A Brief Guide to the Great Equations

by Robert Crease

  Heaviside’s version of Maxwell’s equations were quickly and gratefully adopted by prominent electromagnetic researchers, including Hertz, and the entire scientific community converted by the 1890s. The equations have remained virtually the same ever since; the version at the beginning of the chapter is taken from the standard textbook Classical Electrodynamics by J. D. Jackson.

  Fittingly, Heaviside’s achievement in revising Maxwell’s was once captured in an analogy. Reviewing Heaviside’s work, FitzGerald compared Maxwell to a general who conquered a new territory but had not the time to find the best roads or make a systematic map. ‘This has been reserved for Oliver Heaviside to do’, he wrote. ‘Maxwell’s treatise is cumbered with the débris of his brilliant lines of assault, of his entrenched camps, of his battles. Oliver Heaviside has cleared those away, has opened up a direct route, has made a broad road, and has explored a considerable tract of country.’24

  But Maxwell was a strange kind of general who worked in a strange kind of terrain. The territory he conquered was so potent and extensive that his work, as Feynman noted, would have a far greater impact on human nature than that of any group of generals.

  Interlude

  OVERCOMING ANOSOGNOSIA; OR RESTORING THE VITALITY OF THE HUMANITIES

  Simon Schama’s book History of Britain, at 1,500 pages, is a solid history of that country, and has been made the basis for a multipart documentary film. Yet the book contains no mention of James Clerk Maxwell, nor any mention of the role that this scientist played in laying the foundation for electrification, light, heat, communication, and the electronics revolution of the twentieth century, in Britain or elsewhere. Schama’s book omits any references to the contributions made by British scientists and engineers to transforming Britain and the world.

  The neglect of science, indeed, is common in history books – most disturbingly, even in books that profess to care about the masses, and oppressed and underprivileged peoples. Since it was first published in 1980, for instance, A People’s History of the United States, by Howard Zinn, has sold over a million copies and become one of the most influential works of history in the U.S. A popular textbook in schools and colleges, it claims to focus on ‘hidden episodes of the past when, even if in brief flashes, people showed their ability to resist, to join together, occasionally to win.’

  However, Zinn’s book makes no mention of people resisting, joining together, and winning when it comes to science. It says nothing, for instance, of the struggles to reduce childhood mortality, to increase life expectancy, or to develop systems of mass transportation. There is no mention of Norman Borlaug, who won the 1970 Nobel Peace Prize for leading the ‘green revolution’, and who helped end hunger for millions of people. Another no-show is the microbiologist Maurice Hilleman, whose vaccines saved more lives than were lost in all the wars to which Zinn devotes chapters.

  Mass electrification fails to feature in Zinn’s book, although the unit costs of electricity are discussed in the context of a program to give ‘enough help to the lower classes’ to prevent them rebelling. Steam power is not covered, nor is the internal-combustion engine, although railroads are discussed in relation to racial segregation, unions, strikes, and methods of exploiting American Indians.

  Zinn, in short, considers scientific changes inconsequential to ‘the people.’ History, for him, is a grand pageant of ideologies; if science is at all significant in that pageant it is perhaps only in forging the weapons that the ideological partisans use to beat up each other.

  The omission does not necessarily make the book defective as history. As Zinn notes, historians cannot avoid selecting and emphasizing some facts rather than others, although they have a duty to avoid promoting ideological interests, knowingly or not. But Zinn’s omissions do make the book defective as an account of ‘the people.’ The conquest of dreaded and once-common epidemic diseases, such as polio and encephalitis, have fundamentally affected how all of us view life and death. Developments in astronomy and the discovery of evolution have affected our sense of time and space, and our place in nature. These events all took place within the timeframe of Zinn’s book. Although some of these developments were pioneered by non-Americans, they profoundly altered how human beings seek answers to the questions of what we know, should do, and can hope for.

  Schama and Zinn are not the only ones to ignore the impact of science. Many authors of contemporary fiction fill their books with characters who are nothing more than superannuated children, seemingly unaffected by technological training and devices. Some writers – like Jonathan Franzen, Ian McEwan, Neal Stephenson, and David Foster Wallace – do present protagonists who are interested in and influenced by their technological surroundings. But these writers can be severely criticized by reviewers for their efforts.

  Commenting on McEwan’s Saturday, for instance, John Banville roasted the author for being ‘wearingly insistent on displaying his technical knowledge’ and complains of ‘big words in this book.’ The book indeed has some big words. However, the training that turns people into technically literate professionals not only accustoms them to using big words, but also affects how they speak and act. Technically competent people often delight in their technical competence, and wield this competence when interacting with the world. This is precisely what McEwan so ably captures.

  Dismissing the effect of science on modern life has nothing to do with the ‘two cultures.’ Rather, it shows a blind spot in the work of some writers and scholars whose duty it is to become aware of the world around them. It is more serious than amnesia. We can name the condition with one of the ‘big words’ that McEwan’s protagonist uses in Saturday. It is ‘anosognosia’ – a medical term (derived from a combination of the Greek words agnosia, or ‘without knowledge’, and nosos, or disease) that means a lack of awareness of one’s own diseased condition; that is, not knowing that one is diseased.

