# Chapter 02 Special Relativity Version 110906, 110907, 110908,

Chapter 02 Special Relativity Version 110906, 110907, 110908, 110913 General Bibliography 1) Various wikipedia, as specified 2) Thornton-Rex, Modern Physics for Scientists & Eng, as indicated Outline Galilean Transformations Names & Reference Frames The Ether River

Michelson-Morley Experiments Einstein Postulates Lorentz Transformations Position Velocity Space-Time Diagrams Relativistic Forces & Momentum Relativistic Mass Relativistic Energy CLASSICAL / GALILEAN / NEWTONIAN TRANSFORMATIONS

Galilean Transformations K frame moving with speed v K frame fixed v K & K coincided at t=0. Sketch shown at time t later. How do the position, velocity, acceleration, & time between the 2 frames compare? x y z

t vx vy v z ax ay az ma F

x' y' z' t' v x' ' vy v' z a x' ' ay

a' z ma ' F Galilean Transformations K frame moving with speed v K frame fixed v K & K coincided at t=0. Sketch shown at time t later. How do the position, velocity, acceleration, & time between the 2 frames compare?

K K K K x = x + vt x = x - vt Newtonian Principle of Relativity If Newtons laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system. This is referred to as the Newtonian principle of relativity or Galilean invariance.

Inertial Frames K and K K is at rest and K is moving with velocity Axes are parallel K and K are said to be INERTIAL COORDINATE SYSTEMS The Galilean Transformation For a point P In system K: P = (x, y, z, t) In system K: P = (x, y, z, t) P x

K K x-axis x-axis Conditions of the Galilean Transformation Parallel axes (for convenience) K has a constant relative velocity in the x-direction with respect to K x = x v t y = y z = z t = t is a Time (t) for all observers Fundamental invariant,

i.e., the same for all inertial observers speed of frame NOT speed of object Galilean Transformation Inverse Relations Step 1. Replace with . Step 2. Replace primed quantities with unprimed and unprimed with primed. x = x + v t y = y z = z t = t speed of frame

NOT speed of object Position General Galilean Transformations x x'vt y y' t t ' Velocity t and t ' are the same dx dx' v u x u x ' v dt dt dy dy ' t y t ' y

dt dt dt dt ' 1 1 dt dt inertial reference frame t and t ' are the same dx dx' v v x v' x v dt dt dy dy ' v y v' y dt dt dt dt ' 1 1

dt dt Acceleration frame K t and t ' are the same dvx dvx ' 0 a x a x ' dt dt dv y dv y ' a y a ' y dt dt

ma F frame K ma ' F Newtons Eqn of Motion the same at face-value in both reference frames Classical Reference Frames Inertial Reference Frame Non-accelerating Newtons Laws apply at face-value

Non-Inertial Reference Frame Examples: Rocket during acceleration phase Rotating merry-go-round Rotating Earth Youtube clips (part 1) Galilean/Classical Relativity Part 1 The Cassiopeia Project http://www.youtube.com/watch?v=6rl3Z9yCTn8 The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind. http://www.cassiopeiaproject.com/ To read more about the Theory of Special Relativity, you can start here: http://www.phys.unsw.edu.au/einsteinlight/ http://www.einstein-online.info/en/elementary/index.html http://en.wikipedia.org/wiki/Special_relativity

THE ETHER RIVER HISTORY OF ETHER MICHELSON-MORLEY EXPTS The Ether River v D C A Maximum speed of the boat is c meters/sec The Ether River Time t1 from A to C and back:

d Time t2 from A to D and back: t2 22 c2 v2 22 1 2 c v 1

c2 So that the difference in trip times is: 2 2 t t 2 t1 c 1 v2 2 c 1

2 1 v 2 c r ive r n ow Timeline of luminiferous aether (http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether)

4th cent BC Light propagates in air Aristole 1704 Aetheral medium for light & heat Newton 1818 aether Fresnel wave theory 1830 problems emerge, explained by aether drag, Fresnel & Stokes 1830 ~1955 mixed experimental conclusions Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg Timeline of luminiferous aether (http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether) Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg

1830 ~1955 mixed experimental conclusions 1887 1st Michelson-Morley expt doesnt find aether 1889(1895) Fitzgerald hypothesis (Lorentz) 1902-1904 Refined Michelson-Morley measurements 1905 Trouton-Rankine expt doesnt support Fitz-Loentz hypothesis 1958- nearly all measurements do not find evidence for aether Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg Michelson-Morley Expt the most famous failed experiment

