The Twin Paradox in a Closed UniverseAuthor(s): Jeffrey R. WeeksSource: The American Mathematical Monthly, Vol. 108, No. 7 (Aug. - Sep., 2001), pp. 585-590Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2695267Accessed: 16/10/2008 21:40

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The Twin Paradox in a Closed Universe Jeffrey R. Weeks

In the classical twin paradox, one twin, call him Albert, stays at home, while his sis- ter Betty travels at relativistic speed to a nearby star and back. According to special relativity, traveller Betty measures less time than stay-at-home Albert, and is there- fore younger upon their reunion. But from Betty's point of view, she's at rest, and it's Albert who moves away from her, and later returns, at relativistic speeds. This sug- gests that Albert should be younger upon their reunion. The symmetry of the situation seems to make the impossible demand that each twin be younger than the other. The resolution of the paradox is that Betty experiences an acceleration-and a change of inertial frame-at the turnaround point, while Albert stays in a single inertial frame throughout. The symmetry is broken, and Betty is truly younger at the reunion.

The twin paradox hits harder in a closed universe. Albert stays at home, while Betty takes a trip around the universe! From the traditional point of view, which treats space and time separately, this is a true paradox. The situation is completely symmetrical: Albert sees Betty moving in a straight line at constant velocity from the moment of departure until their reunion, while Betty sees Albert moving in the opposite direction at constant velocity for the same period. Using special relativity, each calculates that the other should be younger at the reunion. Who is right?

Before addressing this new paradox, let us take a closer look at the original twin paradox. Instead of analyzing the time dilations and length contractions separately, let us combine the spatial and temporal information into a single spacetime diagram [1, Ch. 1]. Figure 1 shows the spacetime diagram for the case that Betty travels at 3/5 the speed of light to a star 15 light-years away. From Albert's point of view, each half of the trip takes (15 light-years)/(3/5 light-years/year) = 25 years. But what about Betty's perception? A simple calculation in spacetime will reveal her elapsed time.

50

Betty Albert returning

at home>s (20 years proper time)

t' 25

Betty outbound

(20 years proper time)

15 distance

(light-years)

Figure 1. Albert stays home while twin sister Betty travels to a nearby star and back at 3/5 the speed of light.

August-September 2001] THE TWIN PARADOX 585

The geometry of spacetime differs from the usual Euclidean geometry. In Euclidean geometry the familiar dot product u v = uxv, + uyvy serves to measure lengths and orthogonality via the formulas Iu 2 = u u and u L v X u v = 0. In spacetime the formulas remain the same, but the Euclidean dot product is replaced by the Lorentz inner product u v = ux vx - ut vt. That minus sign throws a new twist into the game: a squared length may now be positive, negative, or zero! For example, the squared length of Betty's outbound trip in Figure 1 is Iu 2 = u . u = (15, 25) (15, 25) = 152 - 252 = _202. When a squared length u u is negative, as it is here, it repre- sents a timelike interval and -u u tells the proper time experienced along the path. In the present case, Iu12 =-202 means that Betty ages 20 years during her outbound voyage. When a squared length u u is positive it represents a spacelike interval and

u u tells the proper distance as measured by an observer who perceives the events along the interval to be simultaneous. When a squared length u u is zero it represents a lightlike interval and represents the potential path of a photon. This interpretation of the Lorentz inner product requires that the units of distance and time be chosen so that the speed of light is one, so we use light-years for distance and years for time. A more elegant approach is to use the same unit, say meters, for time as well as distance, in recognition of the intermingling of space and time both in the spacetime diagram and in the Lorentz inner product [1, pp. 1-3].

To better understand the symmetry-and lack of symmetry-it is helpful to con- sider each twin's lines of constant time. Albert's lines of constant time are the horizon- tal lines t = constant, which are orthogonal to his world line (his path through space- time), as shown in Figure 2a. Betty's lines of constant time are orthogonal to her world line, too (Figure 2b); however, we must judge orthogonality not with our Euclidean intuition, but with spacetime's Lorentz inner product u v = uxvx-utvt = 0. During the outbound portion of Betty's trip her world line has slope +5/3 and so her lines of constant time have slope +3/5, as confirmed by the computation (3, 5). (5, 3) = 15 - 15 = 0. Similarly, on her return trip her world line has slope -5/3 and her lines of constant time have slope -3/5.

