Chapter 26 Relativity. General Physics Relative Motion (Galilean Relativity) Chapter 3 Section 5 jjerrett/relative/relative.html.

Chapter 26Chapter 26RelativityRelativity

General Physics

Relative MotionRelative Motion

(Galilean Relativity)(Galilean Relativity)

Chapter 3 Section 5Chapter 3 Section 5

http://www.physics.mun.ca/~jjerrett/relative/relative.html

General Physics

Michelson InterferometerMichelson Interferometer

Chapter 25 Section 7Chapter 25 Section 7

General Physics

Michelson InterferometerMichelson Interferometer

The Michelson Interferometer is an optical The Michelson Interferometer is an optical instrument that has great scientific instrument that has great scientific importanceimportance

It splits a beam of light into two parts and It splits a beam of light into two parts and then recombines them to form an then recombines them to form an interference patterninterference pattern It is used to make accurate length It is used to make accurate length

measurementsmeasurements

General Physics

Michelson Interferometer, Michelson Interferometer, schematicschematic

A beam of light provided by a A beam of light provided by a monochromatic source is split monochromatic source is split into two rays by a partially into two rays by a partially silvered mirror Msilvered mirror M

One ray is reflected to MOne ray is reflected to M11 and and the other transmitted to Mthe other transmitted to M22

After reflecting, the rays After reflecting, the rays combine to form an combine to form an interference patterninterference pattern

The glass plate ensures both The glass plate ensures both rays travel the same distance rays travel the same distance through glassthrough glass

Active Figure: The Michelson Interferometer

General Physics

Measurements with a Michelson Measurements with a Michelson InterferometerInterferometer

The interference pattern for the two rays is determined The interference pattern for the two rays is determined by the difference in their path lengthsby the difference in their path lengths

When MWhen M11 is moved a distance of is moved a distance of λ/4, successive light and λ/4, successive light and dark fringes are formeddark fringes are formed This change in a fringe from light to dark is called This change in a fringe from light to dark is called

fringe shiftfringe shift The wavelength can be measured by counting the The wavelength can be measured by counting the

number of fringe shifts for a measured displacement of Mnumber of fringe shifts for a measured displacement of M If the wavelength is accurately known, the mirror If the wavelength is accurately known, the mirror

displacement can be determined to within a fraction of displacement can be determined to within a fraction of the wavelengththe wavelength

General Physics

Luminiferous EtherLuminiferous Ether Classical physicists (Maxwell, Hertz, etc.) Classical physicists (Maxwell, Hertz, etc.)

compared electromagnetic waves to mechanical compared electromagnetic waves to mechanical waveswaves Mechanical waves need a medium to support the Mechanical waves need a medium to support the

disturbance (air, water, string, etc.)disturbance (air, water, string, etc.) The The luminiferous etherluminiferous ether was proposed as the was proposed as the

medium required (and present) for light waves to medium required (and present) for light waves to propagatepropagate Present everywhere, even in empty spacePresent everywhere, even in empty space Massless, but rigid mediumMassless, but rigid medium Could have no effect on the motion of planets or other Could have no effect on the motion of planets or other

objectsobjects

General Physics

Verifying the Luminiferous EtherVerifying the Luminiferous Ether Associated with the ether was an Associated with the ether was an absolute frame of reference absolute frame of reference

in whichin which light travels with speed clight travels with speed c The Earth moves through the ether, so there should be an The Earth moves through the ether, so there should be an

“ether wind” blowing“ether wind” blowing If v is the speed of the “ether wind” relative to the Earth, the If v is the speed of the “ether wind” relative to the Earth, the

observed speed of light should have a maximum (a), observed speed of light should have a maximum (a), minimum (b), or in-between (c) value depending on its minimum (b), or in-between (c) value depending on its orientation to the “wind”orientation to the “wind”

General Physics

Michelson-Morley ExperimentMichelson-Morley Experiment

First performed in 1881 by Michelson First performed in 1881 by Michelson Repeated under various conditions by Repeated under various conditions by

Michelson and MorleyMichelson and Morley Designed to detect small changes in the Designed to detect small changes in the

speed of lightspeed of light By determining the velocity of the Earth By determining the velocity of the Earth

relative to the etherrelative to the ether

General Physics

Michelson-Morley EquipmentMichelson-Morley Equipment

An interference pattern was An interference pattern was observedobserved

The interferometer was The interferometer was rotated through 90°rotated through 90°