Michelson-Morley: Ether River - Revisited v D v D C A C A

Measure two orientations because dont know direction of aether river Ether River - Revisited Orientation 1 d 2 2 torient1 t 2 t1 c 1 v2 c2 Orientation 2

r ive r n ow torient 2 d 2 2 t 2 t1 c 1 v2 2 c

r ive r n ow 1 2 v 1 c 2

1 2 1 v 2 c Difference in Orientations torient1 torient 2 2 1 2 c 1 v2 2

c 1 2 1 v2 1 2 2 2 c c 1 v 2 c Michelson-Morley Measurements Apollo 11

Apollo 15 v=30 km/s c=3E8 m/s ~2002 accuracy ~1 mm http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_Experiment l1+l2 to1-to2 c[to1-to2]

(m) (sec) 1887 2*1.5 1E-16 30 nm ~1903 2*30 2E-15

600 nm Earth-Moon 3.8E8 1.3E-8 3.8 m Crises with Reference Frame Xformations Cant find the Ether Maxwells Equations not Galilean Invariant Speed of Light fixed by EM constants c

1 Fitzgerald-Lorentz Hypothesis 1889 (1895) {only a partial explanation} POSTULATE: the null results are due to changes in length in the direction of travel. L 22 torient1 t 2 t1 torient 2 t 2 t1 2

c v 22 2 c v 2 2 1 L 2 v

1 2 2 c 1 v2 c2 2 2 c 1 v2 2 c

c2 1 2 v 1 c 2

1 2 1 v 2 c torient1 torient 2 0 EINSTEINs 1905 POSTULATES All laws for physics have the same functional form in any inertial reference frame Speed of Light (in vacuum) is same in any inertial reference frame. LORENTZ POSITION-TIME

TRANSFORMATIONS Lorentz Transformations K K v x K K P x x-axis x-axis

x' x vt y' y z ' z x t ' t c 1 1 2 v

c Lorentz Transformations K K v x K K P x x-axis x-axis

x x'vt ' y y' z z ' x' t t ' c 1 1 2 v

c K: 3km, 5us K: 2.6km, -1.25us Example As observed from a large asteroid, an explosion occurs at x=3000, y=500, z=-500 and t=5 us. v P A spaceship approaches at a high speed v=0.6c . The reference frames coincided at t=0, t=0 At what position does the spaceship observe the explosion to occur? Example K rear -5km, -10 us

front +5km, +10 us The reference frames coincide at x=0, x=0 & t=0, t=0 A spaceship has indicator lights which are flashed at the same time. At t=0 the lights flash. The locations of the lights are xrear=4km & xfront=+4km. K K v x-axis x-axis The spaceship is observed from the spacestation. The spaceship is observed to move at v=0.6c . At what position does the spacestation observe the lights to flash?

Example 0m, 2.3 s 6.5E8, 3.2 s The reference frames coincide at x=0, x=0 & t=0, t=0 As viewed from the Earth, a meteorite impacts the lunar surface at 3E8m and 2.5s . The impact is observed from 2 passing spaceships, one traveling to the right at 40% c and the other to the left at 40% c. Where do the 2 spaceships observe the impact to occur ? Length Contraction (Lorentz-Fitzgerald) A meter stick, lying parallel to the x-axis, is moving with speed v v

How long does the stick appear to be to a stationary observer who makes the observation of the length at t=0? xleft & tleft=0 xright & tright=0 Moving objects appear shorter Time Dihilation (distinct from the L-F) A clock, located at x=0, makes ticks at t1, t2, v What is the interval between ticks to a stationary observer,

who observes the clock to move at speed v? x1=0 & t1 x2=0 & t2 Moving clocks run slow Distorted Pictures stationary moving to the right Our brain records photographs (frames in a movie) light rays arriving at the same time.

