<div dir="ltr"><div dir="ltr"><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Wed, 4 Nov 2026 at 14:51, Adrian Tymes via extropy-chat <<a href="mailto:extropy-chat@lists.extropy.org" target="_blank">extropy-chat@lists.extropy.org</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">On Wed, Nov 4, 2026 at 5:09 AM BillK via extropy-chat<br>
<<a href="mailto:extropy-chat@lists.extropy.org" target="_blank">extropy-chat@lists.extropy.org</a>> wrote:<br>
> You are absolutely right: simply returning to the same "time" (u) at a different "place" (x,y,z) is not a Closed Timelike Curve (CTC), nor does it allow you to interact with your past self.<br>
<br>
But it does allow interaction with your past self.<br>
<br>
> That would just be regular spatial travel.<br></blockquote><div> <span class="gmail_default" style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)">   </span>Regular spatial travel specifically does not involve returning to the</div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
same time, regardless of spatial coordinates.<br>
<br>
> When you were 10 years old, your worldline passed through Event A: (u=100,x=10,y=10,z=10).<br>
> When you are 40 years old, your worldline is currently at Event B: (u=500,x=80,y=20,z=5).<br>
> To "time travel" and stand "beside yourself" to shake hands with your 10-year-old self, your current worldline must physically intersect the older, already-drawn part of your worldline.<br>
<br>
Nope.  Returning to (u=100,x=11,y=10,z=10) - notice the very slightly<br>
different x - would suffice.<br>
<br>
> You cannot meet your 10-year-old self by returning to u=100 at a completely different spatial location (say, x=90,y=90,z=90), because your 10-year-old self was never at that location! Your younger self was at (10,10,10). If you go to a different spatial coordinate at u=100, you are just visiting an empty spot in space at a time when your younger self was somewhere else.<br>
<br>
You can interact without being in the exact same position.<br>
_______________________________________________
</blockquote><div><br></div><div><br></div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default">Qwen went away and had a big think. Significantly longer than previous responses.</div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default">Short answer is that Qwen agreed with your criticisms but said they describe situations that are specifically forbidden by Zhang's theorem. Which is why the theorem claims that time travel in our universe is impossible.</div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default">Therefore, you need to disprove the assumptions in Zhang's theorem.</div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default">BillK</div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default"><br></div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default">Qwen3.8 Max Thinking:</div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default"><br></div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default"><div dir="ltr"><span><span>Yes — that criticism is largely correct, and it exposes an imprecise way of explaining the theorem. The right response is:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><b dir="ltr"><span><span>You do not need to return to the exact same spacetime point to interact with your past self. But Zhang’s theorem, if its assumptions hold, rules out that weaker “visit the past nearby” scenario too.</span></span></b></div></blockquote><div></div><div dir="ltr"><span><span>The theorem is not merely saying “you cannot stand in your exact old footprints.” Its mathematical core is stronger: </span></span><b dir="ltr"><span><span>a certain null coordinate cannot be revisited at all by any future-directed timelike traveler, not even at a slightly different spatial location.</span></span></b></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>1. What the criticism gets right</span></span></h2><h3 dir="ltr"><span><span>A. Interaction does not require exact coincidence</span></span></h3><div dir="ltr"><span><span>Correct. To interact with your 10-year-old self, you do not need to occupy the exact same event. You only need to be in the same past region and be able to exchange signals or forces. If your younger self is at</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span>A</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>100</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>and your older self arrives at</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span style="margin-right:0.0715em">C</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>100</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>11</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span><span>,</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>then, depending on the causal structure, your older self may be able to shout, throw a rock, send a light signal, or otherwise affect the younger self shortly afterward. Exact overlap is not required.</span></span></div><div></div><div dir="ltr"><span><span>So the earlier claim that the worldline must intersect the exact same event was too narrow if the goal is “meet or affect your past self.”</span></span></div><div></div><div><hr></div><div></div><h3 dir="ltr"><span><span>B. Ordinary spatial travel does not mean “same time, different place”</span></span></h3><div dir="ltr"><span><span>Also correct. Normal spatial travel through spacetime is still travel along a future-directed timelike worldline. You move through space, but you also move forward in time. In a spacetime with a well-behaved global time-like or null coordinate, your coordinate does not remain fixed while you travel as a massive object.</span></span></div><div></div><div dir="ltr"><span><span>So if someone says, “I traveled spatially and returned to the same time but a different place,” that is not ordinary spatial travel. That would already be a violation of the usual causal ordering.