Alternate View Column AV-28
Keywords: paradox, Zeno, Oblers, K mesons, nonlocality, FTL communication
Published in the September-1988 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 2/5/88 and is copyrighted © 1988, John G. Cramer. All rights reserved.
No part may be reproduced in any form without the explicit permission of the author.
This column is about physical paradoxes. A paradox is a sort of logical swindle that leads to a surprising or perhaps incorrect conclusion. It's a chain of plausible physical arguments that lead to an unphysical conclusion.
Paradoxes occupy a very special place in the history of science. They have, on occasion, produced an important scientific breakthrough by leading to a new way of thinking about the universe. Or, with the clear vision of hindsight, we can sometimes see that resolving a paradox might have led to an intellectual breakthrough, but the paradox was ignored and the opportunity wasted.
An example of the second situation is the most famous of the paradoxes of Zeno, the Greek philosopher who lived during the Golden Age of Greece on the island of Elea. Zeno proposed the following "thought experiment". Achilles, a young athlete, runs a race with a tortoise. Achilles can run exactly twice as fast as the tortoise, so to make it fair he gives the tortoise a head start of exactly half the distance from the starting line to the finish line. The starting signal is given and the race begins. Achilles runs to the starting position of the tortoise. In the time it takes to do that, the tortoise has advanced half the distance from his starting position and the finish line. Achilles then advances to the new position of the tortoise. During that time the tortoise again advances half the distance to the finish line. And so on ... Every time Achilles moves ahead by a given distance, the tortoise moves ahead by half that distance. Zeno concluded that Achilles can never catch the tortoise, because in every time interval in which Achilles moves to the tortoise's former position, the tortoise always moves ahead by half that distance.
This conclusion, of course, is absurd. Given enough time it is clear that runner A can always catch and pass runner B, if A can run faster. The flaw in Zeno's logic was the assumption that an infinite number of progressively decreasing time steps will sum to an infinite time interval. This is incorrect. An infinite sum of infinitesimals can add up to a definite finite result. Achilles will catch and pass the tortoise at the finish line, just as common sense would suggest.
Zeno's paradox points to a problem in the mathematical thinking of the classical Greeks. They had a well developed mathematics but had not developed the concept of infinitesimals, the foundation of integral and differential calculus. It is plausible that if the mathematical thinkers of the time had taken Zeno's paradox seriously and devoted the proper thought to its analysis and resolution, they might have been led to invent differential and integral calculus a millennium before its actual formulation by Newton and Leibnitz. This would undoubtedly have altered the progress of science and perhaps changed the course of history.
On the other side, consider the case of Galileo. He was puzzled by the way Aristotle, more than a millennium earlier, had described falling bodies. Aristotle said that a heavy body falls faster than a light body. A feather, for example, clearly falls more slowly than a gold coin. But Galileo considered the following paradox. Suppose that one drops two gold coins. They fall at the same rate, according to Aristotle, because they are equally heavy. But now suppose that the coins are connected with a very light thread. This, according to Aristotle, should make them fall faster, because they are now one object that is twice as heavy. But why? How do they know that the thread is there? Since the coins are falling at the same rate when unconnected, neither can pull on the other through the thread to make it fall faster.
Galileo carefully analyzed this paradox and concluded that Aristotle must be wrong. In the absence of air resistance (which slows the feather more than the coin) all bodies must fall the same, whether they are heavy or light. Galileo's analysis was the foundation stone of modern dynamics. He made the conceptual breakthrough that made it possible for Isaac Newton, about a century later, to discover the laws of motion.
Then there's Oblers' paradox. Heinrich Oblers, an 18th century German astronomer, wondered why at night the sky is dark. That sounds like a really stupid question, right? The sky is dark at night, of course, because the sun has gone down. But Oblers' paradox is not a stupid question; it's quite subtle and profound. There are many other stars besides the sun. Let's consider, as a first approximation, that these stars are distributed uniformly throughout the universe. Divide the universe up like a sort of spherical layer cake with many concentric spherical shells, each with a thickness of, say, 10 light years. In each of these shells there will be a certain number of stars, and each star bathes the earth with a certain amount of starlight. As one includes shell layers further and further away from the earth, the amount of starlight from an average star decreases as the inverse square of the distance. But the number of stars in the shell increases as the square of the distance. Thus the amount of starlight from each shell should be the same, and if we include all the shells in an infinite universe, there should be an infinite amount of starlight.
To put it a slightly different way, if the universe is truly infinite, than in any given direction in the sky the line of sight should eventually intercept a star. Thus, all points in the night sky should be the color and temperature of the surface of a star, or several thousand degrees C. We should be inside a cosmic barbecue pit that should be roasting us, making life impossible on the surface of the Earth. Since life exists on Earth, and the night sky is dark, something is clearly wrong with this argument. But what?
The answer is that the universe is expanding and is only a few billion years old. The more distant stars and galaxies do not bathe the Earth with the same intensity of starlight as those nearby because the energy content of their light is diluted by the Doppler shift of their recession velocities. Further, shells more than a few billion light years away give no light at all, because they contain no stars. This we now know because Edmund Hubble discovered the expansion of the universe in 1929 by studying the Doppler shifts of distant galaxies. But the same conclusion might have been reached almost two centuries earlier, if astronomers had carefully considered Oblers' question and realized its implications.
Albert Einstein devised another paradox at the end of the 19th century, at a time when he was still in high school. Electromagnetic theory describes a light wave as a combination of an electric field and a magnetic field, both at right angles to the direction and to each other, oscillating together from positive to negative as they travel through space. Einstein realized that, as seen by an observer running beside such a wave at the speed of light, this behavior would violate the laws of physics. How is it, he wondered, that the laws of physics that produce such waves in one reference frame, could be violated in another reference frame. He set out on a quest that took him a decade, a quest to find a way of looking a physical law in such a way that the laws of physics work in all reference frames. This led him to his discovery of the special theory of relativity.
