Can this be true? That Albert Einstein, one of mankind’s greatest scientific geniuses, was outsmarted by someone relatively unknown? The answer is both yes and no. No because Einstein is still the sole genius behind the theory of relativity, not to mention many other accomplishments such as the pathbreaking discovery of the photoelectric effect and the physics behind Brownian motion. And Yes because … well, it’s a complicated story. Just read on!
The twist in this story is due to the work of the German physicist, Karl Schwarzschild (1873-1916) who provided the first solution of Einstein’s field equations in the general theory of relativity, published in late November 1915. Remarkably, Schwarzschild derived that solution within one month of receiving a copy of Einstein’s paper. Even more remarkable, he did it in the middle of fighting a war in the German army against the Russians. Sadly, Schwarzschild died shortly after presenting his results, at the age of 42.
Who was Karl Schwarzschild, and what inspired him to close the theoretical gaps left by Einstein?
After earning his doctorate in physics, Schwarzschild worked under two illustrious mathematicians, David Hilbert and Hermann Minkowski, both of whom were leading lights at the famed University of Gottingen in Germany. When the war broke out, Schwarzschild, volunteered out of patriotism to serve in the artillery. He was then 40 years of age.
In November 1915, Einstein published his masterpiece, the General Theory of Relativity, in which he corrected Newton’s theory of gravitation in spectacular fashion. Gravity, according to Einstein, is not a “force” that exists between the Sun and the Earth. Rather, it is warped space-time. What the Sun actually does is create a “valley” in the fabric of space-time, with the Earth orbiting around the upper slopes of this valley like a ball in a roulette wheel. It is this curving of space-time caused by a mass like the Sun that creates gravity. Put differently, “matter tells space-time how to warp, and warped space-time tells matter how to move.”
Schwarzschild obtained a copy of Einstein’s landmark paper in December, a month after its publication. The paper’s beauty and daring moved him deeply. At the same time, he realized that Einstein had not really solved the field equations that underpin his theory. Whereas one equation is sufficient to describe gravity (as a force) in Newton’s theory, Einstein needed ten. As a result, finding the curvature of space-time for a given distribution of matter (technically known as a solution to Einstein’s gravitational field equations) is hard. Einstein himself thought it was impossible to simplify his ten equations into one. Nonetheless, he found an approximate expression for the curved space-time around the Sun that accurately predicted the motion of Mercury – the planet closest to the Sun. That prediction was spot on and it was enough for general relativity to “trounce” Newton’s theory of gravity.
Schwarzschild’s singular ambition was to fill the gap in Einstein’s paper by showing that it is possible to derive one equation in place of ten. To do so, he made three basic assumptions. First, he assumed that the mass that creates the valley such as the Sun is perfectly spherical. Second, he assumed that the curvature of space-time is time-invariant. And finally, he assumed that the curvature of space-time does not depend on the direction, but only on the radial distance from the Sun.
Incredibly, these assumptions allowed Schwarzschild to collapse Einstein’s ten equations into just one. Not only that, he showed that his solution is the only one that exists given the assumptions. In effect, Schwarzschild had achieved what Einstein thought was impossible – all within a month of receiving a copy of Einstein’s hugely complicated paper, and while fighting a war on the Eastern Front!
On 22 December 1915, Schwarzschild wrote up his paper and sent it with a covering letter to Einstein. It concluded: “As you see the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas.”
Einstein immediately replied to Schwarzschild thus: “I have read your paper with the utmost interest. I had not expected that one could formulate the solution to the problem in such a simple way. I liked very much your mathematical treatment of the subject.” He then forwarded Schwarzschild’s letter to the Prussian Academy the following week, and delivered a summary of Schwarzschild’s paper on 13 January 1916. Meanwhile, Schwarzschild was working on another related paper, one that would seed the theory of black holes, those monstrous objects in the universe whose density is so great (due to the extreme warping of space-time) that nothing, not even light can escape out of them. The Prussian Academy presented Schwarzschild’s black hole paper on 13 February 1916. A few months later, Schwarzschild died from illness, at the age of just 42. His ingenious theories, however, survived him and the scientific world is better for it.