Drunk and disorderly:
the rise (and fall) of entropy

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Revised version of an Angelfire page published ca. 2010
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Please see The many worlds of probability, reality and cognition; Part V focuses on the topic of entropy.
http://randompaulr.blogspot.com/2013/11/the-many-worlds-of-probability-reality.html
My thoughts on probability in the current paper should be taken as provisional.

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By PAUL CONANT
One might describe the increase of the entropy (FN 0) of a gas to mean that the net vector -- sum of vectors of all particles -- at between time t₀ and t_n tends toward some constant, such as 0, and that once this equilibrium is reached at t_n, the net vector stays near 0 at any subsequent time.

One would expect a nearly 0 net vector if the individual particle vectors are random. This randomness is exactly what one would find in an asymmetrical n-body scenario, where the bodies are close together and about the same size. The difference is that gravity isn't the determinant, but rather collisional kinetic energy. It has been demonstrated that n-body problems can yield orbits that become extraordinarily tangled. The randomness is then of the Chaitin-Kolmogorov variety: determining future position of a particular particle becomes computationally very difficult. And usually, over some time interval, the calculation errors increase to the point that all predictability for a specific particle is lost.

But there is also quantum randomness at work. The direction that an excited photon exits an atom is probabilistic only, meaning that the recoil is random. This recoil vector must be added to the other electric charge recoil vector associated with particle collision -- though its effect is very slight and usually ignored. Further, if one were to observe one or more of the particles, the observation would affect the knowledge of the momentum or position of the observed particles. Now supposing we keep the gas at a single temperature in a closed container attached via a closed valve to another evacuated container, when we open the valve, the gas expands to fill both containers. This expansion is a consequence of the effectively random behavior of the particles, which on average "find less resistance" in the direction of the vacuum.

In general, gases tend to expand by inverse square, or that is spherically (or really, as a ball), which implies randomization of the molecules.

The drunkard's walk
Consider a computerized random walk (aka "drunkard's walk") in a plane. As n increases, the area covered by the walk tends toward that of a circle. In the infinite limit, there is probability 1 that a perfect circle has been covered (though probability 1 in such cases does not exclude exceptions).

So the real question is: what about the n-body problem yields pi-randomness? It is really a statistical question. When enough collisions occur in a sufficiently small volume (or area), the particle vectors tend to cancel each other out.

Let's go down to the pool hall and break a few racks of balls. It is possible to shoot the cue ball in such a way that the rack of balls scatters symmetrically. But in most situations, the cue ball strikes the triangular array at a point that yields an asymmetrical scattering. This is the sensitive dependence on initial conditions associated with mathematical chaos. We also see Chaitin-Kolmogorov complexity enter the picture, because the asymmetry means that for most balls predicting where one will be after a few ricochets is computationally very difficult.

Now suppose we have perfectly inelastic, perfectly spherical pool balls that encounter idealized banks. We also neglect friction. After a few minutes, the asymmetrically scattered balls are "all over the place" in effectively random motion. Now such discrete systems eventually return to their original state: the balls coalesce back into a triangle and then repeat the whole cycle over again, which implies that in fact such a closed system, left to its own devices, requires that entropy to decrease, a seeming contradiction of the second law of thermodynamics. But the time scales required mean we needn't hold our breaths waiting. Also, in nature, there are darned few closed systems (and as soon as we see one, it's no longer closed at the quantum level), allowing us to conclude that in the ideal of zero friction, the pool ball system may become aperiodic, implying the second law in this case holds.

Maxwell's demon
And now, let us exorcize Maxwell's demon, which, though meant to elucidate, to this day bedevils discussions of entropy with outlandish "solutions" to the alleged "problem." Maxwell gave us a thought experiment whereby he posited a little being controlling the valve between canisters. If (in this version of his thought experiment) the gremlin opened the valve to let speedy particles past in one direction only, the little imp could divide the gas into a hot cloud in one canister and a cold cloud in the other. Obviously the energy the gremlin adds is equivalent to adding energy via a heating/cooling system, but Maxwell's point was about the very, very minute possibility that such a bizarre division could occur randomly (or, some would say, pseudo-randomly).

This possibility exists. In fact, as said, in certain idealized closed systems, entropy decrease MUST happen. Such a spontaneous division into hot and cold clouds would also probably happen quite often at the nano-nano-second level. That is, when time intervals are short enough, quantum physics tells us the usual rules go out the window. However, observation of such actions won't occur for such quantum intervals (so there is no change in information or entropy), and as for the "random" chance of observing an extremely high-ordering of gas molecules, even if someone witnessed such an occurrence, not only does the event not conform to a repeatable experiment, no one is likely to believe the report, even if true.