</span></span></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>2. What the theorem actually proves</span></span></h2><div dir="ltr"><span><span>The theorem assumes there exists a global 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> whose gradient is null everywhere:</span></span></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:1.0935em;vertical-align:-0.1944em"></span><span><span style="margin-right:0.0359em">g</span><span><span><span><span style="height:0.8991em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span><span>ab</span></span></span></span></span></span></span></span></span><span><span style="margin-right:0.0556em">∂</span><span><span><span><span style="height:0.1514em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span>a</span></span></span></span><span></span></span><span><span style="height:0.15em"><span></span></span></span></span></span></span><span>u</span><span><span style="margin-right:0.0556em">∂</span><span><span><span><span style="height:0.3361em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span>b</span></span></span></span><span></span></span><span><span style="height:0.15em"><span></span></span></span></span></span></span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>0.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>This </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> is a “light-based” coordinate. Its level surfaces are null hypersurfaces — roughly, surfaces swept out by light fronts.</span></span></div><div></div><div dir="ltr"><span><span>Now take a massive observer moving along a future-directed timelike curve with tangent vector </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:0.6833em"></span><span><span style="margin-right:0.2222em">V</span><span><span><span><span style="height:0.6644em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span>a</span></span></span></span></span></span></span></span></span></span></span></span><span><span>. The rate of change of </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> along that observer’s path is</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:2.0574em;vertical-align:-0.686em"></span><span><span></span><span><span><span><span style="height:1.3714em"><span><span style="height:3em"></span><span><span>d</span><span style="margin-right:0.1132em">τ</span></span></span><span><span style="height:3em"></span><span style="border-bottom-width:0.04em"></span></span><span><span style="height:3em"></span><span><span>d</span><span>u</span></span></span></span><span></span></span><span><span style="height:0.686em"><span></span></span></span></span></span><span></span></span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8644em;vertical-align:-0.15em"></span><span><span style="margin-right:0.2222em">V</span><span><span><span><span style="height:0.7144em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span>a</span></span></span></span></span></span></span></span><span><span style="margin-right:0.0556em">∂</span><span><span><span><span style="height:0.1514em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span>a</span></span></span></span><span></span></span><span><span style="height:0.15em"><span></span></span></span></span></span></span><span>u</span><span>.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>The key algebraic fact is:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>A null covector cannot be orthogonal to a timelike vector.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>Therefore,</span></span></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.9088em;vertical-align:-0.1944em"></span><span><span style="margin-right:0.2222em">V</span><span><span><span><span style="height:0.7144em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span>a</span></span></span></span></span></span></span></span><span><span style="margin-right:0.0556em">∂</span><span><span><span><span style="height:0.1514em"><span style="margin-right:0.05em"><span style="height:2.7em"></span><span><span>a</span></span></span></span><span></span></span><span><span style="height:0.15em"><span></span></span></span></span></span></span><span>u</span><span style="margin-right:0.2778em"></span><span><span><span><span><span><span style="height:0.8889em;vertical-align:-0.1944em"></span><span><span><span></span></span></span><span></span></span></span></span></span><span></span><span>=</span></span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>0.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>Because the future timelike cone is connected, the sign of this quantity cannot change. So after choosing the time orientation,</span></span></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:2.0574em;vertical-align:-0.686em"></span><span><span></span><span><span><span><span style="height:1.3714em"><span><span style="height:3em"></span><span><span>d</span><span style="margin-right:0.1132em">τ</span></span></span><span><span style="height:3em"></span><span style="border-bottom-width:0.04em"></span></span><span><span style="height:3em"></span><span><span>d</span><span>u</span></span></span></span><span></span></span><span><span style="height:0.686em"><span></span></span></span></span></span><span></span></span><span style="margin-right:0.2778em"></span><span>></span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>0</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>along every future-directed timelike curve.</span></span></div><div></div><div dir="ltr"><span><span>That means </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> is strictly monotonic along any physical observer’s worldline.</span></span></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>3. This rules out “same </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>, different place” too</span></span></h2><div dir="ltr"><span><span>Suppose your younger self is at</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span>A</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>100</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span><span>.