PH 301

Dr. Cecilia VogelLecture 2

Review

Outline

Relativity classical relativity

Einstein’s postulates

Constancy of speed of light consequence: time dilation consequence: Doppler effect

Recall Classical Relativity very close to true when v<<c:

Different observers measure same time Different observers measure same

distance between objects Different observers measure different

position and velocity of each other. Pattern: of another object. Pattern:

Different observers conclude the same laws of mechanics apply

AB BAv vu u v

Postulates Classical relativity not quite right

Einstein's relativity right (so far) Einstein’s postulates

Laws of physics are the same for all inertial (constant velocity) observers

Speed of light is the same, independent of the motion of source or observer.

Postulates Classical relativity not quite right

Einstein's relativity right (so far) Einstein’s postulates

Laws of physics are the same for all inertial (constant velocity) observers

Speed of light is the same, independent of the motion of source or observer.

You Can Hide But You Can’t Run Speed of light is measured to be c =

3X108 m/s by all. Can you catch up? NO! If you chase a light beam, it will still

recede from you at 3X108 m/s Can you run away? NO!If you fly away from a light beam, it will

still catch up to you at 3X108 m/s

What if the source moves?Light from a moving bulb still moves at

3X108 m/s relative to you

Some Consequences Can be derived from constancy of

speed of light:

Time interval between events depends on observers state of motion

Length of object or length of a trip depends on observers state of motion

Recall Classical Relativity Suppose two observers time the pretzel

you throw and catch. One observer on airplane, one on Earth. Same pretzel.

Go-stop. t’=5 s

Go-------------------------stop.t=?Classical relativity says

this is also 5 s.

Recall Classical Relativity

At any point, let the velocity of the pretzel measured by the plane observer be u’.

Then the velocity measured by Earth observer is u = u’ + v , therefore u is faster than u’.

Pretzel goes farther, faster in Earth frame. Same time

Compared to this frame,

in this frame, the pretzel goes…farther

Now Einstein’s Relativity That worked for pretzels, what about light? Person on super-plane shines light at mirror. Suppose two observers time the light that

shines and reflects. One observer on plane, one on Earth. Same light.

Go-stop. t’=5 s

Go----------- stop. t=?

Now Einstein’s Relativity

Compared to this frame,

In this frame, light goes farther At any point, the velocity of the light measured by

the plane observer is c. And the velocity measured by Earth observer is

also c. Light goes further at the same speed in Earth

frame t is longer than t’!!

d d

vt/2

Time Dilation Equation

Eliminating d between these equations:

d d

vt/2

2 od c t

222

2

td v c t

2

21

ottvc

Time Dilation General result:

t and to are both time interval between same two events! measured by two different observers

v is relative velocity of two observers Notice that if v<<c, the times are approximately the

same Hard part: which time is which?

2

1

ottvc

Proper time What’s the difference between t and to?

to is the “proper time” Proper time is always shortest. Def: measured in frame in which the two

events happen at the same place. For example

Person who shines light, since light comes right back.

You measure proper time between your heartbeats. Person who takes a trip measures proper time of trip,

since departure and arrival both happen “right here”

Example Nick travels to a planet 12 light-years away

at a speed of 0.6 c. John stays on Earth. Each measures the trip to take a different amount of time. Note:

A light-year is distance light goes in a year d = (3X108 m/s)(1 yr) = 9.46X1015 m d = ( c )( 1 yr) = 1 c-yr

The values in problem are relative to Earth. Question: How long does the trip take

according to each?

Solution In John’s (Earth’s) frame (in any one

frame), the laws of physics hold, including d = vt, or t = d/v John measures time = (12 c-yr)/(0.6c) = 20yr

To find time in another frame (Nick’s), we need to use time dilation:

2

1

ottvc

Solution

2

1

ottvc

Who measures proper time? Nick – departure and arrival both “right here” John does not – departure is “right here,” but

arrival is way away on another planet. t = 20 yr, to = ?

220c-yr

0.61

ot

cc

220c-yr

1 0.6

ot

16yrot

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 ― ApproachingNow 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