A Brief Guide to the Great Equations

by Robert Crease

  The first prediction was too difficult at the time to confirm, and the third was already contained in the theory when it first appeared. But the second prediction – that part having to do with starlight passing the sun – looked like it could be tested.

  With a little help from nature.

  For experimental science is the art of getting things you understand to tell you things you do not understand. You roll balls down an incline and time them. You time the swing of a pendulum. You measure how oil droplets behave in an electric field. What’s truly marvelous – the wonder of science – is that you get out of such events more than what you put into them. In some cases, you can stage these events as command performances, in the laboratory and entirely under your control. In other cases, you have to wait for nature to set the stage for you.

  And an eclipse is one of these grand cosmic performances.

  Cosmic Performance

  An alien, watching with a powerful telescope from a great distance, would see the earth and the moon, bathed in light from the sun, cast huge cone-shaped shadows as they revolve about each other. These motions can be predicted exactly, thanks to Newton’s laws. Every so often the earth or the moon enters the shadow cast by the other – sometimes completely, sometimes only partially. As seen from the earth, by an incredible coincidence, the moon’s apparent size is the same as the sun’s, so the moon completely blots out the sun’s light, throwing everything in the shadow-cone into darkness, bringing back the stars whose light is ordinarily blotted out during the day. Such eclipses thus set the stage for being able to tell whether this starlight is bent as it passes by the sun.

  Einstein was anxious and excited about the prospect of testing his theory. How could he not have been? It was more than a formula – it was a proposal that the fundamental structure of the universe was different, and stranger, than human beings had ever thought. He had been frustrated by the setbacks of the earlier expeditions. But these had spared him the embarrassment of testing a result he would have had to revise.

  After his monumental 1916 ‘General Theory’ paper outlining the trio of predictions, Einstein began actively promoting again the cause of testing the theory during an eclipse. He sent a copy of the paper to the Dutch astronomer and physicist Willem de Sitter, who in turn passed it to the secretary of the Royal Astronomical Society, Arthur Eddington. Because the war had disrupted communications between the combatants, this was the only copy of Einstein’s paper to reach England. As a physicist, Eddington had been vexed by the Mercury discrepancy, and intrigued by Einstein’s theory, and proceeded to write several review articles about it. Moreover, as a Quaker and pacifist, he was attracted by the opportunity to overcome the hostility between German- and English-speaking scientists. His efforts met resistance. Oliver Lodge, the British physicist who had discovered a form of electromagnetic waves simultaneously with Hertz, and who continued to believe in the existence of ether, claimed that ether drift could account for Mercury’s perihelion. An American opponent of Einstein named Thomas J. J. See, who insisted that gravitation was a real physical force, wrote that ‘the whole doctrine of relativity rests on a false basis and will someday be cited as an illustration of foundations laid in quicksand.’35 Nevertheless, Eddington persisted, and began to drum up interest in an eclipse expedition in this fascinating and fundamental theory.36 Some recent social constructivist work, which tends to see all interest as self-interest, has drawn attention to Eddington’s efforts to garner scientific, media, and public attention to relativity, with the implication that Eddington was engaged essentially in manipulation, public relations, and self-promotion.37 But it is only natural that Eddington was excited by this work and wanted to share his excitement – and that the public and the media responded positively to this news of potentially fundamental significance. To think otherwise is to have an infantile view of scientists, and of the public. Had Eddington not wanted to share his excitement at scientific work that hinted at a revolutionary picture of the world – that would have made him pathological.

  The latest eclipse expeditions fared no better than the earlier ones. The war scotched plans to mount an expedition to Venezuela to observe an eclipse in 1916. Another opportunity arose in 1918, when an eclipse took place in the U.S., but a series of unfortunate developments – including poor weather, not-yet-returned instruments that the University of California’s Lick observatory had lent to the 1914 Russian expedition, and conflicts between group members – ended up with the results not being published.

