CHAPTER TWELVE

INTERSTELLAR TRAVEL AND RELATIVITY

12.1  TIME ENOUGH FOR ANYTHING

He looked me up and down and said wonderingly, “I knew intellectually you would not have changed with the years. But to see it, to realize it, is another thing, eh? ‘The Picture of Dorian Gray.’ ”

—ROBERT A. HEINLEIN, TIME FOR THE STARS

Robert A. Heinlein’s novel Time for the Stars is essentially one long in-joke for physicists. The central characters of the novel are Tom and Pat Bartlett, two identical twins who can communicate with each other telepathically. In the novel, telepathy has a speed much faster than light. Linked telepaths, usually pairs of identical twins, are used to maintain communications between the starship Lewis and Clark and Earth. Tom goes on the spacecraft while Pat stays home; the ship visits a number of distant star systems, exploring and finding new Earth-like worlds. On Tom’s return, nearly seventy years have elapsed on Earth, but Tom has only aged by five [113].

I call this a physicist’s in-joke because Heinlein is illustrating what is referred to as the twin paradox of relativity: take two identical twins, fly one around the universe at nearly the speed of light, and leave the other at home. On the traveler’s return, he or she will be younger than the stay-at-home, even though the two started out the same age. This is because according to Einstein’s special theory of relativity, time runs at different rates in different reference frames.

This is another common theme in science fiction: the fact that time slows down when one “approaches the speed of light.” It’s a subtle issue, however, and is very easy to get wrong. In fact, Heinlein made some mistakes in his book when dealing with the subject, but more on that later. First, I want to list a few of the many books written using this theme:

•  The Forever War, by Joe W. Haldeman. This story of a long-drawn-out conflict between humanity and an alien race has starships that move at speeds near light speed to travel between “collapsars” (black holes), which are used for faster-than-light travel. Alas, this doesn’t work. The hero’s girlfriend keeps herself young for him by shuttling back and forth at near light speeds between Earth and a distant colony world [107].

•  Poul Anderson’s novel, Tau Zero. In this work, mentioned in the last chapter, the crew of a doomed Bussard ramship is able to explore essentially the entire universe by traveling at speeds ever closer to the speed of light [22].

•  The Fifth Head of Cerberus, by Gene Wolfe. In this novel an anthropologist travels from Earth to the double planets of St. Croix and St. Anne. It isn’t a big part of the novel, but the anthropologist John Marsch mentions that eighty years have passed on Earth since he left it, a large part of his choice to stay rather than return home [253].

•  Larris Niven’s novel A World out of Time. The rammer Jerome Corbell travels to the galactic core and back, aging some 90 years, while three million years pass on Earth [182]. ¹

There are many, many others, and for good reason: relativity is good for the science fiction writer because it brings the stars closer to home, at least for the astronaut venturing out to them. It’s not so simple for her stay-at-home relatives. The point is that the distance between Earth and other planets in the Solar System ranges from tens of millions of kilometers to billions of kilometers. These are large distances, to be sure, but ones that can be traversed in times ranging from a few years to a decade or so by chemical propulsion. We can imagine sending people to the planets in times commensurate with human life. If we imagine more advanced propulsion systems, the times become that much shorter.

Unfortunately, it seems there is no other intelligent life in the Solar System apart from humans, and no other habitable place apart from Earth. If we want to invoke the themes of contact or conflict with aliens or finding and settling Earth-like planets, the narratives must involve travel to other stars because there’s nothing like that close to us. But the stars are a lot farther away than the planets in the Solar System: the nearest star system to our Solar System, the triple star system Alpha Centauri, is 4.3 light-years away: that is, it is so far that it takes light 4.3 years to get from there to here, a distance of 40 trillion km. Other stars are much farther away. Our own galaxy, the group of 200 billion stars of which our Sun is a part, is a great spiral 100,000 light-years across. Other galaxies are distances of millions of light-years away.

From our best knowledge of physics today, nothing can go faster than the speed of light. That means that it takes at least 4.3 years for a traveler (I’ll call him Tom) to go from Earth to Alpha Centauri and another 4.3 years to return. But if Tom travels at a speed close to that of light, he doesn’t experience 4.3 years spent on ship; it can take only a small fraction of the time. In principle, Tom can explore the universe in his lifetime as long as he is willing to come back to a world that has aged millions or billions of years in the meantime.