Truly universal?
Can we apply the principle of entropy to the closed system of the universe? A couple of points: We're not absolutely sure the cosmos is a closed system (perhaps, for example, "steady state" creation supplements "big bang" creation). If there is a "big crunch," then, some have speculated, we might expect complete devolution to original states (people would reverse grow from death to birth, for example). If space curvature implies otherwise, the system remains forever open or asymptotically forever open.

However, quantum fuzziness probably rules out such an idealization. Are quantum systems precisely reversible? Yes and no. When one observes a particle collision in an accelerator, one can calculate the reverse paths. However, in line with the Heisenberg uncertainty principle one can never be sure of observing a collision with precisely identical initial conditions. And if we can only rarely, very rarely, replicate the exact initial conditions of the collision, then the same holds for its inverse.

Then there is the question of whether perhaps a many worlds (aka parallel universes) or many histories interpretation of quantum weirdness holds. In the event of a collapse back toward a big crunch, would the cosmos tend toward the exact quantum fluctuations that are thought to have introduced irregularities in the early universe that grew into star and galactic clustering? Or would a different set of fluctuations serve as the attractor, on grounds both sets were and are superposed and one fluctuation is as probable as the other? And, do these fluctuations require a conscious observer, as in John von Neumann's interpretation?

Thinking in terms of computer-like algorithms, Stephen Wolfram writes in A New Kind of Science that it is unclear whether the "basic rules of the universe are really reversible," arguing that it could be that apparent reversibility arises due to effects of an attractor (he does not specify gravitational). He writes that "if pieces of the universe can break off but not reconnect, then there will be inevitably loss of information," thus increasing entropy. Of course, we face such difficulties when trying to apply physical or mathematical concepts to the entire cosmos. It seems plausible that any system of relations we devise to examine properties of space and time may act like a lens that increases focus in one area while losing precision in another. I.e., a cosmic uncertainty principle.

Conservation of information?
A cosmic uncertainty principle would make information fuzzy. As the Heisenberg uncertainty principle shows, information about a particle's momentum is gained at the expense of information about its position. But, you may respond, the total information is conserved.

But wait! Is there a law about the conservation of information? In fact, information cannot be conserved -- in fact can't exist -- without memory, which in the end requires the mind of an observer. In fact, the "law" of increase of entropy says that memories fade and available information decreases. In terms of pure Shannon information, entropy expresses the probability of what we know or don't know.2 Thus entropy is introduced by noise entering the signal. In realistic systems, supposing enough time elapses, noise eventually overwhelms the intended signal. For example, what would you say is the likelihood that this essay will be accessible two centuries from now? (I've already lost a group of articles I had posted on the now defunct Yahoo Geocities site.) Or consider Shakespeare's plays. We cannot say with certainty exactly how the original scripts read.

In fact, can we agree with some physicists that a specified volume of space contains a specific quantity of information? I wonder. A Shannon transducer is said to contain a specific quantity of information, but no one can be sure of that, prior to someone reading the message and measuring the signal-to-noise ratio. And quantum uncertainty qualifies as a form of noise, not only insofar as random jiggles in the signal, but also insofar as what signal was sent. If two signals are "transmitted" in quantum superposition, observation randomly determines which signal is read.

So one may set up a quantum measurement experiment and say that for a specific volume, the prior information describes the experiment. But quantum uncertainty still says that the experiment cannot be exactly described in a scientifically sensible way. So if we try to extrapolate information about a greater volume from the experiment volume, we begin to lose accuracy until the uncertainty reaches maximum. We see that quantum uncertainty can progressively change the signal-to-noise ratio, meaning entropy increases until the equilibrium level of no knowledge.

This of course would suggest that, from a human vantage point, there can be no exact information quantity for the cosmos.

So this brings us to the argument about whether black holes decrease the entropy of the universe by making it more orderly (i.e., simpler). My take is that a human observer in principle can never see anything enter a black hole. If one were to detect, at a safe distance, an object approaching a black hole, one would observe that its time pulses (its Doppler shift) would get slower and slower. In fact, the time pulses slow down asymptotic to eternity. So the information represented by the in-falling object is, from this perspective, never lost.

But suppose we agree to an abstraction that eliminates the human observer -- as opposed to a vastly more gifted intelligence. In that case, perhaps the cosmos has an exact quantity of information at ta. It then makes sense to talk about whether a black hole affects that quantity.

Consider a particle that falls into a black hole. It is said that all the information available about a black hole is comprised of the quantities for its mass and its surface area. Everything this super-intelligence knew about the particle, or ever could know, seemingly, is gone. Information is lost and the cosmos is a simpler, more orderly place, higher in information and in violation of the second law... maybe.

