# 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].