Jump to Light Speed Distorted Pictures Lorentz Transformation - Derivation Light propagates with speed c in all inertial reference frames K K ct Spherical wavefronts in K: Spherical wavefronts in K: ct Derivation see pages 30-31

1) Let x = (x vt) so that x = (x + vt) 2) By Einsteins first postulate: 3) The wavefront along the x,x- axis must satisfy: x = ct and x = ct 4) Thus ct = (ct vt) and ct = (ct + vt) 5) Solving the first one above for t and substituting into the second... Youtube clips (part 2) Galilean/Classical Relativity

Part 2 The Cassiopeia Project http://www.youtube.com/watch?v=WgsKlSnUO0k The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind. http://www.cassiopeiaproject.com/ To read more about the Theory of Special Relativity, you can start here: http://www.phys.unsw.edu.au/einsteinlight/ http://www.einstein-online.info/en/elementary/index.html http://en.wikipedia.org/wiki/Special_relativity LORENTZ VELOCITY TRANSFORMATIONS Lorentz Velocity Transformation see page 40 x x'vt '

dx dx'vdt ' y y' z z ' dy dy ' dz dz ' x' t t ' c dx' dt dt '

c Lorentz Velocity Transformation see page 40 dx dx'vdt ' dy dy ' dz dz ' dx' dt dt ' c dx

dx'vdt ' ux dx' dt dt ' c dy dy ' uy dx' dt dt ' c

Lorentz Velocity Transformation see page 40 dx dx'vdt ' dx' dt dt ' c dx'

dt ' v dt ' dx' dt ' 1 c dt ' dy dy' uy dx'

dt dt ' c dy' dt ' dt ' dx' dt ' 1

c dt ' ux u ' x v 1 u'x c u' y

1 u ' x c Note that because of the time transformation, the y- and z-components get messed up. A spaceship traveling at 60%c shoots a proton with a muzzle speed of 99%c at an asteroid. What is the velocity of the proton as viewed from a stationary space station? MISC. LORENTZ TRANSFORMATION EXAMPLES Cosmic Ray Muon Lifetime

electron mo=9.1e-31 kg halflife = inf muon mo=207 * (mass e) halflife = 1.5e-6 sec http://landshape.org/enm/cosmic-ray-basics/ http://www.windows2universe.org/physical_science/physics/ atom_particle/cosmic_rays.html Cosmic Rays Susan Bailey Nuclear News Jan 2000, pg 32 Cosmic Ray references Cosmic Ray Muon Measurements http://www.youtube.com/watch?v=yjE5LHfqEQI

http://www.ans.org/pubs/magazines/nn/docs/2000-1-3.pdf http://pdg.lbl.gov/2011/reviews/rpp2011-rev-cosmic-rays.pdf http://hyperphysics.phy-astr.gsu.edu ashsd.afacwa.org/ radation cosmic rays Cosmic Ray Muon Lifetime Suppose muon traveling at 0.98c muon mo=207 (9.1e-31 kg) halflife = 1.5e-6 sec Q1. Classically, how far could the muon travel

during a time 1.5e-6 sec ? Q3. How far do we observe the muon to travel during that time ? Q4. How high does the muon think the mountain is? 2000 meters Q2. What do we observe the lifetime to be ? Simultaneity http://www.youtube.com/watch?v=wteiuxyqtoM http://www.youtube.com/watch?v=KYWM2oZgi4E Atomic Clock Measurements http://www.youtube.com/watch?v=cDvmN_Pw96A d

Twin Paradox Video Clip http://www.youtube.com/watch?v=A0jiY-CZ6YA Whats the correct explanation of the paradox? Reliable Discussion at http://www.phys.unsw.edu.au/einsteinlight/jw/module4_twin_paradox.htm Spacetime Diagrams Minkowski Diagrams t In SP211 course: ux x

ux t x Allowed region Forbidden region Two events plotted on a space time diagram P=(x,y,z,t) Simultaneity in a Stationary System #1 Measuring location #2 Measuring location

Watching a moving system Animated Minkowski Diagrams http://www.youtube.com/watch?v=C2VMO7pcWhg (uses Minkowski space diagrams, but with time axis pointing down, opposite from figures in textbook.) Analysis of the Twin Paradox using Minkowski Diagrams Frank sends a signal once a year. Mary sends a signal once a year. Invariant Quantities Define s2 = x2 (ct)2 Then can show s2 = x2 (ct)2 x2 (ct)2 = s2

2 s is independent of choice of reference frame. We can use s2 to discuss the ability for events to impact one another. s2 = x2 (ct)2 If s2 < 0 then x2 < (ct)2 = c2 t2 The distance between events is less than the time required for light to propagate between the spatial locations. RELATIVISTIC MOMENTUM Style 1. Sandins Development Style 2. Rex & Thortons Development The following 4 slides present Sandins treatment of momentum in Special Relativity Forces and Momentum (Sandin version) - a first look v ux , uy , uz ux px , py , pz mo uy ux , uy , uz u ' x v v

