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Einstein and the World Year of Physics

Special Relativity after 100 Years

Special relativity has been phenomenally successful, but its nonintuitive nature has made it difficult for us to absorb Einstein’s central message about time and simultaneity.

JOHN GEOHEGAN

One hundred years after Albert Einstein gave us the theory of special relativity, we have made good progress in applying the equations he gave us, but we have difficulty absorbing his central message about time and simultaneity. Time and again, his predictions have been tested; the Global Positioning System simply wouldn’t give us good results unless Einstein’s equations were hidden in the little handheld GPS instruments used by hunters and hikers to find their way in the woods. Special relativity has been combined with quantum physics to produce quantum electrodynamics, the most accurate physical theory ever devised. Shortly after Einstein finished his famous paper, he used it to derive the famous equation E=mc2, showing that when a body radiates energy, it loses mass.

So what’s the problem? Well, we haven’t yet absorbed his message that “we cannot ascribe absolute meaning to the concept of simultaneity; instead, two events that are simultaneous when observed from some particular coordinate system can no longer be considered simultaneous when observed from a system that is moving relative to that system” (Einstein 1905, 130). As long as we think there’s such a thing as “a point in time” or events that can be described as universally simultaneous, we will be faced with misconceptions such as “moving clocks run slow” and “moving objects shrink in the direction of motion.” The following simple thought experiment shows how such misconceptions can arise.

Imagine two identical space ships, A and B, far out in space, away from any large gravitating bodies, traveling away from each other at a high velocity. Neither one is accelerating, no engines are operating, and the occupants of each ship consider their ship to be motionless while the other ship is moving rapidly away. The two ships have just passed each other, almost colliding, and at their closest positions, they have synchronized their identical clocks at 12:00. The clocks each cause a bright blue light on the outer hull of their respective ships to flash each hour, and the ships are separating so rapidly that the passengers on each see the blue light of the other shifted to half its frequency, into the infrared. This is the famous red shift caused by the Doppler effect, in this case a shift of 100 percent. The same shift means that the light signals sent out by A at 1:00, 2:00, and 3:00 according to A’s clock will be received by B at 2:00, 4:00, and 6:00 according to B’s clock. Similarly, A will receive signals from B at 2:00, 4:00, and 6:00 according to A’s clock.

Now, consider the signal sent by A at 1:00 according to its own clock. B will receive it at 2:00 according to B’s clock, and it will be reflected back to A along with B’s 2:00 signal to be received by A at 4:00 according to A’s clock. A will receive the reflected signal as deep in the infrared, one-fourth the frequency it had when it was sent out. The crew of A can now calculate that this signal, which was sent out at 1:00 and returned at 4:00, must have been reflected at 2:30, halfway between 1:00 and 4:00, because it traveled an equal distance each way. If they learn that B’s clock read 2:00 at the time of reflection, they may think B’s clock must have been thirty minutes slow. The situation is exactly parallel; B will receive its 1:00 signal return at 4:00 and its crew may conclude that A’s clock was thirty minutes slow.

As simple as the above description may be, it’s worth reviewing to see the following points:

A and B are equally justified in considering themselves to be at rest. This is in accordance with Einstein’s postulate that all inertial systems are equivalent.
Both A and B consider light to travel away from the space ship at the same velocity as it returns. This reflects Einstein’s postulate that the speed of light is the same in all inertial reference frames.
A and B observe the same red shift in the light coming from the other spaceship.
Sending and receiving a reflected signal shows how a distant event on a moving body can be timed.
Lack of agreement over the time of reflection is shown to occur even though all clocks are working perfectly. This lack of agreement isn’t normally seen, because we don’t usually deal with such high velocities (in this case, three-fifths the velocity of light) or super-accurate clocks. Disagreement could be avoided by A and B refusing to judge what happened at a distant point as “simultaneous” with the ticking of their own clocks.

To measure the length of a moving object, it is necessary to measure the distance to both its front and rear ends at the same time. Without being able to agree on what’s simultaneous, different observers will measure different lengths of moving objects. This effect is called the Lorentz-Fitzgerald contraction, in recognition of the two scientists who concluded that moving objects are physically shortened in their direction of motion. This was before Einstein explained the same effect as a result of the relativity of simultaneity.

