A version of this article appeared Monday, October 28, 2013. The essay originally
appeared on Angelfire ca. 2005.
This paper was updated Dec. 10, 2009, Oct. 28, 2013. A minor addition citing
Henry Stapp was made Feb. 21, 2019.
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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 (zz2) and science writer Stan Gibilisco (4), who noted that the general theory of relativity resolves the problem. Isaac Asimov showed a good grasp of the paradox (3a), which he resolved by appeal to actual acceleration, while off-handedly exonerating Einstein of an oversight. (Not all of Asimov's arguments regarding the paradox are, however, persuasive.)
Dave Goldberg, a Drexel physics professor, correctly interprets the paradox in his book The Universe in the Rearview Mirror (zz1) and addresses a number of points raised in a previous version of this essay, coincidentally including correction of some (non-crucial) claims that I seem to have got wrong (and which I may or may not have excised).
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. 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).
[It should be noted that Dingle's 1949 attempt at relativistic physics left Einstein bemused (6).]
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 observer 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?
The general theory undermined two claims of the 1905 special theory, according to Henry Stapp[x1], a quantum physicist. One was the Machian and "logical positivist" claim that only verifiable (testable) assertions should be accepted in physics. The other was the absence in nature of a preferred sequence of nows. "Furthermore, the universe we are living in has a global preferred reference frame, which defines instantaneous "nows" empirically. This frame has recently [1990s] been empirically specified to within several parts per million."
In other words, the original concept of "relative" has been somewhat modified. Velocities cannot be strictly relative to one another because, on the basis of a non-eternal Big Bang theory, each velocity must have received an infusion of energy ("true" acceleration) that brought it to its present level. Yet, the constancy of the velocity of light means that motions are still relative with respect to an observer.
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