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Re: orthodox special relativity.

This is the second in a series of posts on the topic of teaching orthodox 
special relativity.  The objective is to treat points at which students 
become confused, usually through a failure to gain a complete, precise, 
and correct understanding of various concepts.  As in previous posts, 
equations are expressed as the ASCII form of LaTeX expressions.  

Back to the beginning: coordinate systems, coordinates, and distances.  

A significant fraction of the student errors encountered on actual 
examinations arise from a failure to distinguish correctly between 
coordinate axes, locations, and distances.  One begins by erecting a 
system of coordinates: orthogonal spatial x, y, and z axes, and a time 
coordinate t.  The discussion is limited to inertial (non-accelerated) 
coordinate systems.  

What, asks the student, is an inertial reference frame?  An inertial 
reference frame is a frame in which ${\bf F} = m {\bf a}$ is correct for 
bodies of constant mass, and for ${\bf F}$ limited to orthodox mechanical 
forces (gravity, friction, Coulomb force, ...).  Frames that accelerate or 
rotate with respect to an inertial reference frame are not inertial 
reference frames.  Frames that experience a uniform gravitational 
acceleration (say, the interior of a sealed, freely-falling elevator) pose 
certain challenges treated by general relativity.  Special relativity does 
handle the reference frame "fixed with respect to the fixed stars", and 
reference frames moving at a constant velocity with respect to the "fixed-
star" frame.  

The coordinate system is used to measure the locations (coordinates) of 
various objects.  For example, one may have a point mass P which at time 
t_{1} was located at (x_{1}, y_{1}, z_{1}), and a second point mass Q 
located at the same time at (x_{2}, y_{2}, z_{2}).  $x_{1}$ is the 
numerical value of the x-coordinate at the location of P, and so.  One 
needs to be sure that students distinguish between "the value of x at P$ 
and "x_{1} is a new coordinate axis defined for P".  There is one coordinate 
axis in the x direction, points then having "coordinates", i.e. values of the 
coordinate axis at their location.

[Aside: (Some students gain a mistaken description of x_{1}, believing 
  that x_{1} is not a position of P along x, but instead that x_{1} is a 
  new coordinate axis, set up to show where P is located. This belief 
  leads to wrong solutions for standard problems.  As an example of this 
  error, consider two masses separated by three springs.   X denotes a 
  pair of rigid walls, the springs are represented -oooooo-, the | denote 
  the equilibrium positions of the two masses, and the : are construction 
  lines.  
                                                                        
        X                                                          X
        X                                                          X
        X-oooooooo-P-oooooooooooooooooooooooooooooooooo-Q-oooooooo-X
        X          :                                    :          X
        X          :<----x_{1}----|      |---x_{2}----->:          X

  It is not unknown for students to propose defining x_{1} and x_{2}, 
  with the positive values being directed as shown, and the zero points 
  being the equilibrium locations..  This pictures makes a great deal of 
  sense if x_{1} and x_{2} are new coordinate axes, but no sense at all 
  if one recalls that x_{1} and x_{2} are values of x at P and Q, 
  respectively.  If x_{1} and x_{2} are the values of x at P and Q, then 
  the locations at which x_{1} and x_{2} are zero must be the same, and 
  the directions in which x_{1} and x_{2} assume positive values must be 
  the same.  In contrst, in the figure the locations x_{1}=0 and x_{2}=0
  are different, and the directions in which x_{1}> 0 and x_{2} > 0 are 
  not the same.
  
  If one is very clever, one can construct a mechanics in which the 
  indicated coordinates work, but for typical students at the freshman-
  sophomore level this construction will fail to give correct laws of 
  mechanics.  The student will, for example, fail to notice that (with 
  coordinates as shown) Newton's Third Law (as commonly stated) does not 
  apply to P and Q, because with these peculiar coordinates an action-
  reaction force pair on P and Q will give two forces having the same 
  sign, not opposite signs.)] 

Let us define the coordinate separations between P and Q to be x = x_{2}-
x_{1}, y = y_{2} - y_{1}, and z = z_{2}-z_{1}.  Within a single reference 
frame, the conventional (3-space) distance is r = (x^{2}+y^{2}+z^{2})^{1/2}, 
as given by Pythagoras.  It is important to emphasize to students that, 
within a single inertial reference frame, special relativity does not 
affect conventional space and time measurements.  Special relativity only 
enters when one is comparing measurements made in different reference 
frames.  So long as one compares positions, times, and velocities 
measured in the same reference frame, the conventional notions of time 
and distance remain correct.  Therefore, in a single reference frame the 
sixth-grade formula $d = v t$, where d is the distance of a trip, v is a 
constant velocity, and t is the duration (the elapsed time) of a trip 
made at constant velocity remains valid.  

[Aside: (If the Pythagorean theorem is correct, space-time is locally 
  flat; general relativistic effects are not included.  The Pythagorean 
  theorem restricts one to an open, uncurved universe, and excludes the 
  closed universe noted by recent posters, and the more complexly 
  connected universes treated as a hypothesis by Fang, among others.)] 