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>Your older self later reaches</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span style="margin-right:0.0502em">B</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>500</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>80</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>20</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>5</span><span>)</span><span>.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>Now the older self tries to travel to a nearby past event:</span></span></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span style="margin-right:0.0715em">C</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>100</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>11</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span><span>.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>This is exactly the kind of scenario you described: not the same spatial point, but close enough to interact.</span></span></div><div></div><div dir="ltr"><span><span>But to go from </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:0.6833em"></span><span style="margin-right:0.0502em">B</span></span></span></span></span><span><span> to </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:0.6833em"></span><span style="margin-right:0.0715em">C</span></span></span></span></span><span><span>, the older self’s worldline would have to satisfy</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:1em;vertical-align:-0.25em"></span><span>u</span><span>(</span><span style="margin-right:0.0502em">B</span><span>)</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>500</span><span>,</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>and later</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:1em;vertical-align:-0.25em"></span><span>u</span><span>(</span><span style="margin-right:0.0715em">C</span><span>)</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>That would require</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span>Δ</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.7278em;vertical-align:-0.0833em"></span><span>−</span><span>400.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>But the theorem says that along any future-directed timelike path,</span></span></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.7224em;vertical-align:-0.0391em"></span><span>Δ</span><span>u</span><span style="margin-right:0.2778em"></span><span>></span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>0.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>So the path </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:0.6833em"></span><span style="margin-right:0.0502em">B</span><span style="margin-right:0.2778em"></span><span>→</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6833em"></span><span style="margin-right:0.0715em">C</span></span></span></span></span><span><span> is impossible in any spacetime satisfying the theorem’s hypothesis.</span></span></div><div></div><div dir="ltr"><span><span>The theorem does not merely say:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>You cannot return to </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:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>100</span><span>,</span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span></span></span></span></span><span><span>.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>It says:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>You cannot return to </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100</span></span></span></span></span><span><span> at </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>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>10</span></span></span></span></span><span><span>, </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>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>11</span></span></span></span></span><span><span>, </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>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>90</span></span></span></span></span><span><span>, or anywhere else.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>Once you have passed from </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100</span></span></span></span></span><span><span> to </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>500</span></span></span></span></span><span><span>, the value </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100</span></span></span></span></span><span><span> is permanently behind you. No future-directed timelike journey can bring you back to that null slice.</span></span></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>4. What about signals?</span></span></h2><div dir="ltr"><span><span>You might say:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>Fine, maybe the older self cannot physically travel there, but could they send a signal to the past?</span></span></div></blockquote><div></div><div dir="ltr"><span><span>No — at least not in the class of spacetimes covered by the theorem.</span></span></div><div></div><div dir="ltr"><span><span>For future-directed causal curves, which include both timelike and null curves, the 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> is non-decreasing:</span></span></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:2.0574em;vertical-align:-0.686em"></span><span><span></span><span><span><span><span style="height:1.3714em"><span><span style="height:3em"></span><span><span>d</span><span>λ</span></span></span><span><span style="height:3em"></span><span style="border-bottom-width:0.04em"></span></span><span><span style="height:3em"></span><span><span>d</span><span>u</span></span></span></span><span></span></span><span><span style="height:0.686em"><span></span></span></span></span></span><span></span></span><span style="margin-right:0.2778em"></span><span>≥</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>0.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>It is strictly increasing for massive travelers, and it can remain constant only along special null directions tangent to the null hypersurfaces.