E=mc2

by David Bodanis

  It was the promise of La Mettrie's L'Homme Machine; the same rush of optimism led to Lavoisier himself suggesting that in the future it would be possible to look inside the brain and see "the efforts required of someone giving a speech or . . . the mechanical energy spent by . . . a scholar who is writing, or a musician who is composing"— a pretty near description of our modern brain scans. The quote is from Lavoisier's Collected Works, vol. II, p. 697.

  This is what Einstein was taught . . . different topics: The division of reality into two parts is something of a default operation of the human mind, seen in the ease with which we create the categories of friend or foe, right or wrong, x or not-x. The particular division bequeathed by Lavoisier, Faraday, and their colleagues was even more compelling, for when one of the divisions is material and physical, and the other is invisible yet still powerful, it's the ancient dichotomy of the body versus the soul that slips into our mind.

  Many other thinkers have been guided in their work by that distinction. Alan Turing seems to have been led by the body-soul division when he came up with his distinction between software and hardware; most users of computers easily think that way, for we can all immediately grasp the notion of a "dead" physical substrate, powered up by a "live" controlling power. The soul-body distinction permeates our world: it's Don Quixote versus Sancho Panza; the cerebral Spock versus the stolid Enterprise; the contrast between the whispered encouraging voice-overs in the running-shoe ads, and the physical body on the screen.

  But these lulling categories only make a suggestive division, not a proof. A young man such as Einstein, always keen to understand the foundation of a field for himself, could readily see that his professors had simply made an induction from a very incomplete data set.

  There are many accounts of how lurking categories pull our thoughts along, as with George Lakoff and Mark Johnson's Metaphors We Live By (Chicago: University of Chicago Press, 1980), or Kedourie's excellent writings on nationalism, yet for some reason this author is especially pulled toward the approach in Bodanis's Web of Words: The Ideas Behind Politics (London: Macmillan, 1988).

  . . . when members . . . in Florence: Galileo's proposal was in the First Day section of his Two New Sciences. The test was over twenty years later, probably around 1660, by the Accademia del Cimento in Florence. Their results are on page 158 of a book with the sort of vivid identifying location publishers no longer have: Essayes of Natural Experiments, made in the Academie del Cimento; Englished by Richard Waller, Fellow of the Royal Society, London. Printed for Benjamin Alsop at the Angel and Bible in the Poultrey, over-against the Church, 1634.

  5. c Is for celeritas

  The effort might be exhausting . . . embarrassing public exposure: Clearly I'm being slightly tongue in cheek about Cassini. From the available evidence, he might have been an insecure man, but as a newcomer to France he had a great deal to be insecure about: At first his appointment was only temporary, and he'd been warned not to try speaking French, but then he'd been told he had to learn French, for the Academy of Sciences couldn't be sullied by being exposed to Latin, let alone his native Italian. It's touching to read his own account of fearfully concentrating to try to develop the crucial language ability—and then his pride when complimented by the king on how much he'd progressed in just a few months. He also had a personal reason to resent Roemer. For Cassini himself had established his reputation by publicly proclaiming, back in July 1665, improved predictions for the transit of Jupiter's satellites. His predictions had been proven right in August and September of that year; his doubters had been humbled; the grand position in Paris had been his reward. He would not have appreciated Roemer trying to use the same twist against him.

  There was also something more than mere pique in his critiques of Roemer for being so confident of the Jupiter observations. Cassini wrote a long poem, "Frammenti di Cosmografia," expressing his humility before the grandeur of space, and his belief that only an unjustified false pride lets humans, isolated on one inconsequential planet, presume they can accurately measure everything that occurs. Even before Roemer arrived, Cassini had applied first-order approximations to try to get rid of Io's anomalies; he was being sincere when he said it would be overhasty to insist on any one new interpretation. The poem and fragmentary autobiography are in Memoires pour Servir a THistoire des Sciences et a Celle de L'Observatoire Royal de Paris (Paris, 1810), compiled by Cassini's great-grandson, also named Jean-Dominique; see especially pages 292 and 321.