But suppose the particle is a twin of an entangled pair. One particle stays loose while the other is swallowed by the black hole. If we measure, say, the spin of one such particle we would ordinarily automatically know the spin of the other. But who's to tell what the spin is of a particle headed for the gravitational singularity at the black hole's core? So the information about the particle vanishes and entropy increases. This same event however means the orderliness of the universe increases and the entropy decreases. So, which is it? Or is it both. Have no fear, this issue is addressed in the next section.

Oh, and of course we mustn't forget Hawking radiation, whereby a rotating black hole slowly leaks radiation as particles every now and then "tunnel" through the gravitational energy barrier and escape into the remainder cosmos. The mass decreases over eons and eons until -- having previously swallowed everything available -- it eventually evaporates, Hawking conjectures. Actually, we needn't invoke tunneling; the condition that the object is rotating means that it has kinetic energy; some quanta of energy associated with rotational acceleration are at the event horizon and are energetic enough to escape the gravity field, assuming they are vectored appropriately.

Hawking's updated black hole view
http://www.nature.com/news/2004/040712/full/news040712-12.html

In 2005, Hawking revived a long-simmering argument about black holes and entropy.

"I'm sorry to disappoint science fiction fans, but if information is preserved, there is no possibility of using black holes to travel to other universes. If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized. It is like burning an encyclopedia. Information is not lost, if one keeps the smoke and the ashes. But it is difficult to read. In practice, it would be too difficult to re-build a macroscopic object like an encyclopedia that fell inside a black hole from information in the radiation, but the information preserving result is important for microscopic processes involving virtual black holes."

Information loss in black holes
http://arxiv.org/pdf/hepth/0507171.pdf

A question: suppose an entangled particle escapes the black hole? Is the cosmic information balance sheet rectified? Perhaps, supposing it never reached the singularity. But, what of particles down near the singularity? They perhaps morph as the fields transform into something that existed close to the cosmic big bang. So it seems implausible that the spin information is retained. But, who knows?

Where's that ace?
There is a strong connection between thermodynamic entropy and Shannon information entropy (FN 0). Consider the randomization of the pool break on the frictionless table after a few minutes. This is the equivalent of shuffling a deck of cards. Suppose we have an especially sharp-eyed observer who watches where the ace of spades is placed in the deck as shuffling starts. We then have a few relatively simple shuffles. After the first shuffle, he knows to within three cards how far down in the deck the ace is. On the next shuffle he knows where it is with less accuracy. Let's say to a precision of (1/3)(1/3) = 1/9. After some more shuffles his potential error has reached 1/52, meaning he has no knowledge of the ace's whereabouts.

The increase in entropy occurs from one shuffle to the next. But at the last shuffle, equilibrium has been reached. Further shuffling can never increase his knowledge of where the ace is, meaning the entropy won't decrease. The runs test gives a measure of randomness (FN 1) based on the normal distribution of numbers of runs, with the mean at n/2, "Too many" runs are found in one tail and "too few" in another. That is, a high z score implies that the sequence is suspected of being non-random or "highly ordered."

What however is meant by order? (This is where we tackle the conundrum of a decrease in one sort of cosmic information versus an increase in another sort.)

Entropy is often defined as the tendency toward decrease of order, and the related idea of information is sometimes thought of as the surprisal value of a digit string. Sometimes a pattern such as HHHH... is considered to have low information because we can easily calculate the nth value (assuming we are using some algorithm to obtain the string). So the Chaitin-Kolmogorov complexity is low, or that is, the information is low. On the other hand a string that by some measure is effectively random is considered here to be highly informative because the observer has almost no chance of knowing the string in detail in advance.

However, we can also take the opposite tack. Using runs testing, most digit strings (multi-value strings can often be transformed, for test purposes, to bi-value strings) are found under the bulge in the runs test bell curve and represent probable randomness. So it is unsurprising to encounter such a string. It is far more surprising to come across a string with far "too few" or far "too many" runs. These highly ordered strings would then be considered to have high information value.

So, once the deck has been sufficiently shuffled the entropy has reached its maximum (equilibrium). What is the probability of drawing four royal flushes? If we aren't considering entropy, we might say it is the same as that for any other 20-card deal. But, a runs test would give a z score of infinity (probability 1 that the deal is non-random) because drawing all high cards is equivalent to tossing a fair coin and getting 20 heads and no tails. If we don't like the infinitude we can posit 21 cards containing 20 high cards and 1 low card. The z score still implies non-randomness with a high degree of confidence.

Negative entropy?
Our discussion should not ignore the impact of Ramsey theory, an important subdiscipline of network theory. "Self-organizing" possibilities are inevitable with sufficient number of nodes in a network. In fact, one might argue that Ramsey theory implies negative entropy. Suppose we had n poker players. The probability that among them there is a royal flush skyrockets quite rapidly. So as n increases, the probability of a specific set of cards increases and the information surprisal value decreases (FN 3).

