By 1915, any list of the world’s greatest living mathematicians included the name David Hilbert. And though Hilbert previously devoted his career to logic and pure mathematics, he, like many other critical thinkers at the time, eventually became obsessed with a bit of theoretical physics.
With World War I raging on throughout Europe, Hilbert could be found sitting in his office at the great university at Göttingen trying and trying again to understand one idea—Einstein’s new theory of gravity.
Göttingen served as the center of mathematics for the Western world by this point, and Hilbert stood as one of its most notorious thinkers. He was a prominent leader for the minority of mathematicians who preferred a symbolic, axiomatic development in contrast to a more concrete style that emphasized the construction of particular solutions. Many of his peers recoiled from these modern methods, one even calling them “theology.” But Hilbert eventually won over most critics through the power and fruitfulness of his research.
For Hilbert, his rigorous approach to mathematics stood out quite a bit from the common practice of scientists, causing him some consternation. “Physics is much too hard for physicists,” he famously quipped. So wanting to know more, he invited Einstein to Göttingen to lecture about gravity for a week.
Before the year ended, both men would submit papers deriving the complete equations of general relativity. But naturally, the papers differed entirely when it came to their methods. When it came to Einstein’s theory, Hilbert and his Göttingen colleagues simply couldn’t wrap their minds around a peculiarity having to do with energy. All other physical theories—including electromagnetism, hydrodynamics, and the classical theory of gravity—obeyed local energy conservation. With Einstein's theory, one of the many paradoxical consequences of this failure of energy conservation was that an object could speed up as it lost energy by emitting gravity waves, whereas clearly it should slow down.
Unable to make progress, Hilbert turned to the only person he believed might have the specialized knowledge and insight to help. This would-be-savior wasn’t even allowed to be a student at Göttingen once upon a time, but Hilbert had long become a fan of this mathematician’s highly "abstract" approach (which Hilbert considered similar to his own style). He managed to recruit this soon-to-be partner to Göttingen about the same time Einstein showed up.
And that’s when a woman—one Emmy Noether—created what may be the most important single theoretical result in modern physics.
Emmy (officially Amalie Emmy) Noether, born 1882, did not stand out in any particular way as a child, although she did, on occasion, attract some notice for her astonishing quickness in providing accurate answers to puzzles or problems in logic or mathematics. Her father, Max, was a fairly prominent mathematician, and one of her brothers eventually attained a doctorate in math. In retrospect, perhaps the Noethers may be another historical example of a family with a math gene.
Germany in the early years of the 20th century was not a convenient place for a woman who wanted to pursue mathematics, or for that matter, any academic field outside of a few considered appropriate for the sex. Luckily for Noether, she had a facility with languages and was allowed to become certified as a language teacher. But Noether recognized her passion was in mathematics, and she decided to chase her dream and find a way to study the subject at the university level.
While women were not permitted to be official students at most German universities then, they were able to audit courses with the permission of the professor. Noether started this way, sitting in on classes at the University of Erlangen. But she also spent a semester in 1903−1904 auditing courses at Göttingen, where she first encountered Hilbert. Rules surrounding enrollment eventually relaxed, and Noether later matriculated at Erlangen to earn her doctorate in mathematics (summa cum laude) in 1907.
However, women were still not accepted as teachers in German universities at the time. Emmy took her fresh doctorate and became an unofficial assistant to her ailing and increasingly frail father, a professor at Erlangen. She also vigorously attacked her own research, forging a personal and original path through abstract algebra. Just a year after her doctorate, Noether's papers and the doctoral research that she was unofficially supervising gained her election to several academic societies, which prompted invitations to speak around Europe. Among those wanting her around, Hilbert reached out to bring Noether to Göttingen in order to tackle Einstein’s theory.
The problem with Einstein’s theory
No one denied it—Einstein’s Theory of General Relativity was undoubtedly beautiful. It was unlike any theory of nature yet imagined by humankind, more surprising and radical even than the special theory of relativity that Einstein had laid out in his revolutionary paper ten years before.
Newton described gravity simply as a force acting over a distance attracting any two masses, whether planets or apples, to each other. The force was proportional to the product of the two masses and inversely proportional to the square of the distance between them. That’s the entire story, and it worked well for over two hundred years.
But there was a mystery embedded in this description of gravity that physicists lived with for those two centuries. This coincidence was impossible to ignore, yet seemingly impossible to explain. The mass that determined the strength of the gravitational force was the same mass that appeared in Newton’s second law of motion, F = ma; gravitational mass was the same as the “inertial mass.” There was no apparent reason this had to be true, it simply was.
Einstein didn’t think this was mere coincidence. He formulated a “principle of equivalence” that can be described in several ways. One way is to insist that the two types of mass are identical because of a fundamental symmetry in nature; that the laws of physics must take the same form whether one is in a gravitational field or in a region of space with no gravity (say, in a spaceship undergoing an equivalent acceleration). Carrying this principle to its logical conclusions eventually led to the equations of general relativity, the theory considered by many (including the great theoretical physicist Lev Landau) to be “probably the most beautiful of all existing physical theories.”
Although Hilbert recognized that general relativity was a tremendous accomplishment, the energy conservation conundrum struck him as unacceptable. To illustrate this idea, let’s draw a circle around a region of space, as in the diagram here.
The circle might contain electric and magnetic fields, water in motion, or something else. If we keep track of the energy flowing out of (and into) the perimeter of the circle (Ef) during a certain interval of time, then that total transfer of energy is equal to the amount that the total energy inside the circle (Ev) has changed. This is local energy conservation. In simple terms, energy is not created or destroyed, just moved around.