Abstract
What is the meaning of general covariance? We learn something about it from the hole argument, due originally to Einstein. In his search for a theory of gravity, he noted that if the equations of motion are covariant under arbitrary coordinate transformations, then particle coordinates at a given time can be varied arbitrarily — they are underdetermined — even if their values at all earlier times are held fixed. It is the same for the values of fields. The argument can also be made out in terms of transformations acting on the points of the manifold, rather than on the coordinates assigned to the points. So the equations of motion do not fix the particle positions, or the values of fields at manifold points, or particle coordinates, or fields as functions of the coordinates, even when they are specified at all earlier times. It is surely the business of physics to predict these sorts of quantities, given their values at earlier times. The principle of general covariance seemed quite untenable.
In honour of Professor Stachel. He will agree with me on points, if not on substance, but I am indebted to him for his guidance on both topics; and on many others besides.
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Saunders, S. (2003). Indiscernibles, General Covariance, and Other Symmetries: The Case for Non-Reductive Relationalism. In: Renn, J., et al. Revisiting the Foundations of Relativistic Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0111-3_7
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