Einstein׳s physical strategy, energy conservation, symmetries, and stability: “But Grossmann & I believed that the conservation laws were not satisfied”

Abstract

Recent work on the history of General Relativity by Renn et al. shows that Einstein found his field equations partly by a physical strategy including the Newtonian limit, the electromagnetic analogy, and energy conservation. Such themes are similar to those later used by particle physicists. How do Einstein׳s physical strategy and the particle physics derivations compare? What energy–momentum complex(es) did he use and why? Did Einstein tie conservation to symmetries, and if so, to which? How did his work relate to emerging knowledge (1911–1914) of the canonical energy–momentum tensor and its translation-induced conservation? After initially using energy–momentum tensors hand-crafted from the gravitational field equations, Einstein used an identity from his assumed linear coordinate covariance xμ′=Mνμxν to relate it to the canonical tensor. Usually he avoided using matter Euler–Lagrange equations and so was not well positioned to use or reinvent the Herglotz–Mie–Born understanding that the canonical tensor was conserved due to translation symmetries, a result with roots in Lagrange, Hamilton and Jacobi. Whereas Mie and Born were concerned about the canonical tensor׳s asymmetry, Einstein did not need to worry because his Entwurf Lagrangian is modeled not so much on Maxwell׳s theory (which avoids negative-energies but gets an asymmetric canonical tensor as a result) as on a scalar theory (the Newtonian limit). Einstein׳s theory thus has a symmetric canonical energy–momentum tensor. But as a result, it also has 3 negative-energy field degrees of freedom (later called “ghosts” in particle physics). Thus the Entwurf theory fails a 1920s–1930s a priori particle physics stability test with antecedents in Lagrange׳s and Dirichlet׳s stability work; one might anticipate possible gravitational instability.This critique of the Entwurf theory can be compared with Einstein׳s 1915 critique of his Entwurf theory for not admitting rotating coordinates and not getting Mercury׳s perihelion right. One can live with absolute rotation but cannot live with instability.Particle physics also can be useful in the historiography of gravity and space–time, both in assessing the growth of objective knowledge and in suggesting novel lines of inquiry to see whether and how Einstein faced the substantially mathematical issues later encountered in particle physics. This topic can be a useful case study in the history of science on recently reconsidered questions of presentism, whiggism and the like. Future work will show how the history of General Relativity, especially Noether׳s work, sheds light on particle physics.