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Relativity

Special Relativity

If we both measure the same object with the same

tools, should we get the same result?

What does it mean to measure something?

Should the laws of physics be the same for

everybody?

What does it mean to know something?

Youre in a spacecraft and a comet zips by. Are

you moving or is the comet moving?

What does it mean to be in motion?

We have this idea that physical reality,

whatever that is, ought to be independent of

who/when/where/how a measurement is made.

Electromagnetic theory was perfected by Maxwell

and others in the late 1800's.

Water waves propagate through water, sound waves

propagate through air.

It is not critical to electromagnetic theory, but

it was believed that electromagnetic waves

propagated through the ether, relative to some

universal reference frame.

The ether, being ethereal, proved very difficult

to detect!

Imagine the ether attached to this universal

reference frame. If you are moving relative to

it, you experience an ether drift.

T1

Newtonian Relativity Theory

You swim 200 meters downstream in a river, turn

around, and swim 200 yards upstream. It takes a

time T1.

You swim 200 meters perpendicular to the river

bank, turn around, and swim 200 yards back. It

takes a time T2.

T2

T1 and T2 are different. Newtonian relativity

theory shows you how to calculate T1 and T2.

river current

If you make the same measurement on light

moving through the ether, you ought to get the

same result, T1 and T2 are different.

Michelson and Morley built an interferometer

capable of making such a measurement.

mirror

partial mirror

A

light source

mirror

?

B

Half the light follows path A.

hypothetical ether drift

Half the light follows path B.

detector

The dashed line portions of the paths are

oriented differently relative to the ether drift.

If the times to travel paths A and B are the

same, the two light beams arrive in phase and

interfere constructively. If the times are

different, the beams interfere destructively. Mea

surement of changes in interference fringe

shifts allow you to deduce the time difference.

A

?

B

hypothetical ether drift

But wait! you object. It is impossible to make

paths A and B exactly the same length. An

observed fringe shift might be due to the path

length difference, or it might be due to the

different orientations of the path relative to

the ether drift.

So you take a measurement, rotate the apparatus

90 degrees in the horizontal plane, and take

another measurement. The difference between the

two measurements allows you to very precisely

measure the time difference due only to the ether

drift.

Michelson and Morley did the experiment in July,

1887. They found nothing.

No ether drift. (Less than 5

km/s current upper limit is 15 m/s.)

No ether.

They tried it again later, in case during the

July measurement the earth was coincidentally at

rest with respect to the ether. They got the

same results.

No universal frame of reference?

Ive seen sources that say this result wasnt

terribly bothersome, because the ether was a

conceptual convenience, and was not required to

make EM theory work. Ive seen other sources

that say this was devastating at the time. It

certainly created a problem.

In fact, if you believe Michelson and Morley and

Maxwell, you are forced to conclude that the

speed of EM radiation is the same in all

non-accelerated reference frames, regardless of

the motion of the radiation source. A bit

difficult to accept!

How do you reconcile the lack of a universal

reference frame with the idea that everybody's

measurement of the same thing ought to produce

the same result?

Einstein, 1905, Special Theory of Relativity

Special theory of relativity treats problems

involving inertial (non-accelerated) frames of

reference.

We believe the laws of physics work and are the

same for everybody.

Experiment demands that the speed of light be

constant.

Lets make these two things our postulates and

see where we are led. The validity of the

postulates will be demonstrated if the

predictions arising from them are verified by

experiment.

Postulates of special theory of relativity

? the laws of physics are the same in all

inertial reference frames

? the speed of light in free space has the same

value for all inertial observers

The first makes sense. The second is required

by experiments but contradicts our intuition and

common sense.

independent of the motion of the source or

relative speeds of observers!

Time Dilation a consequence of the two

assumptions

Lets begin with a definition, and then construct

a clock.

The time interval between two events which occur

at the same place in an observers frame of

reference is called the proper time of the

interval between the events. We use t0 to denote

proper time.

Suppose you are timing an event by clicking a

stopwatch on at the start and off at the end. In

order for the stopwatch to measure the proper

time, the start and stop events must occur at

the same place in your frame of reference.

