Cosmology Essays

Copernicus or Ptolemy: Cosmology Visualized November 16, 1994

[This is the first of four essays.]

Michael's personal logo The geometry of the universe may be visualized as a modified “mesh system,” in two or three dimensions, of polar or spherical coordinates. The familiar system is modified here to the extent of placing each successive radial circle/sphere of the mesh proportionally distant from the center, so as to create “cells” throughout which are geometrically similar. This modification permits the depiction of a four-dimensional or constant-curvature universe, with the fourth spatial dimension quantifying increasing scale with increasing radius. Thus, light would orbit the center of the system at all radii, and periodic motion would be uniform for all identical “clocks” at rest with respect to the mesh.

The justification for this depiction relies upon three fundamental concepts.

The first is the familiar requirement for accommodating the observed large-scale homogeneity and isotropy of the universe, ostensibly restricting the present proposal to a choice among the three F-R-W models. The closed/spherical model above tolerably compromises the perfect isotropy of its orthodox counterpart by replacing a line element for all points (of one radius and three angles) with a line element for one point (of two radii and two angles), so permitting a visualization and rationalization that would be prohibited by a third angular coordinate. One of the two radii proposed is an imaginative projection of a variable “absolute” distance from the origin/center to any given point, but the other is an “infinite” radius, shared by all points in the constant-curvature universe, in consequence of the infinite series of “cells” interposed between the center of the mesh system and any other point therein.

The second supportive concept clarifies an important aspect of general-relativistic/gravitational phenomena, to wit: in order to avoid perpetuity of motion, a photon rising in a gravitational field must therefore lose energy and frequency (not begin with and maintain a lower energy/frequency) as it arrives at a reference clock above. As a simple first hypothesis, then, timekeeping/periodic motion does not vary with gravitational potential, and so the relativistic “slowing” of phenomena in a field is a product of the contracted three-dimensional scale of events, not of the complementary dilation of periodic motion. This diminution of scale with descent into increasing absolute gravitational “density” would be a primary physical attribute of the universe depicted here, as the similarity of the cells of the mesh would therewith maintain uniform proper density and curvature throughout the system.

This attribution of inwardly increasing (absolute) gravitational density to the physical universe relies, factually, upon the quantity and distribution therein of mass-energy, which is radically insufficient in mass form (as now generally reckoned) to account for a closed universe. The third and final concept, therefore, must support positive vacuum energy as a plenary basis for the spherical curvature specified, and must rely upon a tenuous “naturalness” argument, depicting this energy (by analogy with the electron) as distributed with infinitely increasing (absolute) density toward the “center” of the universe. Without quantifying the matter, an energetic vacuum is nominated as that universal plenum which is necessary for a rationalization [to be provided elsewhere], in physical terms, of special-relativistic contractions/dilations/mass-accretions, and which would serve as the physical basis (if “naturally” configured) of the constant gravitational curvature of the model under discussion.

If the foregoing concepts are allowed, the mesh system may be imagined to be expanding, along with space, into an energy-less “void,” while retaining uniform “proper” or radar-principle-determined distance between co-expanding objects. The “Copernican” model thus relieves the otherwise compelling “tired light” thesis of the principle objection to its rationalization of the universal redshift. The summary perspective, here, is that space itself does make the major contribution to the geometry thereof, such that there is no “missing mass” to be accounted for, and that the “mystery of the cosmological constant” remains so, for present failure to reckon scale and (absolute) curvature as inversely proportional (and the Standard Model’s awesomely discrepant prediction as applicable to but one, original, point and not all).

A Pin for the Balloon December 13, 1994

The reader of the following is invited and encouraged first to examine a precedent file, Copernicus or Ptolemy: Cosmology Visualized wherein a pioneering visualization of cosmological geometry is offered with an economy of distracting reference to the state of the frontier left behind. This companion essay now looks back to that state in order to account for the defects in domestic cosmology which burden it with the largest discrepancy between theory and observation in the history of mankind‘s scientific endeavor.

The problem arises in a synthesis of general-relativistic cosmology and the “standard model” of particle physics, implying the origin and retention of the universe in a state of vacuum-energetic hyper-density such as is not within the experience of scientific observation. Relativistic cosmology, as presently interpreted, prevents a rationalization of this remarkable conflict, by attributing an essential isotropy to the whole of the cosmos which entitles any point therein to regard itself (as much as any other point) as the “center” of the universe. Since the reader’s left eyeball, for example, does not seem to enclose even so much as the material of a stellar galaxy, there must be something drastically wrong, hence the “mystery of the cosmological constant.”

