A Budget of Barn Poles

The traditional "barn pole paradox" in special relativity begins with
the observation that a pole whose rest length is (say) twice as great
as the length of the barn could fit entirely within the barn if it was
being carried at great enough speed.  In this condition both the front
and back doors of the barn could be closed, completely enclosing the 
moving rod.  The "paradox" consists of the observation that, with 
respect to the rest frame of the pole, the barn is contracted to just
one fourth the length of the pole, so the pole cannot possibly be fit
inside the barn.  The resolution of this elementary paradox is, of 
course, given by taking into account the relativity of simultaneity,
as illustrated in the figure below.

           

The worldsheet of the barn is described by the two parallel green 
lines, and the lines designated as x and t are the space and time
axes of the barn's inertial rest frame coordinate system.  The 
worldsheet of the pole is indicated by the two parallel blue lines,
and the lines designated as x' and t' are the space and time axes 
of the pole's inertial rest frame coordinate system.  The x,t and
x',t' system are related by a Lorentz transformation for the speed
0.866c.  As shown in this drawing, the spatial extent of the pole
at any time slice t=constant with respect to the barn's inertial 
rest frame coordinates is the segment AB, which is also the barn's 
spatial extent.  Hence the pole just fits inside the barn.  However,
with respect to the inertial rest frame coordinates of the pole, the
barn's spatial extent at any time slice t'=constant is the segment
AD, whereas the pole's spatial extent on this time slice is the
segment AC.  We have the ratio |AD|/|AC| = 1/4, so the pole does 
not fit entirely within the barn at any instant t'=constant.  Of
course, this does not conflict with the fact that it DOES fit entirely
within the barn at one particular instant t=constant.  Naturally if
the back door of the barn is closed when the leading edge of the pole
reaches it, the pole will begin to compress, but the effects of this
compression cannot propagate backwards faster then the speed of light,
so the trailing end of the pole will continue to advance as if nothing
had happenned until it is inside the barn, at which point the pole
will have be at rest (relative to the barn) and compressed to half of
its original rest length.

Trivial examples of relativistic kinematics like this are not difficult
to devise.  Needless to say, the simple Lorentz transformation 

                x'=(x-vt)/g     t'=(t-vx)/g

where g = sqrt(1-v^2) cannot possibly lead to logically contradictory
relations in terms of the (x',t') coordinates if applied to any 
logically consistent situation expressed in terms of the (x,t) 
coordinates.  Nevertheless, with the mentality of "angle trisectors",
some modern paradoxers persist in endlessly trying to invent 
scenarios that will somehow (in defiance of simple algebra) produce
a contradiction from the Lorentz transformation.  All these attempts
are essentially just variations on the barn pole paradox.  Below 
we discuss three such variations, one expressed in terms of 
probabilities, another expressed in terms of electrical currents,
and finally one involving the composition of motion in two (or 
three) spatial dimensions.

For the first "paradox", let (x,t) denote a system of inertial 
coordinates, and suppose an event E occurs at (X,0) where X is 
randomly chosen from a uniform distribution on the range 0 to L.
Also, suppose a rod R1 of length p is at rest relative to this 
coordinate system, and positioned so that the coordinates of its
two end points are (0,0) and (p,0).  The probability of the event
E occuring within the span of R1 is p/L.

Now suppose there is another rod R2 of rest length p moving with
speed v in the positive x direction.  If the trailing edge of R2 is
at x=0 when t=0, then the leading edge is at x=pg when t=0, where
g = sqrt(1-v^2).  The probability of the event E occuring within 
the span of rod R2 is pg/L.

If we consider the same situation in terms of the inertial coordinate
system (x',t') with respect to which R2 is at rest, we will obviously
find the same probabilities for the event E to occur in the spans of
R1 and R2, because these probabilities are just ratios of fixed
segments of the x axis, and are all changed in proportion to each
other by any Lorentz transformation.  

