A Brief Guide to the Great Equations

by Robert Crease

  Few who receive crazy-idea letters reply. It’s assumed to be counterproductive and perhaps even dangerous, reinforcing the authors’ sense of being misunderstood and inviting more urgent appeals. The recipients quickly peruse the letters, then toss them into a ‘crackpot letter’ drawer. Hardly anyone I know throws them out.

  Why not?

  One of my colleagues compares his ‘crackpot letter’ drawer to the neighbourhood art show; if you’re thorough and patient enough you might find something of value, but the search would take so long you never do it. Others offer psychological explanations: We admire and even envy the authors of crackpot letters for their energy and zeal. We feel a closet affinity – don’t we all feel in possession of misunderstood truth? Still darker, we’re thrilled to read them – it’s like watching a mental train wreck. Yet other colleagues keep them because, they tell me, ‘You never know…’

  Crazy-idea letters have interesting philosophical, psychological, and social dimensions. It’s much harder than it seems to characterize crazy-idea letters and their appeal. Great scientists, too, have gone weird: recall the obsessions of Einstein with unified field theories and Pauling with vitamin C. And aren’t scientists fond of and even dependent on ‘crazy ideas’? Haven’t we all heard the famous story about how Wolfgang Pauli charged Niels Bohr with holding a crazy theory, and how Bohr replied that the trouble was his theory was ‘not crazy enough’ and called for a ‘crazier idea’?

  Finally, several cautionary tales suggest we should not be too confident of our ability to recognize crazy-idea letters.

  Cautionary tale 1 is the story of 25-year old Srinivasa Ramanujan, an Indian who in 1913 sent correspondence to several British mathematicians. Like several others, G. H. Hardy tossed it aside initially as the work of a crackpot, then read it, realized its genius, and soon invited Ramanujan to England, where he became recognized as one of the leading mathematicians of all time. Our own prejudices, rather than content, can determine whom we deem a nut.

  But cautionary tale 2 is the story of Nicholas Christofilos, a Greek electrical engineer at an elevator installation company whose hobby was particle accelerators. In 1949, he sent a manuscript proposing a novel scheme to physicists at Berkeley, who wrote back pointing out flaws. Christofilos incorporated corrections, applied for U.S. patents, and sent the revised scheme back to the Berkeley physicists, who this time simply ignored it. In 1952, reading that U.S. physicists had ‘discovered’ a new accelerating method identical in principle to his own, he contacted a legal firm and got his priority recognized. I once asked a physicist who had seen Christofilos’s original papers why they had been ignored. ‘The first violated Maxwell’s equations’, he said, shrugging without elaborating, his body language indicating that this was equivalent to mentioning psychic phenomena, and that he therefore needed to make no apology for ignoring the rest. Bad physics does not necessarily make a crackpot.

  Jeremy Bernstein insists that, had he seen a copy of his correspondent’s recently published paper, ‘The Electrodynamics of Moving Bodies’, he would have been able to tell that this was not a crank paper, and cites two clues. The first clue is ‘connectivity’, or the fact that the theory gave the same answers as Newtonian mechanics when the speeds of bodies are low compared with that of light. Crank theories ‘usually start and end in midair’ without genuinely connecting with the existing body of scientific knowledge. The second clue is the presence of testable predictions.

  I would add two more clues. The first involves the way the authors handle equations. Crazy-idea letters almost always include either no equations or a small number treated like fetish objects. Equations in such letters, indeed, generally illustrate Barthes’ observation about the Gnostic fantasy of knowledge reduced to a formula. The equations appear unaccompanied, as if they were independent nuggets of truth. They are treated as aphorisms, encapsulizing entire philosophies in miniature. The role of the equation in the paper is like that of a musical instrument that someone carries around without ever playing. In genuine scientific papers, by contrast, equations hardly ever occur unaccompanied, but are embedded in a sequence as co-participants in an extensive logical argument, fragments of an extensive intellectual edifice to which they owe their very existence, only a small part of which is reproduced on the page. The equations, that is, are not treated as standing completely on their own.

  But I think that the most important clue to a crackpot is the lack of an engaged attitude coupled with playfulness – the sort of attitude that Einstein displayed in his fear he mentioned to Habicht that God was leading him around by the nose, but in his willingness to go along.

  As evidence I submit the following story. It took place in September 1946 in New York City at one of the first postwar annual meetings of the American Physical Society. At one session, the presentation by the young Dutch theorist Abraham Pais, who was struggling to explain the strange behaviour of a puzzling, recently discovered new particle, was interrupted by Felix Ehrenhaft, an elderly Viennese physicist. Ever since 1910, Ehrenhaft had been claiming to have evidence for the existence of ‘subelectrons’, charges whose values were smaller than the electron’s, and his efforts to advance his claims had long ago exhausted the patience of the physics community. Now approaching seventy, Ehrenhaft was still seeking an audience, and approached the podium demanding to be heard.

