As I was working through the chapters, my children Sam and Sophie received updates at weekly, sometimes daily, and once (sorry, guys) even hourly intervals detailing my progress. Their confidence that this story needed to be told was the most invigorating of motivations.
I dedicated it to Sam because when he was young, and important matters—birthday presents or new computer games—existed too far in the future for mortal souls to wait, I’d explain that if we could simply get into an Einstein rocket, we could reach those future dates in just a few minutes of our time. I loved the way he trusted this. If people do manage to create such devices to accelerate us through time, it will be members of his generation, not mine. And if that generation avoids the hubris that brought Einstein down, I will be delighted.
Appendix
A Layman’s Guide to Relativity
This book stands on its own, but in this appendix you can delve a little more into how relativity works. Readers who skip it won’t find their appreciation of the book affected. Readers who are true gluttons will find a 22,000-word download at davidbodanis.com taking them further.
WHY TIME CURVES: THE CASE OF KING KONG
The idea that it’s not just space that gets curved, but time as well, was first properly developed by one of Einstein’s old professors, Hermann Minkowski, at a lecture in Cologne, Germany, in 1908. He had been thinking about Einstein’s 1905 work and noticed that “Einstein’s presentation [of special relativity] is mathematically awkward—I can say that because he got his mathematical education in Zurich from me.”
To extend Einstein’s work, Minkowski began by laying out an image where space was envisioned as a horizontal plane, and time as a vertical axis sticking up from it. One can think of this as a large table with a spindle or candlestick rising up from it in the middle. Everyone was used to seeing the two realms as separate, but that was what Minkowski wanted to change. It made sense to him, for “the objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place.”
It would be better, Minkowski declared, not to speak of ordinary locations plus a separate time, but rather of a single unitary thing called an “event.” To describe the mixed “space-time” that all possible events fit within, one would just need to make lists of four numbers.
That sounds abstract, but it’s something we do all the time. Suppose your great-grandfather was strolling in New York one crisp evening in the spring of 1933 and saw a large, hirsute creature on the top of the Empire State Building, 1,400 feet up. He wants to alert the press. Once he finds a telephone and rings the New York Herald Tribune, he could say, “It’s—it’s—on the top of the Empire State Building, and, oh God, I see it right now!” But if he and the Herald Tribune reporter both understood Minkowski’s symbolic shorthand, he could more swiftly say, “5th Avenue, 33rd Street, 1,400 feet, 8:30 p.m.!” If they both understood Manhattan’s grid system, he could be even quicker, simply saying, “5, 33, 1,400, 8:30!” The paper’s photographers would know exactly where to head: the corner of 5th Avenue and 33rd Street, way up on the 1,400-foot-high spire, where at least at 8:30 p.m., the city’s largest resident was to be found.
But imagine King Kong is publicity shy and tosses a zip line across central Manhattan to the top of the shinily attractive Chrysler Building. Blond actress in hand, he starts gliding across to that safer refuge. If your great-grandfather were still watching and had the phone, he could tell his Trib contact the new changing coordinates. Five seconds into the gliding journey, he might call out “5, 35, 1,380, 8:30:05”; five seconds later, it might be “5, 36, 1,340, 8:30:10”; and so on. The figures would click along till the pair arrived atop the slightly shorter Chrysler Building on 42nd Street.
That’s what Minkowski meant when he said that every distinct event—every distinct location in space and time—can be identified by a grouping of four numbers. Listing every possible event in the universe would produce an enormous book, one in which all the settings for past or future history are written out. In his 1908 lecture, Minkowski joked about how presumptuous this task was: “With this most valiant piece of chalk I might project [all those locations] upon the blackboard.” It’s precisely what many religions imagine their God to be able to do. But that didn’t stop him from observing that this is how it could be done.
Now for the big question. Does what happens to the first three numbers—the ones describing location in space—have to be linked with the fourth number, the one describing location in time? If it does, then space will not be separate from time, but each will have to be brought in to fully locate what is happening.
To answer that, Minkowski looked at how to work out distances between any two events. For your great-grandfather, the distance between the starting event, where Kong is atop the Empire State Building, and the second event, where Kong has landed atop the 300-foot-shorter Chrysler Building, would be something like “3 avenues, 8 streets, 1,100 feet, 2 minutes.” But remember that time passes at a different rate for objects that are moving relative to you. That is key. Kong isn’t going especially fast past your great-grandfather as he glides over Manhattan, but he’s going to experience a travel time that’s just a very slight amount less than the full minute your great-grandfather sees.
(Why does time vary like that? Well, suppose you’re watching a friend bounce a ball up and down in a stationary car near you. Clearly you and she will agree on how far it’s traveled. Now, however, have her start driving the car while you remain by the roadside watching. She’ll see the ball she’s bouncing continue to go straight up and down beside her. You’ll see it take a longer path as the car moves forward.
