Multi-Dimensional Time

Joe builds a time machine in 2006. He goes back to save Abraham Lincoln. His success in preventing John Wilkes Booth from shooting Lincoln creates an apparent contradiction: On April 14, 1865, Lincoln was killed. Indeed, Joe had read about the assassination in history books. With Joe on the spot, however, Booth is stopped, and so Lincoln’s life is saved on April 14, 1865. Can that be? Yes, if time is multi-dimensional.

The simplest way of thinking about and describing multi-dimensional time is using a “branching”, or “train-track” model. This model, animated below, begins with the original (one-dimensional) time line. At the far left is Lincoln’s birth in 1809, and on the far right is the year 2006. Lincoln’s wedding in 1842, his assassination in 1865 and Joe’s birth in 1967 fall between the two ends. If this time line described the entire temporal history of the universe, then it would be hard to make sense of Lincoln’s death and rescue.

The idea that time is multi-dimensional, however, allows a “branch” or “spur” to emerge from the track in early 1865 upon Joe’s arrival from his trip backwards in time. This spur would include events like Joe’s presence and his stopping Booth, events that are not on the original line. So, though Joe wasn’t at the Ford Theater to save Lincoln in the 1865 of the original time line, he can be there in the 1865 of this spur.

With one-dimensional time, if Joe time travels, he does not make what already has happened not to have happened, or vice versa. With multi-dimensional time, by merely traveling to a time at which he did not exist, he does make something–his arrival, his presence–that didn’t happen (on the original timeline) to have happened (on the spur off the original line).

Philosophers G. C. Goddu (2003) and Jack W. Meiland (1974) have proposed more formal models of multi-dimensional time.1 Let’s take a look at those models in relation to the “train-track” model.


In Goddu’s model, there are two kinds of time: hyper time and normal time. Hyper time steadily progresses. Normal time, however, can skip around. They track together perfectly through the creation of the universe, the building of the pyramids, Napoleon’s exile, and the signing of the Declaration of Independence—so long as there is no time travel. With regard to our story, it is only at the moment when Joe jumps in his machine and travels back to save Lincoln that time starts to behave in unexpected ways. At this point, hyper time keeps moving forward, but normal time reverts to 1865.

Hyper Time T1 T2 T3 T4 T5 T6 T7
Normal Time 1809 1842 1865 1900 1967 2006 1865
Key Events Abe’s birth Abe’s wedding Abe’s death Joe’s birth Joe departs Abe’s rescue

Hyper time works as an index of normal times, which are times as we usually think of times—the times our clocks show. They need indexing on this model because they can recur at more than one hyper time. This is what allows contradiction to be avoided in our story about Joe and Abe. The assassination can take place in 1865 of hyper time T3 and the rescue take place in the 1865 of hyper time T7. While Joe was both present and not present in 1865, that time occurred twice, so there is no contradiction. What is important is that Joe was never both present and not present at the same hyper time/normal time pair.

It is useful to think about Goddu’s model in terms of the train-track model. With Goddu’s understanding of multi-dimensional time travel, it is as if part of the original time line gets overwritten by what on the train-track model happens on a spur. The event of backwards time travel sends the universe back to a time on the original line and then lets the history unfurl a second time, differently than it did originally.

When Joe travels back in time to save Abe, he arrives at a time after Abe’s wedding and before Abe’s death. As a result of his time travel, all the events that occurred after his arrival on the original time line have been erased, including Abe’s death in 1865. Consequently, Joe is able to rescue Abe at Ford’s Theater and essentially rewrite the history he had learned prior to his departure in 2006 on the original time line.

On Goddu’s model, a time traveler to the past is a very powerful individual. The actions of the time traveler are enough to rewind and re-record the history of the universe!


Meiland also argues that time could be multi-dimensional, but has a different take on what the dimensions are like. He uses the following diagram to help explain his model.

diagram of Meiland's model of multidimensional timeThe diagonal dashed line from t1 to t6 represents the typical one-dimensional kind of time line to which we are accustomed. Someone who never time traveled could live an entire life along that diagonal. Assume that the diagonal distance between t1 and t2 represents one year. The distance between t1 and t4 would then represent three years. According to Meiland, the past relative to each present, ti, is represented by the horizontal lines, Pi , up to ti.

