The Special Theory of Relativity

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1 CHAPR 4 h Sial hory of Rlatiity 4-. Substitut q. (4.) into qs. (4.9) and (4.): x x x () From () From () So or x x + x ( ) () x x x x

2 46 CHAPR W introdu osh α y, sinh α y and substitut ths xrssions into qs. (4.4); thn x xosh α tsinh α x t tosh a sinh α x x; x3 x3 Now, if w us osh α os (iα) and i sinh α sin (iα), w an rwrit () as ( α) sin ( α) x x os i + it i it x sin i + it i ( α) os ( α) () () Comaring ths quations with th rlation btwn th rotatd systm and th original systm in ordinary thr-dimnsional sa, x x os θ + x sin θ x x sin θ + x os θ x3 x3 (3) x x x θ x W an s that () orrsonds to a rotation of th x it lan through th angl iα If th quation ψ ( xit, ) ψ ( xit, ) is Lorntz inariant, thn in th transformd systm w must ha whr ψ ( x, it ) t ( x, it ) ψ t + + x y z () () (3) W an rwrit () as 4 ( x, it ) ψ (4) x µ µ

3 H SPCIAL HORY OF RLAIVIY 463 Now, w first dtrmin how th orator W know th following rlations: is rlatd to th original orator x µ µ. x µ µ x λ x (5) x µ µν ν ν λ x (6) ν µν µ µ λµν λµλ δνλ (7) µ hn, xν λµν x x x µ ν ν µ ν xν (8) λ µν λ µλ λµν λ µλ xµ ν xν λ xλ ν λ xν xλ (9) hrfor, µν µλ µ x µ ν λ µ xν xλ δ λ x λ x νλ ν λ ν λ x λ () Sin µ and λ ar dummy indis, w s that th orator Lorntz transformation. So w ha ( x, it ) µ µ x µ is inariant undr a ψ () x his quation mans that th funtion ψ takn at th transformd oint (x,it ) satisfis th sam quation as th original funtion ψ (x,it) and thrfor th quation is inariant. In a Galilan transformation, th oordinats bom x x t x y y yt z z t z () t t Using ths rlations, w ha

4 464 CHAPR 4 hrfor, x t + x x x t x x x t y y y t z y z t t t (3) x y z t x y z t x y z t + + x x t y y t z z t his mans that th funtion ψ (x,it ) dos not satisfy th sam form of quation as dos ψ ( xit, ), and th quation is not inariant undr a Galilan transformation. (4) 4-4. In th K systm th rod is at rst with its nds at x and x. h K systm mos with a loity (along th x axis) rlati to K. K K x x If th obsrr masurs th tim for th nds of th rod to ass or a fixd oint in th K systm, w ha t t x t t x () whr t and t ar masurd in th K systm. From (), w ha

5 H SPCIAL HORY OF RLAIVIY 465 W also ha t t ( t t ) ( x x x ) () x (3) ( t) t (4) ( t) t (5) Multilying () by and using (3), (4), and (5), w obtain th FitzGrald-Lorntz ontration: (6) 4-5. h aarnt sha of th ub is that sha whih would b rordd at a rtain instant by th y or by a amra (with an infinitsimally short shuttr sd!). hat is, w must find th ositions that th arious oints of th ub ouy suh that light mittd from ths oints arris simultanously at th y of th obsrr. hos arts of th ub that ar farthr from th obsrr must thn mit light arlir than thos arts that ar losr to th obsrr. An obsrr, looking dirtly at a ub at rst, would s just th front fa, i.., a squar. Whn in motion, th dgs of th ub ar distortd, as indiatd in th figurs blow, whr th obsrr is assumd to b on th lin assing through th ntr of th ub. W also not that th fa of th ub in (a) is atually bowd toward th obsrr (i.., th fa aars onx), and onrsly in (b). (a) Cub moing toward th obsrr. (a) Cub moing away from th obsrr.

