Hostname: page-component-5c6d5d7d68-thh2z Total loading time: 0 Render date: 2024-09-01T19:00:27.286Z Has data issue: false hasContentIssue false
  • English
  • Français

Conformal deformations, Ricci curvature and energy conditions on globally hyperbolic spacetimes

Published online by Cambridge University Press:  24 October 2008

John K. Beem  and
Paul E. Ehrlich
John K. Beem
Affiliation:
University of Missouri, Columbia, Mo. 65201
Paul E. Ehrlich
Affiliation:
University of Missouri, Columbia, Mo. 65201
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider globally hyperbolic spacetimes (M, g) of dimension ≥ 3 satisfying the curvature condition Ric (g) (v, v) ≥ 0 for all non-spacelike tangent vectors v in TM. This curvature condition arises naturally as an energy condition in cosmology. Suppose (M, g) admits a smooth globally hyperbolic time function h: M → such that for some t0, the Cauchy surface h−1(t0) satisfies the strict curvature condition Ric (g) (v, v) > 0 for all non-spacelike v attached to h−1(t0). Then M admits a metric g′ conformal to g satisfying the strict curvature condition Ric (g′) (v, v) > 0 for all non-spacelike v in TM. If the curvature and strict curvature conditions are restricted to null vectors, the analogous result may be obtained. Similar results may also be obtained for the scalar curvature in dimension ≥ 2 and for non-positive Ricci curvature.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 159 - 175
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Aubin, T.Métriques riemannienes et courbure. J. Differential Geometry 4 (1970), 383–424.CrossRefGoogle Scholar
(2)Cheeger, J. and Ebin, D.Comparison theorems in riemannian geometry (New York: North-Holland, 1975).Google Scholar
(3)Ehrlich, P.Local convex deformations of Ricci and sectional curvature on compact manifolds. Proc. Symp. in Pure Math., Amer. Math. Soc. 27 (1975), 69–71.CrossRefGoogle Scholar
(4)Ehrlich, P.Metric deformations of curvature. I. Local convex deformations. Geometriae Dedicata, 5 (1976), 1–24.CrossRefGoogle Scholar
(5)Ehrlich, P.Metric deformations of curvature. II. Compact 3-manifolds. Geometriae Dedicata 5 (1976), 147–161.CrossRefGoogle Scholar
(6)Geroch, R. P.The domain of dependence. J. Mathematical Phys. 11 (1970), 437–439.CrossRefGoogle Scholar
(7)Hawking, S. W. and Ellis, G. F. R.The large scale structure of space-time (Cambridge: Cambridge University Press, 1973).CrossRefGoogle Scholar
(8)Ihrig, E.The holonomy group in general relativity and the determination of g ii from General Relativity and Gravitation 7 (1976), 313–323.CrossRefGoogle Scholar
(9)Lee, K. K.Another possible abnormality of compact space-time. Canad. Math. Bull. 18 (1975), 695–697.CrossRefGoogle Scholar
(10)Milnor, J. ‘Differential geometry’ from ‘Problems of Present Day Mathematics in Mathematical Developments arising from the Hilbert problems,’ Symp. Pure Math., Amer. Math. Soc. 28 ed. Browder, Felix 1976, 54–59.Google Scholar
(11)Penrose, R.Techniques of differential topology in relativity, Regional Conference Series in Applied Math. 7 (Philadelphia: SIAM, 1972).CrossRefGoogle Scholar
(12)Sachs, R. K.General relativity and cosmology (New York: Academic Press, 1971).Google Scholar
(13)Tipler, F. J.Singularities in universes with negative cosmological constant. Astrophys. J. 209 (1976), 12–15.CrossRefGoogle Scholar
(14)Weinberg, S.Gravitation and cosmology (New York: Wiley, 1972).Google Scholar