The Nature of Time and Space
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Think of Absolute Nothingness. Nothing exists in Absolute Nothingness. Nothing itself does not exist in Absolute Nothingness. But if nothing does not exist, then something does exist. That something has a boundary with Absolute Nothingness and the properties of that boundary will enable us to deduce the fundamental shape of Reality.
We take as axiomatic the statement that Reality consists of multiple objects participating in multiple events. In order to separate and distinguish objects from each other we assert that objects exist in an entity called space. In order to separate and distinguish events from each other we assert that they occur in an entity called time. We discern space and time through objects and events that we use to mark them. We now want to describe time and space in a way that lets us discern their fundamental natures.
We know the nature of a thing when we know what makes it unique, what makes it different from other things, especially things that, in some way, resemble it. To discern the nature of something we need to make comparisons in order to find the relevant differences and properties. To what can we compare space and time so that we may differentiate them from it? The set of the real numbers seems like a good choice, consisting as it does of an infinite set of elements that, in some sense, abut each other. We start by listing our axioms, self-evident statements that form the foundation of our reasoning.
I: Reality consists of all things that differ from Absolute Nonexistence.
II: The Universe consists of multiple bodies/particles occupying positions in space.
III: Reality consists of the Universe enacting multiple events in time.
IV: Reality touches Absolute Nonexistence at a boundary that has zero measure; that is, the boundary exists/nonexists as a single mathematical point.
V: Anything that exists/occurs for one observer necessarily exists/occurs for all other observers, whether they observe it or not. This comes from the law of the excluded middle and gives us a precursor to the first postulate of Relativity.
In order for Reality to exist there must exist something that separates different arrangements of the same objects from each other. We call that something time and we take it as the Prime Parameter of Reality: all the laws of Nature ultimately refer to its elapse. Reality must exist with no gaps in its existence, so time must exist as a continuum, one analogous to the continuum of the real numbers.
The temporal continuum consists of infinitesimal entities that we call instants, the temporal analogue of the points that comprise space. One of the fundamental properties of time is obligation; which means, each and every particle and spatial point in the Universe must associate with each and every one of those instants. Each instant has the nature of a physical point rather than the nature of a mathematical point: we assert the truth of that statement because mathematical points each have zero extent, so multiple points coincide and become confused. Every physical point and, thus, every instant has infinitesimal extent, so multiple instants donít coincide and, consequently, they may be ordered and numbered. We can dismiss the use of mathematical points and their temporal analogues by reference to a humorous definition of time as that which keeps everything from happening all at once.
Time consists of an infinite set of infinitesimal elements ordered in a way analogous to the ordering of the set of the real numbers. Between any two instants, then, we can have a finite elapse of time, a finite duration, just as we can have a finite difference between two real numbers in spite of the infinite set of decimal fractions between those numbers. Time also consists of physical instants that each exist adjacent to two others, one prior and one after. The real numbers comprise a set of ordered names, each of which appears adjacent to two others, one lesser and one greater, that differ from it by an infinitesimal amount. Thus, we can match the instants of time, one-to-one, with the real numbers, using the numbers as indices on the instants. There are no gaps in the sequence, because the instants on one side of the gap would not exist for those on the other side.
The elapse of time consists of something going from one instant to the next in the sequence in a continuous process. We notice the elapse of time because consciousness consists of a series of events, so our awareness is constantly moving through time.
Events mark instants, in essence making them visible to us. We use regularly repeating events, usually in a device that we call a clock, to count time so that we can use it in our calculations. Each and every particle of matter or radiation associates with some point in space at each and every instant of time; otherwise, the particle would not exist. That fact assures continuity in the motions of particles and gives us causality. An event consists of several bodies and/or particles coming to a point in space and changing each otherís properties (usually accidental properties, such as motion). We identify any instant with a lower index number as coming before the event occurring at an instant with a given index number and we identify any instant with a higher index number as coming after the event.
We assert that time had a definite beginning, an instant whose index is perfectly zero. Between that instant and the instant that we call now there must exist a temporal interval of finite duration. We know that statement stands true to Reality because between the beginning of time and now there must exist a finite set of events. If there was an infinite set of events between the beginning of time and now, then at least some of the events would have to be indeterminate, existing for some observers but not for others: without the indeterminacy the set would not be infinite. But events are determinate Ė they either occur or they donít Ė so duration, measured by events, must also be determinate and the duration of time must, therefore, be finite.