(Now suppose it’s not a ball she’s bouncing, but a light beam. You’ll each see it travel at the same speed (for as Einstein made clear, that’s how light works). This is what’s odd. She’ll see the light beam in her car cover a short distance. You’ll see it moving at the same speed—for that’s the only rate light can travel—and cover a longer distance.
(How can something cover two different distances while traveling at the same speed? The only answer, Einstein realized, is if you see time in the moving car slow down, so that there’s more time for the light beam to travel the longer distance. Any object moving relative to you will experience this, be that cars, rocket ships, or even imagined fast-gliding apes.)
The effect stands out if we imagine faster travel. What if Kong doesn’t stay on the Chrysler Building but, worried about the Herald Trib press cars scurrying close by, at 8:32 p.m. he and his date jump into a rocket ship and circle around the galaxy until they land back on top of the Chrysler Building on what you measure as February 8, 2017. You hurry down there, push through the massed photographers and the producers shouting out reality TV offers, and huddle with the great beast and his actress friend. You ask if they’ll help you with the Minkowskian calculation to determine the space-time distances they’ve crossed.
They nod yes and show you the log in which they’ve carefully kept note of their travels. You read it and then look up, puzzled. To you it’s obvious what the “distance” is between the event when Kong was last seen atop the Chrysler Building and the situation today. That event took place at 8:32 p.m. on March 2, 1933, and now you’re standing at the same place, so the difference is “0 avenues, 0 streets, 0 height, 83.9 years.” But Kong’s log shows a much briefer amount of time, due to the distortions of time that took place over the immense distances he traveled in his high-speed journey.
The point is a profound one. Different individuals constantly enter onto separate “tracks” of time. It’s not just you and our imagined far-traveling Kong who won’t agree when it comes to measuring the distance between two events. All of us travel at different rates, and—if one looks closely enough—we will all have these disagreements about what has elapsed between two events.
FINDING YOUR WAY THROUGH TIME AND SPACE
This seems like a recipe for chaos, as if we’re
living in a universe that’s wildly unconnected, with each of us in separate worlds, crashing against one another with no rhyme or reason. But Minkowski showed that although space and time don’t fit together by the simple sort of subtractions between events described in the previous section, they do fit together another way. There is a new kind of distance between any two events—what he termed the “interval”—which everyone will agree on, however they’ve been moving. Although your space and your time might be different from mine, Minkowski found that the curious number x2-c2t2 will always bring us to the same result. (Here c is the speed of light, t is the difference between the time entries in the two events, and x is an elapsed distance between all the space entries—it’s a matter I elaborate on my website. Since x2-c2t2 describes a hyperbola, a number of geometrical diagrams fit there to help.)
Einstein resisted this blending at first, calling Minkowski’s work überflüssige Gelehrsamkeit —“superfluous erudition”—but he soon came around and made Minkowski’s solution integral to his own later work in relativity. It’s a fabulous solution. We no longer have to think of our universe as an ungainly pileup, with three-dimensional space here, and a one-dimensional time sticking out to the side of it at a right angle somewhere else, and everybody whirring along like Magritte characters on their own isolated airport walkways. Instead, we live in a combined “thing” called space-time.
The interval—that strange “distance” x2-c2t2—is at the heart of the trade-off that occurs between space and time. In ordinary space, distances add up, and then, separately, time adds up as well. In space-time, because the components are linked in this particular way (with the time we see of another individual slowing down as his velocity relative to us speeds up), that doesn’t happen. Rather, it’s as if movement in space-time works through two odometers, one of which is constantly subtracting from the other.
The idea of mixing time with space can sound mystical, but think of looking at a circular watch. Viewed face-on, it seems to have an equal amount of “horizontalness” and “verticalness.” Tilt it slightly, however, so you’re viewing it at an angle, and you won’t see a perfect circle anymore, but an ellipse. Some of the verticalness seems to have vanished.
We’re not bothered by this, for we know that the verticalness would still be there if we took a complete measurement of the watch. Its vanishing is simply an artifact of our restricted position in viewing it. The spatial dimensions we’re used to are similar. We know that on earth we can walk due east or due north, but we can also do a bit of each at the same time—that is, we can stroll northeast. The directions north and east might seem distinct, but they fit within a greater unity we can perceive. Similarly for the way that fans in a basketball arena see the hoop foreshortened in different ways. Those whose seats put them precisely ten feet up see it as a horizontal line; those in other positions might see it as an ellipse. But that doesn’t mean they think those distortions are the truth. Once they stand up and walk around, they know they can see the hoop from enough angles to get the full picture.
In the four-dimensional space-time that Minkowski showed we live in, however, it’s impossible for us fragile, carbon-based organisms to step back and see the full arrangement—to see all of space and all of time at once. Yet with his abstract symbols, we can tell it’s there, with all the parts—all space and all time—inextricably linked.
