Relativity Read Your Textbook: Foundations of Astronomy –Chapter 5 Homework Problems –Review...

Relativity• Read Your Textbook: Foundations of Astronomy

– Chapter 5

• Homework Problems– Review Questions: 3-5, 9, 10– Review Problems: 1, 3, 4– Web Inquiries:

Einstein• Published Special Relativity in 1905 (age26).

• Published General Relativity in 1916.

Special RelativitySpecial Relativity: Experimentally verified to work for

all kinds of electromagnetic phenomena,

and for normal fundamental particles as well.

E = mc2 comes from this theory, which means

that energy and mass are related to one another.

Two Postulates of Special Relativity• Observers cannot detect their uniform motion except

relative to other observers. The laws of physics must be the same for observers in any moving or stationary (but always non-accelerated) frame of reference.

• The speed of light is constant in a vacuum and will be the same for all observers, independent of their motion relative to the each other and to the light source.

General Relativity• General Relativity: Ties Gravity in with the

electromagnetic phenomena covered by special relativity.

• The Equivalence Principle: Constant acceleration and gravity are indistinguishable from one another. Observers cannot distinguish locally between inertial forces due to acceleration and/or uniform gravitational forces due to the presence of a massive body.

InertiaSpecial Relativity involves observations and experiments that

are done in an inertial reference frame.

Inertial: Inertia

Newton’s 1st Law:An Object at rest remains at rest,

OR

Travels at constant velocity in a

straight line,

UNLESS

Acted on by a net external force.

(acceleration)

Inertial Reference Frames

An inertial frame of reference is at rest, OR moving at

constant velocity in a straight line with respect to someone

else’s reference frame.

• Special Relativity deals with non-accelerating reference frames.

• General Relativity deals with accelerated reference frames.

Relative Motion

u = relative velocity between inertial reference frames

v = arbitrary velocity of reference frame (X, Y, Z)

v’ = arbitrary velocity of reference frame (X’, Y’, Z’)

u = v + v’

X

Y

Z

X’

Y’

Z’v

v’

River AnalogyFor a fair race, distances D = D| | must be equal.

velocity of your boat = v

Therefore, your true speed is u = v + v’ (boat + current)

Race time = Distance / Speed t = D/u

River Current = v’D

t

D| |

t| |

River SpeedsUp river and Down river u = (v + v’) OR (v - v’)

t| | = {D/(v + v’) + D/(v - v’)}

Straight Across and Back u = (v2 + v’2)1/2

t = 2{D/(v2 + v’2)1/2}

There is a difference in your travel time depending on which

direction you travel with respect to the river current.

t = t - t| |

Light Waves• Light is an electromagnetic wave propagating at speed c.

• Electromagnetic theory predicts c2 = 1/oo

• All waves exhibit reflection, refraction, diffraction and interference. Light waves must also.

• All waves require a medium in which to propagate. Light waves must also?

• What is the medium in which electromagnetic waves propagate?

• The Search: luminiferous ether, (because it was believed that light could NOT propagate in a vacuum!)

Michelson-Morley ExperimentProblem:

Detect absolute motion with respect to luminiferous ether.

Solution:

Upon return, the

light beams will

have traveled

different paths.

Will they interfere

constructively

or

destructively?

Physics FailureOne of the greatest failures in all of physics is the inability

to detect a difference in travel time between two light beams.

Conclusions:• There is NO luminiferous ether.• Light waves propagate in a vacuum at speed c.• There is no preferred frame of reference or absolute rest,

only relative motion between inertial frames.

Curiosity:

If D|| = D [1-(v/v’)2]1/2, then the travel times would be the

same.

Special Relativity

• Special Relativity applies to moving frames of reference that are not undergoing acceleration.

• There is no absolute state of rest, only relative velocities between different frames of reference.

• All observers agree on the fundamental physics, regardless of their frame of reference.

Physics RevisionEinstein’s theory is not a modification to Newtonian mechanics.

It is meant to REPLACE IT COMPLETELY.

Before Newton:

• Galileo studied inertia, falling bodies and projectiles. He developed rules for describing motion. (free-fall, projectiles)

• Kepler studied and described planetary motion. (ellipses, P2 = a3)

• BUT neither explained the WHY. They lacked a synthesis, a complete unifying theory.

After Newton:

• Newton developed his laws of motion and the universal law of gravitiation.

• All of the previous descriptions from Galileo, Kepler and others are explained by and can be derived from Newton’s theories.

• In the same way, Newton’s theories are explained by and can be derived from Einstein’s theory.

Postulates• No Absolute Motion: There exists only relative motion

between inertial reference frames. The laws of physics are the same in any inertial reference frame.

