Regge Calculus

In General > s.a. computational physics, numerical relativity / actions for general relativity; quantum regge calculus.
* Idea: An approach to discretized (or, better, piecewise flat) geometry, used in numerical relativity and quantum gravity; Initially the simplex version was used, then 3+1 and continuous-time ones were developed, which were thought to be more useful for evolution questions and canonical quantization.
* In numerical relativity: Used to be the main application, as a tool related to the finite-element method; 1991, Results usually agree well with continuum ones, except when a bounce occurs in collapse; 2009, its second-order convergence to the continuum makes it non-competitive.
* Variables: The metric for an n-dimensional manifold is given by assigning the lengths (squared) of all sides of all simplices in a fixed triangulation; This gives, for each simplex, n(n+1)/2 parameters (ok with counting of metric components); The kinds of metrics that can be given in this way are limited, because if one imagines the simplexes as superimposed on a smooth manifold, "all the curvature" is contracted at discrete points and the rest is flat; > s.a. simplex (Lorentzian case).
* Hilbert-Einstein action: It can be expressed (up to some coefficient) by the exact formula

∫ R dv = ∑_i η(i) A(i) ,

where the summation is over all (n−2)-simplexes i, of area A(i) (with A(i):=1 if n−2 = 0), and the defect angle η is η(i):= 2π − ∑_{σ ⊃ i} θ(i;σ), where θ(i;σ) is the angle of the (n−1)-simplex σ at i.

References > s.a. curvature [Bianchi identities].
@ Action: Lund & Regge pr(74); Hartle & Sorkin GRG(81) [boundary terms]; Piran & Williams PRD(86); Brewin CQG(88); Miller CQG(97)gq; Khatsymovsky a0804 [Holst action, Immirzi parameter]; Bahr & Dittrich PRD(09)-a0907 [improved and perfect actions]; Romano a1107 [Regge action and equations from statistical mechanics of Ising or Potts models].
@ And general relativity: Cheeger et al in(82), CMP(84) [convergence of curvature]; Friedberg & Lee NPB(84); Barrett CQG(87), & Parker pr(90); Brewin GRG(89), Brewin & Gentle CQG(01)gq/00; Kemmell IJTP(94); Chakrabarti et al CQG(99)gq/98 [geodesic deviation]; Brewin GRG(00)-a1106 [as an approximation to general relativity].
@ Continuum limit: Ôgami CQG(97) [2D action]; Khatsymovsky PLB(02)gq [area variables]; Bittner et al hl/03-in.
@ Initial-value problem: Sorkin PRD(75); Piran & Williams PRD(86); Porter CQG(87); Barrett et al IJTP(97)gq/94 [parallelizable].
@ Simple spaces and formulae: Hartle JMP(85), JMP(86); Piran & Strominger CQG(86).
@ Geometry: Williams & Ellis GRG(81) [geodesics]; Brewin CQG(88) [Riemann and extrinsic curvature], PRD(88) [trapped surface]; Morse CQG(92) [approximate diffeomorphism invariance]; McDonald & Miller a0804-in, CQG(08)-a0805 [dual tessellations and scalar curvature]; Bahr & Dittrich CQG(09)-a0905 [gauge symmetries and constraints]; Ariwahjoedi & Zen a1807 [and SO(3) and SU(2) representations].
@ Affine tensor and exterior calculus: Warner PRS(82); Brewin JMP(86).
@ Related topics: Roček & Williams in(82) [conformal transformations]; Hamber & Williams NPB(97)ht/96 [gauge invariance].

Various Versions > s.a. discrete geometries; dynamical triangulations; lattice gravity.
* Regge approach: Fix the triangulation and vary edge lengths; Typically use R² actions (bouded below) or the Einstein-Hilbert action.
* Canonical: 1991, There is a problem with the closing of the constraint algebra.
* Discrete time: Foliate the spacetime into hypersurfaces (e.g., spacelike), divide each one into convenient blocks in the same way (e.g., by simplices), then join corresponding vertices by edges (obtaining a total tessellation by prisms).
* Null strut calculus: Build simplicial spacetimes with the maximal number of null edges; This reduces the number of variables and simplifies one type of equation, which becomes a linear relation between deficit angles.
* Area variables: Unless restrictions are placed on the variation of the areas, in 4D this version leads to vanishing deficit angles and flat geometries.
@ Regge approach: Hamber PRD(92) [phase transition]; Brewin CQG(98)gq/97 [different implementation of Einstein equation].
@ Canonical, constraints: Friedman & Jack JMP(86); Khatsymovsky CQG(94)gq/93; Mäkelä PRD(94); Tuckey & Williams CQG(90); Bander PRD(87) [d-dimensional, constraint algebra]; Khatsymovsky GRG(95)gq/93, PLB(00)gq/99; Gambini & Pullin IJMPD(06) [and consistent discretization]; Dittrich & Höhn CQG(10)-a0912; Dittrich & Ryan CQG(11)-a1006; Höhn PRD(15)-a1411 [linearized, Pachner moves and lattice gravitons].
@ Connection / Ashtekar variable version: Khatsymovsky CQG(89), CQG(91), gq/93; Immirzi CQG(94)gq; Khatsymovsky CQG(10)-a0912, a1509.
@ Discrete time: Porter PhD(82); Brewin PhD(83), CQG(87); Dubal PhD(87); Tuckey CQG(89), CQG(93)gq.
@ Null strut calculus: Miller & Wheeler NC(85); Miller PhD(86); Kheyfets et al PRL(88), CQG(89), PRD(90), PRD(90).
@ Area variables: Barrett et al CQG(99)gq/97; Mäkelä CQG(00)gq/98, & Williams CQG(01)gq/00; Wainwright & Williams CQG(04)gq [and discontinuous metrics]; Dittrich & Speziale NJP(08)-a0802 [area-angle variables]; Neiman a1308 [ruling our some sectors of area Regge calculus]; Asante et al CQG(18)-a1802; in Asante et al PRL(20)-a2004; Dittrich a2105 [systematic analysis on a hyper-cubical lattice].
@ First-order form: Barrett CQG(94)ht; Gionti gq/98.
@ Linearization: Barrett PLB(87), CQG(88); Christiansen a1106 [3D, around a Euclidean metric].
@ Variations: Brewin PRD(89) [ADM 4-momentum]; Reisenberger CQG(97)gq/96 [left-handed]; Bilke et al PLB(98)hl/97 [U(1) fields]; Schmidt & Kohler GRG(01)gq [dislocations and torsion]; Bahr & Dittrich NJP(10)-a0907 [with constant-sectional-curvature simplices, for non-zero cosmological constant]; > s.a. causality.

