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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

Abstract

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.

Keywords

Orthonormal Frame Manifold Structure Horizontal Lift Frame Bundle Standard Section 
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.

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References

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    Beem, J. K. (1976). “Some Examples of Incomplete Spacetimes,”Gen. Rel. Grav.,7, 501–509.Google Scholar
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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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