The Physics of Time

Comprehensive Exam for California Institute of Integral Studies with Professor Don Salisbury, Spring 2006

 “It is impossible to meditate on time and the mystery of the creative process of nature without an overwhelming emotion at the limitations of human intelligence.”– Alfred North Whitehead (1964, 73)

People seeking to describe time continually face the riddle of its logical inconsistency. How does the past exist in the present after all? Physics is no exception. How does the science of that quantifies motion quantify the paradox of the unmoved mover?

Physics gives a number of different descriptions of time which I explore throughout the course of this paper. Three of physics’ biggest concepts relating to time are entropy, symmetry, and relativity. Entropy implies the unidirectional flow of time. Symmetry implies a reversal of this flow on some level. Relativity implies variations in the rate of time and offers the notion of timelessness as a fundamental aspect of the universe. Each of these ideas seems to describe a different reality. Often people assume that one must be more true or fundamental than another, but in this paper I suggest that each perspective provides a unique and essential take on reality and the truth lies not in which one is more fundamental, but in delineating the scope of each and determining how they fit together.

I begin with entropy, the arrow of time implicit in the second law of thermodynamics, since it corresponds most closely with our everyday experience of time. This container for our experience provides a boundary beyond which it is difficult to see or articulate, but science suggests that this is merely one way time manifests in the universe. Beyond the second law, the rest of physics tells another story by obeying temporal symmetry. I will explore interpretations of several physical scenarios of space time-curvature and particle interaction that suggest possible manifestations of this other direction of time. Then, I consider the implications of relativity, particularly the speed of time as it varies according to velocity, and the notion of a determined, frozen, “block” universe. In the last paragraphs, I will take a brief look at the roles imaginary time and fractals might play in helping to understand and integrate some of these complex ideas. Perhaps attempting a philosophical integration of these ideas might then facilitate their mathematical integration, as well as suggest a model for integrating disparate worldviews across other disciplines in future dialogues.

The Limitations of Time’s Arrow

“The basic objection to attempts to deduce the unidirectional nature of time from concepts such as entropy is that they are attempts to reduce a more fundamental concept to a less fundamental one.” – G.J. Whitrow (1980, 338)

The second law of thermodynamics states that heat flows from hot to cold regions. Another way of saying this is that “entropy” will always increase. An increase in entropy is often defined as an increase in manifest disorder (Penrose 1989, 308), randomness (Penrose 2005, 690), or as a loss of information, (Bohm 1986, 180).   Perhaps the second law of thermodynamics is the scientific way of expressing Buddhism’s first noble truth, “All life is suffering.” The very struggle to keep our bodies fueled and maintained to prevent “wasting away to nothingness” simply prolongs our inevitable demise to maximum entropy. Everyone can relate to entropy in terms of cleaning. No matter how many times you clean it, it will always get dirty again. Even the act of cleaning, an attempt at lowering entropy/ increasing order, releases so much of your energy in heat, that the corresponding entropy increase is positive. There seems to be no winning with this law!

Often hailed as the grand “arrow of time,” the second law of thermodynamics describes, in short, the tendency for energy to spread out, and thus for entropy or manifest disorder to increase. While the second law describes a temporal texture we can relate to, there are some significant limitations to its ability to describe the ultimate reality of time.

While there are many philosophical limitations implicit in the second law, first I will address its more explicit limitations. Specifically, the second law applies to statistical tendencies of macroscopic, closed systems in terms of equilibrium states. Thus, it is not a law describing the entirety of reality, but a specific portion of reality. There are three limitations in here.

First, entropy describes probabilities / tendencies rather than describing every part within the whole. It applies to macroscopic or manifest systems not including the quantum scales of reality, and therefore falls short as a universal law. A universal law would include the domain of entropy, but would also describe how the realms of entropy interact with non-entropic realms. An example of an exception within the rule exists in Prigogine’s work with far from equilibrium dissipative structures where instances of complexity (i.e. life) arise within an overall tendency toward disorder. (Prigogine 1984) Thus, while entropy is the tendency of the entire system, it is not the rule for every micro-system within the larger system.

David Bohm also offers a more nuanced definition, “A state of high entropy is one in which large micro-differences correspond to little or no macro-differences or, in other words to a state in which micro-information is ‘lost’ in the macroscopic context.” (Bohm 1986, 181)   But then there is chaos theory which describes a different reality where small changes in microstates correspond to larges changes in macro-states. Thus we can see how the nature of a statistical law applies only to a limited range of reality and that there is also a reality of the highly improbable to consider.

