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?

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

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 8 c= = 3.0x10 m/ s με 0 o

Measure the Speed of the Ether Wind

Michelson-Morley Experiment

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 v / c 2 2

Project Idea! Explain a paper to us!

Born 1879

On the Electrodynamics of Moving Bodies 1905

Charge Rest Frame (moving with charge) Galilean Relativity The Lorentz Force on a moving charge depends on the Frame. F = qe+ qv B Wire Rest Frame (moving with wire) F = 0 F = qvbsinθ 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 (moving with charge) Wire Rest Frame (moving with wire) F = qvbsinθ F = kqq 1 2 / r 2 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!

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. 3. Months later E = mc 2

General Theory: Gravity Curved Spacetime

Special Theory: Flat Spacetime No gravity.

Space Ship Speed = v Δt 0,v Δt, v

Time Dilation Spaceship Frame Earth Frame:

How far does the light travel? AstroFrame: x = vt => 2D = cδt 0 Space Ship Speed = v Earth Frame: 2 2 2 2 s = D + L = cδt

How far does the light travel? AstroFrame: 2D = cδt 0 Space Ship Speed = v Earth Frame: 2s = cδt

Look at 1/2 time - 1 triangle 0 AstroFrame: D = c Δt 2 Space Ship Speed = v Earth Frame: s = c Δt 2

Look at 1/2 time - 1 triangle 0 AstroFrame: D= c Δt 2 Space Ship Speed = v Δt Δt Earth Frame: s = c, L= v 2 2

AstroFrame: D = c Δt 2 0 Look at 1/2 time - 1 triangle t t Earth Frame: s c Δ Δ =, L= v 2 2 Space Ship Speed = v Pythagorean Thm: s = L + D 2 2 2 2 2 2 t t t0 c Δ v Δ c Δ = + 2 2 2

AstroFrame: D = c Δt 2 0 Look at 1/2 time - 1 triangle t t Earth Frame: s c Δ Δ =, L= v 2 2 Space Ship Speed = v Δ t =Δt / 1 v c 0 2 2

Δ t = Δt 1 0 v c 2 2 = γδt 0 The smallest (proper) time interval Dt 0 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: Space Ship Frame: Δ t = γδt 0 Δ t 0

1 γ =, γ 1 2 v 1 2 c v β =, β < 1 c

γ Factor Table

γ Factor Time dilation is not observed in our everyday lives For slow speeds, the factor of γ is so small that no time dilation occurs As the speed approaches the speed of light, γ increases rapidly

Conceptual Questions 39

Identifying Proper Time The time interval Δt p is called the proper time interval The proper time interval is the time interval between events as measured by an observer who sees the events occur at the same point in space You must be able to correctly identify the observer who measures the proper time interval

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

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 Δt p = 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 Δt p (b) A CERN experiment measured lifetimes in agreement with the predictions of relativity

Quick Quiz 39.4 A crew watches a movie that is two hours long in a spacecraft that is moving at high speed through space. An Earthbound observer, who is watching the movie through a powerful telescope, will measure the duration of the movie to be (a) longer than (b) shorter than (c) equal to two hours

Quick Quiz 39.4 Answer: (a). The two events are the beginning and the end of the movie, both of which take place at rest with respect to the spacecraft crew. Thus, the crew measures the proper time interval of 2 h. Any observer in motion with respect to the spacecraft, which includes the observer on Earth, will measure a longer time interval due to time dilation.

As you approach c, lengths contract.

The 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. L P L= L / γ = L 1 P P v c 2 2 Ruler (Proper) Frame Measures length L P Earth Frame Measures length L

L = L 1 v c 0 2 2 L = L 0 γ Rest (Proper) Frame Distance Earth Frame measures L 0 Moving Frame Distance Rocket Frame measures L

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) 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. Beta Diva

Quick Quiz 39.6 You are packing for a trip to another star. During the journey, you will be traveling at 0.99c. You are trying to decide whether you should buy smaller sizes of your clothing, because you will be thinner on your trip, due to length contraction. Also, you are considering saving money by reserving a smaller cabin to sleep in, because you will be shorter when you lie down. You should: (a) buy smaller sizes of clothing (b) reserve a smaller cabin (c) do neither of these (d) do both of these

Quick Quiz 39.6 Answer: (c). Both your body and your sleeping cabin are at rest in your reference frame; thus, they will have their proper length according to you. There will be no change in measured lengths of objects, including yourself, within your spacecraft.

