Lecture 4 Relativity Ii

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LECTURE 4 RELATIVITY II

Instructor: Shih-Chieh Hsu Galilean trans. vs Lorentz trans.

¨ 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

¨ 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

¨ 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

¨ 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

¨ 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

¨ 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

¨ 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

¨ 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

¨ 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

¨ 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!!

Tatsu Takeuchi Synchronized clocks

¨ 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

¨ 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

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

¨ 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

¨ 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

¨ 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

¨ 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. Question

¨ You are looking into the past when you look at ¤ Distant stars ¤ Distant planes ¤ Your image in the mirror ¤ All of these Clicker Question R-13

¨ The Lorentz transformation for y and z is the same as the classical result: y = y' and z = z'. The relativistic velocity

transformation gives uy = u'y and uz = u'z. ¤ True ¤ False Lorentz Transformation

30 S’ system S system

! vx!$ t = γ #t!+ & t! " c2 %  u' x! x = γ (x!+ vt!) y! y = y! (t’,x’,y’,z’) z! z = z! 

v dx u!x + v dx! u = = u! = x dt v x 1 u dt! + 2 !x c dy ! dy u!y u!y = u = = S’ system moves with respect to S-system dt! y dt " v % γ $1+ u!x ' # c2 & in positive direction x with velocity v; their dz! u! = dz u! z u = = z origins coinciding at t=0 dt! z dt " v % γ $1+ u!x ' # c2 & Example: Velocity Addition

A Spacecraft moving at 0.5c relative to Earth launches a rocket in the forward direction with speed 0.75c relative to the spacecraft. What is the velocity of rocket relative to the Earth frame? =0.5c =0.75c

u! + v 0.75c + 0.5c u = x = = 0.909c x v 0.5c 1+ u! 1+ 0.75c c2 x c2 Relativistic momentum

¨ The relativistic momentum of a particle is defined to have the following properties: 1. In collisions, relativistic momentum is conserved. 2. When the speed of the particle is not comparable to the speed of light, the relativistic momentum approach the classical momentum.

¨ The relativistic momentum is defined as   mu  p = = γ mu u2 1− 2 c ¤ where m is the mass of the particle, and u is the velocity of the particle. Clicker Question R-14

¨ Particle A has half the mass but twice the speed of particle B. If the particle’s momenta are p and Clickerp , then question 2 Set frequency to AD A B

¤ pA > pB

¤ pA = pB

¤ pA < pB Clicker question 2 Set frequency to AD A B

A B

Particle A has half the mass but twice the speed of particle B.

If the particles’ momenta are pA and pB, then Particle A has half the mass but twice the speed of particle B. Classically, both particles If the particles’ momenta are pA and pB, then A. pA > pB have the same momentum. B. pA = pB A. p > p Classically, both particles A B C. pA < pB have the same momentum. γu is bigger for the faster particle. B. pA = pB

C. pA < pB γu is bigger for the faster particle.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 17 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 17 Total relativistic energy

¨ The total relativistic energy is defined as  2  mu  2 mc 2 p = = γ mu E = K + mc = = γ mc u2 2 2 1− 1− u c c2 ( ) ¤ The work done by a net force on a particle at rest increases the energy from the rest energy to the total relativistic energy.

¨ Comparing the relativistic momentum expression, we have a new useful expression: u pc = c E ¨ Energies in atomic and nuclear physics are often expressed in eV or MeV, 1eV = 1.602 × 10-19 J, and masses of atomic particles are often given in the units of eV/c2 or MeV/c2. Energy, momentum, and rest mass

¨ In experiments, often momentum and energy, rather than velocity of particles are measured. So, combine the expressions for energy and momentum, and eliminate u. mc2 mu E = p = 1− u2 c2 1− u2 c2 ( ) ( ) ¨ This yields a useful expression relating energy, momentum, and rest energy. 2 E 2 = p2c2 + mc2 ( ) Clicker Question R-15

¨ A particle of mass M moving at speed u << c has approximately what total relativistic energy? ¤ Mc2 ¤ Mu2/2 ¤ cMu ¤ Mc2/2 ¤ cMu/2 Set frequency to AD ClickerClicker question question 3 3 Set frequency to AD WhichWhich of of the the graphs graphs below below is a is possible a possible representation representation ofof the the total total energy energy of ofa particlea particle versus versus its speed its speed E E EE A.A. B. B. ClickerClickerClicker question question 3Question 3 SetSet frequency frequency R-16 to to AD AD