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0. Taken from a Wikipedia article: The dimension of thermodynamic entropy is energy divided by temperature, and its SI unit is joules per kelvin. In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication."

1. We should caution that the runs test, which works for n1 > 7 and n2 > 7, fails for the pattern HH TT HH TT... This failure seems to be an artifact of the runs test assumption that a usual number of runs is about n/2. I suggest that we simply say that the probability of that pattern is less than or equal to H T H T H T..., a pattern whose z score rises rapidly with n. Other patterns such as HHH TTT HHH... also climb away from the randomness area slowly with n. With these cautions, however, the runs test gives striking results.

2. Taken from Wikipedia: In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication." Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically-distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet. A fair coin has an entropy of one bit. However, if the coin is not fair, then the uncertainty is lower (if asked to bet on the next outcome, we would bet preferentially on the most frequent result), and thus the Shannon entropy is lower. Mathematically, a coin flip is an example of a Bernoulli trial, and its entropy is given by the binary entropy function. A long string of repeating characters has an entropy rate of zero, since every character is predictable. The entropy rate of English text is between 1.0 and 1.5 bits per letter, or as low as 0.6 to 1.3 bits per letter, according to estimates by Shannon based on human experiments.

3. John Allen Paulos on Ramsey theory: 'A more profound version of this line of thought can be traced back to British mathematician Frank Ramsey, who proved a strange theorem. It stated that if you have a sufficiently large set of geometric points and every pair of them is connected by either a red line or a green line (but not by both), then no matter how you color the lines, there will always be a large subset of the original set with a special property. Either every pair of the subset's members will be connected by a red line or every pair of the subset's members will be connected by a green line. If, for example, you want to be certain of having at least three points all connected by red lines or at least three points all connected by green lines, you will need at least six points. (The answer is not as obvious as it may seem, but the proof isn't difficult.) For you to be certain that you will have four points, every pair of which is connected by a red line, or four points, every pair of which is connected by a green line, you will need 18 points, and for you to be certain that there will be five points with this property, you will need -- it's not known exactly - between 43 and 55. With enough points, you will inevitably find unicolored islands of order as big as you want, no matter how you color the lines.'

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Revised Oct. 29, 2013

Einstein, Sommerfeld and the twin paradox

Please notify me of errors at Krypto78 attt gmaaail dottt commm

Topologist Jeff Weeks on the twin paradox
http://www.math.uic.edu/undergraduate/mathclub/talks/Weeks_AMM2001.pdf

Michel Janssen's paper
https://netfiles.umn.edu/users/janss011/home%20page/rel-of-grav-field.pdf

The paradox
Einstein's groundbreaking 1905 relativity paper, "On the electrodynamics of moving bodies," contained a fundamental inconsistency which was not addressed until 10 years later, with the publication of his paper on gravitation.

Many have written on this inconsistency, known as the "twin paradox" or the "clock paradox" and more than a few have not understood that the "paradox" does not refer to the strangeness of time dilation but to a logical inconsistency in what is now known as the special (for "special case") theory of relativity.

Among those missing the point: Max Born in his book on special relativity (1), George Gamow in an essay and Roger Penrose in Road to Reality (2), and, most recently, Leonard Susskind in The Black Hole War (3).

Among those who have correctly understood the paradox are topologist Jeff Weeks (see link above) and science writer Stan Gibilisco (4), who noted that the general theory of relativity resolves the problem.

As far back as the 1960s, the British physicist Herbert Dingle (5) called the inconsistency a "regrettable error" and was deluged with "disproofs" of his assertion from the physics community. (It should be noted that Dingle's 1949 attempt at relativistic physics left Einstein bemused (6). Yet every "disproof" of the paradox that I have seen uses acceleration, an issue not addressed by Einstein until the general theory of relativity. It was Einstein who set himself up for the paradox by favoring the idea that only purely relative motions are meaningful, writing that various examples "suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest." [Electrodynamics translated by Perett and Jeffery and appearing in a Dover (1952) reprint.] In that paper, he also takes pains to note that the term "stationary system" is a verbal convenience only (7).

But later in Elect., Einstein offered the scenario of two initially synchronized clocks at rest with respect to each other. One clock then travels around a closed loop, and its time is dilated with respect to the at-rest clock when they meet again. In Einstein's words: "If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be 1/2tv²/c² slow."

Clearly, if there is no preferred frame of reference, a contradiction arises: when the clocks meet again, which clock has recorded fewer ticks?

