Physics 116
Session 28
Postulates of Relativity
Nov 15, 2011
R. J. Wilkes
Email: ph116@u.washington.edu
Lecture Schedule
(up to exam 3)
Today
2
Examples
From yesterday:
typos corrected
•
A laser emits 2 wavelengths, 420nm and 630nm. At what angle and for what order
will maxima for both wavelengths coincide, for a grating with N lines/mm?
We want same angle to be order number m for one line, and n for the other; both m
and n must be integers:
d sin θ m = mλ1
θ m = sin −1
(
d sin θ n = nλ2
mλ1
= sin −1 mλ1 N
d
(
spacing d (in mm) = 1 / N lines / mm
)
450 lines / mm = 450 × 10−6 lines / nm
)(
(
)
)
(
)
= sin −1 %& m 420 nm 450 × 10−6 nm −1 '( = sin −1 %& m 0.189 '(
θ n = sin −1 %& n 630 nm 450 × 10−6 nm −1 '( = sin −1 %& n 0.2835 '(
* 0.2835 so we want m 0.189 = n 0.2835 → m = n ,
= 1.5n
+ 0.189 /.
Try n=1, 2… and find first value of n that gives integer m
n= 2 gives m=3: so find angle in degrees by putting in m=3 above
)(
(
(
m = -3
n = -2
-2
-1
-1
)
(
)
(
+1
+2
)
)
0
0
+1
+3
+2
3
Clicker quiz question
•  A spy wants to photograph a document through a window
across the street. But his camera produces a blurry image even
when carefully focused. He needs to…
A.  Get a camera with a smaller diameter lens
B.  Get a camera with a larger diameter lens
C.  Get another job
MADtv
D.  It’s physically impossible to do any better
4
Oops: A little problem with Maxwell’s equations
Last time:
•  By 1880s, Maxwell’s work was in everyday technology
–  Every generator, motor, telegraph, telephone proved him right
•  Just like quantum theory today…computer chips, lasers
•  Problem: Maxwell equations don’t obey Galilean relativity !
–  Simple example: imagine Phil holds an electric charge
•  Both standing still: both see only electric field of charge
•  Phil is on moving train, Bill remains at rest:
–  Bill sees moving charge = current → magnetic field B
–  Phil still sees only electric field E of static charge
First: Both are at rest
Both see E field and
no B field
B
P
+
Next: Phil rides on 100 m/s train moving past Bill
Phil sees only E field, but Bill sees a B field
P
B
+
Static charge
Train
Earth
Moving charge = current I
100 m/s
Earth
5
Physics ~ 100 years ago
•  In 1900, it was assumed by almost all physicists that
–  Maxwell’s equations were fundamentally right
•  Some explanation will be found to reconcile them with Gailean relativity
–  EM waves travelled in a medium: the luminiferous ether
•  some explanation will be found to reconcile observations by Michelson
(next topic for us) showing no evidence of “ether wind”
–  No fundamental changes in classical physics concepts are required
•  1905 = Albert Einstein’s “miraculous year”
–  New PhD – couldn’t get a university job, worked in Swiss patent office
–  Published 3 ground-breaking papers
•  Relativity: how to make Maxwell’s equations make sense
Albert Einstein, •  Quantum character of light: explained the photoelectric effect
c. 1900
•  Brownian motion: atoms are real, physical objects, not just a “concept”
Note: our discussion of relativity will be necessarily
brief and limited in depth!
We won’t go into details of correct “relativistic” way
to deal with velocities (29.4), or momentum (29.5).
6
Michelson’s interferometer (c. 1880)
•  Albert Michelson (1852-1931)
–  Nobel Prize 1907 (first American winner) "for his optical precision
instruments and the … investigations carried out with their aid".
•  Simple but effective tool: use interference effects to make ultraprecise length measurements
–  Split a beam of light (waves!) into 2 beams, and let them interfere:
•  If paths differ by λ / 2, get destructive interference: dark fringe
Michelson at US
Naval Academy, c.
1870
•  If shift is by λ, get constructive interference: bright fringe
–  Can measure differences in path length of light beams to accuracy
of λ / 4 (~100 nanometers!)
Michelson at Cal
Tech, c. 1920
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Michelson’s Interferometer
Beamsplitter
•  Use a “beam splitter*” to make 2 light beams from one
–  One part of light beam goes to mirror M1, other to mirror M2
•  Both beams are from the same light waves!
