Kinetic Energy

Relative Motion

Red Car sees Blue Car travel at U Blue Car throws Ball at V' What is V – the speed of the Ball seen by Red Car? Newton’s Principia in 1687.

Galilean Relativity

Velocities add: V= U +V'

V’= 25m/s

V = 45m/s U = 20 m/s TheThe GalileanTransformationsGalileanTransformations

Consider two reference frames S and S'. The coordinate axes in S are x, y, z and those in S' are x', y', z'. Reference frame S' moves with velocity v relative to S along the x-axis. Equivalently, S moves with velocity −v relative to S'. The Galilean transformations of position are:

The Galilean transformations of velocity are: What if instead of a ball, it is a light wave? Do velocities add according to Galilean Relativity? Clocks slow down and rulers shrink in order to keep the speed of light the same for all observers! Time is Relative! Space is Relative! Only the SPEED OF LIGHT is Absolute! James Clerk Maxwell 1860s Light is wave. The medium is the Ether.

1 cxm==3.0 108 / s με0 o Measure the Speed of the Ether Wind Light should travel faster with the wind and slower against it during the year. Michelson-Morely Experiment 1887 The speed of light is independent of the motion and is always c. The speed of the Ether wind is zero. Lorentz Contraction The apparatus shrinks by a factor :

1/− vc22 On the Electrodynamics of Moving Bodies 1905 EinsteinEinstein’’ss PrinciplePrinciple ofof RelativityRelativity

• Maxwell’s equations are true in all inertial reference frames. • Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3.00 × 108 m/s =300,000km/s = 300m/µs . • Therefore, light travels at speed c in all inertial reference frames. Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other. Special Theory: Inertial Frames: Frames do not accelerate relative to eachother; Flat Spacetime – No Gravity.

General Theory: Noninertial Frames: Frames accelerate: Curved Spacetime, Gravity & acceleration. Postulates of Special Relativity 1905

1. The laws of physics are the same in all inertial reference frames.

2. The speed of light in a vacuum is constant in all inertial reference frames, independent of the relative motion of source and observer. We assume a vacuum for this class. 3. Months later……E = mc2 Slower than Light?

Light slows down in a medium like glass as a function of wavelength. Who slows down more, red or blue in glass? Faster than Light?

Cherenkov radiation is electromagnetic radiation emitted when a charged particle passes through an insulator at a speed greater than the speed of light in that medium. The characteristic "blue glow" of nuclear reactors is due to Cherenkov radiation. Light travels at c for all inertial frames Light takes time to travel: Synchronizing Clocks

All clocks in the same frame have the same reading but you have to take into consideration the time it takes for light to travel. Relativity of Simultaneity •Two events occurring simultaneously in one reference frame are not simultaneous in any reference frame moving relative to it. Neither reference frame is ‘correct’: There are no preferred or special reference frames. It’s all relative baby! • Any two observers at rest in the same frame MUST measure events to be simultaneous even if they don’t SEE it that way. WB 37.4 # 11 & 12

Seeing is NOT the same as Being! The time when an event is SEEN is not necessarily when the event actually happens! Example 37.5 on page 1156 Add problem #49 to this weekend’s HW due Tuesday. Δt,v Space Ship 0 Speed = v

Δt, v Time Dilation

Spaceship Frame

Earth Frame: Time Dilation

Δt ' Δ=tt =Δγ ' v 2 1− c2

1 γγ=≥, 1 v 2 1− c2 Δt ' Δ=tt =Δγ ' v 2 1− c2

The smallest (proper) time interval Δt0 between two events is measured in the reference frame in which both events occur at the same place (space ship). In another frame moving at speed v with respect to the first, the measured time interval is increased by a factor of gamma (Earth frame):

Earth Lab Frame: Dilated Time: Δtt=Δγ ' Space Ship Frame: Proper Time: Δt ' TimeTime DilationDilation The time interval between two events that occur at the same position is called the proper time Δτ. In an inertial reference frame moving with velocity v = βc relative to the proper time frame, the time interval between the two events is