  What are the causes of anosognosia? I count four contributing factors.

  One is drama: scientific and technological change tends to lack the exciting settings of other historical turning points. It is not generally heralded by bloody battlefields or by clashes of titanic personalities, and unfolds in a way that makes it difficult to dramatize differences. A second is the hope among even so-called enlightened and progressive scholars that we can reinvent ourselves and remake the world, Marxian-style, achieving liberation at a revolutionary stroke; admitting dependence on science and technology serves to dampen such hopes. A third is fear of specialized knowledge, knowledge that one might take extra training to acquire.

  Finally, and most importantly, scholars in the humanities often see themselves as having a critical function – they see themselves as asking the important questions that help humanity navigate the world’s dangers. But if the fate of ‘the people’ is as tied up with science and technology as it is with ideologies – with who is exploiting whom – this leading role is blunted, or at least shared. For those who identify the humanities with such a critical function, this might even seem threatening. Far safer for its practitioners to circle the wagons, dwelling on what is distinctive about the humanities rather than what is possible! This is what makes so many humanities programs both defendable and lifeless. Moreover, such wagon circling is self-interest in disguise; thus, an ideology – a belief structure lacking empirical support – itself.

  Overcoming anosognosia requires admitting that a truer picture of humanity may be less dramatic than we hope, curbing our fascination with shortcuts to liberation, and accepting that humanity’s important questions are addressed by a variety of disciplines. This will strengthen, not threaten, the humanities. For only when the humanities couple their inquiries into human dimensions and possibilities with an awareness of what science has disclosed of the dimensions and possibilities of the world will the humanities most effectively be able to provide answers to the questions of what we know, should do, and can hope for.

  7

  Celebrity Equation:

 
E = mc2

  DESCRIPTION: Energy and mass can be converted into one another, with the amount of energy being equal to the mass multipled by the speed of light squared.

  DISCOVERER: Albert Einstein

  DATE: 1905

  A while ago I was reading an interview with the actress Cameron Diaz in a movie magazine. At the end the interviewer asked her if there was anything she wanted to know, and she said she’d like to know what E = mc2 really means. They both laughed, then Diaz mumbled that she’d meant it, and then the interview ended.

  – David Bodanis, E = mc2: A Biography of the World’s Most Famous Equation

  E = mc2 is the most famous equation of all time. It has made the cover of Time magazine. It has been the subject of a ‘biography’ that treated the equation as though it were a person. It is the title of a play by Hallie Flanagan, the woman who headed the Federal Theatre Project during the Depression. The Dalai Lama calls it ‘the only scientific equation I know.’1 Poems and pop songs have been written about it; those of a certain age may remember the hit single ‘Einstein A Go-Go’, by 1980s electronic pop band Landscape, the lyrics of which went ‘You’d better watch out, you’d better beware, coz Albert says that E equals mc squared.’ More recently, singer Mariah Carey put out an album entitled, E = MC2, with the right-hand term alluding to her initials. During the so-called science wars of the 1990s, debate raged over the French feminist philosopher Luce Irigaray’s assertion that E = mc2 is a ‘sexed equation’ because it privileges the speed of light.2 The equation has turned up on postage stamps of various lands, in movies (School of Rock), popular fiction with scientific pretensions (Dan Brown’s Angels & Demons), and numerous cartoons and video games.

  The physicist Stephen Hawking was once warned not to include any equations in his writings for a general audience because, or so he was informed, every equation would halve the number of readers. As a result, he was determined not to use any equations in his book, A Brief History of Time. But E = mc2 appears in the book, and in multiple places. This did not dent sales, and it went on to become one of the best-selling science books for a general audience of all time.

  All this might make us wonder whether E = mc2 is not a real equation at all but rather a celebrity. A celebrity is someone everybody knows of, but not about. Similarly, everybody recognizes this equation, and is sure that it is important, but it’s never clear exactly why. We know plenty of gossip about it but still always feel we are seeing it from the outside. We wonder how much work it really does. The status of E = mc2, like that of a celebrity, seems manufactured by some mysterious social process.

  Yet, in the end, celebrities are just human beings, and E = mc2 is just another equation. Like other equations, it sprang from dissatisfaction with the way things were fitting together, its first appearance was different from the form in which we know it today, it reorchestrated the way human beings looked at the world, and it had unexpected consequences.

  How, then, did this equation get to be a celebrity?

  The Collision Between Newton and Maxwell

  Equations can be born from several different kinds of dissatisfactions. Some spring from a scientist’s sense that a confusing heap of experimental data can be better organized. Others arise from the feeling that a theory is too complicated and can probably be simplified, or that its parts are not fitting together properly. Still other dissatisfactions arise from mismatches between a theory’s predictions and experimental results.

