TURBULENCE

Reflections on flying by David Sedaris in The New Yorker:

DavidOn the flight to Raleigh, I sneezed, and the cough drop I’d been sucking on shot from my mouth, ricocheted off my folded tray table, and landed, as I remember it, in the lap of the woman beside me, who was asleep and had her arms folded across her chest. I’m surprised that the force didn’t wake her—that’s how hard it hit—but all she did was flutter her eyelids and let out a tiny sigh, the kind you might hear from a baby.

Under normal circumstances, I’d have had three choices, the first being to do nothing. The woman would wake up in her own time, and notice what looked like a shiny new button sewn to the crotch of her jeans. This was a small plane, with one seat per row on Aisle A, and two seats per row on Aisle B. We were on B, so should she go searching for answers I would be the first person on her list. “Is this yours?” she’d ask, and I’d look dumbly into her lap.

“Is what mine?”

More here.

Fairfield Porter and Myron Stout

Jed Perl in The New Republic:

Nobody knows painting better than painters. So it’s not surprising that they produce some of the finest writing on the subject. But much of this writing remains scattered–in back issues of magazines or in letters and journals that rarely if ever see the light of day. When such work gets into print there’s cause for celebration. And that’s true right now. For new collections of writing by two important twentieth-century American painters, Fairfield Porter and Myron Stout, have just appeared.

Material Witness: The Selected Letters of Fairfield Porter and Selections from the Journals of Myron Stout are terrific books. They’re suffused with the questing, restless, daring intellectual rhythms of a painter’s life.

More here.

Two Stars Poised to Merge

050606_binary_dwarfs_02_1Michael Schirber at Space.com:

Two dense stars whipping around each other at breakneck speed may be the strongest known source of Einstein’s space-trembling gravity waves.

The double star – called RX J0806 – was discovered in 1994 in X-rays. Later shown to be blinking on and off every 5.4 minutes, the two-star setup is believed to be a pair of white dwarfs – the dense ashes of burnt-out stars – rotating around each other.

The implied separation is just 50,000 miles – a mere one-fifth the distance between the Earth and the Moon, making this the closest stellar pair ever observed. The tangled duo should be booming out gravity waves – undulations in the fabric of space and time predicted by Einstein’s theory of general relativity.

More here.

questions and answers about the Supreme Court’s medical marijuana ruling

David Kravets of the AP:

Q: What was the case decided by the Supreme Court?

A: The justices overruled an injunction against federal prosecution of two California women with doctor’s recommendations for marijuana use. The decision clarifies that the federal government can prosecute violators of federal drug laws, even when people are following state law.

Q: Alaska, California, Colorado, Hawaii, Maine, Montana, Nevada, Oregon, Vermont and Washington state allow patients with a doctor’s recommendation to smoke and grow marijuana, or to have it grown for them. Should medical marijuana users in these states now fear federal prosecution?

A: Federal authorities have already made more than 60 medical marijuana arrests in the last five years nationally, almost all of them in California. But such raids remain relatively uncommon. The Justice Department declined to discuss its strategy, but “people shouldn’t panic,” California Attorney General Bill Lockyer said.

Q: What have the 10 states done in reaction to Monday’s ruling?

A: Oregon tentatively stopped issuing medical marijuana identification cards to new patients, but it was business as usual in the other states.

More here.  [Thanks to Winfield J. Abbe.]

A Tragic Grandeur

Jonathan Raban in the New York Review of Books:

Robertlowellbynancycrampton200x350Robert Lowell’s star has waned very considerably since his death in 1977, when his obituarists treated him, along with Yeats, Eliot, Auden, and Wallace Stevens, as one of the handful of unquestionably great twentieth-century poets. The publication two years ago of Frank Bidart and David Gewanter’s massive edition of the Collected Poems did much to restore his work to public and critical view, but even now Lowell’s poems are, I would guess, less widely read, taught, and anthologized than those of his two friends and contemporaries Elizabeth Bishop and John Berryman—a judgment, if that is what it is, that would have astonished serious readers of poetry between the 1950s and the 1970s.

More here.

Ancient Pharaoh’s Statue Found

Rossella Lorenzi at Discovery:

Egyptstatue_gotoA life-sized statue of the 13th Dynasty Pharaoh Neferhotep I has emerged from the ruins of ancient Thebes in Luxor, Egypt’s Supreme Council of Antiquities said on Saturday.

