Pi Day: “Was Einstein Right?” Chuck Adler on the twin paradox of relativity in science fiction

This post is extracted from Wizards, Aliens, and Starships by Charles Adler. Dr. Adler will kick off Princeton’s Pi Day festivities tonight with a talk at the Princeton Public Library starting at 7:00 PM. We hope you can join the fun!

For more Pi Day features from Princeton University Press, please click here.

Tfts56[1]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.

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.
  • 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.
  • 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.
  • 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.

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.


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. 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 ∆t0 go by on his watch, he will see Tom’s watch tick through less time. Letting ∆t 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”. 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. 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 1013. 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. 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. 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!

PUP News of the World, February 21, 2014


Each week we post a round-up of some of our most exciting national and international PUP book coverage. Reviews, interviews, events, articles–this is the spot for coverage of all things “PUP books” that took place in the last week. Enjoy!

Even though this winter may feel like an eternity, we can still begin preparing for all the fun activities spring has to offer, like bird watching. Rare Birds of North America is the first comprehensive illustrated guide to the vagrant birds that occur throughout the United States and Canada. Featuring 275 stunning color plates, this book covers 262 species originating from three very different regions–the Old World, the New World tropics, and the world’s oceans. It explains the causes of avian vagrancy and breaks down patterns of occurrence by region and season, enabling readers to see where, when, and why each species occurs in North America. Detailed species accounts describe key identification features, taxonomy, age, sex, distribution, and status. This week, Parade ran a review of Rare Birds of North America. Want to preview the book? You can view a sample entry.

History is a tricky business. With so much happening all over the world every day, it becomes very easy to unintentionally downplay or overlook important events and natural disasters. One perfect example was the volcanic explosion on the island of Tambora in the East Indies. As volcanic explosions typically go, the explosion greatly  affected the environment and the lives of the island’s residents.  Gillen D’Arcy Wood, professor of English at the University of Illinois, does this disaster a great service by giving it the attention it deserves in his new book, Tambora: The Eruption that Changed the World.

Publishers Weekly’s recent starred review of Tambora said:

“The greatest volcanic eruption of modern times occurred in 1815 on the small island of Tambora in the East Indies. It spawned the most extreme weather in thousands of years. In what contemporaries described as the “year without a summer,” its immense ash cloud encircled and cooled the Earth. While historians have mostly ignored the decades of worldwide misery, starvation, and disease that followed, Wood (The Shock of the Real), professor of English at the University of Illinois, remedies this oversight, combining a scientific introduction to volcanism with a vivid account of the eruption’s cultural, political, and economic impact that persisted throughout the century.”

Interested in Tambora?  You can start reading the introduction here.

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The current state of the economy is always making news. People want to know how their country is surviving and thriving economically. In discussion of the economy, GDP almost always comes into play, but what is the real significance of this economic term.  Diane Coyle’s GDP: A Brief but Affectionate History traces the history of this artificial, abstract, complex, but exceedingly important statistic from its eighteenth- and nineteenth-century precursors through its invention in the 1940s and its postwar golden age, and then through the Great Crash up to today. The reader learns why this standard measure of the size of a country’s economy was invented, how it has changed over the decades, and what its strengths and weaknesses are. The book explains why even small changes in GDP can decide elections, influence major political decisions, and determine whether countries can keep borrowing or be thrown into recession. The book ends by making the case that GDP was a good measure for the twentieth century but is increasingly inappropriate for a twenty-first-century economy driven by innovation, services, and intangible goods.  Dianne Coyle recently wrote an articles for Foreign Affairs and VoxEU in which she explains the practical, or not so practical side of the GDP and elaborates on themes from GDP: A Brief but Affectionate History. You can also read the introduction here.

Just as the economy as a whole is a critical portion of the news cycle, so are the banking systems that play a part in the economy. But why are banking systems unstable in so many countries–but not in others? The United States has had twelve systemic banking crises since 1840, while Canada has had none. Fragile by Design is a revealing exploration of the ways that politics inevitably intrudes into bank regulation. Charles Calomiris and Stephen Haber combine political history and economics to examine how coalitions of politicians, bankers, and other interest groups form, why some endure while others are undermined, and how they generate policies that determine who gets to be a banker, who has access to credit, and who pays for bank bailouts and rescues.

Influential economics blogger Arnold Kling recently reviewed Fragile by Design on his Askblog. Kling critiqued the content of the book from an economics perspective, but ultimately sang its praises with “Everyone, regardless of ideology, should read the book. It offers a lot of food for thought.”  In the review, Kling also referenced attending Russ Roberts’ Econtalk  live interview with the authors, Calomiris and Haber. Kling recommends, “You might look forward to listening–the authors are very articulate and they speak colorfully.”  You can listen to the Econtalk Live podcast here.

Charles W. Calomiris, along with colleague Allen H. Meltzer, recently wrote an op-ed piece for the Wall Street Journal entitled “How Dodd-Frank Doubles Down on ‘Too Big to Fail”, in which he elaborates on a specific act which attempts to undo some of the damage from the 2008 financial crisis. Read the full Wall Street Journal article here.  Howard Davies of Times Higher Education also recently reviewed Fragile by Design, saying “Calomiris and Haber offer a thoughtful counter-argument to the current received wisdom.” You can find the full Times Higher Education review here.    Interested in reading more? Get a head start on Fragile by Design with Chapter 1 found here.

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Also recently reviewed for Times Higher Education was Charles L. Adler’s Wizards, Aliens and Starships: Physics and Math in Fantasy and Science Fiction.  From teleportation and space elevators to alien contact and interstellar travel, science fiction and fantasy writers have come up with some brilliant and innovative ideas. Yet how plausible are these ideas–for instance, could Mr. Weasley’s flying car in the Harry Potter books really exist? Which concepts might actually happen, and which ones wouldn’t work at all? Wizards, Aliens, and Starships delves into the most extraordinary details in science fiction and fantasy–such as time warps, shape changing, rocket launches, and illumination by floating candle–and shows readers the physics and math behind the phenomena.  You can find the Times Higher Education review here and begin reading Chapter 1 of Wizards, Aliens and Starships here.

In last week’s PUP News of the World, we featured Bernard Williams: Essays and Reviews 1959 – 2002 which is the first collection of Williams’s popular essays and reviews, many of which appeared in the New York Review of Books, the London Review of Books, and the Times Literary Supplement. In these pieces, Williams writes about a broad range of subjects, from philosophy and political philosophy to religion, science, the humanities, economics, socialism, feminism, and pornography.  Bernard Williams: Essays and Reviews 1959 – 2002 has since been reviewed in the Telegraph by Roger Scruton.

Scruton’s review says, “This rigorous collection of essays and reviews reveals the brilliant and critical mind of Bernard Williams … In these reviews and essays Williams achieves something that philosophy always promises but seldom delivers: a view from the perspective of reason, on a cultural landscape where reason is only one of the landmarks.

Read the Foreword to this wonderful collection now.

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