The “stretching out” of the time interval is called time dilation. Time Dilation – Generalization • If a clock is moving with respect to you, the time interval between ticks of the moving clock is observed to be longer that the time interval between ticks of an identical clock in your reference frame • All physical processes are measured to slow down when these processes occur in a frame moving with respect to the observer – These processes can be chemical and biological as well as physical The faster you move, the longer you liver – relative to people at rest! Relative to yourself, you age at a normal rate! 1 γγ=≥, 1 v v 2 1− ββ= , < 1 c2 c Time Dilation – Verification • Time dilation is a very real phenomenon that has been verified by various experiments • These experiments include: – Airplane flights – Muon decay –Twin Paradox Time Dilation Verification – Muon Decays

• Muons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron

• Muons have a half-life of Δtp = 2.2 µs when measured in a reference frame at rest with respect to them (a) • Relative to an observer on the Earth, muons should have a lifetime of

γ Δtp (b) • A CERN experiment measured lifetimes in agreement with the predictions of relativity Sample Problem

Beta Diva

As you approach c, lengths contract. Length Contraction The proper length of an object is longest in the reference frame in which it is at rest. In another frame moving parallel to the object, its length is shortened by a factor of gamma.

v 2 LL' ==−/1γ L PPc2

Ruler (Proper) Frame Earth Frame Measures length LP Measures length L LengthLength ContractionContraction The distance L between two objects, or two points on one object, measured in the reference frame S in which the objects are at rest is called the proper length ℓ. The distance L' in a reference frame S' is

NOTE: Length contraction does not tell us how an object would look. The visual appearance of an object is determined by light waves that arrive simultaneously at the eye. Length and length contraction are concerned only with the actual length of the object at one instant of time. Sample Problem

Beta Diva Length Contraction: Distances

v 2 LL' ==−/1γ L PPc2

Proper Length of Ship

Rest (Proper) Frame Distance Moving Frame Distance

Earth Frame measures L0 Rocket Frame measures L Length Contraction Paradox? Twin Paradox? The Spacetime Interval has the same value in all inertial reference frames. The spacetime interval is invariant. It is a value that all experimenters, in all reference frames agree on. It is analagous to the invariance of distance in geometry: the distance between two points is independent of the coordinates system. The interval has the units of meters. Addition of Velocities Galilean Relativity Addition of Velocities vu+ ' .5cc+ u = = = c vu ' .5cc 1+ 1+ c2 c2

If v = .5c TheThe LorentzLorentz VelocityVelocity TransformationsTransformations Consider two reference frames S and S'. An object moves at velocity u along the x-axis as measured in S, and at velocity u' as measured in S' . Reference frame S' moves with velocity v relative to S, also along the x-axis. The Lorentz velocity transformations are:

NOTE: It is important to distinguish carefully between v, which is the relative velocity between two reference frames, and u and u' which are the velocities of an object as measured in the two different reference frames. TheThe LorentzLorentz TransformationsTransformations

Consider two reference frames S and S'. An event occurs at coordinates x, y, z, t as measured in S, and the same event occurs at x', y', z', t' as measured in S' . Reference frame S' moves with velocity v relative to S, along the x-axis. The Lorentz transformations for the coordinates of one event are: Lorentz Transformation Equations

Sxyxt:( , , , ) S ':(xyxt ', ', ', ')

Transform coordinates from S to S’ : ⎛⎞v x''''=−γ ()xvty = y z = zt =−γ ⎜⎟tx2 ⎝⎠c Transform coordinates from S’ to S : ⎛⎞v x =+γ ()xvty'' ' = yz ' ==+ ztγ ⎜⎟tx '2 ' ⎝⎠c Lorentz Transformations, Pairs of Events