1 u 'x c u' y 1 u' y 1 u ' x c px , py , pz If want p to look like mv ;

mu y mou ' y 1 m u ' y mou ' y Then are forced to let m mo BUT Forces and Momentum (Sandin version) Ftot ma Suppose F u

dp Ftot dt d mu du dm Ftot m u dt dt dt speed const const m mo doesn' t change

dm 0 dt du du F m mo mo a ma dt dt So why are we complaining about the phrase relativistic mass? Forces and Momentum (Sandin version) Ftot ma

Suppose F || u dp Ftot dt d mu du dm Ftot m u dt dt dt speed const const dm d

d 2 1/ 2 m 1 m ... dt dt dt du dm F m u ... 3 ma dt dt

BECAUSE Newtons Eqn Motion is different depending on the direction of the force. Forces & Momentum (Sandin version) Definitions Newtons Eqn of Motion if F u Newtons Eqn of Motion if F || u PRORelativistic Mass People ANTIRelativistic Mass People m = mo

No such thing p=mu p=mu F=ma with m mo F=ma with m mo but throw in an extra 2 F=dp/dt F=dp/dt The following 9 slides present Rex & Thortons treatment of momentum in Special Relativity

2.11: Relativistic Momentum Because physicists believe that the conservation of momentum is fundamental, we begin by considering collisions where there do not exist external forces and dP/dt = Fext = 0 Relativistic Momentum Frank (fixed or stationary system) is at rest in system K holding a ball of mass m. Mary (moving system) holds a similar ball in system K that is moving in the x direction with velocity v with respect to system K. Relativistic Momentum If we use the definition of momentum, the momentum of the ball thrown by Frank is

entirely in the y direction: pFy = mu0 The change of momentum as observed by Frank is pF = pFy = 2mu0 According to Mary Mary measures the initial velocity of her own ball to be uMx = 0 and uMy = u0. K In order to determine the velocity of Marys ball as measured by Frank we use the velocity transformation equations: K Relativistic Momentum

Before the collision, the momentum of Marys ball as measured by Frank becomes K Before Before (2.42) For a perfectly elastic collision, the momentum after the collision is After K After (2.43)

The change in momentum of Marys ball according to Frank is (2.44) Relativistic Momentum The conservation of linear momentum requires the total change in momentum of the collision, pF + pM, to be zero. The addition of Equations (2.40) and (2.44) clearly does not give zero. Linear momentum is not conserved if we use the conventions for momentum from classical physics even if we use the velocity transformation equations from the special theory of relativity. There is no problem with the x direction, but there is a problem with the y direction along the direction the ball is thrown in each system.

Relativistic Momentum Rather than abandon the conservation of linear momentum, let us look for a modification of the definition of linear momentum that preserves both it and Newtons second law. To do so requires reexamining mass to conclude that: Relativistic momentum (2.48) Relativistic Momentum Some physicists like to refer to the mass in Equation (2.48) as the rest mass m0 and call the term m = m0 the relativistic mass. In this manner the classical form of momentum, m, is retained. The mass is then imagined to increase at high speeds. Most physicists prefer to keep the concept of mass as an invariant, intrinsic property of an object. We adopt this

latter approach and will use the term mass exclusively to mean rest mass. Although we may use the terms mass and rest mass synonymously, we will not use the term relativistic mass. The use of relativistic mass to often leads the student into mistakenly inserting the term into classical expressions where it does not apply. RELATIVISTIC KINETIC ENERGY The following 5 slides present Rex & Thortons treatment of kinetic energy in Special Relativity 2.12: Relativistic Kinetic Energy Newtonian KE=1/2 m u2 which came from KE = Work = Fds with F = dp/dt = m dv/dt = ma Relativistic Kinetic Energy

Start from rest and accelerate until u Integration by parts u dv uv v du Relativistic Kinetic Energy Start from rest and accelerate until u Integration by parts u dv uv v du u KE m u u m u du 0

u du 2 u 1 c 1/ 2 2 d 2

2 c 2 c 1 1/ 2 1 2 2 imits evaluate

mc 2 1 u 1 KE mu 2 mc 2 mc 2 2 c 2 mc 2 Relativistic Kinetic Energy 1 2 2 KE mu 2 mc mc

2 2 2 1 2 1 2 2 2 2 2 2 u u c u 2 c u 1 c

u c u 2 c 2 2 KE mc mc

KE 1 mc 2 2 Which reduces to the Newtonian expression for u small Comparison Relativistic and Classical Kinetic Energy Formula Relativistic Kinetic

