What does it actually mean to have two time dimensions?

The trivial answer is that it just means "you have two dimensions with inverted sign in your spacetime metric". But the perceptual result of that kind of choice doesn't actually look qualitatively different from our universe.

Consider the 4D case; our 4 dimensional spacetime has a a +,+,+,- metric--according to one convention. If we decide we want two time dimensions, we get +,+,-,-. But, because the choice of which dimensions count as negative and which as positive is in fact purely conventional, this is physically equivalent to a metric with signs -,-,+,+. In other words, physics cannot uniquely distinguish spacelike dimensions from timelike ones. And in fact, Greg Egan wrote a novel in a universe with such a metric (Dichronauts), in which perceptual proper time is still distinctly one dimensional--just as in our universe, time is the length of your (one-dimensional) worldline.

Go the other way and give all dimensions in you metric the same sign, so there are formally zero time dimensions, and you don't get a static universe without time--you get another Greg Egan novel (actually a trilogy, Orthogonal), again with normal one-dimensional proper time as measured along worldlines.

It would seem that actually having two dimensions in a practical perceptual sense would require either

1. Converting proper time into a vector quantity somehow, or
2. Replacing worldlines with worldsheets, such that proper time is proportional to area.

Suffice it to say, I have no idea what either of those things means, from a physics or narrative perspective.

So... ideas?

• Comments are not for extended discussion; this conversation has been moved to chat. – L.Dutch Jul 19 at 19:55
• i was about to ask what it actually means to have a neg/pos sign on a dimension. then i read your last pragraph. Delightful! – bukwyrm Jul 25 at 5:34

First people need to understand how relativity works. There's a thing called proper time which we regard as an interval between two events or points in spacetime. In "ordinary common sense" space you define the distance (the interval) like this :

$$s^2 = (x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2$$

That's the square of the distance between two point. Time (and in particular our human sense of time) has no involvement here. You might also consider defining that interval as $$s^2=c^2(t_2-t_1)^2$$ which is pretty trivial. Either way there is no connection between a human sense of everyday time and the everyday definition of an interval or distance between events or objects.

In relativity in our 3+1 D universe (three spacelike dimensions and one timelike dimension), we define an interval as :

$$s^2 = c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2$$

Two observers have to see that interval as being the same, and the mathematics of that definition is what connects space and time and why we end up with time dilation and length contractions - time and space are connected explcitly and cannot be disconnected.

Timelike means $$s^2$$ is positive. A timelike dimension ($$t$$) has that positive contribution to the interval.

Spacelike means $$s^2$$ is negative. A spacelike dimension has a negative contribution to the interval.

Well that's how your basic universe we live in works. Here's an interval for a universe with two timelike dimensions $$t$$ and $$u$$ :

$$s^2 = b^2(u_2-u_1)^2 + c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2$$

But how is a human time defined in such a space ?

Short version : it's not.

The idea of time as we understand it is apart from these notions of relativity. Our everyday concept of time relates to a boring Newtonian ("classical") universe where space and time are not connected.

It happens we can bridge the Newtonian concept of time with the relativistic concept of time in our universe, but that's not going to work at all in a universe with two timelike dimensions. Whatever perceptions of "time" the inhabitants of such a universe have, it won't relate at all to anything we understand as time.

Now in a deeper sense the human perception of time relates to the concepts of energy and entropy ( "the arrow of time" ). So in your other universe with two or more timelike dimensions their equivalent of human time (as opposed to abstract mathematical timelike dimensions) might (might !) relate to how energy and entropy connect with those dimensions. I'm not at all sure an arrow of time would be expressed as a single scalar value in such a universe - it might require a vector of time along a multidimensional plane of "time" dimensions. There might be multiple entropy values or their equivalent.

Alternatively it's not impossible that such inhabitants might actually not work quite naturally with two time dimensions. For them the idea of one time dimension would be very strange - their common sense (their version of Newtonian mechanics) would have quite a different feeling. In instead of having one velocity vector describing motion, maybe they have two.