6 466 CHAPR K K x x W transform th tim t at th oints and x in th K systm into th K systm. hn, x x t t () x t t From ths quations, w ha ( x x ) t t t x () 4-7. K K x Suos th origin of th K systm is at a distan x from th origin of th K systm aftr a tim t masurd in th K systm. Whn th obsrr ss th lok in th K systm at that tim, h atually ss th lok as it was loatd at an arlir tim baus it taks a rtain tim for a light signal to tral to. Suos w s th lok whn it is a distan from th origin of th K systm and th tim is t in K and t in K. hn w ha t t ( t) t t x t W liminat, t, and x from ths quations and w find ()

7 H SPCIAL HORY OF RLAIVIY 467 t t () his is th tim th obsrr rads by mans of a tlso h loity of a oint on th surfa of th arth at th quator is 8 π R π 6.38 m 4 τ 8.64 s () whih gis m/s m/s 6 β.55 () 3 m/s Aording to q. (4.), th rlationshi btwn th olar and quatorial tim intrals is so that th aumulatd tim diffrn is Sulying th alus, w find hus, t t t + β β t t β t (4) ( 6 ) ( s/yr) ( yr) (5).38 s (6) (3) 4-9. w dm m + dm + d h unsurrising art of th solution to th roblm of th rlatiisti rokt rquirs that w aly onsration of momntum, as was don for th nonrlatiisti as. h surrising, and ky, art of th solution is that w not assum th mass of th jtd ful is th sam as th mass lost from th rokt. Hn ( + )( + )( + ) + w m d m dm d dm w () whr dm is th mass lost from th rokt, dm is th mass of th jtd ful, ( ) w V V is th loity of th xhaust with rst to th inrtial fram, and. On an asily alulat w w d 3 β dβ, ad aftr som algbra on obtains

8 468 CHAPR 4 dm w md+ dm+ w () whr w of ours k infinitsimals only to first ordr. h additional unknown dm is unalarming baus of anothr onsration law m (3) d m dm w dm Subsqunt substitution of dm into () gis, in on of its many intrmdiat forms βw md + dm( w) and will finally om to its dsird form aftr diiding by dt d dm m + V ( β ) (5) dt dt h quantity dt an b masurd in any inrtial fram, but would rsumably only mak sns for th artiular on in whih w masur. Intrstingly, it is not imortant for th jtd ful to ha an sially larg kinti nrgy but rathr that it b nar light sd, a nontriial distintion. For suh a as, a rokt an rah.6 by jting half its mass. (4) 4-. From q. (4.4) Soling () for x and substituting into () gis Soling () for t and substituting into () gis x x t () t t x () x t t + t t t + x t t t t + x or t x x + x ( ) x x + t

9 H SPCIAL HORY OF RLAIVIY θ x From xaml 4. w know that, to an obsrr in motion rlati to an objt, th dimnsions of objts ar ontratd by a fator of omonnt of th stik will b in th dirtion of motion. hus, th x os θ whil th rndiular omonnt will b unhangd: sin θ So, to th obsrr in K, th lngth and orintation of th stik ar ( ) sin θ + os θ or sin θ θ tan os θ os θ sin θ + tanθ tan θ 4-. h ground obsrr masurs th sd to b m.5 8 m/s.4 µ s h lngth btwn th markrs as masurd by th rar is.5 m 55.3 mtrs 3 h tim masurd in th rar s fram is gin by

10 47 CHAPR 4 t t x.4 µ s 8 (.5 m/s) ( m).5 3. µ s h sd obsrd by th rar is t t 8.5 m/s 4-3. t t t.5 µ s hrfor t 34µ s K K sour rir In K, th nrgy and momntum of ah hoton mittd ar hν hν and Using q. (4.9) to transform to K : hν hν ( ) ; hν + hν

11 H SPCIAL HORY OF RLAIVIY 47 So ν ν + + β + β ν ν β β whih agrs with q. (4.3) From q. (4.33) ν β ν + β Sin λ ν λ β λ + β or λ β λ + β 4 4 With λ nm and β, λ nm. 8 3 S o th shift is. nm toward th rd (longr walngth) K K θ star θ arth Considr a hoton snt from th star to th arth. From q. (4.9) also ( ) +

12 47 CHAPR 4 Now Substituting yilds and hus hν hν hν, hν, os θ, os θ ν ν + βosθ ν ν βosθ ( + β os θ)( β os θ ) + β os θ β os θ β os θ os θ β os θ os θ β os θ os θ β Soling for os θ yilds os θ β os θ β osθ whr β θ angl in arth s fram θ angl in star s fram 4-7. From q. (4.33) ν β ν + β Sin ν λ, λ + β λ β W ha λ.5 λ. his gis or 5 β 3 8. m/s

