Today: Review of SR. Einstein s Postulates of Relativity (Abbreviated versions) Let's start with a few important concepts

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1 Today: eiew of Eam: Tomorrow, 7:30-9:00pm, DUANE GB30 You an bring paper (etter format written on both sides with whateer you think might help you during the eam. But you annot bring the tetbook or leture notes. Bring your alulators (but no PCs or ell phones please. Make use of the help sessions! Mon & Tue -5pm. What eatly did we do during the first few weeks Chapter : paetime Galileo transformation: Classial relatiity Mihelson-Morley '' is same in all inertial frames Einstein's postulates: Inompatible with Galilean relatiity ( u! Consequenes are 'time dilation' and 'length ontration' orentz transformation Veloity transformation paetime interal: Inariant under orentz transformation Chapter : elatiisti Mehanis e-definition of important physial quantities to presere onseration laws under T: - Momentum - Fore - Kineti Energy - est Energy - Total Energy Einstein s Postulates of elatiity (Abbreiated ersions et's start with a few important onepts #: Physis (inluding E&M is the same in all inertial frames. #: The speed of light is the same in all inertial frames. Eent Where something is depends on when you hek on it and on the moement of your own referene frame. Time and spae are not independent quantities; they are related by orentz Tr. Definition: An eent is a measurement of where something ours at what time. (, y, z, t Eents are not inariant under orentz transformation! In fat the T onerts the oordinates of an eent from one frame to another; suh as from : (,y,z,t to ' (',y',z',t' What does the orentz tr. do The the orentz (and Galileo transformations onert the oordinates,y,z,t of an eent (,y,z,t in a frame to the orresponding oordinates ',y',z',t' of another frame '. The way T is presented here requires the following: The frame ' is moing along the -aes of the frame with the eloity (measured relatie to and we assume that the origins of both frames oerlap at the time t=0. y y' z z' ' (,y,z,t (',y',z',t' '

2 y = y z = z = t Transformations If is moing with speed in the positie diretion relatie to, then the oordinates of the same eent in the two frames are related by: Galilean transformation (lassial = t orentz transformation (relatiisti = γ ( t y = y z = z emark: This assumes (0,0,0,0 is the same eent in both frames. Veloity transformation (a onsequene of the orentz transformation If is moing with speed in the positie diretion relatie to, and the origin of and ' oerlap at t=0, then the oordinates of the same eent in the two frames are related by: orentz transformation (relatiisti = γ ( t y = y z = z Veloity transformation (relatiisti u u = / t = u / uy y= γ ( u / uz z= γ u / ( Obserer A measures the eloity of a roket as, and a omet as u, both traeling in the same diretion. What is the speed of the omet as measured by Obserer B on the roket u a b d e ( u ( u ( u ( ( u ( u+ ( + u ( u ( + u The Proper Time between two eents is measured by loks at rest in a referene frame in whih the two eents: a Our at the same time. b Our at the same plae. Are separated by the distane a light signal an trael in the gien time interal. d Are ausally related e None of these. Proper time Proper time: Time interal t = t t between two eents (,y,z,t and (,y,z,t measured in the frame, in whih the two eents our at the same spatial oordinates, i.e. time interal that an be measured with one lok. ame spatial oordinates means: = y = y z = z A partile is found to deay µs in its own referene frame. You obsere suh a partile in the laboratory, traeling at a speed of How long does it take to deay in the lab referene frame a µs b 0.8 µs 4 µs d 4.6 µs e 4.6 ms Time dilation: t = γ t proper

3 The proper length of a rigid objet is the length of the objet measured A in any inertial frame. B by the speed of light. C in the one inertial frame in whih both ends of the objet hae the same eent oordinates. D in the frame in whih the objet is not rotating. E in the inertial frame in whih the objet is at rest. Proper length Proper length: ength of objet measured at rest / objet measured in the frame where it is at rest (use a ruler A meter stik moes at 0.99 in the diretion of its length through a laboratory. Aording to measurements in the laboratory, the meter stik length is: Proper mass ( rest mass Proper mass: Mass of an objet measured in its rest-frame. a 0 m b m 0.4 m d 0.8 m e m ength in moing frame = γ ength in stik s rest frame (proper length paetime interal ay we hae two eents: (,y,z,t and (,y,z,t. Define the spaetime interal (sort of the "distane" between two eents as: ( s ( t ( ( y ( z With: = t= t t y= y y z= z z paetime interal In HW3 you showed that the spaetime interal has the same alue in all referene frames! i.e. s is inariant under orentz transformations. Appliation of the pae-time interal Two eents our in the frame. Eent ( = -0.5s*, t =s Eent : ( = 0, t = s What s the proper time between these two eents ( s = ( t - ( = (*s (0.5* = 0.75 A 0 s Proper time: 0 B 0.5 s 0.75 C 0.5 s = (* t proper (0 t proper = 0.87 s D 0.75s E None of the aboe

