Simultaneity. CHAPTER 2 Special Theory of Relativity 2. Gedanken (Thought) experiments. The complete Lorentz Transformation. Re-evaluation of Time!

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1 CHAPTER Speial Theory of Relatiity. The Need for Aether. The Mihelson-Morley Eperiment.3 Einstein s Postulates.4 The Lorentz Transformation.5 Time Dilation and Length Contration.6 Addition of Veloities.7 Eperimental Verifiation.8 Twin Parado.9 Spae-time. Doppler Effet. Relatiisti Momentum. Relatiisti Energy.3 Computations in Modern Physis.4 Eletromagnetism and Relatiity Albert Einstein ( ) Do not worry about your diffiulties in Mathematis. I an assure you mine are still greater. Albert Einstein Young Einstein Gedanken (Thought) eperiments It was impossible to ahiee the kinds of speeds neessary to test his ideas (espeially while working in the patent offie ), so Einstein used Gedanken eperiments or Thought eperiments. Re-ealuation of Time! In Newtonian physis, we preiously assumed that t t. With synhronized loks, eents in K and K an be onsidered simultaneous. Einstein realized that eah system must hae its own obserers with their own synhronized loks and meter stiks. Eents onsidered simultaneous in K may not be in K. The omplete Lorentz Transformation t t y y z z t / Length ontration Simultaneity problems Time dilation + t y y z z t + / t Also, time may pass more slowly in some systems than in others. If <<, i.e., and γ, yielding the familiar Galilean transformation. Spae and time are now linked, and the frame eloity annot eeed. Simultaneity -L Fred Timing eents ourring in different plaes an be triky. Depending on how they re measured, different eents will be pereied in different orders by different obserers. Frank Due to the finite speed of light, the order in whih these two eents will be seen will depend on the obserer s position. The time interals will be: Fred: L/; Frank: ; Fil: +L/ Fil L Simultaneity If light is always the same speed, then obserers do not agree on when two eents are simultaneous Can only tell if lightning hit A and B (A and B ) simultaneously by getting (light) signals from eah! But this obious position-related simultaneity problem disappears if Fred and Fil hae synhronized wathes.

2 Synhronized loks in a frame It s possible to synhronize loks throughout spae in eah frame. This will preent the position-dependent simultaneity problem in the preious slide. But there will still be simultaneity problems due to eloity. So all stationary obserers in the eplosions frame measure these eents as simultaneous. What about moing ones? K Mary -L Simultaneity Compute the interal as seen by Mary using the Lorentz time transformation. L t γ t γ L [ / ] [( ) / ] γ ()( L / )! Mary eperienes the eplosion in front of her before the one behind her. And note that t is independent of Mary s position!.5: Time Dilation and Length Contration More ery interesting onsequenes of the Lorentz Transformation: Time Dilation: Cloks in K run slowly with respet to stationary loks in K. We must think about how we measure spae and time. In order to measure an objet s length in spae, we must measure its leftmost and rightmost points at the same time if it s not at rest. If it s not at rest, we must ask someone else to stop by and be there to help out. Length Contration: Lengths in K ontrat with respet to the same lengths in stationary K. In order to measure an eent s duration in time, the start and stop measurements an our at different positions, as long as the loks are synhronized. If the positions are different, we must ask someone else to stop by and be there to help out. Proper Time To measure a duration, it s best to use what s alled Proper Time. The Proper Time, T, is the time between two eents (here two eplosions) ourring at the same position (i.e., at rest) in a system as measured by a lok at that position. Time Dilation and Proper Time Frank s lok is stationary in K where two eplosions our. Mary, in moing K, is there for the first, but not the seond. Fortunately, Melinda, also in K, is there for the seond. Melinda Mary K Same loation Proper time measurements are in some sense the most fundamental measurements of a duration. But obserers in moing systems, where the eplosions positions differ, will also make suh measurements. What will they measure? Mary and Melinda are doing the best measurement that an be done. Eah is at the right plae at the right time. Frank K If Mary and Melinda are areful to time and ompare their measurements, what duration will they obsere?

3 Time Dilation Mary and Melinda measure the times for the two eplosions in system K as t and t. By the Lorentz transformation: ) T > T : the time measured ( t t ) ( )( ) T t t This is the time interal as measured in the frame K. This is not proper time due to the motion of K :. Time Dilation between two eents at different positions is greater than the time between the same eents at one position: this is time dilation. ) The eents do not our at the same spae and time oordinates in the two systems. Frank, on the other hand, reords in K with a (proper) time: T t t, so we hae: 3) System K requires lok and K requires loks for the measurement. T T 4) Beause the Lorentz transformation is symmetrial, time dilation is reiproal: obserers in K see time trael faster than for those in K. And ie ersa! Time Dilation Eample: Refletion Mirror L Let T be the roundtrip time in K T/ T/ Mirror Time Dilation Consider two obserers again In the train and on the ground How long does it take the light to go from the flashlight to the mirror and bak? For O : Mary K T L / Frank Fred K T / ( T / ) + L d t Time Dilation Consider two obserers again In the train and on the ground How long does it take the light to go from the flashlight to the mirror and bak? For O : For O: d t Time Dilation Consider two obserers again In the train and on the ground How long does it take the light to go from the flashlight to the mirror and bak? t t t t d + t d d 3

