Mathematics – Logic
Scientific paper
Aug 1979
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1979gregr..10..969d&link_type=abstract
General Relativity and Gravitation, Volume 10, Issue 12, pp.969-976
Mathematics
Logic
6
Scientific paper
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.
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