To spell this out explicitly, the locus of possible positions for the
event E is the line from (0,0) to (L/g,-vL/g), and the intersection 
of this line with the worldsheet of R1 is the line from (0,0) to 
(p/g,-vp/g).  Thus R1 covers p/L of the range for event E, and the 
probability of E occuring in the span of R1 is p/L.  The intersection
of the E-locus with the worldsheet of R2 is the line from (0,0) to 
(p,-vp), so it covers pg/L of the range for the event E, and the 
probability of the event occuring within R2 is pg/L.

Incidentally, a naive paradoxer might point out (correctly) that the
intersection of the worldsheet of R2 with the x' axis is of length p,
and the intersection of the worldsheet of the spatial range 0 to L 
(fixed with respect to x,t) with the x' axis has the length gL.  On
this basis he might argue (incorrectly) that the probability of the 
event E falling within the spans of R2 should be p/gL, contradicting
the fact (shown above) that the probability is pg/L.  Needless to say,
the paradoxer's error is failing to account for the fact that the 
specified range for event E is along the x axis, not along the x' 
axis.

For an interesting variation on this problem, suppose we let not only
the location but also the time of occurrence of event E be randomly
selected from a uniform distribution over a finite range.  In other
words, for a system (x,t) of inertial coordinates, suppose an 
event E occurs at (X,T) where X is randomly chosen from a uniform
distribution on the range 0 to L, and T is randomly chosen from a
uniform distribution on the range 0 to D.  Then, as before, consider
the two rods R1 and R2 of rest length p, the former stationary with 
respect to (x,t) and the latter moving in the positive x direction 
with speed v.  In this case the probability of the event E falling 
within the span of R1 (or R2 respectively) is given by the ratios
of the spacetime area swept out by R1 (or R2) to the total spacetime
area of the specified range for event E.  When plotted in terms of
the (x,t) coordinates, the situation is as shown in the left-hand
figure below.



The right-hand figure represents the same situation plotted in terms
of the rest frame coordinates (x',t') of R2.  For this problem the
consistency of the probabilities is, if anything, even MORE obvious
than in the case of an "instantaneous" event E, because of the 
fact that spacetime areas are conserved under arbitrary Lorentz 
transformations of the (x,t) plane.  This follows from the invariance
of all quantities of the form 

                         x1 t2 - x2 t1

under Lorentz transformations, together with the elementary expression
for areas enclosed with arbitrary plane figures in terms of quantities
of this form.  (See the note Net Area and Green's Theorem.)  The
invariance of this form is equivalent to the fact that the determinant
of the Lorentz transformation is unity.

The next paradox that we will discuss involves a sequence of three
light bulbs positioned at uniform intervals along the x axis of an
inertial coordinate system (x,t).  The bulbs are stationary at 
the x coordinates 0, 1, and 2.  Now consider a rod of rest length 
equal to 2 units, but moving in the positive x direction with
velocity v = sqrt(3)/2 so that it's length relative to the (x,t) 
coordinates is just 1 unit.  Also, suppose the lights are triggered 
to turn ON when, and only when, they are in contact with the rod.  
Thus there is always exactly one light bulb illuminated at any given
time t while the rod is passing over the lights.  For example, at 
the time t illustrated below, bulb 0 if OFF, bulb 1 is ON but is 
about to go OFF as the rod proceeds to the right, and bulb 2 if OFF
but will turn ON as soon as the leading edge of the rod reaches it 
(at which point bulb 1 will go OFF as the trailing edge of the rod 
passes over it).

                       rod -> v
                      |--------|
               0--------1--------2---------->
                                          x

Now, just as in the Barn Pole paradox, we are asked to consider the
same situation in terms of the rest frame coordinates (x',t') of 
the rod.  The alleged paradox is due to the fact that, in terms of
these coordinates, the rod has length 2 units whereas the distance
from bulb 0 to bulb 2 is only 1 unit, and hence the rod is in contact
with all three bulbs for a period of time, implying that all three
bulbs are on simultaneously.  

