CHAPTER 15 SPECIAL RELATIVITY

Transcription

1 CHAPTER 5 SPECIAL RELATIVITY 5 Inroduion Why a haper on relaiviy in a book on lassial mehanis? A firs euse migh be ha he phrase lassial mehanis is used by differen auhors o mean differen hings To some, i means pre-relaiviy ; o ohers i means pre-quanum mehanis For he purposes of his haper, hen, I mean he laer, so ha speial relaiviy may fairly be inluded in lassial mehanis A seond euse is ha, apar from one brief foray ino an eleromagnei problem, his haper deals only wih mehanial, kinemai and dynamial problems, and herefore deals wih only a raher resried par of relaiviy ha an be deal wih onvenienly in a single haper of lassial mehanis raher han in a separae book This is in fa a quie subsanial resriion, beause eleromagnei heory plays a major role in speial relaiviy I was in fa diffiulies wih eleromagnei heory ha led Einsein o he speial heory of relaiviy Indeed, Einsein s heory of relaiviy was inrodued o he world in a paper wih he ile Zur Elekrodynamik beweger Körper (On he Elerodynamis of Moving Bodies), Annalen der Physik, 7, 89 (95) The phrase speial relaiviy deals wih he ransformaions beween referene frames ha are moving wih respe o eah oher a onsan relaive veloiies Referene frames ha are aeleraing or roaing or moving in any manner oher han a onsan speed in a sraigh line are inluded as par of general relaiviy and are no onsidered in his haper 5 The Speed of Ligh The speed of ligh is, by definiion, ealy % 8 m s, and is he same relaive o all observers This seemingly simple senene invies several ommens Firs: Noe ha I have used he word speed Some wriers use he word veloiy as if i were merely a more impressive and sienifi-sounding synonym for speed I rus ha all readers of hese noes know he differene and will use he word speed when hey mean speed, and he word veloiy when hey mean veloiy surely no an unreasonable demand To say ha he veloiy of ligh is he same for all observers means ha he direion of ravel of ligh is he same relaive o all observers This is doubless no a all wha a wrier who uses he word veloiy inends o onvey bu i is he lieral (and of ourse quie erroneous) meaning of he asserion Seond: How an we possibly define he speed of ligh o have a erain ea value? Surely he speed of ligh is wha we find i o be, and we are no free o define is value Bu in fa we are allowed o do his, and he eplanaion, briefly, is as follows

2 Over he ourse of hisory, he mere has been defined in several differen ways A one ime i was a speified fraion of he irumferene of Earh Laer, i was he disane beween wo srahes on a bar of plainum-iridium alloy held in Paris Laer sill i was a speified number of wavelenghs of a pariular line in he sperum of merury, or admium, or argon or krypon In our presen sae of ehnology i is far easier o measure and reprodue preise sandards of frequeny han i is o measure and reprodue sandards of lengh Beause of ha, he urren SI (Sysème Inernaional) uni of ime is he SI seond, whih is based on he frequeny of a pariular ransiion in he sperum of aesium, and from here, he mere is defined as he disane ravelled by ligh in vauo in a defined fraion of an SI seond, he speed of ligh being assigned he ea value quoed above Deailed disussion of he ea definiions of he unis of ime, disane and speed is par of he subje of merology Tha is an imporan and ineresing subje, bu i is only marginally relevan o he opi of relaiviy, and onsequenly, having quoed he ea value of he speed of ligh, we leave furher disussion of merology here Third: How an he speed of ligh be he same relaive o all observers? This asserion is absoluely enral o he heory of speial relaiviy, and i may be regarded as is fundamenal and mos imporan priniple We shall disuss i furher in he remainder of he haper 53 Preparaion The raio of he speed v of a body (or a parile, or a referene frame) is ofen given he symbol β: β v / 53 For reasons ha will beome apparen (I hope!) laer, he range of β is usually resried o beween and In our sudy of speial relaiviy, we shall find ha we have o make frequen use of a number of funions of β The mos ommon of hese are γ ( β ) /, 53 k ( β) /( β), 533 z k, 534 K γ, 535 φ ln[( β) /( β)] anh β ln k 536

3 3 θ os γ sin ( i βγ) 537 In figures XV-3 I draw γ, k and φ as funions of β The funions γ and k go from o as β goes from o ; z, K and φ go from o The funion θ is imaginary Many one migh even say mos problems in speial relaiviy (inluding eaminaion and homework quesions!) amoun, when sripped of heir verbiage, o he following: Given one of he quaniies β, γ, k, z, K, φ, θ, alulae one of he ohers Thus I would sugges ha, even before you have any idea wha hese quaniies mean, you migh wrie a program for your ompuer (or programmable alulaor) suh ha, when you ener any one of he real quaniies, he ompuer will insanly reurn all si of hem This will save you, on fuure oasions, from having o remember he ea formulas or having o boher wih edious arihmei, so ha you an onenrae your mind on undersanding he relaiviy γ FIGURE XV β

4 k FIGURE XV β 3 5 φ FIGURE XV β

5 5 Jus for fuure referene, I abulae here he relaions beween hese various quaniies This has involved some algebra and ypeseing; I don hink here are any misakes, bu I hope some reader migh hek hrough hem all arefully and will le me know (jaum a uvi do a) if he or she finds any β γ k k z( z ) ( z ) K ( K ) K anh φ or e φ φ e i an θ γ β k k ( z ) ( z ) K osh φ or φ φ ( e e ) os θ β k γ γ z K K ( K ) β φ e iθ e z β γ γ β k K K ( K ) φ e iθ e K φ ( k ) z ( e ) γ φ β k ( z ) e os θ φ anh β or β ln β osh γ or ln ( ) ( ) γ γ ) ln k ln( z ) ln K K ( K ) iθ θ i ln β β i ln ( ) ( ) [ ] γ γ i ln k i ln z i ln K K ( K ) iφ 54 Speed is Relaive The Fundamenal Posulae of Speial Relaiviy You are siing in a railway arriage (or a railroad ar, if you prefer he erm) The windows and urains are losed and you anno see ouside You are asked o measure he onsan speed of he arriage along is raks You ry a number of eperimens You measure he period of a simple pendulum You slide a puk and roll a ball down an inlined plane You hrow a ball verially up in he air and ah i as i omes down You hrow i up a an angle and you wah i desribe a graeful parabola You ause billiard balls o ollide on he billiards able houghfully provided in your arriage You eperimen wih a orsion pendulum You sand a penil on is end and you wah i as i falls o a horizonal posiion

