Closed Space; Open Time

“…each theory is evidently an abstraction from a totality that is inherently unlimited, both qualitatively and quantitatively. As such, it evidently cannot be entirely self-determined, because it always depends in various ways on what has been left out.”[1]

A Fun Mental Exercise:

It’s pretty unarguable that the basis for any mechanical human creation is our understanding of the physical sciences. In this knowledge base it has emerged that across the decades, certain fundamental constants emerged in our description of the behavior of this material with respect to motion, including heat and time. These fundamental constants, such as Plank’s, Newton’s Gravitational Constant, Einstein’s Speed of Light, etc. weren’t simultaneously conceived, but were so over time. So therefore one could reasonably expect more in the future, and not only because every consistent theory is incomplete, but particularly since it is a given that all knowledge—say, in the study of motion—is not revealed. It’s reasonable to expect that not all fundamental particles have been observed, at will, and in the laboratory. The fundamental particles of gravitation can evade our direct observation, and especially or control in the laboratory, for thousands more years.

In Standard Big-Bang Cosmology, Hubble’s Law plays an important role in measures of length across very large distances.

Apparently, there is a Universe out there that is physical space. According to Einstein, all the information we receive from it is from the past, and the further along we look, and the deeper and deeper we see, and the greater and greater is the magnification of the lens, or other measuring apparatus, the longer ago this information was sent. Doppler Shift measurements of light from deep space indicate a correspondence between the magnification of the lens and the Doppler Shift of the specks of light that gives every indication they are very, very large structures, very far away.

In Standard Big Bang Cosmology, these large structures, can be extrapolated backwards to have existed in a time when the Doppler Shift measurements and our ability to detect light appears to vanish. Increased magnification continues to help; theoretically, there will always be a period of time between the “moment of creation,” and our ability to measure light.

That shouldn’t bother us; what annoys in this cosmology, is the postulate there exists a moment of creation. As a result, we are obliged to speculate on the nature of the spatial origin and a time beforehand: that Unique Singularity, and that incomprehensible time prior.

This is unacceptable to rational thinking. Making a statement on the ultimate properties of matter, that it had been created at one moment and forced to exist the next, that it continues to exist and manipulate the behavior of other moving bodies according to certain laws, will be certain to influence, and be the influence of, the character and the rendering of these laws—because a necessary reciprocity should exist between the theory of the origin of matter with it’s behavior. One cosmology may state that matter had been created in a single moment, while another may hypothesize that matter had always existed in some state over a time-space that is infinite. I like to imagine an infinite time-space as a theoretical basis, more so than a finite one—as the topology is a simpler, more continuous one and precludes all the persistent questions of creation.

It seems impossible, however, to accommodate mechanics with the concept of infinity. Is this a dilemma?

At first impression, the concept of infinite spaces and times are difficult to accept, much less comprehend; they appear to invoke irrational expectations of the physical realm. However, in one real solution presented here, the longer time ago one makes a measurement of scale, the closer are adjacent objects, which it turn correspond to a slower speeds of their communication relative to the speed of communication here on Earth.  Specifically, this correlation function is not linearly related to the present day, but quadratically. So, that as one increases the magnification, the farther one may observe, but also the longer is the time scale between their adjacent spatial intervals.

I read that Standard Cosmology does not like quadratic functions describing the relationship between distance and time (for various reasons), mainly because Hubble’s Law, which dominates standard cosmological observation, is a linear relationship: that is, looking through telescopes strongly suggests this.

Is there a quadratic function that could look linear across vast distance and time scales?

If so, what topology of space and time would look linear to an observer on Earth, not conflict with Hubble’s Law, that is, appear to extrapolate to a finite time in a closed space, with an apparent origin of some spatial singularity, but be descriptive of an actual infinite time scale without origin?

ANSWER: Closed Space; Open TIme:



Measurements of distance, made across vast time scales, may be the result of a composition of functions, such as those above. The slope to the right, pictured above as unity (for visual effect), can be made arbitrarily flat, indistinguishable from y=1, as ℜ⇒0 (see image below).

Image above: What the continuous f(t) may look like, as ℜ approaches 0. This is the file used to generate the plot.


H(t)=f(t) o g(t)

The function f(t) is the Pythagorean sum of two vectors whose cross product is always equal to R, independent of time, t.

Above is an image of what f(t) may look like.

If R is very small relative to numbers for t, then the rate of time is nearly constant for numbers of t. In effect, this would describe a circumstance where any change in the rate of time, if it were to be measured in the present day, would be close to zero and consistent with standard laws of mechanics. If R is close to unity, however, a condition would be described where a change in the rate of time, if it could be measured, would not be as close to zero. In either case, as t approaches zero, one Earth Year of time in the present day would become equivalent in scale to innumerable Earth Years, potentially hundreds of trillions of years, of the “early” Universe. Of course, the word “early” loses it’s meaning here, where the distances of separation between adjacent particles are relatively close to zero, their speed of communication is relatively close to infinity, and which appears as faint as it does stationary from telescopes here near Earth.

[1] D.Bohm and B.J.Hiley, The Undivided Universe: An Ontological Interpretation of Quantumn TheoryRoutledge, London, 1993

Statement of the Problem

This Statement of the Problem is an incomplete paper and so has variations. Mostly corrections of typos and/or sophisms are made, with changes in language to improve clarity. As a result, it is dubious it could even be understood as is, a mere collection of thoughts. The fundamental emphasis of the problem is the following.