•  M2 can be moved back and forth with precision screw
–  Changes path length for beam 2
* Just a “half-silvered” mirror
–  Reflected light is recombined by beam splitter on return trip
•  Observe results on screen or with eyepiece: light, or dark fringe?
•  Can detect λ / 4 differences between path lengths B1 – B2
screen or detector
Sodium lamp:
~ monochromatic
Original
light beam
B2
Light source
(wavelength λ )
B1
beam
splitter
fixed
mirror M1
Michelson interferometer is the basis of many
precision measurement devices used today
movable
mirror M2
What we measure is actually (speed of light)×(time)
for each beam’s travel to and from the beam splitter
8
Michelson-Morley experiment (c. 1887)
Concept: repeatedly measure speed of
light c, in different directions, using
interferometer with equal path
lengths, over one year.
Expect to see variation as Earth’s
direction through the ether changes.
Results: no change in c  no ether?
Ether
“tail-wind”
Interferometer used (at Case Western Reserve Univ.)
http://www.juliantrubin.com/bigten/michelsonmorley.html
Ether
“head-wind”
9
How can we weasel out of this?
•  Fitzgerald contraction hypothesis:
–  Suggested in 1889 by George Fitzgerald (Trinity College, Dublin)
–  Motion through the ether causes physical bodies to contract
in the direction of their motion
"... if the internal forces of a material system arise wholly from
electromagnetic actions … then the effect of imparting … a
uniform velocity of translation is to produce a uniform
contraction of the system in the direction of motion, of
amount
- Joseph Larmor, Aether and Matter, 1900
–  Voila! Perhaps Michelson and Morley’s apparatus shrank just
enough to make speed of light seem equal in all directions!
–  Notice: the amount of contraction becomes an imaginary
number (= physically meaningless) if v is greater than c!
10
Lorentz’s “transform” equation
•  Maxwell’s equations disagree with Galilean relativity only if relative speed
v is near the speed of light (more than few % of c)
•  Maxwell assumed there is a “universal absolute rest frame” = rest frame
of the luminiferous ether
•  H. Lorentz (c. 1895) found a mathematical method to reconcile Maxwell’s
equations with coordinate transformations
–  “Transform” = formula to translate from one coordinate frame to another
•  x = position in fixed, Earth frame (Bill’s coordinates),
•  x’ = position in moving frame (Phil’s coordinates)
•  Formula (1) gives position of bullet in Phil’s frame (on train) vs its position
in Bill’s frame (at rest); v = speed of train
–  Time is “of course” the same in both frames! Or is it?
Notice: if v is small compared to c, v2/c2~0, and Lorentz transform = Galileo’s
11
Einstein’s genius
•  Lorentz did not develop his transform equations into a
complete physical theory
H. A. Lorentz
–  Possible hint to find a way to “fix” Maxwell’s equations?
•  Accepted view among academic scientists by 1900
•  1905: Einstein took them seriously: what if Nature actually
works that way?
–
–
–
–
Maxwell’s equations work fine for v << c
Lorentz’s transform makes them work for v ~ c
They make no sense at all for v > c
So: What are physics implications of Lorentz transform?
•  Astonishing thought: could time actually differ between
fast-moving coordinate frames?
Einstein, c. 1900
c. 1903, with
Mileva Maric
•  Einstein was deeply interested in electrodynamics and light
•  His dad was manager of a dynamo factory
•  As a child he wondered what the world would look like if
you “rode on a beam of light”
12
Einstein’s crazy ideas
•  Einstein’s conclusions: Relativity Theory* (1905)
–  There is no “luminiferous ether”: light has no medium!
»  Explains Michelson-Morley measurements
–  Speed of light c is fastest anything can go: universal speed limit!
»  Explains the denominators in Lorentz transform equations
–  Speed of light is the same for any observer in any frame!
»  This is required for Lorentz transform to make sense
–  Time passes at different rates in different frames!
•  Events that are simultaneous in one frame are not in others!
»  Inescapable conclusion, if you accept Lorentz transform as real
–  These crazy ideas
•  Make Maxwell’s equations work at any speed
•  But: Really more logical than “common-sense” Galilean relativity
…as we shall see
* Called Special Relativity, to distinguish from his General Relativity (1915)
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Step 1: “operational definition” of time
•  Recall: physics is what you can objectively observe and measure
–  “Operational definition” = recipe for measurement of a physical quantity
•  How do we measure time? With a clock, of course!
