer: luminiferous eher (somehing quie srange, presen eerywhere een in he nearly absolue auum of free spae, bu allows planes and oher objes o moe hrough i freely,, and whih is in absolue res

6 No only onsan in auum bu he oher laws by Maxwell s do no obey a Galilean ransformaion, so a las here seemed o be a way of deeing moion, if you do an eleromagnei experimen suh as measuring he speed of ligh in an airplane or on earh, you should ge he relaie speed wih respe o he eher whih is supposed o be a absolue res. 6 Reall sound: raels in air and any kind of body, speeds: 43 s m in air a 93 K, 49 s m in air a 303 K and normal pressure, 3800 m in onree a 93 K, needs aually a medium o propagae, s if you hae a poenial soure of sound in auum you an hear i as he wae an propagae So upwind sound raels faser as i is arried along wih he wind iself, downwind sound rael slower sine he medium (air raels in he opposie direion Galilean ransformaions seem o apply Conundrum: Sine eher seemed o be so speial i should define a ery pariular frame of referene, i.e. he only one in whih Maxwell s equaions are orre, in all oher frames of referene, i.e. our earh, here should be deiaions from Maxwell s laws, on he oher hand, hese laws work quie well, how an his be?

7 Mihelson-Morley Experimen, Designed o dee he eher and earh s relaie moion wih respe o he eher by deeing small hanges in he speed of ligh, i.e. deiaions from Maxwell s onsan law by inerferomery Ligh soure A, semiransparen mirror beam splier B, wo mirrors C and E all mouned on a rigid base Mirrors C and E are plaed a equal disanes L from beam splier, so ha he wo resuling beams hae (apparenly he same pah lengh (L o go in perpendiular direions, reah he mirrors C and D and ge refleed bak o he beam splier where hey are joined ogeher again If ime aken for he ligh o go from B o E and bak is he same as he ime from B o C and bak, emerging beams D and F will be in phase and reinfore eah oher I hese wo imes differ slighly, beams D and F will produe inerferene paern. If apparaus is a res wih respe o he eher, imes should be exaly equal beause he lenghs he ligh mus rael are exaly equal if i is moing owards he righ wih a eloiy u, here should be a differene in he imes, resuling in an inerferene paern. Why should ha be? Time o go from B o E and bak reurn ime E o B (? beause of moemen of apparaus o righ i he apparaus moes, while ligh is on is way o from B o E, he mirror ogeher wih he whole apparaus moes away, his disane is u, i.e. he ligh mus rael wih speed he lengh L + u in order o reah he mirror L + u L / ( - u

8 8 whih means ha eloiy of he ligh wih respe o apparaus is - u for reurn rael eloiy of ligh wih respe o apparaus mus herefore be + u, beause he beam splier B and he ligh beam are moing in opposie direions L / ( + u and L u L oal ime for B o E and bak is + L/( -u ( u now he oher pah: B C and bak, again assumpion is apparaus is moing o he righ (beause we wan o measure his moemen by an aniipaed shif in he inerferene paern during ime 3 mirror C will moe o he righ by a disane u 3 ligh has herefore o rael along he hypoenuse of righ riangle BC ½B ( 3 L + (u 3 L 3 u 3 ( -u 3 So 3 L/ u, L u riangle is symmeri, so ime i akes for he ligh o reurn o B is 3 3 L u? + ( L u

9 differene is jus faor u? (Lorenz faor > 9 denominaors represen modifiaions in ime aused by moion of he apparaus, hey are no he same so we should see an inerferene paern and from his we ould alulae u eloiy of earh wih respe o eher he whole poin of he experimen a minor ehnial poin, we an make he lenghs L exaly equal, we an ompensae for his in he inerferene paern, hen we an urn apparaus around by 90 degrees and should see a shif of inerferene paern beween wo ses of seings arbirary orienaion and 90 roaed wih respe o Bu no shif in inerferene paern was eer obsered, we do know u? 0 3 L u? + L u u so i seems as if lengh of he pah is B o E and bak is onraed by a faor? (Lorenz and Fizgerald. Resul, he speed of ligh (in air a earh is in all direions equal regardless of any relaie moemen of he earh wih