Youve been chosen to be a timer at a track meet,

so you go stand by the finish line. You start

your stopwatch when you see the puff of smoke

from the starters gun at the starting line, and

stop it when the first runner crosses the finish

line. Did you measure the proper time for the

sprint?

Now lets make a clock.

mirror

L0

tick

tock

?

laser with built-in light detector

It's not that I'm so smart , it's just that I

stay with problems longer.A. Einstein

How long does a tick-tock take?

mirror

time distance / velocity

t0 2L0 / c

L0

I measure proper time because the light pulse

starts and stops at the same place.

?

laser

Now put this clock in a transparent spacecraft

and observe as it speeds past.

I dont measure proper time because tick and

tock occur in different places.

L0

tick

tock

?

?

entire clock moves with speed v

How long does a tick-tock take? Let the total

time be t.

distance velocity time

( L02 (vt/2)2 )1/2

( L02 (vt/2)2 )1/2

L0

?

?

vt/2

vt/2

v

According to the second postulate of special

relativity, light travels at a speed c, so D

ct. We also know the proper time from our

stationary clock experiment t0 2L0 / c

Solving (1) and (2) for t and replacing L0 using

(3) gives

Note that (1-v2/c2)1/2 lt 1 so t gt t0. It takes

longer for an event to happen when it takes place

(is timed) in a reference frame moving relative

to the observer than when in takes place in the

observer's reference frame. Time is dilated.

This applies to all clocks.

Everybody chant a moving clock ticks slower, a

moving clock ticks slower, a moving clock ticks

slower If a moving clock ticks slower, it

counts fewer seconds.

How to remember what dilated means. Pupils in

your eye can dilate or contract. Dilate must be

the opposite of contract, so dilate must mean

take on a larger value.

A moving clock ticks slower.

If I time an event which starts and stops in in

my frame of reference, I measure t0. If I use my

clock to time the same event as it takes place in

a reference frame moving relative to me, I

measure tgtt0.

In the latter case, I claim my clock, which

measured t, is correct, so that an identical

moving clock, which would measure t0 in the

moving reference frame, is slow.

An event must be specified by stating both its

space and time coordinates.

Example the Apollo 11 spacecraft that went to

the moon traveled a maximum speed of 10840 m/s.

An event observed by an astronaut in the

spacecraft takes an hour. How long does an earth

observer say the event took?

Problem Solving Step 0. Think first!

Always ask what is the reference frame of the

event? Is the observer in this reference frame

or moving relative to it? The event took place

in the spacecraft. The proper time t0 is the

time measured in the spacecraft. Thus, t0 3600

s. The observer in this problem is the person

on earth, not the astronaut! The earth observer

measures t.

Problem Solving Step 1. Draw (if appropriate) a

fully-labeled diagram. (Include values of known

quantities.)

If it helps you to draw a sketch of the earth, a

spacecraft, and a couple of stick figures, do so!

Include values of known quantities. c 3x108, v

10840, t0 3600. If you use SI units

throughout, your answer will be in SI units, and

I only need to see units with your final

answer. If you mix systems of units, show the

units at each step. You can always show units

at each step if it helps you.

Sooner or later, if you mix units, you will

suffer pain.

Problem Solving Step 2. OSE.

So far, we only have one relativity OSE

Put your hand on a hot stove for a minute, and

it seems like an hour. Sit with a pretty girl for

an hour, and it seems like a minute. THAT'S

relativity. A. Einstein.

Problem Solving Steps 3 and 4. Solve

algebraically first, then substitute values.

The algebra is already done in this case.

t 3600.00000235 s.

Not a big difference, but it is measurable. The

actual experiment has been done with jets flying

around the earth, and the predicted time dilation

has been observed.1

As expected, the earth observer measures a bigger

number for the time. The moving clock on the

spacecraft measured a smaller number. The moving

clock ticks slower.

1J. C. Hafele and R. C. Keating, Science 177, 186

(1972).

What if vgtc? It can't happen. Well see later

that it would take an infinite amount of energy

to accelerate an object up to the speed of light.

What about time running backwards? Sorry, time

always runs forwards.