The Lorentz transformations can be written in a form suitable for describing pairs of events For S to S’ For S’ to S Δ=x' γ ( Δ−Δxvt) Δ=x γ ( Δ+Δxvt'') ⎛⎞v ⎛⎞v Δ=t γ Δ+tx'' Δ Δ=t' γ ⎜⎟ Δ−tx2 Δ ⎜⎟2 ⎝⎠c ⎝⎠c Problem

Superwoman can travel at 0.75c in her glass space ship. She has to fly to Beta Diva, 25.0 light years away as measured from Earth in the Earth frame, to battle alien evil guys. a) What is the total time for the trip for Superwoman? b) What is the total time you measure on Earth? c) How far is Beta Diva as measured by Superwoman? d) As Superwoman leaves Earth, you measure her length as she flies overhead at 0.75c. What is her length you measure? At rest she measures 2.75 m tall. e) If she launches a space pod to Beta Diva while en route at .4c, what is the speed at which it approaches Beta Diva? vu+ ' u = Beta Diva vu ' 1+ c2 RelativisticRelativistic EnergyEnergy The total energy E of a particle is

This total energy consists of a rest energy

and a relativistic expression for the kinetic energy

This expression for the kinetic energy is very nearly ½mu2 when u << c. Kinetic Energy

The kinetic energy is the total energy less the rest energy: 2 KEEE=−00 =−(1γ ) mc Relativistic Momenutm & Mass Imagine you are at rest and a bullet shoots past approaching c. What is the momentum of the particle relative to you? p = γ mu The speed is bounded, is the momentum? NO! As the bullet’s speed increases, the momentum and mass increase: “Relativistic” Mass: mm= γ 0

2 rest energy: E00= mc 2 total energy E = γ mc0 mm= γ 0 Note: We don’t use this notation! Total Energy 2 2 E =γmc E0 = mc pmu= γ uu pcmucmc==γ γγβ22 =() mcE = pcE= β cc

22 2 222mc u 22u 222222 E =γmc = →E(1 −22 ) = ( mc ) → E − E = E00 → E =β E + E u2 cc 1− c2 222 E =+()pc E0

22222 If m = 0 (photon) E =+()pc ( m0 c ) = () pc photon momentum: pEc= / If a photon has momentum, does it have mass? Invariant Mass Invariant mass is independent of frame of reference and is the mass as measured in the proper frame:

2 Rest Energy: E00= mc

2 Invariant mass: mEc00= /

If a mass is moving, than the total energy increases by a gamma factor: Total Energy: E =γmc2 Principle of Invariance!

The Speed of Light! You Try: Sample Problem

Superwoman can travel at 0.75c in her glass space ship. She has to fly to Beta Diva, 25.0 light years away as measured from Earth in the Earth frame, to battle alien evil guys. a) If her rest mass is 65 kg, what is her rest Energy? b) What is her total relativistic Energy you measure? What does she measure? Is it different? c) What is her kinetic energy? 2 rest energy: E00= mc 2 total energy E = γ mc0 2 KE=− E E00 =(1)γ − m c

Beta Diva E = mc2 ConservationConservation ofof EnergyEnergy The creation and annihilation of particles with mass, processes strictly forbidden in Newtonian mechanics, are vivid proof that neither mass nor the Newtonian definition of energy is conserved. Even so, the total energy—the kinetic energy and the energy equivalent of mass—remains a conserved quantity.