  A young physicist named Herbert Goldstein – who told me the story – was sitting next to his mentor and former colleague from the MIT Radiation Laboratory, Arnold Siegert. ‘Pais’s theory is far crazier than Ehrenhaft’s’, Goldstein asked Siegert. ‘Why do we call Pais a physicist and Ehrenhaft a nut?’

  Siegert thought a moment. ‘Because’, he said firmly, ‘Ehrenhaft believes his theory.’

  The strength of Ehrenhaft’s conviction, Siegert meant, had interfered with the normally playful attitude that scientists require, an ability to risk and respond in carrying forward their dissatisfactions. (Conviction, Nietzsche said, is a greater enemy of truth than lies.) What makes a crackpot is not simply our prejudices, nor necessarily the claim, but our recognition of the disruptive effects of the author’s conviction. For conviction tends to wipe out not only the dissatisfaction but also the playfulness, the combination of which produces such a powerful driving force in science.

  8

  The Golden Egg:

  EINSTEIN’S EQUATION FOR GENERAL RELATIVITY

  Gim = −κ(Tim − ½gimT)

  DESCRIPTION: Space-time tells matter how to move, matter tells space-time how to curve.

  DISCOVERER: Albert Einstein

  DATE: 1915

  [O]ne of the greatest achievements of human thought…

  – J. J. Thomson

  The theory appeared to me then, and it still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill. But its connections with experience were slender. It appealed to me like a great work of art to be enjoyed and admired from a distance.

  – Max Born

  Einstein’s field equation of general relativity, expressing the curvature of space-time, is not as instantly recognizable as his formula of special relativity, expressing the interchangeability of mass and energy. But it, too, vaulted to public awareness under dramatic circumstances.

  The date was November 6, 1919, the place a chamber of the Royal Society building in London. The room resembled a small church interior, with rows of pews on either side of a central isle and a row of columns lining the walls. In the back was an anteroom for an overflow crowd. On one wall hung a portrait of the Royal Society’s most famous member, Sir Isaac Newton.

  The event was a joint meeting of the members of the Royal Society of London and the Royal Astronomical Society. The audience had come to hear reports of data collected during an eclipse that had taken place some 6 months earlier, on May 29. Scientists had taken photographs of the stars during the eclipse, trying to see whether the starlight had been bent
as it passed around the sun. Some of those in attendance had alerted the press to the importance of the occasion. The Times of London, in an extensive story, declared, ‘The greatest possible interest had been aroused in scientific circles by the hope that rival theories of a fundamental physical problem would be put to the test, and there was a very large attendance of astronomers and physicists.’1 One of the rival theories was Albert Einstein’s, whose idea that space was ‘curved’ was part of his equation of general relativity; this idea in turn implied the bending of starlight around the sun. British physicist J. J. Thomson, a grand old man of British physics who had discovered the electron, presided. ‘This is the most important result obtained in connection with the theory of gravitation since Newton’s day’, Thomson announced, and described the result as ‘one of the highest achievements of human thought.’2 Philosopher Alfred North Whitehead, an audience member, later wrote:

  The whole atmosphere of tense interest was exactly that of the Greek drama: we were the chorus commenting on the decree of destiny as disclosed in the development of a supreme incident. There was dramatic quality in the very staging: – the traditional ceremonial, and in the background the picture of Newton to remind us that the greatest of scientific generalizations was now, after more than two centuries, to receive its first modification. Nor was the personal interest wanting: a great adventure in thought had at length come safe to shore.3

  Special relativity, too, had been an adventure, but one in which many members of the scientific community – including FitzGerald, Lorentz, Poincaré, and many others puzzled by the contradiction between Newton and Maxwell – had participated. General relativity was different. Einstein embarked on it virtually alone. For 7 years his pursuit of the topic had followed a mazelike path, in which his course was helped by doors that unexpectedly opened, or was blocked by dead ends that forced him to retrace steps and undo years of work. Only at the very end did others realize that a journey of drama and extraordinary significance had taken place.

  Solo Journey

  The path to general relativity, too, was initiated by a thought experiment, which occurred to Einstein sometime in November 1907. He once wrote:

  I was sitting in a chair in the patent office at Bern when all of a sudden a thought occurred to me: If a person falls freely he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.4

  He referred to this insight as ‘the happiest thought of my life.’5

  This thought experiment was an ambitious extension of the thought experiments associated with special relativity. Those had involved situations of uniform motion, and the point was the inability to tell whether they were moving: everything in that smooth and cushy railroad car, for instance, acts exactly the same when it is in motion as when it is at rest. The new thought experiment involved accelerated motion: imagine that the railroad car is lifted high, is released, and falls to the ground. Einstein realized that you could not tell whether you were falling in a gravitational field, or out in empty space in the absence of any gravitational field. If you let go of an object – keys, a ball, coins – they would stay put as if at rest. Einstein called it his ‘happiest thought’ because he realized that that lack of an ability to distinguish the two situations was important.