What about seeing an event before it happens?

Can't, because c is finite.

However, because of time dilation, events which

appear to be simultaneous in one reference frame

may not appear to be simultaneous in another

reference frame.

The only reason for time is so that everything

doesn't happen at once. A. Einstein

On the constancy of c

Recent research suggests that c may not be

constant

Several researchers in Australia have been

studying light absorbed by distant gas clouds

about 12 billion years ago. The fine structure

in spectral lines (i.e., the spacing of multiple

lines close together) depends mathematically on

the fine structure constant

In this formula, e is the charge of an electron,

?0 is a constant you encountered in EM, c is the

speed of light, and h is another constant

important in quantum mechanics.

Everything in the formula for ? is a constant.

However, the data obtained by the Australian

group suggest that ? had a larger value when the

light they observed was emitted than it does now.

? seems to have changed by 0.001 in 12000000000

years. Thats a change of 0.00000000000008 per

year.

If ? has been increasing over time, then either

the charge on an electron has been increasing, or

?0, h, or c have been decreasing.

According to a commentary put out by the

American Physical Society

So it is an admittedly biased opinion of the

commentary author.

Since the effect on the laws of physics of

increasing the electronic charge are too awful to

contemplate, they figure light is going slower.

That kills relativity, but my mail indicates

nobody but physicists believe that stuff

anyway.Bob Park

The Australian researchers have reported on this

work three times in recent years, including 2001

in Physical Review Letters and in 2002 in

Nature. It seems like I am constantly hearing

new reports of findings that c is decreasing,

but it is all essentially this one group

reporting their work as it progresses.

Arguably the most prestigious US physics

journal. The most prestigious science journal

known to man.

To date no one else has reproduced this

result. Some possibilities

? The Australians could have made a mistake

(unlikely). ? Their results could be a

statistical fluke (unlikely). ? A

yet-undiscovered systematic error could have

influenced their results. ? The interpretation

could be wrong. ? They are correct. ? ???

This is important enough that others will be

investigating carefully. We should know the

results within a few years.

Im not picking on Australians. They are as

smart as we are. I am using the country of

origin as a convenient way to identify the

research group.

What if they are right?

Theories you will learn in this class will

supersede theories you learned earlier (e.g.,

Newtonian Mechanics). You should not think of

the earlier theories as being wrong. Rather,

the new theories are better, and incorporate the

old ones within them. Relativistic mechanics

reduces to Newtonian mechanics in the limit of

small relative velocities. Use Newtonian

mechanics when the error introduced is small

(because it is easier to use). Use relativity

only if you must!

"If we knew what it was we were doing, it would

not be called research, would it?A. Einstein

If they are right ? There will be profound

implications for cosmological theories. ?

Someone will have to re-think special relativity.

Someone will have to come up with a new theory

which incorporates all of special relativity but

goes beyond it to include the slowly-changing

value of c. ? This may have profound

implications for mankind (as did special

relativity). It may not. Well see. ?

Newtonian mechanics will still work just fine as

long as velocities are not too big. ? Lots of

physicists will have nice jobs for a long time to

come.

And now, back to our show

Lets consider another problem that time dilation

helps us solve.

Has anyone here ever felt a muon?

Does anybody even know what a muon is?

A muon is an elementary particle with a mass 207

times that of an electron, and a charge of either

e or e. Muons are created in abundance at

altitudes of 6 km or more when cosmic rays

collide with nuclei in the atmosphere. Fortunatel

y, muons interact only very weakly with matter,

which is why it is OK that many of them are

passing through your body right now.

This is in the upper reaches of the troposphere,

the part of the atmosphere in which we live.

Muons travel with speeds of about 0.998 c (fast!)

and have an average lifetime of 2.2 ?s (2.2x10-6

s).

How far can an average muon travel during its

lifetime? d v t d 0.998 3108 2.210-6

0.66 km.

How can muons get through the 6 or more

kilometers of atmosphere between their birthplace

and us if they only live long enough to travel

0.66 km?

OK, a some will go more than 0.66 km, and some

less, but not many, and not by much. So the

question stands.

Time dilation!

