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 d*t*.

* __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-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].

@ __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 20
jul 2017