<div dir="auto"><div class="gmail_quote gmail_quote_container" dir="auto"><div dir="ltr" class="gmail_attr">On Wed, Nov 4, 2026, 10:54 AM BillK via extropy-chat <<a href="mailto:extropy-chat@lists.extropy.org">extropy-chat@lists.extropy.org</a>> wrote:</div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div dir="ltr"><div class="gmail_quote"><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default"><div></div><div dir="ltr"><span><span>That would not be a counterexample to Zhang’s theorem. It would mean that the spacetime does </span></span><b dir="ltr"><span><span>not</span></span></b><span><span> satisfy the theorem’s hypothesis. Specifically, it would mean that there is no globally defined single-valued function </span></span><span dir="ltr" style="display:inline-block;overflow-x:visible;max-width:100%"><span><span></span><span aria-hidden="true"><span><span style="height:0.4306em"></span><span>u</span></span></span></span></span><span><span> with everywhere-null gradient that remains well behaved along the relevant region.</span></span></div><div></div><div dir="ltr"><span><span>That is exactly where the “escape routes” come in:</span></span></div><div></div><ol start="1" dir="ltr"><li dir="ltr"><b dir="ltr"><span><span>Twist:</span></span></b><span><span> the null direction field twists so that no global null foliation exists.</span></span></li><li dir="ltr"><b dir="ltr"><span><span>Topology:</span></span></b><span><span> the spacetime is identified in such a way that the would-be null coordinate is not globally single-valued.</span></span></li></ol></div></div></div></div></blockquote></div><div dir="auto">Put that way, Zhang's theorem seems to reduce to almost a tautology: if no one can bend their future personal timeline to return to the global past, then you cannot return to the global past.  In other words: if no one can go back in time, then you can't either.</div></div>