The Relativity of Perpendicular Distances

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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.

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