41
" One method that Einstein employed to help people visualize this notion was to begin by imagining two-dimensional explorers on a two-dimensional universe, like a flat surface. These “flatlanders” can wander in any direction on this flat surface, but the concept of going up or down has no meaning to them. Now, imagine this variation: What if these flatlanders’ two dimensions were still on a surface, but this surface was (in a way very subtle to them) gently curved? What if they and their world were still confined to two dimensions, but their flat surface was like the surface of a globe? As Einstein put it, “Let us consider now a two-dimensional existence, but this time on a spherical surface instead of on a plane.” An arrow shot by these flatlanders would still seem to travel in a straight line, but eventually it would curve around and come back—just as a sailor on the surface of our planet heading straight off over the seas would eventually return from the other horizon. The curvature of the flatlanders’ two-dimensional space makes their surface finite, and yet they can find no boundaries. No matter what direction they travel, they reach no end or edge of their universe, but they eventually get back to the same place. As Einstein put it, “The great charm resulting from this consideration lies in the recognition that the universe of these beings is finite and yet has no limits.” And if the flatlanders’ surface was like that of an inflating balloon, their whole universe could be expanding, yet there would still be no boundaries to it.10 By extension, we can try to imagine, as Einstein has us do, how three-dimensional space can be similarly curved to create a closed and finite system that has no edge. It’s not easy for us three-dimensional creatures to visualize, but it is easily described mathematically by the non-Euclidean geometries pioneered by Gauss and Riemann. It can work for four dimensions of spacetime as well. In such a curved universe, a beam of light starting out in any direction could travel what seems to be a straight line and yet still curve back on itself. “This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived,” the physicist Max Born has declared.11 Yes, "
― Walter Isaacson , Einstein: His Life and Universe
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" Instead, this time he made what he called a “slight modification” to his theory. To keep the matter in the universe from imploding, Einstein added a “repulsive” force: a little addition to his general relativity equations to counterbalance gravity in the overall scheme. In his revised equations, this modification was signified by the Greek letter lambda, , which he used to multiply his metric tensor gμv in a way that produced a stable, static universe. In his 1917 paper, he was almost apologetic: “We admittedly had to introduce an extension of the field equations that is not justified by our actual knowledge of gravitation.” He dubbed the new element the “cosmological term” or the “cosmological constant” (kosmologische Glied was the phrase he used). Later,* when it was discovered that the universe was in fact expanding, Einstein would call it his “biggest blunder.” But even today, in light of evidence that the expansion of the universe is accelerating, it is considered a useful concept, indeed a necessary one after all.14 During "
― Walter Isaacson , Einstein: His Life and Universe
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" There are historical moments when an alignment of forces causes a shift in human outlook. It happened to art and philosophy and science at the beginning of the Renaissance, and again at the beginning of the Enlightenment. Now, in the early twentieth century, modernism was born by the breaking of the old strictures and verities. A spontaneous combustion occurred that included the works of Einstein, Picasso, Matisse, Stravinsky, Schoenberg, Joyce, Eliot, Proust, Diaghilev, Freud, Wittgenstein, and dozens of other path-breakers who seemed to break the bonds of classical thinking.52 In "
― Walter Isaacson , Einstein: His Life and Universe
47
" Rμv– 1/2 gμv R = 8πTμv The left side of the equation starts with the term Rμv, which is the Ricci tensor he had embraced earlier. The term gμv is the all-important metric tensor, and the term R is the trace of the Ricci tensor called the Ricci scalar. Together, this left side of the equation—which is now known as the Einstein tensor and can be written simply as Gμv—compresses together all of the information about how the geometry of spacetime is warped and curved by objects. The right side describes the movement of matter in the gravitational field. The interplay between the two sides shows how objects curve spacetime and how, in turn, this curvature affects the motion of objects. As the physicist John Wheeler has put it, “Matter tells spacetime how to curve, and curved space tells matter how to move.”83 Thus is staged a cosmic tango, as captured by another physicist, Brian Greene: Space and time become players in the evolving cosmos. They come alive. Matter here causes space to warp there, which causes matter over here to move, which causes space way over there to warp even more, and so on. General relativity provides the choreography for an entwined cosmic dance of space, time, matter, and energy.84 At "
― Walter Isaacson , Einstein: His Life and Universe
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" Einstein’s developmental problems have probably been exaggerated, perhaps even by himself, for we have some letters from his adoring grandparents saying that he was just as clever and endearing as every grandchild is. But throughout his life, Einstein had a mild form of echolalia, causing him to repeat phrases to himself, two or three times, especially if they amused him. And he generally preferred to think in pictures, most notably in famous thought experiments, such as imagining watching lightning strikes from a moving train or experiencing gravity while inside a falling elevator. “I very rarely think in words at all,” he later told a psychologist. “A thought comes, and I may try to express it in words afterwards.”4 "
― Walter Isaacson , Einstein: His Life and Universe