I’ve been spending a lot of time rereading a number of older papers on stable causality. Basically I want to get a handle on the types of techniques needed to manipulate them. For this Seifert’s 1977 paper is a gem.
My particular interest is on the extension of a time function from a compact neighbourhood to all of a stable spacetime, but the material on smoothness is also excellent. In this regard he gives an explicit construction which is must helpful. Working this stuff out myself would have been very hard, time consuming and perhaps not appropriate given the constant push to publish.
As will become usual the links to Seifert’s original paper can be found on my CiteULike page.