10 respe o he eher ha should resul in an eher wind analogous o he wind ha affes he speed of sound 0 apparen 0 eloiy of earh and onsan eloiy of ligh resuls an boh be explained by Lorenz Transformaions (904, moing frame a 0 boh frames oinide x ( x + [m] [m + s ms ] y y, z z x ( + s [s] [s + m ] sm in whih Maxwell s law are inarian, i.e. hae he same form regardless of he moemen of he obserer! reerse ransformaions x y y, z z ( x [m] [m + s ms ] x ( s [s] [s + m ] sm

11 for <<, Lorenz ransformaions go oer in Galilean ransformaions onsequenes (some of whih no fully realized by Lorenz:. spae and ime oordinaes mix, i.e. hey are he same sor of hing,? beer desripion 4 dimensional spae ime definiions: proper lengh: Lengh L 0 of an obje measured in he res frame of he obje is is proper lengh. Measuremens of he lengh from an referene frame ha is in relaie moion parallel o ha lengh are always less han he proper lengh proper ime: When wo eens our a same loaion in an inerial referene frame, he ime ineral beween hem, measured in ha frame, is alled proper ime ineral or he proper ime. Measuremens of he same ime ineral from nay oher inerial referene frame are always greaer

12 . effe on lengh (boh disanes parallel and lengh of moing objes parallel L 0 lengh of an obje a res, proper lengh L 0 x x sine (x + x (same ime of measuremen L 0 (x - x, wih L (x - x L 0 L or L L 0, (les all a?? - inerse Lorenz faor or onraion faor > Resul: L < L 0 lengh onraion for any eloiy, a > 0 say 0., L 0 m L m or m or m? say 00 km h -, L 0 m L m or m or m if a << α - ½ a, so i does no ge noied in eeryday experiene

13 3 3. effe on ime (rae a whih loks and all naural proesses run 0 ime oordinae of a obje a res, proper ime x x ( + sine x x (same plae of een ( ( 0 0, wih?? 0? -? 0,? Lorenz faor > (les all a Resul:? >? 0 ime dilaion for any eloiy, a > 0 say 0 0.,? 0 h 3600 s? h+8 se or h+min+5 se or h+3 min+5 s? say 0 00 km h -,? 0 h 3600 s? h + 5 s or h + 5 µs or h + 8 ps? if a << 5 α + ½ a, so ( does no ge noied in eeryday experiene, bu an be measured

14 4 Lorenz noed dependene of ime on moion in his equaions bu all all hese imes apparen imes in order o disinguish hem from he one rue ime he belieed o be absolue in he Newonian sense 4. relaiisi addiion of eloiies for oneniene:? η α Lorenz faor u x ( ( ( ( ( u u x x x x x x x γ γ / : u y } { } ( ( { ( } { u u x y x y y x y γ γ γ u z } { } ( ( { ( } { u u x z x z z x z γ γ γ reerse ransformaions u x ( ( ( ( ( u u x x x x x x x γ γ / :

15 5 u y } { } ( ( { ( } { u u x y x y y x y γ γ γ u z } { } ( ( { ( } { u u x z x z z x z γ γ γ Resul: eloiies do no add up simply (Galilean say 0.5 super super fa roke sending ou ligh, u x how fas is he ligh going o be?.5 or or 0.75 when measured from earh, u x? say 00 km h - (beween frames, u x 5 km h - (you walk in a super fas rain in he direion of moion 95 km h - or km h - or km h -? does no ge noied in eeryday experiene jus as for <<, Lorenz ransformaions go oer in Galilean ransformaions, relaiisi eloiy addiion goes oer ino Galilean eloiy addiion for u x, and u x << bu for u x u u x u x x x ( ( so hen u x u x as one may easily mix up u x, u x and