12.2  WAS EINSTEIN RIGHT?

This weird prediction—that clocks run more slowly when traveling close to light speed—has made many people question Einstein’s results.² The weirdness isn’t limited to time dilation; there is also relativistic length contraction. A spacecraft traveling close to the speed of light shrinks in the direction of motion. The formulas are actually quite simple. Let’s say that Tom is in a spacecraft traveling along at some speed v, while Pat is standing still, watching him fly by. We’ll put Pat in a space suit floating in empty space so we don’t have to worry about the complication of gravity. Let’s say the following: Pat has a stopwatch in his hand, as does Tom. As Tom speeds by him, both start their stopwatches at the same time and Pat measures a certain amount of time on his watch (say, 10 seconds) while simultaneously watching Tom’s watch through the window of his spacecraft. If Pat measures time Δt₀ go by on his watch, he will see Tom’s watch tick through less time. Letting Δt be the amount of time on Tom’s watch, the two times are related by the formula

where the all-important “gamma factor” is

The gamma factor is always greater than 1, meaning Pat will see less time go by on Tom’s watch than on his. Table 12.1 shows how gamma varies with velocity.

Note that this is only really appreciable for times greater than about 10% of the speed of light. The length of Tom’s ship as measured by Pat (and the length of any object in it, including Tom) shrinks in the direction of motion by the same factor.

Even though the gamma factor isn’t large for low speeds, it is still measurable. To quote Edward Purcell, “Personally, I believe in special relativity. If it were not reliable, some expensive machines around here would be in very deep trouble” [46, p. 134]. The time dilation effect has been measured directly, and is measured directly almost every second of every day in particle accelerators around the world. Unstable particles have characteristic lifetimes, after which they decay into other particles. For example, the muon is a particle with mass 206 times the mass of the electron. It is unstable and decays via the reaction

It decays with a characteristic time of 2.22 µs; this is the decay time one finds for muons generated in lab experiments. However, muons generated by cosmic ray showers in Earth’s atmosphere travel at speeds over 99% of the speed of light, and measurements on these muons show that their decay lifetime is more than seven times longer than what is measured in the lab, exactly as predicted by relativity theory [233]. This is an experiment I did as a graduate student and our undergraduates at St. Mary’s College do as part of their third-year advanced lab course. Experiments with particles in particle accelerators show the same results: particle lifetimes are extended by the gamma factor, and no matter how much energy we put into the particles, they never travel faster than the speed of light. This is remarkable because in the highest-energy accelerators, particles end up traveling at speeds within 1 cm/s of light speed. Everything works out exactly as the theory of relativity says, to a precision of much better than 1%.

How about experiments done with real clocks? Yes, they have been done as well. The problems of doing such experiments are substantial: at speeds of a few hundred meters per second, a typical speed for an airplane, the gamma factor deviates from 1 by only about 10⁻¹³. To measure the effect, you would have to run the experiment for a long time, because the accuracy of atomic clocks is only about one part in 10¹¹ or 10¹²; the experiments would have to run a long time because the difference between the readings on the clocks increases with time. In the 1970s tests were performed with atomic clocks carried on two airplanes that flew around the world, which were compared to clocks remaining stationary on the ground. Einstein passed with flying colors. The one subtlety here is that you have to take the rotation of the Earth into account as part of the speed of the airplane. For this reason, two planes were used: one going around the world from East to West, the other from West to East [252]. This may seem rather abstract, but today it is extremely important for our technology. Relativity lies at the cornerstone of a multi-billion-dollar industry, the global positioning system (GPS).

GPS determines the positions of objects on the Earth by triangulation: satellites in orbit around the Earth send radio signals with time stamps on them. By comparing the time stamps to the time on the ground, it is possible to determine the distance to the satellite, which is the speed of light multiplied by the time difference between the two. Using signals from at least four satellites and their known positions, one can triangulate a position on the ground. However, the clocks on the satellites run at different rates as clocks on the ground, in keeping with the theory of relativity. There are actually two different effects: one is relativistic time dilation owing to motion and the other is an effect we haven’t considered yet, gravitational time dilation. Gravitational time dilation means that time slows down the further you are in a gravitational potential well. On the satellites, the gravitational time dilation speeds up clock rates as compared to those on the ground, and the motion effect slows them down. The gravitational effect is twice as big as the motion effect, but both must be included to calculate the total amount by which the clock rate changes. The effect is small, only about three parts in a billion, but if relativity weren’t accounted for, the GPS system would stop functioning in less than an hour [146, p. 68]. To quote from Alfred Heick’s textbook GPS Satellite Surveying,

