LIGHT and SPECIAL RELATIVITY

Transcription

1 VISUAL PHYSICS ONLINE MODULE 7 NATURE OF LIGHT LIGHT and SPECIAL RELATIVITY LENGTH CONTRACTION RELATIVISTIC ADDITION OF VELOCITIES Time is a relaie quaniy: differen obserers an measuremen differen ime inerals beween he ourrene of wo eens. This arises beause he speed of ligh is a onsan and independen of he moion of he soure of ligh or he moion of an obserer. Moing loks run slow

2 Time Dilaion Effe Proper ime he ime ineral beween wo eens ourring a he same poin in spae w.r.. a lok a res w.r.. ha poin. Dilaed ime ineral ime ineral for een in moing frame as measured on lok by saionary obserer. 1 > sine 1 Lorenz-Fizgerald Conraion Equaion of a moing obje L L onraed lengh L and proper lengh L Conraion akes plae in he direion of moion only Relaiisi addiion of eloiies

3 RELATIVE LENGTH: LENGTH CONTRACTION When measuring he lengh of an obje i may be neessary o be able o deermine he exa posiion of he ends of he obje simulaneously. Howeer, obserers in differen referene frames may disagree on he simulaneiy of wo eens. So, hey may also disagree abou he lengh of objes. In urns ou, he lengh of a moing obje appears o onra in he direion of moion relaie o a saionary obserer. Equaion (1) is known as he Lorenz-Fizgerald Conraion Equaion of a moing obje (1) L L Conraed lengh of obje L in direion of moion as measured by obserer in saionary frame of referene. Res lengh or proper lengh of obje L as measured by obserer in he moing frame of referene. The proper lengh is measured in he frame in whih he obje is a res. Veloiy (magniude) of he obje relaie o he obserer in he saionary frame of referene. Conraion akes plae in he direion of moion only

4 You are a saionary obserer in an inerial frame of referene. A rain was iniially a res in your frame of referene and you measure is lengh. Howeer, when he rain is in moion your measuremen of is lengh is shorer. There is a onraion in is lengh. The rain is shorer in he direion of moion, bu jus as high and wide as i was a res as shown in figure 1. Fig. 1. How long is a rain? I depends on he relaie moion of he obserer and he rain. This is a real differene in lengh of he obje when i is moion relaie o an obserer. For a person in he rain, here is no onraion in lengh (figure ).

5 Fig.. A lengh is measured o be shorer when i is moing relaie o he obserer han when i is a res. Time dilaion effe Lengh onraion L L L L L L 1

6 We will now onsider an alernaie deriaion for he onraion in lenghs parallel o he relaie moion hrough anoher hough experimen. We aah a laser o one end of a rod and a mirror a he oher. The rod is a res in Mary s sysem, and he lengh of he rod is L M (proper lengh sine i is a res w.r. obserer Mary). Mary measures a ime ineral for a pulse of ligh o make he round rip from laser o mirror and bak. This is he proper ime ineral sine he deparure and reurn our a he same loaion in Mary s sysem. Fig. 3. The rod is saionary in Mary s sysem. The ligh pulse raels a disane L from he ligh soure o he mirror. The ime for he round rip is. In See s sysem, he rod is moing o he righ wih onsan speed. See s measures he lengh of he rod as L and he ime for he ligh o rael from laser o mirror as 1. During his ime ineral 1, he rod wih laser and mirror aahed moes a disane 1o he righ.

7 Fig. 4. The rod is moing o he righ wih onsan speed in See s sysem. The ligh pulse raels a disane d 1 from he laser o he mirror in ime 1. The oal lengh of he pah d 1 from laser o mirror is herefore d1 L 1. Bu, he ligh pulse raels wih speed, so i is also rue ha d1 1. Eliminaing d, we find 1 L Noe: he diision of L by -u does no mean ha ligh raels wih speed -u, bu raher he ligh pulse raels in See s frame a disane greaer han L.

8 Repeaing a similar alulaion, he ime ineral from he reurn journey of he ligh pulse from he mirror o he laser is L The oal ime for he for he round rip is L L L The proper ime and he dilaed ime inerals are onneed by Hene, we an onlude L bu (1) L L L L 1 Anoher explanaion There are wo inerial frames of referene, See s and Mary s. From See s poin of iew, Mary s frame in moing wih a onsan eloiy of magniude. They boh make measuremens of he lengh of a rod. The rod is saionary in Mary s frame bu moing wih eloiy in See s frame. They boh measure he lengh of he rod by obsering he ime ineral for a ligh pulse o rael from one end of he rod o he

9 oher. They obsere know ha he speed of ligh is he same w.r. boh frames of referene. The disane measured by Mary is LM M and he disane measured by See is LS S So far eeryhing is sraigh forward, bu now here omes he riky par. We mus idenify he proper and dilaed ime inerals and he proper lengh and he onraed lengh. See s lok is saionary w.r.. his frame, herefore, his lok reords he proper ime and he iew Mary s lok whih is he dilaed ime ineral M S 1 See iews he moing rod, so he measures he onraed lengh, while he rod is saionary in Mary s frame, so i is he proper lengh L L L L M S M L L S S L L L L M Lengh onraion is real. This is no an opial illusion, The rod is really shorer in Mary s sysem han in See s sysem.

