The Lorentz-Fitzgerald Contraction
Something about the principle of time
dilation should bother you. The Principle of Relativity tells us that we can
devise no experiment that two observers with some relative motion between them
can perform that will establish one of them absolutely moving and the other
absolutely at rest. That means, necessarily, that while we see the pulses in a
laser clock mounted on a train tracing a sawtooth pattern in the air, the
passengers on the train must see our clock tracing the same patterns. That fact,
in turn, means that we see the train's clocks running slower than our clock does
and the passengers on the train see our clock running slower than their clocks
do. That statement looks very much like a logical contradiction and it stands as
the basis for the best-known paradox in Relativity.
According to Richard Feynman (and to get
the full effect of one of the Twentieth Century's greatest physicists you must
imagine the following being spoken with a noticeable Brooklyn accent: Professor
Feynman was born and raised in Far Rockaway), "A paradox is a situation which
gives one answer when analyzed one way, and a different answer when analyzed
another way, so that we are left in somewhat of a quandary as to actually what
would happen. Of course, in physics there are never any real paradoxes because
there is one correct answer." He then adds, "A paradox is only a confusion in
our understanding."
So what confusion do we have in our
understanding of time dilation? Our derivation of the description of the effect
from the sawtooth pattern that a laser clock traces seems clear enough to be
free of any logical flaws. And we have clearly applied the Principle of
Relativity properly here. So what have we missed? To answer that question we
need to perform some more imaginary experiments.
From Fresno to Modesto the Southern
Pacific tracks run more or less straight for about 87 miles. I like that
distance because a train that can go from Fresno to Modesto in one hour in our
fantasy world will do so by occupying an inertial frame that has, relative to
the frame occupied by the Fresno and Modesto train stations, a Lorentz factor of
2. We can thus figure easily that for the passengers on that train one half hour
will elapse between their departure from Fresno and their arrival in Modesto.
But that calculation should bring to mind another thought to distract you
momentarily from contemplation of temporal paradoxes.
Imagine that twins John and Jane have come to the Fresno train station and that John takes a train to Modesto while Jane waits in the station's coffee shop for him to return. The Principle of Relativity tells us that if Jane sees John move past her at 87 miles per hour, then John must see her pass him at 87 miles per hour; that is, the ratio of distance crossed to time elapsed must be the same for both of them. If that were not true to Reality, then John and Jane could devise experiments that would establish the existence of a frame of absolute rest, in contradiction to the Principle of Relativity. Thus Jane calculates the ratio of 87 miles crossed by the train to one hour elapsed on the station clocks as 87 miles per hour. John, riding the train, may make a slightly different calculation: he knows that the landscape moves past him at 87 miles per hour and he measures one half hour on his watch from departure to arrival, so he calculates that the distance from Fresno to Modesto in his frame must have been 43-1/2 miles. He has thus detected through his calculation the Lorentz-Fitzgerald contraction, which we describe in
LORENTZ RULE 2B: An
object moving relative to some observer appears to that observer to be shorter
in the direction of relative motion than it does to an observer at rest in the
object's inertial frame, the moving length being equal to the stationary length
divided by the Lorentz factor between the two inertial frames.
In this series of imaginary experiments I will define a straight line drawn down the center of the Southern Pacific track as our x-axis and the direction from Fresno to Modesto, roughly northward, as the positive direction. In the frame occupied and marked by Fresno and Modesto we can calculate the distance between two points on the track by
(Eq'n 1)
in which equation the lower-case vee represents the speed at which the train goes between those points and the lower-case tee represents the time interval that we would measure between the events defined by some feature of the train, such as the front of the locomotive's headlight, passing each point. John would measure the distance between those same two points in the same way; that is, he would calculate
(Eq'n 2)
But John and Jane relate the times that they measure through the time dilation formula, so John calculates
(Eq'n 3)
If John looks out the window of his
carriage when the train has gained its full speed, he will see a landscape
remarkably deformed by the world's adherence to that rule. Objects appear to
have their normal heights but they appear shortened in the direction parallel to
that in which they move. Telephone poles, trees, and houses all appear narrower,
thinner than they ought to be. And cars and trucks on the highway appear
cartoonishly foreshortened. Of course, as you have come to expect, to the people
in those cars and trucks it's the train that appears foreshortened. This is all
quite amusing, but it doesn't help us resolve the problem with time dilation
(though the picture that you have drawn in your mind contains a clue to that
resolution), so let's go back to our contemplation of our clocks and look at the
twin paradox.
So John and Jane have come to the train
station in Fresno. John has an errand he needs to run in Modesto, so while he's
gone Jane will wait for him in the station's coffee shop. She figures that she
will have to wait a little over two hours, but she also figures that for John a
little over one hour will elapse; one half hour each way on the train plus the
few minutes he needs to run his errand once he gets to Modesto. John also knows
about time dilation; thus, when he gets to Modesto and sees that half an hour
has elapsed on his watch, he figures that one quarter hour has elapsed for Jane
and expects that she will have to wait in the coffee shop a little over half an
hour.
Do you see what Feynman meant by a
paradox? Analyzing the situation through Jane's perspective, we figure that Jane
will wait two hours. Analyzing the situation through John's perspective, we
figure that Jane will wait one half hour. We have become stuck in a quandary.
Which answer is the one correct answer that Feynman told us that physics has for
us?
If you should put that question to
physicists, they would chose unanimously two hours as the correct answer. When
asked why they chose that answer, they would all give a reply that involves
pointing out that Jane occupied only one inertial frame while John occupied
three (the frame marked by the Fresno and Modesto train stations, the one moving
north at 87 miles per hour, and the one moving south at 87 miles per hour). But,
we might object, doesn't Jane occupy three inertial frames as seen through
John's perspective? Isn't that what Relativity tells us? We must answer no to
both of those questions: we can have no doubt that John has occupied three
inertial frames and Jane has occupied only one, regardless of whose viewpoint we
use, because we have no doubt that John has undergone accelerations and Jane
hasn't. True, through John's perspective Jane appears to have accelerated, but
that's only an appearance because acceleration is an absolute state of motion,
which means that we can devise experiments that Jane and John can perform that
will determine which of them actually accelerates and which of them does not. If
Jane and John both have cups of coffee sitting before them, for example, then
when the train leaves the station John will see the coffee in his cup tilt but
Jane will see the coffee in her cup remain motionless and level.
And how does the fact of acceleration resolve the twin paradox? It brings into play a distortion of time even stranger than time dilation. Most presentations of Relativity ignore that distortion, even though Einstein regarded it as one of the most fundamental features of his theory, the one that answers the question, "What time is it?" Einstein called it the relativity of simultaneity, noting that events that appear simultaneous for one observer won't necessarily appear simultaneous for other observers.
habg