Chapter 26Chapter 26

Special RelativitySpecial Relativity

http://www.physics.wayne.edu/~alan/2140Website/Main.htm

Lecture 13Lecture 13

Modern Physics1.Relativity

Einstein’s RelativityRelativistic Mechanics

Lightning ReviewLightning Review

Last lecture:

1.1. Modern physicsModern physics IntroductionIntroduction Gallilean relativityGallilean relativity Michelson-Morley ExperimentMichelson-Morley Experiment

Review Problem: Consider the pairs of phasors below, each shown at t = 0. All are characterized by a common frequency of oscillation w. If we add the oscillations, the maximum amplitude is achieved for pair

1. (a). 6. (a), (b), and (c). 2. (b). 7. (a) and (c). 3. (c). 8. (b) and (c). 4. (d). 9. need more 5. (e). information

22L CZ R X X

tan L CX X

R

1, 2

2C LX X fLfC

sin 2mv V ft

Review problem: Gallilean RelativityReview problem: Gallilean RelativityAirplanes in FlightAirplanes in Flight

Two airplanes fly paths I and II as specified in figure below. Both planes Two airplanes fly paths I and II as specified in figure below. Both planes have airspeeds of 100 m/s and fly a distance have airspeeds of 100 m/s and fly a distance L L = 200 km. The wind blows = 200 km. The wind blows at 20.0 m/s in the direction shown in the figure. Find (a) the time of flight at 20.0 m/s in the direction shown in the figure. Find (a) the time of flight to each city, (b) the time to return. to each city, (b) the time to return.

Two airplanes fly paths I and II as specified in figure below. Both planes have Two airplanes fly paths I and II as specified in figure below. Both planes have airspeeds of 100 m/s and fly a distance airspeeds of 100 m/s and fly a distance L L = 200 km. The wind blows at 20.0 m/s in = 200 km. The wind blows at 20.0 m/s in the direction shown in the figure. Find (a) the time of flight to each city, (b) the time the direction shown in the figure. Find (a) the time of flight to each city, (b) the time to return.to return.

Given:

L = 200 kmvp = 100 m/svw = 20.0 m/s

Find:

t = ?

Recall that the ground speed of the plane is

gr air windv v v

Second plane (path OA):

2 2 98.0gr air windv v v m s

33200 10

1.67 10120OB

gr

L mt s

v m s

1100 20 120 , orgrv m s m s m s

33200 10

2.04 1098.0OA

gr

L mt s

v m s

First plane (path OB):

Basic ProblemsBasic Problems

The speed of every particle in the universe The speed of every particle in the universe always remains 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 is contrary to modern experimental resultsIt is contrary to modern experimental results Newtonian Mechanics becomes a specialized Newtonian Mechanics becomes a specialized

case of Einstein’s Theory of Special Relativitycase of Einstein’s Theory of Special RelativityWhen speeds are much less than the speed of lightWhen speeds are much less than the speed of light

Galilean Relativity – LimitationsGalilean Relativity – Limitations

Galilean Relativity does Galilean Relativity does notnot apply to experiments in apply to experiments in electricity, magnetism, optics, and other areaselectricity, magnetism, optics, and other areas

Results do not agree with experimentsResults do not agree with experiments The observer should measure the speed of the pulse as v+cThe observer should measure the speed of the pulse as v+c Actually measures the speed as cActually measures the speed as c

Michelson-Morley EquipmentMichelson-Morley Equipment

Used the Michelson Used the Michelson InterferometerInterferometerArm 2 is aligned along the Arm 2 is aligned along the direction of the earth’s direction of the earth’s motion through spacemotion through spaceThe interference pattern The interference pattern was observed while the was observed while the interferometer was rotated interferometer was rotated through 90°through 90°The effect should have The effect should have been to show small, but been to show small, but measurable, shifts in the measurable, shifts in the fringe patternfringe pattern

