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
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