What does it actually mean to have two time dimensions?

Question

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?

Answer 1

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".

Answer 2

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.

Answer 3

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.)

Answer 4

In general, time could be multidimensional. The reason we work with time being a positive real number (1D) is that we can easily observe (measure) changes of the "magnitude of time" (elapsed time). However, there are a many nuances to be considered:

1. By itself in isolation, time is just a characteristic of events, e.g., order of occurrence, duration, etc. The concept of "time" can certainly be defined, tracked and interpreted in many different (mathematical, philosophical) ways. For instance, "time" can be mathematically defined in multiple dimensions, formulated as spanning the smallest complete field containing the real line where all polynomial equations of order "n" have precisely "n" solutions, i.e., the complex plane (complex time is referred to as "kime"). This is all mathematically well-posed, however, it may draw some criticism from experimental physics, as physical “reality” demands concrete “observable” evidence in support of the (higher dimensional time) math model.

2. The utility, value, and universal relevance of time come from its tight integration with the spatial dimensions into "Minkowski spacetime", a continuous universal model coupling 3D space and 1D time dimensions. The longitudinal order of events in the flat (Euclidean) 4D Minkowski spacetime universe is represented by a continuous positive real number, we refer to as "time".

3. One can of course, entangle the first two scenarios and consider the universe as a higher-dimensional (5D+) space with multi-dimensional time. The easiest approach for that is to introduce complex time (kime) that effectively and completely unifies the spatial and temporal dimensions, i.e., traversal, measurements, trajectory paths, etc. in space are naturally extended in their space-kime counterparts. Below are 3 examples that illustrate specific higher dimensional universal models that agree with the 4D Minkowski spacetime observable universe. There are interesting ramifications of such dimensionality lifts of the classical 4D spacetime to higher-dimensional manifolds using multi-dimensional time.

4. Spacekime theory proposed by University of Michigan researchers defined Spacekime as a 5D universe with 3 spatial and 2 complex-time (kime) dimensions. In this framework, longitudinal measurements (e.g., time-series) are represented as kime-surfaces where time is encoded as the magnitude of the radial displacement, and the second degree of freedom, kime-phase, represents the orientation of the event (cf. mathematical parameterization of the complex time plane via polar coordinates).

5. Itzhak Bars at the University of Southern California proposed a 2T theory, which utilizes an additional time dimension.

6. Another Multi-institutional consortium proposed the 5D space-time-matter (5DSTM) theory to model the universe.

Lifting the dimensions of time results in some profound and about the universe and our interpretation of reality, truth, observations, space travel, and event causality. For instance:

1. Ordered events like classical time-series curves morph into complex kimeseries surfaces where order is relative to parametric descriptions of simple paths/curves, see this animation. There are interesting applications of this generalization to scientific inference or process prediction, and longitudinal data analytics.

2. Increasing the dimensional of time resolves many of the problems of time, like the “arrows of time paradoxes”. Time is actually not just uniformly and automatically marching forward. Our human frame of reference is inertially moving through the 5D+ universe. Time passage is observed as an incremental (time-dimensional) change in this motion. Time traversal is identical to spatial traversal from one location to another (on specific curves/paths). While it's practically impossible, having infinite energy and infinite information allows the theoretical reversal of this spacetime motion (e.g., travel exactly back in time) by applying the inertial motion/transformation in reverse.

3. In higher time-dimensions, and under certain conditions, Heisenberg uncertainty principle does not apply. In other words, in 5D spacekime, one may observe simultaneously and with perfect accuracy both the position and momentum of a particle. The manifestation of the uncertainty principle in 4D spacetime is just due to the presence of a 1 degree of freedom when we project 5D spacekime down to the 4D Minkowski spacetime. In other words, the observer reflection of uncertainty arises by the natural projection of 5D onto the 4D spacetime and is due to the unobserved complex-time (kime) phase direction.

4. Theoretically, complex-time (kime) permits the existence of simple closed 2D kime curves (CKC). However, there is no practical way to exactly go back to one specific (past) time-point and repeat a previously run experience trying to alter a (future) outcome (i.e., there is no "Grandfather paradox"). This theoretical impossibility is due to the fact that the measure of any finite or countable set of elements in R (real line) or C (complex plane) is always trivial. Given sufficient information and energy, we might be able to get near the vicinity (within a neighborhood) of one specific past spacetime location, but we can’t get precisely into one specific, unique, and precise spacetime location (whether “past” or “future”). Just like the rational numbers (Q) are dense in the Reals (R) and the measure of Q is trivial (zero), we can can land somewhere (the “axiom of choice”), but we cant guarantee getting precisely to any specific spacekime locale.

Answer 5

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.

Answer 6

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?

Answer 7

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.

Answer 8

Suppose Dr. Who invites you into the Tardis for a cup of tea while you go backwards in time to the start of the Bronze Age collapse. It takes 10 minutes to drink the tea - exactly the amount of time it takes the Tardis to make the trip back to 1200 BC.

In what dimension of time does the 10 minutes in the Tardis exist?

I've always imagined that the time which passes inside of a time machine is time that exists in a separate time dimension from the time dimension through which the time machine is travelling.