Both in the closed loop scenario and in the polygon-path scenario, Einstein avoids the issue of acceleration. Hence, he does not explain that there is a property of "real" acceleration that is not symmetrical or purely relative and that that consequently a preferred frame of reference is implied, at least locally.

The paradox stems from the fact that one cannot say which velocity is higher without a "background" reference frame. In Newtonian terms, the same issue arises: if one body is accelerating away from the other, how do we know which body experiences the "real" force? No answer is possible without more information, implying a background frame.

In comments published in 1910, the physicist Arnold Sommerfeld, a proponent of relativity theory, "covers" for the new paradigm by noting that Einstein didn't really mean that time dilation was associated with purely relative motion, but rather with accelerated motion; and that hence relativity was in that case not contradictory. Sommerfeld wrote: "On this [a time integral and inequality] depends the retardation of the moving clock compared with the clock at rest. The assertion is based, as Einstein has pointed out, on the unprovable assumption that the clock in motion actually indicates its own proper time; i.e. that it always gives the time corresponding to the state of velocity, regarded as constant, at any instant. The moving clock must naturally have been moved with acceleration (with changes of speed or direction) in order to be compared with the stationary clock at world-point P. The retardation of the moving clock does not therefore actually indicate 'motion,' but 'accelerated motion.' Hence this does not contradict the principle of relativity." [Notes appended to Space and Time, a 1908 address by Herman Minkowski, Dover 1952, Note 4.]

However, Einstein's 1905 paper does not tackle the issue of acceleration and more to the point, does not explain why purely relative acceleration would be insufficient to meet the facts. The principle of relativity applies only to "uniform translatory motion" (Elect. 1905).

Neither does Sommerfeld's note address the issue of purely relative acceleration versus "true" acceleration, perhaps implicitly accepting Newton's view (below). And, a review of various papers by Einstein seems to indicate that he did not deal with this inconsistency head-on, though in a lecture-hall discussion ca. 1912, Einstein said that the [special] theory of relativity is silent on how a clock behaves if forced to change direction but argues that if a polygonal path is large enough, accelerative effects diminish and (linear) time dilation still holds.

On the other hand, of course, he was not oblivious to the issue of acceleration. In 1910, he wrote that the principle of relativity meant that the laws of physics are independent of the state of motion, but that the motion is non-accelerated. "We assume that the motion of acceleration has an objective meaning," he said. [The Principle of Relativity and its Consequences in Modern Physics, a 1910 paper reproduced in Collected Papers of Albert Einstein, Hebrew University, Princeton University Press.]

In that same paper Einstein emphasizes that the principle of relativity does not cover acceleration. "The laws governing natural phenomena are independent of the state of motion of the coordinate system to which the phenomena are observed, provided this system is not in accelerated motion." Clearly, however, he is somewhat ambiguous about small accelerations and radial acceleration, as we see from the lecture-hall remark and from a remark in Foundation of the General Theory of Relativity (1915) about a "familiar result" of special relativity whereby a clock on a rotating disk's rim ticks slower than a clock at the origin.

General relativity's partial solution
Finally, in his 1915 paper on general relativity, Einstein addressed the issue of acceleration, citing what he called "the principle of equivalence." That principle (actually, introduced prior to 1915) said that there was no real difference between kinematic acceleration and gravitational acceleration. Scientifically, they should be treated as if they are the same.

So then, Einstein notes in Foundation, if we have system K and another system K' accelerating with respect to K, clearly, from a "Galilean" perspective, we could say that K was accelerating with respect to K'. But, is this really so?

Einstein argues that if K is at rest relative to K', which is accelerated, the oberserver on K cannot claim that he is being accelerated -- even though, in purely relative terms, such a claim is valid. The reason for this rejection of Galilean relativity: We may equally well interpret K' to be kinematically unaccelerated though the "space-time territory in question is under the sway of a gravitational field, which generates the accelerated motion of the bodies" in the K' system. This claim is based on the principle of equivalence which might be considered a modification of his previously posited principle of relativity. By the relativity principle, Einstein meant that the laws of physics can be cast in invariant form so that they apply equivalently in any unformly moving frame of reference. (For example, |vb - va| is the invariant quantity that describes an equivalence class of linear velocities.)

By the phrase "equivalence," Einstein is relating impulsive acceleration (for example, a projectile's x vector) to its gravitational acceleration (its y vector). Of course, Newton's mechanics already said that the equation F = mg is a special case of F = ma but Einstein meant something more: that local spacetime curvature is specific for "real" accelerations -- whether impulsive or gravitational.

