Announcements: Special relatiity Homework solutions are posted! Remember problem soling sessions on Tuesday from 1-3pm in G140. Homework is due on Wednesday at 1:00pm in wood cabinet in G2B90 Hendrik Lorentz (1853 1928): Today we will derie the Lorentz transformation, addition of elocities, Relatiistic Doppler Shift. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 1
Hyberbolic Sine/Cosine - HW5 Hyperbolic sine: Hyperbolic cosine: Hyperbolic tangent: http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 2
Clicker question 1 Curly runs by real fast with a stick he knows to be of length L C. Larry and Moe are at rest and each measures the stick as it goes by. How are the three measurements related? Set frequency to AD Curly measures L C A. L C < L L < L M B. L C > L L > L M C. L C = L L = L M D. L C < L L = L M E. L C > L L = L M Larry measures L L Moe measures L M http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 3
Clicker question 1 Curly runs by real fast with a stick he knows to be of length L C. Larry and Moe are at rest and each measures the stick as it goes by. How are the three measurements related? Set frequency to AD Curly measures L C A. L C < L L < L M B. L C > L L > L M C. L C = L L = L M D. L C < L L = L M E. L C > L L = L M Larry measures L L Moe measures L M Curly measures the proper length L 0. Larry and Moe measure a shorter length due to length contraction: L = L 0 /γ. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 4
Summary of time dilation and length contraction Factor of gamma always shows up: Time dilation moing clocks run slower: The rest frame time (proper time) is Δt 0 and is the time in the moing system rest frame. It is shorter than the time measured in the frame where the system is moing. Length contraction moing 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) (L 0 ). http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 5
The Lorentz transformation last lecture S S x 0 x A stick is at rest in S. Its endpoints are the eents (position, c time) = (0,0) and (x,0) in S. S is moing to the right with respect to frame S. Eent 1 left of stick passes origin of S. Its coordinates are (0,0) in S and (0,0) in S. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 6
Clicker question from Friday S x Q. In the S frame, the stick s length is x /γ by length contraction. Time t passes. According to S, where is the right end of the stick? A. x = t B. x = t C. x = t + x /γ D. x = t + x /γ E. x = t x /γ t x /γ With a little algebra we can rewrite this as x = γ(x t) This transformation gies the coordinate in the S frame if you hae the coordinate (and time) in the S frame. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 7
Transformations If S is moing with speed in the positie x direction relatie to S, then the coordinates of the same eent in the two frames is related by: In Galilean relatiity In Special relatiity In a minute Note that when 0, then γ 1 and we recoer the Galilean transformations. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 8
The Lorentz transformation S x S 0 A stick is at rest in S. Its endpoints are the eents (position, c time) = (0,0) and (x,0) in S. S is moing to the left with respect to frame S. Eent 1 left of stick passes origin of S. Its coordinates are (0,0) in S and (0,0) in S. x http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 9
Clicker question 2 x/γ Set frequency to AD S t 0 Q. In the S frame, the stick s length is x/γ by length contraction. Time t passes. According to S, where is the right end of the stick? A. x = t B. x = t C. x = t + x/γ D. x = t + x/γ E. x = t x/γ x http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 10
Clicker question 2 S t 0 x/γ Q. In the S frame, the stick s length is x/γ by length contraction. Time t passes. According to S, where is the right end of the stick? A. x = t B. x = t C. x = t + x/γ D. x = t + x/γ E. x = t x/γ http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 11 x Set frequency to AD With a little algebra we can rewrite this as x = γ(x + t ) Can also sole for t : and put in our x transformation to get: After some algebra you can get t ʹ = γ t x c 2
Recall: transformations If S is moing with speed in the positie x direction relatie to S, then the coordinates of the same eent in the two frames are related by: In Galilean relatiity In Special relatiity x = γ( x ʹ + t') y = This assumes (0,0) is the same eent in both frames. Note that is the elocity of the S frame relatie to the S frame and can be positie or negatie. ʹ y z = z ʹ t = γ( t ʹ + x ʹ c ) 2 http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 12