THE EQUATION OF THE UNIVERSE
How did Einstein bring all that together? His 1915 equation looks so different from what’s taught in high school or even basic university math classes that most people on first glance think nothing about it can be understood. Even in its most condensed modern form, the equation comes out as the not especially inviting Gμν=8πTμν. But once we realize that much of that is simply a deft shorthand for listing various mixes of things, it starts to become clearer.
To understand that shorthand, imagine going back to one of the restaurant-cafés that Einstein used to like sitting in when he was a university student. Suppose the menu is extremely small, just schnitzel and beer, and to save time the waiters don’t write down in full their customers’ orders, but instead use little grids printed on their order pads:
If a waiter sends in the order
1 0
0 1
the chef knows to supply a double order of schnitzel (because the first “1” is in the slot where the two schnitzel labels intersect), a double order of beer, and nothing else.
If the chef decides to go wild and offer a third menu item—roasted potatoes!—the restaurant will need to print up fresh pads with a slightly larger grid. The best form of roasted potatoes in Switzerland is called rösti, and so the new grid will read like this:
If a waiter now writes
0 0 1
0 3 0
0 0 0
the chef knows to produce one tray of mixed schnitzel and rösti, and one tray with three double orders of beer. It’s exceptionally unhealthy, remarkably tasty—and very efficiently summarized by that simple pattern of numbers.
Suppose there are dozens of such restaurants in Zurich, and they decide to stop competing with one another. Instead of offering a choice of how much of any mix can be ordered, each restaurant sticks to a particular quantity for every meal it serves. In one restaurant, every customer gets a mixed tray of schnitzel and rösti, and three trays with double orders of beer—and that’s it. Another Zurich restaurant has a sign outside the door with a blowup of the waiters’ order pad filled in like this:
The grid reads
1 0 0
0 0 0
0 0 1
so everyone knows that this restaurant’s offerings are double schnitzels and double rösti, and never anything else. The other restaurants lock in to yet other possibilities. Each simple arrangement of numbers lets you know what culinary treats you will henceforth eat if you enter a restaurant and remain inside forever.
Now back to relativity. Suppose we’re ordering not food, but the shape of a universe. First of all, we need to know what the constituent dimensions are—the equivalents of the schnitzel, beer, and rösti. In the case of a two-dimensional flat surface, like the one our postage-stamp-shaped Mr. A. Square lived on, those constituent parts are the changes in distance in the east-west direction, dx, and the changes in distance in the north-south direction, dy.
That sets up the waiters’ order pad, and to flesh it out we now need to know which particular permutations of those parts will be on offer. Once we have those two sorts of data—what range of possibilities are on offer to set up our grid, and then what particular choices are made from that range to fill in our grid—we know a great deal about the world we are going to enter.
That type of “mixed grid plus entries” is very close to what is called a metric tensor. The name is illuminating. Measures of distance using the Greek root metron, or “meter,” were popularized with the new French measuring system introduced in the eighteenth century: the metric system. Metrics are simply a way of establishing how things fit together. The cluster of numbers that serves as an order in each of our hyperefficient Zurich restaurants defines how the constituent food components fit together. It’s the restaurant’s metric—its way of organizing things. In our physical universe, the cluster of numbers that serves as an “order” defines how our universe’s component dimensions fit together.
CREATING OUR WORLD
In Mr. A. Square’s Flatland world, the background grid will allow for different mixes of dx and dy: different amounts of east-westness or north-southness. That means the empty grid will look like this:
How to fill it in? We know that the very definition of a flat surface is that the Pythagorean theorem holds: that if there’s a right triangle with sides dx and dy, and a hypotenuse ds, then the parts click together so that dx2+dy2=ds2. We can summarize that in the Flatland order form by filling it in as shown:
There are double orders of dx and double orders of dy, and no odd mixes of them. Everything is neat: right triangles fit togethe
r, squares don’t bulge, and it’s fair to think that time will similarly be at “right angles” to space. Stick to combining the parts in this most straightforward way, and you’ll create the Flatland world. In the Bible’s book of Job, God asks, “Where wast thou when I laid the foundations of the earth? . . . Who hath laid the measures thereof . . . or who laid the corner stone?” This is the closest secular equivalent to how that can be done.
What’s great about this approach is that it can easily be expanded. A benevolent deity looks down on his dominion, orders up more constituents, and lo, a restaurant—a universe!—is created with a much-enlarged order form.
Einstein’s equations were built out of analogous grids but allow very different worlds. They were bigger than Flatland’s, of course, for instead of having 2-by-2 boxes, allowing for just two dimensions of space (east-west and north-south), they had 4-by-4 boxes, so that three dimensions of space and one dimension of time could be combined. Also, they weren’t usually going to be filled in with such simple instructions as that of the Flatland grid, where the sequence of 1s on the diagonal meant the only mixes were crisp, neat, straight ones. That would have been like landing in a restaurant world where separate menu items were never mixed: what in four dimensions—i.e., with four ranges of possible items to be ordered—would look like this:
That’s a dull, flat world—like the one on page 74 (left).
Einstein's Greatest Mistake Page 22