Abstract
This chapter further develops Background Independence and the Problem of Time at the classical level for approaches in which spacetime is primary (a position we started in Chap. 10). We consider spacetime diffeomorphisms, and contrast the corresponding Spacetime Relationalism with the dynamically primary approaches’ Temporal and Configurational Relationalism. Given that spacetime diffeomorphisms generators close, we also consider associated notions of spacetime or path (alias Bergmann) observables. We introduce the space of spacetimes and the space of general relativity solutions. Finally, we give the first of two parts of an outline of Bergmann’s work and its expansion by Pons, Salisbury and Sundermeyer. This involves using larger groups than the spacetime diffeomorphisms, such for now the diffeomorphism-induced gauge group.
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Notes
- 1.
One might moreover consider a notion of weak equality instead, now meaning up to terms containing the generators. Also, one is only to select subsets of generators which close algebraically. The Jacobi identity applying to all Lie brackets, these
are also guaranteed to close as a Lie algebra. - 2.
Infinitesimal transformations \(\vec{X} \rightarrow \vec{\widetilde{X}}\) can be written as \(\vec{X} - \vec{\widetilde{X}} = \vec{\epsilon }\). Viewed as solutions in terms of Hamiltonian variables, the right hand side functions here are so-called descriptors: a fairly standard Gauge Theoretic notion, see e.g. [13, 134]). For GR, descriptors are of the particular form \(\vec{\upnu }(\vec{X}; \boldsymbol{\**}(\vec{X})]\). Here \(\boldsymbol{\**}\) denotes the set of dynamical fields \(\boldsymbol{\uppsi}\) and \(\mathbf{h}\); note that this specifically excludes the lapse \(\upalpha \) and shift \(\upbeta^{i}\). Dittrich’s \(\mbox{V}^{\mu }\) in (25.30) and Chap. 27.5’s Weyl scalars can be viewed as particular cases of descriptors.
- 3.
See [721] for the sense in which this is ‘induced’. \(\mathit{Digg}(\mathfrak{m})\) might also be denoted \(BK(\mathfrak{m})\) after Bergmann and Komar [134], though they themselves referred to it as the ‘\(Q\)-group’. Physicists Josep Maria Pons, Donald Salisbury and Kurt Sundermeyer prefer to use the Bergmann–Komar name for the subsequent projected version of this group that introduced in Chap. 32.4. Incidentally, the existence of this larger invariance group does not by itself dictate that it is the gauge group. [This is one example of Sect. 27.7’s matter, as well as the more specific reason to use the projected version.]
- 4.
For suppose that a model, with maximal symmetry group
, fails to capture some physically meaningful features which themselves do not respect 
. Then there is a smaller choice group, ℌ, which happens to be more physical. Moreover, one way of finding it and the missing physically meaningful features is to consider not just 
for the original model, but rather it and all its subgroups. This is out of the possibility that one or more of these are more physically valuable than 
itself. - 5.
Moreover, extending \(\mathfrak{q}\) or its spacetime equivalent may further enlarge the largest symmetry group ℌ, break ℌ or both at once. I.e. extend the subgroup that survives a breaking in a different way from the original ℌ.
are also guaranteed to close as a Lie algebra.
, fails to capture some physically meaningful features which themselves do not respect 
. Then there is a smaller choice group, ℌ, which happens to be more physical. Moreover, one way of finding it and the missing physically meaningful features is to consider not just 
for the original model, but rather it and all its subgroups. This is out of the possibility that one or more of these are more physically valuable than 
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Anderson, E. (2017). Spacetime Relationalism. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_27
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