16 6 wo spaeships A and B are moing in opposie direions obserer on earh measures speed of A as 0.7 and speed of B as 0.85, find eloiy of B wih respe o A so we hae u x A and u x B as boh are measured from res frame a earh and mus assign signs, les A moe o he righ and all i + direion, B moe o he lef and all i - direion we should find u x B he eloiy of B in moing frame of A A moes wih u x A wih respe o earh, i.e. ha is wih respe o earh and i is posiie, so u x B u B x B ux } when numbers are pu in make sure o remember u x B is negaie, resul , seems o be OK wih inuiion, B goes prey fas owards A whih is reeding prey fas as well, bu speed mus be smaller han, so disoun he - sign

17 5: relaiisi Doppler effe, ranserse Doppler effe remember Doppler effe for sound waes? when a ar or ruk is moing while is horn is blowing, frequeny (pih of sound is higher as he ehile approahes you and lower as i moes away from you differen formulae for obserer a res - soure moing; soure a res obserer moing, boh soure and obserer moing for ligh (eleromagnei waes only relaie eloiy is imporan 7 f obs + f soure soure approahing obserer on same axis f obs + f soure soure reeding from obserer on same axis f soure proper frequeny ranserse Doppler effe - soure and obserer on perpendiular axes - is onsequene of relaiiy exiss only for eleromagnei waes f obs f soure analog o T obs T soure nohing else hen ime dilaion formula - wih T soure proper period (proper ime i akes o omplee one osillaion disoered 938 by Iens and Silwell, who did no beliee in relaiiy prior o heir disoery

18 Summary so far Galileo/Newon s lassial relaiiy/mehanis, all mehanial laws are inarian o Galilean ransformaion, work ery well in he realm of our eeryday human experiene aording o Lorenz for elerodynamis, Maxwell s laws are inarian o he Lorenz ransformaions, Lorenz ransformaions onain Galilean ransformaion as limiing ases for small speeds if Maxwell s equaions and Lorenz ransformaion are boh rue, and Mihelson-Morley experimen sugges hey are, hen here all kinds of srange effes o be expeed a high eloiies Lorenz s plae in hisory, besides his 90 Nobel prize for heory of elerons: seing sene for Poinaré (904: Aording o he priniple of relaiiy, he laws of physial phenomena mus be he same for a fixed obserer as for an obserer who has a uniform moion of ranslaion relaie o him, so ha we hae no, nor an we possibly hae, any means of diserning wheher or no we are arried along in suh a moion. so all elerodynamis experimens (e.g. along he lines of Mihelson-Morley are doomed o ge a zero eloiy/no effe resul jus as no mehanial experimen ould dee moion eiher

19 9 Einsein s speial heory of relaiiy, 905 (deals only wih inerial frames herefore speial Einsein general heory of relaiiy deals wih aeleraed referene frames and graiy, 95 (when Einsein proposed boh heories, people would hardly beliee him, een M. Plank, Nobel - laureae himself, hough by 9 ha his an all be righ - i is simply oo weird - when Einsein go his Nobel prize 9 i was for he phooeleri effe - no for relaiiy Einsein s Posulaes: The laws of all physis are he same in all inerial referene r r dp frames. Tha is, basi laws suh as F d hae he same mahemaial form, for all obserers moing a onsan eloiy wih respe o eah oher, his eloiy may be eiher of he order of magniude of our human experiene or lose o he speed of ligh. Ligh propagaes hrough empy spae wih a definiie speed independen of he speed of he soure or obserer. Tha is, all obserers will measure he same speed for regardless of heir frame of referene, here is a definiie speed limi for all objes and his is, anyhing ha has mass will be slower, anyhing wihou mass will rush around a his speed all he ime There is, hene, only one kind of relaiiy in naure, as elerodynamis are no onsisen wih he Galilean ransformaion

