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Congruences of Parallelism of a Field of Vectors by Eisenhart L. P.

By Eisenhart L. P.

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370). Another alternative is to argue along Machian lines that the acceleration is to be judged ‘relative to the fixed stars’. Substantivalism versus relationalism 29 occurs. The most important difference is that the notion of spatial points retaining their identity over time no longer makes sense—there are simply spacetime points whose existence is ‘fleeting’: different spatial points at different times. Thus, we do away with the view attributed to Newton that spacetime is built out of a stack of instantaneous spaces of Euclidean structure, such that there is a definite way to re-identify spatial points over time.

It takes a particularly nasty form in general relativity because, thanks to background independence, there the gauge freedom concerns the points of spacetime and the dynamics of spacetime geometry, rather than fields living on spacetime with the dynamics framed against a fixed background 44 This is true, for example, in the case of general relativity formulated in terms of the metric variables. However, the approach using loop variables is an intrinsically gauge invariant formulation. However, the diffeomorphism invariance is much harder to implement in this approach.

22 Interpretation and Formalism metrical structure. Methods for dealing with the gauge freedom are generally seen as underwriting specific interpretational choices as regards the status of spacetime points, and time and change. However, an underdetermination as regards which method to use leads to a further underdetermination of the interpretational stances by the physics: incompatible interpretations can invoke the same formal methods for dealing with gauge freedom therefore we shouldn’t base interpretative stances on these methods.

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