Relativistic effects are important in GPS surveying but fortunately can be accurately calculated.… [The difference in clock rates] corresponds to an increase in time of 38.3 µsec per day; the clocks in orbit appear to run faster.… [This effect] is corrected by adjusting the frequency of the satellite clocks in the factory before launch to 10.22999999543 MHz [from their fundamental frequency of 10.23 MHz].

This statement says two things: first, in the dry language of an engineering handbook, it is made quite clear that these relativistic effects are so commonplace that engineers routinely take them into account in a system that hundreds of millions of people use every day and that contributes billions of dollars to the world’s commerce. Second, it tells you the phenomenal accuracy of radio and microwave engineering. So the next time someone tells you that Einstein was crazy, you can quote chapter and verse back at him!

12.3  SOME SUBTLETIES

The problem with Heinlein’s Time for the Stars is that when the Lewis and Clark begins to get close to the speed of light, the twins have problems communicating with each other. This is Tom speaking:

At three-quarters of the speed of light [Pat] began to complain that I was drawling while it seemed to me he was beginning to jabber. At nine-tenths of the speed of light it was close to 2 to 1, but we knew what was wrong now and I talked fast and he talked slow.

At 99% of c it was 7 to 1 and all we could do to make ourselves understood. Later that day we fell out of touch entirely. [113, chap. 11, “Slippage”]

This seems reasonable, but unfortunately, it violates one of the underlying principles of the special theory of relativity. Relativity is called relativity because measurements made by an observer are relative to that observer and not to anyone else. The odd thing is that Tom will measure his own time going by normally and Pat’s clock slowed down—by the same gamma factor that Tom’s clock appears slowed down as measured by Pat. This is so odd that when I was teaching this once, one of my students shook her head and stated flatly, “No, that isn’t right.” This is one of the two underlying ideas of Einstein’s special theory of relativity: you can’t tell whether you are moving at constant velocity or standing still by any measurement you can make. If it were true that Pat would measure Tom’s clock as running slow if Tom were moving and he weren’t, and Tom would measure Pat’s clock as running fast, then that would be proof that Tom was moving and Pat wasn’t. Since you can’t do that, both clocks must run slow as measured by the other.³ (How this works and makes sense would take far too long to go into in this book. If you are interested, there are good books on the subject, a few written by Einstein himself [198].) The other principle of relativity, which is what leads to the time dilation effects, is that both Tom and Pat will measure the speed of light as having the same value no matter how fast they are moving in relation to each other.

Ultimately the problem with Heinlein’s novel comes from the fact that the twins are communicating with each other at speeds faster than light, which violates the precepts of the special theory of relativity. If we put v > c into the equation for the gamma factor we get the square root of a negative number, that is, an imaginary quantity—indicating that we can’t do it. Some people have tried to come up with clever ways around this problem, but most physicists think that the speed of light is the ultimate limiting speed in the universe.

This does lead to another issue: why is it that at the end of the voyage, Tom is younger than Pat? If the effect is symmetric, why should either of them be younger than the other?

This has been discussed to death in the past. The answer is straight-forward: yes, time dilation is symmetric, but there is something that isn’t. That is their accelerations. Pat, sitting still on Earth, is not being accelerated. Tom, on the other hand, when he climbs aboard the ship and takes off for the stars, is. Even if the acceleration is for a very short period of time, this difference is what makes Tom younger than Pat. Let me work through an example to show why this is true. Here are the assumptions I’m going to make. This example is based on the discussion in Wolfgang Rindler’s book, Special Relativity [198, pp. 30–31].

•  Tom is going to make a trip to the Alpha Centauri star system, 4.3 light-years away.

•  His spacecraft is capable of traveling at 86% of the speed of light after a very rapid period of acceleration. (We’ll ignore the problem of keeping Tom from getting smashed to jelly during the acceleration.)

At 86% of the speed of light it takes Tom five years to get there and five years to get back as measured by Pat’s clocks. The gamma factor is almost exactly 2, however, so to Tom it should take only 2.5 years out and back, meaning he should be five years younger than Pat on his return. This is from Pat’s perspective. How about from Tom’s perspective?