20 bu agree well wih experimens, he Galilean ransformaion mus be wrong, As Newon s mehanis are onsisen wih he Galilean ransformaion, i an be orre alhough i seems o agree wih experimen well a he ypially enounered speeds on he human experiene sale, so a new kind of mehanis mus be deeloped and was subsequenly deeloped by Einsein! The apparen imes of he Lorenz ransformaion are he real imes, here are in fa many differen real imes depending on he eloiy of he obserer, so ime is no absolue There is no eher o be disoered experimenally, here is no a preferred inerial sysem aahed o he eher, so spae is no absolue eiher Condiion: he new mehanis mus onain he Newonian mehanis as limiing ases for small speeds jus as he Lorenz ransformaions onain he Galilean ransformaions as limiing ase. From his kind of reasoning, Einsein go same onlusions as from Lorenz ransformaion, (he deried and inerpreed Lorenz s equaions independenly : four dimensional spaeime : lengh onraion 0

21 3: ime dilaion 4: relaiisi addiion of eloiies 5: relaiisi Doppler effe, ranserse Doppler effe 6. full blown modern relaiiy no only lengh onraion and ime dilaion as Lorenz bu, relaiisi dynamis 7. res mass, relaiisi mass / mass dilaion 8. relaiisi momenum, fore and aeleraion 9. relaiisi kinei energy 0. res energy, oal energy and mass-energy relaion relaiisi energy and momenum, massless pariles he one nie hing abou speial relaiiy: he mahemaial sheme is jus high shool algebra and alulus, so if you are los by he blun disagreemen beween your eeryday experiene and modern relaiiy, you hae o do he mahs, hey will guide you o he orre onlusions before we pik up he sory again wih poin 7, le s see how leerly Einsein deried he Lorenz equaions

22 direly ranslaed from A. Einsein, Uber die spezielle und allgemeine Relaiiäsheorie 96 exra lines are added in he algebrai deriaions in order o make i easier for you guys o omprehend wha he is doing in figure (Abb. he x axes of boh sysems are oiniding all he ime. We an, hus, died he problem and firs look only a eens ha are loaed on he x-axis a ray of ligh along he x-axis of K obeys x x - 0 ( he same ray of ligh along he x-axis of K obeys x 0 (

23 3 as all spae-ime poins hae o obey ( and ( i mus be rue ha (x? (x (3 where? is a onsan analogously we mus hae x + µ (x + (4 where µ is also a onsan adding or subraing (3 and (4, whereby we replae for simpliiy he onsans? and µ by a λ + µ b λ µ we obain he sysem x a x b (5a a b x (5b

24 4

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26 6

27 7a: relaiisi momenum 7 remember lassi momenum, p lass m, onseraion of momenum in ollisions? Sine impulse is funion of mass has o be reaed relaiisi if no muh smaller han same (Lorenz faor? applies p rela m o say m 0 kg, 00 km h - p lass [kg m s - ] [N s - ] p rela [N s - ] say m 0 kg, 0. p lass [N s - ] p rela [N s - ], esed ounless imes in parile aeleraors

28 7b: res mass, relaiisi mass / mass dilaion 8 same (Lorenz faor as for ime dilaion? mass dilaion : m m o m > m 0 for any > 0 where m 0 is alled res mass proper mass, (as measured in a inerial referene frame a res m > m 0 for any speed, a > 0 say m 0 kg, 0.5 for he sake of i m 30 kg or 5 kg or.5 kg? say m 0 kg, 00 km h - for he sake of geing an idea of he magniude m. kg or.005 kg or kg + 5 pg (pio 0 -? does no ge noied in eeryday experiene relaiisi mass is no a real effe as relaiisi ime, older exs and formulae olleions use i ofen one an replae relaiisi mass in formulae in he same way as one ould relaiisi ime, - in differen formulae - here will be differen faors o aoun for relaiisi effes hek hp:// for he modern one map-wo lok approah