Buried for almost 3,600 years, the six-foot limestone statue shows the “beautiful and good” pharaoh — this is what Neferhotep means — wearing the royal head cloth.

The forehead bears the emblem of a cobra, which pharaohs wore on the crown as a protective symbol: they believed that the cobra would spit fire at enemies.

More here.

Explorer finds sub that may have inspired Verne’s Nautilus

Steven Morris in The Guardian:

A British explorer has discovered an abandoned 19th-century submarine which may have been the inspiration for Captain Nemo’s vessel Nautilus in Jules Verne’s 20,000 Leagues Under the Sea.

Colonel John Blashford-Snell found the cast-iron submarine, named Explorer, half-submerged in three metres of water off the coast of Panama.

Like Nautilus, the craft is cigar-shaped and has a lock-out system, which allows submariners to leave, collect items from the seabed and then return to the vessel.

It was built in 1864, five years before Verne’s classic adventure story was published, and it is thought that the French writer would have read about the sub’s specifications.

More here.

Stalking a Killer That Lurks a Few Feet Offshore

Cornelia Dean in the New York Times:

07ripWhen people think about natural hazards, they usually think about tornadoes or hurricanes or earthquakes. But there is another natural hazard that takes more lives in an average year in the United States than any of those – rip currents.

Each year in American waters, rip currents pull about 100 panicked swimmers to their deaths. According to the United States Lifesaving Association, lifeguards pull out at least 70,000 Americans from the surf each year, 80 percent from rip currents.

Because these drownings and near drownings occur one by one, year-round, up and down the coasts, few people recognize rip currents as a major hazard. Only in recent years have meteorologists and coastal geologists begun to measure rip currents precisely in the field and model them in detail in laboratory wave tanks.

More here.

1599: A Year in the Life of William Shakespeare

From The London Times:

Shakes_1 A heavy snowstorm shrouded London on December 28, 1598. Through it a group of men bristling with swords and axes closed in on a building in the city’s northern suburbs. The building was The Theatre — London’s oldest playhouse, once the scene of full-blooded dramas by Christopher Marlowe and Thomas Kyd, but empty for the past two years since the Chamberlain’s Men whose base it was had quarrelled with their cantankerous landlord, Giles Allen. Now, while Allen unsuspectingly spent Christmas in the country, members of the troupe gathered to dismantle the playhouse which (unlike the leased land it stood upon) technically belonged to them. Carted away and ferried across the Thames, its timbers would be re-erected as a new theatre, The Globe. Among those taking part in this rushed and risky act of reclamation were the company’s star tragedian Richard Burbage, its celebrity-comic Will Kemp and its 35-year-old resident playwright William Shakespeare.

For Shakespeare the next 12 months would be momentous. 1599, James Shapiro compellingly displays, was his annus mirabilis: the year that, deepening and complicating his imagination, took him from outstanding accomplishment to unsurpassed genius. That genius, romantically disposed commentators such as Coleridge have maintained, was “of no age” but arose from “the unfathomable depths of his own oceanic mind”. Shapiro, who can be breathtakingly acute at fathoming Shakespeare’s mind, couldn’t disagree more. Shakespeare’s creativity, he contends, was decisively fuelled and fired by contemporary events — and never more so than in his four great artistic undertakings of 1599: the completing of Henry V, the writing of Julius Caesar and As You Like It, and the drafting of Hamlet.

More here.

Where the wild things still are

A delightful interview with Maurice Sendak, the creator of Max, the child-Hero from “where the wild things are”. At 76, his most treasured possession is a collection of Micky Mouse memorabilia.

Bears500 “Maurice Sendak looks kinda like a Wild Thing,” Ludden notes. “Curly hair on a balding head… a glint in the eye… yet a softening smile around the mouth.”

Read the highlights or simply listen to the interview.

Monkey Hear, Monkey Count

From The Scientific American:

Monkey Rhesus monkeys possess a natural ability to match the number of voices they hear to the number of individuals they expect to see vocalizing, new research concludes. The results indicate that abstract representation of numbers is possible in the absence of language.