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

observedobserved

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

The laws of electricity and magnetism are the The laws of electricity and magnetism are the same in all inertial framessame in all inertial frames

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 observersPostulatesPostulates The The Principle of RelativityPrinciple of Relativity: All the laws of physics are : All the laws of physics are

the same in all inertial framesthe same in all inertial frames The The constancy of the speed of lightconstancy of the speed of light: the speed of light : the speed of light

in a vacuum has the same value in all inertial 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 lightobserver or the velocity of the source emitting the light

The Principle of RelativityThe Principle of Relativity

This is a sweeping generalization of the principle This is a sweeping generalization of the principle of Galilean relativity, which refers only to the of Galilean relativity, which refers only to the laws of mechanicslaws of mechanicsThe results of The results of any kindany kind of experiment performed of experiment performed in a laboratory at rest must be the same as when in a laboratory at rest must be the same as when performed in a laboratory moving at a constant performed in a laboratory moving at a constant speed past the first one.speed past the first one.No preferred inertial reference frame existsNo preferred inertial reference frame existsIt is impossible to detect absolute motionIt is impossible to detect absolute motion

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 A direct demonstration involves measuring the speed

of photons emitted by particles traveling near the of photons emitted by particles traveling near the speed of lightspeed 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 experimentexperimentRelative motion is unimportant when measuring Relative motion is unimportant when measuring the speed of lightthe speed of light We must alter our common-sense notions of space We must alter our common-sense notions of space

and timeand time

Consequences of Special Consequences of Special RelativityRelativity

Restricting the discussion to concepts of Restricting the discussion to concepts of length, time, and simultaneitylength, time, and simultaneityIn 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 Events at different locations that are observed

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

SimultaneitySimultaneity

In Special Relativity, Einstein abandoned In Special Relativity, Einstein abandoned the assumption of simultaneitythe assumption of simultaneityThought experiment to show thisThought experiment to show this A boxcar moves with uniform velocityA boxcar moves with uniform velocity Two lightning bolts strike the endsTwo lightning bolts strike the ends The lightning bolts leave marks (A’ and B’) on The lightning bolts leave marks (A’ and B’) on

the car and (A and B) on the groundthe car and (A and B) on the ground Two observers are present: O’ in the boxcar Two observers are present: O’ in the boxcar

and O on the groundand O on the ground

Simultaneity – Thought Simultaneity – Thought Experiment Set-upExperiment Set-up

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’

Simultaneity – Thought Simultaneity – Thought Experiment Results Experiment Results

The light reaches observer O at the same timeThe light reaches observer O at the same time She concludes the light has traveled at the same She concludes the light has traveled at the same

speed over equal distancesspeed over equal distances Observer O concludes the lightning bolts occurred Observer O concludes the lightning bolts occurred

simultaneouslysimultaneously

Simultaneity – Thought Simultaneity – Thought Experiment Results, contExperiment Results, cont

By the time the light has By the time the light has reached observer O, observer reached observer O, observer O’ has movedO’ has movedThe light from B’ has already The light from B’ has already moved by the observer, but moved by the observer, but the light from A’ has not yet the light from A’ has not yet reached himreached him

The two observers must find The two observers must find that light travels at the same that light travels at the same speedspeed

Observer O’ concludes the Observer O’ concludes the lightning struck the front of lightning struck the front of the boxcar before it struck the boxcar before it struck the back (they were not the back (they were not simultaneous events)simultaneous events)

Simultaneity – Thought Simultaneity – Thought Experiment, SummaryExperiment, 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 firstThat 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

Time DilationTime Dilation

A mirror is fixed to the A mirror is fixed to the ceiling of a vehicleceiling of a vehicleThe vehicle is moving to the The vehicle is moving to the right with speed vright with speed vAn 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 mirrorThe 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)

