The Perfect Theory (13 page)

 

If the Soviet philosophers didn't approve of the mathematical idealism that had gone into the general theory of relativity, they certainly rejected Einstein's later work, for by the time Einstein arrived in Princeton, he had become obsessed with finding a grand unified theory. His general theory of relativity was still dear to his heart, but he wanted to do something bigger and better. He wanted to subsume general relativity into a theory that could bring
all
of fundamental physics into one simple framework. Einstein hoped to show how not only gravity but also electricity and magnetism, and possibly even some of the strange effects that were attributed to the quantum, could arise as the geometry of spacetime. But unlike his journey to general relativity, with his simple physical insights elegantly brought together with Riemannian geometry, Einstein approached his new challenge in a completely different way. He gave up on his formidable physical intuition to follow the math.

Einstein didn't come up with only one grand unified theory. For over thirty years he stumbled from theory to theory, sometimes discarding one possibility to pick it up again years later. One of his attempts extended spacetime into five dimensions instead of four. The additional spatial dimension was wrapped up and almost invisible. Its geometry, or curvature, would play the role of the electromagnetic field, responding to charge and currents exactly as James Clerk Maxwell had proposed in the mid-nineteenth century.

The idea of a five-dimensional universe wasn't originally Einstein's. It came from two young scientists, Theodor Kaluza, a lowly
privatdozent
in mathematics at the University of Königsberg, and Oskar Klein, a young Swedish physicist who had worked under Niels Bohr. Together they had worked out in detail how these five-dimensional spacetimes could mimic electromagnetism almost perfectly. The universes of Kaluza and Klein on which Einstein spent almost twenty years of his life are littered with a strange form of matter, an infinite variety of particles with a wide range of masses that should be all around us, warping the remaining geometry of spacetime. Einstein hoped, but was never able to show, that these extra fields might be inextricably tied to the quantum wave functions that Schrödinger had concocted in his quantum physics. Einstein gave up on these theories in the late 1930s, but interestingly enough the Kaluza-Klein theories would return in the 1970s when the idea of a unified theory took firm hold in theoretical physics.

Einstein devoted much more time to another theory for bringing together gravity and electromagnetism. He took his geometric framework for general relativity, the language that Riemann had proposed many decades before, and loosened it up. The original theory describing the geometry and dynamics of spacetime used ten unknown functions that had to be determined from his field equations. The fact that there were so many unknown functions, and that they were tangled up with each other in his original field equations, was one of the main reasons why general relativity was so hard to work with. But in his new theory, Einstein wanted to extend things by adding another six functions, three of which would describe the electrical part and another three of which would describe the magnetic part. The difficulty was how to bring these
sixteen
functions together in such a way that his theory would still be perfectly well defined and predictable. If he succeeded, the result, just like general relativity, should lead to the remarkable results that come out of
both
general relativity and electromagnetism. He wanted it to be mathematically beautiful, yet for decades he couldn't figure out how to make it so.

Einstein was onto something—the quest for a grand unified theory would come to dominate the physics of the late twentieth century—but during his lifetime he pursued this impossible quest alone. While he cut a solitary figure, working at the coal face of his new and fiendishly difficult theory, the outside world looked on with fascination. Every now and then Einstein would make the front page of one of the main newspapers. In November 1928, a
New York Times
headline hailed,
“Einstein on Verge of Great Discovery,” and a few months later, following a brief interview with Einstein, it reported, “Einstein Is Amazed at Stir Over Theory: Holds 100 Journalists at Bay for a Week.” The level of attention and excited anticipation lasted throughout the next quarter of a century. In 1949, the
New York Times
again declared,
“New Einstein Theory Gives a Master Key to the Universe,” and a few years later, in 1953, it trumpeted, “Einstein Offers New Theory to Unify the Law of the Cosmos.” Despite all this attention in the popular media, among his colleagues Einstein had become somewhat irrelevant and his attempts at unification were widely dismissed.

While Einstein had escaped the torrent of abuse that was being leveled at his work in Germany, he found that general relativity was also disappearing from view in his new home, the United States. Around him, the bright young scientists with the potential to push general relativity ahead were being sucked up into the theory of quantum physics, teasing out its application to fundamental particles and forces.

