International Journal of Theoretical Physics

, Volume 36, Issue 4, pp 815–839 | Cite as

Parallelizable implicit evolution scheme for Regge calculus

  • John W. Barrett
  • Mark Galassi
  • Warner A. Miller
  • Rafael D. Sorkin
  • Philip A. Tuckey
  • Ruth M. Williams


The role of Regge calculus as a tool for numerical relativity is discussed, and a parallelizable implicit evolution scheme described. Because of the structure of the Regge equations, it is possible to advance the vertices of a triangulated spacelike hypersurface in isolation, solving at each vertex a purely local system of implicit equations for the new edge lengths involved. (In particular, equations of global “elliptic type” do not arise.) Consequently, there exists a parallel evolution scheme which divides the vertices into families of nonadjacent elements and advances all the vertices of a family simultaneously. The relation between the structure of the equations of motion and the Bianchi identities is also considered. The method is illustrated by a preliminary application to a 600-cell Friedmann cosmology. The parallelizable evolution algorithm described in this paper should enable Regge calculus to be a viable discretization technique in numerical relativity.


Quantum Gravity Edge Length Bianchi Identity Vertical Edge Spacelike Hypersurface 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Barrett, J. W. (1986). The Einstein tensor in Regge's discrete gravity theory,Classical and Quantum Gravity,3, 203–206.MathSciNetCrossRefADSGoogle Scholar
  2. Barrett, J. W. (1988). A convergence result for linearized Regge calculus,Classical and Quantum Gravity,5, 1187–1192.MATHMathSciNetCrossRefADSGoogle Scholar
  3. Barrett, J. W. (1992). A mathematical approach to numerical relativity, inApproaches to Numerical Relativity, R. d'Inverno, ed., Cambridge University Press, Cambridge, pp. 103–113.Google Scholar
  4. Barrett, J. W., and Parker, P. E. (1994). Smooth limits of piecewise-linear approximations,Journal of Approximation Theory,76, 107–22.MATHMathSciNetCrossRefGoogle Scholar
  5. Barrett, J. W., and Williams, R. M. (1988). The convergence of lattice solutions of linearized Regge calculus,Classical and Quantum Gravity,5, 1543–56.MathSciNetCrossRefADSGoogle Scholar
  6. Brewin, L. (1983). The Regge calculus in numerical relativity, Ph.D. Thesis, Monash University.Google Scholar
  7. Brewin, L. (1987). Friedmann cosmologies via the Regge calculus,Classical and Quantum Gravity,4, 899–928.MathSciNetCrossRefADSGoogle Scholar
  8. Brewin, L. (1989). Equivalence of the Regge and Einstein equations for almost flat simplicial spacetimes,General Relativity and Gravitation,21, 565–83.MATHMathSciNetCrossRefGoogle Scholar
  9. Cheeger, J., Müller, W., and Schrader, R. (1984). On the curvature of piecewise flat spaces,Communications in Mathematical Physics,92, 405–54.MATHMathSciNetCrossRefADSGoogle Scholar
  10. Collins, P. A., and Williams, R. M. (1973). Dynamics of the Friedman universe using Regge calculus,Physical Review D,7, 965–71.CrossRefADSGoogle Scholar
  11. Dubal, M. R. (1987). Numerical Computations in General Relativity, Ph.D. Thesis, SISSA Trieste.Google Scholar
  12. Dubal, M. R. (1989). Relativistic collapse using Regge calculus: I. Spherical collapse equations,Classical and Quantum Gravity,6, 1925–41.MATHMathSciNetCrossRefADSGoogle Scholar
  13. Feinberg, G., Friedberg, R., Lee, T. D., and Ren, H. C. (1984). Lattice gravity near the continuum limit,Nuclear Physics B,245, 343–368.MathSciNetCrossRefADSGoogle Scholar
  14. Friedberg, R., and Lee, T. D. (1984). Derivation of Regge's action from Einstein's theory of general relativity,Nuclear Physics B,242, 145–66.MathSciNetCrossRefADSGoogle Scholar
  15. Galassi, M. (1993). Lapse and shift in Regge calculus,Physical Review D,47, 3254–64.MathSciNetCrossRefADSGoogle Scholar
  16. Hartle, J. B. (1985). Simplicial minisuperspace I. General discussion,Journal of Mathematical Physics,26, 804–18.MathSciNetCrossRefADSGoogle Scholar
  17. Hartle, J. B. (1986). Simplicial minisuperspace II. Some classical solutions on simple triangulations,Journal of Mathematical Physics,27, 287–95.MATHMathSciNetCrossRefADSGoogle Scholar
  18. Kheyfets, A., Miller, W. A., and Wheeler, J. A. (1988). Null-strut calculus: The first test,Physical Review Letters,61, 2042–2045.MathSciNetCrossRefADSGoogle Scholar
  19. Kheyfets, A., LaFave, N. J., and Miller, W. A. (1990a). Null-strut calculus I: Kinematics,Physical Review D,41, 3628–36.MathSciNetCrossRefADSGoogle Scholar
