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Title:
Simple Characterization of the Causal Boundary and Continuous Extension of the Spacetime Metric
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Abstract:
This is an ongoing examination of the Causal Boundary of a spacetime M in terms of a foliation F by observer worldlines (timelike curves), without any assumed symmetries for M. The idea is to reflect spacetime geometry in time-dependent fields (Riemannian metric and drift-form) on the leaf-space Q ( = M/F) of the foliation. In this stage of the set-up (inspired by interior Schwarzschild), the worldlines are assumed to be future-incomplete, and conditions are obtained which will force the Future Causal Boundary of M to be spacelike, diffeomorphic to Q, and attached to M---represented as (a, b) x Q---as {b} x Q in (a, b] x Q. With that in hand, then the continuous extension of the spacetime metric to the Boundary depends simply on an integral condition of sectional curvature along the observer worldlines.
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