Newton’s Principia in 1687.

Velocities add:

Galilean Relativity

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 downand rulers shrink

in order to keep thespeed of light the

same for all observers!

Time is Relative!Space is Relative!Only the SPEED

OF LIGHT isAbsolute!

A Problem with Electrodynamics

Wire Rest Frame(moving with wire)

Charge Rest Frame(moving with charge)

0F =

The force on a moving charge depends on the Frame.

sinF qvB θ=

Einstein realized this inconsistency and could have chosen either: •Keep Maxwell's Laws of Electromagnetism, and abandon Galileo's Spacetime•or, keep Galileo's Space-time, and abandon the Maxwell Laws.

On the Electrodynamics of Moving Bodies1905

EinsteinEinstein’’s Principle of Relativitys Principle of Relativity

• 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 They are moving on inertia alone.

General Theory: Noninertial Frames:Frames accelerate: Curved Spacetime, Gravity & acceleration.

Albert Einstein1916

The General Theory of Relativity

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

83.0 10 / 300 /c x m s m sμ= =

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 in a vacuum at c in all inertial reference frames, regardless

of the relative motion.

Clocks slow downand rulers shrink

in order to keep thespeed of light the

same for all observers!

Time is Relative!Space is Relative!Only the SPEED

OF LIGHT isAbsolute!

0 t ,vΔ

t, vΔ

Space Ship Speed = v

Time Dilation

Spaceship Frame

Earth Frame:

Time Dilation

2

2

' '

1

tt tvc

γΔΔ = = Δ

−

2

2

1 , 1

1 vc

γ γ= ≥

−

, 1vc

β β= <

2

2

1 , 1

1 vc

γ γ= ≥

−

The smallest time interval Δt0 between two events is measured in the reference frame in which both events occur at the same place (space ship) and is call the proper time. The “stretching out” of the time interval is called time dilation. 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). The ‘event’ is the light bouncing in the spaceship.

't tγΔ = ΔEarth Lab Frame: Dilated Time: Space Ship Frame: Proper Time: 'tΔ

2

2

' '

1

tt tvc

γΔΔ = = Δ

−

Proper Time

Sample Problem

Beta Diva

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?

What is the event? Which frame measures the proper time?

2

2

' '

1

tt tvc

γΔΔ = = Δ

−1 c yrly c yr

ly⋅

= ⋅ → =

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!

Time Dilation –Verification

• Time dilation is a very real phenomenon that has been verified by various experiments

• These experiments include:– Airplane flights– Muon decay

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 µswhen 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

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

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.

Quick Quiz 39.4

Dilation Paradox?

Strange but not a paradox. They are not both true at the same time to each other. They are true relative to their own frame!

Fig. 39-9, p. 1124

Twin Paradox?

Strange but not a paradox! Only if George turns around and comes home is there a difference. The situation is no longer

symmetric. If george stops and turns around and speeds us, he is accelerting and special relativity doesn’t apply to

accelerating noninertial frames.

Molly flies her rocket past Nick at constant velocity v. Molly and Nick both measure the time it takes the rocket, from nose to tail, to pass Nick. Which of the following is true?

A. Nick measures a shorter time interval than Molly.

B. Molly measures a shorter time interval than Nick.

C. Both Molly and Nick measure the same amount of time.

A. Nick measures a shorter time interval than Molly.

B. Molly measures a shorter time interval than Nick.

C. Both Molly and Nick measure the same amount of time.

Molly flies her rocket past Nick at constant velocity v. Molly and Nick both measure the time it takes the rocket, from nose to tail, to pass Nick. Which of the following is true?

As you approach c, lengths contract.

Lorentz Contraction in WireMoving 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!

Wire Rest Frame(moving with wire)

Charge Rest Frame(moving with charge)

sinF qvB

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!

The force on a moving charge does NOT depend on the Frame.

θ=

21 2 /F kq q r=

Ruler (Proper) Frame Measures length LP

Earth Frame Measures length L

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.

PL

2'

2/ 1P PvL L Lc

γ= = −

Length Contraction

Length ContractionLength ContractionThe 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.

Rest (Proper) Frame DistanceEarth Frame measures L0

Moving Frame DistanceRocket Frame measures L

2'

2/ 1P PvL L Lc

γ= = −

Length Contraction: Distances

Proper Length of Ship

Sample Problem

Beta Diva

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.

Quick Quiz

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.

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.

Quick Quiz

Beth and Charles are at rest relative to each other. Anjay runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, and LC.

A. LA = LB = LCB. LB = LC > LAC. LA > LB = LCD. LA > LB > LCE. LB > LC > LA

A. LA = LB = LCB. LB = LC > LAC. LA > LB = LCD. LA > LB > LCE. LB > LC > LA

Beth and Charles are at rest relative to each other. Anjay runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, and LC.