Used the Michelson InterferometerUsed the Michelson Interferometer Arm 2 is initially aligned along the Arm 2 is initially aligned along the

direction of the earth’s motion through direction of the earth’s motion through spacespace

Should observe small, but measurable, Should observe small, but measurable, shifts in the fringe pattern as orientation shifts in the fringe pattern as orientation with the “ether wind” changeswith the “ether wind” changes

Active Figure: The Michelson-Morley Experiment

General Physics

Michelson-Morley ResultsMichelson-Morley Results Measurements failed to show any change in the Measurements failed to show any change in the

fringe patternfringe pattern No fringe shift of the magnitude required was ever No fringe shift of the magnitude required was ever

observed observed The addition laws for velocities were The addition laws for velocities were incorrectincorrect The speed of light is a constant in all inertial frames of The speed of light is a constant in all inertial frames of

referencereference

Light is now understood to be an Light is now understood to be an electromagnetic wave, which requires no electromagnetic wave, which requires no medium for its propagationmedium for its propagation The idea of an ether was discardedThe idea of an ether was discarded

General Physics

Relativity IRelativity I

Sections 1–4Sections 1–4

General Physics

Basic ProblemsBasic Problems The speed of every particle of matter in the The speed of every particle of matter in the

universe always remains universe always remains less thanless than the speed of the speed of lightlight

Newtonian Mechanics is a limited theoryNewtonian Mechanics is a limited theory It places no upper limit on speedIt places no upper limit on speed It breaks down at speeds greater than about 10% It breaks down at speeds greater than about 10%

of the speed of light (v > .1c)of the speed of light (v > .1c) Newtonian Mechanics becomes a specialized case Newtonian Mechanics becomes a specialized case

of Einstein’s Theory of Special Relativityof Einstein’s Theory of Special Relativity When speeds are much less than the speed of light When speeds are much less than the speed of light

v<<cv<<c

General Physics

Galilean RelativityGalilean Relativity

Choose a Choose a frame of referenceframe of reference Necessary to describe a physical eventNecessary to describe a physical event

According to Galilean Relativity, the laws of According to Galilean Relativity, the laws of mechanics are the same in all inertial frames of mechanics are the same in all inertial frames of referencereference An inertial frame of reference is one in which An inertial frame of reference is one in which

Newton’s Laws are validNewton’s Laws are valid Objects subjected to no forces will move in straight Objects subjected to no forces will move in straight

lineslines

General Physics

Galilean Relativity, cont.Galilean Relativity, cont.

A passenger in an A passenger in an airplane throws a ball airplane throws a ball straight upstraight up It appears to move in a It appears to move in a

vertical pathvertical path This is the same motion as This is the same motion as

when the ball is thrown when the ball is thrown while standing at rest on while standing at rest on the Earththe Earth

The law of gravity and The law of gravity and equations of motion under equations of motion under uniform acceleration are uniform acceleration are obeyedobeyed

20 2

1

0

gttvy

x

y

General Physics

Galilean Relativity, contGalilean Relativity, cont

There is a stationary There is a stationary observer on the groundobserver on the ground Views the path of the ball Views the path of the ball

thrown to be a parabolathrown to be a parabola The ball has a velocity to The ball has a velocity to

the right equal to the the right equal to the velocity of the planevelocity of the plane

The law of gravity and The law of gravity and equations of motion equations of motion under uniform under uniform acceleration are still acceleration are still obeyedobeyed

20 2

1gttvy

vtx

y

General Physics

Galilean Relativity, finalGalilean Relativity, final The two observers disagree on the shape of The two observers disagree on the shape of

the ball’s paththe ball’s path Both agree that the motion obeys the law of Both agree that the motion obeys the law of

gravity and Newton’s laws of motiongravity and Newton’s laws of motion Both agree on how long the ball was in the airBoth agree on how long the ball was in the air ConclusionConclusion:: There is no preferred frame of There is no preferred frame of

reference for describing the laws of mechanicsreference for describing the laws of mechanics

General Physics

Galilean Relativity – LimitationsGalilean Relativity – Limitations Galilean Relativity does Galilean Relativity does notnot apply to experiments in electricity, apply to experiments in electricity,

magnetism, optics, and other areasmagnetism, optics, and other areas Results do not agree with experimentsResults do not agree with experiments

According to Galilean relativity, the observer S should measure the According to Galilean relativity, the observer S should measure the speed of the light pulse as v+cspeed of the light pulse as v+c

Actually observer S measures the speed as cActually observer S measures the speed as c

What is the problem?What is the problem?

