Chapter 36

The Theory of Relativity

When doing experiments on a boat sailing at constant speed on a straight line on a smooth lake, the results are the same

as on shore!

Without looking outside one cannot tell whether the ship is

moving at all

Velocity is RELATIVE to the reference frame

The idea of relativity

Frames of reference

A set of rods and rulers and clocks that allow us to determine the time

of any event and the position of any object with respect to a

reference point

Reference point

object

An inertial reference frame is a reference frame where isolated bodies are seen to move in straight lines at constant velocity.

Absolute space in its own nature, without relation to anything external, remains always similar and immovable.

The arena whereNature occurs

Space according to Newton

IsJustOne

ThingAfter

Another

Time according to Newton

Absolute and mathematical time, of itself, and from its own nature, flows equally without relation to anything external

Not all frames are inertial:

Roberta is in a non-inertial

reference frame Ingrid is in an inertial

reference frame

Just shoot me

Not moving

Relative and absolute velocities

And I claim everything , including light, will behave like that: absolute space and time imply all velocities (including c) are relative

v=Speed of this bullet with respect to the target

u= its speed with respect to the tank

w= the speed of the tank

v=u+w

Just shoot me…again

The motion of the source and the speed of light

And I claim the speed of light is the same for all observers: c is absolute!

Meet Einstein

Hi

The principle of relativity

0

QdAE

0AB d

dt

dd BsE

dt

dId E

000 sB

Remember these?

Remember this? – the wave eqn2 2

2 2 2

constant constant

1

v

v

( , ) sin( )x t

d D d D

dt dx

A solution is

k

D x t A kx t

Electromagnetic waves If we look at Maxwell’s eqns where there are no charges or currents -

after a bit of math we will get…

constant2

2

constant2

2

00

tx

dx

d

dt

d EE

2 2

2 2 2

constant constant

1

vx t

d D d D

dt dx

8

0 0

1v 2.99792 10 m/sc

Maxwell’s equations: light moves with speed c

Maxwell’s equations are some of the laws of physics

If we accept the principle of relativity, light always moves with speed c

The speed of light

By stating this Einstein alsoimplied that Newtonianmechanics is flawed,

but Maxwell’s equationsare OK

The velocity of the source just adds to the velocity of the projectile…

…even for light

The velocity of the source affects the velocity of the projectile…

But the effect is smaller and smaller as the projectile speed approaches c

The speed of light is unaffected by the motion of the source

And this happens to

be right

This happens to be wrong

The speed of light is independent of the speed of the source:

The speed of light is absolute

The speed of bullets depends on the motion of the source

The speed of bullets is relative

The faster a bullet is shotthe smaller the effect of the motion of the gun

Two mirrors

Time and Special Relativity

Look first at a “light clock”.

We need:

…and a light

Time is relative….In dog-years I’d be dead

At rest with the mirrors

t=0t=1

t=1/2

reflection

Moving with respect to the mirrors

Motion of mirrors

reflection

h

Light always travels at speed c

v Mirrors at rest

vT/2

h2 :mirrors staticfor path ofLength seconds 2every sclock tick static The 0 h/cT

2/2/ 22 cTvTh

02

0

2 /1/1

/2T

cv

T

cv

chT

seconds every sclock tick moving The T

If you ride with the clock the up-down trip takes 1 second

If you watch the clock move the up down trip takes more than 1 second

Moving clocks slow down

v Mirrors at rest

I’m moving with this clock. In my reference frame the trip takes T0

seconds

for trip theframe reference In my

seconds akesclock t thisc2h

T0

seconds akesclock t that

for trip theframe reference In

2

0

cv

1

T

my

He’s at rest with

this mirror

Us

John

boom

Distance that separated us from John

In the reference frame where we are at rest:6 minutes

Us

John

boom

Distance that separated us from John

In the reference frame where John is at rest:

5 minutes

Separation in our rest frame

In the reference frame where we are at rest:

In the reference frame where John is at rest:

Separation in John’s rest frameDifferent

distances

5 minutes

6 minutes

If the principle of relativity is correct

length is relative

The numerical calculations:

In our reference frame:t = fuse-timel = distance traveled by Johnv = John’s speed of approach

In John’s reference frame:t’ = fuse-timel’ = distance traveled by usv’ = our speed of approach