I say the muons clock ticks slow. I say that

while the muon thinks its clock ticks 2.2 ?s, I

observe that it actually ticks

During this time the muon travels a distance d

0.998 3108 34.810-6 10.4 km, so the

average muon will reach me before decaying.

Of course, a muon doesnt think anything, but

we use words like that to help us form a mental

image of the process. If you prefer, imagine a

nano-human riding on the muon and reporting what

he/she sees.

Double-check what is the event, who is the

observer, and who measures the proper time.

The event is the muon living.

The event does not take place at a single

location in my reference frame, so I measure the

dilated time, and the calculation was correct.

One important aspect of relativity is that there

is only one reality. If I see the muon arrive at

the surface of the earth, the muon must agree

that it actually did arrive at the surface of the

earth.

Our average muon says there is no doubt

whatsoever that its lifetime is 2.2 ?s, and

during that time it travels 0.66 km. I say the

muon reaches the surface of the earth. The muon

says it doesnt??

I thought you said time dilation would help us

solve the muon problem. We seem to have created

a new problem. Either we have encountered two

different realities, or else there is

Relativity teaches us the connection between the

different descriptions of one and the same

reality.A. Einstein

Length Contraction

The faster you go, the shorter you are.A.

Einstein

If two observers in relative motion measure

different times for an identical event, what

makes us think they should measure the same

lengths for an identical object?

The formula for length contraction is not

terribly difficult to derive. Ill lend you a

book if you are curious. Here is the formula.

The Proper Length, L0, of an object is its length

as measured in its own rest frame.

An observer measuring the length of an object

moving relative to him will measure a length L

less than the length L0 he would measure if he

were not moving relative to the object.

Let me demonstrate length contraction using a

meter stick

The length contraction occurs only along the

direction of relative motion. A spacecraft

moving past an observer at nearly the speed of

light will seem to be very short in length and

normal diameter.

A muon created at an altitude of 10.4 km would

say that during its lifetime it saw an atmosphere

of length

I say the muon gets to earth because its lifetime

is longer. The muon says it gets to earth

because the atmosphere is shorter. Different

descriptions of the same reality.

Be careful when you talk about the lifetime of a

particle moving with v close to c. You need to

specify the reference frame in which the lifetime

is measured!

The Twin Paradox

A and B are 20 year old twins. A travels on a

spaceship at v 0.8c to a star 20 light years

away and returns. B, left behind on earth, says

the trip takes 220/0.8 50 years. B is 70 years

old when A returns.

B also observes that As clock (which is

identical to Bs) ticks slowly, and records less

time. If the event in question is the ticking of

As clock, then the 50 years calculated above is

the dilated time t (why?).

A light year, y, is the distance light travels

in one year. Thus, y (1 year)(c). If D is a

distance expressed in light years, then the

number of years it takes to travel that distance

at a speed of v is found from time (distance) /

velocity. Thus time in years (distance in

light years) / (velocity expressed as a fraction

of c).

The proper time, which in this case is amount of

time recorded by a clock in the spacecraft, is

is found by solving our time OSE for t0

According to B (who was left back on earth), As

clock only ticked 30 years, so that A is 20 30

50 years old on return to earth.

At the end of the trip, B, left behind, is 70

years old. A, who made the trip, is 50 years

old. Can this be possible?

Yes! Absolutely! and it was verified

experimentally in the jets-around-the-world

experiment mentioned earlier.

Now heres the paradox. A moving clock ticks

slower. This applies to all observers. A, on the

spacecraft, sees B move away and then come back.

A says Bs clock ticks slower. A does the

calculation presented on the last slide and

concludes that at the end of the trip, B is 50

and A is 70.

Thats the famous twin paradox. It would appear

that each twin rightfully claims the other aged

less. Have we discovered an example of the

existence of two different, mutually exclusive

realities?

Remember, there is no absolute reference frame

for specifying motion. Motion is relative! An

observer is free to say I am at rest you are

the one moving!

When you encounter a paradox like this you can be

sure that someone has pulled a fast one on you.

In this case, an unwarranted calculation was made.

