When I began to think about teaching
special relativity for my first modern physics course, I could
usually guess which observer in a given problem would see a
ruler shrink relative to his own. But I wondered, couldn't
we then call the phenomenon "length expansion" from the
vantage point of the frame in which it is shrunk? For the
answer, continue reading...
In my desire to visualize this more concretely, I decided to
illustrate the process with an
x-ct diagram.
Even though the sound byte "length contraction" was familiar to me,
and even though my intuition usually told me how to solve length
contraction problems, I didn't really understand the concept
until this was done. As Jonathon Swift's Gulliver might have
said, "When I could hold the fruit in my hand and taste it, the
inadequacy of my prior wisdom became apparent." This was one of
my first object lessons in the sloppiness of our own language
for dealing with relativistic spacetime effects.
In special relativity terms, "length contraction" problems involve
THREE events - which we might refer to as left measure
event common to both frames, and the right measure at rest
and right measure moving events which take place
simultaneously on the other end of the ruler from the viewpoint
of one observer or another. As I looked at the diagrams I realized
that in length contraction problems, one and only one inertial
frame, namely the "uncontracted one", was accorded and deserved
special status. This special frame, inherent to the statement
of the problem, is the frame in which the ruler is traveling. All
other inertial frames see the rulers in this priviledged frame
as contracted. Thus when you look at any moving ruler, or
equivalently when any yardstick moves, it's length is contracted
in the direction of motion. Moving a ruler never causes
its length to increase or "expand"!
To pose a specific problem, imagine a large ruler, one light-year
in length, and an inertial frame which measures at one of its
"instants" the length of this ruler while traveling at a constant
coordinate velocity of one third of a lightyear per inertial year
(i.e. v = c/3) to the right.
The rest frame of the ruler is chosen as the one with orthogonal axes.
Imagine that the tick marks on the green grid denote times (vertical)
of one year, and distances (horizontal) of one light-year. (This would
work as well with seconds & light-seconds, respectively, if we wanted
the ruler to be shorter.)
The red lines denote the world line (path through time) of the
left and right ends of the ruler. The largest red box denotes the
left side measurement event (i.e. point in time and space when a
frame yardstick is used to measure our ruler). The other two
boxes denote the position of the other end of the ruler at the
instant of the measurement, from the point of view of the frames
in which the ruler is at rest (medium square), and in which it
is moving (smallest square).
The blue grid shows lines of constant maptime t (about
18 degrees up from the horizontal right) and constant position
(about 18 degrees rightward from the vertical) for an inertial
observer moving to the right at one third of a lightyear per
mapyear (i.e. v=c/3). The measurement occurs at
"time zero" for each observer. Note that the point where the
world-line of the right edge of the ruler intersects the blue
(rightward moving observer) time-zero line (denoted by the
smallest red square) is inside of the dotted blue axis
crossing -- hence the measured length of the ruler is less
its resting length of one light-year. Since gamma for the
rightward frame is 1/Sqrt[1-(v/c)^2] = 1/Sqrt[1-1/9] = Sqrt[9/8] = 1.06,
the length measured in the blue frame for the ruler is only
1/1.06 = 0.94 lightyears.
As mentioned on our
x-ct plotting
page, the same information could be obtained for a plot which
uses orthogonal axes for the rightward moving frame. Then of
course the ruler is moving to the left. As in other examples
elsewhere on these pages, the picture looks quite different even
though the information it provides is the same!
