Relativity

Historical Development1600s

Newton discovered his laws of mechanicsApplied to a wide variety of problems over the next two

decadesWorked well

Late 1800sMaxwell’s equations explained the physics of

electromagnetism and lightEarly 1900s

RelativityQuantum Mechanics

Types of RelativitySpecial relativity

Concerned with objects and observers moving at a constant velocity

Topic of this chapterGeneral relativity

Applies to situations when the object or the observer is accelerated

Has implications for understanding gravitation

Relativity

The term relativity arises when a situation is described from two different points of view

When the railroad car moves with a constant velocity, Ted and Alice see different motions of the ball

Section 27.1

Reference FramesA reference frame can be thought of as a set of

coordinate axesInertial reference frames move with a constant

velocityThe principle of Galilean relativity is the idea that

the laws of motion should be the same in all inertial framesFor example, adding or subtracting a constant velocity

does not change the acceleration of an object and if Newton’s Second Law is obeyed in one inertial frame, it is obeyed in all inertial frames

Section 27.1

Interpretation by Ted and AliceTed observes the ball’s motion purely along the vertical

directionAlice sees the ball undergo projectile motion with a

nonzero displacement in both the x- and y-directionsTed would think the ball’s horizontal velocity is zero, but

Alice would disagreeBoth agree that the ball’s acceleration is downward with a

magnitude gBoth agree the ball’s horizontal acceleration is zeroBoth agree that the only force acting on the ball is the

force of gravity and that Newton’s second law is obeyed

Section 27.1

Galilean Relativity and LightAccording to Maxwell’s equations, the speed of light

is a constantHe also showed the speed of light is independent of

the motion of both the source and the observerAssume Ted is moving with a constant velocity v

relative to Alice when he turns on a flashlight Newton’s mechanics predict that speed of the light

wave relative to Alice should be c + vAccording to Maxwell’s theory, Ted and Alice should

both observe the light wave to move with speed c

Galilean Relativity and Light, cont.Galilean Relativity and

electromagnetism predict different results for observers in different inertial frames

Experiments showed that Maxwell’s theory was correct

The speed of light in a vacuum is always cGalilean relativity for how

the speed of light depends on the motion of the source is wrong

Section 27.1

Special RelativityEinstein developed a theory to analyze the Ted and Alice

situation called special relativityHis work was not motivated by any particular experimentHe suspected the speed of light is the same in all reference

frames Maxwell was correct

He then worked out what that implies for all the other laws of physics

The basics of the theory were stated in two postulates about the laws of physicsFor fast-moving objects Newton’s theory breaks down and

Einstein’s theory gives a correct description of motion in this regime

Section 27.2

Postulates of Special RelativityAll the laws of physics are the same in all inertial

reference frames The speed of light in a vacuum is a constant,

independent of the motion of the light source and all observers

Section 27.2

Postulates – Details First postulate is traced to the ideas of Galileo and

Newton on relativityThe postulate goes further than Galileo because it

applies to all physical laws Not just mechanics

The second postulate is motivated by Maxwell’s theory of lightThis is not consistent with Newton’s mechanics

The postulates will lead to a new theory of mechanics that corrects and extends Newton’s Laws

Section 27.2

More About LightOur everyday experience with conventional waves

cannot be applied to lightLight does not depend on having a conventional

material medium in which to travelA light wave essentially carries its medium with it as

it propagatesIn the electric and magnetic fields

The lack of a conventional medium was surprising and hard to reconcile with conventional intuition

Section 27.2

Inertial Reference FramesInertial reference frames play an important role in

special relativityA definition of what it means to be inertial is neededThe modern definition of an inertial reference is one

in which Newton’s First Law holdsYou can test for an inertial frame by observing the

motion of a particle is zero If the particle moves with a constant velocity, the reference

frame is inertialNewton’s other laws should also apply in all inertial

frames

Section 27.2

Earth as a Reference FrameSince the Earth spins about its axis as it orbits the

Sun, all points on the Earth’s surface have a nonzero acceleration

Technically, a person standing on the surface of the Earth is not in an inertial reference frame