Einstein's "equivalence" insight was his recognition that one could express acceleration, whether gravitational or impulsive, as a curvature in the spacetime continuum (a concept introduced by Minkowski). This means, he said, that the Newtonian superposition of separate vectors was not valid and was to be replaced by a unitary curvature. (Though the calculus of spacetime requires specific tools, the concept isn't so hard to grasp. Think of a Mercator map: the projection of a sphere onto a plane. Analogously, general relativity projects a 4-dimensional spacetime onto a Euclidean three-dimensional space.)

However, is this "world-line" answer the end of the problem of the asymmetry of accelerated motion?

The Einstein of 1915 implies that if two objects have two different velocities, we must regard one as having an absolutely higher velocity than the other because one object has been "really" accelerated.

Yet one might conjecture that if two objects move with different velocities wherein neither has a prior acceleration, then the spacetime curvature would be identical for each object and the objects' clocks would not get out of step. But such a conjecture would violate the limiting case of special relativity (and hence general relativity); specifically, such a conjecture would be inconsistent with the constancy of the vacuum velocity of light in any reference frame.

So then, general relativity requires that velocity differences are, in a sense, absolute. Yet in his original static and eternal cosmic model of 1917, there was no reason to assume that two velocities of two objects necessarily implied the acceleration of one object. Einstein introduced the model, with the cosmological constant appended in order to contend with the fact that his 1915 formulation of GR apparently failed to account for the observed mass distribution of the cosmos. Despite the popularity of the Big Bang model, a number of cosmic models hold the option that some velocity differences needn't imply an acceleration, strictly relative or "real."

Einstein's appeal to spacetime curvature to address the frame of reference issue is similar to Newton's assertion that an accelerated body requires either an impulse imputed to it or the gravitational force. There is an inherent local physical asymmetry. Purely relative motion will not do.

Frank Close points out that in the quantum arena, unlike in SR, superconductivity shows that there is an absolute state of rest, That is, he writes, the superconductor is at rest relative to the electron but not the converse (9).

Einstein also brings up the problem of absolute relative motion in the sense of Newton's bucket. Einstein uses two fluid bodies in space, one spherical, S1 and another an ellipsoid of revolution, S2. From the perspective of "Galilean relativity," one can as easily say that either body is at rest with respect to the other. But, the radial acceleration of S2 results in a noticeable difference: an equatorial bulge. Hence, says Einstein, it follows that the difference in motion must have a cause outside the system of the two bodies.

Of course Newton in Principia Mathematica first raised this point, noting that the surface of water in a rapidly spinning bucket becomes concave. This, he said, demonstrated that force must be impressed on a body in order for there to be a change in acceleration. Newton also mentioned the issue of the fixed stars as possibly of use for a background reference frame, though he does not seem to have insisted on that point. He did however find that absolute space would serve as a background reference frame.

It is noteworthy that Einstein's limit c can be used as an alternative to the equatorial bulge argument. If we suppose that a particular star is sufficiently distant, then the x component of its radial velocity (which is uniform and linear) will exceed the velocity of light. Such a circumstance being forbidden, we are forced to conclude that the earth is spinning, rather than the star revolving around the earth. We see that, in this sense, the limit c can be used to imply a specific frame of reference. At this point, however, I cannot say that such a circumstance suffices to resolve the clock paradox of special relativity.

Interestingly, the problem of Newton's bucket is quite similar to the clock paradox of special relativity. In both scenarios, we note that if two motions are strictly relative, what accounts for a property associated with one motion and not the other? In both cases, we are urged to focus on the "real" acceleration.

Newton's need for a background frame to cope with "real" acceleration predates the 19th century refinement of the concept of energy as an ineffable, essentially abstract "substance" which passes from one event to the next. That concept was implicit in Newton's Principia but not explicit and hence Newton did not appeal to the "energy" of the object in motion to deal with the problem. That is, we can say that we can distinguish between two systems by examining their parts. A system accelerated to a non-relativistic speed nevertheless discloses its motion by the fact that the parts change speed at different times as a set of "energy transactions" occur. For example, when you step on the accelerator, the car seat moves forward before you do; you catch up to the car "because" the car set imparts "kinetic energy" to you.

But if you are too far away to distinguish individual parts or a change in the object's shape, such as from equatorial bulge, your only hope for determining "true" acceleration is by knowing which object received energy prior to the two showing a relative change in velocity. Has the clock paradox gone away?

Now does GR resolve the clock paradox?

GR resolves the paradox non-globally, in that Einstein now holds that some accelerations are not strictly relative, but functions of a set of curvatures. Hence one can posit the loop scenario given inElectrodynamics and say that only one body can have a higher absolute angular velocity with respect to the other because only one must have experienced an acceleration that distorts spacetime differently from the other.

To be consistent, GR must reflect this asymmetry. That is, suppose we have two spaceships separating along a straight line whereby the distance between them increases as a constant velocity. If ship A's TV monitor says B's clock is ticking slower than A's and ship B's TV monitor says A's clock is ticking slower than B's, there must be an objective difference, nevertheless.

