LECTURE 4 RELATIVITY II Instructor: Shih-Chieh Hsu Galilean trans. vs Lorentz trans. 2 ¨ A reference frame, S, with coordinate system (x,y,z), time, t, and origin O. ¨ A moving frame, S’, with coordinate system (x’,y’,z’), time, t’, and origin O’. Lecture 4 3 ¨ Tipler chapter 39-3 to 39-6 ¤ Lorentz transformation/Time dilation/Length contraction ¤ The relativistic Doppler effect ¤ Clock synchronization and simultaneity ¤ The velocity transformation ¤ Relativistic momentum Lorentz transformation 4 ¨ When v is close to c, Lorentz transformation must be used since c should be the same in both frames. It is given by ! vx!$ x = γ x!+ vt! , y = y!, z = z!, t = γ t!+ ( ) # 2 & " c % 1 where γ = Lorentz factor 1−(v2 c2 ) ¨ Lorentz factor is close to 1 when v << c, and Lorentz transformation reduces to Galilean transformation. ¨ The inverse transformation is $ vx' x! = γ x − vt , y! = y, z! = z, t! = γ t − ( ) & c2 ) % ( Time dilation 5 ¨ The time between two events that happen at the same place in a reference frame is called proper time, Dtp. vx! vx! t − t = γ(t! + 2 ) −γ(t! + 1 ) 2 1 2 c2 1 c2 x2! = x1! → t2 − t1 = γ(t2! − t1") ¨ The time interval of these events measured in any other reference frame is always longer. This time expansion is called time dilation and given by Δt = γΔtp Length contraction 6 ¨ The length of an object measured in the reference frame in which the object is at rest is called proper length, Lp. ! ! x2 − x1 = γ(x2 − vt2 ) −γ(x1 − vt1) ! ! t2 = t1 → x2 − x1 = γ(x2 − x1) ¨ The length of these events measured in any other reference frame is always shorter. This phenomenon is called length contraction, and the length, L, is given by 1 v2 L = Lp = Lp 1− 2 γ c Light clock 7 ¨ Time is measured using clocks. ¨ Suppose there is a simple clock, consisting of two mirrors and a light beam. 1.5 × 108 m 8 ¨ The mirrors are 1.5 × 10 m apart, so the light makes a round trip once per second. Comparing clocks 8 ¨ Imagine that a boy and a girl both have these clocks. What does he see when she flies by? ¨ He sees her light beam traveling farther than his does between “ticks.” Moving clock runs slow 9 ¨ Since the speed of light is the same for both observers, he observes her clock running slow. distance d d speed = c = girl = boy d > d ⇒ t > t time t t girl boy girl boy girl boy ¤ In the time it takes her clock to tick off one second, his clock has ticked more than that. Time flies slower for moving objects 10 ¨ She sees her own clock as running just fine, but she sees his clock as running slow, for the same reason he saw hers that way. So who is right? They are both right. ¨ They will see the other as aging more slowly. ¤ As long as they both continue in nonaccelerated motion, they are never in the same place at the same time once they separate. ¨ Any phenomenon that varies with time will exhibit the same behavior as the clocks. ¤ the melting of ice ¤ the aging of animals ¤ the decay of unstable nuclei Example 1 11 ¨ The identical twins, Speedo and Goslo, join a migration from Earth to Planet X, 20.0 light-years away in a reference frame in which both planets are at rest. The twins, of the same age (duh), depart at the same moment on difference spacecraft. Speedo’s spacecraft travels steadily at 0.950c and Goslo’s at 0.750c. a) Calculate the age difference between the twins after Goslo’s spacecraft lands on Planet X. Assume the durations of accelerations to launch and decelerations to land on the planet are negligible. b) Which twin is older? Δt = γΔtp Example 1 12 ¨ In the pane rest frame, distance= 20.0 light-years Speedo’s spacecraft travels steadily at 0.950c and Goslo’s at 0.750c. What are ages of two when Goslo ¨ Lands. ¨a) Speedo travel time = 20 light-years/0.95c=21.05 years Goslo travl time = 20 ligh-years/0.75c=26.66 years Δt = γΔtp 2 Δt v 2 Δt = = 1− Δt = 1− 0.95 × 21.05 = 6.57 Speedo γ c2 → Δt! = 6.57+ (26.66 − 21.05) =12.18 2 Δt v 2 t 1 t 1 0.75 26.66 17.63 Goslo Δ Goslo = = − 2 Δ = − × = γ c older Time dilation and Lengh contraction 13 ¨ The time between two events that happen at the same place in a reference frame is called proper time, Dtp. time dilation Δt = γΔtp ¨ The length of an object measured in the reference frame in which the object is at rest is called proper length, Lp. 1 length contraction L = L 0 γ p Clicker Question R-8 ¨ You stand on a corner as a friend drives past in a car with constant velocity. You both note the time interval required for the car to travel one block, and you both measure the distance traveled. The proper time interval is measured by _____, and the proper distance is measured by _____. ¤ you; you ¤ you; your friend ¤ your friend; you ¤ your friend; your friend ¤ your friend; both of you Relativity!! 