–  To study motion of light, use a light clock
•  Lamp flashes briefly
•  Light reflects off mirror
•  Photosensor detects light pulse
1.5 m
•  Photosensor triggers lamp again
For 1.5m spacing, round trip = 3m
c = 300 million m/sec
lamp
Time for round trip = 3m / c = 10-8 sec (10 nanoseconds)
(Neglect the small delay between photosensor and lamp)
So: this clock will “tick” every 10 ns
mirror
(vacuum)
Photosensor
14
Light clock on a moving train
•  Bill and Phil have identical 10 nsec light clocks
–  Both standing still on earth: they agree perfectly
•  Phil takes his clock on a train moving (let’s imagine!)
at 0.1 c
–  Bill sees his clock work normally: 3m path, 10 nanosec ticks
–  Phil also sees his clock work normally: 10 nanosec ticks
•  But what do the say about each other’s clocks?
First: Both are at rest
Next: Phil rides on train (v=0.1c) moving past Bill
P
B
P
B
Train
Earth
0.1 c
Earth
15
Non-Galilean disagreement!
•  Galileo/Newton say physics observations must be identical if two
coordinate frames differ only by a constant relative velocity
•  Bill’s view of Phil’s clock, if Einstein is right
–  Train moves from A to B while light pulse is in flight
•  Light beam’s path length is longer than 1.5m +1.5m
•  Einstein insists light travels at c regardless of vehicle speed
•  So, Phil’s clock tick takes longer than 10 ns according to Bill:
–  “Now Phil’s clock is slower than mine!”
–  “But my clock stayed the same: still 10 ns per tick.”
What Bill sees, as Phil rides on train (v=0.1c) past him
P
P
B
Train
Earth
A
0.1 c
B
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But: Relativity cuts both ways…
•  Now: what is Phil’s view of Bill’s clock, if Einstein is right
–  Phil sees Bill moving backward at 0.1 c
•  According to Einstein, Earth is not a special frame: all are equal
•  Bill’s clock moved backward from C to D (relative to Phil) while
light pulse went from lamp to detector: path is longer than 3m!
•  So, Bill’s clock tick takes longer than 10 ns, according to Phil
–  “Now Bill’s clock is slower than mine!”
–  “But my clock stayed the same: still 10 ns per tick.”
P
B
B
0.1 c
D
Earth
Train
C
You say “But Bill didn’t actually move at all!” Oh yeah? What’s so special about Earth?
Bill moving backward (Phil’s view) is just as valid as Phil moving forward (Bill’s view)
17
Paradox?
•  Each one says the other’s clock is slow: contradiction?
•  Einstein says: both are correct
1899, Henri Poincare:
“…The simultaneity of two events, or the order of
their succession, as well as the equality of two
time intervals, must be defined in such a way that
the statements of the natural laws be as simple as
possible.”
Einstein took Poincare seriously!
•  Paradox does not occur if we reject relativity
Galileo/Newton/Maxwell “classical physics” universe:
–  Time is absolute, and universal (same in all reference frames)
–  Ether rest frame = absolute universal reference frame
•  Speed of vehicle adds/subtracts from light speed
•  So, Q: Why believe in such a silly concept as relativity?
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Einstein’s insight: we live in 4 dimensions, not 3
A: worse contradictions and complications arise if we don’t!
•  The world makes sense only if we treat time in the same way as
space coordinates - then
–  Maxwell’s equations work in any (non-accelerated) frame
–  Michelson’s experiment is explained
(And many, many other things…)
•  3 space dimensions (up-down, N-S, E-W) + time = 4 dimensions
–  Universe occupies a 4-dimensional space-time continuum
–  Time is also relative: it’s just another coordinate
•  We don’t find it peculiar that Bill and Phil measure different x-y-z
coordinates for the same event - why not different times?
–  Objects trace out “world-lines” in 4D spacetime
•  “Event” = something that occurs at some point in spacetime
–  Emission of light pulse, detection of light pulse: some interaction
•  Events are what physics observations must agree upon
–  Not their coordinates! Any coordinate frame is OK
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