Realizing that these facts are easily reconciled by means of the 
relativity of simultaneity, the paradoxer tries to achieve a
contradiction by asking us to consider the current flowing in a 
series circuit supplying the three bulbs from a single battery 
located at bulb 0.  He suggests that, from the standpoint of the 
(x,t) coordinates, the current should just equal one unit for the 
duration of the time that the rod is interacting with the bulbs, 
since only one bulb is ON at any given time t.  On the other hand
(he reasons), from the standpoint of the (x',t') system, the current
must reach 3 units for a period of time, because all three bulbs
are ON.  So, he asks, does the current ever reach 3 units, or
doesn't it?

As always (when dealing with such purported paradoxes), a simple 
spacetime diagram suffices to explain the real situation.  The figure
below shows the worldlines of the three bulbs in red, with their 
periods of illumination shown in yellow.  The worldlines of the
leading and trailing edges of the rod are shown in dark blue, and
the axis of simultaneity for the rod's rest frame is shown in light
blue (marked as "t' = const).

         

For purposes of discussion we have represented each bulb as a source
of current when it is turned on, and the battery at bulb 0 is regarded
as a sink for the current.  We have also assumed the propagation speed
c for the current, and we are ignoring transient fluctuations.  As can
be seen, there are three separate periods of current flow at the 
"source" (i.e., the battery at bulb 0), each with one unit of current.
Also, notice that even though the rod covers only one bulb at a time 
with respect to this coordinate system, it covers all three bulbs 
simultaneously with respect to the (x',t') coordinate system.

Transforming this situation to the (x',t') coordinate system gives
the situation shown in the figure below.

         

Naturally there are "instants" (i.e., loci of constant t') in which 
all three bulbs are ON simultaneously, but of course the current flow
at the source still consists of three disjoint periods with one unit
of current during each period.  Needless to say, it was a foregone 
conclusion that the sequence of distinct changes in current level 
along the worldline of bulb 0 (or along any other worldline) would 
translate into a similar sequence with respect to any other system 
of inertial coordinates.  Straight lines map to straight lines, and
the ratios of intervals along lines are preserved by Lorentz
transformations.  This just illustrates yet again the utter futility
of trying to demonstrate that the Lorentz transformation somehow
leads to logically inconsistent results.

All the examples discussed so far involved just one spatial dimension,
but our last example involves motion in two (or three) spatial 
dimensions.  Suppose we have two rods, one stationary rod U1-U2 of 
length L aligned with the x axis, and another rod A1-A2 with the 
same x-extent but "slanted" and moving in the y direction with speed
u as shown below: 

         y |
           |
           |      ^      A2
           |     u|      .
           |      |  . '  
           |     . '         D
         A1| . '         
        ---***************----------
           |U1            U2       x
           |       L

The ends A1 and U1 are coincident at this "instant", whereas the 
end A2 was coincident with U2 some time in the "past".  Obviously 
the rod A1-A2 has a greater spatial length than the rod U1-U2 in 
terms of this coordinate system.

Now consider these two rods with respect to a coordinate system 
whose origin is moving to the right with velocity v.  The slices of
simultaneity are skewed, so we find (for suitable lengths, speeds, 
etc.) that with respect to this new coordinate system the event [A2 
coincident with U2] is simultaneous with the event [A1 coincident 
with U1].  Hence, the two rods have the same spatial length with 
respect to this coordinate system.  Notice that the rod A1-A2 is 
spatially parallel to U1-U2 with respect to this coordinate system,
whereas the rods are not parallel with respect to the original 
coordinate system.  (This is the same relativistic effect that 
accounts for both aberration and Thomas precession.)

With respect to the first coordinate system, the spatial coordinates
of U1 are x1=0, y1=0, and the coordinates of U2 are x2=L, y2=0.  The
coordinates of A1 are X1=0, Y1=ut, and the coordinates of A2 and X2=L,
Y2=D+ut.  Apply the Lorentz transformation to express these positions
in terms of a coordinate system moving with speed v in the position x
direction.  In units such that c=1, this immediately gives

        x1' = -vt'        y1' =  0   
        x2' = gL - vt'    y2' =  0

        X1' = -vt'        Y1' = ugt'  
        X2' = gL - vt'    Y2' = D + uvL + ugt'

where g = sqrt(1-v^2).  Clearly the ends A1 and U1 are coincident 
when t' = 0.  We wish to see under what condition the ends A2 and 
U2 are coincident at this same "instant".  The x coordinates of 
the A and U rods are the same (in both systems of coordinates), so 
we just need to equate y2' with Y2'.  Noting that t' = 0 at this 
"instant", this gives 0 = D + uvL, so we have uv = -D/L.  Since |v|
can be no greater than 1, it follows that |u| must be at least 
equal to D/L in order for there to exist an inertial reference frame
with respect to which the two rods are parallel and of the same
spatial length.