10 Example 1 A spaeship flies pas Earh a a speed of.99. Mary on board he spaeship measures is lengh o be 4 m. Wha is See s measuremen for he lengh of he spaeship on Earh? See and Ee on Earh are sanding 6 m apar as hey iew he passing spaeship. How far apar are See and Ee from Mary s poin of iew? Soluion The problem relaes o he lengh onraion of a moing rod. Mary in her sysem measures he proper lengh of he spaeship L 4 m See obseres he spaeship moing a uniform eloiy and measures he lengh of he spaeship as L (onraed lengh).99 L?m Lorenz-Fizgerald Conraion L L 56.4 m The answer makes sense, he moing spaeship is obsered o he shorer han in he frame saionary w.r.. o he spaeship. See and Ee are a res in he frame of he Earh. Their separaion disane of 6 m is he proper lengh. From Mary perspeie, See and Ee are moing a a speed. So, Mary will measure a onraed lengh L as he separaion disane: L 6 m.99 L?m L L 8.5 m

11 Example The diameer of our galaxy is 6.x1 m. (a) If he speed of he spaeship was.99999, how long would i ake o ross he galaxy as measured from he frame of referene of he spaeship? (b) How muh ime would elapse on Earh for his journey aross he galaxy? Soluion (a) A raeller in he spaeraf would be a res in he spaeship and would see he galaxy approahing a speed = The raeller would see he galaxy onraed in he direion of moion. The onraed lengh L is L = 6.x1 m = L L m

12 If he ime is measured in he spaeraf, he ime for he journey aross he galaxy is =? s =? years L =.7x1 18 m L = L s 3 years Een a spaeship raelling a speeds i sill akes 3 years o ross he galaxy. (b) An Earh bound obserer will iew he moing lok on he spaeship and measure a dilaed ime ineral Spaeship ime ineral = 3 years (proper ime) Dilaed ime ineral for Earh obserer =? years = years WOW!!! For he Earh obserer, 64 years pass for he spaeship o ross he galaxy bu on he spaeship only 3 years hae passed. Only, spaeships raelling a speeds exremely lose o he speed of ligh an raerse he huge asronomial disanes need for spae rael in reasonable imes.

13 THE RELATIVISTIC ADDITION OF VELOCITIES Suppose you were in a moing ruk and fired a roke. The ruk was raelling a 1 km.h -1 and he roke was launhed wih a muzzle speed a km.h -1 (speed of roke w.r. ruk) in he same direion ha he ruk was moing. Obiously he speed of he roke is 3 km.h -1 w.r. he ground (1 + = 3). Fig. 5. Newonian mehanis: adding eloiies RM RT TM.

14 Mary is in a spaeship raelling a.8 w.r.. See roke is launhed from he spaeship by Mary a a speed of.7 w.r.. Mary RM MS. A. Hene, aording o Newonian mehanis, he speed of he roke obsered by See is RS RS RM MS Bu, our answer is wrong. We know ha by Einsein s posulae ha he speed of an obje mus be less han he speed of ligh. Einsein deried he orre formula for he addiion of eloiies. In our example, he eloiy of he roke w.r.. o he ground is () RS RM RM MS MS relaiisi addiion of eloiies Applying he orreion equaion, he speed of he roke w.r. See is RM MS RS RM MS RS Noe: if he speed of ligh were infinie, he denominaor would be equal o 1, and we would reoer he lassial eloiy addiion equaion. So, i is he finie speed of ligh ha is responsible for relaiisi effes.

15 A New Sandard of Lengh Lengh is one of he fundamenal quaniies in Physis beause is definiion does no depend on oher physial quaniies. The SI uni of lengh, he mere was originally defined as one enmillionh of he disane from he equaor o he geographi Norh Pole. The firs ruly inernaional sandard of lengh was a bar of plainum-iridium alloy alled he sandard mere and kep in Paris. The bar was suppored mehanially in a presribed way and kep in an airigh abine a o C. The disane beween wo fine lines engraed on gold plugs near he ends of he bar was defined o be one mere. In 1961 an aomi sandard of lengh was adoped by inernaional agreemen. The mere was defined o be imes he waelengh of he orange-red ligh from he isoope krypon-86. This sandard had many adanages oer he original inreased preision in lengh measuremens, greaer aessibiliy and greaer inariabiliy o lis a few. In 1983 he mere was re-defined in erms of he speed of ligh in a auum. The mere is now defined as he disane ligh raels in a auum in 1/ of a seond as measured by a aesium lok. Sine he speed of ligh is onsan and we an measure ime more auraely han lengh, his sandard proides inreased preision oer preious sandards. The reason for ha pariular fraion (1/ ) is ha he sandard hen orresponds o he hisorial definiion of he mere he lengh on he bar in Paris. So, our urren sandard of lengh is defined in erms of ime in onras o he original sandard mere, whih was defined direly in erms of lengh (disane).

16 VISUAL PHYSICS ONLINE If you hae any feedbak, ommens, suggesions or orreions please Ian Cooper Shool of Physis Uniersiy of Sydney