Announcements
Today
• Reading for Friday: TZD 1.7 – 1.8
• Homework #1 is posted; due at beginning
of class on Friday next week.
• First midterm is on Tuesday, February 9,
7:30pm-9:15pm. Contact me immediately
if you cannot make this date! Need special
accommodations? See me!
• Office hours: Wed. 1pm – 3pm. Gamow
tower F-527.
• Motivation for special relativity
Compare two reference frames
(now in one-dimension only)
Compare two reference frames
(now in one-dimension only)
• Galilean relativity (cont.)
‘Inertial frames’ in classical physics.
Strange things about ‘Speed of light.’
Frame S has origin here.
x
... -3
-2
-1
0
1
2
... -3
-2
-1
x
3 ...
... -3
-2
-1
0
1
2
... -3
-2
-1
3 ...
x’
0
1
2
3 ...
Frame S’ has origin here, at x=3m
according to reference frame S.
(The frame S’ is drawn below S so
you can read both axes.)
x’
0
1
2
3 ...
Observer in S measures ball at x = 2m.
Observer in S’ measures ball at x’ = -1m.
Inertial reference frames
Inertial reference frames
TZD 1.3
V
Imagine a train car (it’s always a train!) moving
with constant velocity with respect to the ground.
The train runs smoothly, so that you can’t tell it’s
moving by feeling the bumps on the track.
Would you expect the laws of Physics to be different
inside this train compared to the labs here at CU?
Inertial reference frames
Comparing inertial frames
x
V
... -3
-2
-1
0
1
2
... -3
-2
-1
3 ...
v
x’
Now, you’re playing pool on the train. The balls
roll in straight lines on the table (assuming you put
no English on them). In other words, the usual
Newtonian law of inertia still holds. The frame as
a whole is not accelerating.
0
1
2
3 ...
Here are two inertial reference frames, moving with
respect to one another.
According to S, S’ is moving to the right, with v = 1 m/s.
Comparing inertial frames
Comparing inertial frames
v
x
... -3
-2
-1
0
1
2
... -3
-2
-1
3 ...
... -3
-2
-1
0
1
2
3 ...
... -3
-2
-1
0
1
2
3 ...
v
x’
0
1
2
3 ...
Here are two inertial reference frames, moving with
respect to one another.
At time t = 0, the two frames coincide. A ball is at rest in
frame S. Its position is
According to S’, S is moving to the left, with v = -1 m/s.
• x = 2 m in S
• x’ = 2 m in S’
Comparing inertial frames
... -3
-2
-1
0
1
2
... -3
-2
-1
v
3 ...
0
1
2
3 ...
Frame S’ is moving to the right (relative to S) at v=1m/s.
At time t = 3 sec, the position of the ball is
• x = 2 m in S
• x’ = -1 m in S’
Important conclusion
• Where something is depends on when you
check on it (and on the movement of your
own reference frame).
• Time and space are not independent
quantities; they are related by velocity.
• Definition: An event is a measurement of
where something is and when it is there.
( x, y , z , t )
Galilean position transformation
Galilean velocity transformation
u
x
... -3
If S’ is moving with speed v in the positive x direction
relative to S, then its coordinates in S’ are
-2
... -3
-2
-1
v
3 ...
0
1
2
3 ...
Velocity of the object is therefore
u' =
dx ′ d
dx
= ( x − vt ) =
−v = u −v
dt dt
dt
Dynamics
Dynamics
In inertial frame S, we have (in x-direction, say)
F
F = ma
F
S
0
2
x' = x − vt
Note: In Galilean relativity, time is measured the
same in both reference frames; why wouldn’t it
be?
-1
1
If an object has velocity u in frame S, and if frame S’
is moving with velocity v relative to frame S, then the
position of object in S’ is
z′ = z
t′ = t
-2
0
x’
x′ = x − vt
y′ = y
... -3
-1
1
2
3 ...
How about in inertial frame S’? Well,
F' = F
F
since you’re still applying the same forces, and
v
a' =
S’
... -3
-2
-1
0
1
2
du
dv ′ d
= (u − v) =
=a
dt
dt dt
3 ...
since there’s no additional acceleration in an
inertial frame.