4 imultaneity, time dilation & length ontration As a onsequene of Einstein's seond postulate of relatiity ('The speed of light is the same in all inertial frames of referene' we ame to interesting onlusions: - elatiity of simultaneity - Time dilation - ength ontration ome appliations of T All these effets are summarized in a set of equations: The orentz transformation Appliation: orentz transformation A(=0 B A B t= γ ( t' + ' τ 0 = 0 τ = s Two loks (one at A and one at B are synhronized. A third lok flies past A at a eloity. The moment it passes A all three loks show the same time τ 0 = 0 (iewed by obserers in A and B. ee left image. What time does the third lok show (as seen by an obserer at B at the moment it passes the lok in B The lok in B is showing τ = s at that moment. Use orentz transformation! A γ (τ -τ 0 B γ(τ -τ 0 ( / C γ(τ -τ 0 ( + / D (τ -τ 0 / γ E γ(τ -τ 0 ( + '/ Hint: Use the following frames: ' A(=0 B A B t= γ ( t' + ' τ 0 = 0 τ = s Two loks (one at A and one at B are synhronized. A third lok flies past A at a eloity. The moment it passes A all three loks show the same time τ 0 = 0 (iewed by obserers in A and B. ee left image. What time does the third lok show (as seen by an obserer at B at the moment it passes the lok in B The lok in B is showing τ = s at that moment. Use orentz transformation! A γ (τ -τ 0 B γ(τ -τ 0 ( / C γ(τ -τ 0 ( + / D (τ -τ 0 / γ E γ(τ -τ 0 ( + '/ The moing lok shows the proper time interal!! t proper = t / γ Hint: Use the following systems: A(=0 τ 0 = 0 B ' A τ = s B The lok traels from A to B with speed. Assume A is at position = 0, then B is at position = τ, τ=(τ -τ 0 Use this to substitute in the orentz transformation: τ = γ ( τ = γτ ( = τ / γ We get eatly the epression of the time dilation! uy George = γ ( t George has a set of synhronized loks in referene frame, as shown. uy is moing to the right past George, and has (naturally her own set of synhronized loks. uy passes George at the eent (0,0 in both frames. An obserer in George s frame heks the lok marked. Compared to George s loks, this one reads A a slightly earlier time B a slightly later time C same time

5 uy George = γ ( t The eent has oordinates ( = -3, t = 0 for George. In uy s frame, where the lok is, the time t is 3γ = γ (0 ( 3 =, a positie quantity. = slightly later time emember this one uy Ethel uy has a set of synhronized loks in her referene frame, as shown. Ethel is moing to the left past uy, and has her own set of synhronized loks. Ethel passes uy at the eents (='=0,t=t'=3pm. An obserer in uy s frame heks the lok marked. Compared to uy s loks (all showing 3pm at that moment, this one reads slightly after 3pm! = γ ( t Why did uy (at =0 need a helper at =3 to figure that out uy Ethel = γ ( t The eent has oordinates (=3,t=3pm in uy's frame. uy sees Ethel trael to the left with eloity (meaning that is negatie as seen by uy. Therefore, in uy s frame the obserer reads the -lok as: > ( 3γ = γ (3pm 3 = γ 3pm+ Positie Eample from preious eam A high-speed train is traeling at a eloity of = 0.5. The moment it passes oer a bridge it launhes a annon ball with a eloity of 0.4 straight up (as seen by the train ondutor. What is the eloity of the ball right after it was launhed as seen by an obserer standing on the bridge ituation seen by the onlooker on the bridge: Cannon ball right after firing the annon. reads a time after 3pm. Veloity transformation A high-speed train is traeling at a eloity of = 0.5. The moment it passes oer a bridge it launhes a annon ball straight up (as seen by the train ondutor with a eloity of 0.4. What is the eloity of the ball right after it was launhed as seen by an obserer standing on the bridge Veloity transformation = 0.5 y = } = ( + ( y = 0. 6 Attah referene frame to the train: Obserer is in frame ' traeling from right to left ( is negatie!! Now use the eloity transformation: u = u / uy y= γ u / ( y u = 0 u y = 0.4 y' ' = -0.5 ' Veloity transf. u = u / uy y= γ u / ( y u = 0 u y = 0.4 y' ' = -0.5 '