50 50

Albert at home-.:

34 Betty returning

M 25 ZM 2

. / ., ^ /,/t ~~~~~~~~~~~~Betty / 16 // outbound

15 15 distance distance

(light-years) (light-years)

Figure 2a. Albert's lines of constant time are or- Figure 2b. Betty's lines of constant time are orthog- thogonal to his world line. onal to her world line too, relative to spacetime's

Lorentz inner product.

586 ?g THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108

Just before her turnaround, Betty considers herself to be contemporary with an Al- bert who has aged 16 years (Figure 2b), in agreement with special relativity's predic- tion that a brother moving away from her at 3/5 the speed of light should age only 4/5 as fast as she does. But just after her turnaround she finds herself in a new inertial frame. She suddenly considers herself contemporary with an Albert who is 34 years older than he was at her departure. During the second 20 years of her trip, she perceives her brother to age an additional (4/5) (20) = 16 years, making him a total of 50 years older at the reunion. Thus at their reunion the two twins agree that Albert has aged 50 years and Betty only 40, even though each accounts for the elapsed time differently.

The tools are now in place to analyze the new twin paradox, in which Betty takes a constant velocity trip around a closed universe, with no turnaround or acceleration. Viewed in the context of space alone, this looks like a nasty paradox. But, as we shall see, the spacetime diagram resolves it. For simplicity consider a 1-dimensional circu- lar universe. Furthermore, let it be static (non-expanding), so spacetime is a cylinder (Figure 3). If the cylinder has a circumference of 30 light-years, and Betty travels at 3/5 the speed of light as before, then Albert ages 50 years during Betty's trip, while Betty ages only 40. But what happened to the symmetry of the situation? Shouldn't Betty see Albert aging only 4/5 as fast as she is? She does! Betty's line of constant time, drawn in grey in Figure 3, is a helix. At her departure, Betty calculates that she is contemporary not only with her brother standing beside her, but also with an image of her brother 18 years older, which she considers to be 24 light-years in front of her (and another copy of her brother 36 years older and 48 light-years away, and so on).

Betty's helix of

constant time Albert's A-

world line <Z_ ~Bettys

world line

circle rof E p- constant time

Figure 3. In a cylindrical spacetime, each twin stays in a single inertial frame.

Figure 4 shows the spacetime cylinder slit open along Albert's world line and flat- tened to a rectangle. Subjectively, Betty has the impression that she departs from one copy of her brother, call him Albert,, and heads across space to a second copy Albert2 who, at the moment of departure, is already 18 years older than Albert,! As special relativity predicts, Albert2 ages only 4/5 as fast as Betty. While Betty ages 40 years, Albert2 ages only (4/5) (40) = 32 years. Thus at the conclusion of her journey, Betty is greeted by an Albert2 who is 18 + 32 = 50 years older than the Albert, she left behind. The age difference between Albert, and Albert2 exactly compensates for Albert and Betty's differing views about who is aging more quickly. Of course, as the cylindrical spacetime in Figure 3 clearly shows, Albert1 is Albert2. Betty's impression of seeing two copies of her brother is only an illusion.

The statement of the closed universe twin paradox is completely symmetrical- Albert and Betty travel in opposite directions from one another until they reunite- yet the resolution is asymmetrical-Albert's erstwhile twin is now a little sister 10

August-September 2001] THE TWIN PARADOX 587

Albert/ Albert2

Betty's world line

32 40 years

years 50

years

light-years

Figure 4. Betty feels she is travelling toward a copy of Albert who is already 18 years older than the Albert she left behind.

years younger than he is. How can this be? The answer is that we broke the problem's symmetry when we constructed the spacetime diagram. Albert occupies a very special inertial frame. His is the only frame in which lines of constant time are closed circles. In all other frames the lines of constant time are helices. The faster an observer is moving relative to Albert, the steeper the pitch of the helix.