A rather incisive point made by Rindler is that no matter how fast the acceleration period is, when it is over Tom has gone halfway to his destination [198, p. 31]. This is a result of the length contraction effect: from Tom’s point of views when the acceleration ends, the Sun is moving away from him at 86% of the speed of light and Alpha Centauri is moving toward him at that speed. Because of relativistic length contraction, the distance between them has shrunk by 50%. To Tom, his clock is normal, but the distance is contracted by 50%. So it all works out.⁴

12.4  CONSTANT ACCELERATION IN RELATIVITY

There is no such thing as constant acceleration in special relativity, for a simple reason: under constant linear acceleration (with initial condition v = 0 at time t = 0), the velocity after time t is v = at. Because of this, after a sufficiently long time (about one year if a = g), the spacecraft is traveling faster than the speed of light. We therefore must be careful about how we define acceleration.

An astronaut on an accelerating spacecraft will feel the sensation of weight, so this is what we will adopt as our definition of acceleration.

1. Weigh the astronaut on Earth (W₀).

2. Then, once on the spacecraft, put a scale under her and measure her effective weight (W).

The “proper” acceleration of the spacecraft, a, is then

Since g ∼ 10 m/s², if the weight aboard the craft is only 10% of the weight on Earth, the acceleration of the spacecraft is about 1 m/s².

It turns out that we can make life very simple for ourselves in talking about acceleration if we adopt a system of units in which distances are measured in light-years and time is measured in years. That is, the speed of light is c = 1 LY/yr. In this system of units, g works out almost exactly to 1 LY/yr², which makes our calculations very easy. Let’s assume the following:

•  The ship starts from rest and travels out with a constant acceleration of g along a straight line.

•  x is the distance the ship travels.

•  t is the time from the beginning of the trip as measured on clocks on Earth.

•  v is the speed the ship gets to after time t.

•  τ is the time that has passed on the ship.

Then the motion of the ship follows the equation

This is often referred to as “hyperbolic motion” in relativity because if x is plotted against t, the figure is that of a hyperbola.

We can express everything in terms of the time on board the ship:

For those unfamiliar with the terms, “cosh,” “sinh,” and “tanh” are the hyperbolic cosine, sine, and tangents, respectively:

and

12.4.1 A Trip to the End of the Universe

Let’s plan a trip to the great beyond. After all, a trip of a billion light-years begins with a single step, right? The nice thing about relativistic acceleration is that although nothing can go faster than the speed of light. It will take at least a billion years to go a distance of a billion light-years, but time dilation makes it seem much, much shorter for the voyagers. Here’s the idea: we’ll take our ship and accelerate for half the time (i.e., half the distance) of the trip (as measured by shipboard clocks), turn the ship around, and decelerate the other half. How far do we get in this time? How much time has elapsed on Earth? How fast were we going at midpoint? In table 12.2, x_end is the distance the ship travels by the end of the voyage, t_end and τ_end are the times as measured on Earth and on the ship’s clock, respectively, and v_mid is the speed at midpoint.

The results are clear: given a ship capable of accelerating at 1 g continuously, one could reach the nearest stars in a few years, the center of the galaxy in under 25 years, other galaxies in 40 years, and the edge of the universe in 50 years. So the universe is within our grasp! Unfortunately, the energy requirements for a trip like this mount up considerably, but I’ll leave this as an exercise for my readers to work through.

NOTES

1. One point concerning this novel is that much of the time dilation experienced by Corbell is the result of diving near the event horizon of the supermassive black hole at the center of the galaxy. Gravitational time dilation is from the general theory of relativity, which isn’t covered in this chapter.

2. Indeed, the website Crank Dot Net, which specializes in providing links (I quote) to “cranks, crackpots, and loons on the net,” lists no less than 48 sites dedicated to showing that the theory of relativity (really, the two theories of relativity, special and general) are wrong [3].

3. I am being careful to avoid saying “it appears that clocks run slow” because this implies that the rate changes are somehow an illusion. This is wrong. By any test you can make, a clock moving in relation to you runs slow.

4. The readings on Pat’s clocks are harder to explain, but one can show that there are more ticks on Pat’s clock than on Tom’s using a combination of time dilation and what is called the Doppler effect from Tom’s point of view.