Time In Two Dimensions

R.N. Carmona

Lately, I have been thinking a lot about time, both on a personal level and on a philosophical one. Setting aside all of my thoughts on the kind of freedoms and privileges I would need to truly own my time, the phenomenon of time occupies my thoughts, in many ways from the mundane to the complex. When I was younger and far less patient, I would grow incensed having convinced myself that I picked the wrong bus out of the two that showed up at roughly the same time because the other bus eventually drove past mine. As I got older and my philosophical tendencies started to take root ever deeper, I started to notice how whenever the other bus drove by the bus I was on, it would never get too far ahead. In fact, it would arrive at my stop less than a minute before mine did.

So came my idea of virtual simultaneity. From there, I would imagine a parallel me, a sort of ghost, who in another universe got on the other bus and I would imagine how far ahead of me he would be on his walk toward my building. I would then speed up so that I would catch up to my parallel on my way home as means to feel better about not having been on the bus that arrived earlier. This is how I think about time, in a very intimate sense. Nothing stops me in my tracks like an article about time. My mind could be furtherest from time and from philosophy more generally, reading fan theories about a show I am into or looking up the latest sports scores, and when scrolling, an article about time shows up and I am immediately in that headspace. This was the case two days ago when I read Musser’s article “A Defense of the Reality of Time.”

Time is one of the most difficult phenomena to apprehend in the universe. It is as elusive as it is seductive. It is safe to say that no one has won over this lover’s heart. No one has seduced her enough to understand all of her mysteries and secrets. It is because of this that I am often discouraged to write anything about time, despite growing confidence in my philosophical capacities. But just then, Musser said something that got the gears of my mind running.

“Well, what if time had two dimensions?” As a purely algebraic question, I can say that. But if you ask me what could it mean, physically, for time to have two dimensions, I haven’t the vaguest idea. Is it consistent with the nature of time that it be a two-dimensional thing? Because if you think that what time does is order events, then that order is a linear order, and you’re talking about a fundamentally one-dimensional kind of organization.

Musser, George. “A Defense of the Reality of Time”. Quanta Magazine. 16 May 2017. Web.

What if the critical error we continue to make is thinking of time as one-dimensional? What if time has more than one dimension, just like space does? A cursory look at Superstring Theory (ST) will begin to sound exactly like me in a footrace with parallel me. Parallel or possible worlds emerge from the fifth and sixth dimensions of ST. If it were possible to see the fifth dimension, we would be able to take note of similarities and differences between the world we occupy and a parallel world. Perhaps this is what our imagination does. If a runner gets a cramp in her calf a few meters from the finish line, she may envision herself winning the race in a world in where she did not get injured. Perhaps we do, in fact, see the fifth dimension, but I digress.

In the sixth dimension, one would be able to see the universe as it was at the beginning and an array of possibilities. One would, for instance, see universes in where the rate of entropy is lower or higher than in ours. If Boltzmann was correct, and time emerges from basic probabilities, then we would see universes in where .000000001 nanoseconds do not equal a second, but instead, .0.000000001515 nanoseconds equal a second, implying that time moves slightly slower in that universe than it does here. This is to say nothing about the arrow of time, which could move in the opposite direction (see Cartwright, John. “We may have a spotted a parallel universe going backwards in time”. New Scientist. 8 Apr 2020. Web. and De Chant, Tim. “Big Bang May Have Created a Mirror Universe Where Time Runs Backwards”. Nova. 8 Dec 2014. Web.).

In the seventh dimension, one may find themselves in the mirror universe on the other side of the Big Bang, in where the arrow of time travels backwards. The eighth dimension would give one access to exotic universes, with initial conditions so far from our own. In the ninth, one would possess the power to compare and contrast every possible world from worlds very much like our own to worlds decidedly unlike ours. Finally, the tenth dimension exhausts every possibility.

The fact of the matter is that we simply do not fully comprehend our own universe and so, we can be gravely mistaken about time. If one considers ST, and not necessarily whether the theory is true, but rather what the theory can teach us, one quickly notices that ST’s dimensions have consequences for time that are arguably more important than their ramifications for space. If ST is the case or if something close to it is true, time is ultimately nonlinear and therefore, not one-dimensional. So let us walk through what two-dimensional time might look like. First, imagine that the ancestors of homo sapiens in the far off future have invented time machines. If they can travel to and fro in time, time is still one-dimensional. In two dimensions of space, one can move horizontally and vertically. So then the question arises, what would it mean for time to move vertically?