Writing in the June 7 Current Biology, Elizabeth Brannon of Duke University and her colleagues describe their experiment. The researchers played the monkeys “coo” calls made by either two or three unfamiliar conspecifics. They then let the monkeys watch their choice of video images showing either two or three animals. The vast majority of the monkeys selected video images that corresponded to the number of individuals heard on the audio sample. Each monkey was tested only once and did not receive a reward. This allowed the team to observe the animal’s spontaneous behavior, as opposed to skills learned over the course of evaluation. Brannon notes that in the wild, a monkey could conceivably hear various animals calling but not see them. “In a territorial dispute, you could imagine that an animal would want to know, ‘Well, how many animals are really about to encroach on our territory?'”

More here.

Dolphins teach their young to use tools

From MSNBC:

Dolphin A group of dolphins living off the coast of Australia apparently teach their offspring to protect their snouts with sponges while foraging for food in the sea floor. Researchers say it appears to be a cultural behavior passed on from mother to daughter, a first for animals of this type, although such learning has been seen in other species. The dolphins, living in Shark Bay, Western Australia, use conically shaped whole sponges that they tear off the bottom, said Michael Kruetzen, lead author of a report on the dolphins in Tuesday’s issue of Proceedings of the National Academy of Science.

“Cultural evolution, including tool use, is not only found in humans and our closest relatives, the primates, but also in animals that are evolutionally quite distant from us. This convergent evolution is what is so fascinating,” said Kruetzen. Researchers suspect the sponges help the foraging dolphins avoid getting stung by stonefish and other critters that hide in the sandy sea bottom, just as a gardener might wear gloves to protect the hands.

[The photo was taken by Dr. Janet Mann. The dolphin’s name is Dodger and she was taught to sponge by her mother, Demi. Demi’s mom, Half fluke, was also a sponger.]

More here.

The Popularity of first names over the last century

And by way of Steven Levitt:

Namevoyager_3 “The Baby Name Wizard’s NameVoyager is an interactive portrait of America’s name choices. Start with a ‘sea’ of nearly 5000 names. Type a letter, and you’ll zoom in to focus on how that initial has been used over the past century. Then type a few more letters, or a name. Each stripe is a timeline of one name, its width reflecting the name’s changing popularity. If a name intrigues you, click on its stripe for a closer look.”

And there you’ll also find some interesting pieces on name-onomics.

“Levitt’s primary thesis is that fashions which originate with the upper classes gradually trickle down the economic ladder. This, naturally, is no revelation — in fashion-based industries like apparel, it’s an explicit, institutionalized process.  .  . Levitt uses data about California parents’ economic status and name choices to propose a list of names that, ‘unlikely as it seems,’ are candidates to become ‘mainstream names’ ten years from now. . .

In fact, of his 24 predictions for ‘unlikely’ names that could possibly hit the mainstream in a decade, 7 were already top-100 names, including 2 of the top 15 (Emma and Grace). Looking boldly out into the future, he predicted the present. Oops. So much for revelations.”

Monetizing the monkey economy

The first installment of Steven Levitt and Stephen Dubner’s new New York Times Magazine column, “Freakanomics”, looks at what happens to moneys when they monetize exchange.

“The essential idea was to give a monkey a dollar and see what it did with it. . . It took several months of rudimentary repetition to teach the monkeys that these tokens were valuable as a means of exchange for a treat and would be similarly valuable the next day. Having gained that understanding, a capuchin would then be presented with 12 tokens on a tray and have to decide how many to surrender for, say, Jell-O cubes versus grapes. This first step allowed each capuchin to reveal its preferences and to grasp the concept of budgeting.

Then Chen introduced price shocks and wealth shocks. If, for instance, the price of Jell-O fell (two cubes instead of one per token), would the capuchin buy more Jell-O and fewer grapes? The capuchins responded rationally to tests like this — that is, they responded the way most readers of The Times would respond. In economist-speak, the capuchins adhered to the rules of utility maximization and price theory: when the price of something falls, people tend to buy more of it.

. . .

Once, a capuchin in the testing chamber picked up an entire tray of tokens, flung them into the main chamber and then scurried in after them — a combination jailbreak and bank heist — which led to a chaotic scene in which the human researchers had to rush into the main chamber and offer food bribes for the tokens, a reinforcement that in effect encouraged more stealing.