6 6 All your areful work is o no avail None of hem ells you wha speed you are moving a, or even if you are moving a all Afer ehausing all mehanial eperimens you an hink of, you are led o he onlusion: I is impossible o deermine he speed of moion of a uniformly-moving referene frame by means of any mehanial eperimen performed wihin ha frame Frusraed, you open a urain on one side of he arriage You look ou and you see ha here is anoher rain on he line ne o you I appears o be moving bakwards Or are you moving forwards? Or are you boh moving in he same direion bu a differen speeds? You sill an ell You move o he oher side of he arriage and open he urain here This ime you see he saion plaform, and he saion plaform is moving bakwards Or are you moving forwards? (Those of you who have no done muh ravel by rain may no appreiae jus how very srong he impression an be ha he plaform is moving) Wha does i mean, anyway, o say ha i is you ha is moving raher han he plaform? The following sory is no rue, bu i ough o be (I is an aporyphal sory) Einsein was ravelling by rain aross Canada Halfway aross he Prairies he lean aross and apped on he knee of his fellow passenger and asked: Euse me, mein Herr, bie, bu does Regina sop a his rain? You are abou o onlude ha i is no possible by any means, wheher by eperimen or by observaion, o deermine he speed of your referene frame, or even wheher i is moving or saionary Bu no so hasy! I am abou o inven a speedomeer, whih I inend o paen and o use o make myself rih I am going o use my invenion o measure he speed of our rain wihou even looking ou of he window! We shall se up wo long parallel glass rods in he middle of he orridor, parallel o he railway lines and o he veloiy of he rain We shall suspend he rods horizonally, side by side from a ommon suppor, and we shall rub eah of hem wih a silken handkerhief, so ha eah of hem bears an elerosai harge of λ C m They will repel eah oher wih an elerosai fore per uni lengh of λ F e N m, 54 4πε r where r is heir disane apar, and onsequenly hey will hang ou of he verial see figure XV4

7 7 F e * * F e mg mg FIGURE XV4 Now see wha happens when he rain moves forward a speed v Eah rod, bearing a harge λ per uni lengh, is now moving forward a speed v, and herefore eah rod onsiues an eleri urren λv A Therefore, by Ampère s law, in addiion o he Coulomb repulsion, hey will eperiene a magnei araion per uni lengh equal o F m µ λ v 4πr N m 54 The ne repulsive fore per uni lengh is now λ 4πε r ( µ ε v ) 543 This is a lile less han i was when he rain was saionary, so he angle beween he suspending srings is a lile less, as shown in figure XV5 I migh be noed ha he fore beween he srings is redued o zero (and he angle also beomes zero) when he rain is ravelling a a speed / µ We remember from eleromagnei heory ha ε he permeabiliy of free spae is µ 4π % 7 H m and ha he permiiviy ε is

8 % F m ; onsequenly he fore and he angle drop o zero and he srings hang verially, when he rain is moving a a speed of 998 % 8 m s F e * F m F m * F e mg mg FIGURE XV5 To omplee my invenion, I am now going o aah a proraor o he insrumen, bu insead of marking he proraor in degrees, I am going o alibrae i in miles per hour, and my speedomeer is now ready for use (figure XV6) mph * * FIGURE XV6

9 9 You now have a hoie Eiher: i You an hoose o believe ha he speedomeer will work and you an aompany me o he paen offie o see if hey will gran a paen for his invenion, whih will measure he speed of a rain wihou referene o any eernal referene frame If you hoose o believe his, here is no need for you o read he remainder of he haper on speial relaiviy Or: ii You an say ha i defies ommon sense o believe ha i is possible o deermine wheher a given referene frame is moving or saionary, le alone o deermine is speed Common sense diaes ha I is impossible o deermine he speed of moion of a uniformly-moving moving referene frame by any means whaever, wheher by a mehanial or elerial or o r indeed any eperimen performed enirely or parially wihin ha frame, or even by referene o anoher frame Your ommon sense, hen, leads you as i should o he fundamenal priniple of speial relaiviy Whereas some people proes ha relaiviy defies ommon sense, in fa relaiviy is ommon sense, and is prediions (suh as your prediion ha my speedomeer will no work) are ealy wha ommon sense would lead you o epe 55 The Lorenz Transformaions For he remainder of his haper I am aking, as a fundamenal posulae, ha I is impossible o deermine he speed of moion of a uniformly-moving moving referene frame by any means whaever, wheher by a mehanial or elerial or indeed any eperimen performed enirely or parially wihin ha frame, or even by referene o anoher frame and onsequenly I am hoosing o believe ha my speedomeer will no work If i is impossible by any elerial eperimen o deermine our speed, we mus assume ha all he eleromagnei equaions ha we know, no jus he ones ha we have quoed, bu indeed Mawell s equaions, whih embrae all eleromagnei phenomena, are he same in all uniformly-moving referene frames One of he many prediions of Mawell s equaions is ha eleromagnei radiaion (whih inludes ligh) ravels a a speed

10 / µ ε 55 Presumably neiher he permeabiliy nor he permiiviy of spae hanges merely beause we believe ha we are ravelling hrough spae indeed i would defy ommon sense o suppose ha hey would Consequenly, our aepane of he fundamenal priniple of speial relaiviy is equivalen o aeping as a fundamenal posulae ha he speed of ligh in vauo is he same for all observers in uniform relaive moion We shall ake anyhing oher han his o be an ourage agains ommon sense hough aepane of he priniple will require a areful eaminaion of our ideas onerning he relaions beween ime and spae Le us imagine wo referene frames, Σ and Σ Σ is moving o he righ (posiive - direion) a speed v relaive o Σ (For breviy, I shall from ime o ime refer o Σ as he saionary frame, in he hope ha his libery will no lead o misundersanding) A ime he wo frames oinide, and a ha insan someone srikes a mah a he ommon origin of he wo frames A a laer ime, whih I shall all if referred o he frame Σ, and if referred o Σ, he ligh from he mah forms a spherial wavefron ravelling radially ouward a speed from he origin O of Σ, and he equaion o his wavefron, when referred o he frame Σ, is y z 55 Referred o Σ, i also ravels ouward a speed from he origin O of Σ, and he equaion o his wavefron, when referred o he frame Σ, is y z 553 Mos readers will aep, I hink, ha y y and z z Some formal algebra may be needed for a rigorous proof, bu ha would disra from our main purpose of finding a ransformaion beween he primed and unprimed oordinaes suh ha 554 I is easy o show ha he Galilean ransformaion, does no saisfy his equaliy, so we shall have o ry harder Le us seek linear ransformaions of he form whih saisfy equaion 554 A B, 555 C D, 556