You Do It. P39.8 8. An astronomer on Earth observes a meteoroid in the southern sky approaching the Earth at a speed of 0.800c. At the time of its discovery the meteoroid is 20.0 ly from the Earth. Calculate (a) the time interval required for the meteoroid to reach the Earth as measured by the Earthbound astronomer, (b) this time interval as measured by a tourist on the meteoroid, and (c) the distance to the Earth as measured by the tourist. FIRST: Identify PROPER TIME FRAME!

Quick Quiz 39.7 You are observing a spacecraft moving away from you. You measure it to be shorter than when it was at rest on the ground next to you. You also see a clock through the spacecraft window, and you observe that the passage of time on the clock is measured to be slower than that of the watch on your wrist. Compared to when the spacecraft was on the ground, what do you measure if the spacecraft turns around and comes toward you at the same speed? (a) The spacecraft is measured to be longer and the clock runs faster. (b) The spacecraft is measured to be longer and the clock runs slower. (c) The spacecraft is measured to be shorter and the clock runs faster. (d) The spacecraft is measured to be shorter and the clock runs slower.

Quick Quiz 39.7 Answer: (d). Time dilation and length contraction depend only on the relative speed of one observer relative to another, not on whether the observers are receding or approaching each other.

The Twin Paradox The Situation A thought experiment involving a set of twins, Speedo and Goslo Speedo travels to Planet X, 20 light years from the Earth His ship travels at 0.95c After reaching Planet X, he immediately returns to the Earth at the same speed When Speedo returns, he has aged 13 years, but Goslo has aged 42 years

The Twins Perspectives Goslo s perspective is that he was at rest while Speedo went on the journey Speedo thinks he was at rest and Goslo and the Earth raced away from him and then headed back toward him The paradox which twin has developed signs of excess aging?

The Twin Paradox The Resolution Relativity applies to reference frames moving at uniform speeds The trip in this thought experiment is not symmetrical since Speedo must experience a series of accelerations during the journey Therefore, Goslo can apply the time dilation formula with a proper time of 42 years This gives a time for Speedo of 13 years and this agrees with the earlier result There is no true paradox since Speedo is not in an inertial frame

You Try It: P39.16 The identical twins Speedo and Goslo join a migration from the Earth to Planet X. It is 20.0 ly away in a reference frame in which both planets are at rest. The twins, of the same age, depart at the same time on different spacecraft. Speedo s craft travels steadily at 0.950c, and Goslo s at 0.750c. Calculate the age difference between the twins after Goslo s spacecraft lands on Planet X. Which twin is the older?

HW DUE DATE CHANGE Chapter 39 SUPER HW Problems Due 2-15: 21(new),29, 55, 56, 59, 62, 63, 68, 73, 74 These solutions must be FABULOUS and include diagrams and explanations: In which frame is the proper time, length, etc. Use full sentences with good grammar! ALSO Coming up for 2-15: We ll have a long quiz on Ch 39 and then we ll jump into Chapter 44 so read Ch 44 Sections 1-5 and do: P: 9, 15, 25, 27 Due 2-15!!! Turn Ch 44 problems in separately from the Ch 39 super HW set. Next week 2-8 we will finish talking about the Ch 39 and go over the HW problems. Don t wait to do the problems!

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.

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: ƒ obs = 1+ vc 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: ƒ ƒ obs = 1 vc 1+ vc ƒ source

Relativistic Doppler Effect Blue Shift: Moving Toward Red Shift: Moving Away ƒ obs = 1± vc 1 vc ƒ source c =λf c f = λ f = 1 T

39.18. Review problem. An alien civilization occupies a brown dwarf, nearly stationary relative to the Sun, several lightyears away. The extraterrestrials have come to love original broadcasts of I Love Lucy, on our television channel 2, at carrier frequency 57.0 MHz. Their line of sight to us is in the plane of the Earth s orbit. Find the difference between the highest and lowest frequencies they receive due to the Earth s orbital motion around the Sun. ƒ obs = 1+ vc 1 vc ƒ source