Something else happened during that chaotic scene, something that convinced [the researcher Keith] Chen of the monkeys’ true grasp of money. . . What he witnessed was probably the first observed exchange of money for sex in the history of monkeykind. (Further proof that the monkeys truly understood money: the monkey who was paid for sex immediately traded the token in for a grape.)”

You can also read their blog here.

White House Tapes Site

With all the recent interest in Felt and Nixon, it’s a good time to mention an excellent web site, WhiteHouseTapes.org, which acts as a clearinghouse for Presidential audio archives. The project, associated with the University of Virginia, has some timely clips of Nixon on Felt. Of the main page’s features, check out the FBI background check on Janet Leigh, and Nixon discussing one Donald Rumsfeld.

Cold fusion, for real

Michelle Thaller in the Christian Science Monitor:

For the last few years, mentioning cold fusion around scientists (myself included) has been a little like mentioning Bigfoot or UFO sightings.

After the 1989 announcement of fusion in a bottle, so to speak, and the subsequent retraction, the whole idea of cold fusion seemed a bit beyond the pale. But that’s all about to change.

A very reputable, very careful group of scientists at the University of Los Angeles (Brian Naranjo, Jim Gimzewski, Seth Putterman) has initiated a fusion reaction using a laboratory device that’s not much bigger than a breadbox, and works at roughly room temperature. This time, it looks like the real thing.

More here.

Lost Dumas novel hits French bookshelves

From the AFP:

A previously unknown novel by the author of “The Three Musketeers”, Alexandre Dumas — a 1,000-page adventure story about the start of the Napoleonic empire — hit French bookstores.

“Le Chevalier de Sainte-Hermine” (The Knight of Sainte-Hermine) first appeared in serial form in a French newspaper and lacked just a few chapters when Dumas died in 1870.

Claude Schopp, the Dumas expert who found the book at France’s National Library, has added a short section to bring the tale to its conclusion.

The novel completes a trilogy of works set in the aftermath of the French revolution, which begins with “Les Compagnons de Jehu” — written in 1857 — and continues with “Les Blancs et Les Bleus,” completed in 1867.

More here.

Salman’s leap for literary freedom

John Freeman in The Scotsman:

For the past year, however, Rushdie’s professional attentions have been focused on his role as president of PEN/America, which entails not just putting on fancy events but filing legal action. Mention the US government’s attempt to ban literature from countries like Iran to him now and he immediately switches into policy wonk mode.

“PEN has been fighting that particular regulation for a long time,” he says and then explains some of its details. “The US government is just now beginning to plane back on it. The question is whether the damage is already done.”

It seems somewhat ironic that Rushdie should survive a period of life-threatening danger, living in 30 houses in nine years, and wind up in the land of the free only to discover that he must start campaigning for freedom all over again.

If there is resentment, though, he certainly doesn’t show it. Rushdie has lived part-time in New York for more than five years now, and he’s not about to stop. He can at least now freely play table tennis with fellow author Jonathan Safran Foer without first greeting photographers outside.

More here.

Monday Musing: Special Relativity Turns 100

Einst_patOne hundred years ago this month, twenty-six-year-old Albert Einstein published a paper entitled “Zur Elektrodynamik bewegter Körper or “On the Electrodynamics of Moving Bodies”. As we all know by now, 1905 was Einstein’s annus mirabilis, the miraculous year in which he published four papers in the Annalen der Physik. The first was a paper on the photoelectric effect; the second on Brownian motion; and the third, which we have already mentioned, spelled out the ideas which would come to be known as special relativity. In case you are less than sure why Einstein’s name has become a metonym for extreme intelligence, consider that there is broad consensus among physicists that any one of these three papers by itself would have been more than enough to win Einstein a Nobel Prize. The fourth paper, by the way, used the axioms of the third to derive a nice little result equating energy and mass: E = mc2, probably the most famous equation of all time. Not only that, he would certainly have won another Nobel for general relativity, which he published a decade later. In other words, you can safely think of Einstein as someone who, in a fairer world, would have been at least a four-time Nobel winner. As it is, the Nobel committee cited only the photoelectric effect when they awarded him the prize in 1921.

