Spacetime Structure – II: Dynamical Metric  

Einsteinian Model > s.a. complex; cosmological models; embeddings; general relativity; Raumproblem; types of metrics and spacetimes.
* Idea: An equivalence class of Lorentzian manifolds under (active) diffeomorphisms.
* Background structure: Top and diff (not, e.g., time); Spacetime is a 4D connected, C Hausdorff manifold; Notice that E Cassirer tried to negotiate between Kant's and Einstein's positions by saying that the spacetime topology is a priori, while the metric is subject to empirical evidence.
* Dynamical structure: A C2 (or C1,+) time-oriented metric of lorentzian signature, obeying the Einstein equation.
@ Mathematical: Busemann 67; Pimenov 68.
@ Algebraic description: Geroch CMP(72) [Einstein algebras]; Yodzis PRIA(75); Heller IJTP(92); Akofor a0907.
@ Axiomatic: Penrose essay(66); Ehlers et al in(72) [from light propagation; re GRG(12)]; Woodhouse JMP(73); Schröter RPMP(88), & Schelb RPMP(92); Borchers & Sen CMP(90); Rau pr(90); Rodrigues et al FPL(90); Audretsch & Lämmerzahl JMP(91), in(95); Covarrubias IJTP(93); Schelb IJTP(97) [Weyl vs Lorentz structrure]; Nikolić gq/99 [space-time distinction]; Lämmerzahl GRG(01)gq; Kopf & Paschke MPLA(01)mp-in; Schröter gq/07, gq/07, gq/07; Sánchez-Rodríguez AIP(08)-a0803 [geometrical structures]; Krasnikov PRD(09)-a0903 [hole-free spacetimes]; Andréka et al Synth(12)-a1005 [and special relativity]; Rau in(93)-a1009 [Robb's axiomatic approach and Wey's Raumproblem]; Mamone-Capria FP(16)-a1603.
@ Newtonian limit: Ehlers CQG(97).
@ Signature: van Dam & Ng PLB(01)ht; Wetterich PRL(05)ht/04, Darabi gq/04 [spontaneous symmetry breaking]; Yahalom FP(08)gq/06 [from stability]; Kehayias et al PRD(14) [emergent Lorentzian signature, fermions, and the standard model]; > s.a. modified electromagnetic theory.
@ Causal / conformal structure: Jadczyk IJTP(79); Audretsch & Lämmerzahl GRG(95) [axiomatic]; > s.a. causality; conformal structure.
@ Related topics: Segal in(88); Cassa JGP(96) [from world-lines]; Anderson gq/99 [no need for metric]; Bernal et al FP(02)gq/00 [clocks and rods]; Pauri & Vallisneri gq/02-fs [identifying points]; Mitra gq/05 [and dark matter + dark energy]; Jonsson AJP(05)mar-a0708 [visualizing]; Anderson SHPMP(07) [3-space geometrodynamics without spacetime]; Bel a0711 [global structure and meaning]; Madarász et al a0709 [time dilation and equivalence principle]; Fernández & Rodrigues in(10)-a0909 [as plastic distortion of Lorentz vacuum]; news ea(11)apr [visualization through tendex lines and vortex lines]; Braeck & Grøn EPJP(13)-a1204 [river model]; Bacelar a1306 [Einstein's physical geometry]; > s.a. Hole Argument; Splitting Theorem.
> Related topics: see boundaries; Chronogeometry; Compactification; metric decomposition and matching; lorentzian geometry [analogs]; spacetime [geometry].

Classical Generalizations > s.a. branes; kaluza-klein theory; lorentz symmetry violations; special relativity.
* Types: Violations of Lorentz symmetry; More general structures than Lorentzian manifolds (\(\mapsto\) signature change, degenerate metrics, Finsler geometry, area metrics, ...); Higher-dimensional (unified theories); Small-scale topology.
* Two-point functions: A possible generalization is the use of non-local bitensors rather than local tensors like the metric gab, such as Synge's world function and the van Vleck determinant, as they encode the metric properties of spacetime and (de)focussing behaviour of geodesics, respectively.
@ General references: Hájíček JMP(71) [non-Hausdorff]; Caccese et al gq/99 [and inertia]; Bel a1103 [models satisfying Helmholtz's free mobility postulate]; Giesel et al PRD(12)-a1202 [possible tensorial geometries, and gravitational dynamics]; Stargen & Kothawala PRD(15)-a1503 [2-point functions]; Hohmann a1403 [observer-dependent geometries]; Höhn et al a1811 [Lorentzian manifolds, generalized geometries and Mach's principle].
@ More general connections: Beltrán & Koivisto PLB(16)-a1509 [with torsion and non-metricity from vector distortion]; > s.a. connections; torsion.
> Other types: see finsler geometry; modified general relativity [signature change].

Quantum Models and Other Fundamental Aspects > s.a. discrete and quantum spacetime, and dimensionality [including dimensional reduction].
@ And logic: Madarász et al in(06)gq [and first-order logic]; Uckelman & Uckelman SHPMP(07) [modal and temporal logics for abstract spacetime structures].
@ Related topics: Finkelstein & Rodriguez PhyD(86) [levels of description and structures]; Finkelstein IJTP(99) [unification with dynamical laws]; Nakanishi IJMPD(11)-ht/06 [in ultimate theory]; & H B Nielsen; Durham a1106-FQXi, Wharton a1106-FQXi [digital or analog?]; Bradonjić a0905, a1103/FP [structure above the electroweak symmetry-breaking scale]; Ben-Yami a2005 [physical indeterminacy]; Khosravi & Saueressig a1907 [unitarity and quantum information]; > s.a. non-dynamical spacetime models; emergence [including spacetime from entanglement].

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