I've been struggling with this question for a while.

In theorem 4.1 of "Smoothing and extending cosmic time functions" Seifert proves that a time function defined on a compact subset of a stably causal Lorentzian manifold is extendible to a global time function. To do this he constructs a countable collection of non-intersecting stable spacelike boundaries, $C_{\tau_k}$. The boundaries are inductively defined so that $$ C_{\tau_k}=\tilde{J}^-_{\theta_k}(Q_k)=\bigcup_{\eta>\theta_k}J^-(Q_k;g_{\eta}),$$ where $Q_k$ is some compact spacelike set, $\theta_k,\eta\in[a,b]$ for some $a,b\in\mathbb{R}$, $J^-(Q_k;g_{\eta})$ is the causal past `of $Q_k$ with respect to the metric $g_{\eta}$ and for all $\alpha,\beta\in [a,b]$ we have that $g\lt g_\alpha$ and $g_\alpha\lt g_\beta$ if and only if $\alpha$ is less than $\beta$.

Seifert defines a boundary as a set $A$ so that there exists some $W\subset M$ so that $\partial W=A$. It seems to me that $I^-(C_{\tau_k},g)\subset C_{\tau_k}$ implying that $C_{\tau_k}$ has non-empty interior and therefore isn't a boundary. The set $\partial C_{\tau_k}$ seems to fit what Seifert wants, did he just forget a boundary symbol?

Since this paper was published in 1977 and has been cited roughly 15 times I would expect other researchers to comment on this problem. I've looked through the literature, however, and can't see anything that comments on this.

So my question is, "Is $C_{\tau_k}$ a boundary?"

(p.s. sorry for the tick marks ` in strange places, for some reason the latex would only show correctly if I inserted them.)


Looking at Figure 3 and Step B of the proof of the theorem, it looks like the $C_{\tau_k}$ should be of the form $\partial \tilde{J}^-_{\theta_k}(Q_k)$. I am also pretty sure that he chose the symbol $C$ for "cone". Essentially the idea is to construct the surface as a union of a bunch of truncated cones that are wider than the causal cone for the given metric (so that the surface defined is automatically space-like).

  • $\begingroup$ Also, looking on MathSciNet ams.org/mathscinet-getitem?mr=MR0484260 , the reviewer seems to agree with what I just described. $\endgroup$ – Willie Wong May 10 '10 at 13:08
  • $\begingroup$ I'm glad you agree with me. I was hoping for something more definitive, however. I suppose I'm just surprised that in 34 years no one else has commented on this the literature. This seems very strange to me, especially for such an explicit construction. Hope you enjoyed the paper! $\endgroup$ – Ben Whale May 11 '10 at 3:53
  • $\begingroup$ What do you mean by definitive? This is most probably one of those cases where there is a typo in an important paper and everybody knows about it. If you are looking for an actual, published article pointing out the mistake, then don't hold your breath. Most mathematicians are far too lazy (or nice) to "fix" trivially fixable mistakes in others' papers or books. Only fundamentally wrong proofs or claims get that treatment. $\endgroup$ – Willie Wong May 11 '10 at 8:11
  • $\begingroup$ Good point Willie. $\endgroup$ – Ben Whale May 11 '10 at 23:32

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