Einstein writes to a fellow mathematician on his Unified Field Theory research, including his favorite equation: "Rik = 0"
Significant ALS in German, signed “A. Einstein,” one page both sides, 8.5 x 11, April 15, 1950. Handwritten scientific letter to his former assistant, the German-American mathematician Ernst Gabor Straus, incorporating several equations. In part (translated): "I managed to decide the question of the compatibility of the strong system. The answer is: there is compatibility in the sense that the multiplicity of the solutions is not restricted too much, but not in the sense that one could specify conditions for the intersection X4 = constant, the fulfillment of which is a solution obtainable by continuation implied in space. The weaker systems obtainable through the principle of variation are such that in one section (apart from the freedom of choice of coordinates and the resulting freedom of choice of four of the 16 functions Gik) there are still 20 functions freely selectable." In fine condition.
In the center of the page, Einstein pens one of the fundamental equations of his General Relativity Theory: "Rik = 0," the equation for a vanishing Ricci Curvature Tensor. The Ricci Curvature Tensor measures the deviation of a curved spacetime from a Euclidean framework, and it is a concept of central importance for Einstein's General Relativity and Unified Field Theories. When the Ricci Curvature Tensor is equal to a zero value, the equation betokens a spacetime that is stable and static (neither expanding nor contracting). “Rik = 0” has been called 'Einstein’s favorite equation' and Einstein spent the second half of his life and career attempting to fine tune the precise value of this equation.
In 1915, after years of intense struggle, Einstein showed in his Theory of General Relativity that gravity can be described as the curvature of the geometry of four-dimensional “spacetime.” For the last three decades of his life, working either alone or with an assistant—he had a number of them over the years—Einstein tried in vain to construct a “unified field theory,” which would add the electromagnetic force to such a geometrical description.
From 1945 until his death a decade later, Einstein worked on a form of that theory in which the fundamental geometrical elements—the “metric tensor,” the “connection,” and the objects constructed from them—have both a symmetric and antisymmetric part. That is the theory that Einstein is discussing in this letter to Ernst Straus, who served as his assistant from 1944-1948, and who thus was intimately familiar with the work. Einstein excitedly relates here some results he has newly obtained—it is not hard to believe that he and Straus had striven for these same results just a few years before.
What Einstein wouldn’t accept at the time was that the search for any such classical, purely geometrical theory was doomed from the start. From the 1930s onwards physicists knew that the most fundamental description of physics must be quantum mechanical. Not only did Einstein’s work not attempt a quantum description, but it also left out the other two fundamental forces of nature, the strong and weak nuclear forces, which are responsible for holding the atomic nucleus together, and for radioactivity. Thus, Eisntein's long search for a unified theory never came to fruition.