15 Tatsu Takeuchi Synchronized clocks 16 ¨ Since we cannot synchronize two clocks without using traveling light, Two clocks that are synchronized in one reference frame are typically not synchronized in any other frame moving relative to the first frame. ¨ A corollary to this result states: Two events that are simultaneous in one reference frame typically are not simultaneous in another frame that is moving relative to the first.* *true unless the x coordinates of two events are equal where the x axis is parallel to the relative velocity of the two frames. Simultaneity 17 ¨ We define simultaneity as follows: Two events in a reference frame are simultaneous if light signals from the events reach an observer halfway between the events at the same time. ¨ The following video explains how two events that are simultaneous in one frame are typically not simultaneous in another frame. ¤ http://www.youtube.com/watch?v=wteiuxyqtoM Stationary Observer 18 Moving Observer Frame 19 B(x,t) = (−L,0) A(x,t) = (L,0) vx x! = γ(x − vt) t! = γ(t − ) c2 vL A!(x!,t!) = (γ L,−γ ) c2 vL B!(x!,t!) = (−γ L,γ ) c2 Moving Observer 20 Chasing clock shows later time 21 ¨ If two clocks are synchronized in the frame in which they are both at rest, in a frame in which they are moving along the line through both clocks, the chasing clock leads (shows a later time) by an amount v ΔtS = Lp 2 c ¤ where Lp is the proper distance between the clocks. Question 22 ¨ Which statement is not correct. ¤ The time interval between two events is never shorter than the proper time interval between the two events. ¤ Simultaneous events must occur at the same place. ¤ If two events are not simultaneous in one frame, they cannot be simultaneous in any other frame. ¤ None above. They are all correct Relativistic Doppler effect 23 ¨ For EM waves in a vacuum, a distinction between the motions of the source and the receiver cannot be made. ¨ The observed frequency, f’, when a source and a receiver are moving towards/away from each other with speed v is 1+ v c cf 2 ( ) 0 f f approaching f ! = 1− v c ! = 0 c − v ( ) 1− v c ( ) 1− (v c) f ! = f0 receding 1+ v c ( ) ¤ where f0 is the frequency of the source in the reference frame of the source. Light Spectrum Relativistic Doppler effect 25 Relativistic Doppler shift The speed of light is the same for all inertial observers However, the wavelength and frequency change based on relative velocity For a source moving toward an observer: For a source moving away switch + and - It does not matter if it is the source or the observer that is moving; only the relative velocity matters. http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 7 Relativistic Doppler effect Application Relativistic Doppler shift 26 SinceRelativistic c is constant Doppler shift For approaching source, and Relativisticc=λf then Doppler shift λ is shorter – blueshift Since c is constant For approaching source, and c=λf then Relativistic DopplerλFor is shorter receding shift – blueshift source, In 1929Used Hubble to measure showed thevelocity velocity in of λ is longer – redshift galaxiespolice (measured and baseball using redshift) radar guns. was For receding source, In 1929 Hubble showedUsed the velocity to measure of velocity in proportional to distance. First evidence λ is longer – redshift galaxies (measured usingpolice redshift) and baseball was radar guns. for the Big Bang theory. FirstproportionalUsed evidence in Doppler to for distance. the radarBig Bang First theory.evidence for the Big Bang theory. to measure the speedUsed in Doppler radar of the air/rain. to measure the speed of the air/rain. In 1929 Hubble showed the velocity of galaxies http://www.colorado.edu/physics/phys2170/(measured using redshift) was proportional Physics to 2170 distance. – Fall 2013 10 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 10 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 11 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 11 Question 27 ¨ Yellow light emitted by an object traveling at high speed toward the earth would ¤ be shifted toward the blue end of the spectrum. ¤ be shifted toward the red end of the spectrum. ¤ undergo no spectral shifts.