2

2

2

/1'

/1

1

' '

' '

/1

' '

cvll

cvt

t

l

l

lvtlvt

cv

ttvv

Time dilation

Length contraction

Exercise 9.3

In a laboratory frame of reference an observer notes that Newton’s 2nd law is valid (to the accuracy of the instruments). Show that the same is true for an observer moving at constant speed with respect to the first.

dt

d

m

va

aF

:observerFirst

FaaF

avuvuvv

a

uuvv'

' '

)(''

constant ;

:observer Second

mmdt

d

dt

d

dt

d

dt

d

dt

d

An astronomer on Earth observes a meteoroid approaching Earth at a speed of 0.8c. At the time of discovery it is 20 light years away. Find the Earth-time and meteoroid time for impact and the distance to Earth measured in the meteoroid frame of reference at time of discovery. 1 light year ~ 1016m=1year*c

1717 8

Earth 8

2 10 m) 0 8 , 20 light-years=2 10 m 8 10 s~25 years

(0 8)(3 10 m)a v . c d t

.

sccdc year-light 126.020)/8.0(120 ) 2met

met

met metEarth met2

12) 15 years

0 8Alternative: on the meteoroid the Earth clocks are slow so

25 years 15 years0.61 (0.8)

cb t

. c

t tt t

Exercise 9.11

A spacecraft with proper length of 300 m takes 0.75 s to pass an Earthly observer. What is the spacecraft speed with respect to Earth?

c.v

cv

cv

v

cv

80 75.0/

/1

1075.0/1300

2

62

sv

l75.0Earth

m 300 ;/1 02

0Earth lcvll

A meter stick is moving along its long axis near the speed of light. A stationary observer will observe that the meter stick is

• a. longer than one meter.

• b. shorter than one meter.

• c. one meter in length.

• d. moving faster than the speed of light.

• e. moving into the future.

The addition of velocities

22 /1'

/' 1

'

cvu

vuu

cvu

vuu

v=velocity of reference frameu’=velocity of object in reference frame

u=velocity on earth

Exercise

A spacecraft moves at 0.8 c relative to Earth and fires a rocket in the forward direction moving at 0.9 c. What must be the rocket’s speed relative to the Earth?

2 2

v=rocket speed=0.8c

In the spacecraft frame: ' rocket speed=0.9c

' 0.8 0.9On earth: 0.99

1 '/ 1 (0.8 ')(0.9 ) /

u

v u c cu c

vu c u c c

E=mc2

If the principle of relativity is correct,

Energy and mass are equivalent

E=m c2

Exercise 9.39

The power output of the sun is 3.77 x 1026 W. How much mass is converted into energy every second?

kg/s1019.41077.3

time

Mass 92

26

c

second.per nsmillion to 4...about

In a nuclear power plant fuel rods last 3 years. During that time the plant produces 1 GW of power, what is the loss of mass in the rods?

2

2

9

16

Energy produced Power time mass-lost

Power timemass-lost

10 (3 365 24 3600)

9 101.04 kg

c

c

If the radiance of a thousand sunsWere to burst at once into the sky,That would be like the splendor of the Mighty One...I am become Death,The shatterer of Worlds.

Hiroshima:• 66,000 people died and 69,000 injured from the blast, • In a few weeks the number of dead had risen to 130,000• Then to 200,000 in 5 years

Nagasaki • Between 60,000 and 70,000 people died form the blast, • 50,000 more died within 5 years.

The fate of more than 300,000 people was sealed in the fraction of a second it took these bombs to explode.

For comparison, there were about 620,000 casualties in the American civil war that lasted 5 years.

Current atomic weapons are roughly 100 times more potent than those dropped on Japan in 1945.

• 1. What happens to the way a clock measures time as observed by a stationary observer as the clock increases its speed?

• a. It speeds up.• b. It slows down.• c. It stays the same.• d. It stops.• e. The hands move with simple harmonic

motion.

Which of the following is a postulate of relativity theory?

• a. Space and time are invariant.

• b. Mass and energy are equivalent.

• c. The speed of light in a vacuum is the same for all observers no matter what the source velocity is.

• d. Energy is conserved in elastic collisions.

• e. Gravity can bend light.