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The Twin Paradox Revisited

by John G. Cramer

Alternate View Column AV-38
Keywords: special relativity, twin paradox, time dilation, starship, Einstein, Lorentz factor
Published in the March-1990 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 8/20/89 and is copyrighted © 1989, John G. Cramer. All rights reserved.
No part may be reproduced in any form without the explicit permission of the author.

    Six months ago I wrote an Alternate View column about wormholes, faster-than-light travel, and time machines [Analog, June-1989]. It was based on a spectacular recent breakthrough in general relativity from a group at Caltech. The column described how an advanced civilization might construct a stable "wormhole", a trans-spatial shortcut between one region of space and another, and use this shortcut for both faster-than-light travel and time travel.

    Shortly after the column appeared in Analog, I received an irate letter arrived from a regular reader with a good background in physics. He complained that one of the points made in my wormhole column was physically incorrect. He demanded that I correct this serious mistake and set the record be set straight with a printed retraction. This reaction, in itself, isn't unexpected. FTL and time travel are very controversial topics that lie well outside the usual "circle of discourse" of contemporary physics. Even though they are now moving toward this circle of discourse, some objections are to be expected.

    However, the complaint was not about FTL or time travel. The reader objected instead to my reference to the twin paradox of special relativity. In the wormhole column, I had stated (correctly) that if a space ship made a round-trip to a point 0.866 light-years distant while travelling both ways at 86.6% of the velocity of light, the ship's clock at the end of the trip would read one year slow as compared to a clock that remained behind and at rest; astronaut Sam making the trip would age one year while his twin brother Ernest who had remained behind on earth would have aged two years. It was this allusion to the twin paradox which had prompted the letter.

    The relativistic twin paradox, the slowing of biological clocks by relativistic time dilation and the differences in aging thereby produced are not new in science fiction. They provide the premise for many excellent SF works: Joe Haldeman's Forever War, Poul Anderson's Tau Zero, and Ursula LeGuin's Rocannon's World are examples.

    There was no physics goof in the use of the twin paradox in the wormhole column, but the irate letter points up an interesting fact. The twin paradox is still, some 85 years after publication of Einstein's paper on special relativity, not generally understood. Therefore, I've decided to devote this column to an exploration of the twin paradoxes of both special and general relativity.

    The "fixed" Newtonian quantities of mass, length, and time become variables in special relativity at a velocity (v) that approaches the speed of light (c) or when the ratio v/c approached a value of 1. An observer at rest will see three amazing changes in an object moving past him at a velocity v not much less than c: (1) the object's mass increases; (2) its length shrinks along its direction of motion; and (3) its internal clock slows down. All of these variations can be calculated by applying the same "Lorentz factor" from special relativity, often represented by the symbol gamma (g) and defined by g = 1/Sqrt[1 - (v/c)2].

    In considering the twin paradox, let us focus on relativity effects that occur in the particular case when v/c = 0.866 (v is 86.6% of c), a velocity at which the Lorentz factor g is exactly 2. At this velocity, exactly half of the mass-energy of the moving object comes from its mass-energy at rest and the other half is kinetic energy. In other words, acceleration must supply an amount of energy equal to the object's rest mass m0, giving it a kinetic energy equal to m0c2. At this velocity the mass m of the moving object is doubled (m=gm0=2m0), its length L along its direction of motion shrinks by half (L=L0/g=L0/2), and its clock takes twice as much time T between ticks (T=gT0=2T0) and therefore runs slow by a factor of 1/g or half it's speed at rest.

    Much of the confusion surrounding the twin paradox can be traced to two specious arguments:

Argument 1, which might be called the symmetry argument, usually goes like this:
"An observer Ernest who remains behind and at rest will observe that astronaut Sam on board the ship travels away at a speed v and then returns at the same speed. But Sam also observes that the stay-at-home observer Ernest travels away from him at speed v and then returns at that speed. Sam is just as good an observer as Ernest, and since each observes the other having the same velocity relative to himself, there can be no difference in the readings of Ernest's clock and Sam's clock when the two clocks are compared at the end of the trip."

Argument 2, which might be characterized as the know-nothing argument, usually goes like this:
"The Lorentz transformations of special relativity only apply to objects travelling with constant speeds. But space ships must accelerate and decelerate in their operation. From the equivalence principle of general relativity we know that acceleration is equivalent to the effects of gravity and that gravity must be handled with the elaborate mathematical machinery of general relativity. Therefore, the twin paradox cannot be analyzed with special relativity alone, and we can't simply use the Lorentz transformations to say what the two clock readings will be after the trip. If there is any difference in the clock readings, it arises strictly from general relativity effects."

    Both of these arguments are wrong. Let's take the symmetry argument first. The Lorentz transformations of special relativity on which the twin paradox is based assume that the set of observations of mass, length, and time is made from one inertial reference frame, one coordinate system that is either at rest or is moving with an unchanging speed in a particular direction. Clearly, observations do not have to be made from an inertial reference frame, but if they are not, the Lorentz transformations don't directly apply. The significance of this is that the system of observer Ernest is an inertial frame, while the system of observer Sam is not an inertial frame. Sam is not in an inertial frame because in mid trip his ship's engines were used to accelerate him and to reverse the direction of his velocity for the return leg of the trip. If one constructs world lines for the two observers by plotting space position against time position on a piece of graph paper, the world line of Ernest is straight, an indication that he is in an inertial frame. The world line of Sam , however, has a kink at the place where the ship's acceleration takes place, an indication Sam is not in an inertial frame.

