The Riemann and extrinsic curvature tensors in the Regge calculus

A detailed analysis of the Riemann tensor in the neighbourhood of one bone and of the extrinsic curvature in the neighborhood of one triangular face in a simplicial geometry is presented. Unlike most previous analyses this analysis makes no reference to any particular choice of smoothing scheme. Explicit formulae will be presented for both the Riemann and extrinsic curvature tensors. These results are applied, using the formalism developed in an earlier paper, in deriving an exact formula for the integral extrinsic curvature. It is argued that for integrals of R2, R3, . . ., contributions must be expected from the legs, vertices, . . ., rather than just from the bones.

[1]  L. Brewin,et al.  A continuous time formulation of the Regge calculus , 1988 .

[2]  R. Geroch,et al.  Strings and other distributional sources in general relativity. , 1987, Physical review. D, Particles and fields.

[3]  L. Brewin,et al.  Friedmann cosmologies via the Regge calculus , 1987 .

[4]  J Porter,et al.  A new approach to the Regge calculus: I. Formalism , 1987 .

[5]  J Porter,et al.  A new approach to the Regge calculus: II. Application to spherically symmetric vacuum spacetimes , 1987 .

[6]  J. Friedman,et al.  3+1 Regge calculus with conserved momentum and Hamiltonian constraints , 1986 .

[7]  Ruth M. Williams.  Quantum Regge calculus in the Lorentzian domain and its Hamiltonian formulation , 1986 .

[8]  Williams,et al.  Three-plus-one formulation of Regge calculus. , 1986, Physical review. D, Particles and fields.

[9]  Ruth M. Williams,et al.  Higher derivative quantum gravity on a simplicial lattice , 1984 .

[10]  Richard Friedberg,et al.  Derivation of Regge's action from Einstein's theory of general relativity☆ , 1984 .

[11]  R. Sorkin,et al.  Boundary terms in the action for the Regge calculus , 1981 .

[12]  D. D. Sokolov,et al.  The structure of the curvature tensor at conical singularities , 1977 .

[13]  R. Sorkin.  Time-evolution problem in regge calculus , 1975 .

[14]  B. Dewitt,et al.  Relativity, Groups, and Topology , 1964 .

[15]  T. Regge.  General relativity without coordinates , 1961 .

[16]  R M Williams.  Quantum Regge calculus in the Lorentzian domain and its Hamiltonian formulation , 1986 .

[17]  M. Roček,et al.  Introduction to Quantum Regge Calculus , 1981 .

[18]  C. Wong.  Application of Regge Calculus to the Schwarzschild and Reissner‐Nordstro/m Geometries at the Moment of Time Symmetry , 1971 .