Secondly, the second law describes states, not the change between these states. Since time inherently deals with change, a law which does not describe change does not describe the reality of time adequately. While the second law is helpful and accurate in our everyday lives, it falls short of providing an encompassing description of the reality of time.

Philosophical limitations

There are several more philosophical limitations implicit within the second law. First, while the second law may describe one aspect of our experience of time it does not describe the entirety of our experience. Humans experience variations in time when it seems to “drag” or “fly.” Since we, humans, experience time passing at different rates depending on our mental state, it seems plausible that the apparent flow of time could be a function of our perception rather than an external reality independent of perception. Since we experience our lives as unfolding linearly in time, we tend to describe time as unfolding linearly. We must be wary of anthropomorphizing reality. While it is important to describe our experience of time it is also important to situate that experience within the largest picture of reality we can imagine.

Consider the conceptual shift from a flat earth to a round one. The implications for exploration, trade routes, and map making were enormous. Similarly, situating our notion of linear time within a broader vision of a continuum between time and timelessness holds unexplored and undreamt possibilities. The second law locks us into our objective experience of time. A careful interpretation of the scientific concept of temporal symmetry offers an expansion of our idea of time and leads to many profound philosophical implications, as explored later. Through recognizing the limitations of both consciousness and the second law we can release ourselves to explore the broader possibilities of reality.

The second philosophical limitation is in consciousness’ ability to describe anything beyond its own experience. It is difficult to know whether we perceive time as flowing in one direction because it does flow in one direction, or of it seems to flow in one direction because that is how our brains perceive it. Imagine you are on a boat and you see an otter float up next to the boat from behind. Are you moving backward or is the otter moving forward? Well, it depends on what you’re measuring against, like GPS coordinates, or from whose perspective you’re measuring from, yours or the otter’s. The answer changes depending on the internal complexities of the question.

The logic of non-contradiction teaches that in case of conflict there can be only one truth. If two versions of reality conflict with one another, then one is right and the other must be wrong. Kant then draws the distinction between the world as we experience it through our senses and the world beyond our experience, preferring to focus on the former. He claims that we cannot know of the existence or non-existence of an object that exists outside of space and time because it is outside of our experience, but that we can only speak of our own experience. I claim that we do experience the timeless and spaceless, but that we can only communicate and conceptualize it from within time and space. When there are as many realities as there are observing subjects in the world then the task isn’t to discover which is “right,” but how they fit together. A description of reality which encompasses alternative perspectives rather than antagonizing them will surely prove itself superior.

By recognizing the limitations of the second law’s scope of application to the statistical probability of manifest, isolated systems in terms of equilibrium states we can then direct our attention to the potential broader perspectives of time available from the other realms of physics and subjective experience.

Backwards Time

“Gosh that takes me back… or is it forward? That’s the trouble with time travel, you never can tell.” – Doctor Who, The Androids of Tara

Recognizing these limitations, it seems natural that the second law would be an exception or special instance, within the larger reality described by the laws of physics. Contrary to the second law of thermodynamics, most physical laws are time symmetrical. This means that they function equally well forwards and backwards in time. The time symmetrical equations include: Newton’s Laws, Hamilton’s equations, Maxwell’s equations, Einstein’s general relativity, Dirac’s equation, and the Schroedinger’s equation, covering classical mechanics, electromagnetism, and relativity.

While symmetry play an important role in physics, asymmetry is equally important. Symmetry is balanced and static. Motion requires asymmetry. Both are necessary to each other, symmetry for foundation and sustenance, asymmetry for growth and increase. While most of physics is time symmetrical, the asymmetries present within quantum field theory and thermodynamics offer the tension that keeps things moving and interesting. In cosmological evolution it is precisely the symmetry breaking that gives us the something instead of nothing that makes up what we know as reality today. Spontaneous symmetry breaking is the mechanism responsible for the separation of electricity, magnetism, and the weak nuclear force. Might symmetry breaking play a role in the manifestation of time as well?

What I want to explore is the potential for a backwards flow of time as existing simultaneously with forward flowing time. One might see these two aspects of time as symmetrical mirror images of each other or as asymmetrical because of the mirror reversal. Looking at time coming from the future toward your present moment and at the past coming toward your present moment the two are obviously asymmetrical. But if you look at the past coming toward you and the present flowing backwards into the past then the two are symmetrical. The asymmetry of time is important, but does not rule out the existence of its asymmetrical counterpart, backward flowing time.