13 H SPCIAL HORY OF RLAIVIY K K θ light sour obsrr Proding as in xaml 4., w trat th light as a hoton of nrgy hν. hν In K : hν, In K: hν ( + ν ) For th sour aroahing th obsrr at an arly tim w ha hν hus ν ν + ν ν + β β For th sour rding from th obsrr (at a muh latr tim) w ha hν and ν ν β + β So ν ν ν ν + β sour aroahing obsrr β β sour rding from obsrr + β

14 474 CHAPR K K θ sour obsrr Proding as in th rious roblm, w ha In K : hν hν hν os θ K + ν hν In : ( ) β r β + β r t So or hνβ r hν hν β r + β t βr β t βr β + t ν ν ( β ) r β β r t ν λ β r βt ν λ β r For λ > λ, w ha β > β β r r t β > β β t r r β > β β t r r 4-. As masurd by obsrrs on arth, th ntir tri taks 4 lightyars 8 y.3 ars 3 h ol on arth ag 8 3 yars. h astronaut s lok is tiking slowr by a fator of. hus, th astronaut ags

15 H SPCIAL HORY OF RLAIVIY 475 So yars hos on arth ag 6.7 yars. h astronaut ags 5.4 yars. 4-. F ( β) d β m dt β β ( β ) m + 3 () m ββ + β ( β ) 3 If w tak (this dos not man 3 ), w ha m F m m β ( β ) ( β ) () F m β mt (3) F3 m β 3 mt 3 (4) 4-. h total nrgy outut of th sun is whr R.5 m d 3 (.4 W m ) 4π R () dt is th man radius of th arth s orbit around th sun. hrfor, d dt h orrsonding rat of mass dras is dt dt () W dm d kg s (3) 3 h mass of th sun is aroximatly. 99 kg, so this rat of mass dras an ontinu for a tim

16 476 CHAPR yr 4.4 kg s yr (4) Atually, th liftim of th sun is limitd by othr fators and th sun is xtd to xir 9 about 4.5 yars from now From q. (4.67) + + m h minimum nrgy will our whn th four artils ar all at rst in th ntr of th mass systm aftr th ollision. Consration of nrgy gis (in th CM systm) or whih imlis or β 3 4m,CM m o find th nrgy rquird in th lab systm (on roton at rst initially), w transform bak to th lab + () h loity of K (CM) with rst to K(lab) is just th loity of th roton in th K systm. So u. hn Sin, β 3, CM mu m m β Substituting into () 3

17 H SPCIAL HORY OF RLAIVIY lab h minimum roton nrgy in th lab systm is m m 7, of whih 6 is kinti nrgy Lt B B z hn i+ j q B q x x y y i j k B d d F q B m ( ) gis dt dt Dfin ω qb hus m d dt q y B i B qb x ( y i x j ) m ω and ω x y y x j or ω ω x y x and ω ω y x y So Aos ωt+ Bsin ωt x Cos ωt+ Dsin ωt ak,. hn A, C. hn ω x y y y ω ω B, D x x y

18 478 CHAPR 4 hus hn ios ωt jsin ω t r i sin ωt+ j os ωt ω ω h ath is a irl of radius ω From roblm 4- So m r qb m qb qb m+ m + r qb 4-6. Suos a hoton traling in th x-dirtion is onrtd into an and + as shown blow + θ θ Cons. of nrgy gis whr bfor momntum of th hoton aftr Cons. of x gis Diiding gis nrgy of nrgy of + + ( ) os θ momntum of,

19 H SPCIAL HORY OF RLAIVIY 479 or os θ os θ () But >, so () annot b satisfid for os θ. An isolatd hoton annot b onrtd into an ltron-ositron air. his rsult an also b sn by transforming to a fram whr x aftr th ollision. But, bfor th ollision, in any fram moing along th x-axis. So, without anothr x objt narby, momntum annot b onsrd; thus, th ross annot tak la h minimum nrgy rquird ours whn th and ar at rst aftr th ollision. By onsration of nrgy 938 MV Sin.5 MV, 938 MV MV W dsir lassial m rl ( ) m rl rl ( ) lassial lassial. m. m ( ).99 β.98 Putting ( ) β and soling gis

