Paradoxes

If you have read through all of the essays preceding this one, you may have gained a nagging sense of something missing, a feeling that some part of the theory of Relativity has gone AWOL. But I have left no proper part of Relativity out of my presentation. I have, however, left out the infamous paradox of the twins. I have done so by the simple expedient of presenting Special Relativity much as Einstein did and as any presenter should - correctly.

Nonetheless, I want to offer a brief discussion of the twin paradox and a related paradox for the light that they shed upon the use of logic in devising the Map of Physics and also because I refer to them in later essays.

Richard Feynman told us that we have a paradox when we approach a problem from two different directions and come up with two incompatible results. As he put it, "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." Professor Feynman then added, "A paradox is only a confusion in our understanding." Usually we dissolve a paradox by clearing up that confusion in our understanding, by finding what we left out of our statement of the problem and taking it into proper account. But not all paradoxes conform to Professor Feynman's second statement. When we ultimately encounter a paradox that we cannot dissolve in that manner, as we will certainly do, then we will have reached the Gödel Point of the Map of Physics, the point at which the Map no longer correctly describes Reality. Nowhere in Relativity will we reach that point: with suitable diligence we can dissolve all of the paradoxes that we encounter in the theory.

We often call the paradox of the twins the clock paradox and we do so because fundamentally the paradox consists only of our analysis of the behavior of a pair of clocks. Let's return to the Fresno train station in the world where light flies one hundred miles in an hour. A train leaves the station at a speed that will take it to the Modesto train station, 86.6 miles away, in one hour. That speed puts a Lorentz factor of two between the train and the two stations. Thus we have the basic situation that produces the paradox. In this version we will allow two identical clocks to play the roles of the twins: one will stay in the Fresno station and the other will ride the train.

Let's state the problem like this: The train goes from Fresno to Modesto at 86.6 miles per hour and immediately returns to Fresno at the same speed. Between the train's departure and its return observers in the Fresno train station will see two hours elapse on the clock in the station. But the clock on the train counts time dilated by a Lorentz factor of two, so those observers will expect that only one hour elapses on it. For observers on the train, on the other hand, the Fresno station, moving at 86.6 miles per hour, goes 43.3 Lorentz-Fitzgerald contracted miles away and then comes back at the same speed; thus, one hour elapses on the train clock for them. But in the train's frame the Fresno clock counts time dilated by a Lorentz factor of two, so the observers on the train expect that only half an hour elapses on it between the train's departure from and arrival back at the Fresno station.

Looking at that statement, you see how we have a paradox. Though both teams of observers agree on the amount of time that elapses on the train clock, the Fresno observers will claim that two hours elapsed on the Fresno clock while the train observers will claim that only half an hour elapsed on it. So what went wrong in our analysis? How can clear up the confusion in our understanding and set our analysis right?

First, we must see that we misled ourselves with a little semantic trick manifested in the phrase "returns...at the same speed". Our use of the word "same" subtly implies that when the train reverses its motion to return to Fresno nothing of consequence has changed. And we employ the word "same" because we have failed to appreciate the distinction between speed and velocity.

Speed expresses the ratio between distance crossed and time elapsed without regard to any direction in which the distance crossed might point. Velocity expresses the ratio of distance crossed in a specific direction to time elapsed. Had I stated the problem in terms of velocity, rather than of speed, I could not have used the word "same" in the above phrase: I would not have raised that subtle expectation that nothing significant in the situation had changed.

Once we understand explicitly that the train occupied two entirely different inertial frames and that the Fresno station occupied and marked a third, the paradox begins to crumble. When we acknowledge one other defect in our reasoning, the paradox falls apart completely.

In making the standard statement of the twin paradox the presenter refers us to a remarkably truncated version of the Lorentz Transformation, one that comprises only time dilation and the Lorentz-Fitzgerald contraction. We want to analyze a situation involving moving clocks and yet ignore the Relativity of Simultaneity, as expressed in the temporal offset term in the time equation of the full Lorentz Transformation: we cannot rightly wonder that we get a paradox.

That truncated transformation also adds yet another subtle mental nudge into the paradox. It gives us only one relativistic number, the Lorentz factor, and that number happens to be the same between each of the train's inertial frames and that occupied by the Fresno train station. Again we get a "same" that diverts our attention away from the differences that we should discern.

Based on that latter fact, some of my more astute readers may object that they know a way in which we may use that truncated version of the Lorentz Transformation to analyze the situation correctly and dissolve the paradox. If we get into trouble by jumping from one inertial frame to another in the middle of our analysis, as we do by following the observers on the train, then we need only remain in one frame for the full course of the situation we want to analyze.

Imagine that as the train departs the Fresno station a track speeder paces it on a parallel track. When the train stops in Modesto, the speeder continues motoring north at 86.6 miles per hour at least until the train returns to Fresno.