11 We have A B, C D 557 and, by inversion, D C A C 558 Consider he moion of O relaive o Σ and o Σ We have / v and â v B /A 559 Consider he moion of O relaive o Σ and o Σ We have / v and â v B /D 55 From hese we find ha D A and B Av, so we arrive a A( v) 55 and C A 55 On subsiuion of equaions 55 and 55 ino equaion 554, we obain A ( v ) ( C A) 553 Equae powers of o obain A v / γ 554 Equae powers of o obain C vγ 555 Equaing powers of produes no new informaion We have now deermined A, B, C and D, and we an subsiue hem ino equaions 555 and 556, and hene we arrive a γ( v) 556

12 and γ( v / ) 557 These, ogeher wih y y and z z, onsiue he Lorenz ransformaions, whih, by suiable hoie of aes, guaranee he invariane of he speed of ligh in all referene frames moving a onsan veloiies relaive o one anoher To epress and in erms of and, you may, if you are good a algebra, solve equaions 556 and 557 simulaneously for and, or, if insead, you have good physial insigh, you will merely reverse he sign of v and inerhange he primed and unprimed quaniies Eiher way, you should obain γ( v ) 558 and γ( v / ) Bu This Defies Common Sense A his sage one may hear he proes: Bu his defies ommon sense! One may hear i again as we enouner several prediions of he invariane of he speed of ligh and of he Lorenz ransformaions Bu, if you have read his far, i is oo lae o make suh proes You have already, a he end of Seion 54, made your hoie, and you hen deided ha i defies ommon sense o suppose ha one an somehow deermine he speed of a referene frame by some eperimen or observaion You rejeed ha noion, and i was he appliaion of ommon sense, no is abandonmen, ha led us ino he Lorenz ransformaions and he invariane of he speed of ligh There may be oher oasions when we are emped o proes Bu his defies ommon sense!, and i is herefore always saluary o reall his For eample, we shall laer learn ha if a rain is moving a speed V relaive o he saion plaform, and a passenger is walking owards he fron of he rain a a speed v relaive o he rain, hen, relaive o he plaform, he is moving a a speed jus a lile bi less ha V v When we proes, we are ofen presened wih an eplanaion along he following lines: In every day life, rains do no move a speeds omparable o he speed of ligh, nor do walking passengers Therefore, we do no noie ha he ombined speed is a lile bi less han V v Afer all, if V 6 mph and v 4 mph, he ombined speed is % 64 mph The formula V v is jus an approimaion, we are old, and we have he erroneous impression ha he ombined speed is ealy V v only beause we are ausomed, in daily life, o eperiening speeds ha are small ompared wih he speed of ligh This eplanaion somehow does no seem o be saisfaory and nor should i, for i is no a orre eplanaion I seems o be an eplanaion invened for he benefi of he nonsienifi layman bu nohing is ever made easy o undersand by giving an inorre

13 3 eplanaion under he preene of simplifying somehing I is no orre merely o say ha he Galilean ransformaions are jus an approimaion o he real ransformaions The problem is ha i is eeedingly diffiul perhaps impossible o desribe ealy wha is mean by disane and ime inerval I is almos as diffiul as desribing olours o a blind person, or even desribing your sensaion of he olour red o anoher seeing person We have no guaranee ha every person s perepion of olour is he same The bes ha an be done o desribe wha we mean by disane and ime inerval is o define how disanes and imes ransform beween referene frames The Lorenz ransformaions, whih we have adoped in order o make i meaningless o disuss he absolue veloiy of a referene frame, amoun o a useful working definiion of he meanings of spae and ime One we have adoped his definiion, ommon sense no longer omes ino he maer There is no longer a mysery whih our minds anno quie grasp; from his poin on i merely beomes a maer of algebra as o how a measuremen of lengh or of ime inerval, or of speed, or of mass, as appropriaely defined, ransforms when referred o one referene or o anoher There is no impossible fea of imaginaion o be done 57 The Lorenz Transformaion as a Roaion The Lorenz ransformaion an be wrien 3 4 γ iβγ iβγ γ 3 4, 57 where, y, 3 z and 4 i, and similarly for primed quaniies Please don jus ake my word for his; muliply he maries, and verify ha his equaion does indeed represen he Lorenz ransformaion You ould, if you wish, also wrie his for shor: Anoher way of wriing he Lorenz ransformaion is λ 57 3 γ βγ βγ γ 3, 573 where, y, 3 z and, and similarly for primed quaniies

14 4 Some people prefer one version; ohers prefer he oher In any ase, a se of four quaniies ha ransforms like his is alled a 4-veor Those who dislike version 57 dislike i beause of he inroduion of imaginary quaniies Those who like version / 57 poin ou ha he epression [( ) ( ) ( 3) ( 4) ] (he inerval beween wo evens) is invarian in four-spae ha is, i has he same value in all uniformly-moving referene frames, jus as he disane beween wo poins in hreespae, [( ) ( y) ( z) ], is independen of he posiion or orienaion of any / referene frame In version 573, he invarian inerval is [( ) ( ) ( ) ( ) ] / 3 Those who prefer version 57 dislike he minus sign in he epression for he inerval Those who prefer version 573 dislike he imaginary quaniies of version 57 For he ime being, I am going o omi y and z, so ha I an onenrae my aenion on he relaions beween and Thus I am going o wrie 57 as 4 i γ iβγ iβγ γ i 574 and equaion573 as γ βγ βγ γ 575 Readers may noie how losely equaion 574 resembles he equaion for he ransformaion of oordinaes beween wo referene frames ha are inlined o eah oher a an angle (See Celesial Mehanis Seion 36) Indeed, if we le os θ γ and sin θ iβγ, equaion 574 beomes i os θ sin θ sin θ os θ i 576 The maries in equaions 57, 574 and 576 are orhogonal maries and hey saisfy eah of he rieria for orhogonaliy desribed, for eample, in Celesial Mehanis Seion 37 We an obain he onverse relaions (ie we an epress and in erms of and ) by inerhanging he primed and unprimed quaniies and eiher reversing he sign of β or of θ or by inerhanging he rows and olumns of he mari There is a diffiuly in making he analogy beween he Lorenz ransformaion as epressed by equaion 574 and roaion of aes as epressed by equaion 576 in ha, sine γ >, θ is an imaginary angle (A his poin you may wan o reah for your anien, brile, yellowed noes on omple numbers and hyperboli funions) Thus θ os γ, and for γ >, his means ha θ i osh γ i ln( γ γ ) And

