What is a non-eternity? For non-eternity to exist, all time everywhere and for all things that exist, have existed, or will ever exist, must have a beginning and an end. More generally speaking, the extent and influence of time must be bounded at both ends, in a non-eternity. Time bounded at only one end (for example, having a beginning but no end) might be a different type of eternity from time without beginning or end, but it is still an eternity, by definition. Supposing it is possible for time to be bounded, what are the ways in which a boundary might be imposed on it?
A non-eternity can be conceptualized as a line segment. It has a beginning and an end, and time flows from the beginning to the end. Join the opposite ends of the line segment, and you have a circle. In a circular topology of time, identical events are repeated ceaselessly. Although time in a circular flow never begins or ends, the endless perfect repetition places a boundary on time. It is therefore reasonable to label the circle a "non-eternity." In an earlier post, I referred to a "circular eternity" so and am being deliberately inconsistent with that earlier terminology here. This is merely a matter of semantics.
Upon further reflection, it is somewhat inappropriate to call a universe in which only a finite amount of change is possible an "eternity," even if endlessly repeated. On the contrary, it may be logical to define a non-eternity as a universe in which only a finite amount of change is possible. In that case we would have to be very careful about how "an amount of change" is defined, clearly, mere repetition could not increase an amount of change. On the other hand, an eternity might be defined as universe without both an end, and a beginning, of time. Under that definition, circular time might be considered an eternity; however, every moment of a circular time is both an end, and a beginning, to the rest of the time circle. So calling a time circle a non-eternity seems justifiable, under both definitions.
The analogies of a line segment and circle might be extended to additional dimensions with additional details. For example, a sphere or a toroidal helix (see illustration above) might be imagined instead of a circle. However, the topologies of a line segment or a circle represent essential ways that time can be bounded.
The Impossibility Of Distinguishing Linear and Circular Non-Eternities
If the line represents the flow of time, and time can flow only in the forward direction, the two geometries of a line segment and a circle are indistinguishable to any finite being in the continuum. In other words, a circular non-eternity and a linear non-eternity are indistinguishable to beings existing in them. Both are bounded; the linear non-eternity by a beginning and end of time, and the circular non-eternity by infinitely repeating the same sequence of events through a sort of curvature in time. Each transit of the time circle is identical to every other transit, and therefore time in a sense is continually reset or restarted. There can be no change in knowledge from one cycle to the next in a circular eternity, or else the cycles are not identical. Since there can be no change in knowledge between cycles without destroying circularity, a temporal being in a circular eternity can never know whether it is transiting an eternally repeated cycle of a time circle, or just taking a single, unique trip along a time segment. To the temporal being, each experience is and must be exactly the same.
A temporal being in a non-eternal time segment might observe an approach of the end of time. For example, it might be observed that time was gradually slowing everywhere in space, and therefore it may be anticipated that an end of time is drawing near. But the temporal being can never know whether time has ended, or is just temporarily stopped. If time stops and never restarts, the temporal being will never know. If time stops and restarts in a way that preserves at least some information through the stoppage, that is just time behaving eccentrically along its merry way. If time stops in a way that destroys all information in existence, and then restarts after all information is destroyed, no temporal being in a later cycle can recover information from the previous cycle. Such stopping and restarting may be the same, topologically speaking, as time reversing and returning everything to its original state. For the new temporal beings, time will appear to have started at the beginning of the cycle in which the beings exist.
On Different Flow Rates
It has been observed that the speed of light in a vacuum is everywhere the same regardless of the velocity or acceleration of its frame of reference. From this observation, Einstein deduced that time and space form a sort of continuum. One aspect of this continuum is that differences in the momentum history of objects can cause corresponding differences in the relative amounts of time experienced by each object. In other words, time flows at uneven rates. For example, suppose Bob stays on planet Earth, while his brother Ted accelerates away from Earth to visit Sirius, and then accelerates back from Sirius to return to Earth, before Bob dies. When Bob and Ted are reunited, they will discover that Bob has experienced more time waiting on Earth, than Ted has during his journey to Sirius and back. The logical reasons for this odd result are explained clearly here. If we accept the logical conclusion, the fact that objects in space can experience the flow of time at different rates tells us nothing obvious about the end or beginning of time itself. But it might hint at a better understanding of what time is.
