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1
A meter stick is moving with increasing speed
over a horizontal grate with 5-cm slots.
Eventually the meter stick will be moving fast
enough that its length in the rest frame of the
grate is contracted to less than the 5-cm width
of a slot. The situation is different as seen
from the rest frame of the stick, where the
slots are contracted. Would observers in the two
frames disagree about whether it could fall into
a slot?
As with many so-called paradoxes of relativity,
our reasoning breaks down because we have assumed
properties that no real objects can have, in this
case perfect rigidity. A gravitational force
acts on any part of the stick that is
unsupported. If the gravitational force is
strong enough to accelerate the
Lorentz-contracted stick in the grid frame into a
slot before it reaches the other edge, then in
its own rest frame the uncontracted stick will be
deformed so that its leading edge would strike
the grid. There is no physical paradox.
2
Imagine a gigantic pair of scissors with a (more
or less) normal size handle and blades that are
one light-year long. The scissors are open with
a small (few-degree) angle and you squeeze the
handle to quickly close them (in 1 s). The
contact point of the blades moves toward the
scissor tips at a phenomenal speed, a light-year
per second! Does this violate the universal
speed limit?
Again weve mistakenly assumed rigidity. The act
of squeezing the handle stresses the material of
the scissors, sending waves down the lengths of
the blades. We can see the propagation of the
waves as the motion of the point of contact of
the blades, which carries the information that
the scissors are being closed. This information
cannot propagate faster than c (causality!), so
special relativity imposes a real upper limit on
the rigidity of the material of the scissors
blade. This limit is, in fact, far higher than
the rigidity of any real material.
Question How does this case differ from the
laser on the rotating table in Problem 39-2? In
that case the laser spot definitely can sweep
across a cloud at a speed of greater than c. Is
this not a violation of special relativity?
3

• A professor will take a trip on the Photon
  Express, a train with speed 0.8c (?5/3) and
  proper length 150 m. Its route includes a
  100-m-long tunnel. A student recognizes that the
  length of the train in the tunnel frame is
  Lorentz-contracted to 150 m/(5/3) 90 m, so he
  could capture it by closing gates at the ends
  when the train is inside.

• Another student points out that in the trains
  rest frame the tunnel is Lorentz-contracted to 60
  m, so that the train never fits in and could
  never be captured.
• How can these two points of view be reconciled?
  Identify and analyze the key events and use
  spacetime diagrams and L.T.s to show that there
  is no paradox here either.

4
If the speed of light does not depend on the
relative motion of the source and observer, what
does depend on it?

• A lot frequency, wavelength, energy, momentum

Doppler Effect
The medium (air) provides an absolute frame of
reference w.r.t. which v is defined.
No medium, no absolute reference frame w.r.t.
which v is defined. Need L.T.
5

• Light source at O in S. Receiver moves relative
  to S (initial position xr) with velocity u at
  rest in S'.
• Each pulse (wave front) from the source travels
  with c.
• Pulse 1 at t0
• Pulse n1 at tn?, where ?1/f

Intersections of world lines of receiver and
light pulses are arrival events.
Spacetime Diagram
6
The same n pulses must arrive in (t2-t1) in S'
as arrive in (t2-t1) in S.
7

• No preferred reference frame.
• Only relative velocity matters.
• f and ? are the frequency and wavelength in the
  rest frame of the source.
• ugt0 ? recession

f 'gtf Blue Shift for S and R approaching
f ltf Red Shift for S and R moving apart.
Applications meteorology, law enforcement,
astrophysics.
8
Light of f0 2 ? 1015 Hz is reflected back to
its source from a mirror that is moving away at 1
km/s. What is the frequency of the reflected
light?
The mirror sees a redshift as it recedes from the
source. It is reflected at the same frequency,
but is Doppler shifted further into the red as
observed by the source which is receding from the
mirror.
9
Light from quasar Q12081011 shows spectral lines
with wavelengths 4.80 times as large as those
emitted in the same process here on earth. What
is the speed of this objects recession?
Recession speed of 2.75?108 m/s

• Hubbles Law
• Distant galaxies are all observed to be receding.
• Evidence of expansion of the Universe.

10
Mr. Tompkins in Wonderland, by George Gamow
Seeing Relativistic Effects.
For decades physicists imagined that high-speed
objects would be seen as Lorentz-contracted
versions of themselves. The reality is
moreinteresting.
A 3-D object looks rotated rather than simply
contracted as it passes because you get the light
from the F-face at the same time as the light
from the B-face.
11
Geometrical Appearances at Relativistic Speeds
G.D. Scott and H.J. van Driel American Journal of
Physics 38, 971 (1970)

• Used computers to do L.T. and simulated
  light-propagation delays to construct image from
  a particular viewpoint.
• Assume VERY short camera exposures (no blurring)
  and ignore Doppler color change.
• Moving sphere always presents a circular image.
• Apparent rotation backside of the sphere comes
  into view.
• As ? gets bigger, the latitude-longitude grid is
  increasingly distorted

Sphere is tilted forward by 70? so you can see N
pole.
Viewed from 1 diameter away from center