The tricky thing about time symmetry is that we don’t seem to experience backwards time. Our experience of time tends to align with entropy’s arrow of time. In the same way that entropy may be a subset of a larger temporal reality, our subjective experience may too describe only a portion of a larger reality. Often when people try to imagine time running backwards they imagine everything running backwards. Such that they would experience growing younger, etc. Or they are more concerned with time travel. All this talk of backwards time, however, is intended to explore how it already manifests, not to propose any sort of time travel. As we shall discuss in upcoming sections there are definite limitations to our ability to interact explicitly with other realms of time.

What I would like to suggest is a “Merlin” model of backwards time, such that backwards and forwards time occur simultaneously rather than mutually exclusively. In some of the King Arthur legends, the wizard, Merlin, is said to live backwards. This doesn’t alter anyone else’s perception of time; it adds a new possibility for temporal perception. Merlin provides a personification of the simultaneous existence of backwards time with forwards time. I suggest this model in order to entertain the notion that the backwards time of temporal symmetry would not necessarily be distinguishable from a forwards time from the perspective of forwards running consciousness.

One may then ask what the point of entertaining such a notion could be if it is un-testable. The true test, however, may be if such a notion may offer a perspective substantially different enough to reframe and explain some of the current challenges of physics. Take for instance the quandaries of wave/particle duality represented by a photon interacting with itself in the double slit experiment, or the EPR paradox with it’s the action at a distance or faster than light particle interactions. If these particles are indeed participating in a realm of timelessness or reverse causality, our piddling objection to their lack of causal decorum seems irrelevant. Perhaps we could train ourselves, or may naturally evolve, to detect the subtleties of reverse causality, similarly to how we have evolved into our current understanding of time, or to how we gradually grow into time consciousness out of a childhood of timelessness.

Throughout this section we will explore how and where backwards time shows up in physics. One primary example is in Feynman diagrams, which illustrate particle interactions. Imagine a particle and an antiparticle simultaneously emerge from the quantum foam and then when they meet up again annihilate each other. Now imagine that the point of creation is a point of inflection, where an antiparticle, traveling backwards in time, turns around and becomes a particle traveling forward in time. The point of annihilation is another such inflection point, between the two of which the one particle oscillates.

In fact, Feynman diagrams often operate by the convention that the particle is represented by an arrow pointing forward in time while an anti-particle is represented by an arrow pointing backwards in time. This offers a physical counterpart to the Merlin version of time just discussed. It also yields a vision akin to a standing wave when viewed from a perspective of timelessness, which is essentially the vision afforded by a Feynman diagram. Then does our picture of time become one of simultaneous oscillation between past and future rather than a unidirectional flow?

In addition to the particle/anti-particle temporal polarity, we have an understanding of space-time that may account for the points of reversal between backwards and forwards time and thus for their simultaneity as well. These space-time pivot points are singularities, places at which space-time curvature is so great that the time and space coordinates trade places on the inside. Cosmologically, singularities exist at the big bang and within black holes. These two types of singularities are not perfectly symmetrical, but this does not affect their ability to be universal bookends, temporal turnaround points where forwards time turns backwards and backwards time turns forward.

One could say that nothing escapes from a black hole, unless it is going backwards in time. An object exiting a black hole backwards in time would look to us like an object falling into a black hole. We get to the notion of a space-time facilitated time reversal through an extension of the fact that matter bends space-time. This shows up observationally in gravitational lensing and can be conceptualized through the relativistic concept of lightcones.

Gravitational lensing occurs when we look at a massive gravitational object and an object that actually exists behind it appears to be next to it. The more distant object can even appear on two or even four sides of the first object. This occurs because the gravitational well created by the first object bends space time in such a way that the path of light from the second object must bend to go around the first. Light always takes the straightest path between two points. In a vacuum this path is a straight line. A gravitational source curves and stretches space-time in such a way that the straighter path for a photon to travel is around rather than through the gravitational source.                     gravlensing         (Penrose 1994)

This property of light is illustrated with light cones. Light cones are illustrated on a graph where the vertical axis represents time and the horizontal axis represents space. A light cone maps the potential past and future of a photon based on its present position. The photon’s present position is represented by a point. All the places in could have come from in the past are represented by a downward facing cone. All the places it can go to in the future are represented by an upward facing cone. The slope of both of the cones is the speed of light. Normally the light cone sits upright, but in the case of gravitational lensing, it tilts.