Time As The Possibility Of Change
Time is, in a sense, the possibility of change. A photon emitted without any subsequent interaction with matter or energy travels at the speed of light and exists unchanged except for its position in space. It will experience neither time nor change so long as its velocity is exactly 'c'. Unless and until it interacts with the universe and changes state, time for the photon does not pass. Time nonetheless exists, in a sense, for the lonely photon so long as there is a possibility that the photon can change its state. Generally speaking the constant change in its position and the existence of the universe implies that such a possibility -- that the photon will experience time and change at some time in the future -- cannot be ruled out.
Supposing that time flows forward only and is the aspect of the universe providing the possibility of change. It follows that, even if at some time in the past the universe was not experiencing any change, the fact that it is presently experiencing change proves that it has always been possible for change to occur. Therefore time, defined as the possibility of change, has always existed.
If at some future time the entire universe ceases to experience the flow of time and reaches a state in which any further change is utterly impossible, time will have ceased to exist. In such case, time will have passed from eternal existence into oblivion. That is, time might have an end but cannot have had any beginning. It has always existed but might someday end.
The possibility of an End of Time seems unlikely, but perhaps cannot be entirely ruled out logically. Even if this strange topology (an end but no beginning) cannot be proven illogical, time will always have existed and is therefore eternal in the sense of having no beginning. Even supposing it cannot be proved that the universe will never reach a state in which all future change is impossible, logic proves that the past, future change has always been possible and therefore time (in the sense of an enabler of change) has always existed.
Although we can prove that change has always been possible, no one can prove that future change is impossible. Suppose the universe is trending towards absolute stillness, meaning the absence of all change. The trend can be observed while time and change still exist, but nothing can be observed or known once perfect stillness is achieved. Nothing whatsoever can be done. It cannot be known whether absolute stillness in the end of time is unchangeable, or merely unchanging. Unless the stillness is absolutely and necessarily permanent, then time has not ended.
Therefore if we define time as the possibility of change, non-eternity is not a logically coherent idea, because we can deduce that the possibility of change must have always existed. We can also deduce that it is unknowable whether or not time (as possibility of change) will ever end. If it is not impossible for time to have an end but no beginning, it is at the very least quite surprising. Even in that case, however, time as the possibility of change has always existed, so there can be no non-eternity.
However, this definition of time seems a bit like a rhetorical trick. If there was zero change anywhere and forever before the present epoch of change, then the universe was preceded by a changeless void, a nullity in which nothing ever existed or happened. In that case it may be more satisfying to say that time did not exist until the earliest change occurred. Perhaps time is better defined as change itself.
Time As Change Itself
If time is defined not as the possibility of change, but as change itself, we may have to wrestle with more difficult logical conundrums. This definition may require accepting that time is capable of passing into and out of existence more easily than the downstroke beat of a bee's wing. For example, if time did not exist before the earliest change, it sprang into existence with the first change. If in a trillionth to the trillionth power of a nanosecond, no change occurs anywhere in the universe, then time has flashed out of existence for that brief instant. And at countless other instants in every second of time.
If time is change itself and cannot be divided more finely than a certain amount, then time is quantized. If time is quantized, the minimum quantum of time must be non-zero, and each time quantum must be separated from its neighbors by some quantity of no-time or absolute stillness. If there is no stillness between adjacent time quanta, then time is continuous, and not quantized. Quantization by definition requires some separation between adjacent quanta, or else the quanta run into each other and cannot be distinguished from a continuum. Since change cannot occur without passage of time, this would mean that a phase of absolute stillness must, in a sense, occur between each tick of the quantum clock. Quantized time in a sense must wink in, and out, or existence between each tick. This is so, because the state of the universe must be exactly the same at the end of each time quantum as at the beginning of the next, and no change can occur except within a time quantum. So, if time is quantized, each unit of time experiences time extinction events equal in number to the unit of time divided by the time quantum.
If space has some minimum quantum while time does not, there must be some tiny slice of time over which no change within any quanta of space is possible. Within time slices that are sufficiently small, there is not enough time for particles in adjacent but different quanta of space to interact. Thus, no change can occur in countless tiny time slices of any given time interval.
Conversely, if change must occur somewhere in the universe at every instant of time no matter how finely divided, then at least one changeable quality of the universe must subsist in an infinitely divisible matrix. The only apparent candidate for such a matrix is space, alternatives such as "ether" having been shown inconsistent with experimental observations. Change cannot occur without some space for particles to evolve in. Even if it is hypothesized that a particle is a point occupying zero space, it cannot interact with anything else without some space to propagate through.
