A new bundle completion for parallelizable space-times
- 45 Downloads
The Schmidt b-boundary∂M, for completing a space-timeM, has several desirable features. It is uniquely determined by the space-time metric in an elegant geometrical manner. The completed space-time is¯M=M ∼α∂M, where¯M=Ō+M/O+ andŌ+M is the Cauchy completion (with respect to a toplogical metric induced by the Levi-Cività connection) of a component of the orthonormal frame bundle having structure groupO+. Then∂M consists of the endpoints of incomplete curves inM that have finite horizontal lifts inŌ+M, and if∂M=φ we say thatM isb-complete. It turns out thatM isb-complete if and only ifO+M is complete. This criterion for space-time completeness is stronger than geodesic completeness and Beem  has shown that this remains so even for the restricted class of globally hyperbolic space-times. Clarice  has shown that for such space-times the curvature becomes unbounded as theb-boundary is approached.
Now if∂M≠φ, thenŌ+M may contain degenerate fibers; thus the quotient topology for¯M is non-Hausdorff and precludes a manifold structure. Precisely this has been demonstrated by Bosshard  for Friedmann space-time, casting doubt on the physical significance of the completion. The only neighborhood of the Friedmann singularity is the whole of¯M, and in the closed model initial and final singularities are identified in∂M. Similarly, Johnson  showed that the completion of Schwarzschild space-time is non-Hausdorff because of degenerate ibers in¯O+M.
Here we introduce a modification of the Schmidt procedure that appears to be useful in avoiding fiber degeneracy and in promoting a Hausdorff completion. The modification is to introduce an explicit vertical component into the metric forO+M by reference to a standard section, that is, to a parallelizationp∶M→O+M We prove some general properties of thisp-completion and examine the particular case of a Friedmann space-time where there is a fairly natural choice of parallelization.
KeywordsOrthonormal Frame Manifold Structure Horizontal Lift Frame Bundle Standard Section
Unable to display preview. Download preview PDF.
- 1.Beem, J. K. (1976). “Some Examples of Incomplete Spacetimes,”Gen. Rel. Grav.,7, 501–509.Google Scholar
- 2.Bosshard, B. (1976). “On theb-Boundary of the Closed Friedmann Model,”Commun. Math. Phys.,46, 263–268.Google Scholar
- 3.Clarke, C. J. S. (1975). “Singularities in Globally Hyperbolic Spacetimes,”Commun. Math. Phys.,41, 65–78.Google Scholar
- 4.Dodson, C. T. J. (1978). “Space-Time Edge Geometry”Int. J. Theor. Phys. 17, 389–504.Google Scholar
- 5.Geroch, R. P. (1968). “Spinor Structures of Spacetimes in General Relativity. I,”J. Math. Phys.,9, 1739–1744.Google Scholar
- 6.Hawking, S. W., and Ellis, G. F. R. (1973).The Large-Scale Structure of Space-Time, Cambridge University Press, Cambridge.Google Scholar
- 7.Johnson, R. A. (1977). “The Bundle Boundary in Some Special Cases,”J. Math. Phys.,18, 898–902.Google Scholar
- 8.Kobayashi, S., and Nomizu, K. (1963).Foundations of Differential Geometry, vol. 1, Interscience, New York.Google Scholar
- 9.Schmidt, B. G. (1971). “A New Definition of Singular Points in General Relativity,”Gen. Rel. Grav.,1, 269–280.Google Scholar