Speial relaiiy Aording o he Twin Parado, a spae raeler leaing his win broher behind on Earh, migh reurn some years laer o find ha his win has aged muh more han he has, or, if he spen a lifeime in spae, he may perhaps find ha humaniy has eoled ino somehing quie differen, or perhaps ha he Earh is barren and dead! Sudy his arile and find ou why! In his arile we eplore some aspes of Einsein s Theory of Speial relaiiy. This arile inludes quie a bi of algebra, bu hose who do no wan o sudy he mahs an sill learn a lo by skipping he algebra and simply reading he e. The MihelsonMorley eperimen Mihelson and Morley ondued a lassi eperimen o deermine he speed of moion relaie o wha was hen onsidered o be he eher or osmi bakground subsane. In oher words, hey assumed ha he Unierse has some fied eloiy relaie o whih he Earh was moing. To deermine his moion hey onsrued an inerferomeer using laser ligh. When wo ligh beams ha are ou of phase are added ogeher superposed hen an inerferene paern resuls. Two ligh beams ha are originally in phase may beome ou of phase if one of hem raels furher by a disane ha is no a whole number muliple of waelenghs, wih he differene in pah lengh diided by waelengh leaing a remainder. Analysis of he inerferene paern, herefore, ells us how muh furher one beam has raelled han he oher.
The MihelsonMorley Eperimen Obserer S A Earh S O Laser onsider he following wo ligh pahs eah from L laser: Ligh pah P1: AA, from A halfsilered mirror o mirror o A halfsilered mirror Ligh pah P: AA, from A o o A, and on o S amera/obserer. Noe ha disanes LA, L o A, and AS, A o S, are he same and so we an ignore hem when omparing he wo ligh pahs. Noe: disane OA disane A, all his disane L. ase 1. Due o he moion of he Earh, he speed of ligh relaie o he apparaus is naiely epeed o hange, as is he aual pahlengh raeled by he ligh beams. ase. If we hen roaed he apparaus hrough 90 degrees, we would again epe a differene, bu in he opposie sense o ase 1. Adding he resuls ogeher should double he differene, making i quie obious! Resul: No differene in pah lengh was obsered, P1 P! This implies ha he beams mainain a onsan speed in boh direions! The eloiy of he Earh has made no differene!
We epe : The ime aken for he beam o rael A, The lengh raeled by he beam from A o, l Where : A A rel L A rel L L eloiy of beam relaie o apparaus moing wih he Earh, speed of ligh approimaely he speed of ligh in air is he same as ha in a auum, speed of apparaus he eloiy of The ime and lengh of he reurn leg from o A is epeed o be : L A, + L la A, + and he oal pah lengh raeled by he ligh from A o o A : L L laa +, + L + L +, + + L + L L + L so : laa L L, he Earh' s surfae. Defining : g So : g 1 1 1 1,, l AA Lg
For he beam raeling along P AA, he moion of he Earh inreases he pah lengh whih beomes diagonal relaie o he Earh s surfae, as by he ime he ligh reahes he mirror a, his mirror has moed along slighly o he righ o posiion due o he moion of he Earh s surfae. y he ime he ligh beam reurns o A, A has moed wie as far, o A as shown below: Noe ha lengh A lengh L, as measured in he lab. rel q L A A The relaie eloiy of rel osq, so : osq sinq, A using :sin q + os q 1 : l rel A osq 1 L and, as rel A l, A g and also using : eloiy L rel l A Lg : he beam along A is : rel The ime aking for he beam o rael from A o is epeed o be : A 1 sin q disane/ime, and rel : γ l AA l A Lγ
Now omparing he wo pah lenghs we see ha hey are no equal : l AA l AA æ ö L Lç, è ø where we hae used : g 1+ Lg, and Lg L g g 1+, g æ ö æ ö g g ç1 + 1. ç + è ø è ø To summarise, he epeed pahlengh differene raeled by he ligh beams is : L Wha Mihelson and Morley aually found was ha here was no pah differene a all! Tha is: l AA l AA So, wha wen wrong? Nohing! The Priniple of Relaiiy saes: The laws of physis inluding he behaiour of ligh mus be ealy he same for any wo obserers moing wih onsan eloiy relaie o eah oher. Thus, if he eloiy of ligh in a auum is onsan o one obserer, hen i mus be onsan o anoher obserer somewhere else. We were wrong in our alulaions of he relaie eloiy of he ligh beams relaie o he apparaus i an no hange! As we will see laer, here are oher imporan reasons why he speed of ligh in a auum is onsan. Noe: he speed of ligh in differen media is differen, bu under idenial ondiions he speed of ligh is onsan and in a auum i is always, whih is approimaely.998 10 8 m/s.