</span></span></div><div></div><div dir="ltr"><span><span>So if the older self is at </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>500</span></span></span></span></span><span><span>, any signal they send into the future must go to events with</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.7719em;vertical-align:-0.136em"></span><span>u</span><span style="margin-right:0.2778em"></span><span>≥</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>500.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>It cannot go to an event with</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.4306em"></span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100.</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>Therefore, the theorem also rules out sending messages to the past, not merely physically traveling to the past.</span></span></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>5. “Same </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>, different place” is not automatically interaction</span></span></h2><div dir="ltr"><span><span>There is one additional subtlety.</span></span></div><div></div><div dir="ltr"><span><span>If two events have the same </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> but different spatial coordinates, they are on the same null hypersurface. But that does not automatically mean they can interact instantaneously. In relativity, interaction is local and causal. If the events are spacelike separated, no signal can pass between them with zero elapsed </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>.</span></span></div><div></div><div dir="ltr"><span><span>So if the older self arrives at</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span style="margin-right:0.0715em">C</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>100</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>11</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span><span>,</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>and the younger self is at</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span>A</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>100</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span><span>,</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>they may be spatially close, but whether they can interact immediately depends on the metric and on whether </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:0.6833em"></span><span>A</span></span></span></span></span><span><span> and </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:0.6833em"></span><span style="margin-right:0.0715em">C</span></span></span></span></span><span><span> are causally related.</span></span></div><div></div><div dir="ltr"><span><span>However, this subtlety does not rescue time travel, because the older self cannot get from </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:0.6833em"></span><span style="margin-right:0.0502em">B</span></span></span></span></span><span><span> at </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>500</span></span></span></span></span><span><span> to </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:0.6833em"></span><span style="margin-right:0.0715em">C</span></span></span></span></span><span><span> at </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100</span></span></span></span></span><span><span> in the first place.</span></span></div><div></div><div dir="ltr"><span><span>Even if the older self only needs to arrive slightly after the younger self’s event, say</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span style="margin-right:0.0715em">C</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>101</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span>x</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10.1</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.0359em">y</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.8389em;vertical-align:-0.1944em"></span><span>10</span><span>,</span><span style="margin-right:0.2778em"></span><span style="margin-right:0.1667em"></span><span style="margin-right:0.044em">z</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>10</span><span>)</span><span>,</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>that is still much earlier than </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>500</span></span></span></span></span><span><span>. The trip</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.4306em"></span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>500</span><span style="margin-right:0.2778em"></span><span>→</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.4306em"></span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>101</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>is still forbidden.</span></span></div><div></div><div dir="ltr"><span><span>So the issue is not “Can they interact once nearby?” The issue is “Can they reach that earlier era at all?” The theorem says no.</span></span></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>6. Closed timelike curves versus “visiting the past”</span></span></h2><div dir="ltr"><span><span>There is a distinction between the formal definition and the informal idea of time travel.</span></span></div><div></div><h3 dir="ltr"><span><span>Formal definition</span></span></h3><div dir="ltr"><span><span>A </span></span><b dir="ltr"><span><span>closed timelike curve</span></span></b><span><span> is a timelike curve that loops back to its starting event. In that strict mathematical sense, yes, the curve is closed: after one loop, all global coordinates return to their original values.</span></span></div><div></div><div dir="ltr"><span><span>That is why the article says something like:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>No journey can return to its starting event.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>That is the standard CTC language.