In some sense it was understandable. General relativity had delivered a few great successes early on, such as the precession of the perihelion of Mercury and the gravitational bending of light. And it had led to the discovery of an expanding universe, a spectacular change in our worldview. But that was it. From then on, it seemed that it could only serve up somewhat unbelievable,
mathematical
results, like Schwarschild's or Oppenheimer and Snyder's solutions for a collapsing or collapsed star. There was a case for these bizarre solutions existing out there, in space, but no one had seen them, so they really had to be considered mathematical exotica. Quantum physics could be tested in the laboratory and could be used to build things. But it was clear that there were more strange things to be found in general relativity, as the logician Kurt Gödel was able to show.

 

Einstein wasn't always alone on his treks from his house to the institute. Often, this eccentric, rumpled-looking professor, with his straggly hair and kind gaze, would be accompanied by a small figure, always wrapped in a heavy overcoat, his eyes hidden by thick Coke-bottle glasses. While Einstein trundled distractedly up toward Fuld Hall, the other man would trail beside him, quietly listening to Einstein's monologues, responding in a high-pitched voice. Einstein relished these walks with this odd little man, who had been at the institute for as long as he had and confided in him. His friend was Kurt Gödel, the man responsible for dismantling modern mathematics. To Einstein's disbelief, Gödel would also poke a significant hole in his general theory of relativity.

Gödel had come out of the intellectual powerhouse that was Vienna in the early twentieth century. A culture of debate and modernity thrived in the coffeehouses of Vienna, which was home to Ernst Mach, Ludwig Boltzmann, Rudolf Carnap, Gustav Klimt, and any number of brilliant thinkers. The most prestigious of all informal meetings was the world-famous Vienna Circle. To belong to the Vienna Circle, you had to be invited. Gödel was one of the chosen few.

Unlike Einstein, Gödel had flown through his childhood education, obtaining perfect scores in all the subjects he was set and barreling through university, an outstanding student. He had flirted with physics but, unlike Einstein, had been drawn to how mathematics could be brought together into one logical framework. He rapidly mastered the developments that were coming fast and frequently from philosophers and mathematicians alike in their attempts to construct an ironclad theory of mathematics, impervious to irrationality, guesses, and tricks. Such was the plan set forward by David Hilbert, who reigned over mathematics in Göttingen.

David Hilbert firmly believed that all of mathematics could be constructed from a handful of statements, or axioms. With a careful and systematic application of the rules of logic, it should be possible to deduce
every single mathematical fact in the universe
from no more than half a dozen axioms. Nothing would be left out. The verification of any mathematical fact, from 2 + 2 = 4 to Fermat's Last Theorem, should come from logical proof. Hilbert's program was the driving force behind mathematics when Gödel turned his sights on it.

While Gödel immersed himself in Vienna life, quietly attending the meetings of the Vienna Circle and watching the endless debates between the logicians and mathematicians on how to extend Hilbert's program to the whole of nature, he slowly and steadily chipped away at its fundamental premise. Then, in one fell swoop, he completely demolished Hilbert's plans with his own incompleteness theorem.

The incompleteness theorem states something incredibly simple. Whenever you describe a system mathematically, you begin with a set of axioms and rules. Whatever those initial statements are, Gödel showed that there will always be things that you can't deduce from them: true statements that you are unable to prove. If you stumble across a truth you can't prove using your axioms and your rules of logic, you can add it to your set of axioms. But Gödel's theorem showed that there will in fact always be an infinite number of these unprovable true statements. As you meander along picking up truths that you can't prove and adding them to your axioms, your simple, elegant deductive system becomes bloated, gigantic, and yet always incomplete.

Gödel's theorem torpedoed Hilbert's program and threw many of his colleagues completely off-kilter. Hilbert himself grumpily refused to acknowledge Gödel's result at first; eventually he accepted it and tried unsuccessfully to incorporate it into his program. Other philosophers published misguided critiques that Gödel refused to acknowledge. The English philosopher Bertrand Russell was never entirely comfortable with Gödel's result. Ludwig Wittgenstein, who completely dominated philosophical thought during the first half of the twentieth century, simply dismissed the incompleteness theorem as irrelevant. But it wasn't, and Gödel knew it.

