Abstract
Rotation is a challenging riddle for the relationalist. In early versions of the absolute-relational debate for example, Newton’s rotating bucket poured cold water on the relationalist position. While the parameters of the debate have changed, a more recent analysis in 1999 by Belot proclaimed rotation to be “the downfall of relationalism.”In this paper, we provide a relational response to the riddle of rotation. We present a theory that, contrary to orthodoxy, can account for all rotational effects without introducing, as the absolutist does, a fixed standard of rotation. Instead, our theory posits a universal SO(3) charge that plays the role of global angular momentum and couples to inter-particle relations via terms commonly seen in standard gauge theories such as electromagnetism and the Standard Model of particle physics.Our theory makes use of an enriched form of relationalism: it adds an SO(3) structure to the traditional relational description. Our construction is made possible by the modern tools of gauge theory, which reveal a simple relational law describing rotational effects. In this way, we can save all the phenomena of Newtonian mechanics using conserved charges and relationalism. In a second paper, we will further explore the ontological and explanatory implications of the theory developed here.
Highlights
Newton’s rotating bucket pours cold water on the naive relationalist by vividly illustrating how certain rotational effects, those due to non-zero angular momentum, can depend on more than just relations between material bodies
One of the primary technical achievements of this paper is to show that the gauge formalism is powerful enough to extend this symmetry and thereby project the dynamics, given one added choice of constant vector, to a reduced configuration space of inter-particle relations conforming to the intuitions reported in Section §2
Using these properties of the kinematical metric and the known symmetry orbit induced by the action of the gauge group on configuration space, it is possible to treat the configuration space as a principal fibre bundle for this symmetry group and to construct a dynamical connection 1-form on this bundle known as an Ehresmann connection
Summary
Our theory is expressed in terms of two distinct relational force terms: a Lorentzlike force, analogous to the one of electromagnetic theory, describing Coriolis effects; and a force term due to a mass-like quadratic potential describing centrifugal effects. One of the most significant features of the new construction is that it provides an enriched relational interpretation on the rotational bundle when the angular momentum of the system is non-zero. We show that all inertial effects caused by rotational phenomena can be described directly in terms of conserved quantities on relative configuration space This comes about because the motion along the fibres of a curved bundle is conserved if the fibres have a ‘rigid’ geometry.. In the special case where the angular momentum is equal to zero, the motion along the fibres vanishes and is not required to describe the reduced theory While this picture matches the strong relationalism advocated in Barbour & Bertotti (1982), there are notable differences between that work and ours. Our mathematical labours will bear significant conceptual fruit: a compact description of a relational theory of classical (i.e., non-relativistic) rotation in terms of an enticing new proposal for formulating the ontology of such a theory
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