California Institute of Technology
Einstein Papers Project
One Hundred Years of Einstein's Field Equations, and of the Explanation of Mercury's Perihelion

100 years ago Albert Einstein submitted a paper to the Royal Prussian Academy of the Natural Sciences which contained what we today call the Einstein field equations: the new law of gravity that superseded Isaac Newton’s inverse square law of gravity. On the same day, 25 November 1915, that Einstein submitted said paper, his preceeding paper on Mercury's perihelion was published. Here, Einstein correctly calculated an anomoly in Mercury's motion that had remained an unsolved puzzle in the context of the Newtonian theory of gravity. The paper had been submitted only a week earlier in turn, but it did not yet contain the final gravitational field equations that would become the core of the general theory of relativity.

Ten days prior to his submission of the final field equations, Einstein wrote to David Hilbert that he was suffering from exhaustion and abdominal pains. His intense work on general relativity and poor nutrition caused by the ongoing war had clearly taken their toll on Einstein's health. Still, when Einstein submitted the field equations on 25 November 1915, he was aware that he had reached his goal: the discovery of a law of gravity more accurate than Newton's, consistent with the results of special relativity, and indeed a generalisation of the latter theory. From the very beginning of searching for this new law of gravity, Einstein took the lesson from special relativity that mass and energy are equivalent as one of his starting points; or rather the idea that both mass and energy have to produce gravitational fields. The main question was what the resulting gravitational fields would be represented by, what the "left-hand side" of the Einstein field equations would be, given that their "right-hand side" was mass-energy.

The long quest towards the final equations has been deciphered by (in alphabetical order) M. Janssen, J. Norton, J. Renn, T. Sauer and J. Stachel, all of whom have been part of the Einstein Papers Project editorial team. The winding story of the discovery of the Einstein Field Equations has recently been summarized by Janssen and Renn in an article for Physics Today.

That Einstein correctly worked out Mercury's perihelion before arriving at the final gravitational field equations indicates that the latter sit at the center of an intricate theoretical web of mathematical tools, physical assumptions and techniques, which together form general relativity. In the Mercury paper, Einstein used an approximation of what would later be called the Schwarzschild solution to the Einstein field equations to describe the gravitational field of the sun. The main idea he needed was that the sun's gravitational field could be modeled by the so-called metric tensor. He then assumed that the gravitational field of the sun would be static, i.e. not change over time; spherically symmetric; and fall off to zero infinitely far from the sun. With these assumptions, he could find the approximation to a metric tensor that adequately described the gravitational field of the sun. Einstein then assumed that Mercury would move on the geodesics of this metric, i.e. on the straightest possible lines allowed for by the spacetime geometry defined by the metric. Together, these two assumptions made possible one of the most significant empirical confirmations of the new theory of general relativity.

Fortunately, it turns out that the Schwarzschild metric is also a solution of the Einstein field equations, the equations Einstein published a week after the Mercury paper and whose centenary we celebrate today.

Dennis Lehmkuhl