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## Theory of Relativity

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Do Maxwell’s Eqns only work in one reference frame?

If so, this would be the rest frame

of the luminiferous Aether.

If so, the speed of light should change throughout the year

upstream,

light moves

slower

downstream,

light moves

faster

“Aether wind”

Michelson-Morley

No aether wind detected: 1907 Nobel Prize

Einstein’s hypotheses:

1. The laws of nature are equally valid

in every inertial reference frame.

Including

Maxwell’s eqns

2. The speed of light in empty space is

same for all inertial observers, regard-

less of their velocity or the velocity of

the source of light.

All observers see light flashes go by them with the same speed

v

No matter how fast the guy on the rocket

is moving!!

c

Both guys see the light flash

travel with velocity = c

Even when the light flash is traveling in an opposite direction

v

c

Both guys see the light flash

travel past with velocity = c

Time depends of state of motion of the observer!!

Events that occur simultaneously according to one observer can occur at different times for other observers

Prior to Einstein, everyone agreed

the distance between events depends

upon the observer, but not the time.

Same events, different observersy’

y’

y

(x2,t2)

(x1,t1)

x

x

(x1’,t1’)

(x2’,t2’)

t’

t’

x1’

x1’

x2’

dist’

x’

x’

x1

x2

x

t

dist

Time is the 4th dimension

Einstein discovered that there is no

“absolute” time, it too depends upon

the state of motion of the observer

Einstein

Space-Time

Newton

Space

&

Time

completely

different

concepts

2 different aspects

of the same thing

How are the times seen by 2 different observers related?

We can figure this out with

simple HS-level math

( + a little effort)

Catch ball on a rocket ship

Event 2: girl catches the ball

wt

v= =4m/s

w=4m

t=1s

Event 1: boy throws the ball

Seen from earth

V0=3m/s

V0=3m/s

Location of the 2

events is different

Elapsed time is

the same

The ball appears

to travel faster

d=(3m)2+(4m)2 =5m

w=4m

v0t=3m

dt

v= = 5m/s

t=1s

Flash a light on a rocket ship

Event 2: light flash reaches the girl

wt0

c=

w

t0

Event 1: boy flashes the light

Seen from earth

V

V

Speed has to

Be the same

Dist is longer

Time must be

longer

d=(vt)2+w2

w

vt

dt

c= =

(vt)2+w2

t

t=?

How is t related to t0?

t= time on Earth clock

t0 = time on moving clock

wt0

c =

(vt)2+w2

t

c =

ct = (vt)2+w2

ct0 = w

(ct)2 = (vt)2+w2

(ct)2 = (vt)2+(ct0)2

(ct)2-(vt)2= (ct0)2

(c2-v2)t2= c2t02

1

1 – v2/c2

c2

c2 – v2

t2 = t02

t2 = t02

1

1 – v2/c2

t= t0

t= g t0

this is called g

1 – v2/c2

Properties of g =Suppose v = 0.01c (i.e. 1% of c)

1

1 – (0.01c)2/c2

1

1 – (0.01)2c2/c2

g =

=

1

1 – 0.0001

1

1 – (0.01)2

1

0.9999

=

=

g =

g = 1.00005

1 – v2/c2

Properties of g = (cont’d)Suppose v = 0.1c (i.e. 10% of c)

1

1 – (0.1c)2/c2

1

1 – (0.1)2c2/c2

g =

=

1

1 – 0.01

1

1 – (0.1)2

1

0.99

=

=

g =

g = 1.005

1 – v2/c2

Other values of g =Suppose v = 0.5c (i.e. 50% of c)

1

1 – (0.5c)2/c2

1

1 – (0.5)2c2/c2

g =

=

1

1 – (0.25)

1

1 – (0.5)2

1

0.75

=

=

g =

g = 1.15

1 – v2/c2

Other values of g =Suppose v = 0.6c (i.e. 60% of c)

1

1 – (0.6c)2/c2

1

1 – (0.6)2c2/c2

g =

=

1

1 – (0.36)

1

1 – (0.6)2

1

0.64

=

=

g =

g = 1.25

1 – v2/c2

Other values of g =Suppose v = 0.8c (i.e. 80% of c)

1

1 – (0.8c)2/c2

1

1 – (0.8)2c2/c2

g =

=

1

1 – (0.64)

1

1 – (0.8)2

1

0.36

=

=

g =

g = 1.67

1 – v2/c2

Other values of g =Suppose v = 0.9c (i.e.90% of c)

1

1 – (0.9c)2/c2

1

1 – (0.9)2c2/c2

g =

=

1

1 – 0.81

1

1 – (0.9)2

1

0.19

=

=

g =

g = 2.29

1 – v2/c2

Other values of g =Suppose v = 0.99c (i.e.99% of c)

1

1 – (0.99c)2/c2

1

1 – (0.99)2c2/c2

g =

=

1

1 – 0.98

1

1 – (0.99)2

1

0.02

=

=

g =

g = 7.07

1 – v2/c2

Other values of g =Suppose v = c

1

1 – (c)2/c2

1

1 – c2/c2

g =

=

1

0

1

1 – 12

1

0

=

=

g =

g =

Infinity!!!

