🚀 go-pugleaf

This is the first of a series of posts on difficulties encountered while 
teaching special relativity.  My overall theme is that the theory is 
neither that complicated nor that odd in its predictions.  Readers are 
assumed to be familiar with the conventional theory, so I will state 
results without always deriving everything.  I have posted these about a year 
ago, but I see that the twin paradox has once again reared its ugly head.

I employ a few basic pedagogical assumptions.  First, in teaching 
something, you should get it right the first time, because any wrong 
ideas (usually covert approximations) that a student learns are very 
difficult to root out.  Second, physics represents the effort to reduce 
the entirety of the universe to a few basic laws and their applications.  
The number of basic laws is (apparently) small.  Students should 
therefore never be quoted complex results without derivations, because 
to do is a gross perversion of the basic message of physics.  A freshman 
has no way to decide whether F=ma is truly more or less fundamental than 
\omega_c = \omega_{o} \sqrt{ 1 + 1/(4 Q^{2})} (the latter being the 
formula, in power adsorption vs. frequency for a damped driven harmonic 
oscillator, of the midpoint between the two half-maxima) unless the 
latter formula is shown to follow from the former.  The phrase "you'll 
be shown this in a more advanced course" is prima facie evidence of the 
incompetence of the instructor, either for not teaching what should be 
taught, or for pretending to teach material that can't be taught until 
the students are more mature.  

The sermon for today is the misnomer "moving clocks run slowly".  The 
phrase refers to the following experiment: 

"Stationary"
Frame  X         x=0                             x = L
                 t=T_{1}                         t=T_{2}

"Moving"         x'=0                            x'=0
particle         t'= T'_{1}                      t'=T'_{2}
in "Moving"
Frame X'.        The above two                   The above two
                 points are                      points are
                 coincident.                     coincident.

In the "stationary" frame, there are two clocks, one each at x = 0 and 
at x = L.  In the moving frame, there is one clock, always located at 
x'=0 (after all, it doesn't move in the frame in which it is 
stationary).  One observes  experimentally \Delta T' = T'_{2}-T'_{1} 
< T_{2} - T_{1} = \Delta T, an effect known (on this side of the 
Atlantic) as "time dilation".  The effect is sometimes reduced to the 
mnemonic "a moving clock runs slowly".  

The mnemonic ``moving clocks run slowly'' is a cue to the student, a way 
to remember a specific physical configuration and the associated 
calculation.  As a statement of the physics, however, the mnemonic is 
simply wrong, because it posits a class of objects, namely "moving 
objects", whose objective existence is explicitly excluded by special 
relativity, and because it misses the point of the associated 
calculation, in a way that allows people to believe that they have found 
contradictions in special relativity:  

(1) In special relativity, the laws of nature are supposed to be the 
same in all inertial frames of reference.  The labelling of different 
frames as "stationary" or "moving", with a law of nature that 
distinguishes between stationary and moving frames, is inadmissable.  

(2) The actual physical distinction between frames X and X' is not that 
frame X is stationary and frame X' is moving.  We usually treat frame X 
as two clocks on the surface of the earth, and frame X' as a moving 
clock attached to a nuclear particle or an airplane.  However, one can 
equally well say that X' is a single clock bolted to a large mountain, 
while X is a pair of clocks attached to the bow and stern of an airplane 
flying by the mountain.  No matter which description of the three clocks 
is used, no matter whether X or X' is attached to the object that the 
scientifically ignorant would say is "moving", special relativity 
predicts the same relationship between X and X'.  

What, then, is the correct description of the experiment?  The 
fundamental point is that Frame X involves time measurements made with 
two spatially separated clocks, while frame X' involves time measurement 
made using one clock.  A physically accurate statement of the experiment 
is that "one-clock measurements run slower than two-clock measurements". 
In my experience, students who are given a mnemonic in terms of "the 
number of clocks in a measurement" rather than "differences between 
moving and stationary clocks" are much less likely to make incorrect 
applications of what is, after all, a mnemonic, not a law of nature or a 
physical principle.  

What is the difference between a mnemonic and a physical principle?  A 
principle is a statement that can be used in logical reasoning to deduce 
new consequences.  A mnemonic is a short code, which need not be 
accurate, that serves to a label for a calculation.   It is only 
appropriate to teach mnemonics for calculations that are so familiar that 
the student can reproduce them without substantial thought.  "Moving 
clocks run slowly" is a mnemonic, not a principle.  For example, if (see 
above ) the two-clock frame were attached to an airplane, while the one-
clock frame is attached to the mountaintop, special relativity
indicates that the moving clocks run *faster* than the 
stationary clock.  

How can both frames agree that clocks in the other frame are running 
more slowly, without paradox arising?  The answer lies in Einstein's 
important contribution to the Lorentz-Fitzgerald ideas,  a contribution 
playing the same logical role in relativity that Maxwell's displacement 
current played in Maxwell's electromagnetics, a special relativity 
effect summarized loosely as "displaced clocks are out of phase".  I'll 
discuss this in a further posting.  

George D. J. Phillies (phillies@wpi.WPI.EDU   508-831-5334) 
   paper mail: Physics/WPI, Worcester MA 01609