  Eddington helped enlist British Astronomer Royal Frank Dyson. Dyson was the first to propose an expedition to an upcoming total solar eclipse, on May 29, 1919. The path of the eclipse would run from North Brazil across the Atlantic, and pass Africa to the north. It would take place against a background of several bright stars. In November 1917, the Royal Society’s Joint Permanent Eclipse Committee and the Royal Astronomical Society organized two expeditions to two different locations to take photographs. One, led by A. C. D. Crommelin and C. R. Davidson, was to Sobral in northeast Brazil, the other, which Eddington joined, was to Principe, an island about 10 miles long and 4 miles wide, owned by Portugal, about 120 miles off Africa’s west coast. While the eclipse itself could be predicted, the weather could not be – and the weather at Sobral was especially worrisome because May was the last month of the rainy season.

  The eclipse of May 29, 1919, was just another eclipse, yet another time that the earth entered into the cone-shaped shadow of its moon. But this one would turn out to be, scientifically, the most important eclipse in history.

  In March 1919, the two expeditions left Greenwich, England, aboard the steamboat Anselm, and made a brief stop in Lisbon, Portugal, before arriving in Madeira. There the two groups split up. The Sobral group departed for Brazil on the Anselm, while the others stayed on at Madeira for 4 weeks to await a steamer to Principe.

  When the Sobral expedition arrived at their a relatively barren location 80 miles inland, they set up their equipment in the racetrack of the local jockey club, after making sure that no races were planned before the eclipse. The Brazilian government supplied porters, bricklayers, carpenters, and interpreters, and an automobile – the first ever seen in Sobral – was brought all the way from Rio for the expedition’s use. The team built a support structure to shelter their two telescopes against gusts. A minor calamity occurred when a whirlwind suddenly appeared and overturned the structure, but carpenters pillaged beams from elsewhere to fix it. A heavy rain fell on May 25, reminding the members of the expedition of the season. More disturbing was the discovery that the drive mechanism of the larger of the two telescopes was running unevenly, and that both instruments had focusing problems. On May 29, the moon’s shadow began to sweep across the earth’s surface. That morning, at Sobral, the members of the expedition awoke to an overcast sky. When the eclipse began, the sun was still behind clouds. But a minute before the eclipse became total, the sun emerged. For 6 minutes, Crommelin and Davidson took as many photographs as they could, exposing plates for 5 to 6 seconds each. ‘Eclipse splendid’, they telegrammed.

  The other expedition landed in Principe and set up shop at a plantation on the northwest side of the island. The regularly overcast sky made the team members apprehensive. On May 29, the team members awoke to a heavy thunderstorm, and cloud cover hung around the rest of the morning. When the eclipse started, the sun remained completely obscured. About half an hour before totality, the team members spotted glimpses of the now-crescent-shaped sun, raising their hopes. But the cloud cover never cleared completely. Eddington and his colleague, E. T. Cottingham, helpless, took pictures of the clouds and brief glimpses of the stars, hoping against hope that they might show something. ‘Through cloud. Hopeful’, they telegrammed.

  Over the next several months, groups set out computing the amount of deflection, taking into account the many sources of error.

  ‘Joyous News Today’

  In September, Einstein anxiously wrote to Lorentz to ask
if he’d heard of the English results. On September 27, Lorentz telegrammed back:

  Eddington found star displacement at rim of sun, preliminary measurement between nine-tenths of a second and twice that value. Many greetings, Lorentz

  Einstein promptly sent a postcard to his mother, Pauline, deathly ill and with only a few months to live:

  Dear Mother, joyous news today. H.A. Lorentz telegraphed that the English expeditions have actually demonstrated the deflection of light from the sun.38

  Einstein also registered his excitement by sending a note to Naturwissenschaften.39 But now that the anxiety was over, he could collect himself and grew more sangfroid. When his student Ilse Rosenthal-Schneider came to visit him, he showed her the telegram, saying, ‘Here, perhaps this will interest you.’ She later recalled,