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u u u u u u u c u c c c c c c c E. None of above EE. .None None of of them them E E E E

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http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 20 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 20 Velocity transformation

¨ The complete relativistic transformation is

u! + v u! u = x , u = y or z x v y or z # v & 1+ u! γ 1+ u! c2 x % c2 x ( $ ' ¨ The inverse velocity transformation is

u − v u u! = x , u! = y or z x v y or z $ v ' 1− u γ 1− u c2 x & c2 x ) % (

¨ These equations reduce to Galilean transformation when v << c. Example:VelocityVelocity Velocity additionaddition Addition worksworks withwithof light lightlight too!too! 39 Velocity addition works with light too! AA SpacecraftSpacecraft movingmoving atat 0.5c0.5c relativerelative A Spacecraft moving at 0.5c relative A Spacecrafttoto EarthEarth moving sendssends at 0.5c outout relativeaa beambeam to ofof Earth lightlight sends inin out a to beamEarth ofsendsthe thelight forwardforward inout the a forwardbeam direction.direction. of direction. light WhatWhat in What isis thethe is the light velocity the forwardlightlight direction. velocityvelocity Whatinin thethe is EarthEarth the frame?frame? lightin thevelocity Earth inframe? the Earth frame?

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http://www.colorado.edu/physics/phys2170/http://www.colorado.edu/physics/phys2170/ PhysicsPhysics 21702170 –– FallFall 20132013 44 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 4 Relativistic momentum

¨ The relativistic momentum is defined as   mu  p = = γ mu u2 1− 2 c ¤ where m is the mass of the particle, and u is the velocity of the particle. Total relativistic energy

¨ The first term of the relativistic kinetic energy expression depends on the speed of the particle, but the second term does not. mc2 K = − mc2 = (γ − 1)mc2 1− u2 c2 ( ) 2 ¨ The quantity mc is called the rest energy, E0, of a particle. E mc2 0 =

¤ This shows that mass is a form of energy! Energy, momentum, and rest mass

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45 A space repair

During repair of the Hubble Space Telescope, an astronaut replaces a damaged solar panel during a spacewalk. Pushing the detached panel away into space, she is propelled in the opposite direction. The astronaut’s mass is 60 kg and the panel’s mass is 80 kg. Both the astronaut and the panel initially are at rest relative to the telescope. The astronaut then gives the panel a shove. After the shove it is moving at 0.30 m/s relative to the telescope. What is her subsequent velocity relative to the telescope? (During this operation the astronaut is tethered to the ship; for our calculations assume that the tether remains slack.) A space repair (solution)

47 Conservation of Kinetic Energy

Elastic collisions In elastic collisions, the kinetic energy of the system is the same before and after the collision. Elastic collision of Two Blocks

A 4.0-kg block moving to the right at 6.0 m/s undergoes an elastic head-on collision with a 2.0-kg block moving to the right at 3.0 m/s

HINT Conservation of momentum and the equality of the initial and final kinetic energies (expressed as a reversal of relative velocities) give two equations for the two unknown final velocities. Collisions

50 3 ¨ Two lumps of clay, each of mass m , collide head-on at c . They stick together. What is the mass M of the final 5 composite lump? m m

¨ Energy and momentum are conserved.

   Initial Final p = p1 + p2 = 0; 2 2 2   2 2mc 2mc 2mc 5 2 (E1, p1) (E, p) E = E1 + E2 = 2γmc = = = = ⋅ 2mc  1− 0.62 .64 0.8 4 (E2, p2 ) 2 2 2 2 2 5 2 2   (Mc ) = E − (pc) = E → M = ⋅ 2mc > 2mc p1 = −p2 4

€ 51

Backup Twin paradox

¨ Homer and Ulysses are twins. Ulysses travels at high speed, v, to Planet P and returns.

¨ We learned that moving clock runs slow. Homer sees Ulysses moving relative to Homer, but Ulysses sees Homer moving relative to Ulysses. So, when Ulysses returns, who is younger? Twin paradox 2

¨ Ulysses is younger than Homer.

¨ The situation is actually not symmetrical. ¤ The relativity of time only applies to unaccelerated motion. ¤ In order for Ulysses to go away and come back, he must accelerate. ¤ So, Ulysses did not stay in one inertial reference frame while Homer did. Understanding twin paradox with Doppler effect

¨ Ulysses leaves Earth on 1/1/2014 with a relative speed of 0.6c and turns around on 1/1/2019 (according to Homer in the Earth).

¨ The twins send yearly greeting on New Year’s Day according to their own calendars and clocks.

