Einstein's Physical Strategy, Energy Conservation, Symmetries, and Stability: "but Grossmann & I believed that the conservation laws were not satisfied"

TL;DR

This paper explores Einstein's physical strategy in deriving his field equations, focusing on energy conservation, symmetries, and stability issues, and compares it with particle physics approaches and historical perspectives.

Contribution

It provides a detailed analysis of Einstein's use of energy-momentum complexes, symmetries, and stability considerations in the context of the Entwurf theory and its historical critique.

Findings

Einstein's Entwurf Lagrangian modeled on a scalar theory, not Maxwell's.

The Entwurf theory has three ghosts, failing stability tests.

Historical analysis of Einstein's energy conservation and symmetry use.

Abstract

Recent work on the history of General Relativity by Renn, Sauer, Janssen 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? Einstein used an identity from his assumed linear coordinate covariance x'= Mx 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 as on a scalar theory. As a result, it also has 3 ghosts, failing a 1920s-30s a priori particle physics stability test with antecedents in Lagrange's and Dirichlet's stability work. 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. Particle physics also can be useful in the historiography of gravity and space-time. This topic can be a useful case study in the history of science on recently reconsidered questions of presentism, whiggism and the like.