Galilean relativity
The laws of
mechanics
(good old F=ma)
are the same in any
inertial frame of
reference.
Einstein’s
First Postulate of Relativity
If S is an inertial frame and if a second
frame S’ moves with constant velocity
relative to S, then S’ is also an inertial
frame.
Peculiar light-waves
Ideas behind Einstein’s relativity
• A sound wave propagates through air, with
velocity relative to the air (~330 m / sec)
• A water wave propagates in water, with velocity
relative to the water (1..100 m / sec)
• “The wave” propagates through a crowd in a
stadium, with velocity relative to the stadium.
• An electromagnetic wave propagates through...
Is there an ether ?
There where various other
motivations for special
relativity, but for simplicity we
will focus here on the quest
for detecting the ‘ether.’
Frame of reference
Michelson and Morley
Observer on the sun:
‘Ether’
v
Performed a famous
experiment that effectively
measured the speed of
light in different directions
with respect to the “ether
wind.”
Ether ‘viewed’ in the laboratory on the earth:
-v
Ether in the laboratory frame
v
L
-v
Michelson and Morley
v
u-v
-v
-v
If the ether would be a river, we could measure the speed
of the water using a boat that travels at a known speed u.
(u is the relative velocity between the boat and the water.)
If the boat travels the distance L within the time t, then we
know v: L=(u-v)t, therefore v = u – L/t
Very difficult in practice! u = c
t ~ 10ns and v ~ 0.0001*c.
We would have to measure t with an absolute precision of
~0.0000000000001s and we have to know c very precisely!
u+v
L
How can we measure the speed v of the ether?
B
A
u-v
L
Compare the round-trip times tA and tB for paths A and B.
This has the great benefit, that we do not have to
measure the absolute times tA and tB (which are only a
few ns) and we are less sensitive to uncertainties in the
speed of light.
Michelson and Morley
Mirrors
L
-v
Detector
-v
L
Semi-transparent
mirror
“Interferometer”
Light source
The detector measures differences in the position of the maxima or minima
of the light-waves of each of the two beams. (Yes, light is a wave!)
The historic setup (~1887)
-v
u+v
2L
⋅ β
c
-v
(Homework!)
L
u-v
t1 =
t2 =
L
L
2L
+
=
⋅ β , β = 1v 2
c −v
c+v
c
1− 2
c
∆t = t1 − t2 ≈
2
L v
⋅ ,
c c2
∆L ≈ L ⋅
2
v
c2
Michelson-Morley experimental results
Over a period of about 50 years, the Michelson-Morley
experiment was repeated with growing levels of
sophistication. The overall result is a high level of confidence
that the wavelength shift is consistent with zero.
Michelson, 1881
Michelson & Morley 1887
Morley & Miller, 1902-04
Illingworth, 1927
Joos,1930
L (cm)
120
1100
3220
200
2100
Calc.
0.04
0.40
1.13
0.07
0.75
Meas.
0.02
0.01
0.015
0.0004
0.002
Ratio
2
40
80
175
375
Shankland, et al., Rev. Mod. Phys. 27, 167 (1955)
Yes, but...
Michelson and Morley
They thought that the
experiment was a
complete failure because
no effect was found.
Michelson was awarded
the Nobel Prize in 1907!!
True result:
Speed of light is the
same in all directions!
Homework (part of your reading assignment): Work out
the math for this experiment (TDZ, Chapter 1.5)
There is no ether
Q: What if the ether
is “dragged along”
the surface of the
earth, like air flowing
around a tennis ball?
A: If so, this would
require a “viscosity”
of the ether, and
would require rewriting Maxwell’s
equations.
Remark: Lots of effort tried to
save the idea of the ‘ether’, but
none held up.
Einstein’s
Second Postulate of Relativity
The speed of light is the same in
all inertial frames of reference.
This was new in 1905 when Einstein
proposed it. Now it has been
experimentally tested lots of times.
Electromagnetic waves are special. A time-changing
electric field induces a magnetic field, and vice-versa.
A medium (“ether”) is not necessary.
Assignments
• For Friday: Please study TZD 1.7-1.8
• Written homework#1 due next Friday.
Homework assignments are posted on the
course web page
http://www.colorado.edu/physics/phys2130/