Time Dilation, Moving ObserverTime Dilation, Moving Observer

Observer O’ carries a clockObserver O’ carries a clock

She uses it to measure the time between She uses it to measure the time between the events (the events (ttpp)) She observes the events to occur at the same She observes the events to occur at the same

placeplace ttpp = distance/speed = (2d)/c = distance/speed = (2d)/c

Time Dilation, Stationary Time Dilation, Stationary ObserverObserver

Observer O is a stationary observer on the earthObserver O is a stationary observer on the earthHe observes the mirror and O’ to move with speed vHe observes the mirror and O’ to move with speed vBy the time the light from the laser reaches the mirror, By the time the light from the laser reaches the mirror, the mirror has moved to the rightthe mirror has moved to the rightThe light must travel farther with respect to O than with The light must travel farther with respect to O than with respect to O’respect to O’

Time Dilation, ObservationsTime Dilation, Observations

Both observers must measure the speed Both observers must measure the speed of the light to be cof the light to be c

The light travels farther for OThe light travels farther for O

The time interval, The time interval, t, for O is longer than t, for O is longer than the time interval for O’, the time interval for O’, ttpp

Time Dilation, Time Time Dilation, Time ComparisonsComparisons

Observer O measures Observer O measures a longer time interval a longer time interval than observer O’than observer O’

2

2

cv1

1

p

2

2

p

where

t

cv1

tt

Time Dilation, SummaryTime Dilation, Summary

The time interval The time interval Δt between two events Δt between two events measured by an observer moving with respect to measured by an observer moving with respect to a clock is longer than the time interval Δta clock is longer than the time interval Δtpp

between the same two events measured by an between the same two events measured by an observer at rest with respect to the clockobserver at rest with respect to the clock

A clock moving past an observer at speed v runs A clock moving past an observer at speed v runs more slowly than an identical clock at rest with more slowly than an identical clock at rest with respect to the observer by a factor of respect to the observer by a factor of -1-1

Identifying Proper TimeIdentifying Proper Time

The time interval The time interval ttpp is called the is called the proper proper

timetime The proper time is the time interval between The proper time is the time interval between

events as measured by an observer who sees events as measured by an observer who sees the events occur at the same positionthe events occur at the same position

You must be able to correctly identify the observer You must be able to correctly identify the observer who measures the proper time intervalwho measures the proper time interval

Alternate ViewsAlternate Views

The view of O’ that O is really the one The view of O’ that O is really the one moving with speed v to the left and O’s moving with speed v to the left and O’s clock is running more slowly is just as clock is running more slowly is just as valid as O’s view that O’ was movingvalid as O’s view that O’ was movingThe principle of relativity requires that The principle of relativity requires that the views of the two observers in the views of the two observers in uniform relative motion must be equally uniform relative motion must be equally valid and capable of being checked valid and capable of being checked experimentallyexperimentally

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

Time Dilation Verification – Time Dilation Verification – Muon DecaysMuon Decays

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

Muons have a half-life of Muons have a half-life of ttpp = 2.2µs = 2.2µs

when measured in a reference frame at when measured in a reference frame at rest with respect to them (a)rest with respect to them (a)

Relative to an observer on earth, muons Relative to an observer on earth, muons should have a lifetime of should have a lifetime of ttpp (b) (b)

A CERN experiment measured lifetimes A CERN experiment measured lifetimes in agreement with the predictions of in agreement with the predictions of relativityrelativity

v = 0.99c, = 7.1

QUICK QUIZ 26.1

Imagine that you are an astronaut who is Imagine that you are an astronaut who is being paid according to the time spent being paid according to the time spent traveling in space as measured by a clock traveling in space as measured by a clock on Earth. You take a long voyage traveling on Earth. You take a long voyage traveling at a speed near that of light. Upon your at a speed near that of light. Upon your return to Earth, your paycheck will be: return to Earth, your paycheck will be: (a) smaller than if you had remained on (a) smaller than if you had remained on Earth, (b) larger than if you had remained Earth, (b) larger than if you had remained on Earth, or (c) the same as if you had on Earth, or (c) the same as if you had remained on Earth.remained on Earth.