• There is no experiment that can distinguish between events happening in a reference frame at rest and one that is traveling at constant velocity in a straight line with respect to an observer.

• The velocity of light in a vacuum can not depend on motion.

X

Y

Z

X’

Y’

Z’F = m a

a = v/tc = c

F’ = m’ a’a’ = v’/t’

c = cu

Velocity Additionv = velocity of inertial frame #1

v’ = velocity of inertial frame #2

u = relative velocity between reference frame #1 and #2

Velocity Addition (u << c)

A person throws a rock at v = 10 km/hr while riding a

truck traveling at v’ = 60 km/hr. What is the velocity of the

rock as it strikes the idiot in the street?

v’

v

u = v + v’





u = 70 km/hr

Any idiot can calculate that!

v’

v

u = v + v’





v’

v u = v - v’





u = 50 km/hr

v’

v u = v - v’

Velocity Addition (u ~ c)

Now What?! u = must equal c, ALWAYS!

2/3 c

c

u = c + 2/3 c

2/3 c

c

u = c - 2/3 c

Relativistic Velocity Additionu = (v + v’)/(1 + vv’/c)

Consistent with Newtonian velocity when v v’ << c, when

speeds are relatively SLOW.

u = v + v’

When v and v’ = c

u = c

CONSLUSION:• Relativity is IMPORTANT only when speeds approach the

speed of light in a vacuum, c = 3 x 108 m/s.

Transformations of Length, mass and timeFundamental Units are not constant, but variables dependent

upon the velocity through the relativity correction factor ().

L = Lo/Lo is the proper length

m = mo mo is the rest mass

t = toto is the proper time

= 1/(1-2)1/2 the relativistic correction factor,

where = v/c

> 1.0

Relativity Relevance = 1/(1-2)1/2 relativistic correction factor, where = v/c

V % Error between Newton and Einstein

0.1c 1.005 0.5

0.2c 1.021 2.0

0.5c 1.155 13.4

0.6c 1.250 20.0

0.9c 2.294 56.4

0.95c 3.203 68.8

0.99c 7.089 85.9

The correction factor applies to Mass, Length, and Time.

Special Relativity Rules • When viewed by the stationary observer…

– Distances have their maximum lengths• The observer at rest with respect to the object being measured,

measures the proper length Lo




– Masses have their minimum values• The observer at rest with the mass measures the rest mass mo




– Masses have their minimum values• The observer at rest with the mass measures the rest mass mo

– Clocks run the fastest (time required for an event is a minimum)

• The observer that measures an event in the same location measures the proper time to

Proper Length

L = Lo/Lo is the proper length

Proper length is measured by the observer at rest in relation

to the object being measured.

All other lengths which are not at rest with respect to the

observer must be transformed into his reference frame with .

Length Contraction: moving lengths are contracted.

Proper Time

t = toto is the proper time

Proper time is measured by the observer that views events

at the same place.

All other events which are not at the same place with

respect to the observer must be transformed into his reference

frame with .

Time Dilation: moving clocks run slow.

Implications on Physics MeasurementsQuantity Relativistic Correction

mass mo

time to

length Lo/

momentum p = mo v

kinetic energy KE = moc2 (-1)

ApplicationsScientist:

Dear Congressman,

We would like to build a 5 km long accelerator to

study atomic particles. The accelerator will allow us to

study particles that travel approximately 0.99999997c.


Dear Congressman,

We would like to build a 5 km long accelerator to


study particles that travel approximately 0.99999997 c.

Congressman: What type of particles?

Scientist: Mostly electrons and protons but some muons and other short lived particles that have lifetimes of about 2 s.


Dear Congressman,

We would like to build a 5 kilometer long accelerator to


study particles that travel approximately 0.99999997c.

Congressman: What type of particles?

Scientist: Mostly electrons and protons but some muons and other short lived particles that have lifetimes of about 2 s.

Congressman: (back of the envelope) D = v t

D = c (2s) = 600 meters....Ahhhhah! Why do you need 5 km?

AcceleratorScientist:

Einstein says… Moving clocks run slow.

= 1/(1-2)1/2 = v/c

= 1/(1-2)1/2 = 4082

The particles will appear to survive t = (2s) = 0.008 sec

Therefore could travel, D = c (0.008) = 2400 km

AcceleratorScientist:

Einstein says… Moving clocks run slow.

= 4082

t = 0.008 sec

D = 2400 km

The accelerator will cost 10 billion/km so only 50 billion

as opposed to 24,000 billion, a savings of 23,950 billion!

