The Relativity of Perpendicular Distances

We take the law of non-contradiction as
one of the tacit axioms of Relativity. That law of logic declares that two
mutually exclusive propositions cannot both come true to Reality and it provides
us with the fundamental component of the method of reductio ad absurdum
(reduction to an absurdity). Einstein used it in a process of elimination, by
which he deduced which of a set of possible rules actually relates one
observer's measurements of distance and duration between two events to another
observer's measurements of the same two events.

More technically we say that we want to
prove the statement that if P comes true to Reality (or mathematics, or logic,
or etc.), then Q also comes true to Reality (or mathematics, or logic, or etc.),
usually stated more tersely as "P implies Q". We have three basic ways to work a
proof of that statement: 1) direct proof (given P, we deduce Q directly), 2)
contraposition (given not-Q, we deduce not-P), and 3) indirect proof, also known
as reductio ad absurdum (given P and not-Q, we deduce a contradiction).

We work the reductio ad absurdum by
assuming that P and the negation of Q (not-Q) both come true to Reality and then
deduce a contradiction; specifically, that both Q and not-Q must come true to
Reality. But we have tacitly (and now explicitly) assumed that Reality does not
allow a statement and its negation to come true together. When we obtain such a
contradiction, we must discard at least one of the premises from which we
deduced it. For that reason we also commonly call reductio ad absurdum the
process of elimination.

Where would you guess you could find the
best place to start to apply that process of elimination? Like a chess player
plotting strategy, you should want to begin with the simplest moves and work
your way into the more complex moves that stand upon them. It doesn't look
immediately obvious, but after a little thought you would want to start where
Einstein did. Einstein chose to start by considering the two directions in which
he was uninterested; that is, he started out by considering the measurements of
distance that two observers make in the directions oriented perpendicular to the
direction of the relative motion between their respective inertial frames. If,
for example, those observers move relative to each other along a north-south
line, then we want to consider the measurements that the observers make of
distances in the east-west and up-down directions. He then deduced the simplest
of Relativity's rules, which states that we will find those distances invariant
with respect to velocity; that is, that the velocity between the observers will
not make the observers' measurements in those lateral directions different for
the observers.

As an aid to our imaginations let's now
assume that we inhabit a world in which light moves at a speed of 100 miles per
hour. We will thus carry out our imaginary experiments with familiar objects in
a regime of speeds that will seem reasonably familiar and thus ease the
difficulty of imagining unfamiliar concepts. We shall ignore the fact that our
world would not look at all familiar if light flew so slowly and we will imagine
that the tardiness of light makes the only difference between our world and the
world of our thought experiments. And I will point out that we will express the
results of our experiments as proportions, so the rules will remain valid when
we take them from our world of fantasy and apply them in the world in which
light flies halfway to the moon in a heartbeat.

For the purpose of conducting our first
experiment, imagine that we have gone on a visit to the Southern Pacific
Railroad Company's Sierra Yard, a fictitious and absurd marshaling yard set
among walnut orchards east of Visalia, California (and occupying in part land
formerly occupied by the Sierra Drive-In Theater). Freight trains made up of
cars scheduled to go to widely scattered destinations come here to have their
cars reconstituted into trains made up of cars that will all go more or less in
the same direction. After each train arrives, a yard engine pushes it up the
yard's hump, a low knoll built at one end of the yard. As each car reaches the
top of the hump, a brakeman uncouples it from the rest of its train and allows
it to coast freely down the other side of the hump. It picks up speed and goes
through a series of switches that guide it onto one of a number of parallel
tracks, rolling slowly into and coupling to its new consist.

We can see all of this activity from our
position on a catwalk that goes over the track coming down off the hump. We pay
special attention to one car in particular. As we watch the car roll toward us,
our guide tells us that the maximum height of its load above the tops of the
rails comes within a mere hair's-breadth of the lowest part of the catwalk. In a
Newtonian Universe we would have no cause for concern: the car would glide
beneath the catwalk, perhaps brush off some dust, and continue on its way. But
we live in a relativistic Universe and that fact should make us pause to
contemplate the relationship between our inertial frame and the one occupied and
marked, however briefly, by that freight car as it reaches the catwalk. We have
for our consideration three possibilities: A) distances measured perpendicular
to the relative motion by an observer moving through a frame are larger than the
same distances measured by an observer at rest in that frame, B) those distances
are smaller than the same distance measured by the observer at rest, and C)
those distances are the same for both observers.

For our P implies Q we have "If a body
moves relative to some observer, then the dimensions of that body perpendicular
to the direction of relative motion must...." We have identified three possible
statements that complete that proposition. Only one of them can be Q, so two of
them must be not-Q. We now want to deduce contradictions that reveal which is
what.

Assume that Possibility A is true to
Reality. In that case the frame occupied by the freight car as it goes under the
catwalk is larger in the vertical and sideways directions than our frame is in
those directions, which means that in our frame the freight car is also larger
in those directions. Thus expanded the car will hit the catwalk and wreck it.
But a brakeman riding the car down into the yard would disagree. In his frame
it's our frame and the catwalk that are moving. The Principle of Relativity
necessitates that whatever difference we see between our two frames, the
brakeman must see the same difference in the same way in which we see it; thus,
in the brakeman's frame the catwalk will expand and the freight car will pass
under it with room to spare. Such a contradictory state of affairs cannot be
true to Reality, so we must infer that Possibility A is false to Reality: it is
one of the not-Q's.

By similar reasoning we can dismiss Possibility B as false to Reality. That leaves us with only Possibility C, which must necessarily be true to Reality because we have no alternative, having eliminated the only ones available. We then restate Possibility C more formally as

**LORENTZ RULE 1:** In two
inertial frames in relative motion, the distance between any two given points
measured in a direction perpendicular to the relative motion will be the same
for observers in both frames.

Distances measured in directions
perpendicular to relative velocity comprise members of a class of entities
called invariants. Those entities are invariant with respect to (that is,
unchangeable by) shifts from one inertial frame to another. Invariants are
similar to absolute quantities and, in spite of what its name connotes, the
theory of Relativity is deeply concerned with them. Indeed, when Einstein found
out about the moral implications that some people were reading into his theory,
he commented that he should have insisted that the theory be called the Theory
of Invariants.

However, not everything that is the same
for two observers in different inertial frames is an invariant. The speed that
the two observers measure between their respective frames must be the same for
both of them, certainly; for if the observers came up with different speeds, it
would have to be because at least some of the laws of physics were different
between the frames and the Principle of Relativity tells us that such a
difference cannot exist. But the speed between the observers can change. As the
freight car rolls down the hump, its speed increases but its height remains the
same for all observers, whether they be the brakeman riding the car or someone
standing next to the track with a large ruler. Lateral distances are invariants;
velocities are not.

In the logic of physics invariants look like the edge and corner pieces of a jigsaw puzzle. They provide the easiest pieces to find and to put together and they show us the framework into which the other pieces will fit. They provide the starting point from which we proceed to solve the puzzle. So now we have placed one edge piece of the relativistic puzzle in its place. Next we shall use that piece as a guide to finding and placing the next piece, one of the weirder pieces of this puzzle.

habg