However, the Earth’s acceleration is small enough that it can generally be ignored

In most situations we can consider the Earth to be an inertial reference frame

Section 27.2

Light ClockThe two postulates lead to

a surprising result concerning the nature of time

A light clock keeps time by using a pulse of light that travels back and forth between two mirrors

The time for the clock to “tick” once is the time needed for one round trip: 2ℓ / c

Section 27.3

Moving Light Clock

The clock moves with a constant velocity v relative to the ground

From Ted’s reference frame, the light pulse travels up and down between the two mirrors

Section 27.3

Moving Light Clock, cont.The time for the clock to make one tick as measured

by Ted is

Alice sees the light pulse travel a longer distanceThe speed of light is the same for Alice as for TedBecause of the longer distance, according to Alice

the light will take longer to travel between the mirrors

ot c2

Section 27.3

Moving Clock, Alice’s TimeFor Alice, the time for one tick of the clock is

The time for Ted is different from the time for AliceThe operation of the clock depends on the motion of

the observer

ottv

c2

21

Section 27.3

Moving Clocks Run SlowAlice’s measures a longer time than TedPostulate 1 states that all the laws of physics must

be the same in all inertial reference framesTherefore the result must hold for any clock

Special relativity predicts that moving clocks run slow

This effect is called time dilationFor typical terrestrial speeds, the difference between Δt and Δto is negligible

Section 27.3

Time DilationWhen the speed is

small compared to c, the factor is very close to 1

Approximations given in Insight 27.1 may be used in many terrestrial cases

v c2 21

Section 27.3

Speeds Greater the cIf the value of the speed is greater than the speed of

light, Δt / Δto will be imaginary

Speeds greater than the speed of light have never been observed in nature

Experiments have shown that the time dilation predicted by special relativity is correct

The result applies to all clocks, even biological ones

Section 27.3

Proper TimeThe time interval Δto is measured by the observer at

rest relative to the clockThis quantity is called the proper timeThe time interval measured by a moving observer is

always longer than the proper timeThe proper time is always the shortest possible time

that can be measured for a process, by any observer

Section 27.3

Twin Paradox

An astronaut, Ted, visits a nearby star, Sirius, and returns to EarthSirius is 8.6 light-years from EarthTed is traveling at 0.90 c

Alice, Ted’s twin, stays on Earth and monitors Ted’s trip

Section 27.3

Twin Paradox, TimesAlice measures the trip as taking 19 years

Ted’s body measures the proper time of 8.3 years

Alice concludes that Ted will be younger than she isTed calculates the Earth (and Alice) move away from him

at 0.90 cTed concludes Alice will age 8.3 years while he ages 19

yearsTed concludes that Alice will be younger than he is

lyt years

c17.2

190.90

ot t v c years c c years22 2 21 19 1 0.90 / 8.3

Section 27.3

Twin Paradox, ResolutionTime dilation appears to lead to contradictory resultsAlice’s analysis is correct

She remains in an inertial frame and so can apply the results of special relativity

Ted is incorrectHe accelerates when he turns around at SiriusSpecial relativity cannot be applied during this time

spent in an accelerating frame

Section 27.3

Time Dilation and GPSEach GPS satellite

contains a very accurate clock

The satellite clocks are moving in orbit, so they experience time dilationThey run slow by about

7µs per dayTo accurately determine a

position, the effect of time dilation must be accounted for

Section 27.3

Simultaneity

Two events are simultaneous if they occur at the same timeTed is standing the middle of his railroad carHe moves at a speed v relative to AliceTwo lightning bolts strike the ends of the car and leave burn

marks on the ground which indicate the location of the two ends of the car where the bolts strike

Section 27.4

Simultaneity, cont.Did the two lightning bolts strike simultaneously?According to Alice

She is midway between the burn marksThe light pulses reach her at the same timeShe sees the bolts as simultaneous