¨ During the outbound trip with v = 0.6c, the greetings reach the receiver with a frequency of 1− (v c) 1− (0.6c c) f f 1 greeting 1 year ! = 0 = ( ) 1+ (v c) 1+ (0.6c c) 1 = (1 greeting 1 year) 2 Understanding twin paradox with Doppler effect 2

¨ During the inbound trip, with v = -0.6c, the greetings reach the receiver with a frequency of 1+ (v c) 1+ (−0.6c c) f f 1 greeting 1 year ! = 0 = ( ) 1− (v c) 1− (−0.6c c) = 2 1 greeting 1 year ( )

¨ These relativistic Doppler shifts are symmetrical. ¤ It does not matter which twin does the transmitting, and which one does the receiving - it only depends on the relative velocity between them. Ulysses frame

¨ The distance between Earth and Plane is (5 yearx0.6c)= 3 light- year measured by Homer. ¨ Due to length contraction, Ulysses only travels

3 2 L = = 3 1− 0.6c c = 2.4light − year γ ( ) ¨ Ulysses travel time in each way is 2.4 ligh-year/0.6c = 4 years. ¨ Outbound: Ulysses receives 4yearsx(1/2 cards/year) = 2 cards ¨ Inbound: Ulysses receives 4yearsx(2 cards/year) = 8 cards ¨ Ulysses receives 10 cards In 8 years from Homer Homer frame

¨ Outbound: ¤ In Homer’s frame, Ulysses needs 5 years travel time to reach planet P ¤ The last signal sent by Ulysses before it returns needs 3 years to reach Homer. ¤ Homer receives (5+3 years) x (1/2 greetings/year) = 4 cards from Ulysses

¨ Inbound: ¤ In the rest of trip, Homer receives 2 years x (2 greetings/year)= 4 greetings from Ulysses

¨ Summary ¤ Homer receives 8 cards in 10 years from Ulysses Question

When you return from high-speed space travel, you find your stay- at-home friends ¤ From your friends’ frame, you are moving and time runs slower. Your friend is older. ¤ From your frame, your friend is moving in opposite direction and time runs slower. Your friend will be younger. ¤ Both a and b are correct. Therefor, no older or no younger How to Give Talks

Primary objectives of a talk: ●For the audience to learn something about a topic ●For the audience to learn about what you have done ●NOT for you to show off how much you know... The audience will be far more impressed if you can teach them something – and thus prove you understand it well enough to explain it. Keys to Give Effective Talks

An effective presentation achieves: ●Simplicity − Who are your audience? − What do they want to know? ●Brevity − What can you reasonably cover in the given time? − BE REALISTIC!! − Avoid repetition – Say just enough to make your point ●Clarity − Use plots/figures where possible − Don't clutter slides − Topics should be structured and well ordered Slides

●Slides are for emphasizing and supporting ideas presented verbally – not vice versa ●Diagrams, figures, and pictures when possible! ●Use text to stress key points... ●...but don't fill a page with text. No-one will read it. ●Ditto equations, unless necessary or enlightening. ●Avoidridiculous fontsandgratuitous animations at all costs! Effective

62 Plots and Figures

●Make plot large enough for whole audience to see ● Especially plot labels! ●Generally: one full-page plot per slide ● Exception: side-by-side comparison (if needed) ●Should be easy to read and visually appealing: bold lines, and contrasting colours ●Include sources where appropriate Contents

These are progress reports – experiments in progress – but with full physics discussion. In your talk you want to cover: ● An introduction to the topic ● Background, motivation, applications ● Theory – but don't go overboard ● What are you measuring, and how? ● Experimental technique, with apparatus diagrams ● Results and analysis so far ● Current challenges and future plans Delivery

●Don't read slides like a script − Slides are visual aids. Not cue cards or lecture notes ●Do speak clearly and loud enough ●Don't rush. Rule of thumb: ~1 slide/min. ●Do stand next to screen, facing audience ●Do use pointer – but don't wave it around Delivery

Do rehearse!!!

Delivery ● Know how you will say it, not just what ● Determine your timing ● Memorize your opening and closing lines, and any tricky bits Deadline

● Sep 4 – Draft report (5min) ● Sep 17 – Seminar report (15min) Lecture 4

¨ Tipler chapter 39-3 to 39-6

¤ Lorentz transformation/Time dilation/Length contraction

¤ The relativistic Doppler effect

¤ Clock synchronization and simultaneity

¤ Twin paradox

¤ The velocity transformation

¤ Relativistic momentum