Special relativity applies only to observers in

inertial (non-accelerated) reference frames. A

had to accelerate (very rapidly) to leave earth

and get up to speed, and again when turning

around to head home, and a third time when

landing on earth.

A is not allowed to use the equations of special

relativity! B is, and Bs calculation is

correct A comes back 20 years younger.

If you examine the problem carefully, its only

the turning around part that causes A trouble.

Whats poor A to do? Doesnt a moving clock tick

slower? Yes, so evidently during As period of

extreme acceleration, Bs clock (as observed by

A) would tick incredibly fast. Isnt A allowed

to use the laws of physics? Yes, but it would

have to be general relativity.

We wont have completely eliminated the paradox

unless we can find a description for As reality

that agrees with Bs reality.

A, in the spacecraft, needs to reconsider the

distance traveled. During the out portion of

the trip, A will say that the actual distance

traveled was

and that the back portion was also 12 light

years. 24 light years at a speed of 0.8 c takes

30 years so A ages 30 years during the trip, and

comes back at age 50.

B tells A you are younger because your clock

ticked slower.

A says I am younger because the trip covered

less distance than you thought.

Same reality, two different descriptions.

There are a number of famous paradoxes based on

relativistic calculations. Typically, someone

makes an invalid calculation (usually on purpose,

to see if they can trick you).

In another paradox, where a very fast runner

tries to put a 10 meter pole in a 5 meter barn, a

paradox arises because (Ill let you ponder that

and come back to it later).

Electricity and Magnetism

The material weve been studying is fascinating

and thought-provoking, but it is not how

Einsteins theory of relativity came into being.

What led me more or less directly to the special

theory of relativity was the conviction that the

electromagnetic force acting on a body in motion

in a magnetic field was nothing else but an

electric field.A. Einstein.

In other words, Einstein believed that what you

and I might call a magnetic force is really just

an electric force in another inertial reference

frame.

Consider a conducting wire and a positive test

charge.

What force does the test charge feel due to the

charges in the wire?

Repulsion, because there is a closest to the

test ?

No net charge inside the conductor.

No electric field outside the conductor.

No force!

What does the test charge see when an electric

field is applied and current flows?

E

The test charge observes that the space between

the moving electrons is contracted. There are

more electrons in the part of the conductor

nearest the test charge!

The test charge observes that the moving

electrons are closer together than the stationary

protons, and therefore "feels" a Coulomb

attraction.

A human observer is unable to see the electrons,

and attributes the attraction to a magnetic

force generated by the moving charges.

Same reality, two different descriptions! And

both descriptions are incomplete or mildly

troublesome, as we will see shortly

Beisers presentation of this material is

different, but equivalent.

If you think about it, this presentation is

bothersome. To illustrate, I need to talk about

conservation and invariance.

A quantity is relativistically invariant if it

has the same value in all inertial frames of

reference.

The speed of light is relativistically invariant.

Time is not relativistically invariant.

Length is not relativistically invariant.

Electric charge is relativistically invariant.

All observers agree on the total amount of

charge in a system.

A quantity is conserved if it has the same value

before and after some event. Dont confuse

conservation with invariance.

It is a fact that electric charge is both

conserved and relativistically invariant.

Our thought experiment with the conductor and

test charge suggests that a conductor which is

electrically neutral in one reference frame might

not be electrically neutral in another. How can

we reconcile this with charge invariance?

My modern physics textbook author claims there is

no problem, because you have to consider the

entire circuit. Current in one part of the

circuit will be balanced by opposite current in

another part.

Although the explanation is correct, I dont find

it satisfying. Maybe the pole-in-barn paradox

will help us understand.

It seems logical that if moving electrons are

closer in one part of the circuit, they ought to

be closer in other parts of the circuit too, so

that the conductor is no longer neutral and

charge is not conserved.

The Pole-Barn Paradox

A speedy runner carrying a 10-meter pole

approaches a barn that is 5 meters long (short

barn!), with open doors at each end. A farmer

stands nearby, where he can see both front and

back door at the same time.

a) How fast does the runner have to go for the

farmer to observe that the pole fits entirely in

the barn?

b) What will the runner observe?

