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In the Star Trek Voyager Episode "Deadlock" Voyager encountered a "spacial scission" that created two Voyagers. Voyager 1 (unhealthy/damaged) and Voyager 2 (healthy/undamaged) split apart, but only their matter not their anti-matter.

Janeway stated that the problem with the Kent State experiment was that the particles occupied the same space:

JANEWAY 2: Just like original Kent State experiment. The duplicate atoms couldn't occupy the same point in space time for very long before

JANEWAY: Mutual annihilation.

JANEWAY 2: What are our options?

JANEWAY: Let's try a different tact. Instead of trying to merge the two ships, let's try to separate them. Maybe we could divide the antimatter between us.

Moving them out of the same space should have helped the situation, not made it worse.

So why did the ships have more problems when they got further apart?

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I think there are two different definitions of "farther apart" that are being conflated here, but there also a set of different conditions from the Kent State experiment that would have impacted the results.

As Janeways theorized, part of the problem was that the copies occupied the same space. It may well have been that if you could get the Voyagers to stop sharing the same space, the problem would be resolved (at least inasofar as the technobabble problem wouldn't be a danger, though they'd still have a badly damaged Voyager and two crews potentially needing to share a healthy one). But they never actually stop sharing the same space, at least until one of the Voyagers explodes. At which point, the problem is solved.

The problems seemed to get worse, not as the ships got further apart physically, but rather, furthur out of phase. As seen numerous times in Trek, but most notable the TNG episode, "The Next Phase", when two objects are out of phase with each other, one can pass through another with no difficulty, sometimes they're unable to percieve one another, and, in fact, two objects can occupy the exact same physical space. This is what was happening in the episode (and why members of the Voyager crew couldn't see anyone from the other crew, even though they weren't always in the exact same place):

TORRES 2: It's a little harder to establish a comm. link than I thought, Captain. The molecular signature of the second Voyager is slightly out of phase with our own. I can't get visual, audio, nothing.

This may be why individual crew members were less affected than the ship as a whole, because their atoms were often far separated from those of their duplicates.

Now, maybe it would be possible to physically separate the Voyagers as well... except for one thing, the major source of problems for the whole episode: the Antimatter wasn't duplicated, and in fact, was being shared between the two ships (so whatever 'phase' problems apparently doesn't have the same affect on antimatter, just as the subspace divergence field didn't). This complicates any attempt at separation tremendously.

They actually make a reference themselves that provides a good way to visualize it:

JANEWAY 2: Like Siamese twins linked at the chest, with only one heart.

Picture, for example, a hypothetical Voyager crew-member who had an antimatter heart (but with plenty of energy to spare for two of them otherwise they just collapse and die). When they hit the spatial scission, they're duplicated like everyone else... but their heart isn't. And that has certain consequences. Unlike everyone else on the crew, they can't so easily split up and just walk in another direction from their double, because when they try, both sides of the person would be pulling on the antimatter heart, in different directions. Try hard enough and one might at some point break and suddenly be left catastrophically heartless (while the other, who has no idea this has just happened, walks away with the whole bag, because their containment ribcage is still intact... they only know they feel much better), but until then both of them would be causing damage to one another. The same would be true of Voyager, if they pulled apart physically, it might help the quantum-cohesion problem, but they're still sharing antimatter and actually, physically separating then becomes much more difficult.

That doesn't mean it's not doable. It may be possible to do something to give only one Voyager the entire supply of Antimatter and then separate physically.

We don't actually see Janeway ever get to test her theory (except, again, by sudden explosive separation of every atom of Voyager at the end of the episode). Immediately after she proposes it, the other Janeway shoots it down, which prompts her to make a more permanent decision:

JANEWAY: Let's try a different tact. Instead of trying to merge the two ships, let's try to separate them. Maybe we could divide the antimatter between us.

JANEWAY 2: I'm afraid not. We've been studying that theory. B'Elanna tells me that any attempt to disrupt the antimatter supply will destroy us all. What about evacuating your crew to my ship? It might get a little crowded, but we could manage.

JANEWAY: We've been studying that theory. And my B'Elanna tells me that sending any more than five to ten people through the rift would radically alter the atomic balance of the two Voyagers. We'd both be destroyed. Captain, I think you should return to your ship and run a metallurgical analysis. Find out the precise phase modulation of your hull. I'll do the same here. Maybe we can find a way to realign the phase displacement.

JANEWAY 2: You're going to self-destruct your ship.

JANEWAY: What makes you say that?

JANEWAY 2: Because that's what I would do if your Voyager were intact, and my Voyager were crippled, my crew wounded or dead. I'd sacrifice my ship so that yours could survive.

JANEWAY: Then I'm glad we agree. Go back to your ship.

JANEWAY 2: I didn't say I agree, I said I understand. I'm not willing to let you make that sacrifice yet. We haven't explored all the options.

JANEWAY: Yes, we have and we both know it. This is my ship and my decision.

JANEWAY 2: Captain, I'm not going to let you

JANEWAY: I've made my decision! Please don't make me call security and have you escorted off my ship, because you know I'll do it.

JANEWAY 2: All right. I'll go back. But give me fifteen minutes to come up with another solution. After that, it's your decision.

JANEWAY: Fifteen minutes. Good luck.

JANEWAY 2: To both of us.

From this point on, no furthur efforts are made to either merge or divide the ships, there is just an attack by an outside fource. Though in the end, it is the other Janeway who winds up destroying her ship, to save the others all from the Vidiians. Still, you could view the self-destruct option as actually a refinement of her "physically separate the two ships" plan... physically separate it with an explosion that scatters all the pieces of one.

Previous to Janeway proposing this theory, we don't actually see Voyager ever getting farther apart (well, FX shots may show a brief moment where it looks like Voyager's physically pulling apart, but FX shots often show sound in space, too, it's usually best to treat that as a dramatic visualization rather than a literal description of what occurs). What we see is, they try to use a technobabble solution to undo the scission and merge the ships into one:

JANEWAY: My counterpart has suggested we try to merge the two ships. Recreate the subspace divergence field we passed through and then depolarise it.

TORRES: Both ships would have to send out a massive resonance pulse from their deflector dishes at exactly the same time.

Those attempts fail and the ships get pushed further out of phase, and that seems to make the quantum cohesion problem, of occupying the same space, worse.

This is made explicit in lines like the following:

(Both ships are shaking.)

TORRES 2: But we aren't merging. In fact, we're going further out of phase!

JANEWAY: The attempt to merge our ships is throwing us further out of phase. The quantum cohesion on both Voyagers is breaking down.

It's only after this that we see Janeway propose trying to separate the two ships instead of merging them, which suggests that a) that she meant physically separating them, and b) that that's not what was happening before and causing more problems.

So, in short: Janeway was speaking of separating the ships in spacetime, and all the problems we saw resulted from it being separated in phase. Her proposed solution was complicated by other factors not present in the Kent State experiment, and as it turns out the best way to separate the ships in space was to destroy one, which is what happened.

Maybe not the most ideal way to confirm a theory, but any scientific experiment one subspacially-duplicated version of you can limp away from can't be all bad.

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