Special Theory of Relativity

Transcription

1 Speial Theory of Relatiity Introdtion: Galileo was first persaded (oaed) that earth is in otion arond the sn, whih is stationary. Most of his fellows (onteporaries) arged that if it is tre then why birds leaing the earth wold not be left behind by the speeding earth? Galileo reasoned that if a ship oed niforly on the sea, a sailor old not distingish between the sitations either the ship is in otion and the sea is at rest or the ship is at rest and the sea is in otion. In Galileo s iew- the only otion that is easrable is the relatie otion between the ship and the sea; hene the ter Relatiity was anifested first eer. Relatie Motion: The first step is to larify what we ean by otion. Using Cartesian o-ordinate syste in three diensions, we an haraterize any loation by three o-ordinates of that spae point. When we say that soething is oing, what we ean is that its position relatie to soething else is hanging. That is all otion is relatie. In eah ase a frae of referene is part of the desription of the otion. Frae of Referene: In order to desribe the otion of oing bodies, we need to state where the objet is at any gien tie. Bt to state where an objet is, we need to easre its position relatie to soething else, right? So we need a referene point fro whih to define the position of objets. One we hae hosen sh a point, whih is alled the origin, we an speify the position of the objet by saying, for instane, that the objet is distane to the east, distane y to the north, and distane z p fro the origin. We also need a lok so that we an speify at what tie t the objet was at the gien position. The o-ordinate syste, relatie to whih any easreent an be done to speify the state of rest or otion of an objet, is known as the frae of referene. When we hae the origin and the diretions in whih to easre the distane fro the origin set p, and a lok to easre the tie, we say that we hae a frae of referene or siply a frae. Inertial and Non-inertial Frae of referene: Newton's first law, the law of inertia, states that. if an objet is at rest it will stay at rest if no fore is ating on it, and. if an objet is oing it will keep on oing at onstant eloity if no fore is ating on it. This law is atally not always orret! (Srprised?) It depends on whih frae yo are sing to desribe the otion of the objet. For instane, if yo are sing the oing objet itself as the origin of yor frae of referene, it is always at rest no atter what fores are ating on it. So when we talk abot the law of inertia, we are assing that a frae eists in whih the law is orret. Sh a frae is alled an inertial frae. If one sh inertial frae eists, then an infinite nber of other inertial fraes eist sine any frae that is oing at a onstant relatie eloity to the first inertial frae is also an inertial frae.

2 Inertial Frae: An inertial frae of referene is one in whih Newton s first law of otion holds good. In this frae, an objet at rest reains at rest and an objet in otion ontines to oe at onstant eloity (onstant speed and diretion) if no fore ats on it. Any frae of referene that oes at onstant eloity relatie to an inertial frae is itself an inertial frae. Speial theory of relatiity deals with probles that inole inertial fraes of referene that is, where the body oes with a onstant eloity. Non-inertial Frae: A referene frae in whih a body is aelerated withot being ated pon by eternal fore is alled a non-inertial frae of referene. Newton s laws are not alid in sh a frae of referene. General theory of relatiity, whih is pblished by instein a deade later in 97, onerns itself with all fraes of referene inlding the non-inertial fraes of referene whih are aelerated with respet to one another. Failre of Newtonian Mehanis: For an eletron aelerated throgh a MV potential differene, a ale reasonable easy to obtain, the speed eqals If the energy of MeV eletron aboe is inreased by a fator of for (to 4 MeV) eperient shows that the speed is not dobled to.9976 as we ight epet fro the Newtonian relation bt reain below ; it inreases only fro.9988 to.9999, a hane of. perent. M K, Galilean Transforation: Why Galilean transforation introded? Newton s laws of otion does not tell s whether there is one or any inertial fraes of referene, nor, if there is ore than one, does it tell s how we are to relate the oordinates of an eent as obsered fro the point-of-iew of one inertial referene frae to the oordinates of the sae eent as obsered in soe other. In establishing the latter, we an show that there is in fat an infinite nber of inertial referene fraes. Moreoer, the transforation eqations that we derie are then the atheatial basis on whih it an be shown that Newton s Laws are onsistent with the priniple of relatiity. The transforation of o-ordinates of a partile fro one inertial frae of referene to another is alled the Galilean transforation. Let s onsider we are in an inertial frae of referene S and the oordinates of soe eent that ors at the tie t are (, y, z). An obserer loated in a different inertial frae S whih is oing with respet to S at onstant eloity, will find that the sae eent ors at the tie t and has the oordinates (, y, z ). For oneniene, let is in the + diretion as shown in the figre.