According to TedThe light pulse from at B struck before the bolt at A

Since he is moving toward B

The two bolts are not simultaneous in Ted’s view

Section 27.4

Simultaneity, finalSimultaneity is relative and can be different in

different reference framesThis is different from Newton’s theory, in which time

is an absolute, objective quantityIt is the same for all observers

All observers agree on the order of the events

Section 27.4

Length Contraction

Alice marks two points on the ground and measures length Lo between them

Ted travels in the x-direction at constant velocity v and reads his clock as he passes point A and point BThis is the proper time interval of the motion

Section 27.5

Length Contraction, cont.Distance measured by Alice = Lo = v Δt

Distance measured by Ted = L = v Δto

Since Δt ≠ Δto, L ≠ Lo

The difference is due to time dilation and

The length measured by Ted is smaller than Alice’s length

oL L v c2 21 /

Proper LengthTed is at restAlice moves on the

meterstick with speed v relative to Ted

Ted measures a length shorter than Alice

Moving metersticks are shortened

The proper length, Lo, is the length measured by the observer at rest relative to the meterstick

Length Contraction EquationLength contraction is

described by

When the speed is very small, the contraction factor is very close to 1This is the case for

typical terrestrial speeds

o

L vL c

2

21

Section 27.5

Proper Length and Time, ReviewProper time is measured by an observer who is at

rest relative to the clock used for the measurementProper length is measured by an observer who is at

rest relative to the object whose length is being measured

Section 27.5

Experimental SupportA large number of experiments have shown that

time dilation and length contraction actually do occurAt ordinary terrestrial speeds the effects are

negligibly smallFor objects moving at speeds approaching the

speed of light, the effects become significant

Section 27.5

Addition of Velocities

Ted is traveling on a railroad car at constant speed vTA with respect to Alice

He throws an object with a speed relative to himself of vOT

What is the velocity vOA of the ball relative to Alice?Alice is at rest on the ground

Section 27.6

Newton’s Addition of VelocitiesNewton would predict that vOA = vOT + vTA

The velocity of the object relative to Alice = the velocity of the object relative to Ted + the velocity of Ted relative to Alice

This result is inconsistent with the postulates of special relativity when the speeds are very high

For example, if the object’s speed relative to Ted is 0.9 c and the railroad car is moving at 0.9 c, then the object would be traveling at 1.8 c relative to AliceNewton’s theory gives a speed faster than the speed

of light

Section 27.6

Relativistic Addition of VelocitiesThe result of special relativity for the addition of velocities

is

The velocities are:vOT – the velocity of an object relative to an observervTA – the velocity of one observer relative to a second

observervOA – the velocity of the object relative to the second

observer

OT TAOA

OT TA

v vv

v vc2

1

Section 27.6

21cvvvv

vTAOA

TAOAOT

Relativistic Addition of Velocities, cont.When the velocities vOT and vTA are much less than

the speed of light, the relativistic addition of velocities equations gives nearly the same result as the Newtonian equation

For speeds less than approximately 10% the speed of light, the Newtonian velocity equation works well

For the example, with each speed being 0.9c, the relativistic result is 0.994 c

Experiments with particles moving at very high speeds show that the relativistic result is correct

Section 27.6

Relativistic Velocities and the Speed of LightA slightly different result occurs when the velocities

are perpendicular to each otherAgain when vOT and vTA are both less than c, then

vOA is also less than c

In general, if an object has a speed less than c for one observer, its speed is less than c for all other observers

Since no experiment has ever observed an object with a speed greater than the speed of light, c is a universal “speed limit”

Section 27.6

Relativistic Velocities, finalAssume the object leaving Ted’s hand is a pulse of

lightThen vOT = c

From the relativistic velocity equation, Alice observes the pulse is vOA = c

Alice sees the pulse traveling at the speed of light regardless of Ted’s speed

If an object moves at the speed of light for one observer, it moves at the speed of light for all observers

Section 27.6

QuizTed is trying to find your mother on a spaceship

going 0.8c. He sends out a bottle with a love note at 0.7c in front of him. What velocity of the bottle does your mother see as it approaches her?