20 48 CHAPR 4.5 h lassial kinti nrgy will b within % of th orrt 7 alu for 3.5 m/s, indndnt of mass For 9 3 V 6.5 V, 5.88 or β β β.4 4 ( ) A nutron at rst has an nrgy of MV. Subtrating th rst nrgis of th roton (938.3 MV) and th ltron (.5 MV) las.8 MV. Ot hr than rst nrgis.8 MV is aailabl θ θ Consration of nrgy gis π whr nrgy of ah hoton (Cons. of y imlis that th hotons ha th sam nrgy).

21 H SPCIAL HORY OF RLAIVIY 48 hus 35 MV MV h nrgy of ah hoton is 339 MV. Consration of x gis m os θ whr momntum of ah hoton ( 35 M/ ) (.98 ) o s θ MV/ θ os From q. (4.67) w ha With +, this rdus to + Using th quadrati formula (taking th + root sin ) gis Substituting MV gis + ( ltron).5 MV roton 938 MV ltron roton MV 433 MV

22 48 CHAPR n Consration of y gis bfor ν aftr Consration of So x gis sin 6 sin 6 or ν os 6 + ν os 6 ν ν Consration of nrgy gis n + + ν () n Substituting and soling for gis n MV MV.5 MV.554 MV/ ν.554 MV/ Substituting into gis ( ν ) + ν.554 MV 4 MV, or V.5 MV

23 H SPCIAL HORY OF RLAIVIY Using th Lorntz transformation this boms s t + x + x + x 3 x t + xt x + t xt x x3 s x x t t + x + x 3 So t + x + x + x 3 s s Lt th fram of Saturn b th unrimd fram, and lt th fram of th first saraft b th rimd fram. From q. (4.7a) (swith rimd and unrimd ariabls and hang th sign of ) Substituting.9 gis u + u u + u. u Sin w ha d dxµ F and X x, x, x, i d m µ τ dτ µ d dx d x F m m dτ dτ dτ ( 3 t) dx dx3 3 F m F m dτ dτ d dit d t F4 m im dτ dτ dτ

24 484 CHAPR 4 hus dx d F m m ( x t) d d τ τ dx dt m m F + i dτ dτ ( F iβf ) 4 dx iβm ( βf ) 4 dx dx F m m F ; F F dτ dτ d x F4 im t d τ dt im dτ dτ 3 3 h us th rquird transformation quations ar shown From th Lagrangian w omut ( β ) L m kx L kx x L β L β m β β () () (3) hn, from () and (3), th Lagrang quation of motion is from whih Using th rlation w an rwrit (4) as d mβ kx dt + (4) β m β ( β ) 3 + kx d d dx d β (6) dt dx dt dx (5)

25 H SPCIAL HORY OF RLAIVIY 485 his is asily intgratd to gi m β ( β ) m 3 dβ + kx dx kx β + (7) (8) whr is th onstant of intgration. h alu of is aluatd for som artiular oint in has sa, th asist bing x a; β : From (8) and (9), liminating β from (), w ha m m + ka kx m ka β + + (9) () and, thrfor, β 4 ( x ) k a m m + k a x k m + a x 4 k m + a x ( ) + ( ) + ( ) () dx k a x m k a x 4 β () dt m k a x h riod will thn b four tims th intgral of dt dt(x) from x to x a: k a ( a x ) m + m τ 4 dx k (3) k a x + 4 ( a x ) m Sin x aris btwn and a, th ariabl xa taks on alus in th intral to, and thrfor, w an dfin from whih x sin φ (4) a

26 486 CHAPR 4 and a x os φ (5) a W also dfin th dimnsionlss aramtr, Using (4) (7), (3) transforms into Sin a τ dx a x dφ (6) a k κ (7) m ( + κ os φ) π κ + κ os φ ka m for th wakly rlatiisti as, w an xand th intgrand of (8) in a sris of owrs of κ : ( os ) ( + κ os φ) dφ + κ φ κ ( + κ os φ) os φ + os κ φ (8) Substitution of (9) into (8) yilds 3 + os κ φ (9) π a 3 τ κ φ + os κ dφ aπ 3κa + + sin φ φ κ aluating () and substituting th xrssion for κ from (7), w obtain π () or, m 3π a τ π + k 8 k m () τ 3 ka τ + 6 m ()