To vary, continuously over time changes:

1) the function values at some function maximum, an.

2) the derivatives values at an,

While, keeping constant:

3) the integrals, between the upper and lower limits of the domain: set to unity.

4) the intervals -the distance between an and b0, ie. b0-an=1: set to unity.

5) the function values at some b0, ie. f(b0)=0.


Figure: A collection of matter, f(r, t1) contracts to f(r, t2) driving Universe expansion from R1 to R2 .

The purpose of the study is to exploit as many gravitational parameters as reasonably possible, while simultaneously applying as many reasonable constraints in as many possible combinations -of both hypothetical and standard variables- so that ultimately, the apparent orbital motions of galaxies can be most simply explained: “most simply explained” meaning both as 1) a more representative, and corresponding measure of the observables: observed redshifts and observed apparent magnitudes in relation to observed matter with respect to its observed orbital motions, and 2) with a more reasonable, or, at least, equally reasonable, set of assumptions as Big Bang Theory, Inflation, Dark Matter, and other theories.

Functions of Special Interest

Fig 1: Table of Derivatives


In the above example, P=1; signings and constants are assigned for simplicity. Note the continuity and discontinuity of the partial derivatives for a class Ck smooth function..


The Unexplored Abstraction

A change from variable ‘x‘, to variable ‘r‘ in F4(x) to F4(r) yields a function in the form of an inverse square law of force with a positive constant, R4.

A change from variable ‘x‘, to variable ‘t‘ in F3(x) to F3(t) yields a function whose shape is explored elsewhere and whose effects as a cross product with F4(r) are outlined in “Statement of the Problem” [See Article 4].  I ponder the meaning of this cross product of two functions of two variables, one the derivative of the other, one in time units, one in distance units and whether it is merely coincidental or meaningful to the changing structure of space-time.

F4 is the derivative of F3; and F3 is the derivative of F2, F2 is the proposed Universal Expansion function. F2 is the derivative of F1, and F5 is the derivative of F4. Curiously, the shape of F5 resembles a conservative field in 4-space, while the shape of F1 invokes the exponential return of a system to equilibrium after a disturbance, sometimes written:

If the equations above were somehow representative of the dynamics of time and space vectors across cosmological time and distance scales, then how is this change in the “relaxation time” somehow representative of Universal Expansion, can this be useful? What would be the dimensional units, the numbers of dimensions? What is this Meta-Space-Time?

The Unconservative Gravitational Field.

In “Statement of the Problem” the product of F4(r) with F3(t) is defined as a vector field, a gravitational field, that changes with time in such a way that as t approaches infinity, ie. forever (the total life-time of the Universe), R4 approaches zero, and the field approaches a conservative field in 4-space.

Critique of Weyl’s Hypothesis


Weyl’s Hypothesis

Weyl’s Hypothesis, as it is known, is not only a hypothesis, but a postulate at the foundation of Standard Cosmology. What makes the Weyl Postulate unique among other fundamental cosmological principles, such as the Mach Principle, is it’s premise that the motions of large collections of matter, such as galaxies and clusters of galaxies, are groups of particles in a fluid that follow non-intersecting, time-like geodesics. That is to say, these groups don’t collide spatially, but more-or-less maintain their particular set of spatial coordinates throughout all time, and all their metric description would be limited to their location along some time scale.

But galaxies do “collide”, but when so, are so transparent to one another that very few stars actually meet head-on; one measures the ratio between the diameter of a star and the distance between these stars at 1:250,000,000; these galaxies basically pass through one another, almost unchanged in the visible structure of their stars’ motion.

If we ignore for the present argument, the subtle changes in the motion of these stars, and the interactions of the visible gases between these stars, and recent observations of gravitational lensing in regions that are distinct and separated from these collections of stars and gases…

That the centers of these “colliding” galaxies may occupy the same set of spatial coordinates -at either different time coordinates or the same- suggests their is a balance of kinetic energy to be summed in their lateral motion across the hypersurface. Although relatively small, this lateral motion may be contained in the motion of larger groups that follow longer paths. For example, members of the Local Group are moving in relatively the same direction. Is this motion due to the Newtonian gravity of the Great Attractor alone, and only a small irregularity that can be ignored, as postulated by Weyl, or is the motion of the Local Group determined by initial velocity components unrelated to neighboring cosmological structures?


Fig.2  Space-Time diagram of the hypothetical go disassociation


go = g + + g


In the theoretical context hypothesized here these lateral motions are not ignorable: the total motion of these groups, as a sum of directional velocity vectors along the hypersurface of the Universe -with a sign, positive or negative, to some arbitrary origin- although prefer no orientation, and therefore contribute no net force to adjoining groups, have geodesics that are, nonetheless, space-like and would amount to measurable work if summed along finite paths of individual groups.

The source of this work will be discussed in future articles, to paraphrase: a pair of hypothetical particles constitute the go, that follows the non-intersecting time-like geodesic of Weyl, and is merely the midpoint of its previous disassociation into complementary components, the pair gand g+.

The pair separate following intersecting, space and time-like geodesics. See Figure 2.

Their number, respective properties, and behavior would determine the cosmological structure of the Universe. Of these two particles, the g+ and the g, one may merely be the “hole” which remains after the latter has appeared, forming a field [Dirac].