Albert Einstein, together with Boris Podolsky and Nathan Rosen, was also responsible for the famous Einstein-Podolsky-Rosen (or EPR) paradox which burst upon the world of physics in the mid-1930's and has been the center of a raging debate ever since. Einstein distrusted quantum mechanics because he perceived embedded in its formalism what he referred to as "spooky actions at a distance". The characteristic that worried Einstein is called "nonlocality", which means that some quantum relationship is being enforced faster-than-light across space and time. The EPR paradox was carefully constructed to spotlight this peculiarity of the formalism of quantum mechanics, and led to experiments performed in the last decade that demonstrate the effect unambiguously. These measurements of the correlated optical polarizations of oppositely directed photons show that something very like faster-than-light (FTL) hand-shaking is going on within the formalism of quantum mechanics and in nature itself. But a common feature of all the EPR experiments is that the effects demonstrated cannot be used by one observer to send a FTL message to another observer. Nature's FTL telegraph is not available to us.
Or is it? Half a year ago a new quantum mechanical paradox was proposed by Datta, Home, and Raychaudhuri (DHR) of the University of Calcutta, and we will refer to it as the Calcutta paradox. It involves a method of using the peculiarities of neutral K-mesons (kaons) to communicate faster than the speed of light. Faster-than-light communication is considered impossible by most physicists, because it would represent a violation of either special relativity or would be a means of sending messages backwards in time, thereby violating the law of causality.
A meson is one of the particles that populates the world of high energy physics. It has a mass intermediate between that of an electron and a proton and it typically exists for only a few billionths of a second. We now know that mesons are a combination of a matter quark and and antimatter quark in close combination, so that they appear to be a single particle. In a previous AV column I have written about kaons. The Ko meson is made of a "down" quark and an "anti-strange" quark. Its antimatter twin, the anti-Ko, is a strange quark and an anti-down quark. Both Ko's have zero electrical charge and spin, and both have the same mass (about half a proton mass). On the basis of observables they are indistinguishable.
When two quantum states cannot be distinguished, a peculiar thing happens. The two indistinguishable states mix to form two new states of matter that are distinguishable. In the case of neutral kaons, the Ko and anti-Ko combine in two different ways to make the KS particle (K-short) which decays in about 10-10 seconds and the KL particle (K-long) which decays 581 times more slowly. The KL state is unique among mixed states because it shows what is called "CP violation", a preference for matter over antimatter and for one direction of time over the other. The KL demonstrates that systems composed of matter and of antimatter do not behave in precisely the same way, and that if a movie were made of particle reactions involving a KL, one could tell if the film were running forward or backwards through the projector.
The Calcutta paradox uses a process that makes neutral kaons in pairs, so that a KL goes in one direction while a Ks goes in the opposite direction. The calculations presented in the DHR paper indicate that if a detector for anti-Ko particles is placed in one arm of the experiment, it will instantaneously register a change in its counting rate when a chunk of copper is placed in the path of the other kaon, as compared to the counting rate when no copper is in place. In other words, the DHR experiment would allow an observer with a copper block at one arm of the experiment to telegraph a faster-than-light message to an observer watching the counting rate of the anti-Ko detector at the other arm of the experiment. This is the Calcutta paradox.
To illustrate the true content of the Calcutta paradox, lets do something more grandiose. Construct a neutral particle accelerator that can boost the kaons heading for the copper block up to a kinetic energy that is a few trillion times greater than the kaon rest mass. After acceleration to such ultra-relativistic speeds the kaons' decay lifetime is stretched to years rather than nanoseconds. Now send the the kaon beam out into space to travel for a year, make a 180o turn around a conveniently placed black hole, and return to the laboratory two years after leaving. Then decelerate the the kaons back to their former velocity and either let them collide with the copper block or decay in flight.
The FTL communicator has now been transformed into a backwards in time communicator. The experimenter positioning the copper block can send a message backwards in time to the experimenter watching the anti-Ko detector, two years in the past. I will leave it as a problem for the SF reader to figure out what could be done with such a telegraph. Consult the works of Benford, Hogan, and Heinlein if you get stuck.
Is it possible that the fantastic implications of the DHR calculation are correct? Probably not. Nature has up to now erased all the possibilities for using the EPR effect for FTL communication so completely that it seems unlikely that a loophole was left. And yet, if there were such a loophole, it is very plausible that the kaon's CP violation would be just the place where it would appear. It would represent a sort of poetic justice to use the time asymmetry built into the decay of neutral kaons to overcome the time asymmetry of causality, the no-backwards-in-time-communication rule.
The Calcutta paradox is now in the literature of physics. We will just have to wait to see if it is a real paradox, or only a flaw in a chain of logic.
Followup note: The Calcutta Paradox proved to be only a bad quantum mechanical calculation. However, both theoretical and experimental studies of the peculiarities of quantum nonlocality using K mesons (and perhaps B mesons) are continuing.
W. V. Quine, Scientific American 206, #4 (April, 1962), pp. 84-96.
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47 (1935) 777.
A. Datta, D. Home, and A. Raychaudhuri, Phys. Lett. A 123 (1987) 4.
Neutral K Mesons:
Robert Adair, Scientific American 258, #2 (February, 1988).
SF Novels by John Cramer: my two hard SF novels, Twistor and Einstein's Bridge, are newly released as eBooks by Book View Cafe and are available at : http://bookviewcafe.com/bookstore/?s=Cramer .
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