Now assume a person, Jack, were to time travel three years back from t4. He could accomplish this by hopping three years directly left (into the past) of t4, so to Point A. Assume he then lives for two years after reaching this point without time traveling. That is represented by the dashed diagonal line segment from Point A to Point C. At Point C, he is three years in the past of t6. In essence, if a person travels back through time and stays there, he lives normally through events (on a different diagonal) from then on. Jack is living in the past, always a certain amount of time from the presents of his original time line. This allows Meiland to make sense of claims a non-time traveler might make at t6, like, “Right now, Jack is three years in the past.”

What of Joe and his rescue mission? Suppose that t1 and t2 are almost 50 years apart, as are the other times marked on that diagonal. Let t2 be the year 1865 at the time of Lincoln’s assassination along the original time line, a time at which Joe does not exist. Let t5 be the year 2006 along that same line at the time of Joe’s time-travel departure. He travels back almost 150 years, hopping directly left to the 1865 of the past of t5, to Point B. At Point B, Joe does exist and is in position to rescue Abe.

True Time Travel?

In his paper “The Paradoxes of Time Travel,” David Lewis expresses concern that multi-dimensional time travel is not true time travel.

On closer inspection, however, this account seems not to give us time travel as we know it from the stories. When the traveler revisits the days of his childhood, will his playmates be there to meet him? No; he has not reached the part of the lane of time where they are. He is no longer separated from them along one of the two dimensions of time, but he is still separated from them along the other (1976, 145).

Lewis and other philosophers, such as Le Poidevin (2005, 341) are concerned that a time traveler in multi-dimensional time will not be able to interact with the people of his past.

Goddu’s description of multi-dimensional time travel addresses Lewis’s concern. According to Goddu, a time traveler can interact with his childhood playmates via backwards time travel in normal time. For example, Josh, who is thirty years old, wants to visit his childhood playmates at his tenth birthday party in 1986. Therefore, Josh gets into his time machine in 2006 and travels back to the day of his tenth birthday party. As a result of his trip, everything that was after his arrival date (the day of his tenth birthday party) is erased and 30-year-old Josh is able to re-record history. However, 10-year-old Josh and his pals are still on the original time line; 30-year-old Josh has not jumped to another time line. Hence, he is able to interact with his childhood playmates and even his younger self; he is not separated from them as Lewis suggests. Similarly, in our original example involving Lincoln’s assassination, according to Goddu, Joe really rescues Abe, not an Abe counterpart.

Goddu’s answer to Lewis’s concern suggests that the concern is also not troublesome for the “train-track” model. When Josh travels back to his tenth birthday party, he arrives at a point on the original time line and encounters the same childhood friends that he remembered from his childhood years. Immediately, a “branch” or “spur” begins at his arrival point in 1986 just off the original time line. Then, Josh and everyone from the original time line that was living at his arrival point begin to live out their lives on the recently-created “spur.” Since the “branch” started from the original time line, Josh’s childhood friends that are on the “branch” are the same as the ones that were on the original time line. Hence, it is clear that a person on a “spur” or “branch” of the original time line is no different from the same person on the original time line. Josh really time traveled and really interacted with the childhood friends of his memories.

Does Lewis’s concern apply to Meiland’s multi-dimensional model of time? According to Meiland, his two-dimensional time model is not radically different from our ordinary, one-dimensional perception of time. He treats “the past as a continuant, as existing at each of several times” (p. 161). Like an enduring material object or person, the past changes. It grows to include events that were once present or future events, and it could change in other ways too, like with the arrival of a time traveler. Despite the changes, however, there is only one past. When 30-year-old Josh departs on his time travel trip, the past as it is at the time of the departure may include 30-year-old Josh’s arrival, 10-year-old Josh and his playmates. According to Meiland, a time traveler can interact with what is truly his past.

Enriching Meiland’s Model

Meiland wrote his paper before David Lewis’s (1976) and Paul Horwich’s (1975) seminal articles on time travel. So, he did so without the benefit of some now fairly standard terminology, especially terminology marking the difference between proper/personal time and observer/external time. In the preceding section, we have presented Meiland’s model pretty much as he did. But a little thought shows how it can be souped up in a way that makes it more intuitive.

