The geometrodynamic content of the Regge equations as illuminated by the boundary of a boundary principle
Part I. Invited Papers Dedicated To John Archibald Wheeler
In this paper the principle that the boundary of a boundary is identically zero (∂○∂≡0) is applied to a skeleton geometry. It is shown that the left-hand side of the Regge equation may be interpreted geometrically as the sum of the moments of rotation associated with the faces of a polyhedral domain. Here the polyhedron, warped though it may be, is located in a lattice dual to the original skeleton manifold. This sum is related to the amount of energy-momentum (E-p) associated to the edge in question. In the establishment of this equation the ordinary Bianchi identity is rederived by applying the principle that the (∂○∂≡0) in its (1–2–3)-dimensional formulation to polyhedral domain. Steps toward the derivation of the contracted Bianchi identity using this principle in its (2–3–4)-dimensional form are discussed. Preliminary results in this direction indicate that there should be one vector identity per vertex of the skeleton geometry.
“Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.”—Ref. 3, Chap. 1.
KeywordsManifold Bianchi Identity Dimensional Form Regge Equation Vector Identity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Unable to display preview. Download preview PDF.
- 1.J. A. Wheeler, “Physics and Austerity: Law Without Law,” A Working Paper, The University of Texas, (1982), pp. 61–83; published in Chinese as “Physics and Austerity,” Anhui, Science and Technology Publication, China August 1982.Google Scholar
- 2.J. A. Wheeler, “Particles and Geometry,” inUnified Theories of Elementary Particles, P. Breithenlohner and H. Durr, eds. (Springer-Verlag, New York, 1982), pp. 189–217.Google Scholar
- 3.C. W. Misner, K. S. Thorne, and J. A. Wheeler, “Bianchi Identities and the Boundary of a Boundary Is Zero,” inGravitation (W. H. Freeman, San Francisco, 1973), Chap. 16.Google Scholar
- 4.É. Cartan, 1928 and 1946,Leçons sur la Géométrie des Espaces de Riemann (Gauthier-Villars, Paris, France); English translation by Robert Hermann (Mathematical Sciences Press, Brookline, Massachusetts, 1983), Chap. 8.Google Scholar
- 5.W. A. Miller and J. A. Wheeler, “4-Geodesy,” paper presented at the Congress “Galactic and Extragalactic Dark Matter,” Rome, 28–30 June 1983; to appear inNuovo Cimento (1985).Google Scholar
- 6.A. Kheyfets, “The Boundary of a Boundary Principle: A Unified Approach,” to appear,Found. Phys. (1986).Google Scholar
- 7.T. Regge, “General Relativity without Coordinates,”Nuovo Cimento 19, 558–571 (1961).Google Scholar
- 8.I. Ciufolini, “On a Generalized Geodesic Deviation Equation,” Chap. 2 of dissertation, “Theory and Experiments in General Relativity and Other Metric Theories of Gravity,” The University of Texas (1984).Google Scholar
- 9.R. M. Williams and G. F. R. Ellis, “Regge Calculus and Observations. I. Formalism and Applications to Radial Motion and Circular Orbits,”Gen. Relativ. Gravit. 13, 361 (1981).Google Scholar
- 10.J. R. Munkres,Elements of Algebraic Topology (Addison-Wesley, Menlo Park, California, 1984), pp. 26–32.Google Scholar
- 11.H. Bacry,Lectures on Group Theory and Particle Physics (Gordon and Breach, New York, 1977), Chap. 7.Google Scholar
- 12.A. Brøndsted,An Introduction to Convex Polytopes (Springer-Verlag, New York, 1983), p. 64.Google Scholar
- 13.B. Grünbaum,Convex Polytopes (Wiley, New York, 1967), pp. 46, 137, 243, and 285.Google Scholar
- 14.H. S. M. Coxeter,Regular Polytopes (Dover, New York, 1973), 3rd. edn., pp. 6 and 60.Google Scholar
- 15.J. Cheeger, W. Müller, and R. Schrader, “On the Curvature of Piecewise Flat Spaces,”Commun. Math. Phys. 92, 405 (1984).Google Scholar
- 16.T. D. Lee, “Time as a Dynamical Variable,” a talk given at the Shelter Island II Conference, June 2, 1983, Columbia University Preprint CU-TP-266 (1983), p. 18.Google Scholar
- 17.C. W. F. Everitt,James Clark Maxwell (Charles Scribners, New York, 1975), Chap. 10.Google Scholar
- 18.John A. Wheeler, private communication.Google Scholar
- 19.W. A. Miller, “Null-Strut Geometrodynamics and the Inchworm Algorithm,” inDynamical Spacetimes and Numerical Relativity, J. Centrella, ed. (Cambridge Univ. Press, 1986), in press.Google Scholar
© Plenum Publishing Corporation 1986