Arch and scaffold: How Einstein found his field equations Michel Janssen, and Jürgen Renn Citation: Physics Today 68, 11, 30 (2015); doi: 10.1063/PT.3.2979 View online: https://doi.org/10.1063/PT.3.2979 View Table of Contents: http://physicstoday.scitation.org/toc/pto/68/11 Published by the American Institute of Physics Articles you may be interested in The laws of life Physics Today 70, 42 (2017); 10.1063/PT.3.3493 Hidden worlds of fundamental particles Physics Today 70, 46 (2017); 10.1063/PT.3.3594 The image of scientists in The Big Bang Theory Physics Today 70, 40 (2017); 10.1063/PT.3.3427 The secret of the Soviet hydrogen bomb Physics Today 70, 40 (2017); 10.1063/PT.3.3524 Physics in 100 years Physics Today 69, 32 (2016); 10.1063/PT.3.3137 The top quark, 20 years after its discovery Physics Today 68, 46 (2015); 10.1063/PT.3.2749 Arch and scaffold: How Einstein found his field equations Michel Janssen and Jürgen Renn 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. his month marks the centenary of the pothesis he adopted about the nature of matter al- Einstein field equations, the capstone on lowed him to change those equations to equations the general theory of relativity and the that are generally covariant—that is, retain their highlight of Albert Einstein’s scientific form under arbitrary coordinate transformations. In T 1 career. The equations, which relate space- the fourth, he achieved the same end by changing time curvature to the energy and momentum of the field equations of the first paper in a different matter, made their first appearance in a four-page and more convincing way, as shown in figure 3. In paper submitted on 25 November 1915 to the Prus - the third, based on the field equations of the second sian Academy of Sciences in Berlin and reprinted in paper but unaffected by the modification of the The Collected Papers of Albert Einstein (CPAE),2 vol- fourth, he accounted for the 43 seconds of arc per ume 6, document 21. How did Einstein, shown in century missing in the Newtonian account of the figure 1, arrive at those equations? He later insisted perihelion motion of Mercury. that the gravitational equations “could only be found In the first November paper, Einstein made it by a purely formal principle (general covariance).”3 sound as if he had gone from the old to the new field Such statements mainly served to justify his strategy equations by leveling one cathedral and building a in the search for a unified field theory during the new one on its ruins in a completely different style. second half of his career. As a description of how he The former was built according to principles of found the field equations of general relativity, they physics; the latter, Einstein wanted his readers to be- are highly misleading. lieve, according to principles of mathematics. “It is The 25 November paper was the last in a series a real triumph of the method of the general differ- of short communications submitted to the Berlin ential calculus,” he rhapsodized—even before his Academy on four consecutive Thursdays that month theory was generally covariant (CPAE 6; 21). Careful (CPAE 6; 21, 22, 24, 25). In the first paper, Einstein examination of the November papers and his corre- replaced the field equations that he had published spondence at the time suggests a different architec- in 1913 with equations that retain their form under tural metaphor.4 Einstein used the elaborate frame- a much broader class of coordinate transformations work he had built around the field equations of 1913 (see figure 2). In the second, a highly speculative hy- as a scaffold on which he carefully placed the arch stones of the Einstein field equations. Michel Janssen is a professor in the Program in the History of Science, Technology, and Medicine at the University of Covariance lost Minnesota in Minneapolis. Jürgen Renn is a director at the As the box on page 31 shows, Einstein had already Max Planck Institute for the History of Science in Berlin. considered the equations of his first November 30 November 2015 Physics Today www.physicstoday.org General relativity under construction The so-called Zürich notebook contains Albert Einstein’s research notes of 1912–13 as, with the help of Marcel Grossmann, he began looking for field equations for his new theory of gravity.6 At the top of this particular page, 22R, Einstein wrote down the generally covariant Ricci tensor Til under the heading “Grossmann,” presumably because Grossmann had suggested it to him. “If [the determinant of the metric] G is a scalar” (Wenn G ein Skalar ist ), Einstein noted, the first half of Til transforms as a tensor. Coordinate transformations satisfying that determinant condition are now called