The particles say a 5 km accelerator will look like a

D = 5 km/4082 = 0.001 km accelerator. We say the particles lived t = (2s) = 0.008 sec.

Twin ParadoxA trip is taken by one identical twin to alpha centauri in a

rocket ship traveling at v = 0.95c. Alpha centauri is 4.3 LY

distant. What will be the difference in ages between the twin

that stayed on earth and the twin taking the trip?





1. Identify the two events of interest for the time interval.

2. Determine the reference frame in which the events occur at the same place. An observer at rest in this frame measures the proper time.

Twin ParadoxThe events are, departure from the earth and arrival at alpha

centauri. Upon departure, the earth is just outside the rocket

door. Upon arrival at alpha centauri, it is just outside the door.

The passenger of the rocket views the events in the same

place and therefore measures the proper time.

The twin on earth views the events at different places. He was

here, then the ship blasted and he left.

Light travel time to alpha centauri is 4.3 LY. Thus taking

4.3 years traveling at the speed of light c. Therefore,

the twin on earth calculates the time taken as

t = 4.3 years/0.95 = 4.5 years since the speed was v < c.

Twin ParadoxTime is transformed by t = to

The twin on the rocket measures to and the earth bound twin

measures t.

Therefore, to = t/= 1/(1-0.952)1/2 = 3.2

to = 4.5 years / 3.2

to = 1.4 years

Earth twin is 3.1 years older than rocket twin!





1. Identify the length of interest.

2. Determine the reference frame in which the observer is at rest with respect to this length. An observer at rest in this frame measures the proper length.

Twin ParadoxLength is transformed by L =Lo/

Rocket twin says clock is OK, earth flew away at 0.95c.

The twin on the rocket measures L and the earth bound twin

measures the proper length Lo (distance to alpha centauri).

Therefore, L = Lo/= 1/(1-0.952)1/2 = 3.2

L = 4.3 LY / 3.2

L = 1.34 LY

The rocket twin claims to have only traveled 1.34 LY.

Traveling at 0.95c, 1.34 LY takes t = 1.34/0.95 or

1.4 years.

Age difference is still 3.1 years (= 4.5 - 1.4)

Doctor, DoctorA doctor tells me I have a heart condition and will only live

2.0s. I decide to live a little before my untimely death and

jump off a cliff traveling at 0.99c toward the ground.

Everyone needs a little thrill, or at least to go in style.

I want to enjoy every last moment, so I ask you if I need a

parachute.

You say the cliff is 1000 meters high, so…

d = 0.99c x 2.0 s = 600 meters.

You say NO, you’ll be dead before impact, enjoy yourself.

Doctor, DoctorYou say the cliff is 1000 meters high, so…

d = 0.99c x 2.0 s = 600 meters.

I jump and I hit the ground and writhe in pain before I die.

I am very rightfully upset, albeit dead anyway.

You say I lived longer than I said I would.

I say you can’t make a fundamental cliff measurement.

WHO IS RIGHT?

Time DilationYou say I lived longer than I said I would.• Moving Clocks Run Slow.

= 1/(1-0.992)1/2 = 7.09

t = to

= 7.09 (2s) = 14.18 s

You transform the time from my frame to your frame.

Now, d = 0.99c x 14.18s = 4200 meters.

You say I better get a parachute.

Length ContractionI say you can’t make a fundamental cliff measurement.• The cliff rushes past me at 0.99c.

• Moving Lengths Are Contracted.

= 1/(1-0.992)1/2 = 7.09

L = Lo/

= 1000/7.09 = 141 meters

I transform your measurement of the cliff to my frame.

Now, d = 0.99c x 2 ms = 600 meters

I say I need a parachute, we both agree!

mesonsThis is a real phenomenon that is clearly demonstrated by

particles called m mesons that are created when cosmic rays

interact high in the earth’s atmosphere. They produce showers

of particles that travel near the speed of light and can be

detected with sensitive equipment.

In the laboratory, these particles once created, only live 2 ms

before decaying into “other things.” However, number counts

of these particles when compared from mountain top to sea

level confirm that a significant number of m mesons that exist

at high altitudes, live long enough to be detected at sea level.

mesonsWe say the particles lived much longer than they should have.

Time dilation: t = to

The particles say the mountains are merely molehills.

Length contraction: L = Lo/

Measurements of variables (Mass, Length, and Time)

must be transformed when moving between different frames

of reference!

ProblemI fly over a fast food joint in my spaceship at 0.4c. You call

me on my cell phone and say you are at a 6 foot table eating

a 1/4 pound burger that takes you 5 minutes.

What do I say?