15 5 θ sin ( i βγ ) i sinh ( βγ) i ln( βγ β γ ) Eiher of hese epressions redues o θ i ln[ γ( β)] Perhaps a ye more onvenien way of epressing his is θ β i anh β i ln 577 β For eample, if β 8, θ 986i, whih migh be wrien (no neessarily pariularly usefully) as i % 6 o 57 A his sage, you are probably hinking ha you muh prefer he version of equaion 575, in whih all quaniies are real, and he epression for he inerval beween wo evens is [( ) ( ) ( ) ( ) ] / 3 The minus sign in he epression is a small prie o pay for he realness of all quaniies Equaion 575 an be wrien osh φ sinh φ sinh φ osh φ, 578 where osh φ γ, sinh φ βγ, anh φ β On he fae of i, his looks muh simpler No messing around wih imaginary angles Ye his formulaion is no wihou is own se of diffiulies For eample, neiher he mari of equaion 575 nor he mari of equaion 578 is orhogonal You anno inver he equaion o find and in erms of and merely by inerhanging he primed and unprimed symbols and inerhanging he rows and olumns The onverse of equaion 578 is in fa osh φ sinh φ sinh φ osh φ, 579 whih an also (undersandably!) be wrien osh( φ) sinh( φ) sinh( φ) osh( φ), 57 whih demands as muh skill in handling hyperboli funions as he oher formulaion did in handling omple numbers A furher problem is ha he formulaion 575 does no allow he analogy beween he Lorenz ransformaion and he roaion of aes You ake your hoie I may be noied ha he deerminans of he maries of equaions 575 and 578 are eah uniy, and i may herefore be hough ha eah mari is orhogonal and ha is reiproal is is ranspose Bu his is no he ase, for he ondiion ha he deerminan is iniy is no a suffiien ondiion for a mari o be orhogonal The neessary ess are

16 6 summarized in Celesial Mehanis, Seion 37, and i will be found ha several of he ondiions are no saisfied 58 Timelike and Spaelike 4-Veors I am going o refer some evens o a oordinae sysem whose origin is here and now and whih is moving a he same veloiy as you happen o be moving In oher words, you are siing a he origin of he oordinae sysem, and you are saionary wih respe o i Le us suppose ha an even A ours a he following oordinaes referred o his referene frame, in whih he disanes, y, z are epressed in ligh-years (lyr) he ime is epressed in years (yr) y 3 z 7 A ligh-year is a uni of disane used when desribing asronomial disanes o he layperson, and i is also useful in desribing some aspes of relaiviy heory I is he disane ravelled by ligh in a year, and is approimaely 946 % 5 m or 37 parse (p) Even A, hen, ourred a year ago a a disane of lyr, when referred o his referene frame Noe ha, if referred o a referene frame ha oinides wih his one a, bu is moving wih respe o i, all four oordinaes migh be differen, and he disane y z and he ime of ourrene would be differen, bu, aording o he way in whih we have defined spae and ime by he Lorenz ransformaion, he quaniy y z would be he same Imagine now a seond even, B, whih ours a he following oordinaes: 5 y 8 z Tha is o say, when referred o he same referene frame, i will our in wo years ime a a disane of lyr The 4-veor s B A onnes hese wo evens, and he magniude s of s is he inerval beween he wo evens Noe ha he disane beween he wo evens, when referred o our referene frame, is (5 ) (8 3) ( 7) 6 56 lyr The inerval beween he wo evens is (5 ) (8 3) ( 7) ( ) 5 83 lyr, and his is independen of he veloiy of he referene frame Tha is, if we roae he referene frame, i obviously makes no differene o he inerval beween he wo evens, whih is invarian As anoher eample, onsider wo evens A and B whose oordinaes are y 5 z 3

17 7 3 y 7 z 4 6 wih disanes, as before, epressed in lyr, and imes in yr Calulae he inerval beween hese wo evens ie he magniude of he 4-veor onneing hem If you arry ou his alulaion, you will find ha s 58, so ha he inerval s is imaginary and equal o 76i So we see ha some pairs of evens are onneed by a 4-veor whose magniude is real, and oher pairs are onneed by a 4-veor whose magniude is imaginary There are differenes in haraer beween real and imaginary inervals, bu, in order o srip away disraions, I am going o onsider evens for whih y z We an now onenrae on he essenials wihou being disraed by unimporan deails Le us herefore onsider wo evens A and B whose oordinaes are lyr 3 lyr yr 6 yr These evens and he 4-veor onneing hem are shown in figure XV7Even A happened wo years ago (referred o our referene frame); even B will our (also referred o our referene frame) in si years ime The square of he inerval beween he wo evens (whih is invarian) is 63 lyr, and he inerval is imaginary If someone waned o eperiene boh evens, he would have o ravel only lyr (referred o our referene frame), and he ould ake his ime, for he would have eigh years (referred o our referene frame) in whih o make he journey o ge o even B in ime He ouldn oally dawdle, however; he would have o ravel a a speed of a leas 8 imes he speed of ligh, bu ha s no eremely fas for anyone well versed in relaiviy FIGURE XV7 A s B Le s look a i anoher way Le s suppose ha even A is he ause of even B This means ha some agen mus be apable of onveying some informaion from A o B a a speed a leas equal o 8 imes he speed of ligh Tha may presen some ehnial problems, bu i presens no problems o our imaginaion

