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A temporal anomaly in 'Cause and Effect' (TNG)

In general, I tend to argue that people worry far too much about the logical details of Star Trek (not to mention "canon"). However, for stories about time travel and so on the logic is often part of the fun, rather as in detective stories. (Not always: the logic of Back to the Future is a bit dodgy in places but it doesn't matter.)

In "Cause and Effect", the Enterprise becomes stuck in a sort of time loop of perhaps a day. It runs into a rift in space, from which a ship emerges. The other ship collides with them, they can't control the damage, and the Enterprise explodes. This happens repeatedly. However, the reset isn't total: the crew develops slight, deja vu memories, and there are sounds of voices from previous loops. This enables them to discover their predicament and send a message to the next loop.

When they eventually escape from the loop by avoiding the collision, the other ship, the Boseman (also a Federation starship) gets in contact and we learn it has come from at least eighty years in the past. They don't seem to be aware anything is wrong, though. When I first saw the episode, I thought that the Boseman was supposed to have been in the loop for eighty years. However this doesn't add up, because their loop, like the Enterprise's, ends with the collision, so they must have the same number of loops, and even if their loop goes back to when they left space dock, three weeks ago, it won't work.

Let's do the maths. The Enterprise's loop seems to last less than a day, since we start with the night before and the collision happens in the morning. On one loop the collision happens around the end of the meeting, which (we learn from a later loop) apparently started at 0700 hours. Shall we say 12 hours; it can't be too much less than that. When they exit the loop they find they have lost 17.4 days = 17 days 10 hours = 418 hours. So they have been looping (not counting the first loop, which would not put their clocks wrong) for 418/12 = 34.8 times (35 since it must be a whole number).

OK, then for the the Boseman to have spent 80 years in the loop, with 35 loops each loop would be 80/35 years = 2.3 years. Since they left space dock three weeks ago this would imply that their loop took them way back into their past lives: was everyone they encountered in the loop too? That won't work, so the Boseman's loop has to be shorter, let's say no more than three weeks.

For the Boseman to reach 80 years in the loop with a three week loop: 80 years is 80x365x24 = 700,800 hours. Three weeks is 504 hours. So they would have to make (time lost)/(length of loop)= 700,800/504 = 1380 loops. For the Enterprise to have reached only 17.4 days at 1380 loops their loop would be (time lost)/(num loops) = 418/1380 = 0.30 hours long, or less than 20 minutes. This is definitely far too short a time.

Two possibilities: (1) Since the Boseman's loop involves coming through a rift in time, like the Enterprise C in Yesterday's Enterprise, their loop can have the same loop and number of iterations as the Enterprise's. They too are sent back to a time 12 hours ago by their time, but on the other side of the rift. The 80 years would be about the rift rather than the loop. (2) you aren't supposed to worry about this.

I think (2) is more likely.

There is a further problem, actually, which arises because of losing 17.4 days. If the explosion simply sent them into the past, then surely they would be living the same hours again. Captain Picard records 45652.1 as the Stardate as they enter the Typhon Expanse. According to a Stardate converter, this is Monday, 26th August 2368, at 4 o'clock in the afternoon—GMT of course. The collision is next morning, i.e. Tuesday 27th. So they are sent back to Monday 26th, at four o'clock—not just Monday for them but Monday for everyone else. [1] And that keeps happening. After it's happened 35 or so times, they escape, and it's Tuesday 27th August, for them and everyone else. So why are they 17 days out?

Perhaps they are not going back to the same time (in terms of the surrounding universe) but are in some sort of bubble. Within this bubble their own time is resetting, but only their own time. The collision happened on Tuesday morning. And twelve hours later, on Tuesday evening. And twelve hours later, on Wednesday morning... thus when they escape they are 17 days out. Presumably if another ship had turned up on Wednesday it would have found the Enterprise believing it was Monday. Also, where is the Enterprise? Let's call the place where they enter the Expanse "A" and the place where the collision happens "B". If the other ship observed over a period of time it would see the Enterprise cruising along on Monday, then colliding and exploding on Tuesday morning at "B", and then immediately reappearing back at "A" and cruising along on Tuesday...

I have a feeling my neurons have got tangled up.

The most obvious solution is that the 17 days is to show that they have been looping quite a few times, and you aren't supposed to worry about the complications.

A problem with the episode that may be insoluble, however, is the discussion when the crew realize that they are in a loop which will end with a catastrophe. "Maybe we should reverse course," suggests Worf. Riker objects: "For all we know, reversing course may be what leads us into the crash." Captain Picard concludes: "No. We can't afford to start second guessing ourselves. We'll stay on this course until we have reason to change it."

But Worf's suggestion is sound. The Enterprise is currently on a particular course, and (apart from this crisis) was presumably planning to continue on it. The loops have not been going on for ever: there was a first time through. On that first time through, there are two possibilities: (i) the Enterprise stayed on course, and that led to the collision (ii) the Enterprise for some reason changed course, and the new course led to the collision. But if it changed course, it would have been for some different reason. It could not be a course change on the basis that they discovered the loop, because they hadn't been round the loop yet. This other reason, if it exists, has not yet appeared. Suppose the Enterprise now changes course at random. Obviously it will not be travelling on the original course, so if that was the fatal course they now escape. It is possible that they have changed onto the fatal course, but only if the random change happens to produce the same direction as the course produced by the unknown reason. This would be a very unlikely coincidence.

On the other hand, if you continue on the same course until you have some reason to change it, as Picard decides, it seems likely you are reproducing the events which led to the collision the first time through.

So, Picard should go with Worf's suggestion. That one did strike me when I first saw the episode. (There's a classic Youtube video showing how Worf's warnings are always ignored.)

However, let's be fair. The Enterprise has to stay on course, or there is no story. Because the possibility of simply changing course is obvious enough to occur to a lot of viewers, the writers raise it and dismiss it. Their argument is specious—they are simply blowing smoke in the viewers' eyes. But it's convincing enough to get you to set aside the problem. It's well done.

There are a number of interesting logical problems involving loops. Here's one from the late Raymond Smullyan: in order to live for ever, you just have to always tell the truth—not much to ask for immortality—and make just one statement. Here it is: "Tomorrow, I shall repeat this sentence." By the way, how about this version: you can achieve immortality by just making one true statement, namely "Tomorrow, I will truthfully repeat this sentence."

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