# A new bundle completion for parallelizable space-times

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## Abstract

The Schmidt [9]*b*-boundary*∂M*, for completing a space-time*M*, 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*^{+} and*Ō*^{+}*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 group*O*^{+}. Then*∂M* consists of the endpoints of incomplete curves in*M* that have finite horizontal lifts in*Ō*^{+}*M*, and if*∂M*=*φ* we say that*M* is*b*-complete. It turns out that*M* is*b*-complete if and only if*O*^{+}*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 the*b*-boundary is approached.

Now if*∂M≠φ*, then*Ō*^{+}*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 for*O*^{+}*M* by reference to a standard section, that is, to a parallelization*p∶M→O*^{+}*M* We prove some general properties of this*p*-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## Preview

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## References

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