QUICK QUIZ 26.1 ANSWER

(b). Assuming that your on-duty time (b). Assuming that your on-duty time was kept on Earth, you will be was kept on Earth, you will be pleasantly surprised with a pleasantly surprised with a large large paycheck. Less time will have passed paycheck. Less time will have passed for you in your frame of reference for you in your frame of reference than for your employer back on than for your employer back on Earth.Earth.

The Twin Paradox – The The Twin Paradox – The SituationSituation

A thought experiment involving a set of twins, A thought experiment involving a set of twins, Speedo and GosloSpeedo and Goslo

Speedo travels to Planet X, 20 light years from Speedo travels to Planet X, 20 light years from earthearth His ship travels at 0.95cHis ship travels at 0.95c After reaching planet X, he immediately returns to After reaching planet X, he immediately returns to

earth at the same speedearth at the same speed

When Speedo returns, he has aged 13 years, When Speedo returns, he has aged 13 years, but Goslo has aged 42 yearsbut Goslo has aged 42 years

The Twins’ PerspectivesThe Twins’ Perspectives

Goslo’s perspective is that he was at rest Goslo’s perspective is that he was at rest while Speedo went on the journeywhile Speedo went on the journey

Speedo thinks he was at rest and Goslo Speedo thinks he was at rest and Goslo and the earth raced away from him on a and the earth raced away from him on a 6.5 year journey and then headed back 6.5 year journey and then headed back toward him for another 6.5 yearstoward him for another 6.5 years

The paradox – which twin is the traveler The paradox – which twin is the traveler and which is really older?and which is really older?

The Twin Paradox – The The Twin Paradox – The ResolutionResolution

Relativity applies to reference frames moving at Relativity applies to reference frames moving at uniform speedsuniform speedsThe trip in this thought experiment is not The trip in this thought experiment is not symmetrical since Speedo must experience a symmetrical since Speedo must experience a series of accelerations during the journeyseries of accelerations during the journeyTherefore, Goslo can apply the time dilation Therefore, Goslo can apply the time dilation formula with a proper time of 42 yearsformula with a proper time of 42 years This gives a time for Speedo of 13 years and this This gives a time for Speedo of 13 years and this

agrees with the earlier resultagrees with the earlier result

There is no true paradox since Speedo is not in There is no true paradox since Speedo is not in an inertial framean inertial frame

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 objectThe 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

Length Contraction – EquationLength Contraction – Equation

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

2

2

PP

c

v1L

LL

QUICK QUIZ 26.2

You are packing for a trip to You are packing for a trip to another star, to which you will be another star, to which you will be traveling at 0.99traveling at 0.99cc. Should you buy . Should you buy smaller sizes of your clothing, smaller sizes of your clothing, because you will be skinnier on the because you will be skinnier on the trip? Can you sleep in a smaller trip? Can you sleep in a smaller cabin than usual, because you will cabin than usual, because you will be shorter when you lie down?be shorter when you lie down?

QUICK QUIZ 26.2 ANSWER

The answers to both of these questions is The answers to both of these questions is nono. . Both your clothing and your sleeping cabin are Both your clothing and your sleeping cabin are at rest in your reference frame, thus, they will at rest in your reference frame, thus, they will

have their proper length. There will be no have their proper length. There will be no change in measured lengths of objects within change in measured lengths of objects within

your spacecraft. Another observer, on a your spacecraft. Another observer, on a spacecraft traveling at a high speed relative to spacecraft traveling at a high speed relative to yours, will measure you as thinner (if your body yours, will measure you as thinner (if your body is oriented in a direction perpendicular to the is oriented in a direction perpendicular to the

direction of motion relative to him) or will claim direction of motion relative to him) or will claim that you are able to fit into a shorter sleeping that you are able to fit into a shorter sleeping cabin (if your body is oriented in a direction cabin (if your body is oriented in a direction

parallel to your direction of travel relative to the parallel to your direction of travel relative to the other observer).other observer).