Obserers and Inerial Referene Frames i s all relaie! An imporan onep o undersanding relaiiy is he onep of referene frames. If I was sanding sill in a park and you were whizzing around on a merrygoround, hen we would hae ery differen frames of referene. I see you reoling agains he sill bakground of he park. You see he world spinning around you! Of ourse, we would no epe he laws of physis o be any differen on he merrygoround as hey are anywhere else apar from he fa ha he merrygoround is spinning here is nohing speial abou. The laws of angular momenum and enrifugal fore apply equally anywhere in he park. Relaiiy saes ha differen obserers mus see he same laws of physis. Howeer, aeleraion does ause differenes! An obje aeleraes if i alers is speed and/or ourse he merrygoround is aeleraing beause i spins, een if here was no friion and i roaed a a fied speed indefiniely. Of ourse in realiy i will lose energy and momenum hrough friion wih he air and he ais and so gradually slow down. Roaion an aler he apparen fore of graiy. The enrifugal fore of he merrygoround as a bi like graiy eep i is pulling you sideways if he merrygoround spins oo fas hold on igh, oherwise he enrifugal fore migh send you flying aross he playground! Speial relaiiy is speial beause i simplifies hings by assuming ha neiher obserer is aeleraing. We say ha eah obserer is an inerial referene frame. An inerial referene frame is one ha is no aeleraing, hough i an moe a onsan eloiy. For eample, as I sand sill in he park, I am in an inerial referene frame ignoring he fa ha he Earh is roaing and orbiing and so aeleraing as is somebody who passes by on a rain, so long as he rain is no aeleraing. General relaiiy inorporaes he effes of aeleraion and has a lo o do wih graiy. This is a muh more mahemaially omple heory and will be disussed in anoher arile.
Now, bak o our eperimen. We epeed pah P1 AA o be longer han pah P AA. Tha is we epeed he pah in he direion of he moion AA o be longer han he pah perpendiular a righ angles o he moion. Remember, our apparaus was moing o he righ in his ase. The Lorenz onraion Moing Rulers Shoren! Wha has eidenly happened is ha he lengh along he pah AA has shorened, suh ha he wo pahs are he same lengh. If we define he lengh of any pah parallel o he direion of moion as L and any pah perpendiular o he moion as L hen are epeed pahlenghs were: l AA L g l AA L g Howeer, we aually obsered: l AA l AA ha is: L L g The pah parallel o moion onras aording o: L L /g This is he Lorenz onraion: Lorenz onraion : L L g So he pahlengh in he direion of moion, as seen by he lighbeam, has been onraed. Moing objes onra along heir direion of moion, and he faser hey moe, he more hey onra! Moing objes shrink! This is he Lorenz onraion, and g is he Lorenz faor. Howeer, hings are more suble han his. y he Priniple of Relaiiy, if an obserer in he lab sees he pah followed by he ligh beam shoren, hen an obserer moing wih he ligh beam sees he lab in he direion ha he lab appears o be moing pas he obserer shoren! No poin of referene is speial, he laws of physis appear he same o eah obserer and as neiher obserer an be sure ha hey are ruly saionary relaie o wha? all moion mus be relaie.