1 – v2/c2

Other values of g =Suppose v = 1.1c

1

1 – (1.1c)2/c2

1

1 – (1.1)2c2/c2

g =

=

1

1-1.21

1

1 – (1.1)2

1

-0.21

=

=

g =

g = ???

Imaginary number!!!

Moving objects appear shorter

Length measured when

object is at rest

L= L0/g

g >1 L < L0

V=0.9999c

V=0.1c

V=0.86c

V=0.99c

time

mass:F=m0a

=m0

t0

a

time=t0

m0

Ft0

m0

change inv =

Ft0

change inv

m0 =

mass

increases!!

gFt0

change inv

Ft

change inv

= gm0

m=

=

m = g m0

t=gt0

by a factor g

Relativistic mass increase

m0 = mass of an object when it

is at rest

“rest mass”

mass of a moving

object increases

g

as vc, m

m = g m0

as an object moves

faster, it gets

harder & harder

to accelerate

by the g factor

v=c

summary

- Moving clocks run slow
- Moving objects appear shorter
- Moving object’s mass increases

By a factor of g

Twin paradox

4.3 light years

Twin brother

& sister

She will travel to

a-centauri (a near-

by star on a special

rocket ship v = 0.9c

He will stay home

& study Phys 100

Light year

distance light travels in 1 year

dist = v x time

=cyr

1cyr = 3x108m/sx3.2x107 s

= 9.6 x 1015 m

We will just use cyr units

& not worry about meters

Time on the boy’s clock

d0=4.3cyr

v=0.9c

v=0.9c

According to the boy

& his clock on Earth:

d0

v

4.3 cyr

0.9c

tout =

=

= 4.8 yrs

d0

v

4.3 cyr

0.9c

tback =

=

= 4.8 yrs

ttotal = tout+tback

= 9.6yrs

What does the boy see on her clock?

d=4.3cyr

v=0.9c

v=0.9c

According to the boy

her clock runs slower

tout

g

4.8 yrs

2.3

tout =

=

= 2.1 yrs

tback

g

4.8 yr

2.3

tback =

=

= 2.1 yrs

ttotal = tout+tback

= 4.2yrs

So, according to the boy:

d=4.3cyr

v=0.9c

v=0.9c

his clock her clock

out: 4.8yrs 2.1yrs

back: 4.8yrs 2.1yrs

- She ages

less

total: 9.6yrs 4.2yrs

But, according to the girl, the boy’s clock is moving &, so, it must be running slower

v=0.9c

According to her, the

boy’s clock on Earth says:

tout

g

2.1 yrs

2.3

tout =

=

= 0.9 yrs

tback

g

2.1 yrs

2.3

tback =

=

= 0.9 yrs

v=0.9c

ttotal = tout+tback

= 1.8yrs

Her clock advances 4.2 yrs& she sees his clock advanceonly 1.8 yrs,

She should think he has aged less than her!!

A contradiction??

Events in the boy’s life:

As seen by her

As seen by him

She leaves

0.9 yrs

4.8 yrs

She arrives

& starts turn

short time

????

Finishes turn

& heads home

0.9 yrs

4.8 yrs

9.6+ yrs

1.8 + ??? yrs

She returns

turning around as seen by her

According to her, these

2 events occur very,very

far apart from each other

He sees her

finish turning

He sees her

start to turn

Time interval between 2 events depends

on the state of motion of the observer

In fact, ???? = 7.8+ years

as seen by him

as seen by her

She leaves

0.9 yrs

4.8 yrs

She arrives

& starts turn

7.8+ yrs

short time

???

Finishes turn

& heads home

0.9 yrs

4.8 yrs

9.6+ yrs

1.8 + ???yrs

9.6+ yrs

She returns

Ladder & Barn Door paradox

Stan & Ollie puzzle over how

to get a 2m long ladder thru

a 1m wide barn door

???

1m

2m

ladder

Ollie remembers Phys 100 & thetheory of relativity

Stan, pick up the ladder & run very fast

1m

tree

2m

ladder

If Stan pushes both ends of theladder simultaneously, Ollie sees the two ends move at different times:

Too late Stan!

Too soon Stan!

1m

clunk

clank

V=0.9c

(g=2.3)

Stan

Ollie

Stan

status

Einstein’s theory of “special relativity” has been carefully tested in many very precise experiments and found to be valid.

Time is truly the 4th dimension of space & time.

test

g=29.3

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