If we have an event, then there must be other events, both occurring before and after it (except for the event at the beginning of time, which has no prior event). If nothing happens, then nothing else happens; if something happens, then something else happens. If we identify a certain arrangement of particles around a point in space at instant T1 as A and identify a different arrangement of the same particles at a later instant T2 as B, then we say that A caused B if B always follows A. Cause and effect comprise necessarily associated events. A cause precedes its associated effect and is a sine qua non of that effect; otherwise, itís not a cause.
By making an analogy between time and the real numbers we have tacitly assumed that time is one-dimensional. We base our logic on that one-dimensionality, saying that the relation between before and after mimics the relation between cause and effect. If time had more than one dimension, it would not be time because causality, which we assert as a fundamental property of time, could be broken. Time consists of a fixed, ordered sequence of events, marked by real numbers, a fixed, ordered sequence of names.
In a simple model of two-dimensional time, for example, we have radial time and longitudinal time. Radial time elapses on straight lines that radiate away from the beginning of time. Longitudinal time, in this model, is analogous to concentric circles rippling away from the beginning of time. On each radial line history evolves in a way different from history on other lines; the closer the timelines come to each other on the model, the smaller the difference between their histories. The same particles occupy each radial timeline, differing only in their arrangement. This model conforms to the many-worlds interpretation of the quantum theory (presented in 1957 by Hugh Everett III) or the "possibility worlds theory of history" that Andre Norton presented in her 1956 novel "The Crossroads of Time". If we could move in longitudinal time, we could visit worlds in which history differs from what we have experienced: we could go, for example, from a world in which SchrŲdingerís cat died into a world in which the cat is still alive. If objects can move in longitudinal time, then any particle may encounter is doppelganger on another timeline and that encounter may cause problems with logic.
As noted above, the fundamental property that we ascribe to time is change. We mark change at its most fundamental by noting that, as time increments from instant to instant, particles occupy different points in space, always doing so in such a way that as dt approaches zero ds also approaches zero (that is, the path that any particle follows is continuous). That fact is enabled by the fact that space is elective; that is, a particle is not required to occupy each and every point in space. But time is obligatory: each and every particle and spatial point will necessarily occupy each and every instant of time.
If time had two dimensions, call them a-time and b-time, then one of two things would happen. We assume that a-time possesses the property of obligation; that is, each and every particle of matter and radiation and each and every point in space necessarily occupies each and every instant of a-time in proper sequence, as if proceeding up the line of real numbers. If b-time also possesses the property of obligation, then all particles and points must follow the same unique, continuous path through the temporal a-b plane. If b-time does not possess the property of obligation, if particles and points can occupy any instant on the b-timeline and not occupy others, then b-time has one of the fundamental properties of a spatial dimension and we must treat it as such. Thus we must conclude that time has only one dimension.
Nobody can reasonably say that space does not have three dimensions. We need only look at space to see that the maximum number of straight lines that can be mutually perpendicular is three. Alternatively, we know that we can assign three index numbers to each and every point in space in such a way that we can increment or decrement one of them without affecting the other two, but we also know that we cannot do that with four or more index numbers. It seems as though that should all be axiomatic, but how can we know that we are not trapped in an illusion?
Start with the fact that extent is the fundamental property of space. Distance gives us a measure of the degree to which extent separates objects from one another. We can thus identify any point as a place and mark it as such by putting a particle on it. Between any two places there exists an infinite subset of points: there also exists an algebraic function that converts the index numbers on the places into a finite number representing the distance between the places. Note that for that statement to stand true to Reality the physical points that comprise space must have infinitesimal extent (unlike mathematical points, which have zero extent).
The full extent of the Universe, the whole distance across space from one end to the other, is finite relative to the distance between any two places. An end of the Universe, a point where space meets Absolute Nothingness, is a definite place and from that point, in concept at least, we can measure a finite distance to some other point, which is occupied and marked by a particle. We can then measure to another place and then repeat that process as many times as we desire in a procedure perfectly analogous to measuring off a distance by turning a ruler end over end. We assert that in going from one end of space to the other we apply that procedure a finite, if large, number of times.