Mass and energy are not the same thing, but they are equivalent in the sense that mass can be transformed into energy and energy can be transformed into mass as long as the total energy is conserved. Proof: E = mc2

Pure energy converted into Irène and Frédéric Joliot-Curie particles: pair production 1933 Creating Matter From Pure Energy Matter-AntiMatter E = Δmc2 Δm=E/c2 Mass is bound energy. c2 is the conversion factor – the magnitude is 90 quadrillion joules per kilogram. Even a speck of matter with a mass of only 1 milligram has a rest energy of 90 billion Joules!!! cxJkg21= 9106 /

1 GeV/c2- = 1.783 × 1027 kg

The rest mass of a proton is .938 GeV/c2 Electron Volts 1 eV is the energy an electron gains across a 1 volt potential. E = qV

1(1.610)(1/)eV= x−19 C J C 1eV= 1.6 x 10−19 J OR 1 J ~ 1019 eV The Tevatron: 1 Trillion (Tera) Electron Volts E = Δmc2

Solar Flux: Ρ=3.77xW 1026

dm = 4.19xkgs 109 / dt Energy Released: The Mass Defect Atomic Decay: Parent atoms have more mass than product atoms. The difference is released in the form of Kinetic energy. E = Δmc2

Δ=mm( parents − m products ) Atomic Mass Units 1u = 1/12 mass of Carbon-12 1uxk== 1.6605 10−27 g 931.5MeV / c2

238.0508u 234.0436u 4.0026u Δ=mu()238.0508 − 234.0436 − 4.0026 = 0.0046u E =Δmc22 =0.0043 × 931.5MeV / c= 4.3 MeV Most of this energy is Kinetic! Energy released by U decay: ~ 4.3MeV

10−6 eV 10−4 eV 12− eV 40eV KeV MeV

Energy to ionize atom or molecule: 10-1000eV Energy to break 1 DNA strand: ~ 1eV 1eV= 1.6 x 10−19 J OR 1 J ~ 1019 eV The Tevatron: Fermi Lab E = Δmc2 LHC: Large Hadron Collider 2 E = Δmc Higgs Boson ~ 3TeV

The protons will each have an energy of 7 TeV, giving a total collision energy of 14 TeV. It will take around 90 s for an individual proton to travel once around the collider. E = Δmc2 Stanford Linear Accelerator (SLAC)

The International Linear Collider is a proposed future international particle accelerator. It would create high-energy particle collisions between electrons and positrons, their antimatter counterparts. The ILC would provide a tool for scientists to address many of the most compelling questions of the 21st century-questions about dark matter, dark energy, extra dimensions and the fundamental nature of matter, energy, space and time. http://www.slac.stanford.edu/ E = Δmc2

In the ILC's design, two facing linear accelerators, each 20 kilometers long, hurl beams of electrons and positrons toward each other at 99.9999999998 percent of the speed of light (with a few TeV energy). Each beam contains ten billion electrons or positrons compressed down to a minuscule three nanometer thickness. As the particles hurtle down the collider, superconducting accelerating cavities operating at temperatures near absolute zero pump more and more energy into them. The beams collide 2000 times every second in a blazing crossfire that creates a firework of new particles. (Computer animation of the field inside a superconducting accelerator resonator.) Brookhaven National Lab (NY)

E = Δmc2

Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider (Brookhaven National Laboratory). Electrically charged particles are discernible by the curves they trace in the detector's magnetic field. E = Δmc2 Unification of all Forcers High Energy Particle Physics

In grand unification theories (GUT), unification of the three forces occurs around 1016 GeV. To unify gravity with the other forces, energies of 1019 GeV, known as Planck Mass, (~2.17645 × 10−8 kg) must be achieved in particle detectors. http://particleadventure.org/ SR Equations E = Δmc2 Albert Einstein 1916

The General Theory of Relativity Equivalence of Gravitational and Inertial Mass

At rest On Earth In space, acceleration In space, acceleration At rest on Earth

Can’t tell difference Can’t tell difference Principle of Equivalence

• An gravity free accelerated frame of reference is equivalent to an inertial frame in a gravitational field. • No local experiment can distinguish between the two frames. • Einstein proposed that a beam of light should be bent downward by a gravitational field – The bending would be small – A laser would fall less than 1 cm from the horizontal after traveling 6000 km Testing General Relativity

• General relativity predicts that a light ray passing near the Sun should be deflected in the curved space-time created by the Sun’s mass • The prediction was confirmed by astronomers during a total solar eclipse in 1919 by Sir Arthur Eddington. Curvature of Spacetime The curvature of space-time completely replaces Newton’s gravitational theory. According to Einstein there is no such thing as a gravitational field

Einstein specified a certain quantity, the curvature of time- space, that describes the gravitational effect at every point.