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 ƒ obs = 1 vc 1+ vc ƒ source

Cepheids Andromeda Galaxy Hubble V = H o d

Hubble Time: T o = 1/H o T o = 1/70 km/s/mpc ~ 14 Billion Years (assuming constant expansion rate)

Problem 39.20 20. The red shift. A light source recedes from an observer with a speed v source that is small compared with c. (a) Show that the fractional shift in the measured wavelength is given by the approximate expression Δλ λ v source c This phenomenon is known as the red shift, because the visible light is shifted toward the red. (b) Spectroscopic measurements of light at λ = 397 nm coming from a galaxy in Ursa Major reveal a red shift of 20.0 nm. What is the recessional speed of the galaxy?

Addition of Velocities Galilean Relativity

Addition of Velocities u = 1 v+ u' vu ' + 1+ 2 2 c.5c+ c = =.5cc c c If v =.5c

YOU TRY 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? Beta Diva u = v+ u' 1+ vu ' c 2

Quick Quiz 39.8 You are driving on a freeway at a relativistic speed. Straight ahead of you, a technician standing on the ground turns on a searchlight and a beam of light moves exactly vertically upward, as seen by the technician. As you observe the beam of light, you measure the magnitude of the vertical component of its velocity as (a) equal to c (b) greater than c (c) less than c

Quick Quiz 39.8 Answer: (c). Because of your motion toward the source of the light, the light beam has a horizontal component of velocity as measured by you. The magnitude of the vector sum of the horizontal and vertical component vectors must be equal to c, so the magnitude of the vertical component must be smaller than c.

P39.23 A moving rod is observed to have a length of 2.00 m and to be oriented at an angle of 30.0 with respect to the direction of motion, as shown in Figure P39.23. The rod has a speed of 0.995c. (a) What is the proper length of the rod? (b) What is the orientation angle in the proper frame? Conceptual Check: Is the proper length going to be shorter? Longer? The same? Is the proper angle going to be smaller? Larger? The same?

Quick Quiz 39.9 Consider the situation in question 8 again. If the technician aims the searchlight directly at you instead of upward, you measure the magnitude of the horizontal component of its velocity as (a) equal to c (b) greater than c (c) less than c

Quick Quiz 39.9 Answer: (a). In this case, there is only a horizontal component of the velocity of the light, and you must measure a speed of c.

Lorentz Transformation Equations S:( x, y, x, t) S ':( x', y', x', t') Transform coordinates from S to S : v x' = γ ( x vt) y' = y z' = z t' = γ t x 2 c Transform coordinates from S to S : v x = γ ( x' + vt' ) y' = y z' = z t = γ t' + x' 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 Δ x' = γ Δx vδt ( ) v Δ t' = γ Δt Δx 2 c For S to S ( ' ') Δ x = γ Δ x + vδt v Δ t = γ Δ t' + Δx' 2 c

You TRY P39.25 A red light flashes at position x R = 3.00 m and time t R = 1.00 10 9 s, and a blue light flashes at x B = 5.00 m and t B = 9.00 10-9 s, all measured in the S reference frame. Reference frame S has its origin at the same point as S at t = t = 0; frame S moves uniformly to the right. Both flashes are observed to occur at the same place in S. (a) Find the relative speed between S and S. (b) Find the location of the two flashes in frame S. (c) At what time does the red flash occur in the S frame?

E= 2 mc

E = mc 2 Mass is bound energy. c 2 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!!! c 2 = 9x10 16 J / kg

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 = Δmc 2 ( ) parents products Δ m= m m

Energy Released: The Mass Defect Energy-Mass equivalence is true in all processes that give off energy by chemical or nuclear reactions!!! E = Δmc 2 Question: If a nuclear and power plant both produce the same amount of energy in a week how will the change in the mass of the power plants compare? What IS different? SAME! The energy released in each reaction is different the fission of a single Uranium atom releases 100 Million times as much energy as the combustion of carbon to produce a single carbon dioxide molecule.

