8.022 (E&M) Lecture 11

Transcription

1 8.0 (E&M) Leture Topis: Introdution to Speial Relatiit Length ontration and Time dilation Lorentz transformations Veloit transformation Speial relatiit Read for the hallenge? Speial relatiit seems eas but it s not! A new wa of thinking that often goes against intuition It will take some time to digest it, but beliee me: it s worth the effort! Wh do we need it in 8.0? Weren't ou frustrated last time when magneti fores ame out of nowhere? Speial relatiit naturall eplains them in terms of eletri fores seen from in a referene frame in motion This is important for eerbod Phsis majors: first of man iterations on a ruial topi Non Phsis majors: hane to know what ou are missing Don t forget: ou are still in time G. Siolla MIT 8.0 Leture

2 The priniples of speial relatiit Formulated in 905 b A. Einstein Inredible but true: no Nobel Prize for this! Based upon postulates The laws of phsis are the same for all referene frames The speed of light is the same () in all referene frames (Inertial) Referene frame ( Sstem of oordinates in whih the obserer is non aelerating inertial = non aelerating) G. Siolla MIT 8.0 Leture 3 Referene frames: eamples Situation A train is moing with eloit w.r.t. to a station A table is anhored to the train A ball is falling from the table We an identif 3 sstems of referene and 3 obserers: Obserer : sitting on a benh at the station Obserer : sitting on the table on the train Obserer 3: a bug sitting on top of the falling ball Who are the obserers in an inertial referene frame? Obserers and Obserer 3 is not: the ball is falling with aeleration g G. Siolla MIT 8.0 Leture 4

3 Is time the same in all referene frames? These (apparentl) innoent assumptions hae amazing onsequenes suh as time is not absolute! Problem The train is moing with eloit // ais h Obserer : standing in the train Obserer : at the station Obserer flashes a pulse of light ertiall to a photosensor mounted on the floor of the train Both obserers measure the time between when the light is emitted and when the light reahes the sensor Will the obserers measure the same time? G. Siolla MIT 8.0 Leture 5 Time in different referene frames Let s alulate time measured b the obserers Train referene frame (obserer ) Distane traeled b light: h h h t = t = Veloit of light: Station referene frame (obserer ) Distane traeled b light: h'= h + ( t ) h ' t ' = t = Veloit of light: h ' h +( t ) ( t ) = = = t + t t = t Defining γ = t ' = γ t G. Siolla MIT 8.0 Leture h 6 3

4 Time dilation We just deried a er important result! Gamma fator: γ = > with β β β Sine t =γ t t is alwas larger than t t = time measured b the obserer in the station who sees the lok in motion t = time measured b the obserer on the train, at rest wrt the lok Conlusion: Cloks in motion run slower (time dilation) t ' = γ t G. Siolla MIT 8.0 Leture 7 Length in different referene frames Problem Now obserer flashes a pulse of light horizontall from left end of the train The light is refleted b a mirror on the right end wall and deteted b a photosensor on the left wall What is the length of the train measured b eah obserer? G. Siolla MIT 8.0 Leture 8 4

5 Length in train referene frames For obserer in train referene frame Eents we are interested in: emission and reeption of light Time in between the two: t= t train t L Length of the train: L = t = L G. Siolla MIT 8.0 Leture 9 Length in the station referene frame Calulate separatel (L R) and (R L) t ' = ( L ' t ' ) / t ' = ( L ' + t ' ) / t is shorter beause train and light moe in opposite diretions t is longer beause train and light moe in the same diretion L (t ) = length (time) measured from station referene frame Rearrange terms: t ' = t ' = L ' L ' + G. Siolla MIT 8.0 Leture 0 5

6 Length ontration Total time in the station referene frame = sum of t and t : L ' L ' t ' = t ' + t ' = + = + L ' γ = L ' = L ' = ( ) Remember how time dilates: t =γ t L ' γ L = t ' = γ t = γ L ' = L γ Sine γ> Moing objets appear ontrated (length ontration) G. Siolla MIT 8.0 Leture Summar so far Assume Speial Relatiit postulates hold: The laws of phsis are the same for all referene frames The speed of light is the same () in all referene frames Consequenes: Time dilation loks in motion run slower t ' = γ t Length ontration L moing objets appear ontrated L ' = γ REALLY??? Can we hek this eperimentall??? G. Siolla MIT 8.0 Leture 6

7 Appliation: Cosmi Ra Muons Cosmi ra muons: Cosmi ras are energeti partiles (mainl protons) oming from somewhere in the Unierse When the hit the atmosphere the will produe showers of partiles µ are of partiular interest beause the are er penetrating and hae a long lifetime (. µs) Question: Can muons produed in the upper atmosphere reah the ground? Input: Muon s eloit = 99.99% of eloit of light Atmosphere ~0 Km thik G. Siolla MIT 8.0 Leture 3 Appliation: Cosmi Ra Muons () Inputs: = 99.99% of eloit of light, atmosphere ~ 0 Km µ Non relatiisti approah: l = t = 0.6 Km < 0 Km: NO, the annot reah the ground Relatiisti approah Relatiit: same phiss γ = /sqrt(- / ) ~ 7 in all referene frames! Approah : our perspetie τ µ =. µ s in muon s referene frame In our referene frame: τ = τ/ γ = 7. µ s = 56 µs Now muon an trael: l = 4 Km: OK! Approah : muons perspetie The l = 0 Km of atmosphere appear ontrated to a relatiisti µ l = l /γ = 0Km/7 ~ 0.3 Km that an be traeled with τ=. µs: OK! G. Siolla MIT 8.0 Leture 4 7