The Field Equation:

Specifies the curvature Specifies the of spacetime mass/energy content of spacetime

Mass WARPS Space-time Mass grips space by telling it how to curve and space grips mass by telling it how to move! - John Wheeler Confirmation of The General Theory: Precession of the Perihelion of Mercury Confirmation of The General Theory: Precession of the Perihelion of Mercury Gravitational Lensing

p.1274

In the cluster Abell 2218, distant blue galaxies behind the large cluster of galaxies are "squished" into a circular shape around the middle of the foreground cluster. By measuring the amount of distortion in the more distant blue galaxies, we can determine the mass of the cluster. In fact, we can even measure how much mass there is that we can't see -- this galaxy cluster happens to have nearly 400 trillion times the sun's mass in "dark" matter. Gravitational Redshift Gravity Waves

LISA Gravity Probe B Gravitational Frame Dragging Space-Time Twist Gyroscopes on Gravity Probe B

~ millionth degree Black Holes & Worm Holes Michelson-Morley Experiment Galilean Relativity The Lorentz Force on a moving charge depends on the Frame. vvv v FqEqvB= +× Charge Rest Frame Wire Rest Frame (moving with charge) (moving with wire)

F = 0 FqvB= sinθ

Einstein realized this inconsistency and could have chosen either: •Keep Maxwell's Laws of Electromagnetism, and abandon Galileo's Spacetime •or, keep Galileo's Spacetime, and abandon the Maxwell Laws. Lorentz Contraction in Wire Moving charges in the wire cause a Lorentz contraction in the distance between the moving charges in the wire so that from the rest frame of the charge outside the wire (or at rest in the wire) the moving charges bunch up and thus give a net charge to the wire. Eisntein Saves Maxwell! The force on a moving charge does NOT depend on the Frame. Charge Rest Frame Wire Rest Frame (moving with charge) (moving with wire)

FqvB= sinθ

2 Fkqqr= 12/

From the rest frame of the charge, the wire is charged and it feels a Coulomb force. The forces in each frame are equal though they are due to different causes! Relativistic Doppler Effect Relativistic Doppler Effect • Another consequence of time dilation is the shift in frequency found for light emitted by atoms in motion as opposed to light emitted by atoms at rest • If a light source and an observer approach each other with a relative speed, v, the frequency is shifted HIGHER (wavelength shifted SHORTER – Blue) and is measured by the observer to be: 1+ vc ƒƒ= obs 1− vc source • If a light source and an observer move apart from each other with a relative speed, v, the frequency is shifted LOWER (wavelength shifted LONGER – Red) and is measured by the observer to be: 1−vc ƒƒ= obs 1+vc source Relativistic Doppler Effect

Blue Shift: Moving Toward 1±vc ƒƒ= Red Shift: Moving Away obs 1mvc source cf=λ c f = λ 1 f = T Spectral lines shift due to the relative motion between the source and the observer

•Cosmological Redshift: Expanding Universe •Stellar Motions: Rotations and Radial Motions •Solar Physics: Surface Studies and Rotations •Gravitational Redshift: Black Holes & Lensing •Exosolar Planets via Doppler Wobble Extra Solar Planets: Gravitational Doppler Wobble

The newly discovered Neptune-sized extrasolar planet circling the star Gliese 436. Cosmological redshift is caused by the expansion of spacetime Cosmological Red Shift

1−vc ƒƒ= obs 1+vc source Cepheids Andromeda Galaxy

V = H d Hubble o Hubble Time: To= 1/Ho To = 1/70 km/s/Mpc ~ 14 Billion Years (assuming constant expansion rate)