LC                                                                                                      (Penrose 1994)

If a light cone can tilt, one might expect that it could flip over entirely, representing the time symmetrical version of a light cone where the past become the future and the future the past. Actually the graphs of a future moving light cone and a past moving light cone would look exactly the same. Something moving backwards in time would look, to us, exactly the same as something moving forwards in time, because that is the direction we are moving. Even entropy, though it may distinguish between the two directions of temporal flow, does not determine the direction of causality. Were our consciousness different, we might perceive a reversed flow of time with entropy still in tact, just distinguishing between the two directions of time, rather than validating one and denying the other.

LCflipping                                                                                                             (Penrose 1994)

What would it take to flip a light cone? It takes a massive gravitational object to bend the path of light, to tilt a light cone. Is there an object dense enough to flip a light cone over entirely? The most massive objects we know of are black holes, caused by the implosion stars too massive to further support their own gravitational pull. The gravitational pull of a black hole is so great that even light cannot escape. In this scenario light cones lay on their side instead of standing upright or leaning to the side. Inside a black hole, the time coordinate becomes the space coordinate, and the space coordinate becomes the time coordinate. This is a mysterious concept, but it might be analogous to the experience of a photon. A photon, because it moves at the speed of light, experiences no passage of time. Without the experience of temporal separation, there is essentially no spatial separation either. Thus the two (space and time) seem to collapse into one another. What a photon experiences at the extremes of velocity extremes, a black hole experiences at the antithetical extremes of material density.

Back inside the hole, a light cone on its side is not yet a flipped light cone. Only if the light cone were able to escape the black hole might there be a fifty percent chance that it comes out upside-down. If it were able to escape, it would have to be moving into the past. Since an upside-down light cone travels backwards in time, can it escape a black hole? Would it in fact just look like light going into the hole forwards in time? Perhaps the only way something can escape a black hole is by moving backwards in time.   Then, the long sought after white hole is just a black hole, backwards in time.

The notions of particles leaving a black hole backwards in time, anti-particles moving backwards in time combine especially nicely with the idea of Hawking radiation. Hawking radiation is a measure of particles moving forward in time that do actually “escape” from black holes. This happens when the black hole gets tricked into eating an antiparticle and its particle counterpart goes free instead of facing certain annihilation with the now imprisoned antiparticle. Or read in our new language, the antiparticle escapes the black hole by moving out backwards in time only to hit a pivot point and turn into a particle moving forwards in time. From this perspective time seems to simultaneously oscillation between past and future rather than flowing unidirectionally. These speculations may offer fruitful directions for exploring towards a more complete quantum theory.           

The Speed of Time

“Oh! Do not attack me with your watch. A watch is always too fast or too slow. I cannot be dictated to by a watch.” – Jane Austen, Mansfield Park

Beyond backwards and forwards time, there is the issue of timelessness, which relativity requires we take into account. Entropy describes a reality where time flows linearly. Temporal symmetry suggests a universe where time flows both backwards and forwards simultaneously. Relativity describes a temporal continuum between the linear time of entropy, its symmetrical counter part, and a notion of timelessness referred to as the Einsteinian-Minkowski block universe. The continuum between time and timelessness is measured by the speed of time.

On one end of the continuum between time and timelessness is stillness, where an object moves through time without moving through space. On the other end is the speed of light, which is more complicated. Objects moving at the speed of light, like photons, appear to us to move through both time and space. The photon itself, however, does not experience motion through time. The faster something goes the slower time goes for that object. When an object moves at the speed of light (the speed limit of the universe) time stops. Without temporal separation there is no spatial separation. Distance is a moot point when it is traversed instantaneously. The speed of light offers a point of symmetry reunification, like we saw with the singularities, where space and time become indistinguishable and therefore non-existent.

The phenomenon of time moving slower as an object’s velocity increases is referred to as time dilation. The mathematical representation looks like this:

∆t’           ________
∆t     =   √(1-(v/c)^2)

Such that, an observer who experiences a time change, ∆t, sees the time change of a second observer, who is moving at a velocity v, as ∆t’. C is the speed of light.