The answer to a) involves a simple length

contraction calculation.

For the pole to fit in the barn, the farmer must

measure a contracted length L 5 m for the pole

of proper length L0 10 m.

The result is v 0.866 c. If the runner is

going that fast, or faster, the farmer observes

the pole to fit inside the barn.

Length contraction is often called the Lorentz

contraction, named after the scientist who

discovered the mathematical transformations which

lead to the equation for length contraction.

The answer to b) starts with another length

contraction calculation.

The runner is moving

no, the runner isnt moving. The runner sees the

barn moving towards him at a speed of v 0.866 c.

The runner says the speeding barn has a length

equal to

The pole cant possibly fit inside the barn.

How do we explain this paradox? Which

observation is the physical reality?

The answer both observations are correct!

A detailed calculation (I can lend you the book

it is in, if you are interested) shows that the

runner observes the rear end of the barn

arriving at the front end of the pole long

before the front end of the barn arrives at the

rear end of the pole. The pole doesnt fit!

Events which are simultaneous in the farmers

frame of reference (front pole arriving at back

barn and back pole arriving at front barn) are

not simultaneous in the runners frame of

reference.

Remember, the runner sees the barn moving past

him.

Simultaneity is not a universal physical

reality.

Now Im no longer worried about the

test-charge-plus-conductor example. At a certain

instant in time I may observe an excess of moving

negative charge in the portion of the circuit

nearest me, but does not mean I can claim there

is a net excess of moving negative charge in the

entire circuit at that instant in time.

Now where were we before this interruption

started

Because simultaneity is a relative concept and

not an absolute one, physical theories that

require simultaneity in events at different

locations cannot be valid.Beiser, Modern

Physics.

An observer who doesnt know about relativity, or

even one who knows about relativity but invokes

charge invariance, will claim that the conductor

has a neutral charge density and invents a

magnetic force to explain the attraction.

But the magnetic force is present only when

current is flowing. It is not valid to talk

about a separate magnetic force. You must talk

about the electromagnetic force.

What you call magnetic force is just a

manifestation of the Lorentz contraction and

Coulombs law, and is not a separate force of

nature.

The mathematical transformations which lead to

our relativistic equations for length and time

were derived by Lorentz to make Maxwells

equations invariant in inertial reference frames.

Because Maxwells equations are invariant in

inertial reference frames, special relativity

does not demand that we correct them.

On the other hand, when it comes to Newtons Laws

Part of Einsteins genius was realizing that

Lorentz was on to something big!

1.7 Relativistic Momentum

We believe very strongly that momentum is

conserved. Lets see what effect a relativistic

calculation has on momentum.

Heres the essence of the calculation my text

uses Take two observers with identical

(including mass), elastic balls (elastic, so that

kinetic energy is conserved). Have the observers

stand along the y-axis, equal distances away from

the origin, and throw the balls with equal speeds

(call them VA and VB') towards the origin.

B

A

There is no new or exciting physics here. Using

conservation of momentum, you could easily show

that the two balls have equal and opposite

velocities after the collision.

Now, just for kicks, lets put one of the

observers in motion, with a speed v in the x

direction. Call that observer S'. To the other

observer, S, this is what the collision looks

like.

S' throws B

v

S throws A

Let the speeds of the balls as measured by S be

VA and VB and let the y-component of the

(identical) distance each one travels be Y/2.

The travel time (to collision and back) for ball

A as measured by observer S is T0. The travel

time which observer S measures for ball B is the

dilated time T.

Y

The y-components of the distances are identical

because v has no y-component.

According to observer S

According to observer S'

Time dilation

According to observer S, VB is

The two boxed equations give the speeds S

observes for balls A and B.

VBltVA. Same distance, S says B takes longer so B

moves slower.

If we use the classical (Newtonian) definition

for momentum, S says that

If mAmB then pBltpA, as expected -- remember,

VBltVA.

If mA and mB are identical, then momentum is not

conserved.

This analysis is correct, but I find it confusing

because details are left out.

If we could somehow make ball B have more

momentum, then momentum would be conserved.

redmomentum of B

purplemomentum of A

greentotal momentum (not conserved)

Before

After

Non-conservation of momentum is an alarming idea.