General_fig02_1Einstein’s results were disseminated and understood so slowly (especially in the English-speaking world) that when Sir Arthur Eddington lead an expedition to prove general relativity correct by showing that the light from stars near the Sun in the sky would be bent by its gravity (during a solar eclipse in 1919 when you could actually see stars close to the Sun), and a journalist asked him: “Is it true that only three people in the world understand relativity?” Eddington reportedly responded, “Who’s the third?”

Wyp2005_large_logoThis year has been declared the World Year of Physics in commemoration of the centennial of Einstein’s annus mirabilis, and in that same spirit, I would like today to attempt to give you a sense of what the theory of special relativity (SR) is, whose 100th birthday we celebrate this month. Most of you have at least some vague idea of what SR implies: you have heard that time slows down when you start traveling very fast (near the speed of light); that lengths contract; and almost everyone knows the Twin Paradox, where one twin travels out into space at high speed, then returns, say 6 years later to find that her twin on earth has aged 39 years while she was gone. You have probably also heard that Einstein is responsible for inextricably entwining space and time into spacetime. I will explicate all these aspects of SR, and I will not do it by using crude analogies, which tend to confuse more than they illuminate; instead, I will use the actual math so that you fully understand the beauty of this theory. Wait! Don’t stop reading just yet. The math required is no more than simple high school algebra, so if you remember how to do that, stay with me. If the sight of even the simplest equation makes you tremulous, then I can only say: learn some math! Einstein himself famously stated that “The presentation of science to the public must be made as simple as possible, but not more so,” and we cannot but follow his dictum here, where our very aim is to celebrate his science. (The beauty of SR lies in the incredible conceptual leap which Einstein made. The mathematics is relatively (!) straightforward, and this is what makes my elucidation of it possible. The mathematics of general relativity is much more advanced, indeed far beyond the abilities of most people who, like me, are neither mathematicians nor physicists. Even Einstein needed some help from mathematicians to work it all out.)

THE BACKGROUND

Galileo_hist_bigA century ago, this was the situation: Galilean and Newtonian physics said that any descriptions of motion by any two inertial observers (for such observers, bodies acted on by no forces move in straight lines) in uniform (not accelerating) relative motion are equally valid, and the laws of physics must be exactly the same for both of them. Bear with me here: what this means is, for example, if you see me coming toward you at a speed of 100 mph, then we could both be moving toward each other at 50 mph, or I could be still and you could be moving toward me at 100 mph, or I could be moving toward you at 30 mph while you are coming at me at 70 mph, and so on. All these descriptions are equivalent, and it is always impossible to tell whether one of us is “really” moving or not; all we can speak about is our motion relative to each other. In other words, all Newton motion is relative to something else (which is then the inertial frame of reference). So for convenience, we can always just insist that any one observer is still (she is then the “frame of reference”) and all others are in motion relative to her. This is known as the Principle of Relativity. Another way to think about this is to imagine that there are only two objects in the universe, and they are moving relative to one another: in this situation it is more clearly impossible to say which object is moving. (Think about this paragraph, reread it, until you are pretty sure you get it. Just stay with me, it gets easier from here.)

Maxwell_2At the same time, James Clerk Maxwell‘s equations of electricity and magnetism implied that the speed of light in a vacuum, c, is absolute. The only way that this could be true is if Maxwell’s equations refer to a special frame (see previous paragraph) of reference (that in which the speed of light is c) which can truly be said to be at rest. If this is the case, then an observer moving relative to that special frame would measure a different value for c. But in 1887, Michelson and Morley proved that there is no such special frame. Another way of saying this (and this is the way Einstein put it in 1905) is that the speed of light is fixed, and is independent of the speed of the body emitting it. (The details of the Michelson-Morley experiment are beyond the scope of this essay, so you’ll have to take my word for this.)

Now we have a problem. We have two irreconcilable laws: 1) The Principle of Relativity, and 2) The absoluteness of the speed of light for all observers. They cannot both be true. It would be another eighteen years before a young clerk in the Swiss patent office would pose and then resolve this problem. Here’s how he did it: he asked what would happen if they were both true.

Next, I will show how the various aspects of SR fall straight out of the assumption that both of these laws are true. I will focus in greater detail on the slowing down (dilation) of time, and then speak more briefly about length contraction, and the intertwining of space and time.