Prior talks about a quasi change or “what is common to the flow of a literal river on the one hand…and the flow of time on the other” (Oaklander, Nathan. Adrian Bardon ed. “A-, B- and R-Theories of Time: A Debate”. The Future of the Philosophy of Time. New York: Routledge, 2012. 23. Print.). Rivers, in our world, do not rise, but they do fall. Setting aside evaporation, which does speak to water traveling upward, for time to flow vertically downward, we can imagine a powerful waterfall. Though this is completely new territory and may look like a new way of thinking about time, anyone who has ever thought about the past as something long dead and inaccessible implies that the flow of time travels vertically downward, forever making the past impossible to re-experience directly. For instance, Emery, et al. write: “According to presentism, only present objects exist. More precisely, presentism is the view that, necessarily, it is always true that only present objects exist. Even more precisely, no objects exist in time without being present” (Emery, Nina, et al. “Time”. Stanford Encyclopedia of Philosophy. 2020. Web.). Presentists therefore, believe that objects in the past are in a state of oblivion, but if Temporal Parts Theory (TPT) is correct, then objects and events in the past are annihilated and their corresponding intervals of time are also annihilated. A brief review of TPT is in order.

The Temporal Parts Theory of Identity (hereon TPT) is derived from the notion of time being, in some sense or to some degree, like space. One can think of, for instance, a linear timeline depicting the years all 46 U.S. Presidents held office. Or one can think of the x-y axis used in physics. Time is represented by the x-axis whilst space is represented by the y. Or one can think of a space-time diagram containing two axes representing space and another to represent time. These sorts of considerations have led some philosophers and scientists to ask whether time is a dimension. According to some accounts, time is the fourth dimension. Time, however, is not always analogous to space. D.H. Mellor discussed these disanalogies at length. He, for instance, concluded that there is no spatial analogue for our feeling of the passing of time. We cannot, in other words, attribute the passing of time to spatial changes (Mellor, D. H. Real Time II. London: Routledge, 1998. 95-96. Print.).

With respect to parts, however, time and space are analogous. Theodore Sider explains:

Temporal parts theory is the claim that time is like space in one particular respect, namely, with respect to parts. First think about parts in space. A spatially extended object such as a person has spatial parts: her head, arms, etc. Likewise, according to temporal parts theory, a temporally extended object has temporal parts. Following the analogy, since spatial parts are smaller than the whole object in spatial dimensions, temporal parts are smaller than the whole object in the temporal dimension. They are shorter-lived.

Sider, Theodore. “Temporal Parts”. 2008. Web. 

Consider, as an example, the b-moments in Friedrich Nietzsche’s life. Friedrich Nietzsche’s birth on October 15, 1844 is one b-moment and his death on August 25, 1900 is another. The dates of both represent distinct b-times. On TPT, he is spread out from October 15, 1844 to August 25, 1900. If we were to depict him in a space-time diagram, his parts on our diagram will depict his temporal parts. If we were capable of watching Nietzsche in his infancy, we will be observing a temporal part, then another that resembles it, and then another. If one were to watch infant Nietzsche long enough, his later temporal parts will be slightly bigger than the previous ones. That is to say that Nietzsche is no longer an infant; he is now, for instance, a toddler. So on our space-time diagram, Nietzsche grows the further we move away from his birth. It is also worth noting that temporal parts have spatial parts and vice versa. Nietzsche’s hand, like Nietzsche himself, persisted within the interval of time his life occupies. The parts he was comprised of will also be represented on our space-time diagram.

Nietzsche, therefore, has spatio and temporal parts, so if presentism were correct, the interval of time represented by the ~54 years Nietzsche lived would enter oblivion. The flow of time, as it pertains to Nietzsche’s life, ends and yet, continues. Perhaps a better analogy (though in actuality, this is time moving in three dimensions), presentists seem to imagine that time is like a river, at least on the surface, but within it are whirlpools in where intervals of time meet their end. With difficulty, therefore, one can imagine time in more than one dimension even if one is not convinced of presentism. Consider instead the growing blocking theory (GBT). On GBT, the present and the past are real and the future becomes real when the present edge meets it. Similarly, on the moving spotlight theory (MST), only objects within the spotlight are considered present though objects on the peripheries still exist (Emery, Ibid.). How might two-dimensional time look like under GBT or MST?