The above scenario is incomplete because the "real" acceleration prior to the opening of the scene is not given. Yet, GR does not tell us why a "real" acceleration must have occurred if two bodies are moving at different velocities.

So yes, GR partly resolves the clock paradox and, by viewing the 1905 equations for uniform motion as a special case of the 1915 equations, retroactively removes the paradox from SR, although it appears that Einstein avoided pointing this out in 1915 or thereafter.

However, GR does not specify a global topology (cosmic model) of spacetime, though Einstein struggled with this issue. The various solutions to GR's field equations showed that no specific cosmic model followed from GR. The clock paradox shows up in the Weeks model of the cosmos, with local space being euclidean (or rather Minkowskian). As far as this writer knows, such closed space geodesics cannot be ruled out on GR grounds alone.

Jeff Weeks, in his book The Shape of Space, points out that though physicists commonly think of three cosmic models as suitable for GR, in fact there are three classes of 3-manifolds that are both homogenous and isotropic (cosmic information is evenly mixed and looks about the same in any direction). Whether spacetime is mathematically elliptic, hyperbolic or euclidean, there are many possible global topologies for the cosmos, Weeks says.

One model, described by Weeks in the article linked above, permits a traveler to continue straight in a closed universe until she arrives at the point of origin. Again, to avoid contradiction, we are required to accept a priori that an acceleration that alters a world line has occurred.

Other models have the cosmic time axis following hyperbolic or elliptical geometry. Originally, one suspects, Einstein may have been skeptical of such an axis, in that Einstein's abolishment of simultaneity effectively abolished the Newtonian fiction of absolute time. But physicist Paul Davies, in his book About Time, argued that there is a Big Bang oriented cosmic time that can be approximated quite closely.

Kurt Goedel's rotating universe model left room for closed time loops, such that an astronaut who continued on a protracted space flight could fly into his past. This result prompted Godel to question the reality of time in general relativity. Having investigated various solutions of GR equations, Goedel argued that a median of proper times of moving objects, which James Jeans had thought to serve as a cosmic absolute time, was not guaranteed in all models and hence should be questioned in general.

Certainly we can agree that Goedel's result shows that relativity is incomplete in its analysis of time.

Mach's principles
Einstein was influenced by the philosophical writings of the German physicist Ernst Mach, whom he cites in Foundations.

According to Einstein (1915) Mach's "epistomological principle" says that observable facts must ultimately appear as causes and effects. Mach believed that the brain organizes sensory data into knowledge and that hence data of scientific value should stem from observable, measurable phenomena. This philosophical viewpoint was evident in 1905 when Einstein ruthlessly ejected the Maxwell-Lorentzian ether from physics.

Mach's "epistomological principle" led Mach to reject Newtonian absolute time and absolute space as unverifiable and made Einstein realize that the Newtonian edifice wasn't sacrosanct. However, in 1905 Einstein hadn't replaced the edifice with something called a "spacetime continuum." Curiously, later in his career Einstein impishly but honestly identified this entity as "the ether."

By rejecting absolute space and time, Mach also rejected the usual way of identifying acceleration in what is known as Mach's principle: Version A. Inertia of a ponderable object results from a relationship of that object with all other objects in the universe.

Version B. The earth's equatorial bulge is not a result of absolute rotation (radial acceleration) but is relative to the distant giant mass of the universe.

For a few years after publication of Foundations, Einstein favored Mach's principle, even using it as a basis of his "cosmological constant" paper, which was his first attempt to fit GR to a cosmic model, but was eventually convinced by the astronomer Wilem de Sitter (see Janssen above) to abandon the principle. In 1932 Einstein adopted the Einstein-de Sitter model that posits a cosmos with a global curvature that asymptotically zeroes out over eternity. The model also can be construed to imply a Big Bang, with its ordered set of accelerations.

A bit of fine-tuning
We can fine-tune the paradox by considering the velocity of the center of mass of the twin system. That velocity is m1v/m1 + m2. So the CM velocity is larger when the moving mass is the lesser and the converse. Letting x be a real greater than 1 we have two masses xm and m. The algebra reveals there is a factor (x/x+1) > 1/(x+1). The CM velocity for earth moving at 0.6c with respect to a 77kg astronaut is very close to 0.6c. For the converse, however, that velocity is about 2.3 meters per femto-second.

If we like, we can use the equation

E = mc²(1-v²/c²)^1/2

to obtain the energies of each twin system.

If the earth is in motion and the astronaut at rest, my calculator won't handle the quantity for the energy. If the astronaut is in motion with the earth at rest, then E = 5.38*1041J.

