(I didn't update this post on Aug 2, just moved the publication date forward by a two years and a few months, and added this confessional note. But later I'll add more of a substantial something to it.)
Heisenberg’s idea is that only
measurable quantities should enter the theory. Is the length of a moving meter
stick a measurable quantity?
At first we all say, sure, it’s
measured by two measurers who determine where the ends are, simultaneously.
Then we say, how exactly? How do
they locate the two ends of a moving rod simultaneously when they don't
know where the ends are? If there were no length contraction phenomenon, they
would know where the ends of the rod are supposed to be—one meter apart for a
meter stick.
Now there’s a difference in asking
for the measurement to be done in principle, and in actually carrying out that
measurement. Given a repeatable sequence of experiments attempting to measure
the location of the two ends simultaneously, one can zero in on the correct
positions. How would one—two, actually—know when the correct measurement had
occurred?
An air track with gliders provides a
good thought experiment. The measuring devices are the usual photogates, with
one capability added: they must be synchronized. The experiment starts with the
starting of the photogate timers, which are positioned so that they are
separated by a distance equal to the length of the glider—the “rest length.”
Then a glider comes on through. The first photogate is programmed to ignore the
passage of the front of the glider and to stop timing when the back of the
glider passes through it. The second photogate is programmed to stop timing
when the front of the glider passes through it.
A successful length measurement is
achieved when the gates stop timing simultaneously. Which is more of a
detection of the presence of something—the ends of the glider—than a measurement
of some quantity.
In an uncontrolled experiment, where
you’re trying to measure the length of some object whose velocity and rest
length you don’t know and where you only have one chance to make your
measurement—this is basic kinematics—can you find its two ends simultaneously?
Not with just two observers.
One possible realistic way would be
to have two SETS of synchronized photogates placed with arbitrarily precise
separation. Each set would be programmed just like the two single photogates
discussed above. The first set ignores the leading edge of the object, but
stops timing as the trailing edge passes. The second set of photogates stops
timing as the leading edge of the object passes. Here's a drawing of the set-up:
===========================
<<<<<<<<<< >>>>>>>>>>
The double line represents the meter stick, “<” is a trailing-edge
detector and “>” is a leading-edge detector. The total length of this set up
is one meter, corresponding to the longest possible length to be measured. The
10 detectors in each set are spaced at, let’s say, one-millimeter intervals.
Thus, with a precision of one millimeter, lengths ranging from 1.000 (outermost
detectors) to 0.982 meters (innermost detectors) can be measured. (The
separation of the innermost detectors is 0.980 meters, but each one is itself
one millimeter in width, making the smallest measureable length 0.982. See
table below.)
Each detector on the left will
ignore the leading edge of the meter stick and will stop timing as the trailing
edge passes it. Each detector on the right will stop timing as the leading
passes it (and ignore the trailing edge). Length detection is achieved when a
pair (or more than one pair) of detectors is triggered simultaneously. For all
except for the smallest and longest lengths measurable, more than one pair of
detectors will be triggered simultaneously, as shown by the table.
Let the photogates/timers detectors
be numbered 1-20 from left to right. L' is the contracted length.
L' gates triggered simultaneously
1.00
|
1,20
|
.999
|
(1,19) (2,20)
|
.998
|
(1,18) (2,19) (3,20)
|
.997
|
(1,17) (2,18) (3,19) (4,20)
|
.996
|
(1,16) (2,17) (3,18) (4,19) (5,20)
|
.995
|
(1,15) (2,16) (3,17) (4,18) (5,19)
(6,20)
|
.994
|
(1,14) (2,15) (3,16) (4,17) (5,18)
(6,19) (7,20)
|
.993
|
(1,13) (2,14) (3,15) (4,16) (5,17)
(6,18) (7,19) (8,20)
|
.992
|
(1,12) (2,13) (3,14) (4,15) (5,16)
(6,17) (7,18) (8,19) (9,20)
|
.991
|
(1,11) (2,12) (3,13) (4,14) (5,15)
(6,16) (7,17) (8,18) (9,19) (10,20)
|
.990
|
(2,11) (3,12) (4,13) (5,14) (6,15)
(7,16) (8,17) (9,18) (10, 19)
|
.989
|
(3,11) (4,12) (5,13) (6,14) (7,15)
(8,16) (9,17) (10,18)
|
.988
|
(4,11) (5,12) (6,13 (7,14) (8,15)
(9,16) (10,17)
|
.987
|
(5,11) (6,12) (7,13) (8,14) (9,15)
(10, 16)
|
.986
|
(6,11) (7,12) (8,13) (9,14)
(10,15)
|
.985
|
(7,11) (8,12) (9,13) (10,14)
|
.984
|
(8,11) (9,12) (10,13)
|
.983
|
(9,11) (10,12)
|
.982
|
(10,11)
|
In practice, especially
for a very fast meter stick, which is what we’re interested in, it’s likely
there will not be simultaneous times on any pair of (trailing, leading)
detectors. This imprecision in time corresponds to a little "delta-x"
imprecision in length. What does this little delta-x imprecision mean?