A) 0.87cB) 0.96cC) 0.79cD) 0.99cE)1.03c

MomentumAccording to Newton’s mechanics, a particle of

mass mo moving with speed v has a momentum given by

p = mo v

Conservation of momentum is one of the fundamental conservation rules in physics and is believed to be satisfied by all the laws of physics, including the theory of special relativity

The momentum can also be written as

Section 27.7

o

xp m

t

Relativistic MomentumFrom time dilation and length contraction,

measurements of both Δx and Δt can be different for observers in different inertial reference frames

Should proper time or proper length be used?Einstein showed that you should use the proper time

to calculate momentumUses a clock traveling along with the particle

The result from special relativity is o

o oo

m vx xp m m

t t v c v c2 2 2 21 1

Section 27.7

Relativistic Momentum, cont.Einstein showed that when the momentum is

calculated by using the special relativity equation, the principle of conservation of momentum is obeyed exactly

This is the correct expression for momentum and applies even for a particle moving at high speed, close to the speed of light

When a particle’s speed is small compared to the speed of light, the relativistic momentum becomes p = mo v which is Newton’s momentum

Section 27.7

Newton’s vs. Relativistic MomentumAs v approaches the

speed of light, the relativistic result is very different than Newton’s

There is no limit to how large the momentum can be

However, even when the momentum is very large, the particle’s speed never quite reaches the speed of light

Section 27.7

MassNewton’s second law gives mass, mo, as the

constant of proportionality that relates acceleration and force

This works well as long as the object’s speed is small compared with the speed of light

At high speeds, though, Newton’s second law breaks down

Section 27.8

Relativistic MassWhen the postulates of special relativity are applied

to Newton’s second law, the mass needs to be replaced with a relativistic factor

At low speeds, the relativistic term approaches mo and the two acceleration equations will be the same

When v ≈ c, the acceleration is very small even when the force is very large

o

o

mm

v c3

2 2 21

Section 27.8

Rest MassWhen the speed of the mass is close to the speed of

light, the particle responds to a force as if it had a mass larger than mo This “enhancement of the mass” depends on the particle’s

speedThe same result happens with momentum where at high

speeds the particle responds to impulses and forces as if its mass were larger than mo

Rest mass is denoted by mo This is the mass measured by an observer who is moving

very slowly relative to the particleThe best way to describe the mass of a particle is through

its rest mass

Section 27.8

Mass and EnergyRelativistic effects need to be taken into account

when dealing with energy at high speedsFrom special relativity and work-energy,

For v << c, this gives KE ≈ ½ m v2 which is the expression for kinetic energy from Newton’s results

oo

m cKE m c

v c

22

2 21

Section 27.9

Kinetic Energy and SpeedFor small velocities, KE is

given by Newton’s resultsAs v approaches c, the

relativistic result has a different behavior than does Newton

Although the KE can be made very large, the particle’s speed never quite reaches the speed of light

Section 27.9

Rest and Total EnergiesThe kinetic energy can also be thought of as the

difference between the final and initial energies of the particle

The initial energy, moc2, is a constant called the rest energy of the particleA particle will have this much energy even when it is at

restThe total energy of the particle is the sum of the

kinetic energies and the rest energy

Section 27.9

Mass as EnergyThe rest energy equation implies that mass is a

form of energyIt is possible to convert an amount of energy (moc2)

into a particle of mass mo

It is possible to convert a particle of mass mo into an amount of energy (moc2)

The principle of conservation of energy must be extended to include this type of energy

Section 27.9

Speed of Light as a Speed LimitSeveral results of special relativity suggest that

speeds greater than the speed of light are not possible

The factor that appears in time dilation and length contraction is imaginary if v > c

The relativistic momentum of a particle becomes infinite as v → cThis suggests that an infinite force or impulse is

needed for a particle to reach the speed of light

Section 27.9

v c2 21

Speed Limit, cont.The total energy of a particle becomes infinite as

v → c, suggesting that an infinite amount of mechanical work is required to accelerate a particle to the speed of light

