Spacetime Structure – I: Non-Dynamical Metric

In General
* Distinctions: Different philosophers of science have disagreed on which aspects of spacetime structure are a priori (Kant thought that both topology and metric are; Euclidean geometry is necessary because that is how our perception of the world is organized) and which ones are subject to determination by empirical evidence; Also, aspects of spacetime structure can be considered a fixed part of the background, or dynamical.
* Role of models: Spacetime models have often been considered just as "scratchpads" on which the physics of matter is discussed; But they carry structures (including topology, differentiable structure, geometry) which has observable consequences, regardless of whether one considers them to be dynamical or not.
@ References: Brans in(80); Brans GRG(99)gq/98 [and quantum logic, hole argument]; Lo Surdo a1209 [transition from Euclidean-Newtonian spacetime to relativity]; Weatherall a1707-ch [rev].

Aristotelian Model
* Top / Diff and extra structure: A manifold M = \(\mathbb R\)⁴; A function t on M with nowhere-vanishing gradient dt.
* Metric: A preferred metric g with signature (+,+,+,+).
* Symmetry group: Includes time and space translations, space rotations, and space reflection; 7 parameters.
> Online resources: see Visual Relativity page.

Galilean Model > s.a. Galilean Transformations.
* Top / Diff and extra structure: A manifold M = \(\mathbb R\)⁴; A preferred foliation by a time function t (absolute space).
* Metric: A contravariant metric g^ab such that t has a nowhere vanishing null gradient; Of the covariant metric, only the spatial part is defined.
@ Symmetries: Galvan qp/00 [asymptotic, and particle dynamics].
> Online resources: see Visual Relativity page.

Newtonian Model > s.a. newton-cartan theory.
* Top / Diff and extra structure: A manifold M = \(\mathbb R\)⁴; A preferred foliation by a time function t (absolute space).
* Dynamics: After atomic theory, only particles, with extended media/fluids as convenient approximations.
@ References: Marinov PLA(75) [Harress experiment and support for absolute space]; Anderson AJP(90)dec; Arthur BJPS(94); Lynden-Bell & Katz PRD(95)ap [physics without absolute space]; Navarro & Sancho JGP(02) [as limit of Lorentzian geometry]; Bernal & Sánchez JMP(03)gq/02 [comparison with Leibnizian and Galilean]; Saunders PhSc(13)-a1609 [inertial frames are not needed]; Weatherall a1707 [standard of rotation].
> Online resources: see Visual Relativity page.

Minkowskian Model > s.a. field theory; minkowski space; special relativity.
* Background structure: Topology, differentiable, and 4D affine space structure; Spacetime is \(\mathbb R^4\) with a Lorentz-invariant Minkowski metric.
* Dynamical structure: Only matter fields have dynamics, governed by relativistic field theories.
@ Axiomatic: Robb 14, 36 [causality-based]; Alexandrov CJM(67) [chronogeometry]; Goldblatt 87; Darrigol SHPSA(07) [Helmholtzian approach]; Cocco & Babic JPhilL(20)-a2007.
@ From observables: Desloge FP(90) [space and time measurements]; Summers & White CMP(03)ht [quantum].
@ Alternative descriptions: Lanczos JMP(63) [highly curved Riemannian space]; Chappell et al a1205 [Clifford multivectors].

Other Models > see contemporary dynamical-metric models.

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send feedback and suggestions to bombelli at olemiss.edu – modified 27 jul 2020