ProblemI fly over a fast food joint in my spaceship at 0.4c. You call

me on my cell phone and say you are at a 6 foot table eating

a 1/4 pound burger that takes you 5 minutes.

What do I say?= 1/(1-0.42)1/2 = 1.09

You measure both the proper time and the proper length, Lo, to

I say, t = to = 1.09 x 5 = 5.45 minutes

L = Lo/ = 5/1.09 = 5.5 foot table

m = m = 1.09 x 0.25 = 0.27 pound burger

Relativistic MomentumWhat is the difference in momenta for an electron

me = 9.11x10-31 kg, v = 0.99999997c?

Classical Momentum: p = mev

Relativistic Momentum: p = mev

Relativistic MomentumWhat is the difference in momenta for an electron

me = 9.11x10-31 kg, v = 0.99999997c?

Classical Momentum: p = mev

Relativistic Momentum: p = mev

mev / mev =

Relativistic momentum will be greater by a factor ,

4082 times greater.

EnergyEnergy-Mass equivalence: Rest energy Eo = moc2

An object’s Kinetic Energy is its total energy minus its

rest energy.

KE = E - Eo

KE = moc2 - moc2

KE = moc2 { - 1}

How can this be related to 1/2 m v2 ?

Kinetic EnergyKE = moc2 { - 1}

Binomial Expansion: (1-x)n = 1 - nx + (n/2)(n-1)x2 + …

Use binomial expansion for g with x = 2 and n = -1/2.

= 1/(1-2)1/2 = 1 + (1/2)(2) + (3/8)(2) + …

KE = moc2 {1 + (1/2)(2) + (3/8)(2) + … -1}

If v << c then we’ll keep only the first term and get the classical approximation...

KE = moc2 {(1/2)(2)} = 1/2 mov2

Energy Equivalence of a Golf BallA 0.046 kg golf ball lies on the green at rest. What is the rest

energy and how long could the energy in the ball power a 75 Watt light bulb (75 Joules/sec)?

The golf ball’s rest energy Eo = moc2 = 0.046 kg (3x108m/s)2

= 4.1 x 1015 Joules

Since Energy = Power x time

t = 4.1 x 1015 Joules/75 Watts = 5.5 x 1013 seconds

t = 1.7million years!

General Relativity• When Frames of Reference Are Accelerated...

1. Equivalence Principle: There is no way to tell locally, the difference between gravity and an accelerated reference frame.

2. The Relativity Principle: Special Relativity governs local physics. Formulae in physics must be valid locally for every observer. The fastest that information can travel is the speed of light. The global structure of space-time may be warped by gravitation.

GravityObjects are accelerated at a

rate equal to g near the earth’s

surface.

ma = mGM/d2

g = GMearth/Rearth2

EquivalenceGravity = An accelerated reference frame

EquivalenceNo experiment done completely inside the laboratory, can

distinguish between the gravitational influence of matter

(space-time curvature) or an accelerated reference frame.

A ball may accelerate at

less than “g”, we might

interpret this as downward

acceleration.

BUT “g” could be changing.

The Accelerating Lab In SpaceProjectiles will behave like projectiles on the surface of the

earth as long as

a = g.

Light BeamsImagine a photon,

entering on the left.

The laboratory is

accelerating very

very rapidly.

To an observer,

the beam must

appear to be

deflected because

of the motion of

the lab.

EquivalenceEven this experiment

must not be able

to distinguish

between acceleration

and gravity.

CONCLUSION:

Light is bent by

gravity.

Bending of Star Light

Stars

Stars and

Sun

Solar Eclipse Observations• Theoretical deflections and actual 1919 solar eclipse data

from the A. Eddington expedition.

Precession of Mercury

Gravitational Red-Shift• Gravitational Fields Affect Clocks

– Clocks in a larger gravitational field run slower

– Accelerated clocks run slow

• One way to think about this is that atoms radiating photons are essentially clocks. The frequency of photons emitted by atoms in a gravitational field will be lowered. Therefore, any spectrum of this light will be red shifted.

Red-Shifts Encountered

• Doppler Shift (object moving away with some velocity)

• Cosmological Red Shift (caused by expansion)

• Gravitational Red Shift (caused by acceleration, space-time)

Space-Time Curvature• Light travels the shortest distance between two points in the

local curved space-time. Not deflected but following “topography.”

Space-Time Curvature• Possible “geometries” of space-time

Space-Time Curvature• A Closed Universe

c

space

time

Measuring Matter By Gravitation• Lensing Geometry

Gravitational Lensing

Relativity Read Your Textbook: Foundations of Astronomy –Chapter 5 Homework Problems –Review...

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