On the first leg of the journey from Fresno to Modesto, the observer on the speeder carries out the same analysis as does the observer team on the train. As on the train, half an hour elapses on the speeder's clock as the 43.3 miles between Fresno and Modesto go by and one quarter hour elapses on the Fresno and Modesto clocks.

On the second leg of the journey the train moves away from the track speeder at a speed that we must calculate via the relativistic velocity composition formula:

(Eq'n 1)

At that speed the train overtakes Fresno at a relative speed of 12.37 miles per hour, crossing the 43.3 miles in 3.5 hours on the speeder's clock. In that interval 1.75 hours elapse on the Fresno clock, adding to the previously elapsed quarter hour to yield two hours, in agreement with the Fresno observers.

And on the train clock? Because it moves away
from the speeder faster than Fresno does, the train's clock will tick off time
dilated by greater factor than the time ticked off the Fresno clock. For a
compound velocity v_{3} given by Equation 1 we have the Lorentz factor

(Eq'n 2)

In the present example we have L_{1} = L_{2}
= 2, so L_{3} = 7. Thus the observer on the speeder calculates 3.5 ÷ 7 =
0.5 hour elapsed on the train's clock between its leaving Modesto and arriving
in Fresno. Added to the half hour elapsed on that clock on the first leg of the
journey, that answer gives one hour that elapsed on the train's clock between
the train's leaving Fresno and returning thither, again in agreement with the
Fresno observers.

So we have found the truth of the matter and
dissolved the twin paradox without referring to the temporal offset, haven't we?
Well, not quite. In devising the velocity-composition formula of Equation 1 we
divided the space deformation equation of the Lorentz Transformation by the time
deformation equation. But the time equation includes the temporal offset: the
term v_{1}v_{2}/c^{2} in the velocity-composition
formula comes directly from the term describing that offset. So our
track-speeder analysis tacitly acknowledged that clocks running in synchrony in
one frame will not run in synchrony in other frames.

How, then, should we properly take the twin paradox apart? What must we do to ensure that all observers reach the same conclusion?

We don't need to re-analyze the scenario from the viewpoint of the observers who stay in Fresno. We already know how they analyzed the scenario and that they got the analysis correct.

How should observers riding the train analyze the scenario? They will still discern a Lorentz factor of two between the train's frame and the frame occupied and marked by Fresno and Modesto. That Lorentz factor will shrink the Fresno to Modesto distance from 86.6 miles to 43.3 miles and will make the Fresno and Modesto clocks tick off time half as fast as the train's clock does. But in addition to those facts, which they applied in the standard twin paradox, the observers on the train must also acknowledge the fact that on the first leg of the journey the Modesto clock runs ahead of the Fresno clock by an interval

(Eq'n 3)

On the way to Modesto the train observers add that increment to the time that they calculate elapsing on the Fresno clock, doing so in order to determine what time the Modesto clock will tell when the train arrives. Inferring that one quarter hour elapses on the Fresno clock in the half hour that they spend on the train going to Modesto, the train observers thus calculate that the Modesto clock will tell a time one hour later than the time the Fresno clock told when the train departed the Fresno station. On the return journey the train occupies an inertial frame in which the Fresno clock runs ahead of the Modesto clock by three quarters of an hour. In sum, the train observers will assert that between the train's departure from Fresno and its return thither two hours will have elapsed on the Fresno clock and one hour on the train's clock, in perfect agreement with the observers who stayed in Fresno.

Thus the twin paradox dissolves, simply and completely. It really should not have come to us as a paradox. But in the section on General Relativity I will show you a truly ingenious clock paradox. Jack Bennett has devised a paradox that genuinely challenges our understanding of the structure of Reality. In dissolving it we will discover how clocks that run in synchrony in one inertial frame can run out of synchrony in other frames; we will discover the fundamental reason for the term that dissolves the twin paradox.

Now consider this little variation on the clock paradox:

A flatcar loaded with 40-foot rails coasts down a side track at 86.6 miles per hour (127 feet per second: the speed of light in this world, 100 miles per hour, equals 146.7 feet per second). Before the brakeman can stop it, the car goes through a car barn so built that the distance between the insides of the front and rear doors spans 20 feet. The barn's doors open and close instantly and a clock sends electrical impulses that make the rear door open at the same instant that the front door closes. By incredible luck (or, more credible, author's contrivance) the leading ends of the rails reach the rear door at the very instant the door opens. With the rails Lorentz-Fitzgerald contracted to a length of 20 feet, the trailing ends of the rails enter the barn at the same instant that the front door snaps shut. Thus the flatcar and its load of rails pass through the barn, just barely, without striking any part of the doors.