29 aually his one formula aboe and is onsequenes wihin he sheme of high shool algebra/alulus are all here is o modern relaiiy, his modifiaion by Einsein makes Newonian mehanis fully ompaible wih he Lorenz ransformaion, e.g. if E m is orre, wha will be he formula for relaiisi mass? Sar wih body a res, apply a fore o he body, sars moing and gies i kinei energy (sine energy is inreased mass is inreased as well as long as fore oninues, energy and mass boh inrease 9 rae of hange of energy de d r r F ( r r beause W F s done, and now F r r d( m d r ds d, hange in kinei energy (de is equialen o work r r (impulse hange of momenum F d d p r insering in ( d ( m r d( m d d r ( rik o resole for m muliply boh sides by m (m dm d d( m m now d m dm d d( m and d d( m m d d( m d replaing

30 30 d( m d d( m d, (3 if deriaes of wo quaniies are equal, he quaniies hemseles differ by a onsan C m m + C / o define C we onsider speial ase 0 and say m is mass a res m 0, m C m m + m 0 / died by and rearranging m (- / m 0 finally m m 0 he one formula ha is needed o derie relaiisi mehanis whih onfirms o he Lorenz equaions his formula being onsisen wih E m does no mean i is a real effe hek hp:// for he modern one map-wo lok approah so how abou

31 8. relaiisi fore and aeleraion 3 Newon s seond law? dp m d F lass d d a m F m a I F d m d - impulse equals hange of momenum Sine impulse, fore and aeleraion are funions of mass hae o be reaed relaiisi if no muh smaller han If fore, aeleraion, eloiy parallel x-axis F 3 ( m a a ( F 3 m NOTE THAT THE FACTORS ARE NEITHER? NOR? Consequenes: onsan fore no longer auses onsan aeleraion Newon s nd law is o be reaed relaiisially in ase no <<

32 as eloiy inreases, aeleraion aused by gien fore onsanly dereases, 3 if lose o, a goes o zero, i s impossible o aelerae an obje wih mass o (howeer hard one may ry 9. relaiisi kinei energy remember kinei energy KE lass ½ m an be righ if m m( KE? m 0 ½ naure is no ha simple, I am afraid KE rela m - m 0 ( - m 0 faor is neiher? nor? LET S ompare KE lass and KE rela Say m 0 kg, 00 km h - KE lass [kg m s - ] [Nm], [Ws], [J] KE rela [J]

33 33 Say m 0 kg, 0. KE lass [TJ] KE rela [TJ] [Terra Joule] WHOW!!! World onsumpion of elerial energy is only TJ per hour 0: res energy, oal energy and mass-energy relaion, massless pariles rearranging KE rela m - m 0 m KE rela + m 0 is oal energy, here is no poenial energy around wih m 0 res energy, energy of body a res m E? m 0 mos famous formulae of physis a las E oal energy kinei energy + res energy so mass is in effe a form of energy, an be onered ino oher forms of energy (is aually done in nulear fission, fusion, any hemial reaion, e.g. in your somah when you diges your food

34 energy released in Hiroshima bomb equialen o abou gram, ha gram was mass loss/energy gain 34 law of onseraion of energy has o be expanded: in all proesses he ombinaion of energy and mass are onsered Consering energy, energy/mass, and momenum for proesses suh as ollisions, radioaie deay, hemial reaions A+B? C A+B? C+D A? B+C sum of momena of he iniial objes is equal o sum of momena of he final objes sum of energies of iniial objes is equal o sum of energies of final objes here is no independen onseraion law for mass, sum of masses of reaans is differen from sum of masses of produs, here is no independen onseraion law for kinei energy, bu oal energy kinei energy + res energy (m 0 is onsered produs are ligher han reaans if reaion is releasing energy, exohermi, g in Hiroshima bomb produs are heaier han reaans if reaion onsumed energy, endohermi