3 y y S S y O y O p z t z Fig. 6: Frae S oes in the X-diretion with speed relatie to frae S. Let s sppose that the loks in S and S are set sh that when the origins of the two referene fraes O and O oinide, all the loks in both fraes of referene read zero i.e. t = t =. Aording to oon sense, if the loks in S and S are synhronized at t = t =, then they will always read the sae, i.e. t = t always. This is the absolte tie onept. We onsider an eent of soe kind, i.e. an eplosion ors at a point (, y, z, t ) aording to S. The otion is in the + diretion and there is no relatie otion in the y and z diretions, and so the eent ors in S at the point t y y z z t t () These eqations together are known as the Galilean transforation. They tell s how the oordinates of an eent in the inertial frae S, whih oing with a nifor eloity with respet to S, are related to the oordinates of the sae eent as easred in S whih is in rest. Aording to the syetry of spae, if S is oing with a eloity with respet to S, then S will be oing with a eloity - with respet to S so the inerse transforation shold be obtainable by siply ehanging the pried and npried ariables, and replaing by. Ths the inerse transforation, i.e., the transforation fro S to S is t y y z z t t () The tie-interal and spae-interal easreents are absoltes aording to the Galilean transforation. 3

4 Speial relatiity is based on two postlates whih are ontraditory in lassial ehanis: It was instein who, in 95, pointed ot that the only way to nderstand this was to hange or notion of siltaneity. This was his faos speial theory of relatiity.. Priniple of eqialene of physial laws: The laws of physis are the sae in all inertial frae of referene. No preferred inertial syste eists.. Priniple of onstany of eloity of light: The speed of light in free spae (a) has the sae ale 3 8 s in all inertial fraes of referene, independent of the relatie eloity of the sore and obserer. Conseqenes of instein s postlates: There are for iportant onseqenes of instein s postlates: Relatiity of siltaneity: Two eents that appear siltaneos to an obserer A will not be siltaneos to an obserer B if B is oing with respet to A. Relatiity of Tie (Tie dilation): Moing loks tik slower than an obserer's "stationary" lok. Relatiity of Length (Length ontration): Objets are obsered to be shortened in the diretion that they are oing with respet to the obserer. Mass-energy eqialene: Aording to the relationship = ², energy and ass are eqialent and transtable. Failre of Galilean transforation: The seond postlate alls for the sae ale of the speed of light when deterined in S or S. This ontradits with Galilean transforation. If we easre the speed of light in the -diretion in the S syste to be, howeer, in the S syste it will be Clearly a different transforation is reqired if the postlates of speial relatiity are to be satisfied. Lorentz Transforation: instein's speial theory of relatiity says the speed of light is onstant; we hae to odify the way in whih we translate the obseration in one inertial frae to that of another. The set of Galilean transforation t, t t () is wrong and are not onsistent with the eperiental reslts. The orret relations whih are onsistent with the eperiental reslts are t t t () These eqations are alled the Lorentz transforation. 4