What can we do to fix this situation?

Making ball B have more mass would conserve

momentum!

then the problem would be fixed. Kind of.

If

I say kind of because both balls A and B would

be moving. We would really like to compare the

mass of a moving object with the mass of an

identical, non-moving object. If we let VA?0

then we have the proper condition for comparison,

and you can show that,

where m is the mass of the ball at rest, and m(v)

is the mass it needs to have when it is moving,

if we believe in conservation of momentum.

In the old days we then said OK, m is the mass

of the object at rest and m(v) is its mass when

it is moving. Lets call m0 the rest mass and m

the mass when it is moving (relativistic

mass). This notation is consistent with our

equations for time dilation and length

contraction, so we have

Not an OSE this semester. Dont use it!

This was Einsteins original approach, but later

he said it is not good.

When I studied relativity in college, and in the

previous edition of our text.

The new approach is to say Look, mass is mass.

We believe it is something fundamental. If we

believe in conservation of momentum, we had

better change our definition of momentum.

If we define momentum as

where

Then mass is mass, momentum is conserved in our

thought experiment (and in real life), and

relativistic momentum reduces to classical

(Newtonian) momentum in the limit v?0.

More satisfying than saying mass changes with

velocity.

So in this most recent version of our text, we

are always going to use the symbol m for mass.

Its what we called proper mass or rest mass

in the old days.

In the old days, rest mass was relativistically

invariant. Now mass is relativistically

invariant. Same reality, just different use of

words.

Lets make this new notation official.

More consequences of this new definition of

momentum

For finite F, a?0 as v?c. No finite force can

accelerate an object having nonzero mass up to

the speed of light!

When can I use rest mass, and when do I have to

use

Object v v/c m(v)/m

jogger 10 km/h .000000009 1

space shuttle 104 m/s 0.000033 1.0000001

electron 106 m/s 0.0033 1.001

electron 108 m/s 0.333 1.061

1.8 Mass and Energy

From your first-semester physics course

Use the definition of ? and integrate by parts to

get

Assuming potential energy is zero (we can always

choose coordinates to do this), we interpret ?mc2

as total energy.

The box indicates an OSE.

When an object is at rest KE 0, and any energy

that remains is interpreted as the objects rest

energy E0.

When an object is moving, its total energy is

This is really just a variation of the OSE on the

previous slide.

This is the closest youll come to seeing Emc2

in this class. In the old days, E?mc2 would

have been written Emc2.

These equations have a number of interesting

implications.

Mass and energy are two different aspects of the

same thing.

Conservation of energy is actually conservation

of mass-energy.

The c2 in E0mc2 means a little mass is worth a

lot of energy.

Your lunch an example of relativity at work in

everyday life.

Total energy is conserved but not

relativistically invariant. Rest (or proper) mass

is relativistically invariant. Mass is not

conserved! (But it is for the purposes of

chemistry.)

Example when 1 kg (how much is that?) of

dynamite explodes, it releases 5.4x106 joules of

energy. How much mass disappears?

This is actually a conservation of mass-energy

problem. If this material were presented in

Physics 23, I would make you start with your

conservation of mass-energy OSE and derive the

appropriate equations from there.

For Physics 107, it is sufficient to realize that

the problem is just asking what is the mass

equivalent of 5.4x106 joules of energy?

Conservation of mass is a very good approximation!

If we are to claim relativistic mechanics as a

replacement theory for Newtonian mechanics, then

relativistic mechanics had better reduce to

Newtonian mechanics in the limit of small

relative velocities.

Our text shows that for vltltc,

When can I get away with using KE mv2/2, and

when do I have to use KE ?mc2 - mc2?

Use Newtonian KE every time you can get away with

it! Use relativistic KE only when you must!

If v 1x107 m/s (fast!) then mv2/2 is off by

only 0.08. Probably OK to use mv2/2. If v

0.5 c, then mv2/2 is off by 19. Better use

relativity.

Energy and Momentum

Total energy and magnitude of momentum are given

by

With a bit of algebra, you can show

The quantities on the LHS and RHS of the above

equation are relativistically invariant (same for

all inertial observers).

