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).
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 24 may 2021