5 Deriation of Lorentz Transforation qations: For the onstany of eloity of light we hae to introde the new transforation eqations whih flfill the following reqireents:-. The speed of st hae the sae ale in eery inertial frae of referene.. The transforations st be linear and for low speeds <<, they shold approah the Galilean transforations. 3. They shold not be based on absolte tie and absolte spae. A reasonable gess abot the natre of the orret relationship between and is k( t) () Here k is a fator that does not depend on either or t bt ay be a fntion of. Bease the eqations st hae the sae for in both S and S, we need only hange the sign of (in order to take into aont the differene in the diretion of relatie otion) to write the orresponding eqation for in ters of and t: k( t) () The fator k st be the sae in both fraes of referene sine there is no differene between S and S other than in the sign of. Let s onsider that a light plse that starts at the origin of S at t =. Sine we hae assed that the origins are oinident at t = t =, the plse also starts at the origin of S at t =. instein s postlates reqire that the eqation for the -oponent of the wae front of the light plse is t in frae S and t in frae S. Sbstitting t for and t for in q. () and (), we get and t k( t t) k( ) t (3) t k( t t) k( ) t (4) t Fro eqation (3), t k( ) Siilarly fro eqation (4), t k( ) and ths we get t k( ) k k( ) k k k (5) ' ' Sbstitting k( t) for in k( t ), we get k[ k( t) t] ( t) kt 5

6 k ( t) or t k ( t) ( ) t t t t t k( t ) (6) The oplete relatiisti transforation is k( t).... (7) y y.... (8) z z.... (9) t k( t ).... () These are the Lorentz transforation eqations. (i) Lorentz transforation eqation is linear in and t. (ii) Redes to Galilean transforation for. The obserer in S will obsere that the frae S is oing to the right with a eloity with respet to it. Ths, when we sole qs. (7)-() for, y, z, and t in ters of the pried oordinates, we obtain k( t).... () y y.... () z z.... (3) t k( t ).... (4) whih are idential in for of qs. 7- and are known as Inerse Lorentz transforations. Relatiisti Veloity Transforation: The oplete set of relatiisti Lorentz transforation is k( t), y y, z z and t k( t ).. () Let, y, and z are the eloity oponents of a partile with respet to the S frae along, y and z diretions respetiely, while, y, and z are the eloity oponents of that partile with respet to S frae along, y, and z diretions respetiely. By differentiating the Lorentz transforation eqations for, y, z and t, we get d k( d dt), dy dy, dz dz and dt k( dt d ) 6

7 7 Therefore, dt d dt d d dt k dt d k dt d ) ( ) ( (5) We ay write the oplete set of relatiisti eloity transforation as z z y y and,, The inerse eloity transforation eqations are z z y y and,, (i) If and are ery sall opared to, the eloity transforations satisfy the lassial reslts. z z y y and,, (ii) If a ray of light is eitted in the oing frae S in its diretion of otion relatie to S, then, and no atter what the ale of of the obserer, an obserer in frae S will easre the speed Ths the obserers in the ar and on the road both find the sae ale for the speed of light, as they st. aple #: Two eletrons leae a radioatie saple in opposite diretions, eah haing a speed of.67 with respet to the saple. What is the speed of one eletron relatie to the other? Soltion: Here 67. and =.67 The speed of one eletron relatie to other This is eans that speed of one eletron relatie to the other is less than.