Another common misunderstanding concerns the so-called “clock paradox” (or “twin paradox”—see the sidebar on page 35), which says that if B’s clock is transported to be compared with A’s, it will show that less time has elapsed than will A’s. This is not a paradox, because the two clocks have no longer had identical experiences. Specifically, B’s clock will have been accelerated so that it could be returned to A’s position. The behavior of clocks has been accurately tested, and Einstein’s equations have been verified. More troublesome, though, is trying to understand what “really” happens to B’s clock, or more accurately, what we “really” mean by time. We have an intuitive concept of time that resists accurate definition, and we have seen that the timing of a moving clock by sending light signals back and forth gives nonintuitive answers in experiments involving high speeds and very accurate clocks. Our intuitions work perfectly well with everyday clocks and everyday velocities, so it’s a slow process trying to give up the idea of absolute simultaneity.

Perhaps the best approach belongs to Stephen Hawking: “I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation” (Hawking and Penrose 1996). We have the model that gives accurate predictions. Perhaps a new concept of time will gradually work itself into our collective intuition as more and more practical applications of special relativity are realized.

References

Bondi, Hermann. 1962, 1964. Relativity and Common Sense. New York: Dover Publications. Chapters VII, VIII, and IX are recommended for learning the method of tracing light beams on simple space-time diagrams.

Darwin, C.G. 1957. The clock paradox in relativity. Nature 180 (November), 976–977. This is the ultimate short-and-sweet explanation of the clock paradox.

Einstein, Albert. 1905. On the electrodynamics of moving bodies. In John Stachel (ed.). 1998. Einstein’s Miraculous Year: Five Papers that Changed the Face of Physics. Princeton, New Jersey: Princeton University Press. This is a recent translation of Einstein’s famous paper.

Einstein, Albert. 1961 [1916]. Relativity: The Special and General Theory, 15th edition. New York: Crown Publishers. Pp. 21–27. These pages clearly show that simultaneity is not an absolute.

Hawking, Stephen and Roger Penrose. 1996. The Nature of Space and Time. Princeton, New Jersey: Princeton University Press. Pp. 3–4.

The Twin Paradox
by Dave Thomas

The “twin paradox” is not a paradox in the sense of a logical contradiction that falsifies relativity but rather a very curious puzzle. Traditionally, the twin paradox is concerned with the strange result that if one of two twin brothers leaves the other and embarks on a high-speed journey to a remote point and back again, the twins will no longer be the same age. Let’s call these hypothetical twin brothers A and B. For this discussion, we’ll stipulate that A stays home while B travels away from his brother at a speed of 60 percent of the speed of light (0.6c, where c is the speed of light, nearly 300 million meters per second). B travels for fifteen years by A’s reckoning then quickly decelerates to a stop, turns around, and quickly accelerates back to 0.6c in the direction toward his brother, A. After another fifteen years (again, by A’s reckoning), B arrives home, decelerates, and rejoins his brother, who has aged thirty years since he last saw B. The “paradox” is that, even though A’s velocity relative to B is the same as B’s velocity relative to A, B will have experienced only twenty-four years of travel and find himself six years younger than his twin brother, A.

While this is indeed puzzling, it is not a logical flaw in relativity. The twins do not have similar experiences during B’s long journey, and that resolves the “paradox.” (While the fiction of very short deceleration/acceleration periods is useful to keep this discussion from getting into general relativity theory, it should be noted that such accelerations would almost certainly reduce twin B to a thin red puddle. It would take weeks to make the velocity changes at tolerable accelerations, say 5 to 10 g. See my accompanying sidebar “On Problems with Near-light-speed Travel” for more on this type of difficulty.) The journey of B, as viewed by twin A, is depicted in figure 1.

Figure 1: The journey of twin B, as observed by twin A.

The workings of the “Twin Paradox” can be explained with the aid of space-time diagrams. A space-time diagram for the stay-at-home twin, A, appears in the left half of figure 2. The grid marks show years on the vertical axis and distance in light-years on the horizontal axis. The thick lines represent A’s and B’s positions over time, while the thin lines with arrows represent the paths of light beams sent between the twins. During the fifteen years (in A’s frame of reference) of outbound travel by twin B, B gets out to a distance of nine light-years (0.6c315 years) from twin A. However, signals or light rays sent from B’s turnaround point won’t even reach A for another nine years, or until twenty-four years (15+9) after B’s departure. That is, A will see his brother B recede for twenty-four years, and then approach for just six years, arriving thirty years after his initial departure.