Classically, it can be improved upon indefinitely. Quantumly, the limit is set
by (delta-x)(delta-p) > h-bar/2, the Heisenberg uncertainty principle for
simultaneous position and momentum measurements.
How precise the length
measurement can be depends on how the fast the meter stick is traveling. Linear
momentum, p, is mass times speed in Newtonian physics. In relativistic physics,
the "gamma factor" of g
= (1-v2/c2)-1/2 must be included, so p = g mv. And we are assuming or at least desiring a
very fast-moving meter stick so we need the gamma factor and we expect a length
contraction.
We know the rest length,
L. The contracted length we're trying to measure is given by L’ = L/g. This is one equation in two unknowns. Do you
(we, I, or whoever) have some other equation to use? I don't know, but when or
if we do measure L', then v can be calculated to the same precision as L' was
measured. The uncertainty principle seems to be violated by successful length
detection.
Wait. The gamma factor in
relativistic momentum and the gamma factor in length contraction cancel each
other when multiplied together in the uncertainty principle inequality! What
does this mean!? More musing will be required.
_____________________
_____________________
But, in the meantime, let us not
forget Dr. W.G.V. Rosser, who says there ain’t no conception of “length
contraction” unless or until you make a transformation to another reference
frame! The length is whatever you measure, or as Dr. WgV Spunk says, whatever
you “detect.” However, what perforce does that actually tell you,
information-wise? You got data, which you assume to be the times that ends of
stick passed these locations.
What are these hierarchies of
motion, anyway? The speed, acceleration, rate of acceleration, etc, things? And
then there’s a hierarchy within each of these: rest frames moving at constant
relative velocities with respect to each other; frames moving with constant
acceleration with respect to each other, etc. Imaginary, mostly. And what about
rotating frames? Can we imagine a hierarchy of these? The easiest hierarchy to
imagine is concentric rotating frames—all centered about the same point,
rotating at different angular velocities. BUT, then we have isolated a special
point in space, the center of rotation. Hmmm. So, immediately, the rotation
situation is dramatically different in one simple way.
But we don’t have a conception of
space to begin with, except a space that exists with respect to a given mass.
Hoo wee. Can space itself rotate? Ha Ha. Happens all the time.
What is the connection between
spaces and mathematics? We know this. Topology. But what is the relation
between real space and mathematics? Undiscovered. Doglegs and cateyes. Hornye
toad surfaces.
Start the ajp paper pl;ease, okay?
Kay.
Length measurement is a
straightforward process when the object whose length is to be measured is at
rest. Length measurement for a moving object is not a determination of distance
from end to end that can be characterized as having the property of length.
This property is for sale now, since private companies have bought out the
metric system.
No, really.
Another conundrum: If you measure
the time an object takes to pass a single detector, what information do you
have? Time! How do you interpret this time? If you know the rest length of the
object, and you don’t worry about length contraction, then you have a speed
measurement—an average speed measurement: length/time. If you are worried about
length contraction—if you’re the worrying type, which might be good in this
case—then you have two unknowns: the contracted length and the velocity. And
you only have the one time measurement as your experimental data. What’s a
worrywart to do?
| |
1<------------ x ---------------->2
The average speed of the meter stick can be calculated thusly: distance x divided by Δt, the time it takes the leading edge of the meter stick to get from 1 to 2, or
v = x / Δt.
There's also a time ΔT that the stick takes to pass by a single point, such as timer 1 or timer 2. This is the time I was discussing above in the "conundrum" paragraph. But now we have an independent measurement of the average velocity, and so we can find L', the assumed-contracted length, by a simple calculation. The timers are set up so they can measure both Δt and ΔT . Then we can use
L' = v ΔT = (x/Δt ) ΔT
= x (ΔT/Δt).
The right-hand side contains three measured quantities, so this is another way to find the length of the moving meter stick. Has it ever been done for real.....?
| |
1<------------ x ---------------->2
The average speed of the meter stick can be calculated thusly: distance x divided by Δt, the time it takes the leading edge of the meter stick to get from 1 to 2, or
v = x / Δt.
There's also a time ΔT that the stick takes to pass by a single point, such as timer 1 or timer 2. This is the time I was discussing above in the "conundrum" paragraph. But now we have an independent measurement of the average velocity, and so we can find L', the assumed-contracted length, by a simple calculation. The timers are set up so they can measure both Δt and ΔT . Then we can use
L' = v ΔT = (x/Δt ) ΔT
= x (ΔT/Δt).
The right-hand side contains three measured quantities, so this is another way to find the length of the moving meter stick. Has it ever been done for real.....?
