General Relativity and Gravitation

, Volume 10, Issue 12, pp 969–976 | Cite as

A new bundle completion for parallelizable space-times

  • C. T. J. Dodson
I: Singularity Definitions and Existence


The Schmidt [9]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 [1] has shown that this remains so even for the restricted class of globally hyperbolic space-times. Clarice [3] 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 [2] 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 [7] 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.


Orthonormal Frame Manifold Structure Horizontal Lift Frame Bundle Standard Section 
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Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • C. T. J. Dodson
    • 2
    • 1
  1. 1.International Center for Theoretical PhysicsTriesteItaly
  2. 2.Department of MathematicsUniversity of LancasterEngland

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