Gödel loved Vienna, but he eventually found himself drawn to what Einstein called
“a wonderful piece of Earth and . . . ceremonial backwater of tiny spindle-shanked demigods.” Over a series of visits in the 1930s, he slowly began to feel comfortable at the Institute for Advanced Study, befriending Einstein, discussing with von Neumann, coming to realize just how high the intellectual caliber of émigrés ensconced in Princeton was. Following a particularly nasty incident in Vienna in which he was beaten up for looking like a Jew, he made the jump.

Einstein and Gödel hit it off immediately. As Einstein said, he would go into the office “just for the privilege of walking home with Kurt Gödel.” When Gödel fell ill, Einstein came and looked after him. When Gödel applied for American citizenship and was about to be sworn in, he found what he perceived was a logical inconsistency in the American Constitution that could allow the country to descend into tyranny. Einstein stepped in and went with Gödel to prevent him from sabotaging his own citizenship ceremony.

While Gödel's obsession was mathematics, he enjoyed physics and would often spend hours discussing relativity and quantum mechanics with Einstein. Both of them found the randomness in quantum physics hard to accept, but Gödel wouldn't stop there: he thought there seemed to be a crucial flaw in Einstein's general theory of relativity.

Gödel threw himself into Einstein's field equations, and, just like Friedmann, Lemaître, and many others before him, he tried to simplify them, looking for a manageable solution that might still represent the real universe. You may remember that Einstein assumed that the universe was full of stuff—atoms, stars, galaxies, whatever might take your fancy—evenly distributed everywhere. At any moment in time, you could move around in the universe and it would look the same, completely featureless, with no center or preferred place. Friedmann and Lemaître had each in his own way followed Einstein's lead, and both of them had found simple solutions
in which the geometry of the whole universe evolved with time. Gödel decided to add a small complication, small enough that he would still be able to solve the field equations but significant enough that something interesting might happen. He assumed that the whole universe was rotating around a central axis, like a merry-go-round, spinning around and around over time. The spacetime in the new universe that Gödel found, just like the universe that had been proposed by Friedmann and Lemaître, could be described in terms of time, three space coordinates, and the geometry at each point in spacetime. But there were differences. For a start, Friedmann and Lemaître's universe had the redshift effect that Slipher and Hubble had shown to be there in the real universe. Gödel's universe didn't. Quite clearly, it couldn't explain the expansion that had been measured by Slipher, Hubble, and Humason. But that wasn't the point. It was still a valid solution, a possible universe in Einstein's general theory of relativity.

However, Gödel's solution differed dramatically from all the universes that had come before in one unusual way. An observer in the Friedmann and Lemaître universe could roam around, exploring different parts of spacetime, and as time moved on, she would get older, leaving her past life behind her. There would be a clear sense of past, present, and future. This was not so in Gödel's universe. There, if an observer was moving around fast enough, she could coast along the rotating spacetime and loop back on herself. With enough accuracy, she could intercept herself when she was much younger, before she had set on her trek. In other words, in Gödel's universe, it was possible to travel back in time.

In Gödel's fantastical universe, it was possible to zoom backward and forward in time, revisit the past, correct youthful mistakes, apologize to long-departed relatives, warn yourself about future bad decisions. But it also meant that it was possible to do things that don't make sense, giving rise to some troubling paradoxes. Suppose you speed up and go back in time to meet your grandmother when she was a young girl, and through some terrible act, you kill her. You erase her existence from the face of the Earth so she can't give birth to your father or mother. You have also negated the possibility of your own existence, which means there wouldn't be a you to go back and do the dreadful deed. Yet if you lived in Gödel's universe, there was nothing to stop you from doing so, technological limitations and moral quandaries aside. Gödel's result showed that Einstein's general theory of relativity allowed for a solution in which it was possible to travel back in time and hence where paradoxes like this would be allowed, greatly at odds with our experience of the world. If Einstein's theory truly reflects nature, Gödel's absurd universe is a real physical possibility.