PH 301

Dr. Cecilia VogelLecture 2

Review

Outline length contraction Doppler Lorentz

Consequences of Einstein’s postulates

Constancy of speed of lighttime dilation

Recall Trip Example John saw Nick’s 12 light-year trip

to take 20 years. Nick saw the trip to take 16

years … saw the Earth receding and the other planet approaching for 16 yrs.

Nick’s Frame In Nick’s frame

Earth is at x=vt = -0.6c*16yr = -9.6 c-yr

How far apart are the Earth and the planet in Nick’s frame? 9.6 c-yr

Recall the distance from Earth to planet is 12c-yr according to John!

Lengths and distances are not the same for all observers!

Thus length contraction John measures proper length,

because he is at rest relative to Earth & planet

Nick measures length-contracted distance

Just How Proper is it? If there is a proper time and a proper

length, is there a proper reference frame?

NO!!!! Proper time of trip in example: Nick Proper length of trip in example: John Proper time of astronaut’s heartbeat:

Astronaut’s heartbeat looks ____ to you. Proper time of your heartbeat:

Your heartbeat looks _____ to astronaut.

slow

slow

Astronaut

you

Time Dilation Plus Light source with frequency fo (in its

own frame) Emits N cycles of EM waves

in time to.

N = fo to.

to is the proper time to emit N cycles, since in source’s reference frame all cycles are emitted at same place, “right here”

Additional Effect In another reference frame, the light

source is moving toward the observer.

Time to emit N cycles is given by time dilation equation t = to.

There is a second effect due to the fact that the light takes time to arrive And in that time, the source has moved

ct

vt N’

Doppler Effect Geometry

With this geometry

ct

vt N’

N

tvc

tvtcN

)(

Doppler Effect ― Approaching

Now plug in

21

)(

)(

cv

o

oo

o

f

vc

tf

tvc

Since ’ =c/’, 21

)(

cv

of

vc

f

c

cv

cv

off

1

1 Holds if source and observer approaching

Doppler Effect ― Receding

Can repeat the previous derivation for receding source or observer

cv

cv

off

1

1 Holds if source and observer receding

Holds if source and observer approaching Higher frequency ― blue shift

Lower frequency ― red shift

cv

cv

off

1

1

Doppler Effect ― Evidence Hydrogen absorption spectrum:

moving H-atoms absorb different frequencies than H-atoms at rest in lab. Because they “see” a Doppler-shifted

freq.

Application Laser cooling

Aim a laser with a slight lower freq than an (at-rest) absorption line.

Atoms at rest won’t absorb the laser light. Approaching atoms will “see” a slightly

higher freq such atoms can absorb the laser light this will slow the atoms (head-on)

At-rest atoms unaffected, moving atoms slowed (on average)

Overall effect – slower atoms -- COOLER

Lorentz Transformations Relates time and position of an event

in one frame to those in another frame Event is something that happens at one

place at one time. our class is not an event, because it lasts 75

min New Years day is not an event, because it

happens all over Earth

x and t from x’ and t’ Can be used generally for any event

Recall Classical Relativity

x x vt

recall t t

x x vt

Postulates and Assumptions Postulate: Both the primed and

unprimed observer measure the speed of light to be c.

Assumption: The primed and unprimed frames have agreed on an origin, so that x=0, t=0 corresponds to x’=0, t’=0.

Definition The primed frame moves at velocity v

relative to the unprimed frame (defines v). Which means that the origin of the x’-axis

separates from the origin of the x-axis at velocity v in the positive-x direction

x=0x’=0

t=0

x’=0

vt

More Assumptions The relationships between x, t

and x’, t’ are linear.x x t So that a constant velocity in

one frame will be a constant velocity in the other (Law of inertia). The relationships between x, t

and x’, t’ are symmetric. since neither frame is special the only difference is the sign of

v

Finding and Plug x’=0, t=t, and x=vt into

x=0x’=0

vt0 vt t x x t

v

So the relationship becomes ( )x x vt

Symmetry We found

The symmetric equation, just changing the sign of v is:

( )x x vt

( )x x vt

Timing a Flash of Light If a flash of light occurs at t=t’=x=x’=0,

it will travel away from the origin at speed c. according to both observers.

x=0x’=0

light’swavefront

x=ct x’=ct’

Finding

Solve for :

( )x x vt ( )x x vt Plug x = ct and x’ = ct’ into

( )ct ct vt ( )ct ct vt

2

1

1vc

Lorentz Transformation Eqns

Can also combine x’=(x-vt) and x=(x’+vt’), and solve for t’:

( )x x vt where 2

1

1vc

2

vt t x

c

Example As a Vulcan spacecraft passes planet Bolth

in Jan 2001 at a speed of 0.7 c, it synchronizes its origin of time and position with the Bolthians. Both agree that time is zero and that place is zero. Where and when did we begin lecture #1, relative to the Vulcans on the ship? Earth and Bolth are at rest relative to each other, 53 light-years apart.B E

V

More Example According to Bolthians, our class

started at t=3.75 (Sep 2004), and x= 53 c-yr

( )x x vt

(1.4) 53c-yr (0.7 )(3.75yr)x c

2 2

1 11.400

1 0.71

vc

70.5c-yrx

More Example

The Vulcan’s say our class started 70.5 light-years from them, 46.7 years before they passed Bolth!

2

vt t x

c

2

0.7(1.4) 3.75yr (53c-yr)

ct

c

46.7yrt

Vulcan Timeline x=0 is Vulcan ship

time place Event

-46.7 yr 70.5 c-yr Our class begins

0 0

0

0

Pass planet Bolth

+23.8 yr Light from our class beginning arrives

+23.8 yr + 1day

Vulcans calculate that our class began 70.5 yrs

ago, i.e. t = 23.8 – 70.5 = -46.7

yr