8 Relatiity of Siltaneity: Two eents are said to be siltaneos if they or at the sae tie. Aording to the relatiity of siltaneity, if two obserers are in relatie otion, they will not agree as to whether two eents are siltaneos. If one obserer finds the to be siltaneos, the other generally will not, and onersely. The siltaneity of the two eents is not an absolte onept and depends on the frae of referene. In fat eah obserer is orret in his own frae of referene. Two eents that appear siltaneos to an obserer A will not be siltaneos to an obserer B if B is oing with respet to A. Let s onsider an eaple to larify the aboe stateent: Figre: Spreading of light signals in a toy ar as obsered (a) by an obserer on the ar itself and (b) by an obserer standing on the grond. Iagine a trolley traeling at a onstant speed along a sooth, straight trak. In the entre of the trolley there a light blb is hanged. When it is swithed on, the light spreads ot in all diretions at a speed. Bease the lap is eqidistant fro the two ends, an obserer on the trolley will find that the light reahes the front and the rear ends at the sae tie, i.e., the two eents of light reahing the front and the rear ends or siltaneosly (Fig. a). Howeer, to an obserer on grond these two eents do not appear to be siltaneos. As the light traels ot fro the blb, the trolley itself oes forward, so the bea going to the rear end has a shorter distane to trael than the one going forward. Aording to this obserer, therefore, the seond eent appears to happen before the first eent (Fig. b). Therefore, we an onlde that the two eents that are siltaneos in one inertial frae are not, in general, siltaneos in another frae. In fat, easring ties and tie interals inole the onept of siltaneity and fro the aboe disssion it follows that the tie interal between two eents ay be different in different fraes of referenes. Proof of this stateent: Let s onsider two fraes of referene S and S. The frae referene S is oing with eloity relatie to the frae of referene S along +e diretion of -ais and the two eents or siltaneosly in S. Sine the eents are siltaneos in frae S, therefore we hae t = t. If t and t are the orresponding ties of the sae two eents with respet to syste S, then we hae fro Lorentz transforation eqations:- t k( t ) and t ( ) k t 8

9 t t k( t ) k( t ) ( ) k sine t t Ths if the eents are siltaneos in frae S, t st be eqal to t or t t st be eqal to zero, bt it is not so bease is not eqal. Therefore, the sae two eents are not siltaneos in frae S. Length Contration: The length of a body is easred to be greatest when it is rest relatie to the obserer. When it oes with a eloity relatie to the obserer its easred length is ontrated. In siple ters: The length of an objet to be shorter when it is oing then when it is rest. Sitation : A stik fied in S bt obsered in S. Let s onsider a stik at rest in the syste S and the oordinates of two ends are and so that its length as easred by an obserer in S is gien by L = -. The frae S is oing with a eloity with respet to S. Fro S, length st be easred at the sae tie t. Aording to the Lorentz transforation k( t) and k( t). L Hene L ( ) k( ) L L This sarizes the effet known as length ontration. The frae S is at rest with respet to the objet so the easred rest length is L and L >L. As the frae S is oing with respet to S, all obserers otside the frae S are in otion relatie to S and easre a shorter length, bt only along the diretion of otion; length easreents transerse to the diretion of the otion are naffeted. For ordinary speeds (<<), the effets of length ontration are too sall to be opared (L L). Faster eans shorter Length ontration sggests that objets in otion are easred to hae a shorter length than they do at rest. 9

10 Tie Dilation: Moing loks tik slower than an obserer's "stationary" lok. A lok is easred to go at its fastest rate when it is at rest relatie to the obserer. When it oes with a eloity relatie to the obserer, its rate easred to hae slowed down by a fator. Proof: Let there be two fraes of referene S and S ; S is oing with a eloity relatie to S along (+e) diretion of -ais. Let a lok to be at rest at the point in the oing frae S and another lok be rest at the point in the frae S. Let two eents ors in S at : one ors at tie t and other at tie t. The orresponding ties easred by the lok at S are t and t respetiely. Then the tie interal between the two eents as noted on the lok in the oing frae S is gien by t t t and the tie interal between the sae two eents as noted by the lok in the stationary obserer S is gien by t t t. Aording to the Inerse Lorentz transforation t k( t ) and t k( t ) Therefore, fro these eqations, we hae Now sine t t t k( t t) is greater than nity as <, therefore t t t. Ths the tie interal t between two eents orring at a gien point in the oing frae S appears to be longer or dilated to the obserer in the stationary frae S. The relation is tre only when t represents the tie interal between two eents in a referene frae where the two eents or at the sae point in spae. The tie dilation effet has been erified eperientally with deaying eleentary partiles as well as with preise atoi loks arried aboard airraft. Meson Deay: -eson Proded in the pper reah of the atosphere as a reslt of ollision between fast osi ray partiles, arriing the earth fro spae, and the air oleles. It deays into eletron and two netrinos eah. -eson ass = 5 Mass of an eletron Lifetie =. -6 se Veloity =.998 Trael Distane = 66 Obsered Trael Distane = k Callated Trael Distane = 479 (Using Length Contration) Obsered Lifetie = se Callated Trael Distane = 455 (Using Tie Dilation)