35 hypoheial fission een, exohermi of ourse, so we will lose some mass, one nuleus (M 0 a res splis ino wo idenial fragmens (m 0 ha head of wih equal momenum in differen direions 35 p? m 0 p -? m 0 M 0? m 0 rearranged? M 0 m 0 > and whih aken ogeher are ligher han he original nuleus in a sense: energy and mass an be reaed from eah oher how abou m 0 0, a massless parile? Does no iolae he E m law, i is jus all pure kinei energy KE rela m - m 0 all oher objes in he uniersiy are a mixure of impure energy, i.e. mass, and pure energy eeryhing in he unierse is in a sense energy lose o Greek philosopher Heralius, BC: pa?a s?? all hings are in flux, eeryhing is onsanly moing and hanging, only hange is permanen, you an sep ino he same rier wie

36 Pariles/objes wih mass a res hae erain res energy, if you diide res energy by you ge res mass 36 Eleron (a res Proon (a res Alpha (a res u aomi mass uni / of mass C (0 0-7 kg Human being, 80 kg (a res 0.5 MeV MeV MeV (8 MeV 93.5 MeV MeV if wo pariles of exaly equal eloiy (wih opposie signs, i.e. equal kinei energy, and mass ollide and sik ogeher, hey form a new parile ha parile will hae zero kinei energy, i.e. will be a res, bu i will be heaier [by summand m (? -] han he sum of he wo original pariles beause he kinei energy is onered ino mass ombining rearranged relaiisi oal energy and relaiisi momenum formulae E ( m 0 and ( p m 0 (afer subraing he seond from firs and some rearranging f p E ( m0 + ( p (alernaiely E

37 if here is a parile a res, p 0: E m 0 37 bu an parile may hae energy and momenum een if i has no m 0, in oher words no res mass suh a parile, e.g. a phoon, has E p hf hn where h: Plank s onsan: Js [Ws ] f: frequeny λ and? waelengh and is doomed o rush around wih speed of ligh Noa Bene: p anno be alulaed by p m sine m 0 0

38 Experimenal erifiaions 38 Lengh onraion: Mihelson-Morley and similar experimens, muon deay muon is formed in amosphere a a few km heigh a res in laboraory lifeime. µs, 0.99 in. µs lifeime (as measured in frame a res wih respe o he muon i an only rael 0.99 imes. µs 653 m bu i is obsered on earh, wha s happens is he 653 m are a onraed lengh, orresponding o a lengh of 653 m imes? 7., i.e m for he obserer a earh Time dilaion: muon deay - alernaie iew as he muon is moing so fas, is life ime, i.e.. µs, is dilaed by a faor 7., i.e. is really 5.6 µs wih respe o an obserer on earh, in his ime i will oer 4636 m in he obserers frame oher experimenal eidene: flying aomi loks around he earh, appliaion in global posiioning sysem jus as he muon lies longer due o is rapid moemen, if a person moes around lose o he speed of ligh, she/he will lie longer han always

39 saying a res, bu i won exend ha person s life span, if he is desined o reah age 85, he will reah i anyhow, her/his win sibling whih we assume o be desined o reah age 85 and one day as well will die earlier relaie o him as her (win paradox, see your book 39 Relaiisi Doppler Effe: used by polie o ah speeding ars Relaiisi mehanis: eery single day a a parile aeleraor, eery single day in an eleron mirosope so now be areful, elerons are small, i.e. ligh, an be aeleraed o high eloiies, if he eloiies are high we hae o make relaiisi alulaions

40 40 aeleraion olage 00 kv, wha is he waelengh of he eleron wae? p m h /? boh lassis m m 0 and relaiisi m m 0

