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 ― 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
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
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
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