Recall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0

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1 1/27 Day 5: Questions? Time Dilation engt Contraction PH3 Modern Pysics P11 I sometimes ask myself ow it came about tat I was te one to deelop te teory of relatiity. Te reason, I tink, is tat a normal adult neer stops to tink about problems of space and time. Tese are tings wic e as tougt about as a cild. But my intellectual deelopment was retarded, as a result of wic I began to wonder about space and time only wen I ad already grown up. - Albert Einstein Net Week: pacetime 1 elatiistic Momentum & Energy ast time: Galilean relatiity Micelson-Morley Eperiment & Postulates of Today: Time dilation, lengt contraction eminder: HW2 due, beginning of class; HW3 assigned Net week: pacetime, relatiistic momentum & energy è E=mc 2!! Eam I Tursday, Feb. 1 2 ecall from last time: Eents are recorded by local obserers wit syncronized clocks. Eent 1 (firecracker eplodes) occurs at = = and t=t = : in te train Eent : Eent : (=-3, t=3s) (=+3, t=3s) says: imultaneous! : on te platform Eent : Eent : ( =-2, t =2s) ( =+5, t =4s) says: Not simultaneous! Proper Time refers to te time measured by a clock in an inertial frame were it is at rest. Eample: Any gien clock neer moes wit respect to itself. It keeps proper time for itself in its own rest frame. Any obserer moing wit respect to tis clock sees it run slow (i.e., time interals are longer). Tis is. ame location Matematically: Eent 1: ( 1,y 1,z 1,t 1 ) Eent 2: ( 1,y 1,z 1,t 2 ) à Proper time is te sortest time tat can be recorded between two eents. Δτ = t 2 t 1 peed of ligt Comparing inertial frames An obserer and a ball are at rest in reference frame. At t =, te obserer in flases a ligt pulse to be receied at = 3 m. At Δt = 1 ns, te ligt is receied. Obserer measures a distance Δ = 3 m, so te speed of ligt in frame is: c = Δ Δt = 3m =.3m / ns 1ns is moing wit respect to at =.2 m/ns. At t =, obserer in flases a ligt pulse to be receied at = 3 m. 1

2 Ten nanoseconds later Ten nanoseconds later is moing wit respect to at =.2 m/ns. At Δt = 1 ns, te ligt is receied. In Galilean relatiity, ow far does te obserer in tink te ligt as traeled? a) 3 m b) 2 m c) 1 m d) m is moing wit respect to at =.2 m/ns. At Δt=1 ns, te ligt is receied. In Galilean relatiity, (Δt=Δt ) te obserer in would terefore measure te speed of ligt as c = Δ' Δt ' = 1m =.1m / ns 1ns U-o! U-o! If we are to beliee Einstein s postulate, ten: Anoter argument for In frame Mirror In frame Conclusion: ince we accepted Einstein's postulate of relatiity ( c is te same in all inertial frames) and we found tat, we conclude tat. I.e., time passes at different rates in te two frames of reference!! measures te time interal: Δt = 2 / c (Not a big surprise!) Anoter argument for Mirror Anoter argument for c Δt / 2 Δt / 2 Etel Note: Tis eperiment requires two obserers. (Δt ') 2 c 2 2 Etel 2 + Δt ' 2 ( ) 2 = ( c Δt ' 2) 2 ( ) = (2) 2 (Δt ') 2 = c 1 c ( ) 2 2

3 Anoter argument for c Δt / 2 Time dilation in moing frames measures: Δt Etel and : Δt = γδt, wit Δt / 2 Etel Etel and measure te time interal: But measured Δt = 2/c!! Te γ-factor can take on wat alues? < γ < < γ 1 γ < A) B) C) 1 γ < D) E) ometing else Time dilation in moing frames measures: Δt Etel and : Δt = γδt, wit Wat we found so far: imultaneity of two eents depends on te coice of te reference frame Δt = γδt Δt For time seems to run slower! ( is moing relatie to Etel and ) concludes: igt its bot ends at te same time concludes: igt its left side first. Time Dilation: Two obserers (moing relatie to eac oter) can measure different durations between two eents. measures: Wat we found so far: Δt = 2 / c Here: Δt =Δτ is te proper time Etel Etel and : Δt = γ 2 / c wit Are your clocks really syncronized? (I know mine are!) 3

