Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-08T19:28:06.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Hodge Theory on Compact Two-Dimensional Spacetimes and the Uniqueness of gij with a Specified Rij

Published online by Cambridge University Press:  20 November 2018

Edwin Ihrig
Edwin Ihrig*
Affiliation:
Arizona State University, Tempe, Arizona
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main question we wish to address in this paper is to what extent does the Ricci curvature of a spacetime determine the metric of that spacetime. Although it is relatively easy to see that the full Riemann curvature uniquely determines the metric for a generic choice of curvature tensors (see [4], [10], [11], [14] and [15], and the references contained therein), very little has been discovered about whether, if ever, Ric (or the stress energy tensor in Einstein's equations for that matter) determines g. Most exact solution techniques for Einstein's equations look only for solutions that have the same symmetries as Ric. It is not true in general that g must inherit the symmetries of Ric. It is not even clear that there is a Ric such that every g with this Ricci tensor is known.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 6 , 01 December 1987 , pp. 1459 - 1474
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. DcTurck, D. and Koiso, N., Uniqueness and non existence of metrics with a prescribed Ricci curvature, Ann. Inst. H. Poincaré, Sect. A (N.S.) 1 (1984), 351–359.Google Scholar
2. Filimonova, A., Estimates in the C metric and uniqueness of a convex surface in II with a given Gaussian curvature, that is homeomorphic to a sphere, 64-68 in Questions of global geometry, (Leningrad Gos. Ped. Inst., Leningrad, 1979).Google Scholar
3. Guillemin, V. and Pollack, A., Differential topology, (Prentice Hall, Englewood Cliffs, N.J., 1974).Google Scholar
4. Hall, G. and Mcintosh, C., Algebraic determination of the metric from the curvature in general relativity, Inter. J. Theor. Phys. 22 (1983), 469–476.Google Scholar
5. Hamilton, R., The Ricci curvature equation, MSRI Report, Berkeley, Ca. (1983).Google Scholar
6. Hicks, N., Notes on differential geometry, (Van Nostrand, New York, 1971).Google Scholar
7. Hilton, P. and Wylie, S., Homology theory, (Cambridge U. Press, Cambridge, 1962).Google Scholar
8. Hirsch, M., Differential topology, (Springer Verlag, New York, 1976).CrossRefGoogle Scholar
9. Hirsch, M. and Smale, S., Differential equations, dynamical systems and linear algebra, (Academic Press, New York, 1974).Google Scholar
10. Ihrig, E., An exact determination of the gravitational potentials gij in terms of the gravitational fields , J. Math. Phys. 16 (1975), 54–55.Google Scholar
11. Ihrig, E., The uniqueness of gij in terms of , Inter. J. Theor. Phys. 14 (1975), 23–35.Google Scholar
12. Kazdan, J., Prescribing the curvature of a Riemannian manifold, N.S.F.-C.B.M.S. 57 (A.M.S., Providence, R.I., 1984).Google Scholar
13. Kobayashi, S. and Nomizu, K., Foundations of differential geometry., Vol. 1 (Interscience, New York, 1963).Google Scholar
14. Mcintosh, C. and Halford, W., The Riemann tensor, the metric tensor and curvature collineations in general relativity, J. Math. Phys. 23 (1982), 436–441.Google Scholar
15. Mcintosh, C. and Van Leeuver, E., Spacetimes admitting a vector field whose inner product with the Riemann tensor is zero, J. Math. Phys. 23 (1982), 1149–1154.Google Scholar
16. Milnor, J. and Stasheff, J., Characteristic classes, (Princeton U. Press, Princeton, N.J., 1974).CrossRefGoogle Scholar
17. Peixoto, M., Structural stability on 2-dimensional manifolds, Topology 1 (1962), 101–121.Google Scholar
18. Smale, S., Differentiable dynamical systems, Bull. A.M.S. 73 (1967), 747–817.Google Scholar
19. Spanier, E., Algebraic topology, (McGraw Hill, New York, 1966).Google Scholar
20. Steenrod, N., The topology of fibre bundles, (Princeton U. Press, Princeton, 1951).CrossRefGoogle Scholar
21. Wallich, N. and Warner, F., Curvature forms for two manifolds, Proc. Amer. Math. Soc. 25 (1970), 712–713.Google Scholar