  Today, thanks to countless news clips of weightless conditions in spacecraft, in high-altitude planes, and so forth, we find this thought much less startling than did the 28-year-old Einstein. The idea that free fall in a gravitational field was indistinguishable from the absence of force suggested to him that the presence of gravitation was identical to the presence of acceleration – that is, force – itself. For you could reverse the situation: If the railroad car were on the ground, could you tell whether you were in a gravitational field or being accelerated? These thoughts focused Einstein’s dissatisfaction, and he set out to explore the implications of what not being able to tell the difference might mean.

  This thought experiment involved a different sort of dissatisfaction than the one that had led to special relativity. Instead of bringing to a head a contradiction born from trying to combine two complete systems – Newton’s and Maxwell’s – this one was produced by an apparent identity between two things thought to be vastly different. It is as if, out of tradition and habit, we were used to dealing with two different governmental offices for different things, discovered that their actions were identical, and then had to figure out how this could possibly be true. The two things that seemed different were inertial mass and gravitational mass. According to Newton, gravitation was a special kind of force that tugged at heavier objects more strongly than at lighter objects – but by sheer coincidence, the inertial mass of the heavier ones made them resist the tug by just the right amount so that everything accelerated at the same rate. Einstein thought: let’s assume that this is no coincidence, and see what happens.

  Another way of putting this is to say that he was probing for an even deeper covariance than the one he had discovered in his earlier theory of relativity. Covariance is simply part of what we mean by objectivity: to say that something is a real part of the world is to say that it looks different from different ‘angles’ – including not only things like lighting conditions but also spatial locations and speeds – in a way that you can spell out precisely through ‘transformations.’ Covariance thus seems to draw a difference between how a thing appears to us, and what it really is. In Einstein’s 1905 theory, he had discovered transformations that would make the same description apply no matter how the object moved – so long as it was moving uniformly – even though the object would ‘look’ different if it were traveling at close to the speed of light. If something did not behave that way – did not look different if it were traveling at close to the speed of light – we were entitled to say it was not a real object, not a part of our world. Now Einstein was trying to extend covariance to accelerated systems: How would the description of a ‘real’ object change if the reference frame were accelerated? This would lead him to such a radical reformulation of his work that it would be called ‘general relativity’ to distinguish it from its precursor, ‘special relativity.’

  First Step: The Principle of Equivalence (1907)

  Einstein’s thought experiment inaugurating the solo journey had occurred to him while working on the paper summarizing his theory of relativity for the Jahrbuch der Radioaktivität. He added a final, ten-page section to the paper with a ‘novel consideration.’ Up to now, he wrote, he has applied the principle of relativity to uniformly moving systems, but: ‘Is it conceivable that the principle of relativity also applies to systems that are accelerated relative to each other?’ What if there were two systems, one accelerated at a certain rate and the other at rest in a uniform gravitational field exerting the same force? As far as we know, he says, the physical laws of the two systems are the same, and ‘we shall therefore assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.’6 Einstein wrote that he did not know whether this ‘equivalence principle’ is true – he just means to see what would happen if it were.

  The next few pages contain many key ingredients of what would become general relativity. He draws surprising conclusions, such as that people higher in a gravitational field will perceive clocks farther down (toward the source) as moving slower, and that gravitational fields affect the path of light – though ‘unfortunately, the effect of the terrestrial gravitational field is so small…that there is no prospect of a comparison of the results of the theory with experience.’7 Still, he hoped for testable predictions.

  Einstein, for instance, had his eye on the longstanding problem of Mercury’s orbit. In the mid-nineteenth century, astronomers had noted that the point at which it made its closest pass by the sun – called the perihelion – was not staying put but slowly moving around the orbit. At first astronomers assumed that this was due to the gravitational influen
ce of other planets, but when these influences were all scrupulously added in, a tiny amount – 43 seconds of arc per century – still could not be accounted for. The discrepancy was not small enough to ignore, but not big enough to cast doubt on the Newtonian system within which it appeared as a problem. Fixes were proposed. Some involved postulating phenomena such as a hidden planet named Vulcan or a lenslike layer of nebulous matter near the sun, but these could not be found. Another approach was to tinker with Newton by introducing slight modifications of the inverse square law, but these had undesirable side effects. For half a century the precession of Mercury’s orbit had been one of the great unsolved mysteries of astronomy.

  Einstein realized that this final piece of the precession was something his theory might explain from first principles thanks to certain terms in his theory that were absent from Newton’s, so that astronomers would not have to hang their hopes for a predictable universe on undiscovered phenomena, tinkering with constants, or adjusting formulas. ‘At the moment’, he wrote his friend Habicht on Christmas Eve 1907, ‘I am working on a relativistic analysis of the law of gravitation by means of which I hope to explain the still unexplained secular changes in the perihelion of Mercury.’8 But for several years these changes remained mysterious.