The Lorenz onraion an also be wrien as : L L 0 1 where L is he onraed lengh as measured by our ' saionary' L 0 is he lengh as measured by our moing obserer, who does no noie he onraion. obserer;
Time Dilaion Moing loks Run Slow! Wha if we measured ime insead of lengh? There is no denying ha he perpendiular pah AA beame lenghened due o he earh s moion he beam followed diagonal pahs o reah he mirror and as he speed of ligh is onsan we would epe he ligh o ake longer o raerse his pah han if he earh was saionary relaie o he ligh beam. The aual disane beween A and does no aler. Howeer, from he poin of iew of he ligh beam i is raelling in a sraigh line, sine i is moing wih he apparaus whih o an obserer on he ligh beam would appear saionary. This would amoun o an apparen slowing down of he lighbeam as measured by an obserer moing wih i and his iolaes he Priniple of Relaiiy! The soluion o his parado is ime dilaion. Moing loks run slow, so ime from he poin of iew of he ligh beam has slowed. This means ha when one seond of ime, as measured by someone riding wih he beam, elapses, he elapsed ime ineral aording o our saionary obserer in he lab is muh greaer. Time dilaion ensures ha he Priniple of relaiiy is upheld he ime slows by he ea amoun o oneal any hange in pah lengh due o he Earh s moion and no Lorenz onraion for pah AA so ha he speed of ligh remains unhanged. Time dilaion is relaie. If wo spae ships were drifing oward one anoher hrough he oid of spae and unable o measure heir speeds, hen hey ould no ell if hey were boh moing oward oneanoher or if one was saionary as he oher flies pas i. All hey ould measure is heir relaie eloiy. If he rew on one spaeship hink hey are saionary, hen hey will onlude ha he oher spaeship is moing, and ie ersa. Wha Mihelson and Morley s eperimen proed was ha here was no way o measure he absolue eloiy for he Earh relaie o he bakground of spae he Lorenz onraion and ime dilaion preen is measuremen! We do no know wha saionary means. All we an do is measure he earh s eloiy relaie o he Sun or o oher sars, all of whih are moing relaie o oneanoher! No referene frame is priileged by being a res and eloiy is only relaie! Like Lorenz onraion, i depends on perspeie. If eah ship onsiders iself a res, hen i will appear as if ime on he oher ship whih is moing has slowed. Eah rew obsere he same phenomena happening o he oher spae ship. I is he relaie moion of he ships, raher han he moion of any one in pariular, whih will ause Lorenz onraion and ime dilaion o be obsered o he oher moing spaeship. We will see ha General Relaiiy inrodues a suble differene by aking aeleraion ino aoun see The Twin Parado seion.
The ime dilaion is gien by : Time dilaion: g where : is he ime as measured by an obserer S in a referene frame moing a onsan eloiy relaie o he ' res frame'; is he ime as measured by an obserer S in he res frame. Of ourse we an no say ha a frame is really a res, raher we are alking in relaie erms! The ime dilaion may also be wrien as : D D 0 1 where D D he dilaed ime,,as obsered by our 'saionary' obserer; 0, he ime ineral as obsered by our moing obserer who does no noie he dilaion. This an seem onfusing, as i almos appears he firs ime dialion equaion, o onradi bu i does no one jus needs o undersand who is obsering wha!
The Twin Parado Sine ime dilaion, like Lorenz onraion, depends on he relaiisi Lorenz faor g gamma, i beomes more imporan a higher eloiies. A he kinds of eeryday eloiies we deal wih less han ~100 m/s in mos ases and up o abou 500 m/s in a fas passenger airraf like onorde, whih is only abou 0.000% of he speed of ligh! he Lorenz onraion and ime dilaion are no noieable. Alhough an aomi lok on board onorde was shown o lose a fraion of a seond in aordane wih ime dilaion. Howeer, for a spaeraf raelling a a signifian fraion of he speed of ligh boh effes beome signifian. An eample of he win parado is alulaed and ploed below. In his eample one win goes on a spae oyage and raels a 99.99% he speed of ligh. The graph shows he ime passed in spae, aording o he raeler erial ais agains ime elapsed aording o he win lef on Earh horizonal ais. As you an see, he spaeraeling win spends en years of heir ime in spae bu when hey reurn o Earh hey find ha more han 00 years hae elapsed on Earh! The win who wen ino spae has aged muh less! In effe he win in spae has raelled forward in ime. This migh seem odd, sine o he win in he spaeship i is he one on earh who is raelling and so ime should slow more for he win on Earh righ? Wrong! Aeleraion impars an asymmery: he win in he spaeship uses energy fuel o aelerae and deelerae during heir oyage. Speial relaiiy assumes ha boh wins remain a onsan eloiy and ha neiher aeleraes. The win parado is eplained fully by General Relaiiy whih aouns for he effes of aeleraion 9and energy ependiure. A slighly more inoled alulaion shows ha if our spae raeler aeleraed a a onsan aeleraion equal o Earh s graiy g, assuming limiless fuel, hen wihin a years heir speed would be ery lose indeed o he speed of ligh hough i an neer quie reah i. In his ase: A 5 yearround rip for our asronau would see 1,300,000 years elapse on Earh!