  It was Eddington’s cable with the results of measurement of the eclipse expedition. When I was giving expression to my joy that the results coincided with his calculations he said, quite unmoved, ‘But I knew that the theory is correct’, and when I asked what if there had been no confirmation of his prediction, he countered: ‘Then I would have been sorry for the dear Lord – the theory is correct.’40

  On November 6, 1919, the joint meeting of the Royal Society of London and the Royal Astronomical Society was presided over by Sir Joseph J. Thomson, who had discovered the electron about a quarter-century earlier. He opened the meeting by saying, ‘I call on the Astronomer Royal to give us a statement of the result of the Eclipse Expedition of May last.’41

  Dyson:

  The purpose of the expedition was to determine whether any displacement is caused to a ray of light by the gravitational field of the Sun, and, if so, the amount of the displacement. Einstein’s theory predicted a displacement varying inversely as the distance of the ray from the Sun’s centre, amounting to 1.75’ for a star seen just grazing the Sun. His theory or law of gravitation had already explained the movement of the perihelion of Mercury – long an outstanding problem for dynamical astronomy – and it was desirable to apply a further test to it.

  Any bending, Dyson continued, would have the effect of ‘throwing the star away from the Sun’; that is, making it look farther away. He briefly reviewed the events of the expedition, mentioning the defect discovered in the larger instrument. A good scientist, Dyson knew that all data are not created equal – some instruments perform better than others, in a way that one can evaluate independently of the result. Emphasizing the smaller instrument’s measurements, Dyson concluded: ‘A very definite result has been obtained that light is deflected in accordance with Einstein’s law of gravitation.’ Crommelin followed, offering a brief explanation of the defect in the larger instrument.

  Eddington, a little outclassed by Dyson and by the quality of Crommelin’s results, recounted his expedition and its frustrations with the weather. He painted his results as positively as possible, finding good in bad, pointing out that the cloud cover and uniform temperature at Principe had a beneficial side effect in reducing the mirror distortion that had adversely affected the larger Sobral mirror. Gamely admitting that he was ‘making the most of a small amount of material’, Eddington noted that his result of 1.6’ value for displacement at the limb ‘supports the figures obtained at Sobral.’ Disposing of the refracting matter explanation, Eddington concluded that the results support Einstein’s law – the statement that light bends – though not necessarily Einstein’s theory, the ideas about curvature of space behind it.

  Thomson then took the floor again. He called the reports a ‘momentous communication.’ He said, ‘This is the most important result obtained in connection with the theory of gravitation since Newton’s day, and’ – alluding to the portrait of this early member hanging nearby – ‘it is fitting that it should be announced at a meeting of the Society so closely connected with him.’

  Some discussion lamented the mathematical complexity of the theory, which seemed out of the reach of most physicists. ‘I cannot believe’, one said, ‘that a profound physical truth cannot be clothed in simpler language… Cannot Prof. Eddington translate his admirable treatise from the tensor notation into some such form?’ A skeptic, citing the continuing absence of evidence of a spectroscopic shift, argued that the deflection result – ‘an isolated fact’ – did not necessarily confirm Einstein’s theory. ‘We owe it to that great man’, he said, dramatically pointing to Newton’s portrait – ‘to proceed very carefully in modifying or retouching his Law of Gravitation.’ But the arguments against Einstein’s theory would all but die out within 2 years.

  The experimental differences between his theory and those of Newton were tiny – small shifts in the positions of a few stars and spectral lines, and of a minuscule wobble in Mercury’s orbit. But the differences could hardly be more profound, for they implied a fundamental difference in the way the universe was structured.

  In Newton’s theory, gravitation involves an attractive force – a tug – that each mass exerts on all others at a distance. That force operates instantly and everywhere, and is inversely proportional to the square of the distance between the masses. Masses experiencing that force respond by accelerating toward its source. Different masses are accelerated at the same rate, because the force pulls them in proportion to their mass: a small mass experiences a small pull, a greater mass a greater pull. In Einstein’s theory, by contrast, gravitation involves in effect a curvature of space. That curvature is structured by the masses around it. When matter and energy move through space they follow the paths that are open to them.