General Physics

Albert EinsteinAlbert Einstein

1879 – 19551879 – 1955 1905 published four papers:1905 published four papers:

Brownian motionBrownian motion Photoelectric effectPhotoelectric effect 2 on Special Relativity2 on Special Relativity

1916 published theory of 1916 published theory of General RelativityGeneral Relativity

Searched for a unified theorySearched for a unified theory Never found oneNever found one

General Physics

Einstein’s Principle of RelativityEinstein’s Principle of Relativity

Resolves the contradiction between Galilean Resolves the contradiction between Galilean relativity and the fact that the speed of light is relativity and the fact that the speed of light is the same for all observersthe same for all observers

PostulatesPostulates The Principle of RelativityThe Principle of Relativity: All the laws of physics : All the laws of physics

are the same in all inertial framesare the same in all inertial frames The constancy of the speed of light:The constancy of the speed of light: The speed of The speed of

light in a vacuum has the same value in all inertial light in a vacuum has the same value in all inertial reference frames, regardless of the velocity of the reference frames, regardless of the velocity of the observer or the velocity of the source emitting the observer or the velocity of the source emitting the lightlight

General Physics

The Principle of RelativityThe Principle of Relativity

The results of The results of any kindany kind of experiment performed in of experiment performed in one laboratory at rest must be the same as when one laboratory at rest must be the same as when performed in another laboratory moving at a performed in another laboratory moving at a constant velocity relative to the first oneconstant velocity relative to the first one

No preferred inertial reference frame existsNo preferred inertial reference frame exists It is impossible to detect absolute motion with It is impossible to detect absolute motion with

respect to an absolute frame of referencerespect to an absolute frame of reference

General Physics

The Constancy of the Speed of The Constancy of the Speed of LightLight

Been confirmed experimentally in many waysBeen confirmed experimentally in many ways A direct demonstration involves measuring the speed of A direct demonstration involves measuring the speed of

photons emitted by particles traveling near the speed of lightphotons emitted by particles traveling near the speed of light Confirms the speed of light to five significant figuresConfirms the speed of light to five significant figures

Explains the null result of the Michelson-Morley Explains the null result of the Michelson-Morley experiment – relative motion is unimportant when experiment – relative motion is unimportant when measuring the speed of lightmeasuring the speed of light We must alter our common-sense notions of space and timeWe must alter our common-sense notions of space and time

General Physics

Consequences of Special Consequences of Special RelativityRelativity

In relativistic mechanicsIn relativistic mechanics There is no such thing as absolute lengthThere is no such thing as absolute length There is no such thing as absolute timeThere is no such thing as absolute time Events at different locations that are observed to Events at different locations that are observed to

occur simultaneously in one frame are not observed occur simultaneously in one frame are not observed to be simultaneous in another frame moving uniformly to be simultaneous in another frame moving uniformly past the firstpast the first

In Special Relativity, Einstein abandoned the In Special Relativity, Einstein abandoned the assumption of simultaneityassumption of simultaneity

General Physics

Thought experimentThought experiment A boxcar moves with A boxcar moves with

uniform velocity uniform velocity vv Two lightning bolts strike Two lightning bolts strike

the endsthe ends Flashes leave points A’ Flashes leave points A’

and B’ on the car and and B’ on the car and points A and B on the points A and B on the ground at speed cground at speed c

Simultaneity – Thought ExperimentSimultaneity – Thought Experiment

• Observer O is midway between the points of Observer O is midway between the points of lightning strikes on the ground, A and Blightning strikes on the ground, A and B

• Observer O’ is midway between the points of Observer O’ is midway between the points of lightning strikes on the boxcar, A’ and B’lightning strikes on the boxcar, A’ and B’

General Physics

The light signals reach observer O at the same timeThe light signals reach observer O at the same time He concludes the light has traveled at the same speed over equal He concludes the light has traveled at the same speed over equal

distancesdistances Observer O concludes the lightning bolts occurred simultaneouslyObserver O concludes the lightning bolts occurred simultaneously

Simultaneity – ResultsSimultaneity – Results

General Physics

Simultaneity – Results, contSimultaneity – Results, cont

By the time the light has reached observer O, observer O’ on the car By the time the light has reached observer O, observer O’ on the car has movedhas moved