11 instein s Mass And nergy Relation: Let a fore F is ating on a body so that there is a displaeent ds along the diretion of the fore. The work done is then gien by W = Fds If no other fores at on the objet and the objet starts fro rest, all the work done on it beoes kineti energy K. and is gien by K s Fds dp d( ) If the fore is hanging with tie and is gien by F. Where is the eloity of the dt dt body and the relatiisti oent is p. So the kineti energy K beoes K S Fds S ( d d) ( d d) d( ) ds dt d( ) in whih and are ariable. These qantities are related as ( ) (3) () ds dt () Differentiating eqation (3), we get d d d d d d (4) Sbstitting this ale in eqation () and integrating, we get K K d d d d (3) This is the relatiisti eqation for kineti energy of a body oing with a eloity. Also, if we take = as the total energy of the body then the aboe eqation ay be written as K (4) in whih is alled the rest energy of the body. The rest energy is energy of the body at rest when =, and K = and the body has an aont of Rest energy (5)

12 Ths we an write K (6) If the body is oing, its total energy is (7) This is the instein faos ass-energy relation. (i) For low speeds,, Then we an write K (8) Using the binoial approiation ( ) n n with, we get K (9) At low speed, the relatiisti epression for the K.. of a oing body redes to the lassial one. (ii) Mass-less partiles: We know, total energy, () and relatiisti oent p () When = and <, i.e. = p = i.e. A ass less partile with a speed less than that of light an hae neither energy nor oent. (iii) When o = and =, = and p = whih are indeterinate: and p an hae any ales. The eqations [] & [] are onsistent with the eistene of ass-less partiles that possess energy and oent proided that they trael with the speed of light. (i) There is another restrition on ass-less partiles. 4 we an write Also we an write p, or, p Sbtrating p fro yields

13 3 () p 4 p Therefore, for all partiles, we hae 4 p p o o (3) Aording to this forla, if a partile eists with =, the relationship between its energy and oent st be gien by p - less partiles ass (4) In fat, ass-less partiles of two different kinds - the photon and the netrino hae indeed been disoered and their behaior is as epeted.. Find the ass of an eletron ( o =9. -3 kg) whose eloity is.99. Soltion: Gien =.99, so. 64 (.99) kg whih is 7 ties greater than the eletron s rest ass. If =, =, fro whih we an onlde that an neer eqal :. A stationary body eplodes into two fragents eah of rest ass. Kg that oe apart at speeds.6c relatie to the original body. Find the rest ass of the original body. Soltion: The total energy of the original body st eqal the s of the total energies of the fragents. Hene (.6) ) (. (.6) ) (.. kg kg K Therefore, 5kg. Mass an be reated or destroyed bt when this happens an eqialent aont of energy siltaneosly anishes or oes into being and ie ersa. No aterial objet an trael as fast as light Mass and energy are different aspets of the sae thing