18 8 You ll noie ha, in his ase, he inerval beween he wo evens ie he magniude of he 4-veor onneing hem is imaginary A 4-veor whose magniude is imaginary is alled a imelike 4-veor There is quie a long ime beween evens A and B, bu no muh disane Now onsider wo evens A and B whose oordinaes are lyr 7 lyr yr 3 yr The square of he magniude of he inerval beween hese wo evens is 9 lyr, and he inerval is real A 4-veor whose magniude is real is alled a spaelike 4-veor I is shown in figure XV8 B A s FIGURE XV8 Perhaps I ould now ask how fas you would have o ravel if you waned o eperiene boh evens They are quie a long way apar, and you haven muh ime o ge from one o he oher Or, if even A is he ause of even B, how fas would an informaionarrying agen have o move o onvey he neessary informaion from A in order o insigae even B? Maybe you have already worked i ou, bu I m no going o ask he quesion, beause in a laer seion we ll find ha wo evens A and B anno be muually ausally onneed if he inerval beween hem is real Noe ha I have said muually ; his means ha A anno ause B, and B anno ause A A and B mus be quie independen evens; here simply is oo muh spae in he inerval beween hem for one o be he ause of he oher I does no mean ha he wo evens anno have a ommon ause Thus, figure XV9 shows wo evens A and B wih a spaelike inerval beween hem (very seep) and a hird even C suh he inervals CA and CB (very shallow) are imelike C ould easily be he ause of boh A and B; ha is, A and B ould have a ommon ause Bu here an be no muual ausal onneion beween A and B (I migh be noed parenheially ha Charles Dikens emporarily nodded when he hose

19 9 he ile of his novel Our Muual Friend He really mean our ommon friend C was a friend ommon o A and o B A and B were friends muually o eah oher) Disane B A C FIGURE XV9 Time Eerise The disane of he Sun from Earh is 496 % m The speed of ligh is 998 % 8 m s How long does i ake for a phoon o reah Earh from he Sun? Even A: A phoon leaves he Sun on is way o Earh Even B: The phoon arrives a Earh Wha is he inerval (ie s in 4-spae) beween hese wo evens? 59 The FizGerald-Lorenz Conraion This is someimes desribed in words somehing like he following: If a measuring-rod is moving wih respe o a saionary observer, i appears o be shorer han i really is This is no a very preise saemen, and he words ha I have plaed in invered ommas all for some larifiaion We have seen ha, while he inerval beween wo evens is invarian beween referene frames, he disane beween wo poins (and hene he lengh of a rod) depends on he oordinae frame o whih he poins are referred Le us now define wha we mean by he lengh of a rod Figure XV shows a referene frame, and a rod lying parallel o he -ais For he momen I am no speifying wheher he rod is moving wih respe o he referene frame, or wheher i is saionary

20 y FIGURE XV Le us suppose ha he -oordinae of he lef-hand end of he rod is, and ha, a he same ime referred o his referene frame, he -oordinae of he righ-hand end is The lengh l of he rod is defined as l Tha ould sarely be a simpler saemen bu noe he lile phrase a he same ime referred o his referene frame Tha simple phrase is imporan Now le s look a he FizGerald-Lorenz onraion See figure XV y Σ y Σ v FIGURE XV The are wo referene frames, Σ and Σ The frame Σ is moving o he righ wih respe o Σ wih speed v A rod is a res wih respe o he frame Σ, and is herefore moving o he righ wih respe o Σ a speed v In my younger days I ofen used o ravel by rain, and I sill like o hink of railway rains whenever I disuss relaiviy Modern sudens usually like o hink of spaeraf, presumably beause hey are more ausomed o his mode of ravel In he very early days of railways, i was usomary for he saionmaser o wear op ha and ails Those days are long gone, bu, when hinking abou he FizGerald-Lorenz onraion, I like o

21 hink of Σ as being a railway saion in whih here resides a saionmaser in op ha and ails, while Σ is a railway rain The lengh of he rod, referred o he frame Σ, is l, in wha I hope is obvious noaion, and of ourse hese wo oordinaes are deermined a he same ime referred o Σ The lengh of he rod referred o a frame in whih i is a res is alled is proper lengh Thus l is he proper lengh of he rod Now i should be noed ha, aording o he way in whih we have defined disane and ime by means of he Lorenz ransformaion, alhough and are measured simulaneously wih respe o Σ, hese wo evens (he deerminaion of he oordinaes of he wo ends of he rod) are no simulaneous when referred o he frame Σ (a poin o whih we shall reurn in a laer seion dealing wih simulaneiy) The lengh of he rod referred o he frame Σ is given by l, where hese wo oordinaes are o be deermined a he same ime when referred o Σ Now equaion 556 ells us ha / γ v and / γ v (Readers should noe his derivaion very arefully, for i is easy o go wrong In pariular, be very lear wha is mean in hese wo equaions by he symbol I is he single insan of ime, referred o Σ, when he oordinaes of he wo ends are deermined simulaneously wih respe o Σ) From hese we reah he resul: This is he FizGerald-Lorenz onraion l l /γ 59 I is someimes desribed hus: A railway rain of proper lengh yards is moving pas a railway saion a 95% of he speed of ligh (γ 36) To he saionmaser he rain appears o be of lengh 3 yards; or he saionmaser hinks he lengh of he rain is 3 yards; or, aording o he saionmaser he lengh of he rain is 3 yards This gives a false impression, as hough he saionmaser is under some sor of misapprehension onerning he lengh of he rain, or as if he is labouring under some sor of illusion, and i inrodues some sor of unneessary mysery ino wha is nohing more han simple algebra In fa wha he saionmaser hinks or assers is enirely irrelevan Two orre saemens are: The lengh of he rain, referred o a referene frame in whih i is a res ie he proper lengh of he rain is yards The lengh of he rain when referred o a frame wih respe o whih i is moving a a speed of 95 is 3 yards And ha is all here is o i Any phrase suh as his observer hinks ha or aording o his observer should always be inerpreed in his manner I is no a maer of wha an observer hinks I is a maer of whih frame a measuremen is referred o Nohing more, nohing less