If space is infinitely divisible, every particle no matter how small encompasses an infinitely divisible volume of space, unless the particle occupies no space at all. Suppose, for example, that a "quark" is the fundamental particle. If each quark encompasses an infinitely divisible volume of space, it must either be perfectly homogenous throughout its entire volume, or have a structure that includes some smaller, more fundamental thing. If the quark must be perfectly homogenous, it encompasses a volume of space in which no change that would disrupt its internal homogeneity can occur; such that it must change instantaneously or not at all. If the quark has internal structure, either there is some smaller fundamental, necessarily homogenous thing that is too small for us to detect making it up, or there is no fundamental particle at all and every structure is infinitely complex. Some theorize that quarks encompass no space at all, and exist as "point-like" particles. Point-like particles occupy no space, and thus allow existence of arbitrarily small structures. An infinite number of points can be packed in an arbitrarily close arrangement, if there is no lower limit to the scale of space.
Thus, if space is infinitely divisible, there is no scale to the universe, which is infinitely complex and extends from from the infinitely tiny to the infinitely large. If there is no lower limit to the divisibility of space, an infinite amount of change can occur during what seems, at larger scales, to be a finite amount of time, due to this infinite complexity. Each subatomic particle can encompass a universe of smaller changes, each sub-atomic particle of that smaller universe can encompass its own universe of smaller changes, and so on up and down the scale of space. Tiny universes with tiny persons can be born, evolve, and die in the span of a heartbeat in our world -- while even tinier universes similarly pass during heartbeats of the tiny persons. Both time and space would have no discernible scale. Such a reality is not consistent with our experience, because the universe as we know it does exhibit scale. Subatomic particles exhibit different structures and behaviors than atoms, molecules, collections of atoms and molecules, cells, animals, worlds, solar systems, galaxies, and on up the scale. There do seem to be upper and lower limits to this scale. Perhaps the scale we observe in the universe is an illusion peculiar to our particular location in a sort of infinite fractal. If so, we might reasonably be deluded into believing that space and time are not infinitely divisible.
Based on observations of universal scale, but not to a logical certainty, our apparent universe includes periods of change surrounded by timeless instances of changelessness ("no-time"). If time is defined as change itself, time is continually stopping and starting. When stopped universally, time (defined as change itself) does not exist at all.
Instantaneous Change
Can change occur instantaneously, therefore proving that time cannot be defined as change or the possibility of change? Logically, all change requires the passage of time. If something exists in a first state, and then changes into a second state, the question of how much time the change required is immaterial. It is sufficient that at an earlier time the thing existed in a first state, and at a later time it existed in a different state. The presence of time is needed to enable a thing to change from one state into a different state.
If a thing merely exists in two or more different states simultaneously, it is not changing. Existing in two states simultaneously does not require any time. For example, an unchanging photon may exist as both a particle and a wave without any need for time. As Relativity teaches, light can exist unchanging in no-time, such as when moving at velocity 'c' in a vacuum. It cannot be observed without interacting with something else, and thus, being changed and made subject to time. Once observed and made to experience time, it can and does change into one or the other of a particle or a wave.
Schrodinger's cat cannot really be both alive and dead at the same time, because it observes itself at some conscious or subconscious level, during macro-time. But all living things passing in and out of no-time may exist in superimposed states of life and death during those timeless leaps. Also, it may be that information can be preserved by a thing entering no-time in two or more simultaneous states. If so, it may be possible to prove that non-eternity does not exist, even in a universe with scales of space and time.
Conclusion
If we define time as change itself, and accept our observation that the universe has scale as valid, we can deduce that periods of change in the universe are like islands rising from an ocean of absolute stillness in which no change occurs.
We cannot deduce how our epoch will end, or whether other epochs existed prior to our own. Surmising that our own epoch is like a vast chain of islands representing change in an ocean of changelessness, we might reasonably imagine that our own "island chain" (epoch) may be separated from other epochs by an ocean of timelessness. But we will never find any evidence to support such an imagining, unless we can discover a record that can be proved to have been created in an earlier epoch.
Until then, we live in a universe of unknown age and origin, progressing towards the unknown. If time is change, non-eternity is not logically impossible, but neither is it required.
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