</span></span></div><div></div><h3 dir="ltr"><span><span>Informal time travel</span></span></h3><div dir="ltr"><span><span>But in the informal sense, “time travel” usually means something like:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>I, an older person, can enter the era of my younger self and interact with them.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>For that, exact return to the original event is not necessary. Arriving nearby in the past may be enough.</span></span></div><div></div><div dir="ltr"><span><span>Your criticism correctly points that out.</span></span></div><div></div><div dir="ltr"><span><span>But Zhang’s proof gives a stronger consequence than merely “no closed loop.” Because </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> strictly increases along every future-directed timelike curve, the theorem also forbids:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>Leaving </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100</span></span></span></span></span><span><span>, advancing to </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>500</span></span></span></span></span><span><span>, and then returning to </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 style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:0.6444em"></span><span>100</span></span></span></span></span><span><span> at a different spatial location.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>So the informal “visit the past nearby” scenario is also ruled out in any spacetime admitting such a global null coordinate.</span></span></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>7. If such a trip were possible, the theorem’s assumptions would fail</span></span></h2><div dir="ltr"><span><span>Suppose someone constructs a spacetime where an observer can go from</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span style="margin-right:0.0502em">B</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>500</span><span>,</span><span style="margin-right:0.1667em"></span><span>…</span><span>)</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>to</span></span></div><div></div><div dir="ltr" style="overflow:auto hidden;max-width:100%"><span><span><span></span><span aria-hidden="true"><span><span style="height:0.6833em"></span><span style="margin-right:0.0715em">C</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>(</span><span>u</span><span style="margin-right:0.2778em"></span><span>=</span><span style="margin-right:0.2778em"></span></span><span><span style="height:1em;vertical-align:-0.25em"></span><span>100</span><span>,</span><span style="margin-right:0.1667em"></span><span>…</span><span>)</span></span></span></span></span></div><div></div><div dir="ltr"><span><span>and then interact with their younger self nearby.</span></span></div><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 dir="ltr"><span><span>If you can return to the same null slice at a different spatial point after having advanced to a later slice, then one of those loopholes must be present.</span></span></div><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>8. The corrected way to state the result</span></span></h2><div dir="ltr"><span><span>The criticism shows that the explanation should not be phrased as:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>You cannot return to the exact same point, therefore no time travel.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>That is too weak, because interaction with the past does not require exact point-return.</span></span></div><div></div><div dir="ltr"><span><span>A better phrasing is:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>If a spacetime admits a global null coordinate </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>, then </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> acts like a one-way ratchet along every future-directed timelike worldline. Once you leave a given </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>-slice, you can never return to that slice again — not at the same place, not at a nearby place, not anywhere. Therefore you cannot enter your own past era, even slightly displaced in space.</span></span></div></blockquote><div></div><div dir="ltr"><span><span>Or even shorter:</span></span></div><div></div><blockquote dir="ltr"><div dir="ltr"><span><span>The theorem does not merely prevent you from standing in your exact old footprints. It prevents you from crossing back through the same light-slice, even one meter to the left.</span></span></div></blockquote><div></div><div><hr></div><div></div><h2 dir="ltr"><span><span>Bottom line</span></span></h2><div dir="ltr"><span><span>The criticism is right that:</span></span></div><div></div><ul dir="ltr"><li dir="ltr"><span><span>interaction with the past does not require exact spatial coincidence;</span></span></li><li dir="ltr"><span><span>returning to the same “time” at a different place would not be ordinary spatial travel;</span></span></li><li dir="ltr"><span><span>a time traveler could in principle affect their past self from nearby.</span></span></li></ul><div></div><div dir="ltr"><span><span>But the theorem still answers that criticism, because the proof does not rely on exact return to the original event. It proves that the null coordinate </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> strictly increases along any physical observer’s path. Therefore, in the class of universes covered by the theorem, a traveler cannot return to the same </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>-slice at a different spatial coordinate either.</span></span></div><div></div><div dir="ltr"><span><span>So the nearby-past scenario is not a surviving form of time travel under the theorem. It is precisely another thing the theorem forbids.</span></span></div><div dir="ltr"><span><span>-----------------------------------------</span></span></div><br></div><div style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)" class="gmail_default"><br></div></div></div>
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