Causal Set Kinematics  

In General > s.a. Alexandrov Sets; causal structure and spacetime; spin-foam models.
* Hauptvermutung: (Original version) If a causal set can be faithfully embedded in two Lorentzian manifolds (M, g) and (M', g'), then those two manifolds are close down to scales of the order of the embedding density.
* Coarse graining: A random coarse-graining procedure consists in starting with a causal set C and removing each point with probability p.
* Feature: Causal sets can implement the notion that spacetime topology may be scale-dependent; No known continuum approach can do this.
* Definitions of curvature: One definition of curvature at an element x of a causal set with a minimal element 0 is Gudder's, K(x) = number of \(\cal L\)-geodesics between 0 and x, where \({\cal L}(x):= \big[\sum_i(l(a_{i+1})-l(a_i))^2\big]{}^{1/2}\).
@ And posets: Low JMP(00); Droste JMP(05)gq [universal past-finite causal set]; Vatandoost & Bahrampour JMP(11) [and sphere orders].
@ Dimension: Meyer PhD(88), Ord(93); Reid PRD(03)gq/02; Eichhorn & Mizera CQG(14)-a1311 [spectral dimension]; Carlip CQG(15)-a1506, Belenchia et al PRD(16)-a1507 [reduction to spectral dimension = 2].
@ Closeness of causal sets: Yazdi & Kempf CQG(17)-a1612 [spectral distance from the Feynman propagator].

Relationship with the Continuum > s.a. emergence.
@ General references: Bombelli & Meyer PLA(89); Daughton CQG(98) [symmetric case]; Brightwell & Gregory PRL(91); Filk CQG(01)gq [time]; Requardt JMP(03)gq/01 [renormalization group]; Ilie et al CQG(06)gq/05 [longest paths and geodesics]; Henson CQG(06)gq [manifoldlike causal sets]; Surya TCS(08)-a0712 [topology]; Rideout & Wallden CQG(09)-a0810, JPCS(09)-a0811 [lengths]; Benincasa & Dowker PRL(10)-a1001 [scalar curvature]; Krugly a1006 [unfaithful embeddings and matter]; Glaser & Surya PRD(13)-a1309 [proposed definition of locality]; Saravani & Aslanbeigi CQG(14)-a1403; Bolognesi & Lamb a1407; Gudder a1502 [and the emergence of 4D]; Gudder a1507 [c-causets and curvature].
@ 2D causal sets: Brightwell et al CQG(08)-a0706, JPCS(09); Glaser et al a1706 [scaling and transition between non-continuum and continuum phases].
@ Properties of manifoldlike causal sets: Eichhorn et al CQG(17)-a1703 [manifestations of asymptotic silence].

Hypersurfaces, Foliations
@ Thickened spatial hypersurfaces: Major et al CQG(06)gq/05; Major et al JMP(07)gq/06; Major et al CQG(09)-a0902 [stable homology and manifoldlikeness]; Jubb CQG(17)-a1611.
@ Related topics: Bleybel & Zaiour a1508 [results on temporal foliations].

Special Type of Metrics
@ Specific types of metrics: He & Rideout CQG(09)-a0811 [Schwarzschild]; D'Ariano & Tosini a1008, SHPMP(13)-a1109, Cristina CQG(16)-a1603 [Minkowski].

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