41 kev e U poenial energy (PE KE 4 KE lass ½ m 0 p lass ( m 0 KE lass kgms -? lass h / p-lass Nms / Ns.94 pm bu lass ( KE lass /m m / s so lassial mehanis definiely no apply, all our resuls will be off kev e U PE KE rela ( m ( rik o simplify: β / 0 so p rela m0 β β e U ( m β ( eu + m0 β eu + m 0 0 eu resoling for ß yields ( eu m ( in ( p rela m 0eU ( kgms -? rela h / p-rela Nms / Ns.508 pm 0

42 4 and rela m / s, i.e ha is wha is obsered in experimens a he mirosope so les summarize deiaions p (p lass p rela 00 % / p rela 9.4 %? (? lass? rela 00 % /? rela -.6 % ( lass rela 00 % / rela 7. % New onep of ime; presen is no a momen, i is a ime ineral, i inludes all he eens from he pas ha we an in priniple know abou from differen plaes and all he eens from he fuure ha we an in priniple influene, e.g. if sun goes ou, we won know for 8 minues, if a far away sar ligh years away has died some years ago we won know abou his for a year New onep abou spae: spae iself is no a frame of referene

43 Quesions a brigh suden may ask 43 Is Earh an inerial referene frame? Is Newon s s law alid on Earh? Is Newon s nd law alid on Earh? No only approximaely Yes, for small Yes, for small Is he res of physis, elerodynamis, quanum mehanis alid on Earh? Yes Why is ha? Einsein s heory of general relaiiy, aeleraion is equialen o graiaional field, Einsein himself (afer some mediaion on his desk a he paen offie in Zurih 907 here ourred o me he happies hough of my life, in he following form. The graiaional field has only a relaie exisene, beause for an obserer falling freely from he roof of a house here exis a leas in his immediae surroundings no graiaional field. Indeed, if he obserer drops some bodies hen hese remain relaie o him in a sae of res or of uniform moion, independen of heir pariular hemial or physial naure. The maer independene of he aeleraion of fall is a powerful argumen for he fa ha he relaiiy posulae has o be exended o oordinae sysems whih, relaie o eah oher, are in non-uniform moion.

44 in oher words: relaie moion does no maer in physis a all, be i linear wih onsan eloiy or irularly wih onsan speed, or aeleraed in whaeer oher manner 44 so i does no maer ha Earh is no an inerial referene frame, as a maer or fa, inerial referene frames do no exis in our unierse - bu hey are quie helpful o explain basi physis o undergraduaes jus as a physiis in a losed laboraory an no ell if he is in moion or a res by all kind of experimens he is doing if here is a ground for him o walk on, he an no ell by any of his experimens if he is in a graiaional field or being aeleraed in he opposie direion (good for us, so we are on Earh in wha we onsider a graiaional field and we do physis and our physis is appliable anywhere else in he unierse general relaiiy is a generalizaion of speial relaiiy in a leas 3 senses: i s a new heory of graiy ha replaes Newon s heory of graiy, ha explains he Perihelion of he orbi of Merury, all planes hae i merury has i mos as i is loses o he sun where spae-ime is more seerely ured obsered effe an no be explained by Newon s heory ligh is bend by graiy, alhough i does no hae mass, i.e. ould no be subjeed o a fore of graiy in Newon s sense, aually he ligh follows a sraigh line in a ured spae