But the paradox is restored as soon as we set m1 equal to m2.

Notes on the principle of equivalence
Now an aside on the principle of equivalence. Can it be said that gravitational acceleration is equivalent to kinematic acceleration? Gravitational accelerations are all associated with the gravitational constant G and of the form g = Gm/r2. Yet it is easy to write expressions for accelerations that cannot be members of the gravitational set. If a is not constant, we fulfill the criterion. If in rx, x =/= 2, there will be an infinity of accelerations that aren't members of the gravitational set.

At any rate, Einstein's principle of equivalence made a logical connection between a ponderable object's inertial mass and its gravitational mass. Newton had not shown a reason that they should be exactly equal, an assumption validated by acute experiments. (A minor technicality: Einstein and others have wondered why these masses should be exactly equal, but, properly they meant why should they be exactly proportional? Equality is guaranteed by Newton's choice of a gravitational constant. But certainly, min = kmgr, with k equaling one because of Newton's choice.)

Also, GR's field equations rest on the premise (Foundation) that for an infinitesimal region of spacetime, the Minkowskian coordinates of special relativity hold. However, this 1915 assumption is open to challenge on the basis of the Heisenberg uncertainty principle (ca. 1925), which sets a finite limit on the precision of a measurement of a particle's space coordinate given its time coordinate.

Einstein's Kaluza-Klein excursion
In Subtle is the Lord Pais tells of a period in which Einstein took Klein's idea for a five-dimensional spacetime and reworked it. After a great deal of effort, Einstein offered a paper which took Klein's ideas presented as his own, on the basis that he had found a way to rationalize obtaining the five-dimensional effect while sticking to the conventional perceptual view of space and time denoted 3D+T (making one wonder what he thought of his own four-dimensional spacetime scheme).

A perplexed Abraham Pais notes that a colleague dismissed Einstein's work as unoriginal, and Einstein then quickly dropped it (7). But reformulation of the ideas of others is exactly what Einstein did in 1905 with the special theory. He presented the mathematical and physical ideas of Lorenz, Fitzgerald and Poincare, whom he very likely read, and refashioned them in a manner that he thought coherent, most famously by rejecting the notion of ether as unnecessary.

Yet it took decades for Einstein to publicly acknowledge the contribution of Poincare, and even then, he let the priority matter remain fuzzy. Poincare's work was published in French in 1904, but went unnoticed by the powerful German-speaking scientific community. As a French-speaking resident of Switzerland, it seems rather plausible that the young patent attorney read Poincare's paper.

But, as Pais pointed out, it was Einstein's interpretation that made him the genius of relativity. And yet, that interpretation was either flawed, or incomplete, as we know from the twin paradox.

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Footnotes

Apologies for footnotes being out of order. Haven't time to fix.

1. Einstein's Theory of Relativity by Max Born (Dover 1962).

2. Road to Reality by Roger Penrose (Random House 2006).

3. The Black Hole War by Leonard Susskind (Little Brown 2009).

4. Understanding Einstein's Theories of Relativity by Stan Gibilisco (Dover reprint of the 1983 edition).

7. In his biography of Einstein, Subtle is the Lord (Oxford 1983), physicist Abraham Pais mentions the "clock paradox" in the 1905 Electrodynamics paper but then summarily has Einstein resolve the contradiction in a paper presented to the Prussian Academy of Physics after the correct GR paper of 1915, with Einstein arguing that acceleration ends the paradox, which Pais calls a "misnomer." I don't recall the Prussian Academy paper, but it should be said that Einstein strongly implied the solution to the contradiction in his 1915 GR paper. Later in his book, Pais asserts that sometime after the GR paper, Einstein dispatched a paper on what Pais now calls the "twins paradox" but Pais uncharacteristically gives no citation.

5. Though Dingle seems to have done some astronomical work, he was not -- as a previous draft of this page said -- an astronomer, according to Harry H. Ricker III. Dingle was a professor of physics and natural philosophy at Imperial College before becoming a professor of history and the philosophy of science at City College, London, Ricker said. "Most properly he should be called a physicist and natural philosopher since his objections to relativity arose from his views and interpretations regarding the philosophy of science."

6. Dingle's paper Scientific and Philosophical Implications of the Special Theory of Relativityappeared in 1949 in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp. Dingle used this forum to propound a novel extension of special relativity which contained serious logical flaws. Einstein, in a note of response, said Dingle's paper made no sense to him.

8. See for example Max Von Laue's paper in Albert Einstein: Philosopher-Scientist edited by Paul Arthur Schilpp (1949).

9. The Infinity Puzzle: Quantum field theory and the hunt for an orderly universe by Frank Close basic books 2011

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This paper updated Dec. 10, 2009, Oct. 28, 2013