Abstract
Foliations of spacetime play major roles, firstly in dynamical and canonical formulations, and secondly in modelling the different possible fleets of observers within approaches in which spacetime is primary. Background Independent Physics is moreover to possess Foliation Independence. If this cannot be established, or fails, a Foliation Dependence Problem facet of the Problem of Time has been encountered. For classical general relativity itself, this has the status of a solved problem, due to Teitelboim’s demonstration that it follows from the Dirac algebroid formed by the Hamiltonian and momentum constraints.
In this chapter, we review the previous half-century’s significant works on foliations of spacetime, covering the theory in this chapter and some applications in the next. The theory involves more modern geometrical notions and formulations than Dirac and Arnowitt–Deser–Misner’s earlier works on canonical general relativity, as developed by e.g. Kuchař, Teitelboim and Isham. This approach places emphasis on spacetime primality followed by splitting spacetime. We also outline spaces of embeddings and of foliations, bubble time and many-fingered time.
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Notes
- 1.
This can be formulated in terms of a fibre bundle \(\boldsymbol{\mathfrak{T}}(\boldsymbol{\mathfrak{R}}\mbox{iem}(\boldsymbol{\Sigma}))\): the tangent bundles space of extrinsic curvatures with over the base space \(\boldsymbol{\mathfrak{R}}\mbox{iem}(\boldsymbol{\Sigma})\). Another formulation involves using \(\mathbf{p}\) in place of \(\mathbf{K}\). In the latter case, the \(\underline{\boldsymbol{\mathfrak{s}}\mbox{ym}}\) involved is a space of symmetric 2-tensor densities and the corresponding fibre bundles space is \(\boldsymbol{\mathfrak{T}}^{*}(\boldsymbol{\mathfrak{R}}\mbox{iem}(\boldsymbol{\Sigma}))\).
- 2.
They furthermore arm this with the topological structure that it inherits naturally as an open subset of \(\mathfrak{c}^{\infty }(\boldsymbol{\Sigma}, \mathfrak{m})\). This space can be considered [426] as a manifold in the sense of Fréchet (see Appendix H.2).
- 3.
This notion of deformation indeed coincides with that of hypersurface deformation and of deformation algebroid.
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Anderson, E. (2017). Embeddings, Slices and Foliations. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_31
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