45 a onex lens shaped galaxy will a like a fousing lens, we speak abou graiaional lens, here an be blak holes, objes so massie ha een ligh anno esape from hem, hese holes suk eeryhing around hem in ime depends also on he srengh of graiaional fields, he higher he graiaional poenial, he slower he lok wo idenial ulrahigh preision loks (radioaie?-ray emiers 4 meers apar in a building a Harard Uniersiy, he lok in he basemen is slower by abou a seond in a million years, bu he effe is signifianly large ha i mus be orreed for in he global posiioning sysem i s an imporan ool in osmology, big bang,. he fronier fuure heory of eeryhing? besides he new effes due o relaiiy: any physial law (o be disoered in he fuure or any sensible hypohesis mus by inarian o he Lorenz ransformaions (speial relaiiy and inarian o anoher ransformaion of he so alled Gaussian oordinaes (general relaiiy ha desribe bend spae-ime so spae-ime is ured (non Eulidian by masses, hen here is no fore of graiy needed a all, i is all geomery, eeryhing moes in a sraigh line in ured spae-ime a lower dimension analogue: onsider wo expediions heading o he Norh pole, one saring in Siberia, one in he USA, boh of hem follow exaly a meridian and don know abou eah oher and don feel any fore exered on hem by he oher eam, bu hey ge loser and loser ogeher by geomery ha looks like he effe of a fore bu i is pure geomery 45

46 Spae rael for human beings??? 46 Say m 0 00 kg, 0. KE rela J WHOW!!! World onsumpion of elerial energy is only J per hour sine ha is (a leas he energy needed o ge 00 kg mass o 0. speed, spae rael is hardly feasible as a pasime for he many, bu, ery nex solar sysem, a-enaury, is 4.6 ligh years away i.e. o ge here akes (a years earh ime wihin hese 46 years almos all of he guys who designed he hardware/sofware of he spaeship are reired hink abou he sae of he ar of ompuers 46 years ago!!! he asronaus mus by hen be prey bored and hae reiremen on heir mind as well as he has aged (46 0.5% years

47 so wha lessons are o be learned from speial relaiiy? 47 Einsein s suggesing and numerous experimenaliss onfirming ha Newon s laws and wih hem all of lassial mehanis are only alid for small speeds should make us humble ha all physials laws may be wrong one way or he oher, een if we do no like srange ideas suh as ime dilaion and lengh onraion, we hae o deal wih hem beause hey desribe ery well wha is seen in experimens I ook mankind so long afer Newon o figure our his laws do no desribe naure auraely beause all of mankind s experiene before say 900 was for speeds ha are no a all omparable o he speed of ligh, so mankind s experiene was sor of inomplee in quanum mehanis, here will be een weirder oneps beause we are dealing wih hings ha are muh muh smaller han ourseles and eeryhing in our pereiable surrounding, so we an pereie/imagine orrely wha is down here, we hae o rely on mahemaial heories ha are in agreemen wih experimens his is going o be he new ruh here is as inuiion misleads us in regions where we do no hae dire experiene, e.g. he ery fas, he ery small, one has o sik o he mahemaial sheme see Heisenberg s opening saemen speial relaiiy also explains why eleriiy and magneism are one and he same hing

48 48 remember: moing elerial harge gies rise o magnei fore i.e. an eleri moor is based on a relaiisi effe alhough speed of elerons is small ompared o i.e. is abou mm / s, bu eleri fore beween eleron and proon in hydrogen aom is 0 39 more powerful ha graiaional fore and here may be 0 9 elerons / mm example fore beween parallel urren urrying wires, eleri harge is (like relaiisially inarian (a hare of magniude q is he same in all referene frames fas: wo parallel wires, elerially neural for boh urren zero, urren? zero disane beween moing harges undergoes lengh onraion by inerse Lorenz faor urrens parallel, wires ara eah oher (remember definiion r r µ 0µ rel I I l of Ampere Fmag π r bu why??? In laboraory frame of referene, boh wires are neural, o us here appears o be an magnei ineraion beween he urrens, o eah of he harges, here is only an elerial ineraion adding up o F ele Q Q 0 4π ε r so speial relaiiy is a grea unifier!!!

49 Feynman s ligh lok 49

50 50

51 5

52 5 u earh 0 4 in Mihelson-Morley experimen Binominal expansion n( n ( ± x n ± nx ±! ± muh smaller erms. If x <<, ( ± x n ± nx + x ± ½ x + x -+ ½ x, so faor u + ½ ß

53 53

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