P39.14

The identical twins Speedo and Goslo join a migration from the Earth to Planet X. It is 20.0 lyaway 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?

Length Contraction Paradox?

Strange but not a paradox. They are not both true at the same time to each other. They are true relative to their own frame!

Quick Quiz

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

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.

Quick Quiz

Quick Quiz

Consider the situation in the last question 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

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.

Quick Quiz

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. 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?

P39.20

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

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

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.

Quick Quiz

Addition of Velocities

Galilean Relativity

Addition of Velocities

Addition of Velocities

2

''1

v uu vuc

+=

+ 2

.5.51

c c cccc

+= =

+

If v = .5c

The Lorentz Velocity The Lorentz Velocity 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.

Problem

Beta Diva

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?

2

''1

v uu vuc

+=

+

Lorentz Transformation Equations

( ) 2' ' ' ' vx γ x vt y y z z t γ t xc

⎛ ⎞= − = = = −⎜ ⎟⎝ ⎠

( ) 2' ' ' ' ' 'vx γ x vt y y z z t γ t xc

⎛ ⎞= + = = = +⎜ ⎟⎝ ⎠

Transform coordinates from S to S’ :

Transform coordinates from S’ to S :

: ( , , , )' : ( ', ', ', ')

S x y x tS x y x t

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( )

2

'

'

x γ x v t

vt γ t xc

Δ = Δ − Δ

⎛ ⎞Δ = Δ − Δ⎜ ⎟⎝ ⎠

( )

2

' '

' '

x γ x v t

vt γ t xc

Δ = Δ + Δ

⎛ ⎞Δ = Δ + Δ⎜ ⎟⎝ ⎠

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.

Seeing is NOT the same as Being!The time when an event is SEEN is not necessarily when the event actually happens!

O: sees strikes simultaneous

O’: front is struck before back

A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is flying her rocket at v = 0.5c in the direction from the tree toward the pole. The lightning hits the tree just as she passes by it. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2?

A. before event 2B. after event 2C. at the same time as event 2

A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is flying her rocket at v = 0.5c in the direction from the tree toward the pole. The lightning hits the tree just as she passes by it. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2?

A. before event 2B. after event 2C. at the same time as event 2

P39.22A red light flashes at position xR = 3.00 m and time tR = 1.00 × 10–9 s, and a blue light flashes at xB = 5.00 m and tB = 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? 

Relativistic EnergyRelativistic EnergyThe 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.

20 0(1 )KE E E m cγ= − = −

Kinetic EnergyThe kinetic energy is the total energy less the rest energy:

Relativistic Momenutm & MassImagine you are at rest and a bullet shoots past approaching c.What is the momentum of the particle relative to you?

=p muγThe speed is bounded, is the momentum? NO!

“Relativistic” Mass: 0=m mγ

20 0

20

0

rest energy:

total energy

=

==

E m c

E m cm m

γγ

As the bullet’s speed increases, the momentum and mass increase:

Note: We don’t use this notation!

Total Energy

2 2 2 2 20( ) ( ) ( )= + =E pc m c pc

2= γE mc p muγ=

If m = 0 (photon)

/p E c=photon momentum:

20 =E mc

2 2( )u upc muc m c mc Ec c

γ γ γ β= = = =

22 20( )= +E pc E

p Eβ=2 2 2

2 2 2 2 2 2 2 2 2 2 20 02 22

2

(1 ) ( )1

= γ = → − = → − = → = β +

−

mc u uE mc E mc E E E E E Ec cu

c

Invariant MassInvariant mass is independent of frame of reference and is the mass as measured in the proper frame:

20 0 /=m E c

20 0=

Invariant mass:

Rest Energy: E m c

Total Energy: 2E mc= γ

If a mass is moving, than the total energy increases by a gamma factor:

An electron moves through the lab at 99% the speed of light. The lab reference frame is S and the electron’s reference frame is S´. In which reference frame is the electron’s rest mass larger?

A. Frame S, the lab frameB. Frame S´, the electron’s frameC. It is the same in both frames.

A. Frame S, the lab frameB. Frame S´, the electron’s frameC. It is the same in both frames.

An electron moves through the lab at 99% the speed of light. The lab reference frame is S and the electron’s reference frame is S´. In which reference frame is the electron’s rest mass larger?

Sample Problem

Beta Diva

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.

f) If her rest mass is 65 kg, what is her rest Energy?g) What is her total relativistic Energy you measure? What does she measure? Is it

different?h) What is her kinetic energy? 2

0 0

20

rest energy:

total energy

E m c

E m cγ

=

=2

0 0( 1)KE E E m cγ= − = −

E =mc2

Δm=E/c2Mass 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!!!