  The equation E = mc2 resulted from a special and rare case of dissatisfaction felt by many physicists at the end of the nineteenth century and the beginning of the twentieth. The dissatisfaction was created by a troubling experimental result that highlighted an inconsistency between two great, comprehensive, and venerable scientific systems: Newton’s and Maxwell’s. More exactly, the result highlighted an inconsistency between two principles – the principle of the relativity of motion, and the principle of the constancy of the speed of light – each basic to one system.

  The inconsistency involved an idea called invariance. In its loosest sense, invariance simply means that something can appear in two different ways but nonetheless be the same thing. Two people standing in different parts of a room, for instance, can see a chair as to the right or to the left of the television – but once we take the difference in positions into account, it becomes clear not only that they are seeing the same chair, but also how and why that chair appears differently to each. If we cannot account for the difference in appearance, then one or both of these people is hallucinating or seeing some sort of illusion. Real things, we might say, are supposed to look different from different perspectives. Reality therefore necessarily involves a difference between how something appears to us, and what it really is. We can put this point another way, in terms of the difference between local effects and global properties. When I see an object I see only one profile of it – one that changes if I move, if the light changes, and so forth. As I change my position, so does this ‘local’ effect. Yet all the time I am seeing the same object. Invariance therefore involves understanding unity as it shows itself in changing appearances. Philosophers call this the noetic-noematic correlation; physicists call it invariance under transformation, or covariance. Covariance is simply part of the definition of objectivity; to say that something is a real part of the world is to say that it looks different from different perspectives, though the descriptions flow together in an orderly way when described by the right set of transformations.

  Newton’s mechanics supposed the existence of an absolute time and an absolute space as the arena in or stage on which events happen, and that there is no privileged time or space on that stage. This is called translation invariance, and all it means is that if we move about in time or space the laws are still the same. But Newtonian mechanics implies a further kind of invariance, that according to a ‘principle of relativity of motion’, there is no privileged movement, whether in motion or at rest. The laws of physics are the same for anyone moving at constant speed regardless of direction, no matter how fast or slow. This is a familiar experience. So long as a train, say, is traveling smoothly and doesn’t jostle, anything we do – drink a glass of water, play cards or handball, dance – happens exactly the same as if the train were at rest in the station. The water stays in the glass and doesn’t slosh, the ball bounces at the equivalent point on the floor, and the dancer confidently executes the same gestures and winds up in the same spot as if the train were still. We could perform no experiment to tell how fast the train was moving, or even whether it was moving. Even people on a second train at rest in the station down the track would see the same laws of physics in play on ours, once the difference in speed between the trains was taken into account. And that difference in speed is a matter of simple addition and subtraction.

  Scientists would call such a train, from whose position we describe events, a reference frame, and one moving at uniform speed an inertial reference frame. They call transformations the equations used to change the mathematical description of an event – its x, y, and z of position and its time t – from one reference frame into another. They call the equations that connect the properties of a description in one inertial reference frame with those of another Galilean transformations, for these express a principle of relativity of motion already present before Newton in Galileo’s mechanics, in the latter’s thought experiments involving dropping cannonballs from the masts of sailing ships. The Galilean transformations are quite simple. On board the moving train, for instance, the only thing about events that changes is their distance down the track (let’s make that the x-axis). Any x position on that train, call it x’, differs from the x position for an observer on the ground by the distance the train has traveled in a time t: x’ = x − vt. All the other coordinates – y and z – remain the same, and things continue to happen at the same time t.

  A physicist’s definition of reality and objectivity depends on Galilean transformations. A ‘real’ thing or event is on
e with the same physical description in different inertial frameworks, once you use the appropriate transformations to take the differences in speeds and directions into account. The notion of reality requires drawing a difference between how something appears to us, and how we describe it; the variability to observation is built into the objectivity of the object that I see. In developing the notion of transformations, scientists were merely elaborating the conditions of objectivity – of what is the same regardless of which inertial frame it is seen from.

  Thus the principle of the relativity of motion was at the core of Newtonian mechanics. But according to a ‘principle of the constancy of the velocity of light’ central to Maxwellian mechanics, light introduces a new element into this neat picture. Light acts more like sound. Sound always travels at the same speed (about 1,100 feet a second in air), regardless of how fast its source travels. The reason has to do with the properties of the medium (air molecules, say) that propagate the sound waves that make it impossible to push sound waves any faster than a certain speed. According to Maxwell’s equations, light also always travels at the same speed (about 186,000 miles a second) regardless of the speed of the source. Physicists assumed this stemmed from the fact that light moves in a medium called ether, whose properties governed how fast light could travel. If so, this principle implied that there was a favored inertial reference frame in the ‘stage’ of absolute space and time, provided by the ether. In moving around the sun, the earth moves through the ether, and while it might ‘drag’ some small amount with it, its speed with respect to the ether could be detected by measuring the speed of light in different directions. For the ether moves the light, by an amount that involves the Pythagorean theorem.