And they might even use both of these concepts (or something like them) simultaneously. We do. It just happens it's easy for us because in this universe we only have to reconcile one timelike dimension with the arrow of time and as it happens the maths works out that the relativistic effects aren't ones we normally have to experience, so the simplest view is easy to use. In the other universe these inhabitants will presumably develop their own perceptions of before and after and maybe have a concept like before-after and before-before and after-before and after-after with different combinations of when things happen in the different timelike dimensions.

So a question like "when were you born ?" could have an answer like "Dec 1965, January 1831". A question like "which came first ?" might be meaningless and you might need to say "which came first-first ?" or "Which came anytime-last ?". It might make perfect sense to them and be impossible for us to cope with.

As perceptions of time and space are something your brain invents to make sense of the world, that's likely to happen in such a universe (if anything like inhabitants can exist at all). They might commonly have quite different perceptions.

Extra dimensions

Let's go back to that interval :

$$s^2 = b^2(u_2-u_1)^2 + c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2$$

In our universe there's no $$b$$ part of this, but depending on the relative values of $$b$$ and $$c$$ it's not impossible that one timelike dimension dominates the other in "everyday human" common sense events. The effect of the other time dimension could be important in cosmology in that universe and irrelevant to ordinary everyday physics. It's quite possible you could have values of $$b$$ and $$c$$ such that they can discover physics exactly like our 3+1 D universe even down to advanced quantum field theory long before they even realize there's more timelike dimensions.

So this universe might actually "look" quite similar to us with particular parameters for these kind of values. We actually use physics theories with way more than 3+1 dimensions ourselves. String theories some in a variety of flavors including ones with 26 dimensions (!), 10 dimensions and 11 dimensions. You can have these theories in such a way that the extra dimensions don't impact "normal" physics because there effect is small (in more sophisticated ways than my rather simplistic suggestion above).

But again, remember that these abstract physical theories don't necessarily easily connect to a human (or "universe inhabitant") idea of "time" or "time's arrow".

• I do not want to be the programmer who writes the datetime library in that universe! – Tim Jul 17 at 16:28
• It might be far worse than you describe with the after-before. If the two time dimensions are indistinguishable and not separable like our 3 spatial dimensions are, after-before or before-after might only be true relative to some other point in 2-d time in the same way that up and down are relative to a particular planet and where you are on said planet. – Shufflepants Jul 17 at 17:17
• It's worth noting that even in 26-D or 10-D string theory, there's only one time dimension; the rest are spatial. – Michael Seifert Jul 17 at 17:32
• Also: with 1 time dimension, there is a clean separation between future-directed timelike vectors and past-directed timelike vectors. With 2D, there's not. In mathematical terms, the set of all timelike vectors is isomorphic to S^0 x R^3 in a Universe with 1 time dimension and 3 spatial dimensions, while it's S^1 x R^3 if there are 2 time dimensions & 3 spatial dimensions. This means that if there are 2 dimensions you can continuously change your coordinates to turn the "past" into the "future" and vice versa. – Michael Seifert Jul 17 at 17:51
• In natural (Planck) units, there is in $c^2$ in the time interval; it is merely a conversion factor between our units of time and space. – Yakk Jul 17 at 18:09

Imagine you are a single dimensional creature, You go along and encounter an object. You hit said object and it disappears out of your universe, because it took a vector outside your perceptional ability to comprehend. but if you change your comprehension to a 2D universe, you notice it moved in a vector perpendicular to the 1D field of view, moved to the edge of your universe and disappeared. Change it up again to a 3D universe and you notice it was a ball that rolled across the table a fell off the side, and bounced away.

Imagine time having the same vector capabilities. We perceive time in a 1D sense. We interact with objects, things happen and the event ceases to exist in your universe. Add a 2d or 3d to time, then you can perceive events occurring in vectors outside of your 1D point of view.

So if you can freely travel through time on the 1D vector, you can see an event play out, reverse and play it again or you can speed it up to see the event in conclusion. If you move freely in multiple dimensions of time, you can do the same, but then you can watch the even in any and all possible initiations and conclusions that could come out of an event.

Effectively, perceiving time in multiple dimensions could be what you call alternate realities.

