1Lecture 7Reminders:Reading prior to this lecture TZD 1.11-1.13Reading for next lecture TZD 2.1-2.3Homework #2 due a few minutes ago.Homework #3 to be posted and due next WednesdayExam #1 will be Friday, September 25 in class.Today’s topic: Lorentz TransformsSummary of Time Dilation and Length ContractionFactor of gamma always shows up:2)/(11cv−=γTime dilation – moving clocks run slower:0 tt Δ=ΔγThe rest frame time (proper time) is Δt0and is the time in the moving system rest frame. It is shorter than the time measured in the frame where the system is moving. Length contraction – moving objects are shorter (in the direction of motion):The length of an object in motion (L) is less than the rest frame length (proper length) (L0).γ/0LL =Clicker questionγ0LL =0ttγΔ=ΔFor time dilation and length contraction, the subscript ‘0’does not refer to a specific frame, but rather denotes proper time and proper length.Are proper time and proper length:A) Always in the same inertial reference frameB) Always in different inertial reference framesC) Always well defined for any object and time measurementD) None of the above* Proper time and Proper length can be in the same or different frames. Sometimes, a time measurement has no frame with proper time.Luke stays on Earth and twin Leia departs for the star Sirius, 8 light-years away, traveling at a speed v = 0.8 c (γ = 5/3). We found this journey takes 10 years each way according to Luke (8 c·years/0.8 c = 10 years). Due to time dilation, only 6 years elapse for Leia each way:vLukeLeiaTwin Paradoxyears 63/5years 100==Δ=ΔγttLeia has clearly measured proper time in her rest frame. No problems.Now Leia flies back to Earth at the same speed. So she claims to measure a proper time of 12 years.However, now a single Luke on earth can measure both start and end times at the same spatial location in his frame, and he believes he is measuring proper time too!vLukeLeiaTwin Paradox ResolutionvLuke claims a proper time of 20 years, so he says Leia’sclock should read years 33 5/3years 200=×=Δ=Δ ttγLeia claims a proper time of 12 years, so she says Luke’s clock should read years 20 5/3years 120=×=Δ=Δ ttγWhile observers can measure different time and length intervals, there can be only one answer for what Leia’s clock reads (12 years or 33 years).Answer: One of the two (Luke or Leia) could not have been in an inertial reference from for this round trip, and thus they have incorrectly determined the time. Needs General Relativity.Space-TimexctA useful way to visualize things in relativity. Think of events as (x,y,z, ct) coordinates.Suppose something is moving to the right in frame S. It starts at x=0 at t=0.It moves to positive x at positive time.Connect the dots – this is the world line.2The Lorentz Transformation0 vS’SA stick is at rest in S’. Its endpoints are the events (position, c*time) = (0,0) and (x’,0) in S’. S’ is moving to the right with respect to frame S. Event 1 – left of stick passes origin of S. Its coordinates are (0,0) in S and (0,0) in S’.x’What is the proper length?vSAs viewed from S, the stick’s length is x’/γ. Time t passes. According to S, where is the right end of the stick?a) x = vt b) x = -vt c) x = vt + x’/γd) x = -vt + x’/γ e) x = vt – x’/γxClicker questionThe Lorentz TransformationvSx = vt + x’/γ . This relates the coordinates of an event in one frame to its coordinates in the other.Algebrax’ = γ(x-vt)Transformationsttzzyyvtxx=′=′=′−=′If S’ is moving with speed v in the positive x direction relative to S, then the coordinates of the same event in the two frames is related by:In Galilean relativity)()(2xcvttzzyyvtxx−=′=′=′−=′γγIn a minute…Remark: this assumes (0,0) is the same event in both frames.In Special relativityThe Lorentz Transformation 20 vS’SA stick is at rest in S. Its endpoints are the events (position, c*time) = (0,0) and (x,0) in S. S is moving to the left with respect to frame S’. Event 1 – left of stick passes origin of S’. Its coordinates are (0,0) in S and (0,0) in S’.x’As viewed from S’, the stick’s length is x/γ. Time t’ passes.According to S’, where is the right end of the stick?a) x’ = vt’ b) x’ = -vt’ c) x’ = vt’ + x/γd) x’ = -vt’ + x/γ e) x’ = vt’ – x/γ0 vS’Sx’Clicker question3The Lorentz Transformation 2x’ = -vt’ + x/γ . This relates the coordinates of an event in one frame to its coordinates in the other.Algebra0 v2vttxcγ⎛⎞′=−⎜⎟⎝⎠()xxtvvxvtxvvγγγ′′=−−=−Transformationsttzzyyvtxx=′=′=′−=′If S’ is moving with speed v in the positive x direction relative to S, then the coordinates of the same event in the two frames is related by:In Galilean relativityIn Special relativity)()(2xcvttzzyyvtxx−=′=′=′−=′γγRemark: this assumes (0,0) is the same event in both frames.2()()xxvtyyzzvtt xcγγ′′=+′=′=′′=+Transformationsttzzyyvtxx=′=′=′−=′We now have the tools to compare positions and times in different inertial reference frames. NOW we can talk about how velocities, etc. compare.:In Galilean relativityIn Special relativity)()(2xcvttzzyyvtxx−=′=′=′−=′γγ2()()xxvtyyzzvtt xcγγ′′=+′=′=′′=+Newton worked with these…but needs re-working of momentum and energy to work with