  It was one of the great rearrangements of fundamental concepts in the history of science. As Eddington wrote:

  The Newtonian framework, as was natural after 250 years, had been found too crude to accommodate the new observational knowledge which was being acquired. In default of a better framework, it was still used, but definitions were strained to purposes for which they were never intended. We were in the position of a librarian whose books were still being arranged according to a subject scheme drawn up a hundred years ago, trying to find the right place for books on Hollywood, the Air Force, and detective novels.42

  By 1921, the only prominent figure who continued to be disappointed in the theory was Einstein himself. To his exacting eyes, which wanted symmetry throughout the theory, the left-hand side was solid, for it was expressed in the geometry of space-time, while the right-hand side was not. He once compared it to a poorly planned building, one half of which was ‘fine marble’, the other ‘low-grade wood.’43 Dissatisfied, he would spend much of the rest of his life in a futile effort to fix that building. Though he would work for over three decades on that repair, he would never succeed.

  Interlude

  SCIENCE CRITICS

  The equations of the gravitational field which relate the curve of space to the distribution of matter are already becoming common knowledge.

  – Italo Calvino, Cosmicomics

  Hence the pathetic paradox that Einstein’s discoveries, the greatest triumph of reasoning mind on record, are accepted by most people on faith.

  – Time, July 1, 1946

  The process by which the public comes to understand new scientific developments often appears in the form of what might be called the Moses and Aaron model. Like Moses, the scientist seems to have one foot in the sphere of the divine, bringing into the human world some discovery from beyond. The meaning of this primordial activity is then communicated by some Aaron, who translates for the public via images and popular language.

  The Moses and Aaron model is essentially a two-step process. We see it reenacted all the time – in the nightly TV news, for instance, when a spokesman or talking head tries to explain some novel development before the allotted 60 seconds run out. Some Aarons are more effective and entertaining than others.

  Einstein’s general theory of relativity poses a particular burden on would-be Aarons. A one- or two-step translation process is difficult to achieve. The theory involves complex mathematics, and unfa
miliar ways of thought, that take physicists years to master. Learning the theory is like acclimatization into a culture, with no shortcuts. Sometimes – as The New York Times famously did after the November 6, 1919, announcement – journalists simply throw up their hands and say that a discovery cannot be explained to nonscientists. And physicist Hermann Bondi once said that members of the public would not understand relativity until they had relativistic toys to play with.

  But talking about science to outsiders is like talking about a city to noninhabitants; what you say depends on the interests of your audience. If they intend to become inhabitants, you give them one kind of talk, focusing on regulations, institutions, laws, and so forth, that may take them awhile to master. If your listeners are just tourists with no intention to become inhabitants, on the other hand, you can focus on the public attractions, not go into too much detail, and safely condense a lot.

  Many interesting attempts have been made to make general relativity accessible to a tourist audience. One way is by selecting and elaborating clever illustrations that show its implications in an accessible way: for instance, the twin paradox, wherein one twin who travels in space at near the speed of light ages differently from the other twin who remains on Earth; or the astronaut who cannot tell whether he or she is accelerating or in a gravitational field. Another way is through biographies: witness the phenomenal popularity of Walter Isaacson’s recent book Einstein: His Life and Universe, or of Abraham Pais’s earlier, more difficult but American Book Award–winning ‘Subtle is the Lord’: The Science and the Life of Albert Einstein. Still another way to convey the meaning of general relativity to outsiders is by dramatic images, such as that of weights on a rubber sheet. A weight on a rubber sheet bends the sheet, by an amount that depends on the heaviness of the object, in a way that affects the path of marbles rolling past – in an analogous manner that an object distorts space, by an amount that depends on its mass, in a way that affects the paths of objects, including light, that pass through it. And some authors use all three methods at once, as Brian Greene in his wonderful books The Fabric of the Cosmos and The Elegant Universe.