22 i Σ i Σ l θ l FIGURE XV I is possible o desribe he Lorenz-FizGerald onraion by inerpreing he Lorenz ransformaions as a roaion in 4-spae Wheher i is helpful o do so only you an deide Thus figure XV shows Σ and Σ relaed by a roaion in he manner desribed in seion 57 The hik oninuous line shows a rod oriened so ha is wo ends are drawn a he same ime wih respe o Σ Is lengh is, referred o Σ, l, and his is is proper lengh The hik doed line shows he wo ends a he same ime wih respe o Σ Is lengh referred o Σ is l l/osθ And, sine os θ γ, whih is greaer han,, his means ha, in spie of appearanes in he figure, l < l The figure is deepive beause, as disussed in seion 57, θ is imaginary As I say, only you an deide wheher his way of looking a he onraion is helpful or merely onfusing I is, however, a leas worh looking a, beause I shall be using his onep of roaion in a forhoming seion on simulaneiy and order of evens Illusraing he Lorenz ransformaions as a roaion like his is alled a Minkowski diagram 5 Time Dilaion We imagine he same railway rain Σ and he same railway saion Σ as in he previous seion eep ha, raher han measuring a lengh referred o he wo referene frames, we measure he ime inerval beween wo evens We ll suppose ha a passenger in he railway rain Σ laps his hands wie These are wo evens whih, when referred o he referene frame Σ, ake plae a he same plae when referred o his referene frame Le he insans of ime when he wo evens our, referred o Σ, be and The ime inerval T is defined as Bu he Lorenz ransformaion is γ( v / ), and so he ime inerval when referred o Σ is T γt 5

23 3 This is he dilaion of ime The siuaion is illusraed by a Minkowski diagram in figure XV3 While i is lear from he figure ha T T os θ and herefore ha T γt, i is no so lear from he figure ha his means ha T is greaer han T beause os θ > and θ is imaginary i i T T θ FIGURE XV3 Thus, le us suppose ha a passenger on he rain holds a -mere measuring rod (is lengh in he direion of moion of he rain) and he laps his hands a an inerval of one seond apar Le s suppose ha he rain is moving a 98% of he speed of ligh (γ 55) In ha ase he saionmaser hinks ha he lengh of he rod is only 99 m and ha he ime inerval beween he laps is 55 seonds I deliberaely did no word ha las senene very well I is no a maer of wha he saionmaser or anyone else hinks or assers I is no a maer ha he saionmaser is somehow deeived ino erroneously believing ha he rod is 99 m long and he laps 55 seonds apar, whereas hey are really mere long and seond apar I is a maer of how lengh and ime are defined (by subraing wo spae oordinaes deermined a he same ime, or wo ime oordinaes a he same plae) and how spaeime oordinaes are defined by means of he Lorenz ransformaions The lengh is 99 m, and he ime inerval is 55 seonds when referred o he frame Σ I is rue ha he proper lengh and he proper ime inerval are he lengh and he ime inerval referred o a frame in whih he rod and he lapper are a res In ha sense one ould loosely say ha hey are really mere long and seond apar Bu he Lorenz onraion and he ime dilaion are no deermined by wha he saionmaser or anyone else hinks Anoher way of looking a i is his The inerval s beween wo evens is learly independen of he orienaion any referene frames, and is he same when referred o wo referene frames ha may be inlined o eah oher Bu he omponens of he veor

24 4 joining wo evens, or heir projeions on o he ime ais or a spae ais are no a all epeed o be equal By he way, in seion 53 I urged you o wrie a ompuer or alulaor programme for he insan onversion beween he several faors ommonly enounered in relaiviy I sill urge i As soon as I yped ha he rain was ravelling a 98% of he speed of ligh, I was insanly able o generae γ You need o be able o do ha, oo 5 The Twins Parado During he lae 95s and early 96s here was grea onroversy over a problem known as he Twins Parado The onroversy was no onfined o wihin sienifi irles, bu was argued, by sieniss and ohers, in he newspapers, magazines and many serious journals I goes somehing like his: There are wo -year-old wins, Alber and Bey Alber is a sedenary ype who likes nohing beer han o say a home ending he family vineyards His win siser Bey is a more advenurous ype, and has rained o beome an asronau On heir wenieh birhday, Bey waves a heery au revoir o her broher and akes off on wha she inends o be a brief spaefligh, a whih she ravels a 9998 % of he speed of ligh (γ 5) Afer si monhs by her alendar she urns bak and on her s birhday she arrives bak home o gree her broher, only o find ha he is now old and sere and has laboured, by his alendar for 5 years and is now an aged man of 7 years If we aep wha we have derived in he previous seion abou he dilaion of ime, here would seem o be no pariular problem wih ha I has even been argued ha ravel beween he sars may no be an impossibiliy Whereas o an Earhbound observer i may ake many deades for a spaeraf o ravel o a sar and bak, for he asronaus on board muh less ime has elapsed And ye a parado was poined ou Aording o he priniples of he relaiviy of moion, i was argued, one ould refer everyhing o Bey s referene frame, and from ha poin of view one ould regard Bey as being he saionary win and Alber as he one who ravelled off ino he disane and reurned laer Thus, i ould be argued, i would be Alber who had aged only one year, while Bey would have aged fify years Thus we have a parado, whih is a problem whih apparenly gives rise o opposie onlusions depending on how i is argued And he only way ha he parado ould be resolved was o suppose ha boh wins were he same age when hey were re-unied A seond argumen in favour of his inerpreaion ha he wins were he same age when re-unied poins ou ha dilaion of ime arises beause wo evens ha may our in he same plae when referred o one referene frame do no our in he same plae when referred o anoher Bu in his ase, he wo evens (Bey s deparure and re-arrival) our a he same plae when referred o boh referene frames