  once the underlying mathematics . . . could describe: Maxwell's later success has meant that other researchers of the time have tended to be overlooked. Weber at Gottingen is an especially interesting intermediate figure, for he too computed the speed of light in his efforts to link electricity and magnetism; but as it was masked by an extra factor of v2, he didn't recognize what he'd found, and left it aside. Weber's story is nicely described in M. Norton Wise's article "German Conceptions of Force . . . ," pp. 269-307 in Conceptions of Ether: Studies in the History of Ether Theories 1740-1900, ed. G. N. Cantor and M. J. S. Hodge (Cambridge: Cambridge University Press, 1981). Weber's caution is similar to that of the early Ampere; his hybrid equations—extending almost into the Maxwellian world of fields, but not quite making it—resemble a battleship forlornly loaded with antiaircraft guns.

  "Aye, I suppose . . .": I fear this one is apocryphal too. All that's certain is that Maxwell liked making fun of his own literalness; also that he did experiment with staying up extremely late as a Cambridge student, to the astonishment and bemusement of his fellow students. See e.g., Goldman, Demon in the Aether, p. 62.

  "They never understood me, but . . .": Ivan Tolstoy, James Clerk Maxwell: A Biography (Edinburgh: Canongate, 1981), p. 20.

  "As I proceeded with the study . . . " : Treatise on Electricity and Magnetism, James Clerk Maxwell (Oxford: Clarendon Press, 1873); Maxwell's preface to his first edition, p. x.

  When a light beam starts going forward . . . : Ordinary language is inherently inexact here, for what we're really describing are the properties of electrical and magnetic fields, a specification of what "would" happen at any given location. The subjunctive mood in grammar, and especially the subjunctive conditional, comes close to matching the idea: you might not be able to say what's happening at a particular street corner in a bad neighborhood right now, but you could tell what's liable to happen if a Rolex-laden tourist sauntered past. In the case of physics, think of the swirling curves of iron filings you can see around a bar magnet. Now take away those filings, and instead, where each one was, write down a number or a group of numbers that tell you how any filing placed there is likely to respond.

  To someone who hadn't seen how you started, there would only be a cold list of numbers. But to someone who knew about the whirling power a magnet can have on iron filings, your list is a vivid description—and to Maxwell and Faraday, with their religious beliefs, your list would be a direct readout of the holy power that created that field in the first place.

  The electricity and magnetism . . . "mutual embrace": It doesn't take much power to send out a wave. Play a piano key and the string simply vibrates back and forth, otherwise unmoving, while it's the pattern of those vibrations that moves along and carries the sound. There can be hundreds of gallons of air between two people standing a few yards apart in a corridor, yet they don't have to blow all that air forward to call out a hello. Each need only puff the tiny amount that can be shot up from their throats, and that will start a rippling compression wave that gets the job done.

  Light and electromagnetic waves generally can be as easy. Switch on the ignition of your car, and your spark plug sends out an electromagnetic wave that has several frequencies that will pass through the metal around it and make it streak to the orbit of the moon within two seconds of your hearing the engine catch; the wave continues outward and will have reached the distance of Jupiter a few hours later.

  Maxwell's equations: Maxwell's work was a tremendous achievement—and would ha
ve been even more tremendous if he had ever written the four equations that bear his name. But he didn't. It's not just a matter of notation changing, for even the details Hertz later looked at, which led to the realization that radio waves could be transmitted and received like light waves, weren't in the equations Maxwell conceived.

  The story of how Maxwell's equations were finally completed, by a group centered around three physicists in England and Ireland in the two decades after his death, is thoroughly recounted in Bruce J. Hunt's The Maxwellians (Ithaca, N.Y.: Cornell University Press, 1991).

