It’s more exciting—and, Einstein realized, more realistic—to allow for mixes of the different dimensions to take place. That way, for example, a bit of east-westness could be ordered in combination with a bit of up-downness. This is like one of the restaurants where the chefs ventured away from the standard offering of double orders of beer or double schnitzels. Instead, moving off the diagonal line, they offer variable mixes of food. That’s closer to the universe we exist in. The result is like the one we saw on page 74 (right), where at some places up-downness is mixed with east-westness, and other mixes occur as well.
Words get unwieldy when you use them to fully describe what goes in all the many possible slots. Instead of saying, for instance, “Here’s the value that will go in the box where the third column intersects the fourth row,” it’s quicker just to say “Here’s the value for box34.” Einstein and Grossmann went further, and instead of the word “box,” they began to use the letter g. Then, when they wanted to show that their boxes would work through a full range of subscripts, they switched from numbers to Greek subscripts. When Einstein wrote g34, he was referring to the value that would go in the box at the intersection of the third column and the fourth row; when he wrote gμν, he was referring to the whole range of sixteen boxes in his menu grid. It’s much like the way we refer to the boxes in spreadsheets today.
Thus the equation Einstein came up with in 1915, what in its neatened form is Gμν=8πTμν. The capital G on the left side is still a bit more complicated than what can be described here, but it has at its heart the expression gμν: a set of values that fit within a 4-by-4 grid and lay out how space and time at a particular location are ordered. The Tμν is similar and has at its heart another 4-by-4 grid whose values describe what’s in that location in space-time: the mixes of energy and momentum we’d find there.
Most of all, magnificently, Einstein recognized the deep connection between the two sides. One didn’t have to go to every possible location in space-time and then measure all the mass and energy there to fill in the two sides. That would be—and it is difficult to emphasize how much of an understatement this is—a lengthy task to complete. Instead, through Einstein’s genius, half the work is already done for us. Identify some particular arrangement of space and time on the left, and you will have a great start in knowing how mass and energy are operating there. Or you can start on the right, measuring what’s in the T grids, and then by the magic of his equation you will immediately be able to travel to the left side and start describing the geometrical configuration of space and time there. And of course if the values on the left side are so great that they would pull the entire universe into collapse, you can subtract some fraction of that left side so that everything balances without collapsing—and that is what Einstein did with his additional lambda term in 1917.
Solving this relation is hard, for the entries in the boxes on either side don’t just sit still. They vary from different perspectives. For example, if I see an object as stationary, you who are moving relative to me will not see it as stationary. But since it’s moving for you, it now has kinetic energy for you, and by the equivalence of energy and mass—remember E=mc2—you’ll genuinely experience it having greater gravitational attraction than I would.
Similarly, a mass that has one length when seen by a relatively stationary observer will be contracting when seen by a moving observer. But since the mass doesn’t change, while the volume does, its density will be higher, and that, too, will have to be incorporated. How to write such matters out so our differing perspectives each hold true? That’s what took Einstein and Grossmann so long.
Luckily, there are ways to make the calculations a little bit easier. For example, each side of Einstein’s relation has deep symmetries around the diagonal axis that goes from the top left to the bottom right, because everything on one side of that line has a duplicate in a matching position on the other (just the way an order of schnitzel plus beer yields the same result as an order of beer plus schnitzel). This means that instead of there being sixteen independent slots, and so sixteen separate equations, there are only the four along the middle diagonal, and then the six others above it. This makes G=T a mere ten tangled equations—which is, small mercies, at least easier to deal with than sixteen.
WHAT EINSTEIN SAW
It’s often possible to get rich insights without having to solve the equations at all. To understand how time bending depends on gravity, for example, imagine the space entrepreneur Elon Musk wants to check out one of his rocket ships before launch. He climbs into the bottom of the rocket ship, looks at the watch on his wrist, and then peers up to the top of his rocket—the inside having been hollowed out enough that he can see all the way—where another clock is waiting. He can tell the two clocks are synchronized because flashes of light are coming down from the top one and arriving at what his wristwatch shows are rock-steady intervals.
Everything seems fine.