The Lorenz or relaiisi faor gamma, g is so imporan ha i is useful o look a is properies, le us see how i s alue aries wih eloiy,. Noie he minimum alue of 1 a a eloiy u in he graph of zero. A zero eloiy here is no relaiisi effe a all. As speed inreases ploed here as inreasing in eiher direion from he origin hen gamma rapidly inreases, in fa i inreases asympoially, ending o +infiniy as speed ends o he speed of ligh se equal o one in his graph, simply by hanging unis of measuremen. Gamma beomes infinie a he speed of ligh!
Spaeime Graphs and WorldLines We an plo he moion of a parile on an ordinary spaeime graph. To keep hings simple le us sik o onsidering jus one spaial oordinae, ignoring y and z for he momen, hen a spaeime diagram is a plo of ersus ime, : 45 o 45 o O + I hae pu on he horizonal ais and ime on he erial. We se an arbirary origin O 0,0 for our oordinae sysem a whih we se 0 and 0. This ould, for eample, be a erain posiion on a lab benh and ime zero ould be when we se an obje ino moion. I hae ploed wo pahs aken hrough spae and ime by wo objes in blue boh of whih began a he origin. One obje has moed away a onsan speed and moed o he righ +, he oher has aken a more onolued pah, aeleraing in arious ways, as i winds off o he lef. These pahs are he worldlines of he objes he rajeories ha hey rae ou in spae and ime. Also shown are he pahs of wo ligh beams as he wo red doed lines, raelling away from he origin in eiher direion a onsan speed. Seing up he ais sales aording o onenion, hese ligh beams are a 45 degrees o he and aes. If I added in anoher ais, say he y ais and draw i going ino and ou of he page, hen our allowed ligh pahs would form a ligh one. We shall see laer ha Speial Relaiiy forbids any normal obje any signal from raeling faser han ligh, so no worldline an hae an angle less han 45 degrees in our diagram. If we were saionary, hen our worldline would simply be a sraighline along he ais our posiion would no hange!
The Simulaneiy Shif onsider one saionary obserer, S. heir worldline is he ais. onsider anoher obserer moing away from S a a onsan eloiy,, all his moing obserer S Sprime. The worldline of S is a slaned line on our graph, whih is really he graph ha S sees: S S O To S i is S ha is moing away o all inens and purposes and heir graph looks idenial bu wih he S and S labels swihed. They hae passed eah oher in ouer spae and who an be sure wha heir aual eloiies are? Veloiy is only relaie! Now suppose obserer S shines a ligh beam a poin A whih is a he origin sraigh off o heir side and oward a mirror a poin and hen hey inerep he refleed beam a boh lighbeams rael a speed and a 45 degrees o he aes and are shown in red: S S The line is a simulaneiy line for obserer S. A
When does he ligh signal arrie a? oh S and S assume hey are a res. Looking a he graph, whih is ploed from he perspeie of S, he ligh akes longer o rael from A o han i does o rael from o. To S he ligh reahes a ime. To S he ligh akes an equal ime hey hink hey are res and he disane o he mirror is fied from heir perspeie, so A aording o S. S sees he ligh reah he mirror a ime ealy halfway along her worldline from A o. alulaion of he slope of line The slope of a line is he erial disane diided by he horizonal disane. D Slope of D D The inlinaion 1/slope of line D The inlinaions of he oher lines are as follows: 1 A : A : 3 : his is essenially, speed disane/ime. This gies : 4 From he graph : 5 or and 6
and + + + ø ö ç è æ + + + + 9 Sine : 1 and so : 1 and : Subsiuing in 1 : From 3 8 7 Also from he graph :
he simulaneiy shif. whih is and 1 1 : using 9 so and D D + + ø ö ç è æ + ø ö ç è æ + D D ø ö ç è æ + + ø ö ç è æ + + D D Wha does he simulaneiy shif mean? I means ha he imeineral beween wo eens appears differen o differen obserers moing wih differen eloiies. Noe: The line is he line along whih eens appear simulaneous o S, i is parallel o heir ime ais. Noie ha heir ais, runs along heir worldline on heir own ersion of he spaeime graph, sine hey are saionary relaie o hemseles. Apparenly simulaneous eens appear o our a he same ime and so are on a line parallel o he ime ais, boh aes hae beome iled for our raeler:
We shall represen spaeime eens in differen ways. Ploing eeryhing on he aes aording o our saionary obserer, he aes of our moing obserer appear ompressed: Aboe: he spaeime aes of S are shown in blue, hey are ompressed as S is moing relaie o S whose aes are a righangles and shown in blak. The higher he relaie eloiy,, he more he aes ompress, and a hey boh oinide wih he lighpah red doed line. We an also represen an een, suh as obserer S moing away from obserer S along he ais a speed : z z The referene frame of S shown in red is moing away from he frame of S shown in blak a speed, so he disane beween he frames a any ime afer S firs passed S is. We hae arranged hings so ha boh frames oinided a he origin a 0, when S and S were ogeher. y y