This isotropy is typically “depicted,” by analogy rather than diagrammatic representation, as characteristic of the surface of an expandable balloon, which obviously knows no singular center thereupon. No problem inheres in a “flat,” Minkowskian universe, which easily sustains a three-dimensional image of the transferability of its familiar coordinates, but there are immediately evident pictorial and physical problems in reconciling an omnipresent and omnidirectional cosmological curvature, whether positive or negative in any degree. Cosmology takes license, for now, from the apparent “paradoxes” of special relativity and from the “weirdness” of quantum mechanics to satisfy itself with mere “analogies,” and so forsakes a needed rationalization for a faith in the doctrinal premise of the relativity or equivalence, hence centrality, of all observers, inadvertently and illegitimately imported from the otherwise valid Special Theory.

The “faith,” it may be conceded however, is an unwitting victim of an understandable misunderstanding concerning the geometry of constant positive curvature. Schwarzschild’s interior solution, the classic Einstein line element, and the F-R-W spherical metric, all characterize an idealized body or space filled with a “perfect" fluid of uniform proper density. Until there was “Copernicus,” there had not been a recognition (to the knowledge of the author) that in the instance of a sufficiently mass-energetic sphere, uniform proper density and inwardly increasing absolute density needed to be reconciled by the imperfectly isotropic mesh system proposed in that essay. It had been taken, quite naturally, that the homogeneity implied by the uniform density specified implied essential isotropy as well, so the “faith” and a degree of thoughtlessness have since conspired to produce the present insistence upon nonsense.

It should be emphasized, in summary, that the resort had by the balloon analogy, to what seem to be metaphysical coordinate dimensions, is not rigorously necessitated by empirical considerations, but is, rather, the dearly held implication of the failure to recognize that uniform proper density of sufficient mass/energy points to a singular center and not to the lack thereof. If, therefore, the hyper-dense origin and center of our universe can be thought, legitimately, to be elsewhere than in our own neighborhood, then an apostasy has purchased a measure of progress that may be worth the price paid therefor.

Comments on ‘Copernicus’ December 24, 1994

[This essay is offered in response to requests for clarification from readers of the companion essays, Copernicus or Ptolemy: Cosmology Visualized and A Pin for the Balloon.]

Please imagine a round-faced “clock” of unit three-dimensional size, with a single “hand” of unit period. Place this clock at one end of a “rod” of clock-face-circumferential length. Move the clock slowly across the rod to the other end in unit time period. Now, duplicate this clock and rod, but condense and contract both objects of this second set, such that each spatial dimension is “absolutely” (but not nominally) one-half the unit value of the first. The second clock thus occupies one-eighth its former absolute volume and is correspondingly more dense, but it is geometrically similar to its predecessor (with the second rod likewise reduced).

Please move the second clock slowly across its companion rod in unit time period. Note that the time-keeping properties of the two clocks, large and small, are identical, but that the second clock and hand have nevertheless gone “slower” in transit of a rod and face, since that rod and face were of diminished “absolute” length. Obviously, the first set could have gone as “slowly” as the second, by using twice the period of its larger clock, which illustrates that any combination of 3-D spatial contraction and time period dilation may account for phenomenal “slowing” in "complementary" fashion.

Now, please imagine a static sphere of unspecified mass-energy. This sphere, as do all such, becomes increasingly dense towards its center. If this sphere is of sufficient mass-energy [and beyond 9M/4 radius] it will have “crystallized” into four-dimensionality, constant curvature, and uniform proper density. Uniform proper density and the non-uniform absolute density mentioned may be reconciled, by absolutely (but not nominally) contracting the spatial dimensions of the increasingly dense inward areas, via the “Copernican” mesh system introduced in that essay.

This integration is permitted, physically, by the characterization of gravitational “slowing” as "contractional" rather than “dilational.” Readers of “Copernicus” will recall the brief argument therein contending that, since a rising photon must lose energy and frequency (rather than beginning with and maintaining such) in order to avoid perpetuity of motion [see Schutz], gravitational “slowing” must be accounted for with other than time dilation. Thus, light may orbit the center of our mesh-system/universe at all radii with the same passage of time (according to clocks of universally variable scale but uniform time period), so saving the velocity of light and our ability to visualize a four-spatial-dimensional universe.