We see a perfect analogy to that situation when we look at the numbers from zero to one hundred. Between zero and one we find an infinite set of decimal fractions; between one and two we find another infinite set of decimal fractions; and so on. So between zero and one hundred we have a finite set of integers and an infinite set of real numbers. In like manner space gives us a finite distance across the width of the Universe on a path that consists of an infinite set of points.
The above comments apply to space, regardless of how many dimensions it has. But the geometry underlying the laws of physics depends on the number of dimensions available, so we need to determine that number. Letís start by noting that the boundary between space and Absolute Nothingness has zero measure, because it must have the same measure on both sides; therefore, it has the character of a mathematical point.
We assert that space consists of an infinite set of ordered points and that particles of matter and radiation can only occupy those points in their proper sequence. The beginning of the sequence, the point bearing the index number zero, must lie adjacent to the boundary between space and Absolute Nothingness. The end of space is indeterminate, so it cannot touch the boundary. Because space cannot simply end, it must have some way to return to the boundary. In some sense, then, space must extend away from the boundary and then return to it. We note in passing that, in accordance with their index numbers, an object can only go from occupying one point to occupying another by occupying the intervening points in the sequence of their indices: in that way we obtain the illusion of distance.
Space consists of an infinite set of points, which set, like the set of the real numbers, has a definite beginning (which we label with zero) and an indefinite end. The definite end of space, conceived as a straight line, must touch the boundary, but the indefinite end cannot touch the boundary (otherwise it would mark a definite place). The indefinite end cannot simply terminate in nothingness, so we must assert the existence of at least one more line whose definite end touches the boundary and whose indefinite end merges with the indefinite end of the first line.
That model gives us a boundary existing at two places separated from each other by a compound line. But we cannot have two boundaries: the boundary must be continuous in order to be one object. We assert, then, that the boundary appears as a continuous closed curve on which the compound lines of an infinite set rest their zero points while all of the indefinite ends merge together at a common indefinite place. To an observer in that two-dimensional space the boundary would appear as a circle enclosing all of space.
The circle is, of course, an illusion: it has no actual extent. That fact means that an object lying some distance from one part of the circle must necessarily lie the same distance from all parts of the circle. The object lies at the center of the Universe and nothing can change that fact. If the object were to move toward the circle in one direction, the circle would shrink in such a way as to keep the object at the center of space. If the object were to reach the boundary, all of the points of space would merge into one and the boundary would show its true extent. It would also be clear that space exists inside the boundary and Absolute Nothingness lies outside (to the extent that inside and outside have meaning in this situation).
If we look at the boundary from the center of space and pick what appear to be two points not opposite each other, we will infer that those points are separated from each other along the boundary by two different distances. Of course, the distance is actually zero, but the situation presents the illusion that the distances are nonzero. The sum of the two distances equals the full length of the boundary. Because the actual distance equals zero, the illusory distance must be indeterminate, so the boundary must have a form that gives the distances indeterminate values on a continuum. The boundary must be perfectly symmetrical, so it must have the form of a spherical shell and the space that it encloses must have three dimensions.
I have assumed that if an entity has an actual value of zero and an apparent (or illusory) value of nonzero, then the nonzero value must be indeterminate and Reality must be so shaped as to make it so. Whence did that come? When we look at the illusion we see two infinite subsets of the infinite set of zero points spread along the boundary between the two designated points. Those two subsets appear to have definite length, which we can determine through surveying techniques. But the infinite cannot be definite: the lengths of those lines must, in some way, be indefinite. If two observers were to disagree over the length of those lines, we would satisfy that criterion. If the boundary is so shaped that it presents different perspectives on the distance between the points, then it will achieve the desired indefiniteness.
We can add a fourth index number to each point by associating that point with an instant of time, thereby converting the point and the instant into a spatio-temporal locus, a place/time where/when an event can occur. The full set of loci gives us a universe with an history.
So far we have conceived space as a single entity, which means that we have also tacitly assumed the existence of a universal state of rest, against which all motion is to be measured. Define the set of all points that remain motionless relative to each other as an inertial frame of reference. Through any given point other points move at different speeds in all directions. Those points belong to other inertial frames, so we assert that space consists of an infinite set of inertial frames of reference. We thus interpret the linear motion of an object as that object occupying an inertial frame different from the one we occupy.
