Abstract
John Earman’s recent proposal that a substantive version of general covariance consists in the requirement that diffeomorphism invariance be a gauge symmetry is critically assessed. I argue that such a principle does not serve to differentiate general relativity from pre-relativistic theories. A model-theoretic characterization of two formulations of specially relativistic theories is suggested. Diffeomorphisms are symmetries of only one such style of formulation and, I argue, Earman’s proposal does not provide a reason to deny diffeomorphisms the status of gauge transformations relative to this formulation. Carlo Rovelli’s distinction between “passive” and “active” diffeomorphism invariance is also clarified.
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Notes
- 1.
As discussed below, this is but one of a number of closely related properties that go by the name “general covariance”.
- 2.
I gloss over the fact that the claim that “all physical observations consist in the determination of purely topological relations” formed part of Kretschmann’s argument. See Norton (1993, 818), from where the translation is taken.
- 3.
- 4.
The terminology is Earman’s (2006a, 446).
- 5.
This matches the definition given by Earman (1989, 47). Diff(M) is the group of M’s automorphisms; i.e., the group of all invertible maps from M onto itself that preserve its differentiable structure. GC3 is the requirement that Diff(M) be a subgroup of T’s covariance group in Anderson’s sense.
- 6.
Uncontroversial, that is, amongst those who classify GR as a generally covariant theory. Maudlin’s metrical essentialist (Maudlin 1988, 1990) denies that both (M, O 1, O 2, …, O N ) and (M, d ∗ O 1, d ∗ O 2, …, d ∗ O N ) represent genuine possibilities. But even the metrical essentialist can admit that (M, O 1, O 2, …, O N ) and (M, d ∗ O 1, d ∗ O 2, …, d ∗ O N ) are on a par as models of T. They should claim only that, relative to the choice of (M, O 1, O 2, …, O N ) as the representation of a genuine possibility, (M, d ∗ O 1, d ∗ O 2, …, d ∗ O N ) does not represent a possibility (compare Bartels 1996).
- 7.
For an illuminating discussion of various connections between Noether’s theorems and general covariance, see Brown and Brading (2002).
- 8.
The following is, very loosely, based on the much more sophisticated material in Belot (2007, §4).
- 9.
I believe this fits with a more recent characterisation that Rovelli has given (2004, 62–5).
- 10.
Compare Earman’s definition of a dynamical symmetry (Earman 1989, 45).
- 11.
The former need not entail the latter. For example, consider the, admittedly somewhat contrived, theory whose field equations are g ab ∇ a ∇ b Φ − m 2 Φ = 0 and R ab (g) = 0. The metric in this theory has a non-trivial dynamics and constrains, but is unaffected by, the evolution of Φ.
- 12.
David Wallace has pointed out to me that, in the vacuum case, there is no difficulty in principle in obtaining the flatness of the metric via the extremization of an action whose only dependent variables are the components of the metric. What needs to be investigated is whether this observation can be extended to theories involving matter fields. (In vacuum GR the extremization of the gravitational action imposes Ricci flatness but spacetime is certainly not Ricci flat when the theory includes a non-trivial matter Lagrangian.)
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Acknowledgements
I have previously given talks related to the topic of this paper in Oxford, Konstanz, Les Treilles, Barcelona and Montreal. I am grateful to numerous members of those audiences for useful discussion. Support for this research from the Arts and Humanities Research Council (UK) Research Leave Scheme (grant ID No: AH/E506216/1) and from the Spanish government via research group project HUM2005-07187-C03-02 and MICINN project FI2008-06418-C03-03 is also gratefully acknowledged.
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Pooley, O. (2010). Substantive General Covariance: Another Decade of Dispute. In: Suárez, M., Dorato, M., Rédei, M. (eds) EPSA Philosophical Issues in the Sciences. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3252-2_19
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