But then suddenly, his good friend Jeff Bezos, watching from outside, punches a red button. Musk feels that his rocket is shooting away from the earth. He’s flung back against the bottom, and—delighted that Bezos has given him this opportunity to experience general relativistic effects—notes something curious. Somehow the light flashes from the top, or front, are arriving more quickly now than they did before. He’s puzzled by this. He knows that the length of his rocket hasn’t changed. Nor has the speed of light.
So why are these flashes reaching him more quickly?
He puzzles some more and then realizes what is happening. Because he’s accelerating, the back of the rocket, where he is, is approaching where the front had been ever more quickly than it had before. (That’s what acceleration means, as opposed to constant velocity.) Musk, at the back, is intercepting the light flash from the front before it’s gone the full length of his rocket ship: before, that is, his wristwatch has had a chance to mark a full second.
He can draw only one conclusion. The light flash from the front is arriving too “soon.” It used to send a flash every second by his wristwatch’s time. Now it’s sending a flash sooner than that. The clock must be running fast.
If this only happened inside rocket ships, the effect might be considered as due to the rumbling vibrations of the engines. But remember Einstein’s insistence that if there are no windows, the passenger can’t know for sure if he has been flying away from the earth. Maybe he’s been tricked and is still down on the ground, and it’s gravity that is holding him to the floor. (This is, as we’ve noted, like being pressed back in an accelerating sports car. If your eyes are closed and there’s no shaking, it will feel just the same as if you’re being pulled backward by a huge gravitational source behind you.)
Since no observer in these circumstances can tell whether he’s on the ground or flying away from the earth, this means that the different rates of time—the different speeds that the clocks tick—will occur in a gravitational field as much as in an accelerating vehicle. Identical clocks show time passing more quickly “high up,” where gravity is weaker. Time passes more slowly “low down,” where gravity is stronger.
This sounds preposterous, but it’s true. When GPS satellites whiz overhead, then in accord with special relativity their high velocity contributes to making time on board slow down. But since satellites also orbit at 12,500 miles up, where gravity is several times weaker than on the ground, the effect that Musk’s imagined experience demonstrates also comes into play. From this second effect, in accord with general relativity, time on the satellites flows more quickly than it does for us on earth, where our denser gravitational field slows time down.
Which factor dominates? In the case of our GPS satellites, the speeding of time due to the lesser gravity in their high orbits adds 45,000 nanoseconds of extra time to their existence each day, while the slowing of time due to their great speed takes away only about 7,000 nanoseconds each day. The net difference is a 38,000 nanosecond gain. That’s the figure engineers use to “reset” our G
PS systems each day, to keep time in our world in synch with time on the satellites. Without this correction, we’d soon end up miles off course.
There’s more. The greater the difference between gravity at two places, the greater the general relativistic effect. Time just above the surface of the sun proceeds a minute slower each year than it does on earth. Time very near a black hole proceeds many millions of times slower. We would see (subject to certain dimming effects) an astronaut who is falling into the black hole seem to move in incredibly slow motion, while to him time is normal, and it’s the galaxy outside that has sped up, with life speeding along at millions of times its usual rate. In his last moment, he could, in theory, see entire civilizations rise and fall.
It would be hard for him to actually observe such sights, however, and not just because of the limitations of any telescope he’s brought along. A gravitational gradient strong enough to produce time rates that are so different will also produce very different pulls on his body. His raised hand will receive a certain amount of gravitational pull; his foot, if it’s closer to the hole, will receive a greater—a VERY much greater—pull. Other effects are going on, but this alone is enough to create what’s called “spaghettification,” where even the strongest material is pulled apart. However his investments back on earth might have turned out, he would soon not be in any condition to enjoy them.