If we cut the spacetime cylinder open along Albert's world line, shift it 18 years, and reglue (Figure 5), then the situation is reversed. Betty's lines of constant time are now circles, and Albert's are helices! Upon their reunion, Betty is now Albert's big sister, having aged 40 years while Albert aged only 32. From Albert's point of view the cut-shift-and-reglue makes Betty arrive home 18 years sooner than she ought to. Betty, however, feels that 32 years is the right amount for Albert to age, and the cut- shift-and-reglue serves to correct the previous lack of synchronization between the images Albert, and Albert2. Away from the cut line, the contents of spacetime do not change at all. All that has changed is how spacetime connects up with itself. The global connectivity of spacetime determines which twin occupies the preferred rest frame.

4+\

Betty's circle of

constant time

Albert's helix of

constant time

Figure 5. Cut the cylinder open, shift by 18 years, and reglue. Betty's line of constant time becomes a circle, while Albert's becomes a helix.

588 ( THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108

preferred preferred observers' _ observers'

world lines circles of - simultaneity

Figure 6. The real universe began with a big bang, so spacetime is approximated better by a cone than a cylinder. Nevertheless, it too has a preferred inertial frame at each point.

We do not yet know whether the real universe is closed, but we do know that it began with a big bang. Thus our spacetime is more accurately modelled by a cone (Figure 6) than by a cylinder, and the real universe, whether finite or infinite, has a preferred rest frame. The preferred spacelike slices are the constant time slices, with time measured from the big bang. The preferred inertial frame at each point is the frame whose time axis is orthogonal to the preferred spacelike slice.

If we assume that the primordial plasma was, on average, at rest relative to the preferred inertial frame, then we may empirically determine the sun's "absolute velocity" by measuring the dipole component of the 2.7K Cosmic Microwave Background (CMB) radiation [2]. Using the four-year COBE-DMR data, researchers have determined that the sun is moving in the direction of the constellation Leo at 369.0 ? 2.5 km/sec relative to the CMB (and thus, if our assumption is correct, rel- ative to the preferred rest frame). The Milky Way galaxy as a whole moves at about 600 kmlsec, but the sun rotates around the galaxy with a speed of about 225 km/sec, and is now moving in a direction nearly opposite the galaxy's motion. Finally, the earth rotates about the sun with a speed of 30 kmlsec.

We have seen that the traditional lesson of special relativity-that all inertial frames are equivalent-applies only locally. Globally the symmetry is broken in any universe that is finite, or began with a big bang. Assuming that the primordial plasma of the big bang was at rest relative to the preferred frame, researchers calculate the modem sun's velocity to be about 369 km/sec. As in the classical twin paradox, an observer at rest relative to the preferred (local) inertial frame measures the longest proper time between any two events on his or her world line; moving observers always measure less.

To introduce students to the concept of a closed universe, please see the Torus and Klein Bottle Games at http://www.northnet.org/weeks/TorusGames .

ACKNOWLEDGMENTS. I thank Jeffrey Kochanski for correspondence leading to this note, Neil Comish, David Spergel and Glenn Starkman for their ongoing collaboration, Daniel Koon for suggesting improvements, and the MacArthur Foundation for its financial support.

REFERENCES

1. E. Taylor and J. Wheeler, Spacetime Physics, W.H. Freeman and Co., San Francisco, 1963. 2. C. Lineweaver, L. Tenorio, G. Smoot, P. Keegstra, A. Banday, and P. Lubin, The dipole observed in the

COBE DMR four-year data, Astrophysical J. 470 (1996) 38-42.

JEFF WEEKS works in the broad intersection of geometry and physics. After a decade as a free-lance geome- ter, he now pursues his writing and research with the low administrative overhead of a MacArthur Fellowship.

August-September 2001] THE TWIN PARADOX 589

His most recent work is Exploring the Shape of Space, a 2-week unit on geometry and the universe for niiddle school and high school students. He enjoys spending time with his wife and son, and gets out on his bicycle or cross-country skis whenever possible. 15 Farmer Street, Canton, NY 13617-1120 weeks@northnet.org

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