QUICK QUIZ 26.3

You are observing a rocket moving away You are observing a rocket moving away from you. Compared to its length when it from you. Compared to its length when it was at rest on the ground, you will was at rest on the ground, you will measure its length to be (a) shorter, (b) measure its length to be (a) shorter, (b) longer, or (c) the same. Now you see a longer, or (c) the same. Now you see a clock through a window on the rocket. clock through a window on the rocket. Compared to the passage of time Compared to the passage of time measured by the watch on your wrist, you measured by the watch on your wrist, you observe that the passage of time on the observe that the passage of time on the rocket's clock is (d) faster, (e) slower, or rocket's clock is (d) faster, (e) slower, or (f) the same. Answer the same questions if (f) the same. Answer the same questions if the rocket turns around and comes toward the rocket turns around and comes toward you.you.

QUICK QUIZ 26.3 ANSWER

(a), (e). The outgoing rocket will appear (a), (e). The outgoing rocket will appear to have a to have a shorter shorter length and a length and a slower slower clock. The answers are the same for clock. The answers are the same for the incoming rocket. Length the incoming rocket. Length contraction and time dilation depend contraction and time dilation depend only on the magnitude of the relative only on the magnitude of the relative velocity, not on the direction. velocity, not on the direction.

Relativistic DefinitionsRelativistic Definitions

To properly describe the motion of To properly describe the motion of particles within special relativity, Newton’s particles within special relativity, Newton’s laws of motion and the definitions of laws of motion and the definitions of momentum and energy need to be momentum and energy need to be generalizedgeneralized

These generalized definitions reduce to These generalized definitions reduce to the classical ones when the speed is much the classical ones when the speed is much less than cless than c

Relativistic MomentumRelativistic Momentum

To account for conservation of momentum in all To account for conservation of momentum in all inertial frames, the definition must be modifiedinertial frames, the definition must be modified

v is the speed of the particle, m is its mass as v is the speed of the particle, m is its mass as measured by an observer at rest with respect to the measured by an observer at rest with respect to the massmass

When v << c, the denominator approaches 1 and so p When v << c, the denominator approaches 1 and so p approaches mvapproaches mv

mvcv1

mvp

22

Relativistic Addition of VelocitiesRelativistic Addition of Velocities

Galilean relative velocities cannot be applied to Galilean relative velocities cannot be applied to objects moving near the speed of lightobjects moving near the speed of lightEinstein’s modification isEinstein’s modification is

The denominator is a correction based on length The denominator is a correction based on length contraction and time dilationcontraction and time dilation

2dbad

dbadab

cvv

1

vvv

Relativistic CorrectionsRelativistic Corrections

Remember, Remember, relativistic corrections relativistic corrections are needed because are needed because no material objects no material objects can travel faster than can travel faster than the speed of lightthe speed of light

Relativistic EnergyRelativistic Energy

The definition of kinetic energy requires The definition of kinetic energy requires modification in relativistic mechanicsmodification in relativistic mechanics

KE = KE = mcmc22 – mc – mc22

The term mcThe term mc22 is called the is called the rest energyrest energy of the of the object and is independent of its speedobject and is independent of its speed

The term The term mcmc22 is the is the total energytotal energy, E, of the , E, of the object and depends on its speed and its rest object and depends on its speed and its rest energyenergy

Relativistic Energy – Relativistic Energy – ConsequencesConsequences

A particle has energy by virtue of its mass A particle has energy by virtue of its mass alonealone A stationary particle with zero kinetic energy A stationary particle with zero kinetic energy

has an energy proportional to its inertial masshas an energy proportional to its inertial mass

The mass of a particle may be completely The mass of a particle may be completely convertible to energy and pure energy convertible to energy and pure energy may be converted to particlesmay be converted to particles