Given this integration of physics and geometry, the mesh is obviouslynon-transferable and, in that sense, non-isotropic. Further, if a “pie slice” of the mesh is analyzed, an inwardly infinite series of similar cells of finite size can be created, pointing unambiguously toward a singularity. Does this not conflict with a valid derivation of the line element from the field equations? The surprising answer is no, given the following analysis: the orthodox derivation of 4-D line elements proceeds unexceptionably to the equation for density or homogeneity, then assumes the premise of uniform proper density or homogeneity and integrates. The emergent equation involves a constant of integration (expressed in ratio with the radius) which, for all non-zero values that might be attributed to it, implies singularity at the origin. Traditional derivations, in order to “prove” what has been assumed of isotropy, simply eliminate this implication by setting the constant of integration to precisely zero (whereas the field equations are creating a powerful presumption of singularity and anisotropy). A closing note might anticipate an objection at this point by observing that antithetical “proofs” of isotropy, which rely upon uniformity of interval and equivalence of Lorentz frames, do not, of course, take account of the properly-uniform/absolutely-variable intervals which characterize the “Copernican” mesh system discussed above.

[The author hopes that he has herewith achieved a successful balance of explicitness in argument and a regard for the reader’s patience in being informed of familiar or obvious matters and implications.]

[If a problem has been resolved by the foregoing, it has, of course, created a new one. Eager students might address themselves to identifying the value of the constant of integration, given what is known about minimal lengths and times.]

The Copernican Quantification January 17, 1995

This is the fourth of four essays, which began with Copernicus or Ptolemy: Cosmology Visualized and proceeded through A Pin for the Balloon and Comments on ‘Copernicus’. The last of these essays concluded with a verbal description of a mildly novel derivation of a cosmological line element, to which will now be given formal presentation in order to attempt a quantified resolution of the “mysteries” of both the "cosmological constant" and the "missing matter" of our unsettled theoretical cosmos:

ds2 = - dt2 + dr2/((1 - r2/R2(t)) + (A/r)) + r2 d(theta)2 + r2sin2(theta) d(phi)2

where r is a variable “absolute” radius, and the cosmological radius R is a function of the momentary time t. A is the constant of integration introduced and rationalized in the previous essay, “Comments on ‘Copernicus’.”

The cosmological constant (lambda) is a function of vacuum energy density and may be expressed as the inverse of the square of the cosmological radius. For purposes of approximating the desired result, the constant A will be taken to be the Planck length, which permits a cosmological radius/constant of eight hundred kilometers to produce a “proper” radial interval of approximately ten billion light years. This interval is the desired marginal “closure” radius/density, which accords with the observed “flatness” of the universe and which permits maximum spherical radius to the "Copernican" cosmos.

The “missing matter” difficulty is more than accounted for by taking the geometry of the universe as a product of vacuum energy density alone. Given the Planck length as our constant, we have had to over-/underestimate the absolute cosmological radius/constant by three/six orders of magnitude [see Abbott] so as to avoid too small a calculated proper radius. We have, in compensation however, achieved a CC discrepancy reduction of some forty of forty-six orders of magnitude, and some hope of relief from tortuously contrived theories attempting to dispense with the otherwise troublesome vacuum energy altogether. The surviving discrepancy does not, at this early date, discourage continuing efforts at discovering “cancellations” in the particle field contributions to the vacuum energy, or even at some convenient fudging in order to bring the involved numbers into mutual compliance.

An obvious question and objection regarding the foregoing is to the combination of a seemingly absurd “absolute” radius of geographic-scale kilometers with a “proper” interval reckoned in terms of billions of light years. The two may be reconciled by imagining a “rod” of a few hundred kilometers length being “ejected” from the highly vacuum energetic sphere of our universe into the presumed surrounding void and “decompressing” (amidst its evanescence) to the equivalent of the cosmological radius. Alternative estimations of the universal curvature radically reduced to the equivalent of the Planck length may be taken to be the particular victims of the isotropy confusion discussed in “A Pin for the Balloon,” the second of the essays in this series.

[The acute student will have realized that the foregoing equation is unorthodox only to the extent of recovering the expression “(A/r).” Otherwise, it retains the premise of progressive reduction to “flat” space toward the origin and a hyperbolic density distribution consistent therewith. A formula consistent with a progressive reduction to singularity at the origin and “Copernican” energy density is suggested by the following:

ds2 = - dt2 + (R/A)(dr2/(r2/R2(t)) + R2 d(theta)2 + R2sin2(theta) d(phi)2)]

Email me to discuss: Michael Ray Laurence