25 5 The argumen over his poin raged quie furiously for some years, and a pariularly plausible ool ha was used was somehing referred o as he k-alulus an argumen ha is, however, faally flawed beause he rules of he k-alulus inherenly inorporae he desired onlusion Two of he prinipal leaders of he very publi sienifi debae were Professors Fred Hoyle and Herber Dingle, and his inspired he following leer o a weekly magazine, The Lisener, in 96: Sir: The ears of a Hoyle may ingle; The blood of a Hoyle may boil When Hoyle pours ho oil upon Dingle, And Dingle old waer on Hoyle Bu he dus of he wrangle will sele Old sars will look down on new soil The po will lie down wih he kele, And Dingle will mingle wih Hoyle So wha are you, he reader, epeed o believe? Le us say his: If you are a suden who has eaminaions o pass, or if you are an unenured professor who has o hold on o a job, be in no doub whaever: The original onlusion is he anonially-aeped orre onlusion, namely ha Alber has aged 5 years while his asronau siser has aged bu one This is now firmly aeped ruh Indeed i has even been laimed ha i has been proved eperimenally by a sienis who ook a lok on ommerial airline flighs around he world, and ompared i on his reurn wih a say-a-home lok For myself I have neiher eaminaions o pass nor, alas, a job o hold on o, so I am no bound o believe one hing or he oher, and I ele o hold my peae I do say his, however ha wha anyone believes is no an essenial poin I is no a maer of wha Alber or Bey or Hoyle or Dingle or your professor or your employer believes The real quesion is his: Wha is i ha is predied by he speial heory of relaiviy? From his poin of view i does no maer wheher he heory of relaiviy is rue or no, or wheher i represens a orre desripion of he real physial world Saring from he basi preeps of relaiviy, wheher rue or no, i mus be only a maer of algebra (and simple algebra a ha) o deide wha is predied by relaiviy A diffiuly wih his is ha i is no, srily speaking, a problem in speial relaiviy, for speial relaiviy deals wih ransformaions beween referene frames ha are in uniform moion relaive o one anoher I is poined ou ha Alber and Bey are no in uniform moion relaive o one anoher, sine one or he oher of hem has o hange he direion of moion ie has o aelerae I ould sill be argued ha, sine moion is relaive, one an regard eiher Alber or Bey as he one who aeleraes bu he response o his is ha only uniform moion is relaive Thus here is no symmery beween Alber and Bey Bey eiher aeleraes or eperienes a graviaional field (depending on wheher

26 6 her eperiene is referred o Alber s or her own referene frame) And, sine here is no symmery, here is no parado This argumen, however, admis ha he age differene beween Alber and Bey on Bey s reurn is no an effe of speial relaiviy, bu of general relaiviy, and is an effe aused by he aeleraion (or graviaional field) eperiened by Bey If his is so, here are some severe diffiulies is desribing he effe under general relaiviy For eample, wheher he general heory allows for an insananeous hange in direion by Bey (and infinie deeleraion), or wheher he final resul depends on how she deeleraes a wha rae and for how long mus be deermined by hose who would akle his problem Furher, he alleged age differene is supposed o depend upon he ime during whih Bey has been ravelling and he lengh of her journey ye he porion of her journey during whih she is aeleraing or deeleraing an be made arbirarily shor ompared wih he ime during whih she is ravelling a onsan speed If he effe were o our solely during he ime when she was aeleraing or deeleraing, hen he oal lengh and duraion of he onsan speed par of her journey should no affe he age differene a all Sine his haper deals only wih speial relaiviy, and his is evidenly no a problem resried o speial relaiviy, I leave he problem, as originally saed, here, wihou resoluion, for readers o argue over as hey will 5 A, B and C A, B and C were hree haraers in he Canadian humoris Sephen Leaok s essay on The Human Elemen in Mahemais A, B and C are employed o dig a dih A an dig as muh in one hour as B an dig in wo We an ask A, B and C o ome o our aid in a modified version of he wins problem, for we an arrange all hree of hem o be moving wih onsan veloiies relaive o eah oher I goes like his (figure XV4): * B The senario is probably obvious from he figure There are hree evens: B passes A B mees C 3 C passes A * A FIGURE XV4 * C

27 7 A even, B and A synhronize heir wahes so ha eah reads zero A even, C ses his wah so ha i reads he same as B s A even 3, C and A ompare wahes I shall leave he reader o ogiae over his The only hing I shall poin ou is ha his problem differs from he problem desribed as he Twins Parado in wo ways In he firs plae, unlike in he Twins Parado, all hree haraers, A, B and C are moving a onsan veloiies wih respe o eah oher Also, he firs and hird evens our a he same plae relaive o A bu a differen plaes referred o B or o C In he win parado problem, he wo evens our a he same plae relaive o boh frames 53 Simulaneiy If he ime inerval referred o one referene frame an be differen when referred o anoher referene frame (and sine ime inerval is merely one omponen of a fourveor, he magniude of he omponen surely depends on he orienaion in four spae of he four aes) his raises he possibiliy ha here migh be a ime inerval of zero relaive o one frame (ie wo evens are simulaneous) bu are no simulaneous relaive o anoher This is indeed he ase, provided ha he wo evens do no our in he same plae as well as a he same ime Look a figure XV5 i i Σ Σ l θ FIGURE XV5 I have drawn wo referene frames a an (imaginary) angle θ o eah oher Think of Σ as he railway saion and of Σ as he railway rain, and ha he speed of he railway rain is an θ (You may have o go bak o seion 53 or 57 o reall he relaion of θ o he

28 8 speed) The hik line represens he inerval beween wo evens ha are simulaneous when referred o Σ, bu are separaed in spae (one ours near he fron of he rain; he oher ours near he rear) (Noe also in his e ha I am using he phrase ime inerval o denoe he ime-omponen of he inerval For wo simulaneous evens, he ime inerval is zero, and he inerval is hen merely he disane beween he wo evens) While he hik line has zero omponen along he i ais, is omponen along he i ais is l sin θ Tha is, i ( ) lsin θ l iβγ β γ l Hene: 53 For eample, if he evens ook plae simulaneously, km apar in he rain (i is a long rain) and if he rain were ravelling a 95% of he speed of ligh (γ 33; i is a fas rain), he wo evens would be separaed when referred o he railway saion by seonds The even near he rear of he rain ourred firs 54 Order of Evens, Causaliy and he Transmission of Informaion Maybe i is even possible ha if one even preedes anoher in one referene frame, in anoher referene frame he oher preedes he one In oher words, he order of ourrene of evens may be differen in wo frames This indeed an be he ase, and Minkowski diagrams (figure XV6) an help us o see why and in wha irumsanes i i i i Σ Σ Σ Σ θ (a) In par (a), of he wo evens and, ours before in eiher Σ or Σ (from his poin on I shall use a shor phrase suh as in Σ raher han he more umbersome when referred o he referene frame Σ Bu in par (b), even ours before even in Σ, bu FIGURE XV6 θ (b)