Types of Spacetimes > s.a. FLRW spacetime; bianchi I [Kasner]; quantum regge calculus.
@ 2D: Beirl & Berg NPB(95)hl; Hamber & Williams NPB(95); Hartle & Perjés JMP(97)gq/96 [CP²]; Rolf PhD(98)ht [quantum].
@ 2D, random triangulations: Holm & Janke PLB(97)hl/96; Carfora et al CQG(02)gq.
@ 3D, in general: Roček & Williams CQG(85); Boulatov & Krzywicki MPLA(91); Boulatov JHEP(98); Ariwahjoedi & Zen a1709 [curvatures].
@ 3D, related topics: Waelbroeck CQG(90) [constraints]; Durhuus & Jonsson NPB(95) [entropy]; Hartle et al CQG(97)gq/96 [simplicial superspace].
@ Black holes: Williams & Ellis GRG(84), Brewin CQG(93) [Schwarzschild]; Káninský a2012 [with quantum matter fields].
@ Taub universe: Tuckey & Williams CQG(88) [3+1, continuous time].
@ Other types: Collins & Williams PRD(74) [Tolman]; Porter CQG(87) [spherical symmetry], CQG(87) [model stars]; Dubal CQG(89) [radiation], CQG(89), CQG(90) [collapse]; Clarke et al CQG(90) [cosmic strings]; Bilke et al PLB(97) [topology and free energy]; Gentle CQG(99) [Brill waves, initial data]; Khatsymovsky PLB(00)gq/99, PLB(00)gq/99 [simple model, path integral and canonical], PLB(03)gq, a0808 [discontinuous]; Liu & Williams PRD(16)-a1502, a1510-proc [lattice and Λ-FLRW universes].

Other References > s.a. numerical general relativity; quantum spacetime.
@ Intros, reviews: Regge NC(61); in Wheeler in(64); in Misner et al 73 [clear; good introduction]; Williams NS(86) [I]; Williams & Tuckey CQG(92); Immirzi NPPS(97)gq [comments]; Gentle & Miller gq/01-MG9; Regge & Williams JMP(00)gq; Gentle GRG(02)gq/04 [numerical relativity]; Larrañaga gq/03-ln [en español]; Cuzinatto et al a1904 [intro].
@ Geometrical aspects: Caselle et al PLB(89) [dual lattice]; Miller FP(86); Jourjine PRD(87) [in terms of cell complex]; Brewin CQG(88).
@ Convergence, consistency: Brewin GRG(00)gq/95; Miller CQG(95)gq.
@ Scaling, renormalization: Martellini & Marzuoli in(86); Mitter & Scoppola CMP(00)ht/98; Hamber PRD(00)ht/99.
@ And matter: Weingarten JMP(77) [electromagnetism]; Berg et al PRD(96) [SU(2) gauge theory]; Khatsymovsky PLB(01)gq/00; Bittner et al PRD(02) [Ising spins]; McDonald & Miller CQG(10)-a1002 [lattice action for scalar, vector, and tensor particles].
@ For similar theories: Sorkin PhD(74), JMP(75) [electromagnetism]; Pereira & Vargas CQG(02)gq [teleparallel gravity]; D'Adda a2007 [higher-order theories, with coordinates associated to vertices]; > s.a. higher-order gravity.
@ Gauge theory style: Weingarten NPB(82); Kawamoto & Nielsen PRD(91).
@ Entropy: Ambjørn & Varsted PLB(91); Bartocci et al JGP(96)ht/94; Carfora & Marzuoli JGP(95).
@ (No) exponential bound, thermodynamic limit: Ambjørn & Jurkiewicz PLB(94); Catterall et al PRL(94); Carfora & Marzuoli JMP(95).
@ Random simplices: Ambjørn et al NPB(93) [Euclidean]; David in(95)ht/93; Carfora et al gq/03-in [2D].
@ Random surfaces: Kostov & Krzywicki PLB(87); Polyakov 87; Ambjørn NPPS(95)hl/94, et al 97; Thorleifsson NPPS(99)hl/98.
@ Random / polymerized manifolds: Gabrielli PLB(98); Harris & Wheater PLB(99)ht/98.
@ Phase structure: Hamber PRD(92); Catterall et al PLB(94).

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