The light from B’ has already moved by observer O’, but the light from The light from B’ has already moved by observer O’, but the light from A’ has not yet reached himA’ has not yet reached him The two observers must find that light travels at the same speedThe two observers must find that light travels at the same speed Observer O’ concludes the lightning struck the front of the boxcar before it Observer O’ concludes the lightning struck the front of the boxcar before it

struck the back (they were not simultaneous events)struck the back (they were not simultaneous events)

General Physics

Simultaneity – SummarySimultaneity – Summary

Two events that are simultaneous in one Two events that are simultaneous in one reference frame are in general not simultaneous reference frame are in general not simultaneous in a second reference frame moving relative to in a second reference frame moving relative to the firstthe first

That is, simultaneity is not an absolute concept, That is, simultaneity is not an absolute concept, but rather one that depends on the state of but rather one that depends on the state of motion of the observermotion of the observer In the thought experiment, both observers are correct, In the thought experiment, both observers are correct,

because there is no preferred inertial reference framebecause there is no preferred inertial reference frame

General Physics

Time Dilation, Moving ObserverTime Dilation, Moving Observer The vehicle is moving to the The vehicle is moving to the

right with speed vright with speed v A mirror is fixed to the ceiling A mirror is fixed to the ceiling

of the vehicleof the vehicle An observer, O’, at rest in An observer, O’, at rest in

this system holds a laser a this system holds a laser a distance d below the mirrordistance d below the mirror

The laser emits a pulse of The laser emits a pulse of light directed at the mirror light directed at the mirror (event 1) and the pulse (event 1) and the pulse arrives back after being arrives back after being reflected (event 2)reflected (event 2)

General Physics

Time Dilation, Moving ObserverTime Dilation, Moving Observer

Observer O’ carries a clockObserver O’ carries a clock She uses it to measure the time between the events (She uses it to measure the time between the events (ΔΔttpp))

The p stands for properThe p stands for proper

She observes events 1 and 2 to occur at the same placeShe observes events 1 and 2 to occur at the same place Light travels distance 2d = cΔLight travels distance 2d = cΔttpp

The time interval The time interval ΔtΔtpp is called the is called the proper timeproper time The proper time is the time interval between events as The proper time is the time interval between events as

measured by an observer who sees the events occur at the measured by an observer who sees the events occur at the same positionsame position You must be able to correctly identify the observer who You must be able to correctly identify the observer who

measures the proper time intervalmeasures the proper time interval

General Physics

Time Dilation, Stationary ObserverTime Dilation, Stationary Observer

Observer O is a stationary Observer O is a stationary observer on the Earthobserver on the Earth

He observes the mirror and He observes the mirror and O’ to move with velocity O’ to move with velocity vv

By the time the light from By the time the light from the laser reaches the the laser reaches the mirror, the mirror has mirror, the mirror has moved to the rightmoved to the right

The light must travel farther with respect The light must travel farther with respect to O than with respect to O’to O than with respect to O’

General Physics

Observer O carries a clockObserver O carries a clock He uses it to measure the time between He uses it to measure the time between

the events (the events (ΔΔt)t) He observes events 1 and 2 to occur at He observes events 1 and 2 to occur at

different placesdifferent places Events separated by distance vEvents separated by distance vΔΔtt Light travels distance cLight travels distance cΔΔtt

Time Dilation, Stationary ObserverTime Dilation, Stationary Observer

General Physics

Time Dilation, ObservationsTime Dilation, Observations

O and O’ must measure the same O and O’ must measure the same speed of lightspeed of light

The light travels farther for OThe light travels farther for O The time interval, The time interval, Δt, for O is longer Δt, for O is longer

than the time interval for O’, Δtthan the time interval for O’, Δtpp

Observer O measures a longer Observer O measures a longer time interval than observer O’ by time interval than observer O’ by the factor gammathe factor gamma

222

222

ptctvtc

Active Figure: Time Dilation

General Physics

Time Dilation, ExampleTime Dilation, Example

The time interval The time interval Δt between two events Δt between two events measured by an observer moving with measured by an observer moving with respect to a clock is longer than the time respect to a clock is longer than the time interval Δtinterval Δtpp between the same two events between the same two events measured by an observer at rest with measured by an observer at rest with respect to the clockrespect to the clock

