Causal structure and smooth or piecewise smooth curves

Question

I am currently working through Wald's GR book and in the chapter about causal structure, he defines a future directed causal/timelike curve to be differentiable. Now in his remark after Theorem 8.1.2 he says one could deform any curve from $p$ to $q in J^+(p) - I^+(p)$ to a null geodesic. $J$ and $I$ are all points that can be reached by causal or timelike curves respectively. I have the impression that I could make it to a piecewise null geodesic, by choosing points on the the causal boundary along the curve an connecting them by null geodesics but this would not lead to a single null geodesic but a curve which is a piecewise null geodesic. On the other hand it would make sense that all points of $J^+(p) - I^+(p)$ could be reached by a single null geodesic, since then really light could reach them without a weird change of direction along the way.
I also saw that Penrose defines the causal parts via piecewise curves, so I thought maybe they are sufficient.
Evey comment is welcome!

Jerry Schirmer · Answer

I'm pretty sure that "null geodesic" and not "piecewise null geodesic" is correct.
The reason why is that $J^{+}(p) - I^{+}(p)$ is a null hypersurface (being the boundary of the timelike future of p), which means at any point on that surface, there is exactly one null direction that remains on the surface, so, at the "kinks", you'd either have to leave the surface or have spacelike segments on the curve.

Causal structure and smooth or piecewise smooth curves

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