Time travel is a very old and popular theme used in stories, especially in science fiction. It is also one of the most difficult plot devices to get right, in terms of logical consistency, simply because how weird the nature of time is. There are many different ways for the mechanism of time travel to work, but from a high level, they can be roughly divided into two types: single-world-line and multi-world-line. In single-world-line style time travel, there is only one version of the universe, and as a result, objects, events, and people can only have one timeline. If someone goes back into the past and paints a black ball in white, then the ball becomes white from that point on; the “previous” state when the ball remained black ceases to exist. In multi-world-line style time travel, there are multiple (potentially infinite) versions of the universe, and thus everything has many possible timelines. If someone goes back into the past and paints a black ball in white, under the multi-world-line model of time travel, there are now (at least) two versions of the ball, in two different world-lines: one that remained black, and one changed into white.

Single-world-line style time travel is extremely tricky to do in a logically consistent way. Many paradoxes, such as the famous Grandfather Paradox, can easily arise, causing any story or event to make little sense. Because of this, many fictional works involving time travel choose to use the multi-world-line version, since by comparison, this version is a lot easier to make sense of logically. However, due to how unnatural messing with time can be, even multi-world-line time travel doesn’t always make for a consistent story, at least not without some rules. So here, I want to examine some of the conditions that must be met, in order for multi-world-line time travel to be logically consistent.

First, a couple of definitions.

• Time travel: any movement by an object or person (and more generally, information, but I’ll touch more on that later), in the opposite direction of the natural direction of time passage relative to the rest of the world. The reason that only “traveling into the past” is considered here is that time travel into the future is trivial. In fact all of us “time travel” 1 second per second into the future, all the time. In addition, due to relativistic effects, the relative rate of flow of time between two entities varies based on their relative velocities, so anyone can effectively “travel into the future” by traveling at near the speed of light relative to the rest of the world
• World-line: a single, logically consistent universe, with all the objects, events, people, and information in it, across all time. Technically in physics, a world-line is the property of an object, tracing out (i.e. fully encapsulating) its path in 4-dimensional spacetime. But here, we will expand this term to refer to the collective world-lines of everything that exists in a universe, thought of was a single entity

Now, on to the actual rules and conditions of time travel. We start with the most important condition for logically consistent multi-world-line time travel, which is that there can be no single-world-line time travel. If single-world-line time travel is allowed in any form, then we are susceptible to all the consistency problems there. Technically we could allow some logically consistent forms of single-world-line time travel, but since this can get very complicated, for simplicity’s sake, we will just impose the strict ban on single-world-line time travel here.

An obvious corollary of the above rule/condition is that whenever any time travel takes place, the traveller (could be a person or an object) will always end up in a different world-line, or universe. This is regardless of whether the traveller is time travelling in its original world. For example, suppose a person, John, originally from world-line X in the year 2000, time travels to the year 1900. This will land him in another world-line’s year 1900, let’s call this world-line Y. If he time travels again in this world-line Y, say, to the year 1800, then he will “hop” world-lines again, and end up in the year 1800 of a new world-line, Z.

But what about returning to previously visited world-lines? Not being able to return to at least the universe that the time traveller originally came from would be pretty bad. But it turns out, returning to any previously visited world-lines can be logically consistent, as long as some rules are followed. From the first rule and its corollary above, we can see that the moment you visit any world-line, you will be marked with a timestamp in that world-line, as the most into-the-past you can ever go there. The first rule, that no single-world-line time travel is allowed, cannot be violated under any circumstances, including from actions that are not technically single-world-line time travels but result in the equivalent outcome.

Suppose John is originally from world-line X, year 2000 (let’s denote this as X-2000, for short). He travels to Y-1900, then Z-1800. At this point he cannot travel back to X-1900, even though this is technically a multi-world-line “hop”, otherwise we would end up with the effective result of a single-world-line time travel in world-line X. John’s latest timestamp (presumably kept track by some cosmic logical consistency enforcer) in world-line X is the year 2000, so he cannot go to any point in time prior to that in this world-line. He can however, return to X-2000 (his original world-line and time), as well as Y-1900. Spending time in a world-line would push his timestamp in that world-line forward, so the time period that John can travel to is constantly shrinking in his current world-line. The timestamps in any of his other previously visited world-lines are frozen when he is not in it.

These conditions are actually still not quite strict enough to prevent logical inconsistencies. The no single-world-line time travel allowed rule is not just to prevent physical time travel within a single-world-line, but rather to prevent the time travel of any information within a single-world-line. This causes a big issue when you begin to interact with the people and objects of another world-line. Again, let us illustrate this with an example.

Suppose John, originally from X-2000, travels to Y-1900. He interacts with Jane, an original resident of Y-1900. With just the rules we have outlined above, Jane could technically time travel to any point in time in world-line X, since she has never traveled to X. But because of her interactions with John, she now carries information from X-2000 (if nothing else, at least the mere existence of John in X-2000), and traveling to any point prior to the year 2000 in world-line X would actually violate our first rule. To keep things logically consistent, Jane must be restricted from traveling to anywhere prior to the year 2000 in world-line X, just like John. Similarly, all other time travel restrictions that John had (from other past time travel experiences to other world-lines) must also be inherited by Jane, in order to keep things consistent globally.

Two things we can further expand from this condition. The first is the scope of this restriction inheritance. Since there is no way to safely contain the interactions that a time traveller has, once entered into a new world-line, the entire universe must inherit the traveller’s time travel restrictions. Sure, John may have only explicitly interacted with Jane, but by being in world-line Y, his potential influence can spread to the entire world. John’s presence could technically be detected by anyone, and Jane could also simply tell everyone about him. The second thing to note is the direction of the restriction inheritance. Interactions are bi-directional, so by interacting with Jane, John must have inherited all of her time travel restrictions as well. And not just John, if he travels to another world-line (including returning to his original one), he would be carrying that inheritance from Jane’s world-line with him, spreading it there while simultaneously picking up new inherited time travel restrictions.

So it seems that the act of time traveling between two world-lines actually has the effect of somehow intertwining the two world-lines, with each inheriting the other’s time travel restrictions. More than two world-lines can be intertwined like this, should the traveller goes on trips through multiple world-lines. The more times a time traveller “hops” between world-lines like this, and the more people that time travel, the more intertwined the world-lines become, and the more restrictions will spread. Eventually the world-lines might end up become deadlocked: their time travel restrictions are so intertwined that traveling between any of them are no longer possible at all. Travellers might be unable to return to their original world-line in this case. Of course, traveling to new world-lines are always possible, as long as there are an infinite number of possible world-lines.

This is far from a comprehensive list of conditions to ensure the logical consistency of multi-world-line time travel. But even from this rough look, we can see that time travel rules are hard to get right, if we want things to make sense. The conditions for logically consistent single-world-line time travel is even more difficult to lay out. I might get to it in a future post.