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 we’ve 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 train’s 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

' r

s

v vf f

v v

For sound:

The medium (air) provides an absolute frame of reference w.r.t. which v is defined.

For light :

??????

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 t=0

:

Pulse #n+1 at t=n, where =1/f

ct

xO xr

1st P

ulse

(n+1)

th Puls

e

(x1,t1)

(x2,t2)

nc

Intersections of world lines of receiver and light pulses are “arrival events.”

Spacetime

Diagram

1 1 r 1

2 2 r 2( )

x ct x ut

x c t n x ut

6

1 1 r 1 2 2 r 2 ( )x ct x ut x c t n x ut

2 1 2 1 cn ucn

t t x xc u c u

2 12 1 2 1 2

2 1 2

1 2

( )( )

:

x xt t t t u

c

cn u ucnt t

c u

t

u c c

t

Lorentz transform and into S

The same n pulses must arrive in (t’2-t’1) in S' as arrive in (t2-t1) in S.2

2 12

1t t c u

n c u c

c u

c u

11

1

ucf fuc

1

1

ucuc

7

f ‘<f Red Shift

for S and R moving apart.

f '>f Blue Shift

for S and R approaching

1

1

ucf fuc

• No preferred reference frame.

• Only relative velocity matters.

• f and are the frequency and wavelength in the rest frame of the source.

• u>0 recession

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 Q1208+1011 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?

14.8

1

ucuc

24.8 1- 1u u

c c

0.92u

c

Recession speed of 2.75108 m/s

• Hubble’s Law:– Distant galaxies are all

observed to be receding.– Evidence of expansion

of the Universe.

18 -12.3 10 s

Hubble Constant

uD

H

H

10

“Seeing” Relativistic Effects.

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.

For decades physicists imagined that high-speed objects would be “seen” as Lorentz-contracted versions of themselves. The reality is more…interesting.

Mr. Tompkins in Wonderland, by George Gamow

11

Geometrical Appearances at Relativistic Speeds

• 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

G.D. Scott and H.J. van Driel American Journal of Physics 38, 971 (1970)

Sphere is tilted forward by 70 so you can see N pole.

Viewed from 1 diameter away from center