TIME DILATION

As I have mentioned, Einstein began by assuming that the following two postulates always hold true:

1) The Principle of Relativity, and

2) The speed of light will always be measured as c by all observers

Einstein1_1Now, keeping these in mind, let us consider a simple mechanism that we will call a light clock (shown in Fig. 1). The way it works is this: the top and bottom surfaces are perfect mirrors. The distance between the top and bottom mirror is known exactly. The light clock’s period is the time that it takes light to go from the bottom to the top, and then to come reflected back. Since the mirrors are perfect, light will keep on bouncing back and forth like this forever. All observers can build identical clocks with exactly the same distance between the two mirrors, ensuring the same period. And since the speed of light is always c, and the distance between the two mirrors can be measured precisely, we know exactly how long one “tick” or period of the clock is in seconds.

Einstein2   

Let us now say that there are two observers, each of whom has such a clock. If one of them is moving past the other at a velocity v, something close to the speed of light, then the first observer, F, will see the second observer S’s clock as something like what is shown in Fig. 2. Of course, by symmetry, and the principle of relativity, S will see F’s clock the same way. Take a little time to look at Fig. 2 and convince yourself of this. (This is basically like thinking about a man moving a flashlight vertically up and down on board a train; if the train is stationary relative to you, you will see what is shown in Fig. 1; if the train is moving by you, you will see what is shown in Fig. 2. It should be quite obvious once you try to imagine it.)

Since S’s clock seems to be moving to F, it will seem to F that the light travels a longer path than just the vertical distance between the two mirrors, because after the light leaves the bottom mirror, the top mirror keeps moving to the right, and the light beam travels a diagonal path up to where the top mirror has moved to. (Imagine the whole apparatus moving to the right as the light beam goes up from the bottom mirror, or look at Fig. 2.) Since the speed of the light must still be measured as c by both observers, and according to F, the light beam is traveling a greater distance, it must be taking longer to make the trip to the top mirror and back. Therefore, according to F, S’s clock is ticking more slowly, and vice versa!

Someone might object that this is a special kind of clock, and maybe we could construct a different type of clock that would not slow down when seen speeding along relative to some other observer. This cannot be true. The reason is that if we were able to construct such a clock, it would violate our first postulate, the Principle of Relativity. Remember that in saying that only relative motion is physically significant, we are insisting that nothing done by S can tell her whether it is she who is moving past F, or vice versa. Suppose that two different types of clock were synchronized (one of them a light clock of the type we have been describing), then both of them are sent off with S at high speed,  if they do not behave exactly the same way and were to fall out of synchronization, this would tell S that it is she who is really moving, and this contradicts the first postulate. All clocks must therefore slow down in the same way when they are observed in relative motion close to the speed of light. In other words, this time dilation is not a property of any particular type of clock, but of time itself.

Einstein3So how much exactly is S’s time seen to be slowing down by F? To answer this question, consider the situation in Fig. 3. As shown, the light clock is moving to the right at a velocity v. At rest, the light clock has a period T. In half that time, when stationary, the light beam travels the distance A, but when moving, it has a slower period T’ because it travels the distance C at the same speed. This means that the ratio of the distances A/C is the same as the ratio of the times taken to traverse them, T/T’. (At the same speed, if you travel twice the distance, it will take you twice as long.) So,

(1) A/C = T/T’

Now while the light beam travels the distance C, the mirrors have moved a distance B to the right. The ratio of these two distances B/C, traveled in the same amount of time, is the same as the ratio of the speeds with which the distances are covered. (Moving for a given time at double the speed just doubles the distance covered.) So,

(2) B/C = v/c

Look at Fig. 3 again, and notice that the sides A, B, and C form a right triangle, so by the Pythagorean Theorem:

(3) A2 + B2 = C2

Dividing both sides by C2 we get:

(A/C)2 + (B/C)2 = 1

Subtracting (B/C)2 from both sides we get:

(A/C)2 = 1 – (B/C)2

Taking the square root of each side we get:

A/C = sqrt ( 1 – (B/C)2 )

Now if we substitute for A/C and B/C from equations (1) and (2) above, we get:

T/T’ = sqrt ( 1 – (v/c)2 )

And finally, inverting both sides, we get:

(4) T’/T = 1 / sqrt ( 1 – (v/c)2 ) = γ (gamma — the relativistic time dilation factor)

Since the speed of light is so high (186,000 miles per second or 300,000 kilometers per second), gamma is not significant at speeds that are common to our experience. For example, even at the speed which the space shuttles must attain to escape Earth’s gravity (11 km/sec), gamma is 1.000000001. At fifty percent of the speed of light (0.5 c), gamma is 1.155. You can confirm these values by plugging in the speeds into the time dilation equation above. One way in which we know that Einstein was correct about time dilation is that particles with known half-lives decay much more slowly when they are accelerated to near the speed of light in particle accelerators. For example, muons, which have a half-life of 1.5 microseconds, are observed to decay in 44 microseconds on average in a CERN experiment which accelerated them to 0.9994 c, at which speed gamma can be calculated using the equation above to be 28.87. In perfect agreement with the theory, 1.5 microseconds multiplied by 28.87 comes out to 44 microseconds, exactly what is seen in the experiment. There are countless other very exact confirmations of relativistic time dilation effects.

LENGTH CONTRACTION

This time, imagine the light clock lying on its side (in other words, Fig. 1 rotated by 90 degrees counter-clockwise). Now the motion of the light pulse is back and forth in the same direction that the whole clock is moving. What happens this time? Well, as the light pulse leaves one mirror and heads toward the other, that mirror advances forward to meet it. This trip is shorter than when the clock is stationary. On the way back, though, the light pulse is chasing a retreating mirror, and the trip takes longer than it would in a stationary situation. This round trip period, T”, is longer than T’ by the factor 1/γ. (This is similar to the case where an airplane traveling across the Atlantic with a steady headwind against it, and then returning with the same wind at its back, will take a longer time for the round trip than if there were no wind at all. I leave the simple math here as an exercise for the reader.)

Now if this were all there is to the story, the amount of time dilation would depend on the orientation of the clock relative to the direction of motion, but then this would violate the Principle of Relativity. What prevents this violation is a shortening of lengths along the direction of motion. The distance between the two mirrors would thus contract by the factor 1/γ, reducing T” to the correct value T’ as it should be. So, lengths are observed to contract along the direction of motion by a factor of 1/γ. Again, this only becomes noticeable at very high speeds, approaching c.

SPACETIME

Newton’s notion of absolute space and absolute time are no longer valid for us. We have seen that measures of time are relative to the observer, as are measures of space. The good news is that different observers of the same reality can agree on something. And this is what it is: we know that

T/T’ = 1/γ

Substituting for 1/γ from equation (4) and squaring both sides we get:

(T/T’)2 = 1 – (v/c)2

Multiplying both sides by (cT’)2 we get:

(cT)2 = (cT’)2 – (vT’)2

Here, vT’ is just the distance L’ that the moving clock travels in time T’. Meanwhile the stationary clock doesn’t go anywhere in time T, so L = 0, and by substituting that L2 = 0 and L’2 = (vT’)2 into the equation above, we get:

(cT)2L2 = (cT’)2L’2

Here, finally, is a quantity that is the same for both observers. It is not a measure of time or a measure of space; instead, it is a spacetime measure. So we find that in the end, though observers cannot agree about measures of space or time by themselves, it is possible to weave them together into a spacetime measure that everyone does agree on. This is what is meant when it is said that space and time have become interwoven after Einstein.

The account I have followed in explaining special relativity is essentially that used by Richard Feynman (who invented light clocks as a way of explaining SR) and Julian Schwinger. The two of them shared the 1965 Nobel in physics with Sin-Itiro Tomonaga.

Thanks to Margit Oberrauch for all the light clock illustrations.

Have a good week!

My other recent Monday Musings:
Vladimir Nabokov, Lepidopterist
Stevinus, Galileo, and Thought Experiments
Cake Theory and Sri Lanka’s President

The DNA of Literature

Pity (and praise) the poor intern or assistant whose job it is to put The Paris Review author interviews online. Then settle in for the fine experience of what TPR rather dramatically calls “The DNA of Literature,” a vast pdf archive of material stretching from the 1950s to the present, from Algren to Auster. Yet another nice feature of the TPR site – the Audio Index feature which allows you to hear work read by the author.