Lorenz Transformaion We would like o be able ake oordinaes from one referene frame and map hem o he oher frame, ha is ake an een,,y,z in frame S and map i o he oordinaes as obsered by S and ie ersa. This is a Lorenz ransformaion. Suh a ransformaion is no as simple as alulaing Lorenz onraions and ime dilaions when we plo he moing referene frame on he aes of he saionary frame as before, now we mus ake he relaie moemen of he frames ino aoun. We will ransform he and oordinaes, keeping y and z he same, implying relaie moion along he ais only. Using all 3 spaial dimensions adds mahemaial ompleiy wihou produing new physis, and we an always arrange aes so ha he moion ours along he ais we are onsidering onsan eloiy moion, so moion a onsan speed in a sraigh line. The Lorenz Transformaion Equaions : 1 y y z z g 1 g Here, he primed oordinaes,,y,z are he oordinaes of he een as seen by he obserer, S, we are onsidering o be moing e.g. a person on a rain and,,y,z are he oordinaes for he een as seen by he obserer, S, whom we onsider o be saionary e.g. someone waiing on he plaform. Of ourse moion is relaie, and we an no really say ha S is saionary hey are on he earh whih is moing! bu so long as we undersand wha we mean he physis is unhanged.
Where do hese ransformaion equaions ome from? Equaion for refer o he graph aboe: From he graph and he Lorenz onraion, S measures aording o he Lorenz onraion as: + 1 + g Equaion for refer o he spaeime graph below: S P A Een P is reorded by S as ourring a S,S,; AP is he simulaneiy line for S and is parallel o. O S Aording o he simulaneiy shif, A ours a ime, A : A Howeer, his is aording o he ime as read by S, aording o S his ime, A, is : A γa hanging A for he equialen symbol : æ ö g ç1 è ø as required.
So now we hae he orre equaions for he ransformaion. Galilean Transformaion efore he disoery of speial relaiiy, physiiss used he Galilean ransformaion o oner oordinaes from a saionary frame o a moing one: The Galilean Transformaion Equaions : y y z z Noie ha ime was onsidered absolue and no relaie! If is ery small ompared o, g is almos 1, or if is infinie, hen g 1 and he Lorenz ransform beomes almos idenial o he Galilean ransform. The Galilean ransform is sill a good approimaion for ypial daily speeds! ausaliy and he Ulimae Speed Limi of he Unierse elow is a worked eample of a Lorenz ransform, wih oordinaes hosen o show a ery imporan poin abou simulaneiy and ausaliy. Don worry abou he srange unis gien by Mahad in he answers, hey are odd bu quie orre!.
Noie how in his eample, he order of eens obsered by obserer was he opposie of he order of eens obsered by obserer A! This implies ha eens 1 and an no be ausally linked! One een an no be he ause of he oher, oherwise ime has gone wrong, wih he eplosion ourring before he buon is pressed aording o frame. This is beause of he simulaneiy shif. Eens whih our simulaneously in one frame may no our simulaneously in anoher, or eens ha our in one order may our in reerse order in anoher frame! Wha hen is o sop obserer from sending a quik message o A elling hem no o push he buon beause jus saw he eplosion ha would ause! If you look a he disane and ime separaing he eens in eah frame, and hen look a how long i akes ligh o rael from one een o he oher, hen you will see ha ligh is no fas enough o hae raersed he spae beween he eens in he ime ineral. This means ha if no signal an rael faser han ligh hen here is no way hese wo eens an be ausally linked, one an no be he ause of he oher! They are quie separae eens, so he order in whih hey are winessed is immaerial. Noe ha also ould no relay a signal fas enough o A. Thus, ausaliy is presered IF no signal an rael faser han ligh. Sine mass, and hene ineria, inrease as an obje speeds up, we an neer aelerae an obje o he speed of ligh simply by applying a fore i would ake an infinie fore! The obje an reah ery lose o he speed of ligh bu neer quie reah i. Massless pariles, howeer, do almos always rael a he speed of ligh, sine hey hae no ineria. If i is possible o rael faser han ligh, and more imporanly, o send a signal faser han ligh, hen aording o Speial Relaiiy all hao an break loose! One obserer of a horse rae ould obsere i and hen ell somebody else far away who he winner was and he message ould reah he reipien before he rae aually ook plae and hen hey would know in adane whih horse was going o win! This is why i is belieed ha no signal an rael faser han ligh. essenially, faser han ligh rael inoles raeling o he pas and ha an ause a whole hos of paradoes!