4 Refletion (ontinued) T / ( T / ) + L The time in the rest frame, K, is: T ( / ) ( T / ) + L But T L / or L T / T ( / ) ( T / ) + ( T / ) or or or T (/ ) T + T T (/ ) T + T T [ (/ ) ] T or T γ T So the eent in its rest frame (K ) ours faster than in the frame that s moing ompared to it (K). Time stops for a light wae Beause: T And, when approahes : T Proper Length When both endpoints of an objet (at rest in a gien frame) are measured in that frame, the resulting length is alled the Proper Length. For anything traeling at the speed of light: T In other words, any finite interal at rest appears infinitely long at the speed of light. We ll find that the proper length is the largest length obsered. Obserers in motion will see a ontrated objet. 4

5 Length Contration L Length ontration is also reiproal. Frank Sr., at rest in system K, measures the length of his somewhat bulging waist: L r l Proper length Now, Mary and Melinda measure it, too, making simultaneous measurements ( t t r l ) of the left, l, and the right endpoints, r Frank Sr. s measurement in terms of Mary s and Melinda s: ( r ) ( t r t ) L l + l L r γ L l where Mary s and Melinda s measured length is: L L / γ L L r l Moing objets appear thinner! Frank Sr. So Mary and Melinda see Frank Sr. as thinner than he is in his own frame. But, sine the Lorentz transformation is symmetrial, the effet is reiproal: Frank Sr. sees Mary and Melinda as thinner by a fator of γ also. Length ontration is also known as Lorentz ontration. Also, Lorentz ontration does not our for the transerse diretions, y and z. Lorentz Contration %.6: Addition of Veloities A fastmoing plane at different speeds. 8% 99% 99.9% Suppose a shuttle takes off quikly from a spae ship already traeling ery fast (both in the diretion). Imagine that the spae ship s speed is, and the shuttle s speed relatie to the spae ship is u. What will the shuttle s eloity (u) be in the rest frame? d γ ( d + dt ) Taking differentials of the Lorentz transformation [here between the rest frame (K) and dy dy the spae ship frame (K )], we dz dz an ompute the shuttle eloity in the rest frame (u d/dt): dt γ[ dt + ( ) d ] The Lorentz Veloity Transformations Defining eloities as: u d/dt, u y dy/dt, u d /dt, et., we find: d γ ( d + dt ) u + u dt γ[ dt + (/ ) d ] + u / with similar relations for u y and u z: dy dy u y u dt γ[ dt + (/ ) d ] γ (+ u / ) y dz dz u z u dt γ [ dt + (/ ) d ] γ (+ u / ) z Note the γ s in u y and u z. The Inerse Lorentz Veloity Transformations If we know the shuttle s eloity in the rest frame, we an alulate it with respet to the spae ship. This is the Lorentz eloity transformation for u, u y, and u z. This is done by swithing primed and unprimed and hanging to : d u u dt u dy uy u y dt γ ( u ) dz uz u z dt γ ( u ) 5

6 Relatiisti eloity addition plot Eample: Lorentz eloity transformation As the outlaws esape in their really fast getaway ship at 3/4, the polie follow in their pursuit ar at a mere /, firing a bullet, whose speed relatie to the gun is /3. Question: does the bullet reah its target a) aording to Galileo, b) aording to Einstein? Speed, u pg / bp /3 og 3/4 polie bullet outlaws Speed, u pg eloity of polie relatie to ground bp eloity of bullet relatie to polie og eloity of outlaws relatie to ground Galileo s addition of eloities In order to find out whether justie is met, we need to ompute the bullet's eloity relatie to the ground and ompare that with the outlaw's eloity relatie to the ground. In the Galilean transformation, we simply add the bullet s eloity to that of the polie ar: bg bp + pg bg + Therefore, > justie is sered! Einstein s addition of eloities Due to the high speeds inoled, we really must relatiistially add the polie ship s and bullet s eloities: u + + u + + bp pg bg u bp pg + bg + / ( )( ) < 4 justie is not sered! Eample: Addition of eloities We an use the addition formulas een when one of the eloities inoled is that of light. At CERN, neutral pions (π ), traeling at %, deay, emitting γ rays in opposite diretions. Sine γ rays are light, they trael at the speed of light in the pion rest frame. What will the eloities of the γ rays be in our rest frame? (Simply adding speeds yields and!) Parallel eloities: u u + u + + (/ )( + ) Anti-parallel eloities: u + u + u (/ )( ) Aether Drag In 85, Fizeau measured the degree to whih light slowed down when propagating in flowing liquids. Fizeau found eperimentally: u / n + n This so-alled aether drag was onsidered eidene for the aether onept. 6