Let the diagonal through t1 be the original time line, quite analogous to the original line of the train-track model. Every other diagonal line that could be drawn parallel to the first one is a potential time line. (The original time line of course is also (trivially) a potential time line, but it is always also an actual time line.) Think of the vertical axis (insofar as there is one) as diagonally indexing the potential spurs; don’t think of the vertical axis as horizontally indexing the past relative to different presents.2

We need a way to represent a traveler’s proper time, the temporal progression of his life. This is was done implicitly in Meiland’s model, Goddu’s model, and even in the train-track model as accompanying descriptions always made clear what happens first and what happens subsequently from the perspective of the traveler. Let’s make the representation of the progression of the traveler’s life explicit. The traveler’s proper time shall be tracked by a line (or some other plot of points) tracing his journey through the two-dimensional temporal plane. We can do it here with a bit of animation:

This plot of points above represents the life of a person who never time travels. To show someone like our hero Joe time traveling, the plot will need to be a little more complicated.

When Joe lands at coordinate point (P5, Pt2), a spur of the original line is started, one that includes his arrival and does not include the assassination. With a simple rescue trip like Joe’s, it becomes clear why he can expect to find Abe in serious trouble. Joe’s arrival starts a spur off of the original time line in 1865, the time line familiar to Joe from his history books. Don’t be misled by the fact that the graphical representation of the spur is not continuous with the initial portion of the original timeline. Events along the spur should still be thought of as caused by earlier events along the original line.

The mechanics of the enriched Meiland model will take a little getting used to and the math can be a little complicated. But, here is the rough idea: Every time a time traveler arrives at a destination off all previously actualized time lines, a new diagonal time line through that destination is actualized. The circumstances and events present on this newly actualized time line are based on the first actual time line reached when drawing a vertical line towards the original line from the earliest (in normal time) arrival of the traveler along that newly actualized line. The histories of those two lines match up to that earliest moment.

For example, at age 30, Sam travels back twenty years to visit his pet frog. He stays on the new diagonal line for 10 years, then travels back another twenty years. Will he see a past based on the original time line, or his modified time line which lags the original by ten years? By dropping a vertical line towards the original line, it is apparent that Sam’s new adventure will be based only on the original time line. He will arrive to find circumstances similar to the ones he experienced at infancy.

Had he only traveled back 5 years on his second trip, his final time line would have been a spur off the line he had just been on.

Our enriched Meiland’s model is really just a slightly more formal version of the “train-track” model. Though Meiland’s graphs appear to have separate, unrelated time lines, these apparently separate time lines can be thought of as “spurs” of the original time line. Therefore, just like with Goddu’s model and Meiland’s original model, a time traveler can interact with real people of the past and not people that are just similar to the real ones.

References and Further Reading

Goddu, G.C. “Time Travel and Changing the Past: (Or How to Kill Yourself and Live to Tell the Tale).” Ratio 16 (2003): 16-32.

Horwich, Paul. “On Some Alleged Paradoxes of Time Travel.” Journal of Philosophy 72 (1975): 432-444.

Le Poidevin, Robin. “The Chesire Cat Problem and Other Spatial Obstacles to Backwards Time Travel.” Monist 88 (2005): 336-352.

Lewis, David. “The Paradoxes of Time Travel.” American Philosophical Quarterly 13 (1976): 145-152.

Meiland, Jack. “A Two-Dimensional Passage Model of Time for Time Travel.” Philosophical Studies 26 (1974): 153-173.

van Inwagen, Peter. “Changing the Past.” Oxford Studies in Metaphysics, Volume 5. Ed. D. Zimmerman. Oxford: Oxford University Press, 2010: 3-28.

For page credits, see the “Topics-Page Credits” page.

1. Peter van Inwagen (2010) proposes a model with some similarities to Goddu’s. The differences between these two models will not be important for our introductory-level discussion.

2. This is a break from Meiland’s model. Meiland (pp. 154, 161-162) was very concerned to make sense of claims like, “Right now, Jack is three years in the past.” Lewis (pp.146-147) has shown how to do that without introducing two-dimensional time. So, our inclination is to ignore the element of Meiland’s model that indexes horizontal lines to represent pasts of present moments. The focus should be the diagonals.