Using this equation we can see just how time changes with velocity. If the second observer’s velocity is the speed of light[1], c, then the right hand side of the equation becomes zero. The right hand side of the equation is the part to watch because it defines the ratio of ∆t’ to ∆t. The “speed of time” is this ratio between two different rates of temporal passage.   If the right hand side equals zero, then either ∆t or ∆t’ must equal zero. So each observer will experience their subjective passage of time normally but to each it will appear that the other observer experiences no time change.   Essentially, something moving at the speed of light is moving through space without moving through time at all. Time ceases to proceed externally for the observer traveling at the speed of light. And the stationary observer no longer perceives any temporal flow within the vicinity of the speedy observer.

The trick comes when the two observers meet up again at the same speed. This conundrum is referred to as the twin paradox. Alice and Betty have agreed to assist in the demonstration. Alice stays on earth while Betty takes a space flight at a speed very close to that of light and is gone for 30 earth years. When Betty returns to earth she feels that she has only been gone for a few years and is actually correspondingly younger than her twin sister Alice.   This is what makes time dilation real rather than just an observational illusion.

Temporal Landscape

“Eternity is not something that begins after you are dead. It is going on all the time. We are in it now.” – Charlotte P. Gilman

What does it mean for time to slow down, or for it to stop? We have an intuitive sense of this feeling, but is it the same as the relativistic sense? The slowing of time, in the relativistic sense, is tricky, because the photon does not make time go slower for all frames of reference. Time does not go any slower for us slow movers who are not moving at the speed of light. The photon does not feel like it’s moving in slow motion either. The slowness emerges in our interaction. So, as something approaches the speed of light, its internal speed of time doesn’t change, but the speed of time compared to its external relationships does. Of course time is invariant from the perspective of general relativity. That means that any observer will calculate the same proper time that has elapsed for a given event.

There seems to be two conflicting effects of slowing time. On one hand, the slowing of time seems to indicate a swelling of the moment, like a pupil expanding when dilated allowing for greater peripheral vision. There the object is able to interact with larger mounts of space in smaller amounts of time, similar to the regular effects of an increase in velocity.

On the other hand there is a sort of detachment from the realms through which the object passes, such that though the realm of interaction is expanded, the actual ability to interact is inhibited. In fact one of the actual physical effects of humans enduring sustained rapid acceleration, as evidenced by fighter pilots, is tunnel vision, a decrease in peripheral vision. Rindler coordinates also point to the phenomena of decreased view of the universe when operating within accelerating reference frames.   Both of these phenomena seem to clearly describe the approach to a boundary, beyond which we can not venture, and within which our interactions with the outside world are limited.

It is not uncommon to encounter the realm of paradox, when approaching the extremities of the universe, as seen with wave particle duality or the Heisenberg uncertainty principle. So I encourage the reader to try to attempt to entertain both of these effects simultaneously rather than trying to choose one over the other.

Through the lens of relativity, a moment for a photon encompasses all of eternity.

If a moment for a photon covers all of eternity, what, actually, is a moment? What does it mean for it to expand?   I tend to think of this like a hot-air balloon ride, the higher you get, the more you can see. When standing on the ground, your horizon is much smaller than when you’re 1,000 ft in the air. Now imagine a timeline under your feet in place of the landscape, and imagine the vertical dimension as velocity, so greater height corresponds to greater speed. When standing on the ground / not moving, you can only see the landscape / timescape of your immediate surroundings. Your moment is normal sized, containing only the present. When you hop in your hot air balloon and travel up, your horizon expands. The higher you go, the more you can see. Your horizon expands as you move higher, in the same way a moment expands as you move faster. Our view from within time, when stationary / on the ground, is a limited perspective of a greater whole existing simultaneously and visible from greater speeds / heights.

Extend the analogy a bit further, the higher you go the more you can see, but the less you can interact with those surroundings. When you’re in the present moment / on the ground, you can interact with all the things that are immediately present to you. The further you get away from something the less you can interact with it (present technology excluded for the sake of the analogy). Touch only works within a very immediate sphere of influence, about as far as your arms can reach. Smell extends our sphere of influence a bit further. Sound certainly travels much further than touch and smell, but also reaches a distance beyond which you can not hear someone calling your name.

In a hot air balloon, you’re out of range of all of these levels of interaction, sight is the only resource left to you. And the further away you are the bigger the message had better be if you want to actually communicate something. Thus not only does moving faster expand our perspective on the external world, but it also prevents our interaction with it and erases small details.