Answer 9

Without jumping ahead to alternative metrics, consider a similar situation in normal 3+1 spacetime. Though we can successfully model our personal experiences as a 1d worldline, one can also imagine zooming in so that each person is made up of a 3-d volume of constitutive elements. For their own proper times, these constituent elements are occasionally space-like separated. Owing to the obvious success of the person's 1d worldline model, we would like to call this 4d collection of particles (worldtube?) a person, despite the spacelike separation of it's components. The intriguing question is where does the person's 4-velocity come from, out of these components? I think the answer lies from a causal perspective.

Basically, imagine a discretization of spacetime so that there is a discrete 'grid' of event points. If two points are timelike separated, draw an arrow from the one with the earlier time to the one with the later time (i.e. draw arrows respecting light cones). Now, each point of this discrete spacetime can be in one of many states, and let the future evolution of point $p = f_p(parents(p), noise)$, for some deterministic function $f_p$ (which can change from point to point) and independent noise (n.b. quantum mechanics might somewhat complicate this assumption, so given it lets keep it classical :P). This is the same as Judea Pearl's causal model.

Now, we can imagine grouping the state space of this finite collection of points into 'macro-causal equivalent,' coarsened states (as is the case in thermodynamics, where we group together identical macrostates). Ideally, at the correct coarsening level we'd see that the states of the points that were in the worldtube above all get grouped together such that its one world line of points, with the state at every point corresponding to the person's experience at that time. That is, $p_{t+\varepsilon} = f_{t+\varepsilon}(p_t, noise_{t+\varepsilon})$, note the recursive deterministic function application. Assuming each such function is computable, we can move any unique behavior of the each point's particular function into the noise term, and then treat each $f$ as an executing Turing machine (with a quite large alphabet haha).

Therefore, I think that treating a consciousness in this fashion creates the perceptual concept of proper time, which is intricately tied to the computational model used and the appropriate causal aggregation, rather than the actual spacetime geometry. Though, 1 time dim sure helped making sure that at a high enough coarsening level, there wouldn't be weird arrows jumping all over the place! Now, what would a 2d perceptual time look like? Note that this time, it's not anymore the case that 'most' of the arrows are going in the time direction anymore, but causal influences can now spread out in multiple directions (e.g. a full circle in ++--, since $[\cos(\tau), \sin(\tau), 0, 0]$ are all timelike separated points from $[0,0,0,0]$). This allows for CTC's, and I'm not sure how Egan deals with that... Note that subsequent macro-level coarsenings will also likely have to contend with largely 2d 'chunks' of events, due to this 2 dimensional spread of arrows.

I believe the right subjective lens to interpret this is again through a computational model. There are plenty of computational models using combinatory like logic which can extend in multiple space dimensions with no time dimension. For example, look into Wang tiles (generally there's rules about how tiles can fit next to each other, and the question is whether from a given starting position, any added tiles will eventually tile the plane). These ideas are used in practice in self-assembling DNA computational models (see Computability and Complexity in Self-Assembly by James I. Lathrop, Jack H. Lutz⋆, Matthew J. Patitz, and Scott M. Summers for more detail).

One could imagine that an 'instant' in perceived time corresponds to the shape 'growing' along it's perimeter. Now, this looks very similar to a 1d time increment, however, the corresponding 1d time increment for the Turing machine would be moving along the entire perimeter and adding new tiles one at a time, at a constant rate. So, in essence I think the big difference is that someone with 2 dimensions of time perception would notice that as time went on, the complexity of the 'update' of their experienced state would increase at a linear rate, while for 1d time it would stay at a constant rate forever. Essentially, information appears to increase at a quadratic rate in a 2d time universe but at a linear rate for the 1d time universe.

Perhaps a more coherent view of this last part is that in >1d time, the difference in perception is the difference of a tree of coarsenings vs a line of coarsenings. So, at the perimeter one could see (and be affected by) multiple versions of 'oneself' which are all moving in 1d time.

Answer 10

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}.

Answer 11

If you think of 1D time as reversing backwards down a road so as you can only see what you have passed, this means you can travel in both directions but you can only know that you are facing (Example of 1D time). (Given a complete timeline, you could reference any point and from there only know what had happened in the relative past from that point)

In a 2D time plane then this fact would apply to both temporal dimensions meaning you would see everything behind you in either direction. This would mean the more temporal dimensions the universe has in it the more of the complete temporal picture you can see from inside the temporal plane (Example of 2D time). (Given a complete timeplane, you could reference any point and from there know both histories that led to converge on that point). The two temporal dimensions would not have to progress in sync nor would either be forced to only progress forwards but any backwards progression would not be remembered.

Answer 12

H. Beam Piper, in his Paratime series describes time as having two dimensions. The first everyone understands: go forward in time or go backwards in time. The second dimension of time, called "paratime", is perpendicular to the first. By moving in paratime, one travels to parallel realities rather than before and after now.

Paratime is measured by looking at the point in the past where the two realities were unified. Suppose in 2001, the World Trade Center was not brought down. Now for the sake of discussion, now is September 11, 2021. The parallel reality wherein the major difference is whether or not 9/11 happened is 20 parayears from our reality. Flip a coin and it lands heads. The reality where it landed tails is a few paraseconds from here.

Philosophically speaking, this is an axiom if you subscribe to "many worlds".

In thinking about this over the years, I extended this idea to say that there could be more than two dimensions of time. What phenomena would happen through these additional dimensions? Perhaps things like two different frames of reference running at different speeds? Time loops? Marvel's cosmology of megaverses and the omniverse certainly would require at least three dimensions.

Answer 13

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.