Mie's Theories of Matter and Gravitation

Abstract

Unifying physics by describing a variety of interactions---or even all interactions---within a common framework has long been an alluring goal for physicists. One of the most ambitious attempts at unification was made in the 1910s by Gustav Mie. Mie aimed to derive electromagnetism, gravitation, and aspects of the emerging quantum theory from a single variational principle and a well-chosen world function (Hamiltonian). Mie's main innovation was to consider nonlinear field equations to allow for stable particle-like solutions (now called solitons); furthermore he clarified the use of variational principles in the context of special relativity. The following brief introduction to Mie's work has three main objectives.1 The first is to explain how Mie's project fit into the contemporary development of the electromagnetic worldview. Part of Mie's project was to develop a relativistic theory of gravitation as a consequence of his generalized electromagnetic theory, and our second goal is to briefly assess this work, which reflects the conceptual resources available for developing a new account of gravitation by analogy with electromagnetism. Finally, Mie was a vocal critic of other approaches to the problem of gravitation. Mie's criticisms of Einstein, in particular, bring out the subtlety and novelty of the ideas that Einstein used to guide his development of general relativity. In September 1913 Einstein presented a lecture on the current status of the problem of gravitation at the 85th Naturforscherversammlung in Vienna. Einstein's lecture and the ensuing heated discussion, both published later that year in the Physikalische Zeitschrift, reflect the options available for those who took on the task of developing a new theory of gravitation. The conflict between Newtonian gravitational theory and special relativity provided a strong motivation for developing a new gravitational theory, but it was not clear whether a fairly straightforward modification of Newton's theory based on classical field theory would lead to a successful replacement. Einstein clearly aimed to convince his audience that success would require the more radical step of extending the principle of relativity. For Einstein the development of a new gravitational theory was intricately connected with foundational problems in classical mechanics, and in the Vienna lecture he motivated the need to extend the principle of relativity with an appeal to Mach's analysis of inertia. According to Einstein Mach had accurately identified an ``epistemological defect'' in classical mechanics, namely the introduction of a distinction between inertial and non-inertial reference frames without an appropriate observational basis.2 The special theory of relativity had replaced Galilean transformations between reference frames with Lorentz transformations, but the principle of relativity still did not apply to accelerated motion. Extending the principle of relativity to accelerated motion depended on an idea Einstein later called ``the most fortunate thought of my life,'' the principle of equivalence. This idea received many different formulations over the years, but in 1913 Einstein gave one version of this principle as a postulate: his second postulate requires the exact equality of inertial and gravitational mass. He further argued that this equality undermines the ability to observationally distinguish between a state of uniform acceleration and the presence of a gravitational field. The principle of equivalence gave Einstein a valuable link between acceleration and gravitation, tying together the problem of gravitation and the problem of extending the principle of relativity. At the time of the Vienna lecture Einstein was in the midst of an ongoing struggle to clarify the connections among Mach's insight, a generalized principle of relativity, and the formal requirement of general covariance, a struggle that would continue for several more years. Although he also drew heavily on classical field theory in his work, he was convinced that this cluster of ideas would provide the key to a new theory of gravitation.