For example, when observer O’, moving at For example, when observer O’, moving at v = 0.5c, claims that 1.00 s has passed on v = 0.5c, claims that 1.00 s has passed on the clock, observer O claims that Δt = the clock, observer O claims that Δt = ΔtΔtpp= (1.15)(1.00s) = 1.15 s has passed – = (1.15)(1.00s) = 1.15 s has passed – Observer O considers the clock of O’ to be Observer O considers the clock of O’ to be reading too low a value – “running to slow”reading too low a value – “running to slow”

A clock in motion runs more slowly than an A clock in motion runs more slowly than an identical stationary clockidentical stationary clock

v

O’

pt

O

General Physics

Time Dilation – Equivalent ViewsTime Dilation – Equivalent Views Initial ViewInitial View: Observer O views O’ moving with : Observer O views O’ moving with

speed v to the right and the clock of O’ is running speed v to the right and the clock of O’ is running more slowlymore slowly

Equivalent ViewEquivalent View: Observer O’ views O as the one : Observer O’ views O as the one who is really moving with speed v to the left and the who is really moving with speed v to the left and the clock of O is running more slowlyclock of O is running more slowly

The principle of relativity requires that the views of The principle of relativity requires that the views of the two observers in uniform relative motion must be the two observers in uniform relative motion must be equally valid and capable of being checked equally valid and capable of being checked experimentallyexperimentally

General Physics

Time Dilation – Generalization Time Dilation – Generalization

All physical processes slow down relative All physical processes slow down relative to a clock when those processes occur in to a clock when those processes occur in a frame moving with respect to the clocka frame moving with respect to the clock These processes can be chemical and These processes can be chemical and

biological as well as physicalbiological as well as physical Time dilation is a very real phenomena Time dilation is a very real phenomena

that has been verified by various that has been verified by various experimentsexperiments

General Physics

Time Dilation – VerificationTime Dilation – Verification

Muons are unstable particles that have the Muons are unstable particles that have the same charge as an electron, but a mass 207 same charge as an electron, but a mass 207 times more than an electrontimes more than an electron

Muons have a half-life of Muons have a half-life of ΔtΔtpp = 2.2 µs when = 2.2 µs when measured in a reference frame at rest with measured in a reference frame at rest with respect to them (a) – unlikely to reach the respect to them (a) – unlikely to reach the Earth’s surface.Earth’s surface.

Relative to an observer on earth, muons Relative to an observer on earth, muons should have a longer lifetime of Δtshould have a longer lifetime of Δtpp = = Δt Δtpp (b) – likely to reach surface(b) – likely to reach surface

A CERN experiment measured lifetimes in A CERN experiment measured lifetimes in agreement with the predictions of relativityagreement with the predictions of relativity

General Physics

Length ContractionLength Contraction

The measured distance between two points The measured distance between two points depends on the frame of reference of the depends on the frame of reference of the observerobserver

The The proper lengthproper length, L, Lpp, of an object is the length , of an object is the length of the object measured by someone at rest of the object measured by someone at rest relative to the objectrelative to the object

The length of an object measured in a reference The length of an object measured in a reference frame that is moving with respect to the object is frame that is moving with respect to the object is always less than the proper lengthalways less than the proper length This effect is known as This effect is known as length contractionlength contraction

General Physics

Length Contraction – EquationLength Contraction – Equation

Length contraction Length contraction takes place only along takes place only along the direction of motion the direction of motion

2

21PP

L vL L

c

Active Figure: Length Contraction

General Physics

Length Contraction, ExampleLength Contraction, Example

The length between two points L The length between two points L measured measured by an observer moving with respect to a by an observer moving with respect to a ruler is shorter than the length Lruler is shorter than the length Lpp between between the same two points measured by an the same two points measured by an observer at rest with respect to the rulerobserver at rest with respect to the ruler

For example, when observer O’, moving at For example, when observer O’, moving at v = 0.5c, claims that a length of 1.00 m is v = 0.5c, claims that a length of 1.00 m is measured by a ruler, observer O claims that measured by a ruler, observer O claims that L = LL = Lp /p / = (1.00 m)/(1.15) = 0.87 m is the = (1.00 m)/(1.15) = 0.87 m is the measured length between the two points – measured length between the two points – Observer O considers the length of O’ to be Observer O considers the length of O’ to be “contracted”“contracted”

A ruler in motion is contracted compared to A ruler in motion is contracted compared to an identical stationary ruleran identical stationary ruler

v

O’

O

pL