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Two examples are presented in this paper, the first is unfavorable to the c-boundary construction given by Geroch, Kronheimer, and Penrose [Proc. R. Soc. London Ser. A 327, 545 (1972)] but in favor of that given by Budic and Sachs [J. Math. Phys. 15, 1302 (1974)], while the second plays an opposite role. The second example is also an example of a causally continuous space-time with a ‘‘really big gap,’’ contrary to what was believed in the literature.

The structure of the phase boundary between degenerate and nondegenerate regions in Ashtekar's gravity has been recently studied by Bengtsson and Jacobson who conjectured that the phase boundary should be always null. In this paper, we reformulate the reparametrization procedure in the mapping language and distinguish a phase boundary ∂M 1 from its image φ[∂M 1 ]. It is shown that φ[∂M 1 ] has to be null, while the nullness of ∂M 1 requires some more suitable criterion.

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Previously, we gave two classes of plane-symmetric line elements associated with semi-planesymmetric null electromagnetic fields. Different functions J ( V ) in those line elements may represent the same geometry. Two criteria on this "gauge freedom" are proven in the present paper. The line elements of class B are identified with the conformally flat plane-fronted gravitational waves with parallel rays (pp waves), and the gauge freedom on the function, W v ) , appearing in the line element of the pp waves is also derived.