“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 world views 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 microsystem 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 untestable. 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
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.
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:
Δt'/∆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 light1, 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 timelessness2. 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.
Now the reader might notice that I
have described a sequential process where I have also claimed the
non-existence 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. 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.
Fractals
“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.
Δt'/∆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.
Conclusion
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.
References:
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New York Press.
Gleick, J. 1987. Chaos: Making a New
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