29 9 afer even in Σ One an see ha here is reversal of order of evens if he slope of he line joining o wo evens is less han he angle θ The angle θ, i may be realled, is an imaginary angle suh han an θ iβ iv /, where v is he relaive speed of he wo frames In figure XV7, for simpliiy I am going o suppose ha even ours a he origin of boh frames, and ha even ours a oordinaes (v, i) in Σ The ondiion for no reversal of evens is hen evidenly i v an θ iβ iv ; or v 54 Now suppose ha evens and are ausally onneed in he sense ha even is he ause of even For his o be he ase, some signal arrying informaion mus ravel from o However, if even is he ause of even, even mus preede even in all referene frames Thus i follows ha no signal arrying informaion ha ould ause an even o our an ravel faser han he speed of ligh This means, in effe, ha neiher mass nor energy an be ransmied faser han he speed of ligh Tha is no quie he same hing as saying ha nohing an be ransmied faser han he speed of ligh For eample a Moiré paern formed by wo ombs wih slighly differen ooh spaings an move faser han ligh if one of he ombs is moved relaive o he oher; bu hen I suppose i has o be admied ha in ha ase nohing is aually being ransmied and erainly nohing ha an ransmi informaion or ha an ause an even An almos idenial eample would be he modulaion envelope of he sum of wo waves of slighly differen frequenies A wellknown eample from wave mehanis is ha of he wave represenaion of a moving parile The wave group (whih is he inegral of a oninuous disribuion of wavelenghs whose een is governed by Heisenberg s priniple) moves wih he parile a a sub-luminal speed, bu here is nohing o preven he waveles wihin he group moving hrough he group a any speed These waveles may sar a he beginning of he group and rapidly move hrough he group and einguish hemselves a he end No informaion is ransmied from A o B a a speed any faser han he parile iself is moving 55 Derivaives We ll pause here and esablish a few derivaives jus for referene and in ase we need hem laer We reall ha he Lorenz relaions are γ( v ) 55

30 3 and β γ 55 From hese we immediaely find ha ; ; ; γ βγ γ γ v 553a,b,,d We shall need hese in fuure seions Cauion: I is no impossible o make a misake wih some of hese derivaives if one allows one s aenion o wander For eample, one migh suppose ha, sine, / γ hen obviously γ / / - and indeed his is orre if is being held onsan However, we have o be sure ha his is really wha we wan The diffiuly is likely o arise if, when wriing a parial derivaive, we negle o speify wha variables are being held onsan, and no grea harm would be done by insising ha hese always be speified when wriing a parial derivaive If you wan he inverses raher han he reiproals of equaions 553a,b,,d, he rule, as ever, is: Inerhange he primed and unprimed symbols and hange he sign of v or β For eample, he reiproal of is, while is inverse is For ompleeness, and referene, hen, I wrie down all he possibiliies: / ; ; / ; / γ βγ γ γ v 553e,f,g,h ; ; ; γ βγ γ γ v 553ijkl / ; ; / ; / γ βγ γ γ v 553m,n,o,p Now le s suppose ha, ), ( ψ ψ where and are in urn funions (equaions55 and 55) of and Then ψ βγ ψ γ ψ ψ ψ 554 and ψ γ ψ γ ψ ψ ψ v 555

31 3 The reader will doubless noie ha I have here ignored my own advie and I have no indiaed whih variables are o be held onsan I would be worh spending a momen here hinking abou his We an wrie equaions 554 and 555 as equivalen operaors: β γ 556 and γ v 557 We an also, if we wish, find he seond derivaives Thus, β ψ ψ β γ ψ ψ 558 from whih we find β β γ 559 In a similar manner we obain ( ) β β γ v 55 and γ v v 55 The inverses of all of hese relaions are o be found by inerhanging he primed and unprimed oordinaes and hanging he signs of v and β 56 Addiion of veloiies A railway rain rundles owards he eas a speed v, and a passenger srolls owards he fron a speed v Wha is he speed of he passenger relaive o he railway saion? We migh a firs be emped o reply: Why,, v v of ourse In his seion we shall

32 3 show ha he answer as predied from he Lorenz ransformaions is a lile less han his, and we shall develop a formula for alulaing i We have already disussed (in seion 56) our answer o he objeion ha his defies ommon sense We poined ou here ha he answer (o he perfely reasonable objeion) ha a he speeds we are ausomed o we would hardly noie he differene is no a saisfaory response The reason ha he resulan speed is a lile less han v v resuls from he way in whih we have defined he Lorenz ransformaions beween referenes frames and he way in whih disanes and ime inervals are defined wih referene o referene frames in uniform relaive moion Figure XV7 shows wo referenes frames, Σ and Σ, he laer moving a speed v wih respe o he former A parile is moving wih veloiy u in Σ, wih omponens u and u y ( in Σ referred o he referene frame Σ ) Wha is he veloiy of he parile in Σ? Le us sar wih he -omponen We have: d d d d u u d d u 56 We ake he derivaives from equaions 553a-d, and, wriing v / for β, we obain / u u u v v 56 The inverse is obained by inerhanging he primed and unprimed symbols and reversing he sign of v y y Σ FIGURE XV7 Σ u * v

33 33 The y-omponen is found in an ealy similar manner, and I leave is derivaion o he reader The resul is u u y y γ( u v / ) 563 Speial ases: I If u u and u y, hen u u v and u u v / y 564a,b II If u and u y u, hen u v and u y u / γ 565a,b Equaions 564a as wrien is no easy o ommi o memory, hough i is raher easier if we wrie β v, β u / and β u / Then he equaion beomes / β β β 566 β β Σ Σ β β FIGURE XV8 β β FIGURE XV9

34 34 In figure XV8, a rain Σ is rundling wih speed β (imes he speed of ligh) owards he righ, and a passenger is srolling owards he fron a speed β The speed β of he passenger relaive o he saion Σ is hen given by equaion 566 In figure XV9, wo rains, one moving a speed β and he oher moving a speed β, are moving owards eah oher (If you prefer o hink of proons raher han rains, ha is fine) Again, he relaive speed β of one rain relaive o he oher is given by equaion 566 Eample A rain rundles o he righ a 9% of he speed of ligh relaive o Σ, and a passenger srolls o he righ a 5% of he speed of ligh relaive o Σ The speed of he passenger relaive o Σ is 95% of he speed of ligh The relaion beween β, β and β is shown graphially in figure XV β 8 6 β FIGURE XV β If I use he noaion β /β o mean ombining β wih β, I an wrie equaion 566 as β β β β 567 β β