  . . . nothing can go faster: Or more precisely, nothing which started slower than the speed of light can end up going faster. What if there were particles—or perhaps an entire parallel world—located permanently on the "other side" of the light speed barrier? It sounds like science fiction, but physicists have learned to keep an open mind. (These postulated superluminal particles were labeled tachyons by Gerald Feinberg.) Another proviso is that we're discussing the speed of light in a vacuum, and light's velocity is lower in other substances. This is why diamonds sparkle: light skimming above the surface will go faster than the light that has dipped inside.

  There are more significant exceptions—due to the effects of varying space-time curvatures on relative speeds; also there can be effects due to the role of negative energy, and there have also been intriguing results about pulses of light that exceed our velocity "c" (albeit in ways that keep added information from being transferred in the process). These, however, take us beyond the technical level of this book. I suspect future scientists will either look back at us in amazement that we ever took this seriously—or that we took so long to realize that this was the way to open up the first Disneyland in Andromeda.

  . . . the solid mass of the shuttle starts to grow: None of our ordinary words apply well to this realm, and the term swelling has to be thought of as only a metaphor. The shuttle—or a proton, or any other object—doesn't get fatter in all directions. Rather, this is where the seemingly fussy distinction between conservation of matter and conservation of mass from the Lavoisier chapter comes into its own. With mass defined as the property of resistance to an acceleration— which is what we reflexively try to assess whenever we heft an object, to estimate its weight—then it's possible for mass to increase without matter swelling outward. So long as there is an increased resistance to an applied acceleration, the requirements are fulfilled.

  In the slow speeds and ordinary realms we're used to, the amount of mass increase will not be enough to notice— this is why Einstein's predictions were so startling—but as an object moves away from us at rates approaching the speed of light, the effect gets clearer. The predictions are very precise.

  The way to compute how much a given mass will increase is to take its velocity, square that, divide it by the square of the speed of light, subtract the result from 1, take the square root of that, then take the inverse, and then multiply that final result by the mass you're interested in. It's easier in shorthand: if a mass is traveling at the speed of "v," then to work out how much it will appear to swell, you multiply the original mass "m" by .

  It usually helps in getting a feel for an equation to "tweak" it by examining its properties at various extreme values. If v is much less than c—i.e., if the shuttle is moving slowly—then (1—v2/c2) is almost the same as one, for v2/c2 will be so slight. It doesn't matter if you then go on to take its square root and the inverse: you'll still get a number extremely close to 1. For the actual space shuttles launched from Florida, maximum v is about 18,000 mph. That's such a small percentage of the speed of light that its mass only expands by much less than a thousandth of a percent, even when it's screaming out of the atmosphere at top speed. But if a shuttle or anything else is moving really fast, so that v is close to c, then (1—v2/c2) is close to zero. That means the square root is small also, and when you divide 1 by a small fraction, the result is huge. Watch an object streak past at 99 percent of the speed of light—and its mass will to you have increased several times.

  There's a temptation to think this is just some quirk, and that although we might be confused in our measurements, a moving object won't "really" be more massive in any way that counts. But the magnets around the accelerator rings at CERN really do have to raise their power this much to keep a proton speeding at that rate on track, for otherwise the momentum of its increased mass will send it into the accelerator's walls. At 90 percent of the speed of light, the power needed to control 2.5 times more mass will have to be pumped in to avoid a skid out of the flight path and a crash. If the speed goes up to 99.9997 percent of the speed of light, then gives a mass increase of 430 times—whence the problems of the accelerators at CERN, having to find some way of drawing their extra energy, without disturbing the good citizens of Geneva.

  Simply to assert, however, that the expression gives a rule you must follow, would put us in the same category as the obedient rule-following instructors Einstein had so resented. At davidbodanis.com we'll explore why it's true.