Credits
Illustrations on pages 60, 65, 66, 74, 76, 87, 89, 160, 244, 245, 246, 247, and 248 are by Michael Hirschl, © 2016.
x: Esther Bubley, The LIFE Images Collection/Getty Images
2: SPL/Science Source®
10: Besso Family, American Institute of Physics, Emilio Segrè Visual Archives
11: American Institute of Physics, Emilio Segrè Visual Archives/Science Source®
12: ullstein bild/Pictures from History
42: ullstein bild/AKG
45: From Flatland: A Romance of Many Dimensions, Edwin Abbott Abbott, 1884
46: From Flatland: A Romance of Many Dimensions, Edwin Abbott Abbott, 1884
82: Keystone-France/Getty Images
96: © UPPA/Photoshot
114: © Jiri Rezac
124: Sergey Konenkov, Sygma/Corbis
129: RIA Novosti/Science Source®
138: Harvard College Observatory/Science Source®
144: AP Images
149: Margaret Bourke-White, Time Life Pictures/Getty Images
151: SPL/Science Source®
155: Mondadori Portfolio/Getty Images
170: Albert Einstein, Courtesy of the University of New Hampshire
173: ullstein bild/Getty Images
179: ullstein bild/Rainer Binder
187: Francis Simon, American Institute of Physics, Emilio Segrè Visual Archives/Science Source®
194: Science & Society Picture Library/Getty Images
202: American Institute of Physics, Emilio Segrè Visual Archives/Science Source®
210: Bettmann/Getty Images
Bibliography
Here are a handful of especially worthwhile books, mostly for the general reader. In each section, I’ve marked two of my favorites with asterisks. There’s also a long annotated version of this list at davidbodanis.com. In addition to these is the fundamental The Collected Papers of Albert Einstein (Princeton: Princeton University Press, 1987–), now at fourteen volumes and counting.
LETTERS, ESSAYS, AND QUOTATIONS
Albert Einstein–Michele Besso Correspondance, 1903–1955. Translated and edited by Pierre Speziali. Paris: Hermann, 1972.
*Born, Max. The Born-Einstein Letters, 1916–1955: Friendship, Politics and Physics in Uncertain Times. Translated by Irene Born. London: Macmillan, 2005. First published 1971.
Calaprice, Alice, ed. The Ultimate Quotable Einstein. Princeton: Princeton University Press, 2011.
*Einstein, Albert. Ideas and Opinions. London: Folio Society, 2010.
Solovine, Maurice. Albert Einstein: Letters to Solovine. New York: Philosophical Library, 1987.
BIOGRAPHIES (AUTHORS WHO KNEW HIM)
*Frank, Philipp. Einstein: His Life and Times. New York: Da Capo Press, 2002). First published 1947.
*Hoffmann, Banesh. Albert Einstein: Creator and Rebel. New York: Viking, 1972.
Pais, Abraham. Subtle Is the Lord: The Science and Life of Albert Einstein. New York: Oxford University Press, 1982.
Seelig, Carl. Albert Einstein: A Documentary Biography. London: Staples Press, 1956.
BIOGRAPHIES (MORE RECENT)
Folsing, Albrecht. Albert Einstein: A Biography. Translated and abridged by Ewald Osers. New York: Viking, 1997.
*Isaacson, Walter. Einstein: His Life and Universe. New York: Simon & Schuster, 2007.
*Neffe, Jurgen. Einstein: A Biography. Translated by Shelley Frisch. New York: Farrar, Straus and Giroux, 2007.
Renn, Jürgen. Albert Einstein: Chief Engineer of the Universe. Hoboken, N.J.: Wiley, 2006.
REFLECTIONS, AND SPECIAL TOPICS
French, A. P., ed. Einstein: A Centenary Volume. Cambridge, Mass.: Harvard University Press, 1979.
Galison, Peter. Einstein’s Clocks, Poincaré’s Maps. New York: Norton, 2003.
Gutfreund, Hanoch, and Jürgen Renn. The Road to Relativity: The History and Meaning of Einstein’s “The Foundation of General Relativity.” Princeton: Princeton University Press, 2015.
Holton, Gerald, and Yehuda Elkana, eds. Albert Einstein: Historical and Cultural Perspectives. Mineola, N.Y.: Dover, 1997. First published 1982.
*Levenson, Thomas. Einstein in Berlin. New York: Bantam Books, 2003.
Miller, Arthur I. Einstein, Picasso: Space, Time, and the Beauty That Causes Havoc. New York: Basic Books, 2001.
*Schilpp, Paul Arthur. Albert Einstein: Philosopher-Scientist. LaSalle, Ill.: Open Court Press, 1949.
Stachel, John. Einstein from “B” to “Z.” Boston: Birkhäuser, 2002.