2 169 10 /c x J kg=

The rest mass of a proton is .938 GeV/c2

2 -271 GeV/c = 1.783 × 10 kg

Electron Volts

E qV=1 eV is the energy an electron gains across a 1 volt potential.

191 (1.6 10 )(1 / )eV x C J C−=19 191 1.6 10 OR 1 ~ 10eV x J J eV−=

The Tevatron: 1 Trillion (Tera) Electron Volts

610 eV− 410 eV− 1 2eV− 40eV KeV MeV

Energy to ionize atom or molecule: 10-1000eVEnergy to break 1 DNA strand: ~ 1eV

19 191 1.6 10 OR 1 ~ 10eV x J J eV−=

Energy released by U decay: ~ 4.3MeV

Proof: E = mc2

Irène and Frédéric Joliot-Curie1933

Pure energy converted into particles: pair production

Matter-AntiMatterCreating Matter From Pure Energy

E = Δmc2

E = Δmc2

Solar Flux: 263.77 10x WΡ =

94.19 10 /dm x kg sdt

=

E = Δmc2

Energy Released: The Mass DefectAtomic Decay: Parent atoms have more mass than product atoms.

The difference is released in the form of Kinetic energy.

( )parents productsm m mΔ = −

E = Δmc2Question: 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!

Energy Released: The Mass DefectEnergy-Mass equivalence is true in all processes that give off

energy by chemical or nuclear reactions!!!

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.

Compare Reactions

n + U-235 -> Ba-143 + Kr-91 + 2 n

C + O2 -> CO2Chemical @ 700K

H-2 + H-3 -> He-4 + n

Fission@ 1000K

Fusion@ 108K

Energy Released per kg of Fuel (J/kg)

3.3 x 107

2.1 x 1012

3.4 x 1014

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? 

( )( )( )) 1.20 J s 50 min 60 s min 3 600 JΔ = = =Pa E t

( )14

2 28

3 600 J 4.00 10 kg3 10 m s

−ΔΔ = = = ×

×

Emc

1412

34.00 10 kg) 1.60 10

25 10 kg

−−

−Δ ×

= = ××

mbm

Atomic Mass Units1u = 1/12 mass of Carbon-12

27 21 1.6605 10 931.5 /u x kg MeV c−= =

238.0508u 234.0436u 4.0026u

( )238.0508 234.0436 4.0026 0.0046m u uΔ = − − =2 20.0043 931.5 / 4.3= Δ = × =E mc MeV c MeV

Most of this energy is Kinetic!

19 191 1.6 10 OR 1 ~ 10eV x J J eV−=

The Tevatron: Fermi Lab E = Δmc2

LHC: Large Hadron Collider

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.

Higgs Boson ~ 3TeVE = 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.)

E = Δmc2

Brookhaven National Lab (NY)

Thousands of particles explode from the collision point of two relativistic (100 GeVper 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

E = Δmc2

Unification of all ForcersHigh 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

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:

• 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:

obs source11

ƒ ƒv cv c

+=

−

−=

+obs source11

ƒ ƒv cv c

Blue Shift: Moving TowardRed Shift: Moving Away

Relativistic Doppler Effect±

=mobs source

11

ƒ ƒv cv c

c f= λ

1fT

=

cf =λ

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. 

obs source11

ƒ ƒv cv c

+=

−

•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

Spectral lines shift due to the relative motion between the source and the observer

Extra Solar Planets: Gravitational Doppler Wobble

The newly discovered Neptune-sized extrasolar planet circling the star Gliese 436.

−=

+obs source11

ƒ ƒv cv c

Cosmological Red Shift

Cosmological redshift is caused by the expansion of spacetime

Hubble

Cepheids Andromeda Galaxy

V = Ho d

Hubble Time: To= 1/HoTo = 1/70 km/s/Mpc~ 14 Billion Years

(assuming constant expansion rate)

Albert Einstein1916

The General Theory of Relativity

Equivalence of Gravitational and Inertial Mass

In space, acceleration At rest on Earth

Can’t tell difference

At rest On Earth In space, acceleration

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 SpacetimeThe 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 of spacetime

Specifies the mass/energy content of spacetime

Mass WARPS Space-time

Mass grips space by telling it how to curve andspace 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

p.1274

Gravitational Lensing

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 BGravitational Frame Dragging

Space-Time Twist

Gyroscopes on Gravity Probe B

~ millionth degree

Black Holes & Worm Holes

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: =i fE E

= γp muConservation of momentum: =∑ ∑i fp p

2= γtotalE mc