For simplicity’s sake, I will consider GBT. Since presentists deny the spatio-temporal parts of the past and thus, bury the past, we can imagine that for presentists, time moves to the right on the x-axis (that is, along the first quadrant of the axis) and also downward on the y-axis (along the fourth quadrant). Growing block theorists, on the other hand, since they believe in the future emerging at the present edge, see time moving in the same direction as presentists on the x-axis, but moving along the y-axis in the opposite direction, upward in the first quadrant. In other words, if the death of temporal parts is a downward movement, the birth of temporal parts is an upward movement.

To pursue a brief tangent, not all theories of time think of time as a line. Eternalists see time in one dimension, but their view of time is more in keeping with a circle. Emery, et al. state:

One version of non-presentism is eternalism, which says that objects from both the past and the future exist. According to eternalism, non-present objects like Socrates and future Martian outposts exist now, even though they are not currently present. We may not be able to see them at the moment, on this view, and they may not be in the same space-time vicinity that we find ourselves in right now, but they should nevertheless be on the list of all existing things.

Ibid.

Another eternalist conception that has been entertained can be called recurrentism. Nietzsche probably explains it best:

What if some day or night a demon were to steal after you into your loneliest loneliness and say to you: “This life as you now live it and have lived it, you will have to live once more and innumerable times more; and there will be nothing new in it, but every pain and every joy and every thought and sigh and everything unutterably small or great in your life will have to return to you, all in the same succession and sequence—even this spider and this moonlight between the trees, and even this moment and I myself. The eternal hourglass of existence is turned upside down again and again, and you with it, speck of dust!”

Would you not throw yourself down and gnash your teeth and curse the demon who spoke thus? Or have you once experienced a tremendous moment when you would have answered him: “You are a god and never have I heard anything more divine.” If this thought gained possession of you, it would change you as you are or perhaps crush you. The question in each and every thing, “Do you desire this once more and innumerable times more?” would lie upon your actions as the greatest weight. Or how well disposed would you have to become to yourself and to life to crave nothing more fervently than this ultimate eternal confirmation and seal? 

Nietzsche, Friedrich W, and Walter Kaufmann. The Gay Science: With a Prelude in Rhymes and an Appendix of Songs. New York: Vintage Books, 1974. 373. Print.

On the cosmological interpretation of eternal return (see here), eternalism simply is the idea that time is like a circle. There is a sense in which recurrentism is already implied and though eternalists are usually not committed to the idea of time literally repeating itself, if Socrates and Martian outposts exist, then time and causation are a loop. This implies that some future event will jumpstart the Big Bang and every event in the universe will play out in identical ways all over again. This is precisely the sort of thinking entailed in the presentist’s response to eternalism. Heather Dyke explains:

According to this argument, any theory that assigns ontological privilege to the present moment while also recognizing the existence of non-present times faces an insurmountable problem: it is unable to account for our knowledge that we are located in the present. If anything is certain, surely our knowledge that we are present is! But if past times exist as well as the present time, what is to say we are not located in one of those past times, mistakenly believing ourselves to be present? We might insist that our experience of presentness is so compelling that it must be veridical. But what about Queen Elizabeth I’s experience of presentness? That’s pretty compelling too, yet she is in the past, so her experience misleads her. Perhaps our experience misleads us too.

Dyke, Heather. Presentism and eternalism. Time, metaphysics of, 2011, doi:10.4324/9780415249126-N123-2. Routledge Encyclopedia of Philosophy, Taylor and Francis.

We tend to give the present an ontological status greater than that of the past and the future. It appears that the only escape we have is the idea of spatio-temporal oblivion. Queen Elizabeth I is not somewhere in the past under the mistaken impression that she and not us is present. Time, therefore, probably does not have one dimension. Forward implies backward and up implies down, so any linear or circular view of time runs into problems of recurrence and all sorts of time-related paradoxes like the well-known Grandfather Paradox. Without straying too far in quantum mechanics, all of this is already implied in Bell’s Theorem: “Most models of nature are reversible in time; we can run the basic equations backwards in time as easily as forwards in time. This implies that theories with causality forwards in time must also have causality backwards in time; this was ignored by Bell” (t’Hooft, Gerard. “Time, the Arrow of Time, and Quantum Mechanics”. Frontiers. 29 Aug 2018. Web.).