• Although this is an interesting answer, I don't believe this is a correct description of a 3D+2 spacetime. – forest Jul 17 at 7:51
• This is what I imagined. where a second time axis means sliding through alternate timelines infinitely. after all a square could be described as an infinite set of lines the same length and start/stop points between axis1 and axis 2 between axis A and axis B – IT Alex Jul 17 at 18:22
• @MrLister It's not an accurate description of any spacetime with two time dimensions. – forest Jul 18 at 0:00
• This is how I understand it by reading up on it. The concept pf "alternate realities" would only exist on how a 1 temporal perceiving being would see the effects of a multi-vector time event. a 2D temporal being would perceive "alternate realities" as just a continuation of an event. – Sonvar Jul 18 at 2:55
• I should add, multiple people have attempted to determine how a multiple temporal universe would work, but due to their math, they state a 3D+2 world would hinder how gravity works and make the universe unstable. This is based on the math that cannot perceive a universe in anything other than 3D+1. – Sonvar Jul 18 at 2:58

I'm going to post a bit of an off-beat answer: you've probably already experienced a 2-time-dimension universe. Because that's exactly what you experience when you play a computer game with pauseable time controls.

Think about it - you've got two time dimensions: real-time and game-time. More than that, those different 'times' are orthogonal. Real time can run while game time is stilled whenever you pause the game; Real time and game time can occur in differing ratios when the game is running under its various speeds; real time can freeze while game time runs during any sort of "Auto Resolve" function. You as a person are experiencing a single time dimension, but within the context of the game, you're experiencing and controlling two separate time dimensions. The game itself is experienced in two time dimensions.

Literally, the only differences between your question and someone playing a computer game is that they can't comprehend events along the scale of game-time (because their brain runs solely in realtime) and the fidelity of the game during the autoresolution (because the computer running it is also operating solely in realtime.) So a great makeshift way of mentally picturing it is to simply imagine playing a computer game, with the tweak that when you "autoresolve" something, your brain still somehow knows everything that happened during the autoresolution (even if no real time passed during it.)

• Game time is more like a treadmill (comparing spatially) you can choose if you're moving in that direction or not but it doesn't create any extra directions for you to move in and isn't orthogonal to any other dimensions (you can't move in game time without moving in real time). – Lio Elbammalf Jul 25 at 5:37
• -1 This is not what a 3D+2 spacetime is. – forest Aug 17 at 7:25
• @forest, I'm not saying "that's what 3D+2 spacetime is" - the question's not about physics. The question is about Worldbuilding, about what the denizens of that sort of universe would experience from a narrative perspective. The OP had two main thoughts - either they'd simply experience a single composite time vector or would have a "time sheet" instead of a "time line". Well, my answer is a way of constructing/visualizing two separate time vectors that's neither of those options. – Kevin Aug 19 at 5:29

I haven't read Dichronauts, but unless you can time travel, I don't see how it would change anything.

Time is not entirely comparable to a space dimension, because you can't move freely in it. It goes in only one direction, and and at the same rate by everyone (except for relativity, I'll come to that later).

If we see time as a vector (typically in these kinds of graphs), having two orthogonal time vectors will just make one vector that is the sum of both, and that will be perceived as the one time vector by everyone.

The only thing that would make things different from a universe with one time dimension is if there's a special kind of gravity that distorts one time dimension more or less than the other. However, there's no reason for gravity to act differently on one of them. If you're close to a black hole or travel near the speed of light and both dimensions are affected equally, there won't be any difference.

So unless an object or entity has the power to travel in time or bend one of the time dimensions, it should feel like a universe with one time dimension.

• I'd advise understanding the question before answering. Both Dichronauts and Orthogonal cover the difference between relativistic and entropic (perceptual) time heavily, and you don't seem to be keeping that in mind. – Yakk Jul 17 at 18:16

From a world building perspective, this is how i would describe 2 time dimensions:

1D + time_alpha: A dot moving in {x} space and time{a}. The time line will always be straight.

2D + time_alpha: A dot in {x,y} space and time{a}. The line will now be curved, but only in one dimension.

3D + time_alpha: A dot moving in {x,y,z} space{a}. The line is now free to form in any spacial dimension, but it will always be continuous.

3D + time_alpha + time_beta: a dot moving in {x,y,z} space and {a,b} time.