  20. Kheyfets, A., LaFave, N. J., and Miller, W. A. (1990b). Null-strut calculus II: Dynamics,Physical Review D,41, 3637–51.MathSciNetCrossRefADSGoogle Scholar
  21. Kurki-Suonio, H., Laguna, P., and Matzner, R. A. (1993). Inhomogeneous inflation: Numerical evolution,Physical Review D,48, 3611–24.CrossRefADSGoogle Scholar
  22. Lewis, S. M. (1982). Regge calculus: Applications to classical and quantum gravity, Ph.D. Thesis, University of Maryland at College Park.Google Scholar
  23. Miller, W. A. (1986a). Geometric computation: Null-strut geometrodynamics and the inchworm algorithm, inDynamical Spacetimes and Numerical Relativity, J. Centrella, ed., Cambridge University Press, Cambridge, pp. 256–303.Google Scholar
  24. Miller, W. A. (1986b). The geometrodynamic content of the Regge equations as illuminated by the boundary of a boundary principle,Foundations of Physics,16, 143–169.MathSciNetCrossRefGoogle Scholar
  25. Miller, W. A. (1990). Unpublished.Google Scholar
  26. Miller, W. A., and Wheeler, J. A. (1985). 4-Geodesy,Nuovo Cimento,8, 418–434.MathSciNetGoogle Scholar
  27. Misner, C., Thorne, K. S., and Wheeler, J. A. (1973).Gravitation, Freeman, San Francisco, 505–8.Google Scholar
  28. Piran, T., and Strominger, A. (1986). Solutions of the Regge equations,Classical and Quantum Gravity,3, 97–102.MATHMathSciNetCrossRefADSGoogle Scholar
  29. Porter, J. D. (1987a). A new approach to the Regge calculus: I. Formalism,Classical and Quantum Gravity,4, 375–89.MATHMathSciNetCrossRefADSGoogle Scholar
  30. Porter, J. D. (1987b). A new approach to the Regge calculus: II. Application to spherically-symmetric vacuum space-times,Classical and Quantum Gravity,4, 391–410.MATHMathSciNetCrossRefADSGoogle Scholar
  31. Regge, T. (1961). General relativity without coordinates,Nuovo Cimento,19, 558–571.MathSciNetGoogle Scholar
  32. Roček, M., and Williams, R. M. (1981). Quantum Regge calculus,Physics Letters,104B, 31–7.ADSGoogle Scholar
  33. Roček, M., and Williams, R. M. (1982). Introduction to quantum Regge calculus, inQuantum Structure of Space and Time, M. J. Duff and C. J. Isham, eds., Cambridge University Press, Cambridge.Google Scholar
  34. Roček, M., and Williams, R. M. (1984). The quantization of Regge calculus,Zeitschrift für Physik C,21, 371–81.CrossRefGoogle Scholar
  35. Rourke, C. P., and Sanderson, B. J. (1982).Introduction to Piecewise-Linear Topology, Springer, Berlin, Proposition 2.9.Google Scholar
  36. Sorkin, R. D. (1974). Development of simplicial methods for the metrical and electromagnetic fields, Ph.D. Thesis, California Institute of Technology [available from University Microfilms, Ann Arbor, Michigan].Google Scholar
  37. Sorkin, R. D. (1975a). Time-evolution problem in Regge calculus,Physical Review D,12, 385–96.MathSciNetCrossRefADSGoogle Scholar
  38. Sorkin, R. D. (1975b). The electromagnetic field on a simplicial net,Journal of Mathematical Physics,16, 2432–40 [Erratum,19, 1800 (1978)].MathSciNetCrossRefADSGoogle Scholar
  39. Sorkin, R. D. (1991). A finitary substitute for continuous topology?International Journal of Theoretical Physics,30, 923–47.MATHMathSciNetCrossRefGoogle Scholar
  40. Thorne, K. S. (1987). Gravitational radiation, in300 Years of Gravitation, S. Hawking and W. Israel, eds., Cambridge University Press, Cambridge, pp. 330–458.Google Scholar
  41. Thorne, K. S. (1990). Sources of gravitational waves and prospects for their detection, Caltech preprint GRP-234.Google Scholar
  42. Tuckey, P. A. (1993). The construction of Sorkin triangulations,Classical and Quantum Gravity,10, L109–13.MATHMathSciNetCrossRefADSGoogle Scholar
  43. Wheeler, J. A. (1964). Geometrodynamics and the issue of the final state, inRelativity, Groups and Topology, B. DeWitt and C. DeWitt, eds., Gordon and Breach, New York, pp. 463–500.Google Scholar
  44. Williams, R. M., and Tuckey, P. A. (1992). Regge calculus: A brief review and bibliography,Classical and Quantum Gravity,9, 1409–22.MATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • John W. Barrett
    • 1
  • Mark Galassi
    • 2
  • Warner A. Miller
    • 3
  • Rafael D. Sorkin
    • 4
    • 5
  • Philip A. Tuckey
    • 6
  • Ruth M. Williams
    • 7
  1. 1.Mathematics DepartmentUniversity of NottinghamNottinghamUK
  2. 2.Space Data Systems GroupLos Alamos National LaboratoryLos Alamos
  3. 3.Theoretical DivisionLos Alamos National LaboratoryLos Alamos
  4. 4.Physics DepartmentSyracuse UniversitySyracuse
  5. 5.Departamento de Gravitacion y Teoria de Campos, Instituto de Ciencias NuclearesUNAMMexicoMexico
  6. 6.Laboratoire de Physique MoléculaireUniversité de Franche-ComtéBesançonFrance
  7. 7.DAMTPCambridgeUK

Personalised recommendations