Energy and Relativistic Energy and Relativistic MomentumMomentum

It is useful to have an expression relating total It is useful to have an expression relating total energy, E, to the relativistic momentum, penergy, E, to the relativistic momentum, p EE22 = p = p22cc22

+ (mc+ (mc22))22

When the particle is at rest, p = 0 and E = mcWhen the particle is at rest, p = 0 and E = mc22

Massless particles (m = 0) have E = pcMassless particles (m = 0) have E = pc This is also used to express masses in energy unitsThis is also used to express masses in energy units

mass of an electron = 9.11 x 10mass of an electron = 9.11 x 10-31-31 kg = 0.511 Me kg = 0.511 Me

Conversion: 1 u = 929.494 MeV/cConversion: 1 u = 929.494 MeV/c22

QUICK QUIZ 26.4

A photon is reflected from a mirror. A photon is reflected from a mirror. True or True or falsefalse: :

(a) Because a photon has a zero mass, it does (a) Because a photon has a zero mass, it does not exert a force on the mirror. (b) Although not exert a force on the mirror. (b) Although the photon has energy, it cannot transfer any the photon has energy, it cannot transfer any

energy to the surface because it has zero energy to the surface because it has zero mass. (c) The photon carries momentum, and mass. (c) The photon carries momentum, and when it reflects off the mirror, it undergoes a when it reflects off the mirror, it undergoes a change in momentum and exerts a force on change in momentum and exerts a force on the mirror. (d) Although the photon carries the mirror. (d) Although the photon carries

momentum, its change in momentum is zero momentum, its change in momentum is zero when it reflects from the mirror, so it cannot when it reflects from the mirror, so it cannot

exert a force on the mirror.exert a force on the mirror.

QUICK QUIZ 26.4 ANSWER

(a) False (b) False (c) True (d) False (a) False (b) False (c) True (d) False A reflected photon does exert a force on the surface. Although A reflected photon does exert a force on the surface. Although

a photon has zero mass, a photon does carry momentum. a photon has zero mass, a photon does carry momentum. When it reflects from a surface, there is a change in the When it reflects from a surface, there is a change in the momentum, just like the change in momentum of a ball momentum, just like the change in momentum of a ball

bouncing off a wall. According to the momentum bouncing off a wall. According to the momentum interpretation of Newton’s second law, a change in interpretation of Newton’s second law, a change in

momentum results in a force on the surface. This concept is momentum results in a force on the surface. This concept is used in theoretical studies of space sailing. These studies used in theoretical studies of space sailing. These studies

propose building nonpowered spacecraft with huge reflective propose building nonpowered spacecraft with huge reflective sails oriented perpendicularly to the rays from the Sun. The sails oriented perpendicularly to the rays from the Sun. The large number of photons from the Sun reflecting from the large number of photons from the Sun reflecting from the

surface of the sail will exert a force which, although small, will surface of the sail will exert a force which, although small, will provide a continuous acceleration. This would allow the provide a continuous acceleration. This would allow the

spacecraft to travel to other planets without fuel.spacecraft to travel to other planets without fuel.

Pair ProductionPair Production

An electron and a positron An electron and a positron are produced and the are produced and the photon disappearsphoton disappears

A positron is the antiparticle A positron is the antiparticle of the electron, same mass of the electron, same mass but opposite chargebut opposite charge

Energy, momentum, and Energy, momentum, and charge must be charge must be conserved during the conserved during the processprocess

The minimum energy The minimum energy required is 2mrequired is 2mee = 1.04 = 1.04

MeVMeV

Pair AnnihilationPair Annihilation

In pair annihilation, In pair annihilation, an electron-positron an electron-positron pair produces two pair produces two photonsphotons The inverse of pair The inverse of pair

productionproduction

It is impossible to It is impossible to create a single create a single photonphoton Momentum must be Momentum must be

conservedconserved