Especially for people high on the idea of the universe being conscious or that there is a god(s), the universe would therefore offer us clues to help us better understand time. Think of the notion of space and time changing places within a black hole. Going back to the paradigm, namely the four-dimensional concept of space-time, time would then have three dimensions and space would have one. As Pösel explains:

Only outside the cylinder does the intersection with a plane at constant height (“at constant time” as seen from the outside) correspond to a snapshot. Inside the cylinder, time and space have switched places. Inside, the intersection image doesn’t show a snapshot – it shows something much more weird: a caleidoscopic combination of many different times. After all, inside, time is not the axial, but the radial coordinate, and all the different distances from the “center” which you see in the sketch correspond to different moments in time. Instead of the spatial structure of the black hole, the sketch shows a strange mix of space and time!

Pössel, Markus. “Changing places – space and time inside a black hole”. Einstein Online. 2010. Web.

I would highly recommend getting a handle of Pössel’s illustrations if you would like to better understand how time and space trade places within a black hole. For our purposes, the fact that this happens in a black hole might offer a clue. Perhaps the lesson to be learned from the inner workings of blacks holes is that time has more than one dimension. Maybe the relativity of time inhered in General and Special Relativity has to do with the fact that time is superimposed or even supervenes on space in a different way relative to one’s direction in space. In other words, time has a horizontal behavior, so to speak. It has a slightly different vertical behavior. Further still, it has another behavior along curved or edged spaces within the dimension of depth. On Earth, we commonly move as though we lived in two-dimensions, so thinking of moving in a third dimension is difficult to conceptualize. We would have to travel in space to get a better idea of how the effects of gravity, over long distances, look like traveling along a curved surface. Or, we can simply think of the difference between Mario on Super Nintendo versus Mario on the Nintendo Switch. When Mario performs any movements in the depth dimension, or to add to our x- and y-axes, a z-axis, he is now covering diagonal domains analogous to the width of a cube, for instance.

To exhaust a well-known example, time moves faster at higher altitudes, the further one is from the Earth’s center. This might be suggestive of time behaving differently along the vertical axis. This also readily explains the relative experiences of someone falling into a black hole and another person seeing this happen. Recall the following:

1. The light coming from the person gets redshifted; they’ll start to take on a redder hue and then, eventually, will require infrared, microwave, and then radio “vision” to see.

2. The speed at which they appear to fall in will get asymptotically slow; they will appear to fall in towards the event horizon at a slower and slower speed, never quite reaching it.

3. The amount of light coming from them gets less and less. In addition to getting redder, they also will appear dimmer, even if they emit their own source of light!

4. The person falling in notices no difference in how time passes or how light appears to them. They would continue to fall in to the black hole and cross the event horizon as though nothing happened. (“Falling Into a Black Hole Sucks!”ScienceBlogs. 20 Nov 2009. Web.)

If the observer never quite sees the faller fall into the back hole while the faller notices no difference in the passage of time and ultimately, crosses the event horizon, time here can be taking on a different behavior along the z-axis, in the dimension of depth along a curved plain. The 3:1 relation of space-time can be simplified with a 3:3 relation which is virtually the same as 1:1 or colloquially, one-to-one. To my mind, if simplicity is the aim, then this is our next recourse. ST was borne out of an attempt to condense the three dimensions of space into one dimension. That unfortunately did not work, at least not as intended. We should therefore attempt to think of time in three dimensions and see how this might help the project of unifying physics.

Ultimately, our theories of time, along with physics, may be impeded by the idea of one-dimensional time. Maybe it is time for us to begin thinking about the behavior of time in two or three dimensions. ST already shows that there are peculiarities about time across multiple dimensions, implying further that the ten dimensions of ST are not dominated by space. In other words, it is not that nine out of the ten dimensions belong to space and still one to time. There could be a disproportion, like a space-time ratio of 6:4, but if ST entails a few dimensions of time, we should begin conceptualizing how time behaves in a multi-dimensional setting. This might help us to finally unravel all of her mysteries and secrets. Until then, I leave you with a mind-bending interpretation of what happens inside of black hole from Christopher Nolan’s “Interstellar.”

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