7 Aether Drag u + / n + + n/ u + u / + ( / n)/ n + / n ( + n/ )( / n) ( + n/ / n) + n n n n + n n Armand Fizeau (89-896) Let K be the frame of the water, flowing with eloity,. We ll treat the speed of light in the medium ( u, u ) as a normal eloity in the eloity-addition equations. In the frame of the flowing water, u / n whih was what Fizeau found..7: Eperimental Verifiation of Time Dilation Cosmi Ray Muons: Muons are produed in the upper atmosphere in ollisions between ultra-high energy partiles and air-moleule nulei. But they deay (lifetime.5 µs) on their way to the earth s surfae: ( ) t τ N t N No relatiisti orretion With relatiisti orretion Top of the atmosphere Now time dilation says that muons will lie longer in the earth s frame, that is, τ will inrease if is large. And their aerage eloity is.98! Deteting muons to see time dilation It takes 6.8 ms for the -m path at.98, about 4.5 times the muon lifetime. So, without time dilation, we epet only muons at sea leel. Sine.98 yields γ 5, instead of moing 6 m on aerage, they trael 3 m in the Earth s frame. In fat, we see 54, in agreement with relatiity! And how does it look to the muon? Lorentz ontration shortens the distane!.8: The Twin Parado The Set-up Mary and Frank are twins. Mary, an astronaut, leaes on a trip many lightyears (ly) from the Earth at great speed and returns; Frank deides to remain safely on Earth. The Problem Frank knows that Mary s loks measuring her age must run slow, so she will return younger than he. Howeer, Mary (who also knows about time dilation) laims that Frank is also moing relatie to her, and so his loks must run slow. The Parado Who, in fat, is younger upon Mary s return? 7

8 The Twin-Parado Resolution Frank s lok is in an inertial system during the entire trip. But Mary s lok is not. As long as Mary is traeling at onstant speed away from Frank, both of them an argue that the other twin is aging less rapidly. But when Mary slows down to turn around, she leaes her original inertial system and eentually returns in a ompletely different inertial system. Mary s laim is no longer alid, beause she doesn t remain in the same inertial system. Frank does, howeer, and Mary ages less than Frank. t There hae been many rigorous tests of the Lorentz transformation and Speial Relatiity. Partile Auray Eletrons -3 Neutrons -3 Protons -7 Quantum Eletrodynamis also depends on Lorentz symmetry, and it has been tested to part in..9: Spae-time When desribing eents in relatiity, it s onenient to represent eents with a spae-time diagram. In this diagram, one spatial oordinate, speifies position, and instead of time t, t is used as the other oordinate so that both oordinates will hae dimensions of length. Spae-time diagrams were first used by H. Minkowski in 98 and are often alled Minkowski diagrams. Paths in Minkowski spae-time are alled world-lines. Partiular Worldlines Slope of worldline is /. Stationary obserers lie on ertial lines. A light wae has a 45º slope. 8

9 Worldlines and Time Moing Cloks Obserers at and. see what s happening at 3 at t simultaneously. Alternatiely, an eent ourring at 3 an be used to synhronize loks at and. Obserers in a frame moing at eloity,, will see the eent happening at 3 at t at different times. The Light Cone The past, present, and future are easily identified in spae-time diagrams. And if we add another spatial dimension, these regions beome ones. Spae-time Interal and Metri Reall that, sine all obserers see the same speed of light, all obserers, regardless of their eloities, must see spherial wae fronts. s + y + z t ( ) + (y ) + (z ) (t ) (s ) z y This interal an be written in terms of the spae-time metri: [ ] s y z t y z t Spae-time Inariants The quantity Δs between two eents is inariant (the same) in any inertial frame. Δs is known as the spae-time interal between two eents. There are three possibilities for Δs : Δs : Δ Δt, and the two eents an be onneted only by a light signal. The eents are said to hae a light-like separation. Δs > : Δ > Δt, and no signal an trael fast enough to onnet the two eents. The eents are not ausally onneted and are said to hae a spae-like separation. Δs < : Δ < Δt, and the two eents an be ausally onneted. The interal is said to be time-like. 9

10 w w w w w w + w w w + w w w w w + w w + w w w w w w w w P w ( ) P w w (w ) P (, w ), w P (, w ), w w w Shemati Representation of the Lorentz Transformation Frame F t Frame F t Frame F t t Frame F L t t L Length ontration L<L Rod at rest in F. Measurement in F at fied time t, along a line parallel to -ais Time dilatation: t< t Clok at rest in F. Time differene in F from line parallel to -ais Q R R Q