27 H SPCIAL HORY OF RLAIVIY F ( mu) d dt d dt d m u m dt ( ) (for onstant) hus d u m dt u m ( u ) u ( u ) ( u ) ( ) 3 m u u du dt du dt du F m u dt ( ) h kinti nrgy is For a momntum of MV/, + m m () 4 () 4 roton MV In ordr to obtain and β, w us th rlation (3) 4 ltron MV m m m β (4) so that and (5) m β (6) ltron (7).5

28 488 CHAPR 4 his is a rlatiisti loity. his is a nonrlatiisti loity. β ltron β (8) 936 roton.54 (9) 93 roton.53. () 4-4. If w writ th loity omonnts of th ntr-of-mass systm as j, th transformation of α, j into th ntr-of-mass systm boms j α α, j α, j () whr or, j α j α. Sin in th ntr-of-mass systm,, must b satisfid, w ha α j α α, j α, j α () j α α α, j α (3) 4-4. W want to omut m m whr and rrsnt th kinti and total nrgy in th laboratory systm, rstily, th subsrits and indiat th initial and final stats, and m is th rst mass of th inidnt artil. h xrssion for in trms of is m () () an b rlatd to (total nrgy of artil in th ntr of momntum rfrn fram aftr th ollision) through th Lorntz transformation [f. q. (4.9)] (rmmbring that for th inrs transformation w swith th rimd and unrimd ariabls and hang th sign of ):

29 H SPCIAL HORY OF RLAIVIY 489 ( β ) + θ os (3) whr mβ and m : hn, from (), (), and (4), ( os ) m + β θ (4) + β os θ For th as of ollision btwn two artils of qual mass, w ha, from q. (4.7), and, onsquntly, hus, with th hl of (6) and (7), (5) boms + (6) β (7) ( ) + osθ + osθ W must now rlat th sattring angl θ in th ntr of momntum systm to th angl ψ in th lab systm. Squaring q. (4.8), whih is alid only for m m, w obtain an quation quadrati in os θ. Soling for os θ in trms of tan ψ, w obtain os θ + tan ψ ± + + tan On of th roots gin in (9) orrsonds to θ π, i.., th inidnt artil rrss its ath and is rojtd bak along th inidnt dirtion. Substitution of th othr root into (8) gis + os + ( + ) sin ψ os ψ + tan ψ ψ ψ () An lmntary maniulation with th dnominator of (), namly, (5) (8) (9)

30 49 CHAPR 4 ( ) os ψ + + sin ψ os ψ + os ψ + sin ψ + sin ψ + os ψ os ψ + os ψ roids us with th dsird rsult: os ψ os ψ + + os ψ + () os ψ ( + ) ( ) os ψ () Noti that th sha of th ur hangs whn > m, i.., whn > GV GV GV ψ 4-4. y hν m φ x θ hν From onsration of nrgy, w ha h + m m + h () ν ν Momntum onsration along th x axis gis hν hν os θ + m os φ () Momntum onsration along th y axis gis hν m sin φ sin θ (3)

31 H SPCIAL HORY OF RLAIVIY 49 In ordr to liminat φ, w us () and (3) to obtain hn, hν hν os φ os θ m hν sin φ sin θ m (4) Sin hν hν hν hν os φ + sin φ os θ m + + (5) and w ha Substituting from () into (6), w ha h From (5) and (7), w an find th quation for ν : ( ) (6) h ( ν ) ( m ν + ν m ν ) (7) or, hn, or, hν hν hν hν h + osθ hm ( ν ν ) + ( ν ν m h + ν ( os θ ) ν ν h m ν ν hν + ( osθ ) m + os ( θ) m ) (8) (9) () () h kinti nrgy of th ltron is

32 49 CHAPR 4 m m hν hν + os m ( θ ) m osθ + os m ( θ ) ()

Problem 22: Journey to the Center of the Earth

Problem 22: Journey to the Center of the Earth Problm : Journy to th Cntr of th Earth Imagin that on drilld a hol with smooth sids straight through th ntr of th arth If th air is rmod from this tub (and it dosn t fill up with watr, liquid rok, or iron