This explains how and why two rods in motion with respect to each 
other can have the same spatial length with respect to one system of
inertial coordinates and different spatial lengths with respect to 
another, just as they can be parallel with respect to one system 
and not parallel with respect to another.  This is an immediate 
consequence of the relativity of simultaneity, i.e., the fact that
two events can be simultaneous with respect to one system of 
coordinates but not simultaneous with respect to another.  

The rest length of the U1-U2 rod is simply L, whereas to find the
rest length of the A1-A2 rod we must transform the coordinates of
it's endpoints to the rest frame of the rod, which is moving with
speed u in the positive y direction relative to the (x,y,t) 
coordinates.  Therefore, letting G denote the factor sqrt(1-u^2),
the rest length of the A1-A2 rod is sqrt[L^2 + D^2/G^2].  So, if 
the rods were originally constructed with the same rest length, we
see that the A1-A2 rod must have been materially stretched in 
order for it to be in its current state of motion and positions.

In other words, whether or not a rod fractures depends on its 
rest length, not on its length relative to some moving system of 
coordinates.  In the situation described here the two rods have
two different rest lengths, namely, L and L sqrt[1 + (D/(LG))^2] 
where D/L = -uv (assuming the rods are parallel with respect to the
coordinate system that is moving with speed v in the x direction). 
So, the ratio of the two rest lengths is sqrt[1 + (uv/G)^2].  
Needless to say, if both rods originally had rest length L, then 
the A1-A2 rod has been stretched, and would fracture (if it was 
sufficiently brittle).

The paradoxer's confusion in this scenario is based on his neglect 
of the relative motion u between the two rods (since he tends to 
focus on just the motion in the x direction).  In an attempt to
impose this confusion on everyone else, the paradoxer sometimes
alters the situation so that the rod A1-A2 is attached to a cylinder
of radius R, whose axis is parallel to the x axis, as illustrated
below.
           

Of course, this doesn't substantially change the situation, because
for sufficiently brief intervals of time and small angular offsets,
we can simply "unroll" the cylinder (which is an intrinsically flat
surface anyway), and reduce this to a two dimensional problem again,
letting "y" denote the circumferential position around the surface 
of the cylinder.  Thus we set y = theta R and u = wR where theta is
the angular position of the cylinder, w is the angular velocity, and
R is the radius.  The spatial length of the cylinder is again just 
sqrt[L^2 + (D/G)^2] where D = R theta and theta is the angular offset
of the attachment point A2 relative to the attachment point A1 (with
respect to the (x,y,z,t) system).

The only new feature of this rotating version is that the attached
rod is not strictly in inertial motion, so it has no single rest 
frame.  To describe it's kinematics in full generality (assuming an
ideal kinematic cylinder), we can give it's (x,y,z,t) coordinates 
in terms of a parameter q ranging uniformly from 0 to 1 along its 
length.  We have

     x = qL      y = R sin(q theta)       z = R cos(q theta)

so again the spatial length is sqrt[L^2 + D^2] where D = R theta and
theta is the total angular offset of the two attachment points.  In
this case we can't transform to the rest frame of the rod, because
the rod has no single rest frame, but we can consider each 
infintessimal span of the rod and evaluate its length with respect
to the momentarily co-moving inertial frame.  This is the length
that determines the mechanical response for each incremental part 
of the rod.  For each increment the problem reduces to the simple
two-dimensional case discussed previously, so the net effect is a
total "local rest length" of sqrt[L^2 + (D/G)^2] for the rod, just
as before.