Having established that x_{1}, x_{2}, ... are locations of points, whether 
points in space or reference points on material bodies, one can now 
introduce lengths/intervals.  If x_{1} and x_{2} are the locations of the 
two ends of an object, then x_{12} = x_{2} - x_{1} is the length of an 
object.  The definition requires that x_{1} and x_{2} be measured at the 
same moment in time.  The final qualification "at the same moment" is 
often not part of a student's mental apparatus.

[Aside:  Virtually all students know perfectly well how to measure the 
  length of an object by taking a meter stick, finding the coordinates of 
  the two ends of an object, and subtracting.  The constraint that x_{1} 
  and x_{2} be measured simultaneously is usually not recognized.  In 
  class, I demonstrate this by using a toy truck and a meter stick.  
  First I do the rational measurement of the truck's length.  Then I read 
  off coordinate locations of the two ends of the truck as it "drives 
  along the highway".  That is, I move the truck slowly along the meter 
  stick,and announce the coordinate locations of the two ends of the 
  truck, quoting one number, pointing out to the students that the truck 
  is being moved, and then reading out the second number after the truck
  has visibly been displaced through some distance.  By choosing which 
  end's coordinate I determine first, I can get  "lengths" for the truck 
  that are much longer or shorter than the truck's actual length.  One or 
  two demonstrations are usually enough to impress on students the need 
  for simultaneous measurements of r_{1} and r_{2}.] 

It is important to emphasize that r_{1} and r_{12} are quantities 
fundamentally different in nature.  r_{1} and r_{12} have different 
responses to translation.  Namely, if one moves the origin from 0 to A, 
r_{1} becomes r_{1}-A, while r_{12} becomes (r_{2}-A-(r_{1}-A)) = r_{2}-
r_{1} = r_{12}.  Translation changes coordinates, but does not change 
lengths.  While it is true that r_{1} is numerically equal to the length 
L_{1} between the Origin and particle P, r_{1} and L_{1} are fundamentally 
different in their physical nature.  The Lorentz transforms calculate 
coordinates, not lengths, in new reference frames.

Students often become confused about the difference between 
lengths and coordinate locations because instructors and texts sloppily 
put one end of an object at the origin, so that the length of the object 
is numerically equal to the coordinate location of the other end of the 
object.  While a mature physicist understands the senses in which the 
origin of the coordinate system is arbitrary, freshmen are not mature 
physicists.  The shortcut of setting x_{1} = 0 or x_{2} = 0, while it 
saves the instructor a lot of work, risks destroying the student's 
understanding of the length concept.   

Furthermore, the assertion that x_{1} = 0 and x_{2} = L implies that the 
length of the object is L is incorrect.  Lengths are intrinsically 
positive numbers, while L < 0 is allowed as a value of x_{2}.   Only if 
one sketches the diagram to incorporate a covert assumption L > 0 can one 
claim that x_{1} = 0, x_{2} = L proves that an object whose ends are 
x_{1} and x_{2} must have length L.  One commonly finds freshmen and text 
solutions that treat algebraic quantities x, L,...  as though they may 
always be assumed to be positive, even though the underlying physical 
quantities are signed and can have either sign.  Failure to enforce clear 
thinking about signs at the beginning level is very difficult to correct 
at later times, especially when students have been brought up to view 
sign errors as trivial. 

Why do we dwell at such length on the distinction between the coordinates 
of a single point and the length of an object?  We are trying to give 
students a mindset that prevents them from making a certain class of 
error.  The error arises as follows:  First, if one wishes to compute the 
length of an object, it is mathematically convenient to put one end of the 
object at the origin (say, x_{1} = 0) , since one can then write L = 
x_{2}, and equate the length of an object to position of one of its ends 
(subject to the difficulties noted in the previous paragraph).  Second, 
the Lorentz transformations of coordinates are x'= \gamma(x-vt) and 
t'=\gamma(t-vx/c^{2}), so the origin (x,t)=(0,0) is at the same point in 
every inertial reference frame.  Third, if the length of an object is L = 
x_{2}, it incorrectly appears to the naive  that the transformation of L 
from one frame to the next is the same as the transformation of x_{2} 
from one reference frame to the next.  

If the distinction between lengths and coordinates is not rigorously 
maintained, students thus enter the error of computing the length of an 
object in a new reference frame by computing the position L' of one end of 
the object in the new reference frame at an arbitrary time, without 
asking where the other end of the object is, in the new reference frame, 
when the first end of the object was at L'.  A student who confuses a 
length with a coordinate implicitly assumes (without recognizing that an 
assumption was made) that when the first end of the object is at L', the 
other end is at 0.  If a student always checks that the positions of both 
ends of an object are obtained at the same time as measured in the new 
reference frame, a variety of errors are avoided.  Contrariwise, 
students who always transform lengths from one reference frame to another 
by invoking  "length contraction" --- a mnemonic for a particular set of 
conditions --- without understanding those conditions, will get the 
transformation backwards at least half the time.  

Next Discussion: Simultaneity and displaced times.

George D. J. Phillies (Professor of Physics, Worcester Polytechnic 
  Institute, Worcester MA 01609; phillies@wpi.WPI.EDU; 508-831-5334) 
"Getting the general physical idea" is the enemy of "getting it right".