Abstract
This is a historical-critical study of the hole argument, concentrating on the interface between historical, philosophical and physical issues. Although it includes a review of its history, its primary aim is a discussion of the contemporary implications of the hole argument for physical theories based on dynamical, background-independent space-time structures.The historical review includes Einstein’s formulations of the hole argument, Kretschmann’s critique, as well as Hilbert’s reformulation and Darmois’ formulation of the general-relativistic Cauchy problem. The 1970s saw a revival of interest in the hole argument, growing out of attempts to answer the question: Why did three years elapse between Einstein’s adoption of the metric tensor to represent the gravitational field and his adoption of the Einstein field equations?The main part presents some modern mathematical versions of the hole argument, including both coordinate-dependent and coordinate-independent definitions of covariance and general covariance; and the fiber bundle formulation of both natural and gauge natural theories. By abstraction from continuity and differentiability, these formulations can be extended from differentiable manifolds to any set; and the concepts of permutability and general permutability applied to theories based on relations between the elements of a set, such as elementary particle theories.We are closing with an overview of current discussions of philosophical and physical implications of the hole argument.
Highlights
This is a historical-critical study of the hole argument, concentrating on the interface between historical, philosophical and physical issues
Consider the objects defining the geometry of a general-relativistic space-time with n = 4: Again, if one wants to preserve the four-volumes of space-time, which are needed to formulate meaningful physical averages, one must restrict these transformations to SL(4), the group of 23 special linear transformations with unit determinant
Sophisticated substantivalism may be compatible with taking seriously physicists’ concerns, but does it have a coherent motivation? The obvious thing to be said for the position is that one thereby avoids the indeterminism of the hole argument
Summary
“Hole argument” is the English translation of the German phrase “Lochbetrachtung,” used by Albert Einstein to describe his argument against the possibility of generally-covariant equations for the gravitational field, developed in late 1913 and accepted until late 1915. This article is a historical-critical study, in Ernst Mach’s sense.. This article is a historical-critical study, in Ernst Mach’s sense.2 It includes a review of the literature on the hole argument that concentrates on the interface between historical, philosophical and physical approaches. Recounting the history of the hole argument, the primary purpose is to discuss its contemporary significance – in both physics and philosophy – for the study of space-time structures. In the philosophy of space-time, this leads me to advocate a “third way” that I call dynamic structural realism, which differs from both the traditional absolutist and relationalist positions
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