Energy Equation (2.58) does not seem to resemble the classical result for kinetic energy, K = mu2. However, if it is correct, we expect it to reduce to the classical result for low speeds. Lets see if it does. For speeds u << c, we expand in a binomial series as follows: where we have neglected all terms of power (u/c)4 and greater, because u << c. This gives the following equation for the relativistic kinetic energy at low speeds: (2.59) which is the expected classical result. We show both the relativistic and classical kinetic energies in Figure 2.31. They diverge considerably above a velocity of 0.6c. Total Energy KE 1 mc 2 2 KE mc mc 2 2 mc KE mc 2 Tot Energy Etot mc 2 Relationship between Total Energy & Momentum p mu

Square, mult c2, convert u, use =(1-2)1/2 to subst 4 2 2 2 2 4 2 4 2 2 2 tot 2 4 2 2

2 4 p c m c m c p c E m c 2 tot E p c m c an invariant Youtube clips (part 3) Galilean/Classical Relativity Part 3 The Cassiopeia Project http://www.youtube.com/watch?v=W6o_-yTa168 The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.

http://www.cassiopeiaproject.com/ To read more about the Theory of Special Relativity, you can start here: http://www.phys.unsw.edu.au/einsteinlight/ http://www.einstein-online.info/en/elementary/index.html http://en.wikipedia.org/wiki/Special_relativity Examples Example 2.11 mc2 = 0.511 MeV m = 9.1e-31 kg |q| = 1.6e-19 Coul Electrons in a television set are accelerated by a potential difference of 25000 Volts before striking the screen. a). Calculate the speed of the electrons and

b). Determine the error in using the classical kinetic energy result. http://express.howstuffworks.com/exp-tv1.htm http://www.o-digital.com/wholesale-products/2227/2285-4/LCD-TV-LDT32-225837.html Example 2.13 A 2-GeV proton hits another 2-GeV proton in a head-on collision in order to create top quarks. http://www.fnal.gov For each of the initial protons, calculate Speed v Momentum p Rest-mass Energy

Kinetic Energy KE Total Energy Etot mc2=938 MeV Example 2.16 The helium nucleus is built from 2 protons and 2 neutrons. The binding energy is the difference in rest mass-energy of the nucleus from the total rest mass-energy of its component parts. Calculate the nuclear binding energy of helium. mHe = 4.002603 amu mp = 1.007825 amu mn = 1.008665 amu Hints: 1 amu = 1.67e-27 kg or

c2 = 931.5 MeV/amu http://www.dbxsoftware.com/helium/ Example 2.17 The molecular binding energy is called dissociation energy. It is the energy required to separate the atoms in a molecule. The dissociation energy of the NaCl molecule is 4.24 eV. Determine the fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl. mNa = 22.98976928 amu Average mCl = 35.453 amu http://www.ionizers.org/water.html Hints: 1 amu = 1.67e-27 kg or

c2 = 931.5 MeV/amu Sandin 5.30 A spaceship has a length of 100 m and a mass of 4e+9 kg as measured by the crew. When it passes us, we measure the spaceship to be 75 m long. What do we measure its momentum to be? RHIC The diameter of an gold nucleus is 14 fm. If a Au nucleus has a kinetic energy of 4000 GeV, what is the apparent thickness of the nucleus in the laboratory? Length contraction mc2=197*931.5 MeV http://www.bnl.gov/rhic/

Sandin 5.22 At the Stanford Linear Accelerator, 50 GeV electrons are produced For one of these electrons, calculate Speed v Momentum p Rest-mass Energy Kinetic Energy KE Total Energy Etot http://www.flickr.com/photos/kqedquest/3268446670/ http://www.daviddarling.info/encyclopedia/L/linear_accelerator.html mc2 = 0.511 MeV Sandin 5.25

A cosmic ray pion (rest mass 140 MeV/c2) has a momentum of 100 MeV/c. http://www.mpi-hd.mpg.de/hfm/CosmicRay/Showers.html Calculate Speed v Momentum p Rest-mass Energy Kinetic Energy KE Total Energy Etot http://www2.slac.stanford.edu/vvc/cosmicrays/cratmos.html Sandin 4.26 Spaceship A moves past us at 0.6c followed by Spaceship B in the same direction at 0.8c B

What do they measure as their relative speed of approach? What do we measure as their relative speed of approach? A Sandin 4.28 Spaceship A approaches us from the right at at 0.8c Spaceship B approaches us from the left at 0.6c B A What do they measure as their relative speed of approach? What do we measure as their relative speed of approach?

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