The brakeman riding the flatcar has a different perspective on the situation. In his frame the rails have their full length of 40 feet and the Lorentz-Fitzgerald contraction has so shrunk the car barn that the distance between the doors spans 10 feet on the brakeman's ruler. If the rear door opens and the front door closes at the same instant, then the rails will certainly collide with one or both doors.

Again the Relativity of Simultaneity dissolves the paradox. In the brakeman's frame the rear door opens before the front door closes and does so by an interval of

(Eq'n 4)

In that interval the barn moves an additional 30 feet (at 127 feet per second) and thus brings the trailing ends of the rails into the barn just as the front door closes. In its essentials, what the brakeman sees agrees with what observers on the ground see, except for the timing of the events. Again we can only dissolve the paradox that we would otherwise have if we take into account this bizarre phenomenon of relative motion pushing upstream clocks into the future relative to downstream clocks.

This particular paradox gives us an example, in one spatial dimension, of the Rindler-Shaw paradox, which offers us a similar scenario in two spatial dimensions. You might want to take note of how little space I needed to dissolve this paradox and compare it to how much I will need to tackle Rindler-Shaw.

Suppose that we had been unable to dissolve one of those paradoxes. Imagine that no matter what technique we applied, what knowledge we brought to bear, our observers would nonetheless derive mutually exclusive descriptions of the same scenario. If we ever find ourselves in that predicament, we will have reached what I call the Gödel point.

We tacitly assume that the laws of physics comprise a set that is consistent and complete. That is, we assume that

1) the laws never conflict with each other, that they never instruct Reality to make any part of the Universe do two mutually exclusive things, and

2) the laws are necessary and sufficient to the operation of Reality, neither too many nor too few.

Indeed, we can take those statements as axiomatic; that is, as self-evidently true to Reality. We know that Reality cannot include too few laws of physics: if it did, then it would include circumstances in which, to use an illegitimate but properly metaphorical anthropomorphism, bodies participating in certain events would not know what to do. Reality cannot include too many laws of physics because, if it did, the superfluous laws would conflict with other laws over the control of phenomena of Reality, acting to make bodies do things inconsistent with the demands of other laws. And we know that what we call Reality cannot show us two faces. If we observe some phenomenon in which we see the same bodies participate in a single event, then the laws of physics cannot so guide the outplay of the phenomenon that I will see this and you will witness that.

In creating the Map of Physics we seek to deduce all of the laws of physics and only the laws of physics in order to satisfy that necessary and sufficient criterion. But we have included nothing in our assumptions about Reality that obliges those laws to connect to each other in an axiomatic-deductive way. Nonetheless, what we have seen so far gives us probable cause to believe that existence does so connect the laws of physics. That belief will lead us eventually to a truly interesting problem; for Kurt Gödel has told us that predicate logic, essentially the logic of the syllogism, cannot yield a structure that is both consistent and complete. As we extend the Map of Physics via predicate logic we must eventually reach a point where the Map no longer matches Reality. What we do then depends upon what we find.

In essence Gödel presents us with a computer that will answer any question put to it and will answer the question with the truth, the whole truth, and nothing but the truth. Gödel's computer contains all knowledge, so it has a lot of truth to tell. After programming the computer Gödel gives it one more piece of knowledge, the statement "Gödel's computer will never say that this statement, called Gödel's statement, is true". Properly skeptical, we turn the computer on and ask, "Is Gödel's statement true?"

Gödel's computer cannot answer that question. If it tells us that the statement is true, it lies. If it tells us that the statement is false, then it has committed itself to saying at some other time that the statement is true and thus it lies. The statement is true if and only if the computer cannot answer the question we put to it. But Gödel programmed the computer to answer all questions put to it and to do so truthfully. So the computer must either lie to us or refuse to answer the question: in either case it violates its programming. Thus we infer that the program in either inconsistent with itself or is incomplete.

The science of mathematics is a computer and mathematicians put questions to it via predicate logic. Via the process of deduction and proof mathematicians have spun out arithmetic, number theory, algebra, the calculus, trigonometry, and so much more. But in its technical form Gödel's theorem can be translated into any system generated by predicate logic and containing simple arithmetic. So now we know that mathematics contains at least one proposition that it cannot tell; that is, a proposition that no one can ever deduce or prove. Attempts to deduce or to prove that proposition must necessarily yield an undissolvable paradox. Thus we know that our knowledge of mathematics cannot be complete without being inconsistent; cannot be consistent without being incomplete.

Reality is a computer, one whose output comprises the Universe and its history. In that light we can see the Map of Physics as a description of the program running on that computer. Gödel's theorem assures us that we will eventually come across an undissolvable paradox in the course of our effort to deduce the Map. But how would Reality manifest a proposition that it can know but cannot tell? What physical phenomenon could correspond to such a thing? At this stage of drawing the Map of Physics I just don't have a clue, but now we know what we need to find.

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