We see the photon’s interior as frozen and indivisible. The interesting thing is — the photon likely sees us in the same way, frozen and indivisible. After all, the faster you go, the less detail you see, the expanse of space and time becomes unified, undifferentiated, and point-like. Here is another important nuance — the photon doesn’t interact with us at only the present moment in time, it interacts with all of time simultaneously. It participates within the temporal realm but is rooted outside of time. The fact that a photon experiences no temporal separation between two points, since it travels along null lines, gives it a special relationship to time, such that it simultaneously participates in all time. This lack of travel through time suggests a position outside of time which allows for entrance into time at any point. This is important to understand as a mechanism to explain the observed effects that manifest in reality, like in the twin paradox, such that by increasing speed, one decreases participation in time.

To a certain extent, communication from the “slow world” of matter into the “fast world” of energy, and visa versa, is blocked. So a photon’s expanded moment over the block universe does not seem to afford it omniscience. The details of the immanent are lost in the transcendent; the specifics are lost in the abstractions. When we try to probe the depths of a photon and its experience, our imagination is our only guide. But the imaginary is often what is required in order to see to the next level of reality.

Similarly one can posit that when a photon looks into the temporal realm from the realm of timelessness the view is equally muddied. This boundary, demarcated by the speed of light, functions as a boundary of the universe or a horizon of our experience in a similar way to the boundaries of the plank scale, a black hole event horizon, and the horizon of the universe’s beginning as delineated by background radiation.

Deepening Time

“Time does not change us. It just unfolds us.” – Max Frisch

One way to tie together all these layers of temporal reality – the forward flow, the backward flow, the atemporal, and our own diverse experiences of time – is a framework I refer to as deepening time. The concept is best understood intuitively as the experience of subjective time as it differs from objective time as you age. For instance, the older you get the faster time seems to flow. This can be explained by thinking about time in proportion to the rest of your life. As a five-year-old, one year is twenty percent of your life. Whereas when you’re one hundred, it’s a mere one percent of you total life, so by virtue of comparison, naturally seems shorter.

If the photon’s moment is infinitely large then, when it slows down or “drops out” of that moment, like when it is absorbed by the leaf of a plant, then the moment doesn’t change completely. It deepens, dividing into the past, present and future, like a higher octave in music. Time is redefined by its new interactions. With slowness comes the differentiation of time and space into smaller, more defined portions of the previously experienced infinite moment. Perhaps, in addition to flowing continually from the past to the present and on into the future, time also divides the eternal moment over and over again in a reiterative process. So, like the perception of age: the more time passes, the smaller your moment gets, the more of your moments fit into the total lifetime, and the longer the total lifetime seems to be. Thus the illusion of temporal flow is created by successive division. Division begets an appearance of linearity.

A child is not born with a notion of time, it must be taught. The conceptualization of time relies on repetition and memory. The repetition of events establishes a temporal structure of cyclicity. This cyclicity is not necessarily successive, but may be conceived of spatially as returning to the same place. When fall rolls around again one can think of   it as a participation in the eternal fall which always exists in a particular segment of the earth’s orbit. It is only when we focus on the differences between one fall and the next that we divide one from the other, establishing them as two distinct entities which occur in succession. Thus, a notion of linear time flow emerges from cycles divided from one another through their own repetition.

As current cosmological theories put it, the universe began as pure energy. Pure energy, made up of massless photons, travels along null lines, which have no temporal length and thus exist in a state of timelessness. Only as the universe expanded and cooled was energy able to “freeze out” into particles of matter, into time from timelessness[2]. I think of matter as bound energy, like a photon wrapped up on itself into the cycle of an electron, such that a previously free flowing structure now cycles within itself. This interior cycle then facilitates a structure of repetition and thus temporality. It seems that just as the curvature of space-time is intimately tied to gravity and mass, so the curvature of space-time, or perhaps of just space, might actually be as intimately tied to the emergence of time.   Time is, literally, built out of matter and gravity, out of bound energy. On a macrocosmic scale, it is the cycles of the mass and gravity of our solar system that give us our days, seasons, and years from which we come to know time.

Scientists have proof of the Big Bang through the observed background radiation in which we are immersed. Even though the event itself occurred 13.7 billion years ago, it is continually occurring, and we are literally deepening into it. When we look at the sky we can still see the matter-less energy from which we came. It doesn’t exist on one side of us so that we’re moving away from it as linear time would suggest. It surrounds us, at the distance of the age of the universe, 13.7 billion light years away. For, to look across space is to look back in time, even to the time of our birth as a universe.