Stern, Fritz. Einstein’s German World. Princeton: Princeton University Press, 1999.
RELATIVITY IN PARTICULAR
Einstein, Albert. Relativity: The Special and the General Theory (A Popular Account). Translated by Robert W. Lawson. New York: Random House, 1995. First published 1916.
Ferreira, Pedro G. The Perfect Theory: A Century of Geniuses and the Battle over General Relativity. New York: Houghton Mifflin Harcourt, 2014.
*Geroch, Robert. General Relativity, from A to B. Chicago: University of Chicago Press, 1978.
*Susskind, Leonard. General Relativity. Online course. The Theoretical Minimum, Stanford Continuing Studies, http://theoreticalminimum.com/courses/general-relativity/2012/fall.
Taylor, Edwin, and J. Archibald Wheeler. Spacetime Physics: Introduction to Special Relativity. New York: W. H. Freeman, 1992.
Thorne, Kip. Black Holes and Time Warps: Einstein’s Outrageous Legacy. New York: Norton, 1995.
Wald, Robert M. Space, Time, and Gravity: The Theory of the Big Bang and Black Holes. Chicago: University of Chicago Press, 1992.
Will, Clifford M. Was Einstein Right?: Putting General Relativity to the Test. Oxford: Oxford University Press, 1993.
QUANTUM MECHANICS
Fine, Arthur. The Shaky Game: Einstein, Realism, and the Quantum Theory. Chicago: University of Chicago Press, 1996.
Kuhn, Thomas S. Black-Body Theory and the Quantum Discontinuity, 1894–1912. Chicago: University of Chicago Press, 1978.
*McCormmach, Russell. Night Thoughts of a Classical Physicist. Cambridge, Mass.: Harvard University Press, 1982.
Polkinghorne, John. Quantum Theory: A Very Short Introduction. New York: Oxford University Press, 2002.
*Stone, A. Douglas. Einstein and the Quantum: The Quest of the Valiant Swabian. Princeton: Princeton University Press, 2013.
OTHER PLAYERS
*Cassidy, David. Uncertainty: The Life and Science of Werner Heisenberg. New York: W. H. Freeman, 1992.
Halpern, Paul. Einstein’s Dice and Schrödinger’s Cat: How Two Great Minds Battled Quantum Randomness to Create a Unified
Theory of Physics. New York: Basic Books, 2015.
Heilbron, John. The Dilemmas of an Upright Man: Max Planck and the Fortunes of German Science. Cambridge, Mass.: Harvard University Press, 2000. First published 1986.
Moore, Walter. Schrödinger: Life and Thought. New York: Cambridge University Press, 2015. First published 1989.
Pais, Abraham. Niels Bohr’s Times in Physics, Philosophy, and Polity. New York: Oxford University Press, 1991.
*Rozental, Stefan, ed. Niels Bohr: His Life and Work as Seen by His Friends and Colleagues. Hoboken, N.J.: Wiley, 1967.
ASTRONOMY
Christianson, Gale E. Edwin Hubble: Mariner of the Nebulae. Chicago: University of Chicago Press, 1995.
Douglas, Vibert. The Life of Arthur Stanley Eddington. London: Thomas Nelson, 1956.
Ferris, Timothy. Coming of Age in the Milky Way. New York: Perennial, 2003. First published 1988.
Johnson, George. Miss Leavitt’s Stars: The Untold Story of the Woman Who Discovered How to Measure the Universe. New York: Norton, 2005.
Levenson, Thomas. The Hunt for Vulcan . . . and How Albert Einstein Destroyed a Planet, Discovered Relativity, and Deciphered the Universe. New York: Random House, 2015.
*Miller, Arthur I. Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes. New York: Houghton Mifflin, 2005.
*Singh, Simon. Big Bang: The Origin of the Universe. New York: HarperCollins, 2004.
Notes
Prologue
xii building card castles: The Collected Papers of Albert Einstein, vol. 1, The Early Years, 1879–1902, trans. Anna Beck (Princeton: Princeton University Press, 1987), p. xix (hereafter cited as CPAE1). Princeton University Press has been bringing together all of Einstein’s papers in a collection that sets the standard for scholarly editions.
Einstein's Greatest Mistake Page 23