Ne we gie a few oher imporan resuls of Speial relaiiy, hough by no means all of hem. Relaiisi Mass In he sense ha mass is a resisane o aeleraion, or ineria, i s harder o push a heaier obje! mass inreases wih speed! This is why an obje wih mass an no reah he speed of ligh a leas no by sandard means known o Earh physiiss beause as i ges faser i ges harder and harder o push i faser! We hae a mss funion, mass as a funion of speed, someimes alled he relaiisi mass: M mg momenm, p M and M0 is he ' res mass' of a parile. Proper Time This is he ime as gien by he raeler s lok. Now, a raeler may no be moing a onsan eloiy, bu a any insan hey hae a erain eloiy relaie o he Earh. Wha we do is onsider an infiniesimal insan of ime aording o he raeler a his poin, his is he proper ime and is gien by: d d g
Relaiisi energy, res energy and E m For a non relaiisi obje moing a ' normal' speeds, he kinei energy, K is : K m g 1 For low eeryday speeds, ha are no quie zero, we an approimae gamma : g» 1+ and hen : 1 K m whih is he usual epression gien, e.g. for he kninei energy of a moing ball. Noe ha for massless pariles, like phoons, K p, where p is he momenum of he phoon. These pariles hae zero res mass and moe a speed. For hese pariles, p h/l, where h is Plank s onsan and l is he waelengh. If E m g M hen we an wrie : where we hae defined : E 0 as we define he oal energy of E m E E 0 m + K he ' res mass' energy. + m g 1 a parile as : This famous equaion gies us an equialene beween energy and maer, wih one being ompleely onerible ino he oher. When your ells respire he fuels from he food you hae eaen, hey oner a ery iny fraion of he mass of hose sugars e. ino energy ia his equaion! Nulear power oners slighly more of he mass of he reaing aoms ino energy. Maer/animaer annihilaion an resul in oal onersion of mass o energy, releasing a remendous amoun of energy due o he ery high alue of, energy hidden and loked away in een a small amoun of maer!
Eidene for Speial Relaiiy Speial relaiiy is no jus a heory, bu a sienifi heory, meaning ha i makes prediions ha hae been ried and esed eperimenally and here is a huge amoun of eperimenal eidene supporing relaiiy. I shall menion jus wo here. 1. For eample, highly aurae aomi loks onboard ommerial airlines slowed by some 39 nanoseonds during a fligh from London o Washingon and bak! learly, a hese relaiely low speeds he effe is small, bu neerheless his effe was almos ealy he same as he prediions due o Speial Relaiiy and General Relaiiy aouning for Earh s graiy and idenial o wihin eperimenal error.. Anoher eample is gien y muon radiaion. Energei proons in he osmi rays olliding wih aoms in he Earh s amosphere, someime sprodue muons, a ype of heay eleron whih is unsable, deaying ino an eleron, anieleron neurino and a muneurino, wih a halflife of.19710 6 seonds. Muon deay : m e + n e + n m Now, a deeor on a mounain and a deeor a sea leel an boh monior he densiy of muons raining down upon he Earh from he upper amosphere. Sine muons deay ery quikly, we would epe o see fewer a sea leel han a mounain leel, and we do. Howeer, we don see as few as we migh naiely epe, we see raher more, so fewer deay han epeed! The reason is ha aording o Speial relaiiy, ime is relaie! The halflife we obsere for a muon is no he same as ha seen by muons raining down o Earh hese muons rael a abou 98% he speed of ligh! Thus, by ime dilaion, a fasmoing muon will seem o las longer and rael furher. Thus, hey deay more slowly han we epe beause hey are moing fas and hus muonime dilaes and slows down! Lorenz alulaions predi he obsered deayrae! General Relaiiy Speial relaiiy assumes onsan eloiy, bu wha if one or more obserers are aeleraing? Speial Relaiiy an be used o approimae hese siuaions, sine a any insane in ime, he hange in eloiy will be small, and we an arrange a series of inerial referene frames along he pah of he obserer. Howeer, fores, whih ause aeleraion, are really he subje of General Relaiiy whih aouns for he fore of graiy. We shall look a General Relaiiy in a fuure arile.