Rearranging

Is it possible for a particle to have no mass?

If m 0, what are E and p?

For a particle with m 0 and v lt c, then E0 and

p0. A non-particle. No such particle.

But if m 0 and v c, then the two equations

above are indeterminate. We cant say one way or

the other.

If m 0 and v c, we must use

The energy of such a particle is E pc. We

could detect this particle! It could exist.

Do you know of any massless particles?

? photon

? neutrino

? graviton

graviton is to gravity as photon is to EM field

Maybe. Nobel prize for you if you show

mneutrino 0.

Maybe. Nobel prize for its discoverer.

Problem gravitational fields much, much weaker

than EM fields.

- Looking ahead
- Particles having KE gtgt E0 (or pc gtgt mc2) become

more photon-like and behave more like waves. - The momentum carried by massless particles is

nonzero (E pc).

Could you stop a freight train with a flashlight?

Could you stop a beam of atoms with a laser beam?

A note on units. We will use the electron volt

(eV) as an energy unit throughout this course.

Variations on the eV 1 meV 10-3 eV (milli) 1

keV 103 eV (kilo) 1 MeV 106 eV (mega) 1 GeV

109 eV (giga)

Because mass and energy are convenient, we

sometimes write masses in energy units.

An electron has a rest mass of 9.11x10-31 kg. If

you plug that mass into E0 mc2, you get an

energy of 511,000 eV, or 511 keV, or 0.511 MeV.

We sometimes write the electron mass as 0.511

MeV/c2.

It is also possible to express momentum in

energy units. An electron might have a

momentum of 0.3 MeV/c.

If you are making a calculation with an equation

like

and you want to use 0.511 MeV/c2 for the electron

mass, please do. It often simplifies the

calculation. But watch out

What is the total energy of an electron that has

a momentum of 1.0 MeV/c?

Notice the convenient cancellation of the cs in

the 2nd step.

Avoid the common mistake dont divide by an

extra c2 or multiply by an extra c2 in the 2nd

step.

General Relativity

Something to think about. Is the mass that goes

in F ma (or the relativistic version) the same

thing as the mass that goes in F Gm1m2/r2?

Not necessarily!

Experimentally, the two kinds of mass are the

same to within better than one part in 1012, and

must of us believe they are the same anyway, so

An observer in a closed laboratory cannot

distinguish between the effects of a

gravitational field or an acceleration of the

lab.

The principle of equivalence.

The principle of equivalence leads one to

conclude that light must be deflected by a

gravitational field.

Experimental observation of this effect in 1919

was one of Einsteins great triumphs.

We investigate more about light and gravity in

modern physics classes.

"If A equals success, then the formula is

AXYZ. X is work. Y is play. Z is keep your

mouth shut.A. Einstein

More Einstein quotes As far as the laws of

mathematics refer to reality, they are not

certain, and as far as they are certain, they do

not refer to reality." "Relativity teaches us the

connection between the different descriptions of

one and the same reality". "I sometimes ask

myself how it came about that I was the one to

develop the theory of relativity. The reason, I

think, is that a normal adult never stops to

think about problems of space and time. These are

things which he has thought about as a child. But

my intellectual development was retarded, as a

result of which I began to wonder about space and

time only when I had already grown up." "The

secret to creativity is knowing how to hide your

sources." "The important thing is not to stop

questioning. "Only two things are infinite, the

universe and human stupidity, and I'm not sure

about the former." "Things should be made as

simple as possible, but not any

simpler." "Sometimes one pays most for the things

one gets for nothing." "Common sense is the

collection of prejudices acquired by age

18. "Strange is our Situation Here Upon

Earth" "If you are out to describe the truth,

leave elegance to the tailor." "I never think of

the future. It comes soon enough. "Not

everything that counts can be counted, and not

everything that can be counted counts." "The

faster you go, the shorter you are." "The

wireless telegraph is not difficult to

understand. The ordinary telegraph is like a very

long cat. You pull the tail in New York, and it

meows in Los Angeles. The wireless is the same,

only without the cat. "

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