unimodular trans- formations. If the first half transforms as a tensor under unimodular transformations, the second half must too, since their sum transforms as a tensor under arbitrary transformations. Underneath the second half of the second equation, Einstein wrote, “probable gravitation tensor” (Vermutlicher Gravitationstensor). Thus in 1912–13 Einstein had already considered the field equations he proposed in the first of the four papers he submitted to the Prussian Academy of Sciences in Berlin in November 1915 (see figure 2). Near the middle of the page, Einstein imposed the condition μν Σκ∂γκα /∂xκ =0, in modern notation ∂μ g =0. That is the same condition he imposed in the first November 1915 paper to show that those field equations reduce to their Newtonian counterpart in the case of weak static fields. (Courtesy of the Albert Einstein Archives, Hebrew University of Jerusalem.) paper three years earlier in the course of his collab- that is not very practical) you can turn the route be- oration with mathematician Marcel Grossmann,5 tween the two cities into a straight line. That ability shown in figure 4. The two of them had been class- is the analogue of general covariance. Nonetheless, mates at what is now ETH Zürich and were reunited no matter what it looks like on any given map, the at their alma mater in July 1912. Notes of their col- segment of the great circle remains the shortest route. laboration have been preserved in the so-called Similarly, general covariance cannot make all trajec- Zürich notebook.6 In the notebook, as Einstein re- tories in spacetime equivalent, and it is simply not the called in the introduction of his first November paper, case that all states of motion are on the same footing. they had given up the search for field equations In a sense, therefore, the formulation of gener- based on the Riemann tensor “with a heavy heart.” ally covariant field equations in November 1915 was What had defeated them were problems with the an empty victory. Despite its misleading name, physical interpretation of such equations. Eventu- however, general relativity was a powerful new the- ally they adopted field equations specifically engi- ory of gravity, based on the equivalence principle, neered to avoid those problems and published them which in the mature form Einstein gave it in 1918 in a joint paper in June 1913 (CPAE 4; 13). The theory states that both the spacetime geometry and gravity and the field equations they presented are known, should be represented by the metric tensor field (see based on the title of that paper, as the Entwurf (out- reference 7 and CPAE 7; 4). line or draft) theory and the Entwurf field equations. Except for its field equations, the Entwurf the- Coordinate restrictions ory has all the basic elements of the mathematical Why did Einstein reject the field equations of the formalism of general relativity. Einstein nonetheless first November paper when he and Grossmann first cautiously referred to it as a generalized theory of considered them in 1912–13? For a long time, the an- relativity—not a general theory—because of the swer given by historians was that the two did not limited covariance of the Entwurf field equations. know about coordinate conditions. To compare field Part of Einstein’s difficulty was that throughout equations of broad covariance with field equations the period from late 1912 to late 1916, he conflated of limited covariance, such as the Poisson equation general covariance and general relativity of motion. of Newtonian theory, the former need to be consid- A map analogy illustrates the difference. The short- ered in a similarly restricted class of coordinate sys- est route between two cities is a segment of the great tems. That is done with the help of a coordinate con- circle connecting them. Suppose you have a map on dition consisting of four equations for the metric which that route does not correspond to a straight tensor field (hereafter, metric for short). Imposing a line. By switching to a different map (probably one suitable coordinate condition, one eliminates various www.physicstoday.org November 2015 Physics Today 31 Arch and scaffold John Norton suggested that it may have been because the four-divergence of the metric of Minkowski spacetime in rotating coordinates does not vanish.9 For ease of reference, we rephrase Nor- ton’s suggestion using some shorthand introduced in The Genesis of General Relativity1 (hereafter Gene- sis): Einstein rejected the November tensor (the Rim introduced in figure 2) because the rotation metric (the metric of Minkowski spacetime in rotating coordinates) does not satisfy the Hertz condition (the vanishing of the four-divergence of the metric).