Traeling o he Sars Faser han ligh? Due o he problems of ausaliy ha we hae highlighed, mos physiiss beliee ha i is impossible for any signal, inluding a massie obje like a spaeship, o rael faser han ligh. Howeer, Speial Relaiiy does enable spae rael oer ery grea disanes, by raeling forwards in ime a a differen rae, as in he Twin Parado. The rew of a spaeship aeleraing ery lose o he speed of ligh ould rael o he edges of he known osmos in heir lifeimes, in priniple. The only rouble is, by he ime hey go here, billions of years would hae elapsed! A spaefairing rae would hen be far from onneed, bu raher saered in ime and a sarship would be unable o reurn home afer any lenghy oyage! Oher heories hae been and are being deeloped, howeer, whih inorporae faserhanligh rael in ways whih would no iolae ausaliy. Sring Theory predis he eisene of Tahyons, pariles ha rael faser han ligh. Howeer, we perhaps ould neer dee ahyons and so neer use hem o send a message. Some of hese heories assume ha he Lorenz symmery, whih gies us he ransformaions beween differen frames, is only approimae. Under erain ondiions, perhaps a ery high energies, here is some iolaion of his heory whih would also lead o PT iolaion in parile physis. PT symmery whih is disussed in anoher arile on Symmeries saes ha a mirrorimage of he Unierse, in whih all posiions and momena were reersed and in whih maer would be replaed by animaer would follow idenial laws. erainly, no differenes beween pariles and anipariles oher han heir eleri harges hae been irrefuably obsered; eep ha maer is muh more abundan. Howeer, here is some suggesion ha anineurinos may hae slighly differen masses han neurinos, implying a iolaion. If his symmery is no ea, bu only approimae, hen Lorenz inariane is also neessary iolaed. This would make relaiiy only an approimae heory. In he Superfluid Vauum Theory SVT, he auum of spae onains a sea of bosons ha are ondensed ino a oseeinsein ondensae E. This heory auses Lorenz symmery o be only approimae, due o small auumfield fluuaions. This ould allow massie pariles o reah he speed of ligh a finie energy! General relaiiy also opens doors o new possibiliies. In General relaiiy he presene of energy auses spaeime o bend or warp, aouning for graiy. The meri of a loal region of spaeime is a mahemaial law defining he disane beween wo poins in spae and ime in spaial erms, alulaing he disane beween wo poins on a fla surfae and wo on he surfae of a sphere require a differen meri. The Shwarzshild meri, for eample, desribes he spaeime uraure around a spherial nonroaing mass, suh as a saionary sar or blakhole. The Alubierre meri desribes a warp bubble, ahead of whih spaeime onras, and af of whih spaeime epands. Warping spaeime signifianly requires massie amouns of energy, bu perhaps some form of warp drie is possible. Some heries also inrodue addiional spaial and emporal dimensions. Sring Theory may inrodue 5 spaial dimensions and one of ime, supersring heory has 10 or 11 dimensions, bu sill only one of ime. Heim heory uses era imelike dimensions of spae and an apparenly suessfully predis QED. The erdi is sill ou hee, perhaps waiing for he ne Einsein!?
ibliography 1. Spae, Time and Quana: An inroduion o onemporary physis, Rober Mills, 1994, Freeman and ompany New York. [An eellen book for firsyear ollege sudens and lay people, assumes lile prior knowledge, jus a mahemaial apiude.]. The Open Uniersiy, ourse ebooks for S71 Disoering Physis and S357 Spae, Time and osmology. [Eellen maerials, ery omprehensie and wellwrien.]