In a finite space all of the inertial frames must have a common boundary with Absolute Nothingness. That boundary can have no properties whatsoever, so the frames that touch it must be perfectly indistinguishable from one another. Thus, there can be no preferred frame and no universal state of rest.
We thus infer that space consists of an infinite set of three-dimensional inertial frames of reference, of finite extent, that are indistinguishable from one another.
The Boundary of Reality
Coordinates or features of any kind cannot exist on the boundary: Absolute Nothingness "exists" on the far side of the boundary, so nothing exists on the near side. That fact means that on the boundary there can be no elapse of time, no area, and no orientation (angular coordinates). Those facts, in turn, determine the shape of the Universe.
No area on the boundary necessitates that the boundary consist of a single mathematical point, an entity with perfectly zero extent. Because the physical points of space have infinitesimal extent, tending toward zero as a limit, the size of the boundary does not affect the fundamental structure of space.
No time elapsing on the boundary necessitates that no object or phenomenon can ever reach the boundary. If something did reach the boundary, its arrival would mark an elapse of time (think of pulses of light reaching the boundary, one after another, like the ticks of a clock). But if thereís no possibility of anything reaching the boundary, then there must be no possibility of anything reducing its distance from the boundary. Objects can move in space, so the boundary must move away from all objects at a speed that no object can achieve.
Because the boundary appears spread across the whole sky, it must recede from all objects in all directions. That fact means that space is constantly expanding. That expansion can occur because the physical points that space comprises have infinitesimal extent. Any volume of space thus contains an infinite set of points: the actual number of points in the set is indeterminate, so that volume can expand or contract freely.
No orientation means that the boundary is absolutely featureless. Nothing can exist on the boundary that would indicate a preferred direction, something that would solicit the establishment of something like a latitude-longitude grid. That proposition must also apply to the straight lines of space that appear to emanate from the boundary: measured from any object, they must have the same length in all directions. That fact necessitates, in turn, that the boundary recede from any object at the same speed in all directions.
The perfect symmetry of the boundary necessitates no possibility of an asymmetry. But if the recession speed of the boundary is infinite, it has an indefinite value: it could have different values in different directions. Thus we must infer that the recession speed has a finite value. By the same reasoning we must also infer that the distance to the boundary has a finite, if large, value. So now we also know that a finite elapse of time in the past space had zero extent.
Of course, we have tacitly assumed that the distance to the boundary is measured from some given point. What can we say about the distance measured from other points?
When space had zero extent every point that came into existence moved away from the boundary at the same speed in all directions, in accordance with the requirement of absolute symmetry. Thus every point floats equidistant from the boundary in all directions: every point appears, to any observers who occupy it, to occupy the center of the Universe. This would lead observers at different points each claiming to occupy the center of the Universe, an obvious absurdity.
The absurdity evaporates when we recall to mind the fact that the interval between any two points comprises an infinite set of points: the distance between the endpoints is actually indefinite. The distance only takes a definite value when itís measured from a specific location. Thus, observers at two different locations may measure different distances between the points A and B, subject to the proviso that there exists a transformation that will convert one observerís measurement into the other observerís measurement and do so in such a way that each observer, transforming the right distances, will infer that the other observer appears to occupy the center of the Universe. Both observers can then legitimately claim that the boundary lies equidistant from them in all directions.
Now imagine that our two observers, instead of being separated from each other by a distance, are separated from each other by a relative motion. We know that the boundary must recede from one observer at the same speed in all directions. Is that statement true for the other observer?
Our intuition, conditioned by our existence at a certain scale, tells us that it cannot be so. Surely, we think, the second observer must see the boundary flying away from them more slowly in one direction and more rapidly in the opposite direction. But a closer look at the principle of absolute symmetry of the boundary tells us that our intuition is wrong.
At the instant when space had zero extent, which instant we may call T=0, space came into being at the boundary. Possessing perfect symmetry, space expanded in order that the boundary would appear to move away from each and every point at the same speed in all directions, thereby making each and every point appear to occupy the center of the Universe. Because space is expanding, those points move relative to each other. If observers occupy a point A and more observers occupy a point B that floats some distance from A and moves away from A due to the expansion of space, then each team will nonetheless see the boundary of space moving away from them at the same speed in all directions. Further, symmetry demands that the recession speed have the same value for both teams. Now imagine a point C that remains stationary relative to the point B and is just passing point A. Observers occupying that point will see the boundary as the observers at point B see it, so they also see the boundary as the observers at point A see it.