The idea that c is a “speed limit” is not one of the postulates of special relativity

Combining the two postulates of special relativity leads to the conclusion that it is not possible for a particle to travel faster than the speed of light

Section 27.9

Mass-Energy ConversionsConversion of mass into energy is important in

nuclear reactions, but also occurs in other casesA chemical reaction occurs when a hydrogen atom

is dissociatedThe mass of a hydrogen atom must be less than the

sum of the masses of an electron and protonThe energy is lower by 13.6 eV when bound in the

atomMass is not conserved when a hydrogen atom

dissociatesSection 27.9

Conservation PrinciplesConservation of mass

Mass is a conserved quantity in Newton’s mechanics The total mass of a closed system cannot change

Special relativity indicates that mass is not conserved The principle of conservation of energy must be extended to

include mass

Momentum is conserved in collisionsUse the relativistic expression for momentum

Electric charge is conservedIt is possible to create or annihilate charges as long as

the total charge does not change

Section 27.9

General RelativityA noninertial reference frame is one that has a

nonzero accelerationPhysics in noninertial frames is describe by general

relativityGeneral relativity is based on a postulate known as

the equivalence principleThe equivalence principle states the effects of a

uniform gravitational field are identical to motion with constant acceleration

Section 27.10

Equivalence ExampleTed stands in an elevator

at rest (A)He feels the normal force

exerted by the floor on his feet

He concludes that he is in a gravitational field

The elevator is not in a gravitational field and has an acceleration of g (B)Since there is an

acceleration, Ted feels the same force on his feet

Equivalence Principle, cont.According to the equivalence principle, there is no

way for Ted to tell the difference between the effects of the gravitational field and the accelerated motion

The equivalence principle has the following consequencesInertial mass and gravitational mass are equivalentLight can be deflected by gravity

Experiments in 1919 verified light passing near the Sun during an eclipse was deflected by the predicted amount

Section 27.10

Black HolesBlack holes contain so

much mass that light is not able to escape from their gravitational attraction

A black hole can be “seen” by its effect on the motion of nearby objects

Stars near a black hole move in curved trajectories and so the mass and location of the black hole can be determined

Section 27.10

Gravitational Lensing

If the black hole is between the star and the Earth, light from the star can pass by either side of the black hole and still be bent by gravity and reach the Earth

The black hole acts as a gravitational lensLight from a single star can produce multiple imagesAnalysis of the images can be used to deduce the mass of the

black hole

Section 27.10

Relativity and Electromagnetism

Alice is at rest with the charged line and the point charge

Ted sees the line of charge and the point charge in motionThe moving charged line acts as a current

Section 27.11

Relativity and EM, cont.Ted says that there is an electric force and a

magnetic force on the particleAlice says there is only an electric forceBoth are correctThey will agree on the total force acting on the

particleThe larger electric force seen by Ted due to his motion

is canceled by the magnetic force producedMaxwell’s equations were already consistent with

special relativitySection 27.11

Importance of RelativityThe relation between mass and energy and the

possibility that mass can be converted to energy (and energy to mass) mean that mass is not conservedInstead we have a more general view of energy and its

conservation The three conservation principles in physics are

Conservation of energy Conservation of momentum Conservation of charge

It is believed that all the laws of physics must obey these three conservation principles

Section 27.12

Importance of Relativity, cont.The rest energy of a particle is huge

This has important consequences for the amount of energy available in processes such as nuclear reactions

Relativity changes our notion of space and timeTime and position are two primary quantities in

physics but it is not possible to give precise definitions of such quantities

Our everyday intuition breaks down when applied to special relativity

Section 27.12

Importance of Relativity, finalRelativity plays a key role in understanding how the

universe was formed and how it is evolvingBlack holes can’t be understood without relativity

Relativity shows that Newton’s mechanics is not an exact description of the physical worldInstead, Newton’s laws are only an approximation that

works very well in some cases, but not in othersWe shouldn’t discard Newton’s mechanics, but

understand its limits

Section 27.12