As {a} and {b} are perpendicular, motion through one is inversely proportional to motion through the other(assuming motion through {a} is constant).

In this case, motion through {a} is no longer guaranteed to be constant, therefore, perceived speed would change depending on one's motion through those temporal dimensions. High velocity through {b} would reduce velocity through {a}, leading to discrepancies in observed and experienced velocity through {x,y,z}.

The result could be large, sudden changes in spatial velocity or position over very little {a}.

e.g. A space ship sets velocity(v) to the 0.5c. It should take 8 years{a} to reach Proxima Centauri. But from the perspective of Earth, it would take 17.4 years.

So, the ship changes it's temporal velocity to move in equal amounts {a,b} equal to the amount that we currently move through {a}. Thusly, motion through {a} is reduced by about 30% (The unit vector ratio of two equidistant perpendicular axes is 0.707).

The journey, from Earth's point of view, now appears to take 12.18 years. From Earth's perspective, The ship has increased in speed, but it has been travelling at 0.5c all this time.

If the ship further increases it's motion through {b}, eventually you could have near lightspeed travel with no appreciable time dilation.

Of course, this is all actually bunkum, but if you are writing a scifi story, it could be just believable enough to the layman to suspend disbelief.

You could add some kind of side effect of moving through {a,b} instead of {a}, like slipping into very close-by alternate realities with only subtle changes. high {b} velocity can take you far from your "home" reality, where the changes are more profound. In such a way there could be some fictional calculus, which describes the rate of change between realities depending on v{b}.

See Heinlein's "The Number of the Beast" There it's 3D+3T. Time dimensions are orthoganal, and a given universe only experiences one of the dimensions.

However this means that if you switch to the Tau axis instead of the Tee axis no time passes on the Tee axis (for you) while you are gone. You can become wealthy selling holiday vacations that take no time at home. Possibilities for cramming for exams; illicit affairs; Reconciling large age differences between would be lovers.

Without separate universes, the everyday physics gets difficult. How does air resistance work? Gravity? (If you have multiple time dimensions are inverse square laws still inverse square?

I'm fairly sure that what I am about to explain is not what you're going for (as multiple time dimensions in the spacetime relativity sense of that phrase are utterly beyond my comprehension), but multidimensional time is a concept that already exists in the philosophy of time travel.

The idea is that every usage of a time machine creates a branch in the timeline at the moment the machine is scheduled to arrive, with the machine landing only in the new branch.

In other words, time travel invariably lands the traveler in a parallel universe. A parallel universe identical to the universe they departed from right up until the moment of arrival.

This neatly solves all the classic time travel paradoxes- nothing you can do can ever affect your own history, because after you turn on your time machine, you'll forever lose access to the branch of the timeline (timetree?) in which your own history exists. As such, if you try to murder your grandfather, you'll at best manage to murder his alternate universe counterpart- a man with no causal relation to your actual grandfather or to yourself.

If we consider the 4th dimension to be the ordinary, familiar dimension of time, then these separate branches exist in different locations in the 5th dimension, which could be called "meta-time". How the different timelines are arranged in meta-time doesn't really matter, since there's no way to actually observe them.

1 dimension in time is what we perceive as past(-), present(0), and future(+) like a number line. For any given tx, you have exactly one point. 2 dimensions in time would be like a grid paper. At any given time tx, you ca have infinite values for ty. Basically, infinite parallel instances for past, present and future aka Parallel universes or timelines or branch realities.

Humans(us) are 4 dimensional beings, with the ability to move in and perceive three dimensions. We cannot see or interact with the past or future. As a rule of thumb, Any being has freedom of movement and perception in n-1 dimensions, where n is the number of dimensions they exist in, bcoz otherwise, there will be some serious paradoxes. As such a being capable of traversing through time or our 4 dimensions (x,y,z,tx) must exist in 5 dimensions. What this means that even if this guy travels to past or future, he is never in the same timeline i.e. the ty value uniquely defines him and always grows in one direction, like tx does for us. If he goes back in time, he is not going back in his own timeline, but rather a different one, and no matter what he does there will be no change in his history.

P.S.:The real mindbender is when you try to imagine 3 dimensions in time.