The general theory of relativity was born on November 25th, 1915, when Albert Einstein delivered the final lecture that presented his equations of gravity to the world. The theory turns one hundred this month, but it’s just as graceful as ever, so formidable and elegant that it still bewitches the scores of ardent scientists who study it.
I should know. I had a youthful fling with relativity. The affair was brief -- or I should say, rather, that its extent in the time dimension was woefully truncated. But while it lasted, our love was rich in tensors and matrices.
Our relationship began three decades ago when I was a naïve twenty-year-old, a wide-eyed romantic in Princeton University’s astrophysics department. At the time, I was struggling with the baffling math of quantum mechanics, the partial differential equations that govern the physics of atoms, molecules and subatomic particles. But then a sympathetic professor named J. Richard Gott III introduced me to the Einstein field equations, the beating heart of the general theory of relativity. The equations are commonly summarized in the formula:
Gμν = 8πTμν
It was so beautiful, so fundamental. On the right side of the equation is the stress-energy tensor, which describes the density and flow of energy, whether it’s in the form of matter (stars, planets, lovers) or energy-carrying fields (microwaves, x-rays, breathtaking sunsets). On the left side is the Einstein tensor, which shows how energy warps the space and time it occupies, bending the paths of any objects nearby. The more energy, the greater the curvature.
The formula is so concise it can fit on a T-shirt, and yet it’s powerful enough to explain the movements of galaxies. Einstein discovered that gravity isn’t an instantaneous force exerted by all massive objects, as Isaac Newton assumed, but a distortion of the spacetime around them. As I stared at the equation’s lovely symbols, with their oh-so-classical Greek-letter subscripts, I could almost hear it whisper its meaning into my ear: “Matter tells spacetime how to curve; spacetime tells matter how to move.”
I was smitten. The allure of Einstein’s equations is their simplicity -- each tensor is just an array of sixteen components, represented mathematically by a box-like matrix with four rows and four columns (because our universe’s spacetime has four dimensions, three of space and one of time). And doing math with matrices is actually kind of fun; there are easy-to-understand rules for adding and multiplying them. Trust me, it’s a lot less grueling than solving the equations of quantum mechanics.
Still, it’s rare for a callow undergraduate to fool around with relativity. Professor Gott made things a little easier for me by suggesting that I work on a fairly straightforward problem: What would happen if you applied the general theory of relativity to a hypothetical universe with only two dimensions of space instead of three -- that is, a universe with length and width, but no height?
This imaginary cosmos is called Flatland, named after the 1884 novella by English schoolteacher Edwin Abbott, who chronicled the adventures of a free-thinking square in a repressive polygonal society. As far as Professor Gott and I knew, no one had ever tried to solve the Einstein field equations for Flatland. (We later discovered, though, that Polish theorist Andrzej Staruszkiewicz had tackled the problem back in 1963. Ah relativity, you heartbreaker!)
Because I had fewer dimensions to worry about, I knew the math of general relativity would be simpler for Flatland. Each of the tensors in the field equations would hold only nine components instead of sixteen. Nevertheless, I was full of anxiety that evening when I sat behind the desk in my dorm room and started scribbling calculations in my notebook. I worked for hours, making mistakes and crossing them out, filling page after page with penciled matrices. I muttered and moaned and grew lightheaded. I lurched back and forth between despair and exaltation.
By dawn I had a solution, but I had no idea what it meant. The formula scrawled in my notebook was utterly bewildering. Later that day I brought it to Professor Gott’s office, and he spent several minutes staring at the symbols. Then he gave me the greatest compliment that one theorist can give another: “This solution is non-trivial!”
As it turns out, the general theory of relativity would work very differently in Flatland than it does in our familiar four-dimensional universe. Massive objects in Flatland wouldn’t attract each other gravitationally, because the spacetime between them would be locally flat. And yet Flatland wouldn’t be flat overall. Each mass would bend the surrounding spacetime into a cone, altering the paths of nearby objects without attracting them. You can picture the effect by making an incomplete tear in a sheet of paper and sliding one jagged torn edge over the other. Imagine the Flatland object at the apex of the paper cone you’ve made. The greater the mass, the sharper the cone would be.
Because the solution was odd and unexpected, Professor Gott wrote a journal article about the results and listed me as a coauthor. But by the time the article appeared in General Relativity and Gravitation, my affair with relativity was over. The relationship ended the same way that many youthful dalliances do: because something else caught my eye. I fell in love with modern poetry, and then with journalism. As I moved from job to job, though, I held on to my notebook of scribbled calculations. I stored it at the bottom of a cardboard box like a cache of perfumed letters.
Twenty years later I became an editor at Scientific American, and while working on a story about Einstein I learned something surprising. Physicists involved in the study of quantum gravity -- a theory that would mesh the Einstein field equations with the laws that govern the subatomic world -- had taken notice of the work that Professor Gott and I did so long ago. Because it’s useful to test theories of quantum gravity in the simpler Flatland universe, dozens of researchers have looked up our article and cited it in their own papers. Apparently, my brief fling with the field equations had yielded a love child.
But I shouldn’t have been surprised. The field of general relativity is remarkably fertile. A century of study has only heightened the ardor of physicists, who are eagerly fashioning new theories about black holes and the Big Bang, as well as groping for the connections between gravity and quantum theory. I’m a little jealous, of course, but all is fair in love and theoretical physics. Or as Einstein himself noted: “Falling in love is not at all the most stupid thing that people do -- but gravitation cannot be held responsible for it.”
Mark Alpert writes science thrillers. His latest novel, The Six, is about teenagers who turn into robots. His website is www.markalpert.com