  Energy that's pumped [in] . . . will turn into extra mass: The shuttle example is only heuristic; we'll see as the book goes on that energy is mass: the unified thing called "mass-energy" just happens to take on different aspects, depending on how we're viewing it. The restrictions of our fragile bodies means that we hardly ever change our speeds substantially, and so we view mass from a highly "skewed" angle. That distortion is the reason the "released" energy seems to be so high. (A significant proviso, however, is that this equivalence between mass and energy only holds when an object is viewed from the one particular view in which it's at rest. This is especially important in general relativity, for an object's gravitational attraction stems from its total energy, and not just from its rest mass. Page 199 of the main text touches on this reasoning in connection with black holes; the point is developed at more length on our Web site.

  6. 2

  "Supervising the workers": Voltaire et la Societe Francaise au XVIII e Siecle: Volume I, Lajeunesse de Voltaire, by Gustave Desnoiresterres (Paris: Dider et Cie, 1867), p. 345.

  . . . a new concept in the air . . . : Arouet didn't need Newton's work to make him aware of France's faults. If anything, it wasn't abstract ideas, but seeing England's working parliament—and the tradition of at least semi-independent judges and citizens' rights—which helped in showing up the lackings in France. But it was sweet to have the backing of the world's most-renowned analytic system in these critiques. See Voltaire's English Letters.

  Newton had created a system of laws . . . : Curiously enough, seeing an apple fall does seem to have helped Newton take the final step. William Stukeley recorded the elderly Newton reminiscing in his account published two centuries later as Memoirs of Sir Isaac Newton's Life (London: Taylor & Francis, 1936), pp. 19-20.

  After dinner, the weather being warm, we went into the garden [of Newton's last residence, in London's Kensington] and drank thea, under the shade of some appletrees, only he and myself Amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion 'd by the fall of an apple, as he sat in a contemplative mood. Why should that apple always descend. . . constantly to the earths centre? Assuredly, the reason is, that. . . there must be a drawing power in matter . . . like that we here call gravity, which extends its self thro' the universe.

  That was how Newton could be so sure that the forces on Earth are the same as those operating up in space. It's easy enough to measure the speed at which an object on Earth falls. In a single second, a dropped apple—or any other object—will fall about 16 feet. But how to measure the speed at which the moon "falls" to compare with that?

  The way to do this is to recognize that the moon constantly falls downward, at least a little bit. (If it didn't fall, and only moved in a perfectly straight line, then it would soon shoot away from our planet.) The amount that it "falls" is just enough to keep it curving around the earth. Knowing the lengt
h of its orbit, and the amount of time it takes to make one circuit, one can conclude that the moon is tumbling earthward at about 1/20 inch every second.

  At first that seems like a failure of Newton's guess. If there's some force making rocks fall 16 feet in one second down on Earth, then one might think that only a very different sort of force, out in distant space, would make giant rocks such as the moon fall a scant 1/20 inch in every second. Even taking into account the greater distance of the moon, it doesn't seem to work. The Earth is about 8,000 miles thick, so Newton, as well as his mother's apple trees, existed about 4,000 miles above the center of the Earth. The moon is in orbit about 240,000 miles from the center of the Earth, i.e., about 60 times farther. Even if you weakened a rock's fall by 60 times, it would still not flutter downward as slowly as the moon. (1/60 of 16 feet is about 3 inches—still far more than the scant 1/20th inch the moon falls each second.)

  But what if you imagine a force that weakened by 60 times 60 times as it stretched up and away from our planet? It's an interesting idea—that gravity acts in accord with the square of the distance between objects—but how could you verify such a thing? You would have to prove somehow that gravity produces a force 3,600 times (60 x 60) stronger on Earth than out in space. No one in the seventeenth century— not even from Cambridge—could rocket up to the moon and compare the force of gravity there with what it is on Earth. But no one needed to. The power of equations is immense. Newton had the answer all along. "Why should that apple," he'd asked, "always descend . . . constantly to the earths centre?" In one second on the Earth's surface, a rock or an apple or even an astonished Cambridge don will fall 16 feet. But the moon in that time will fall just 120th inch. Divide the two numbers, and you have the ratio: how much stronger gravity's tugging power is on the Earth's surface than up on the moon.