Clicker question 3? Gracie Set frequency to AD... -3-2 -1 0 1 2 3... George George has a set of synchronized clocks in reference frame S, as shown. Gracie is moing to the right past George, and has her own set of synchronized clocks. Gracie passes George at the eent (x,t) = (x,t ) = (0,0) in both frames. The eent is the clock at x=0 striking 3 o'clock. An obserer in George s frame checks the clock marked? at t=0. Compared to George s clocks, this one reads A. slightly before 3:00 B. slightly after 3:00 C. 3:00 D. Impossible to tell http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 13
Clicker question 3? Gracie Set frequency to AD... -3-2 -1 0 1 2 3... George Let George be the unprimed frame and Gracie be the primed frame which is moing to the right with elocity. For George, the eent is at (x,t)=( b,0) where b is positie. In Gracie s frame, the eent in question is at (x,t ) gien by: (not what we are looking for) where b>0 so t >0 A. slightly before 3:00 B. slightly after 3:00 C. 3:00 D. Impossible to tell http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 14
Velocity An object moes from eent A to eent B. B A S... -3-2 -1 0 1 2 3... S... -3-2 -1 0 1 2 3... As seen from frame S the elocity is: As seen from frame S the elocity is: http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 15
Velocity transformation Using the Lorentz transformation for x and t we get: Cancel the γ and diide top and bottom by Δt to get The elocity addition formula in special relatiity is: Galilean result New in special relatiity Note that u and u are the elocities of an object in a frame while is the elocity of one frame to another (S elocity relatie to S). http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 16
Clicker question 4 Set frequency to AD A spacecraft traels at speed =0.5c relatie to the Earth. It launches a missile in the forward direction at a speed of 0.5c. How fast is the missile moing relatie to Earth? A. 0 B. 0.25c C. 0.5c D. 0.8c E. c http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 17
Clicker question 4 Set frequency to AD A spacecraft traels at speed =0.5c relatie to the Earth. It launches a missile in the forward direction at a speed of 0.5c. How fast is the missile moing relatie to Earth? A. 0 B. 0.25c C. 0.5c D. 0.8c E. c This actually uses the inerse transformation: Hae to keep signs straight. Depends on which way you are transforming. Also, the elocities can be positie or negatie! Best way to sole these is to figure out if the speeds add or subtract and then use the appropriate formula. Since the missile if fired forward in the spacecraft frame, the spacecraft and missile elocities add in the Earth frame. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 18
Velocity addition works with light too! A Spacecraft moing at 0.5c relatie to Earth sends out a beam of light in the forward direction. What is the light elocity in the Earth frame? What about if it sends the light out in the backward direction? It works. We get the same speed of light no matter what! http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 19
Vicki Vicki stays on Earth and twin Carol departs for the star Sirius, 8 light-years away, traeling at a speed = 0.8 c (γ = 5/3). We found this journey takes 10 years each way according to Vicki (8 c years/0.8 c = 10 years). Due to time dilation, only 6 years elapse for Carol each way: As Carol goes to Sirius at 0.8c in her reference frame, the distance to Sirius is which she coers in 6 years. Since she sees the Earth receding at 0.8c she figures time is running slow there so less than 6 years pass on Earth: Twin paradox Carol http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 20
Vicki Twin paradox resolution Carol Multiplying by 2 for the return trip we would come to following conclusions: Vicki says she ages 20 years and Carol ages 12 years Carol says she ages 12 years and Vicki ages 7.2 years Both statements cannot be true so someone must be wrong! Simple answer: Vicki is in a single inertial reference frame during the entire trip. Carol is in at least 2 inertial reference frames during the trip (out and back). Therefore the problem is not symmetric. Carol s reasoning is incorrect. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 21
Twin paradox analysis using Lorentz transformations Vicki Carol Three inertial reference frames: S=Earth, S =trip out, S =return trip 3 eents: Leaing Earth (1), reaching Sirius (2), reaching Earth (3) Let T be the time to reach Sirius in Earth frame (10 years). Then T is the distance to Eent 1 (leaing) Eent 2 (Sirius) Sirius in the Earth frame. S frame will also hae x =0 and start at time t =T/γ. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 22
Twin paradox analysis using Lorentz transformations Vicki Carol In the S frame the transformations will be This is because in Earth frame we are starting at x=t and t=t so x becomes x T and t becomes t T. Also, at t=t we are at t =0. Eent 1 (leaing) Eent 2 (Sirius) Eent 3 (return) http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 23