Additionally, since every point in the universe was once the center of the universe, according to the Big Bang model, then the universe is its own center expanding away from itself. Thus we recognize the omnicentricity of the universe. In other words, the center and the beginning of the universe do not exist in some distant place and time. They are both immediately present everywhere and at all times, continually occurring. The universe is not only omnicentric, but also omnigenetic (always beginning). Time is continually reborn at every point of its unfolding from timelessness. Time exists within timelessness, as matter exists within a bath of energy. Timelessness also exists within time, in the infinity of each moment, as the energy is still present even in the bounded state of matter.   The beginning of the universe is timelessness, not one point in time, but every point in time continually beginning. Time is the continual deepening of the eternal timeless moment.

One can think of the universe as oscillating between matter and energy, and between time and a-temporality, as it continually manifests the meeting between forward and backward time, between the past and future flowing into one another.   We’re the multiplicitous variations on the themes of its oscillations. Every moment that passes divides this one grand eternal moment again and again – creating an exponential deepening, making the initial moment seem ever larger and larger as the universe’s accelerating expansion.


“The universe is full of magical things patiently waiting for our wits to grow sharper.”  – Eden Phillpots

When I think of time as deepening rather than flowing, and try to imagine how that might be represented mathematically, I think of two mathematical tools that are intricately intertwined, fractals and complex numbers. Complex or imaginary numbers involve a factor of i = √-1. By utilizing the complex number plane the fractal patterns emerge and display four remarkable qualities which are especially applicable to the notion of deepening time that I have just described. These qualities are: self-similarity, the infinite within the finite, embedded fractional dimensionality, and a relationship with complex numbers that is especially pertinent to our understanding of time.

First, fractals are self-similar, which means the pattern of the whole is repeated within each of its parts, the microcosm reflects the macrocosm. Mathematically this is the result of a reiterative process, where a formula is applied to its product and then to that product etc. When we compare this to time, we recognize the essential nature of reiteration in the cycles which we use to demarcate temporal progress – repetitions of years, of seasons, of days, of lifetimes, of historical patterns.

Secondly, fractals provide a mathematical description for an infinitely increasing surface area within a finite space. “A fractal is a way of seeing infinity.” (Gleick 1987, 98) Time always runs into the notion of infinity, whether through the notion of eternity or the timeless depths of the ever-present moment. Thus fractals may offer a way to visualize this infinite expanse within a potentially finite space or within a bounded moment.

Third, fractals have their own version of dimensionality which expresses their interior complexity and links to infinity. These fractal dimensions exist between our regular spatial dimensions as fractional dimensions, like 1.7 instead of 1 or 2. “Fractional dimensions become a way of measuring qualities that otherwise have no clear definition: the degree of roughness or brokenness or irregularity of an object.” (Gleick 1987, 98). The speed of time may be just such a quality that could benefit from a description in fractal dimensions.

Here we might have an appropriate alternative to the spatialization of time by allowing time to deepen into spatial dimensions rather than trying to figure out exactly where time extends orthogonally to our three spatial dimensions. This directly addresses the unique way in which time invisibly interweaves with the spatial dimensions. This dimension of interiority also has possible links to the curled dimensions of string theory as well as to the interior invisible dimensions of consciousness which play an essential role in our perception of time.

Fourth, fractals utilize the complex plane. It is precisely the complex axis that opens up an interior space embedded within the real number plane. Taking a square root is like turning a number inside out and seeing what makes it up. When you take the square root of a negative number you find imaginary numbers, which have no manifest reality, but are essential to orchestrating the way reality manifests and is described by real numbers. Fractals are the same way in that they describe and unseen order which hides behind the apparent disorder.

And this interior space facilitates deepening into the spatial dimensions. The number of reiterations performed corresponds to the fractal dimension in that each reiteration is cycle, turning back on itself creating more complexity at ever-increasing scales of intricacy. As fractal dimension increases, the surface area of the fractal increases. As we know from biology, an increase in surface area corresponds to an increase in efficiency and diversity. For example the permaculture principle of edge effect recognizes that the greatest species diversity exists on the boundaries between two ecosystems, like the edge between a pond and a field. Similarly as the edge between matter and energy simultaneously becomes more fruitful and increases its surface area, like the interface of the earth and the sun.

Complex numbers are particularly relevant to the study of time, especially in their roles in relativity and quantum mechanics. For example, the time dependent Schroedinger equation requires a complex part. And in relativity, the metric is simplified by taking the time variable to be imaginary.

Additionally by taking the equation for time dilation one step further we find a new link to the realm of complex numbers.