This is in marked contrast to B’s observations: B will see his stay-home brother recede for twelve years. After B turns around, he will see A approaching for twelve years and will return a total of twenty-four years after his departure. However, the same interval is thirty years by A’s calendar. The difference is that, during the short but intense accelerations experienced by B, B’s velocity relative to the universe (and to A) is changing. Twin B effectively “loses synch” with the rest of the universe, including his twin brother, A. Twin B is not in an inertial reference frame over the entire trip—and his bouts with intense accelerations will certainly remind him of that fact. Of course, A won’t be aware of B’s velocity changes until many years later.

Figure 2: Twin paradox space-time
diagrams for stay-at-home
twin A (left) and traveler B (right).

The space-time diagrams for B’s journey appear on the right of figure 2. These can’t be represented as a single diagram, because they are views of two different inertial frames (B outbound versus B inbound). The twin that undergoes acceleration will be the one who returns home younger than his stay-at-home brother. The loss of synchronization due to acceleration is the key and the reason it’s not a logical “paradox.”

This point is crucial: the time discrepancies between the twins are absolutely real. Here is a quick example, presented with the “radar method”: since any radar beams sent from A meet the target (B) at only one point in space-time, those beams must spend equal times outbound and inbound with respect to the sender. Figure 2 shows that a radar beam emitted by twin A at his time of two years will be reflected from B at some unknown time, and received again by A when his (A’s) calendar reads eight years. Likewise, a beam emitted by twin A at four years will be reflected from B and received by A when his calendar reads sixteen years.

Twin A can calculate the time and distance (in A’s frame of reference) of reflections from B, knowing only his own sending and receiving times and that the signals propagate at the speed of light. Since A’s two-year pulse returns at eight years, the reflection occurred (by A’s calendar) at the midpoint of the send/receive times, (2+8)/2=5 years. Since A’s four-year pulse returns at sixteen years, the reflection occurred at (4+16)/2=10 years by A’s calendar. Therefore, A measures the interval between these reflections (at five years and ten years) as being five years long.

Because the twins are separating rapidly, there will be a delay in B’s receipt of A’s transmissions. In particular, while A’s transmissions were sent two years apart by his clock, they were received by B over an interval longer than two years, say, K32 years, where K is a factor greater than 1. However, the same must hold true for B’s “transmissions” back to A: whatever period separates the reflections from B’s craft, A’s measurement of receiving times will be longer—in fact, precisely K times longer—since B is moving away from A exactly as fast as A recedes from B (“relativity”). So, A’s original pulses were sent two years apart; these were received by B at K32 years apart and received again by A at K3K32 years apart, or eight years. Clearly, K must equal 2, and B’s interval between receipt of A’s two signals must be 232=4 years, while A’s measurement of the time for the pulses to return from B is K34=8 years, as required. This is how “Time Dilation” comes to be measured by twin A: the five-year interval that A experiences in his own frame of reference takes only four years in B’s frame of reference.

Dave Thomas, a CSICOP fellow, is a physics and mathematics graduate of New Mexico Institute of Mining and Technology and is currently a senior scientist at Quasar International, Inc., in Albuquerque, New Mexico. He is the president of New Mexicans for Science and Reason.

On Problems with Near-light-speed Travel

Forget Star Trek-style warp-speed (greater than the speed of light) travel and its attendant problems (like the possibility of warping through a sun). Just traveling at near-light speed could bring a host of serious problems. Take a grain of interstellar dust, for example. A tiny grain of silicon dioxide (quartz, or sand) just one micron wide (a millionth of a meter, fifty times smaller than the width of a hair) would present no problem to travelers at normal speeds. But if a spacecraft were going along at 90 percent of light speed, the innocent sand grain would appear like a high-energy missile. In fact, the relativistic calculation of the micron-sized grain’s kinetic energy, as viewed by the approaching craft, would be close to 170 joules, which is about the energy of a 22-caliber bullet (40 grains, 64.8 mg/grain) traveling over the speed of sound (about 1,200 feet per second, or 366 meters per second). At such energy levels, the sand particle might even explode into a shower of protons and neutrons when it collides with the spacecraft. And a proton, traveling at 0.9c, can penetrate a stainless steel hull about 74 cm (about 2 and a half feet) thick. I don’t want to bum out all the Trekkies out there, but it’s worth pondering: near-light-speed travel is going to be hard.

—Dave Thomas

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