8 More on Cosmi Ra Muons The number of osmi muons deteted at sea leel and on the top of Mount Eerest are different. B how muh? Hpotheses: Muons are produed in the upper atmosphere: ~ 0 Km β = γ = /sqrt(- / ) ~ 7 N(t) Mount Eerest ~ 8 Km Muons dea eponentiall N(t) = N0ep(-t/τ) Choose RF and sta with it t τ µ = 56 µ s in our R.F. At sea leel: L=0Km T=66 µs N sea =N 0 ep(-66/ 56)=0.65 N0 On Mount Eerest: L=Km T=40 µs N Eerest =N 0 ep(-40/ 56)=0.77 N0 At sea leel epet ~5% less osmi µ than on Mount Eerest: OK! G. Siolla MIT 8.0 Leture 5 How do lengths perpendiular to transform? Thought eperiment Train moing towards a tunnel with eloit =0.9 Height of train in train s RF: h tra in= 3.5 m Height of tunnel in tunnel s RF: h tunnel = 4.0 m If we hae Lorentz ontrations: L =L/γ γ = /sqrt(-0.9 )=.9 In tunnel s referene frame: the train moes with β=0.9 h train = h train /γ = 3.5/.9 =.5m no problem: it will fit! In train s referene frame: tunnel moes with eloit β=0.9 h tunnel = h tunnel /γ = 4/.9 =.7m < h train the will smash! Different obserers ome to different onlusions against relatiit priniple! Lorentz ontration annot happen G. Siolla MIT 8.0 Leture 6 8

9 Lorentz transformation Time dilation and Length ontration are onsequenes of the so alled Lorentz transformation Consider inertial referene frames: O and O O is moing w.r.t. O with eloit // ais where (,,z,t) the oordinate in the O referene frame (,,z,t ) the oordinate in the O referene frame z z Lorentz transformation: Linear transformation that relates the oordinate in the R.F. Wh linear? Beause referene frames are inertial G. Siolla MIT 8.0 Leture 7 Lorentz transformation () The most general form for a l near transformation: z and do not hange beause // ignore them in the following i z z ' = A + Bt () ' = z ' = z t ' = C + Dt () Goal: alulate oeffiients A,B,C,D First requirement: O and O oerlap at t=0: At t=t =0, = =0 For O, the origin of O moes awa with eloit ' = A ( t ) (3) Substitute in () 0 = At + Bt B = - A t ' = C + Dt (4) For O, the origin of O moes awa with eloit - Substitute in (3) : ' = A ( t ) = - At. From (4): ' = t ' = ( C + Dt ) Dt ' = A ( t )(3) D= A t ' = C + At (5) G. Siolla MIT 8.0 Leture 8 9

10 Lorentz transformation (3) z z Seond requirement: Send a light pulse along the diretion at t=0 After a time t the oord nates of the light pulse are =t and =t. Substitute in (3) and use (5): i t ' = ' = A ( t ) = ( A t t ) Ct ( + At ) = At ( ) t C = A t ' = ( C + At ) = ( C t+ At ) ' = A ( t ) (3) t ' = A t (6) G. Siolla MIT 8.0 Leture 9 Lorentz transformation (4) z z Third requirement: Send a light pulse along the diretion at t=0 After a time t the oordinates of the light pulse are (=0; =t) in O; in O the total displaement is: + = (t ). Substitute (3) and (6): ' + ' = ( t ') A ( t ) + = A t Sine = 0 and = t A ( ) t + ( t ) = A t ' = γ ( t ) A = γ t ' = γ t G. Siolla MIT 8.0 Leture 0 0

11 Lorentz transformation: summar Summarizing: when O moes wrt O with eloit +// ais To go from O (at rest) to O (in motion): ' = γ ( t ) t ' = γ t z z To go from O (in motion) to O(at rest), just hange the sign of the eloit: = γ ( ' + t ') t = γ t ' + ' The other oordinates ( and z) are not affeted G. Siolla MIT 8.0 Leture Transformation of eloit z z Consequene of Lorentz transformations Obserer in motion O shoots a bullet with eloit u What is the eloit of the bullet u measured b O? d d ( γ ( ' + t ')) d ' + dt ' u = = = dt d ( γ t ' + ' ) dt ' d' + d '/ dt ' + u ' + = = u ' + '/ ' d dt + // + ais Conlusion: u u ' + u = and u ' = u ' u + G. Siolla MIT 8.0 Leture

12 Veloit not // to z z How do we sum eloit not // to the relatie motion of the R.F.? Obserer in motion O shoots a bullet with eloit u What is the eloit of the bullet u measured b O? perpendiular to d d ' d ' u = = = dt d ( γ t ' + ' ) ( ' ') γ dt + d d '/ dt ' u ' = = u ' γ ( dt ' + d ') / dt ' γ ( + ) Conlusion: u ' u u = an d u ' = u ' u γ ( + ) γ ( ) G. Siolla MIT 8.0 Leture 3 Summar and outlook Toda: Priniple of Speial Relatiit and its amazing onsequenes Length ontration and Time dilation Lorentz transformations Veloit transformation ( alwas < ) Net time: More on Relatiit: How to transform eletri fields and fores Proe that E and B are intimatel onneted G. Siolla MIT 8.0 Leture 4