6 How fast and in whih diretion relatie to you would a soure of iolet light (400 nm hae to be traelling in order to hae you see it s emitted light as red (650 nm a 0.65 towards you b 0.65 away from you 0.45 towards you d 0.45 away form you e 0. away from you λ = λ soure obs β = = = + β β + Now let's do some relatiisti mehanis! (Chapter elatiisti momentum: elatiisti fore: Definitions: We redefined seeral physial quantities to maintain the onseration laws for momentum and energy in speial relatiity. dr p= m dt dp F= = dt proper d dt γ = u dr =γ m =γ mu dt ( γ mu elatiisti Energy: E = γm = K + m Important relations: (These relations result from the preious definitions est energy of an objet: E rest = m Kineti energy of an objet: K = (γ-m Energy momentum relation: E =(p + (m Momentum of a mass-less partile: p =E/ Veloity of a mass-less partile: u = (K: kineti energy elationship of Energy and momentum eall: Total Energy: E = γm Momentum: p = γmu Therefore: p = γ m u = γ m 4 u / u γ use: = γ p = γ m 4 m 4 =E This leads us the momentum-energy relation: or: E = (p + (m E = (p + E 0 Appliation: Massless partiles From the momentum-energy relation E = p + m 4 we obtain for mass-less partiles (i.e. m=0: E = p, (if m=0 p=γmu and E=γm p/u = E/ Using E=p leads to: u=, (if m=0 Massless partiles trael at the speed of light!! no matter what their total energy is!!

7 Do neutrinos hae a mass Neutrinos are elementary partiles. They ome in three flaors: eletron, muon, and tau neutrino (ν e,ν µ, ν τ. The standard model of partile physis predited suh partiles. The predition said that they were mass-less. The fusion reation that takes plae in the sun (H + H He produes suh ν e. The standard solar model predits the number of ν e oming from the sun. All attempts to measure this number on earth reealed only about one third of the number predited by the standard solar model. Do neutrinos hae a mass (ont. Bruno Ponteoro predited the neutrino osillation, a quantum mehanial phenomenon that allows the neutriono to hange from one flaor to another while traeling from the sun to the earth. Why would this imply that the neutrinos hae a mass Massless partiles trael at the speed of light! i.e. γ, and therefore, the time seems to be standing still for the neutrino: t Earth = γ t neutrino( proper In the HW: muon or pion eperiments. The half-lie time of the muons/pions in the lab-frame is inreased by the fator γ. A kg ball is heading towards you at a speed of 0.8. How muh momentum does it hae ompared to what it would hae in Galilean relatiity A proton (mass.6e-7 kg is traelling at How muh Kineti energy does it hae a It is the same b 5/3 times as muh p=γ mu a 6e- Joules b e-0 Joules K = (γ-m 3/5 times as muh 9e-0 Joules d None of the aboe d e-9 Joules e 4e-8 Joules How does nulear power work The total rest energy of the partile equals the sum of the rest energy of all onstituents plus the total internal energy, suh as binding energy ( E B Appliation: Nulear Fusion: M = Σ(m i E B ength ontration (Consequene of time dilation and ie ersa Isotope mass: Deuterium: u ( u kg Helium 4: u kg of Deuterium yields ~0.994 kg of Helium 4. Energy equialent of 6 grams: E 0 = m = (0.006 kg (3 0 8 m / s = J Enough to power ~0,000 Amerian households for year!

8 ength of an objet ength of an objet The length, measured in the stik s rest frame, is its proper length Eent Origin of passes left end of stik. Another obserer omes whizzing by at speed. This obserer measures the length of the stik, and keeps trak of time. ength of an objet Eent Origin of passes left end of stik. Eent Origin of passes right end of stik In frame : length of stik = (this is what we re looking for time between measurements = t (this is proper time speed of frame is - = - / t In frame : length of stik = (this is the proper length time between the two eents = t speed of frame is = / t 0 Conlusion: peeds are the same (both refer to the relatie speed. And so = = = t γ t' = γ paetime diagrams ength in moing frame ength in stik s rest frame (proper length ength ontration is a onsequene of time dilation (and ie-ersa.

9 paetime Diagrams (D in spae paetime Diagrams (D in spae In PHY 0: In PHY 30: t t = / t t t paetime Diagrams (D in spae eall: uy plays with a fire raker in the train. Ethel wathes the sene from the trak. In PHY 30: t objet moing with 0<<. Worldline of the objet objet moing with 0>>- t objet at rest at = t objet moing with = -. =0 at time t= uy Ethel Eample: uy in the train t Eample: Ethel on the traks t ight reahes both walls at the same time. ight traels to both walls Ethel onludes: ight reahes left side first. In uy s frame: Walls are at rest uy onludes: ight reahes both sides at the same time In Ethel s frame: Walls are in motion