14 The onersion fator between the nit of ass (the kg) and the nit of energy (the jole, J) is. So, kg of atter has an energy ontent of 8 6 ( kg) * ( 3 s) 9 J This is enogh to send a payload of a illion tons to the oon. 3. An eletron ( =.5 MeV ) and a photon ( =) both hae oenta of. MeV. Find the total energy of eah. Soltion: The eletron s total energy is 4 4 p (. 5MeV ) (. MeV ) 64 MeV. 4. What is the perentage inrease in the ass of an eletron aelerated to a K. of 5 MeV? Use rest ass of eletron =.5 MeV. 5. What is the speed of a partile if its kineti energy is % larger than o? Conept of ther: When we say that the speed of sond in dry air at C is 33.3 se, we hae in ind an obserer, and a orresponding referene syste, fied in the air ass throgh whih the sond wae is oing. It is howeer known that sond waes are ehanial ibrations whih are longitdinal in natre annot propagate in a and reqire a edi haing soe non-zero density. Howeer, when we say that the speed of light is = s, it is not lear at all what referene syste is iplied. A referene syste fied in the edi of propagation of light presents diffilties bease no edi sees to eist in ontrast to sond. Aording to Mawell, light waes are eletroagneti wae and the physiists pto the 9 th entry felt qite sre that the M waes reqire a propagation edi like all other kinds of waes. For light, they assed this edi naed as Liniferos ther. Bt probles arose when ther is attribted to soe strange properties: i) Sine light passes throgh a, so ther st be assless and of zero density and st hae perfet transpareny to aont for its ndetetability. ii) Wae propagation reqires shearing fores and these fores an or in solid only. It eans that ther st be a rigid solid filling the whole spae. As the eloity of wae propagation depends on the elastiity of the edi, ther st be highly elasti. Ths, the whole free spae st be filled p with sh an elasti edi to sstain the ibration of light waes. Thogh these are diffilt to predit the people belieed the ther edi. A soltion of Mawell s eqation gies the speed of light, = 3 8 se, this is in agreeent with eperiental reslt. This is now well known that light waes are transerse eletroagneti waes in origin and do not need any edi. Mihelson and Morley onted the interferoeter on a assie stone slab for stability and floated the apparats in erry so that it old be rotated soothly abot a entral pin. In order to ake the light path as long as possible, irrors were arranged on the slab to reflet the beas bak and forth throgh eight rond trips. 4

15 The arrangeent was apable of easring th of a fringe shift. Bt, atally no fringe shift was fond dring this easreent. The eperient was perfored at different seasons and different plaes with the sae reslt eery tie and no fringe shift was deteted. Ths, it was onlded that the relatie eloity between the earth and the ther is zero and there is no fringe shift at all. The reslt of Mihelson-Morley eperient was eplained by instein and he onlded that: i) There is no ether frae in spae, and ii) The speed of light in free spae is inariant i.e. it is onstant and is independent of the otion of sore, edi or the obserer. This fat lead to n =. Downstrea and ross-strea speed of light is, not +. Conlsion: I hope I hae seeded in giing yo a flaor of instein's speial theory of relatiity. Key points I wold like yo to take hoe with yo are:. The speed of light is onstant regardless of the inertial frae in whih it is easred. All of the theory's onlsions are deried fro this siple eperiental fat.. Siltaneity is a relatie onept. The hronologial order in whih two eents orred ay depend on the frae of the obserer. 3. Faster than light speed trael or oniation is ipossible. (Otherwise, asality will be broken.) 4. Interesting phenoena like tie dilation and Lorentz ontration hae been predited and obsered. 5. qialene of ass and energy Faos People: The following is a list of faos people whose naes I hae entioned in y letre notes. Galileo Galilei (564-64) Sir Isaa Newton (643-77) Jaes Clerk Mawell (83-879) Hendrik Antoon Lorentz (854-98) Albert instein ( ). Why the nierse is belieed to be epanding?. The epansion of nierse began abot 3 billion years ago. How do yo beliee the epansion of entire nierse? 3. What do yo ean by Doppler ffet in light? Find the epression of obsered freqeny of light in the ase of longitdinal Doppler ffet. 4. State Hbble s law? 5. The relatiisti eqation for the kineti energy is K, where the ters hae their sal eaning. Find the kineti energy of the body when it is oing with a ery low speed i.e.,. 5