So now we know that if some phenomenon moves at the speed of the boundary, it will pass all observers at the same speed, regardless of any distance or velocity between the observers. If we identify the phenomenon as light, then we have the content of Einsteinís second postulate of Relativity.
We now gather together all of the points that remain stationary relative to one another and call them an inertial frame of reference. Space then consists of inertial frames of reference all moving relative to one another. We can then say that light travels at the same speed in all inertial frames.
Imagine a spherical surface on which each point represents an inertial frame. An observer on any point sees a symmetrical arrangement all around and concludes heís at the center of space. But the center is not on the surface; itís inside the sphere. For the Universe the center is the instant of creation. If space has the nature of an expanding sphere, we have a proof of inertial frames. An inertial frame is one in which distances remain constant, so each frame slides over the sphere and through other inertial frames.
We might think that the cosmic background would serve as a reference for a universal state of rest. But if I teleport myself a billion lightyears in any direction without changing my momentum, I will find that I appear to be moving toward my origin. The light that passed my origin a billion years ago is hotter and the light that will pass it a billion years hence is colder, so the CBR appears to me to be Doppler shifted. If I accelerate away from my origin to an extent that cancels the Doppler shift, I will see the CBR uniform in all directions, so I will once again appear to exist in a frame of absolute rest, even though Iím moving relative to my original "frame of absolute rest".
In any given inertial frame particles of matter and radiation (whose existence we have yet to deduce) behave in certain ways. We encode those ways in mathematical formulae, smooth and continuous functions of the spatio-temporal coordinates. We call those ways and the mathematical expressions the laws of physics. Absolute symmetry requires that no point in space differ from any other point in any conceivable way. Thus the laws of physics must have the same mathematical form in all inertial frames. Thatís the principle of Relativity, the content of Einsteinís first postulate of Relativity.
From those two postulates, following Einstein, we can deduce the Lorentz Transformation and then Minkowskiís theorem. We extend Minkowskiís theorem to create a set of Lorentz invariants; in particular, we find that multiplying a particleís momentum-energy four-vector by the particleís space-time location four-vector as an inner product gives us a number thatís invariant under a Lorentz Transformation. That fact lets us deduce the principle of least action. From that principle we can deduce (via the Euler-Lagrange equations) classical dynamics and relativistic dynamics, the second law of thermodynamics (as the principle of least action density), and the quantum theory. Thus we get the basic laws of physics. All we need to do now is the deduce the existence and form of the particles of matter and radiation and of the forces that act on them.
Appendix: Newtonís Laws of Motion
Motion consists of some object or particle changing its location in space as time elapses. Motion cannot exist in Absolute Nothingness, so the Universe itself has no motion. The amount of a property possessed by the whole equals the sum of the amount of that property possessed by each of all its parts; the Universe does not move if its motion equals zero; therefore, the motions of the Universeís parts always add up to a net zero.
If the motion of an object changes, the motion of some other object must change in the opposite way by the same amount in order to keep the total motion of the Universe at zero. Those changes must take place at the same location in order that they occur at the same time. In that way the Universe does not acquire any temporary nonzero motion. Thus we have Newtonís third law of motion, also known as conservation of linear momentum.
We mark motion with velocity and say that a change in motion occurs when the velocity of an object changes, in magnitude, direction, or both. Thus we say that an object continues to move in a straight line at a uniform speed until interaction with some other object compels it to change its motion. This statement gives us Newtonís first law of motion.
Newtonís second law gives us more of a definition. We define force by saying that the force acting on a free body equals the rate at which the bodyís motion changes. F=dp/dt=ma+vdm/dt comes from this.
Quantity of motion is also known as linear momentum. By Newtonís definition, linear momentum equals mass and velocity taken together. Mass simply provides us with a measure of how much matter we have present in a body and we generally determine the mass of an object through measurement. Velocity comes to us as the ratio of two measurements Ė distance crossed and time elapsed.
Thus we have Isaac Newtonís basic three laws of motion.
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