4 Now and eac ae a set of clocks. s are syncronized in er frame (te train), wile s are syncronized in is frame (te tracks). How do te clocks of one frame read in anoter frame? At 3 o clock in s frame, two firecrackers go off to announce te time. It so appens tat tese firecrackers are at te left and rigt ends of te train, in s frame. Eent 1: firecracker 1 eplodes at 3: Eent 2: firecracker 2 eplodes at 3: Eent 1: firecracker 1 eplodes at 3: Eent 2: firecracker 2 eplodes at 3: Wen sees eac flas from te two firecrackers, se sees tat s clocks bot read 3: wen eac of te firecrackers went off. ometime later, te waefronts meet. Te meeting point is alfway between te firecrackers in s frame, but is somewere toward te left in te train car, in s frame. Eent 3: two ligt pulses meet, sortly after 3:. Te situation as seen by In s frame, ligt left first from te rigt end of te car. Te ligt pulses bot sow clocks reading 3: in s frame. According to s reference frame, wic of te following is true: A) s clock on te left reads a later time tan s clock on te rigt. B) s clock on te rigt reads a later time tan s clock on te left. C) Bot of s clocks read te same time. 4

5 In s frame: Important conclusion Clocks in (syncronized in ) moing to te left wit respect to If tis clock reads 3: in, ten: Tis clock reads a little after 3: in Clocks in as seen by obserer in Tis clock reads een a little later in Een toug te clocks in are syncronized (in ) te obserer in sees eac clock sowing a different time!! engt of an object Proper lengt Tis lengt, measured in te stick s rest frame, is its proper lengt. Tis stick is 3m long. I measure bot ends at te same time in my frame of reference. ame time or not doesn t actually matter ere, because te stick isn t going anywere. Proper lengt: engt of object measured in te frame were it is at rest (use a ruler) emember proper time Proper time: Time interal Δτ = t 2 t 1 between two eents measured in te frame were te two eents occur at te same spatial coordinate, i.e. a time interal tat can be measured wit one clock. engt of an object Eent 1 Origin of passes left end of stick. Obserer in measures te proper lengt of te blue object. Anoter obserer comes wizzing by at speed. Tis obserer measures te lengt of te stick, and keeps track of time. 5

6 engt of an object A little mat In frame : (rest frame of te stick) lengt of stick = (tis is te proper lengt) time between eents = Δt speed of frame is = /Δt Eent 1 Origin of passes left end of stick. Eent 2 Origin of passes rigt end of stick. In frame : lengt of stick = (tis is wat we re looking for) time between eents = Δt = Δτ speed of frame is - = - /Δt Follow te proper time! Q: a) Δt = Δt b) Δt = γ Δt c) Δt = γδt A little mat Te orentz transformation peeds are te same (bot refer to te relatie speed). And so engt in moing frame engt in stick s rest frame (proper lengt) engt contraction is a consequence of (and ice-ersa). A stick is at rest in. Its endpoints are te eents (,t) = (,) and (,) in. is moing to te rigt wit respect to frame. Eent 1 left of stick passes origin of. Its coordinates are (,) in and (,) in. orentz transformation An obserer at rest in frame sees a stick flying past im wit elocity : As iewed from, te stick s lengt is ' γ. Time t passes. According to, were is te rigt end of te stick? (Assume te left end of te stick was at te origin of at time t=.) Te orentz transformation = t + ' γ Tis relates te spatial coordinates of an eent in one frame to its coordinates in te oter. A) = γ t B) = t + ' γ C) = t + ' γ Algebra ' = γ ( t) D) = t ' γ E) ometing else 6

7 Transformations If is moing wit speed in te positie direction relatie to, ten te coordinates of te same eent in te two frames are related by: Galilean transformation (classical) orentz transformation (relatiistic) A note of caution: Te way te orentz and Galileo transformations are presented ere assumes te following: An obserer in would like to epress an eent (,y,z,t) (in is frame ) wit te coordinates of te frame ', i.e. e wants to find te corresponding eent (',y',z',t') in '. Te frame ' is moing along te -aes of te frame wit te elocity (measured relatie to ) and we assume tat te origins of bot frames oerlap at te time t=. y ' y' (,y,z,t) (',y',z',t') ' Note: Tis assumes (,,,) is te same eent in bot frames. z z' Transformations If is moing wit speed in te positie direction relatie to, ten te coordinates of te same eent in te two frames are related by:? George Galilean transformation (classical) Δ = Δ Δt Δ y = Δy Δ z = Δz Δ t = Δt orentz transformation (relatiistic) Δ = γ (Δ Δt) Δ y = Δy Δ z = Δz Δ t = γ (Δt c Δ) 2 George as a set of syncronized clocks in reference frame, as sown. is moing to te rigt past George, and as (naturally) er own set of syncronized clocks. passes George at te eent (,) in bot frames. An obserer in George s frame cecks te clock marked?. Compared to George s clocks, tis one reads A) a sligtly earlier time B) a sligtly later time C) same time? George Te eent as coordinates ( = -3, t = ) for George. In s frame, were te? clock is, te time t is, a positie quantity.? = sligtly later time 7

Spacetime Diagrams (1D in space)

Spacetime Diagrams (1D in space) PH300 Modern Phsics P11 Last time: Time dilation and length contraion Toda: pacetime ddition of elocities Lorent transformations The onl reason for time is so that eerthing doesn t happen at once. /1 Da