    Therefore, the seeming "symmetry" between systems Ernest and Sam is illusory. Their observations are not equivalent because Ernest observes from an inertial frame while Sam does not. Can we analyze the problem from the point of view of Sam? We can for part of Sam's trip by using the point of view of an observer Albert who goes out with the ship on which Sam travels, but continues to travel on at the same speed in the original direction after Sam's ship turns around and comes back. Albert sees Ernest moving away at v/c=0.866. Half a year later, after Ernest has traveled .433 light year (because for Albert the 0.866 light-year distance is contracted by a factor of 2), Albert sees the ship with Sam aboard accelerate to v/c=0.9897 (corresponding to g=7) and race after Ernest. [This velocity comes from relativistic velocity addition: v=(v1+v2)/(1+v1v2/c2).] It takes 3.5 years for Sam to catch up with Ernest. The whole episode takes 4 years according to Albert's clock. However, Ernest's clock runs slow by a factor of 2, so it will register only 2 years. Sam's clock read 0.5 years when it accelerated to catch up with Ernest. Catching up with Ernest required 3.5 years, and during that time Sam's clock was running slow by a factor of 7, so that it recorded only 0.5 years of elapsed time. Therefore, Sam's clock registered a total elapsed time of 1 year as compared to 2 years on Ernest's clock. This is the same result as before, even though it was observed from a very different perspective.

    Notice, however, that all the clock readings do not match. From Albert's point of view, Ernest's clock reads only ¼ year when the ship accelerates to turn around. From Ernest's point of view, his own clock reads ½ year when that acceleration event occurs on the ship. But Ernest's clock was at a large distance from Sam when his ship accelerated, so Sam's clock and Ernest's clock cannot be placed side-by-side for comparison. In special relativity separated clocks need not have the same relative readings in all systems. This is the reason that in relativity two separated events can be said to happen at the same time only in reference to a particular reference frame. There was a ¼ year difference in relative clock readings in the scenario above depending on whether Albert or Ernest did the observing.

    The "know-nothing" argument is wrong because its basic premise is incorrect. The formalism of general relativity is not required to deal with acceleration. Special relativity works fine for accelerating objects, as long as the Lorentz transformations are based in a reference frame that is not accelerating. One simply allows object's velocity to change with time. You can think of the process as breaking up the acceleration into little time intervals during which the velocity has a certain average value, and then apply the value of gamma appropriate to that velocity in that time interval. This may require some calculus, but it doesn't require general relativity, which deals with the effects of gravity fields.

    However, general relativity does provide us with another way of thinking about the twin paradox. According to general relativity, a clock runs slower in a gravitational potential V by a factor of (1-V/c2). This means that a clock on the surface of the Earth runs slower that an identical clock in gravity free space by a factor of (1 - 6.60 × 10-10). In one year of running, the earth-based clock would lose 20 milliseconds.

    Now consider a thought experiment. Suppose that for a period of one hour, a cosmos-wide 1000 g gravitational field switches on over the whole observable universe, and then switches off again. Everything in the universe will free-fall in this field, gaining enormous velocity and energy. What are the observable consequences of such a cosmic event? The answer is: none. Since all objects in the universe would accelerate together, there would be no observable clues that the gravitational field is even there. It would be as if it had never happened.

    Because switching on and off such a field has no observable consequences, we can pretend it is happening to deal with acceleration. In the case of Sam and Ernest, suppose that a universe-wide gravitational field switches on, but astronaut Sam, determined to maintain his ship's position, uses it's engines to counteract the force. Sam stays at a fixed point. He experiences a force pulling him in the direction of the field, but he remain at rest. Then, after the his twin brother Ernest who is freely falling in the field reaches v/c=0.866, the force switches off, and Ernest continues to move away from Sam at that speed. After half a year, the force switches on again, this time acting in the opposite direction for twice as long, and Ernest's velocity is reversed, bringing him back to Sam's ship. Sam always uses the engines to hold his fixed spatial position. Finally, the force switches on one last time to bring Ernest to rest with respect to Sam at the same position. Ernest has never observed any acceleration, since he was always in free fall. From his point of view, Sam used the ship's engines to accelerate away from him, travel 0.866 light years, and return.

    Do their clocks match now? During the initial acceleration and deceleration, Sam and Ernest were in the same gravitational potential, so no clock differences were produced. During the Ernest's trip away and back, he was moving at v/c=0.866 and so his clock is slowed down by a factor of ½. But at the distant turnaround point, when Ernest is decelerating and reversing his velocity, Sam was in a gravitational potential that is larger than Ernest's by an amount aL, where a is the acceleration produced by the uniform gravitational field and L is the distance between Sam and the Ernest. During the time the uniform gravitational field is switched on, Ernest's clock runs faster than the Sam's clock by a factor of (1 + aL/c2). The result of this gravitational speedup is that it puts the Ernest's clock precisely one year ahead of Sam's clock when he returns, just as before.

    It's comforting, but not particularly surprising, to find that special and general relativity give the same answer for the twin paradox. The conclusion is that Sam, the twin who feels the force pushing him against the floor, ages less than his twin Ernest, who can remain in free fall or zero-g the whole time. So when you feel the pull of gravity on your aging bones and muscles, consider that because of it you're aging less (if only by 20 milliseconds per year).


References:

Special Relativity Twin Paradox:
G. Builder , "The Resolution of the Clock Paradox", Phil. Sci. 26, 135 (1959).

General Relativity Twin Paradox:
C. Møller, The Theory of Relativity, Clarendon Press, Oxford (1955).


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 .

AV Columns Online: Electronic reprints of about 177 "The Alternate View" columns by John G. Cramer, previously published in Analog , are available online at: http://www.npl.washington.edu/av.


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This page was created by John G. Cramer on 7/12/96.