60. A rechargeable AA battery with a mass of 25.0 g can supply a power of 1.20 W for 50.0 min. (a) What is the difference in mass between a charged and an uncharged battery? (b) What fraction of the total mass is this mass difference? ( )( )( ) a) Δ E = Pt = 1.20 J s 50 min 60 s min = 3 600 J ΔE 3600 J 14 Δ m = = = 4.00 10 kg c 2 2 ( 8 3 10 m s) 14 Δ m 4.00 10 kg b) = = 1.60 10 3 m 25 10 kg 12

Compare Reactions Energy Released per kg of Fuel (J/kg) Chemical @ 700K C + O 2 -> CO 2 3.3 x 10 7 Fission @ 1000K n + U-235 -> Ba-143 + Kr-91 + 2 n 2.1 x 10 12 Fusion @ 10 8 K H-2 + H-3 -> He-4 + n 3.4 x 10 14

E = Δmc 2 Solar Flux: 26 Ρ= 3.77x10 W dm dt = 9 4.19x10 / kg s

Electron Volts 1 ev is the energy an electron gains across a 1 volt potential. E = qv 19 1 ev = (1.6x10 C)(1 J / C) 1 1.6 10 OR 1 ~ 10 19 19 ev = x J J ev

Atomic Mass Units 1u = 1/12 mass of Carbon-12 1u = 1.6605x10 kg = 931.5 MeV / c 27 2 238.0508u 234.0436u 4.0026u Δ m= 238.0508 234.0436 4.0026 u = 0.0046u ( ) 2 2 E =Δ mc = 0.0043 931.5 MeV / c = 4.3MeV Most of this energy is Kinetic!

Energy released by U decay: ~ 4.3MeV 6 10 ev 10 ev 4 1 2eV 40eV KeV MeV Energy to ionize atom or molecule: 10-1000eV Energy to break 1 DNA strand: ~ 1eV 1 1.6 10 OR 1 ~ 10 19 19 ev = x J J ev

1 1.6 10 OR 1 ~ 10 19 19 ev = x J J ev

Proof: E = mc 2 Pure energy converted into particles: pair production Irène and Frédéric Joliot-Curie 1933

Creating Matter From Pure Energy Matter-AntiMatter E = Δmc 2

E = Δmc 2 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 = Δmc 2 In the ILC's design, two facing linear accelerators, each 20 kilometers long, hurl beams of electrons and positrons toward each other at 99.999999999 8 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 = Δmc 2 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

The Tevatron: Fermi Lab E = Δmc2

LHC: Large Hadron Collider E = Δmc 2 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 ms for an individual proton to travel once around the collider.

Unification of all Forcers High Energy Particle Physics In grand unification theories (GUT), unification of the three forces occurs around 10 16 GeV. To unify gravity with the other forces, energies of 10 19 GeV, known as Planck Mass, (~2.17645 10 8 kg) must be achieved in particle detectors.

http://particleadventure.org/

E= 2 mc

Invariant Mass Invariant mass is independent of frame of reference and is the mass as measured in the proper frame: Rest Energy: E = mc 2 Invariant mass: m = E/ c 2 If a mass is moving, than the total energy increases by a gamma factor: Total Energy: E =γmc 2

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: rest energy: E total energy m = γ m 0 E = mc 0 0 0 2 = γ mc Note: We don t use this notation! 2 m = γ m 0

Total Energy E 2 =γ mc p = γ mu If u = 0 (rest or proper frame) E = pc + ( mc) 2 2 2 2 2 Invariant mass: m = E/ c 2 If m = 0 (photon) photon momentum: p = E/ c

If a photon has momentum, does it have mass?

Kinetic Energy The kinetic energy is the total energy less the rest energy: KE = E E = (1 γ ) mc 0 2

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: E0 = mc 0 total energy E = γ mc KE = E E0 = (1 γ ) mc 0 2 2 Beta Diva

41. An unstable particle with a mass of 3.34 10 27 kg is initially at rest. The particle decays into two fragments that fly off along the x axis with velocity components 0.987c and 0.868c. Find the masses of the fragments. Conservation of Energy: E = Etotal i =γmc 2 E f Conservation of momentum: pi = p = γmu p f

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 timespace, that describes the gravitational effect at every point. The Field Equation: Specifies the curvature of spacetime Specifies the 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

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

LISA Gravity Waves

Gravity Probe B Gravitational Frame Dragging Space-Time Twist

Gyroscopes on Gravity Probe B ~ millionth degree

Black Holes & Worm Holes