Finding Field Equations...

Figure 1. Albert Einstein (1879–1955), in this 1916 photograph, poses in his study at Wittelsbacherstraße 13 in Berlin-Wilmersdorf. (Courtesy of the Leo Baeck Institute, New York.)
Citation: Phys. Today 68, 11, 30 (2015); http://dx.doi.org/10.1063/PT.3.2979

Topics: Einstein, History, General Relativity, Research

In his later years, Einstein often claimed that he had obtained the field equations of general relativity by choosing the mathematically most natural candidate. His writings during the period in which he developed general relativity tell a different story.

This month marks the centenary of the Einstein field equations, the capstone on the general theory of relativity and the highlight of Albert Einstein’s scientific career.¹ The equations, which relate spacetime curvature to the energy and momentum of matter, made their first appearance in a four-page paper submitted on 25 November 1915 to the Prussian Academy of Sciences in Berlin and reprinted in TheCollected Papers of Albert Einstein (CPAE),² volume 6, document 21. How did Einstein, shown in figure 1, arrive at those equations? He later insisted that the gravitational equations “could only be found by a purely formal principle (general covariance).”³ Such statements mainly served to justify his strategy in the search for a unified field theory during the second half of his career. As a description of how he found the field equations of general relativity, they are highly misleading.

The 25 November paper was the last in a series of short communications submitted to the Berlin Academy on four consecutive Thursdays that month (CPAE 6; 21, 22, 24, 25). In the first paper, Einstein replaced the field equations that he had published in 1913 with equations that retain their form under a much broader class of coordinate transformations (see figure 2). In the second, a highly speculative hypothesis he adopted about the nature of matter allowed him to change those equations to equations that are generally covariant—that is, retain their form under arbitrary coordinate transformations. In the fourth, he achieved the same end by changing the field equations of the first paper in a different and more convincing way, as shown in figure 3. In the third, based on the field equations of the second paper but unaffected by the modification of the fourth, he accounted for the 43 seconds of arc per century missing in the Newtonian account of the perihelion motion of Mercury.

Physics Today: Arch and scaffold: How Einstein found his field equations
Michel Janssen and Jürgen Renn