∆t’           ________
∆t     =   √(1-(v/c)^2)

If the observer’s speed is greater than c, the lower term on the right hand side of the equation becomes imaginary, or complex, by taking the square root of a negative number. What does it mean for the ration of ∆t’ to ∆t to be complex? What does it mean to have a velocity faster than the speed of light? Obviously these realms are unseen by us and traditional mathematics and physics have neglected them as non-existent because they are outside of our experience. But although these realms may be beyond our experience, as abstract mathematical entities they point to a larger reality. Complex numbers, for example, are essential for solving many practical equations   Additionally, it would seem that, if an object of increasing velocity moves increasingly slowly through time, to the point that, at c, it ceases to move through time all together, then logically, if it were to continue in the same vein, it might then proceed to move backwards in time, first slowly, then with greater speed. This would offer an interesting explanation as to why we don’t experience anything moving faster than light, because it is moving backwards in time!

Perhaps the imaginary component of time carries the effects of the future, incorporating the subtle variables of reverse causality. The mysterious random occurrences which we cannot describe with traditional causality might very well be influenced by interior dimensions of hidden order and complexity from the fractal realms that just may serve to interweave our connections to the past and future in ways we have not yet postulated.

Just as fractals provide a link between the realm of the seen and unseen, the real and the imaginary, finite and the infinite, the random and the ordered, so might they, in their partnership with complex numbers, offer clues as to how we can describe the intricate intertwining of the temporal and the timeless.


As we approach the end of this particular journey through a variety of perspectives on time, hopefully the beginnings of a more integrally woven view of time have begun to emerge. The asymmetry of entropy distinguishes the forward from the backward flow of time, but does not necessarily deny the latter nor does it offer the final description of time. Relativity brings the broader perspective of the atemporal, which is perhaps the most difficult and the most revolutionary to really consider in the attempt to understand the universe. Then there is the role of complex numbers and fractal patterns of reiteration that may serve as the best mathematical tools we have to somehow describe the interactions of these aspects of time in such a way that we may yet describe integral time, and might just solve a few other mysteries of physics along the way.


Barbour, J. 1999. The End of Time: The Next Revolution in Physics. New York: Oxford University Press.

Bohm, D. 1986. Time, the Implicate Order, and Pre-Space. in Physics and the Ultimate Significance of Time. edited by D. R. Griffin, 177-208. Albany: State University of New York Press.

Gleick, J. 1987. Chaos: Making a New Science. New York: Viking.

Grifin, D. R. Introduction: Time and the Fallacy of Misplaced Concreteness. In Physics and the Ultimate Significance of Time. edited by D. R. Griffin, 177-208. Albany: State University of New York Press.

Peat, F.D. 1989. Superstrings and the Search for the Theory of Everything. Chicago:

Contemporary Books.

Penrose, R. 1989. Emperor’s New Mind: Concerning Computers, Minds, and the Laws of  

     Physics. New York: Oxford University Press.

Penrose, R. 1994. Shadows of the Mind: A Search for the Missing Science of

     Consciousness. New York: Oxford University Press.

Penrose, R. 2005. The Road to Reality: A Complete Guide to the Laws of the Universe.

     New York: Alfred A. Knopf.

Prigogne, I and Stengers, I. 1984. Order Out of Chaos: Man’s New Dialogue with Nature. New York: Bantam Books.

Thorne, K. 1994. Black Holes and Time Warps. New York: W.W. Norton & Co.

Whitehead, A. N. 1964. The Concept of Nature. Cambridge: Cambridge University Press.

Whitrow, G.J. 1980. The Natural Philosophy of Time. 2nd Ed. Oxford: Clarendon.

[1]           In reality, only things without mass can move at the speed of light. Anything with mass requires more and more energy to sustain increasing acceleration. It would take an infinite amount of energy to accelerate even the smallest amount of mass to c. Only photons and energetic fields, which are massless, propagate at the speed of light. Whether or not these entities can be considered “observers” is a matter for your own discretion. Here, we allow it as another perspective on the universe that offers some food for thought!

[2]           Now the reader might notice that I have described a sequential process where I have also claimed the nonexistence of time. This seems paradoxical, and it is. It is a result of the edge effect of looking at and trying to describe a state of timelessness from within the process of time. One can only approximate the “other” from an experience of what it is not. So we can describe timelessness to the best of our ability from within a temporal perspective, but must keep in mind the limitations of this description.


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