DISTORTIONAL EXTREMA AND HOLES IN THE GEOMETRIC MANIFOLD

It is shown in the present work that the distorted-space model of matter as extended to extreme curvature limits results in characteristics mimicking those of galactic-holes. The distorted-geometry structures exhibit non-Newtonian features wherein the hole or core-region fields of the structure are energetically-repulsive (negative pressure), do not behave functionally in an r -4 manner and terminate at zero at the radial origin (no singularity). Of particular interest is that of r -6 energy-density behavior at structural radial distances near the core of the distortion, a region also displaying potential-well behavior.


Summary
The work is a non-conventional mathematically-geometric approach to describing "black-hole" structures.
We have produced a description of the "black hole" as a geometric-mimic, a "distorted geometry" structure, formulated from a solution of Riemann's geometric equations. The model is essentially the "Curved empty space as the building material of the physical world" supposition of Clifford 1 in 1876 and is the conceptual basis for this "distorted-geometry" modeling. The resulting geometric description of matter (mass-energy) mimics the classical-physics electromagnetic and gravitational-field models at large radii but departs significantly at small radii to produce a magnetic-field (spin) mimic as well as a weak-field mimic (beta decay and the Fermi constant) and a strong-field mimic without an infinity at the origin (no singularity) 2 . The structure is constituted by a core-region within which the propagation-velocity, by virtue of the distorted metrics, is greater than c and exhibits a "partial light trapping phenomenon", a "black hole".
Distorting the geometry in our spatial-manifold requires energy but with limits as to the degree of distortion thereby predicting and describing fundamental-electromagnetic-particle structures as well as gravitational (dark-matter, black-hole) structures. Such a geometric description of the spacetime manifold would seem (?) to constitute a "first-principle" model of the universe.

Introduction
A "Curved empty space as the building material of the physical world" supposition of Clifford 1 in 1876 is the conceptual basis for this "distorted-geometry modeling" 2, 3 We maintain and expand the geometrical perspectives inherent in the earlier work 2 and building on that work, we apply the geometric concepts to produce a distortional-geometric extremum, a "stability-based minimum-energy-density" condition or "maximum gravitational radius" condition. Additionally, we showed in 2,3 that the propagation velocity in the core region of these distorted-geometry structures was approximately 1.5 times that external to the core (see Fig.1). This feature, which is present for all such structures, is equivalent to a "partial light trapping" phenomenon (a black hole core?).
A "geometric maximum-energy-density" feature, in the EM (electromagnetic) energy-density realm, was successfully exploited to geometrically explain and quantify the Fermi constant 2 in addition a "stabilitybased minimum-energy-density" condition was fundamental to describing the structure of the "stable distortional-geometry electron" feature.
In this perspective, the distorted-geometry model is a departure from the classical geometry model where the Einstein Curvature tensor is the stress-energy-tensor describing the "material contents" of the energy distribution. This distorted-geometry model is rather viewed with the energy-content residing in the warping or distorting of the manifold and therefore in its geometric-tensors, and the "curved empty space" 2,3 referred to above is a "localized curved or distorted space" devoid of an "external or foreign" causative matter-entity. The "distorted metrics" and the core propagation velocity are displayed for example, for the distorted-geometry electron-mimic in Fig. 1.
We have used a modified coupling-constant definition by omitting the factor 8π and retaining the factor in the enegy-density equations; conventionally, the coupling-constant definition would be 8πκ.
The following mathematical development is taken from 2 and is reproduced here to render the manuscript more readable without constant attribution to the earlier manuscript.
Allowing space itself to be material in nature, we constrain the modeling by requiring that the descriptive stress-energy tensors satisfy a "constitutive relation" or an "equation-of-state" between the temporal and spatial tensor-curvature elements, namely Td 4 4 = -(Td 1 1 + Td 2 2 + Td 3 3 ). (2) We have introduced the explicit distortional-tensor symbolism Td for the geometric quantities. Contrast this perspective with cosmological renditions of geometric curvature structure resulting from "matter" causation, wherein several "equations of state" relating to the "matter" variables ρ (density) and p (pressure) have been forthcoming 4 where p = σ ρ and where σ varies from -1 to +1.
Inherent in the geometric "equation-of-state" constraint is the requirement that the descriptive stressenergy tensor, Td, be Maxwellian in nature; the mimicking process is therefore limited to asymptotically flat-space regions of the manifold since 1/r 2 field behavior does not adequately describe elementary-particle structural-detail 5 . The field equations, in both the EM realm and the gravitational realm (Q = 0), exhibit r -6 geometric behavior which we have interpreted as constituting a "magnetic monopole" mimic (what is a "magnetic monopole" ?).
where the metric quantity coupling-constant appears in the geometric quantities γ (Eq.13) and the geometric "transformation radius" R0 (Eq.13) both determined from the "distorted spatial volume" with electromagnetic and/or gravitational energy-density components.
A radial zero in the field quantity (Fd 14 ) 2 , namely r0 ≡ Rs geo = R0 geo /u(r0), with u(r0) = γ(grav)/2 = 3.27512/2, is the geometric manifestation of the Schwarzschild "metric-radial-zero", the radial singularity classically interpreted as a "black-hole" radius. The core-radius is a fundamental feature of the "distortedspace" structures; it is the radial point at which the energy-density-distortion transitions from a positive shell-like value to a negative core-like value. The structures inherently illustrate r -4 , r -6 and repulsive-radial energy-density behavior (relative to the shell energy-density behavior), thereby accounting for Newtonian, weak and strong field-attributes.
In discussions of the negative energy-density core-regions of this universal (EM as well as gravitational) distorted-geometry structure, it should be emphasized that a negative energy-density gravitational feature (a repulsive gravitational force or negative pressure) is non-Newtonian. The hole or core region-fields of the structures are repulsive (relative to the extra-core, or shell, region), do not behave functionally in an r -4 manner and terminate at zero at the radial origin (no singularity). See Fig. 2 where the magnetic and radial field energy-densities are graphed for quantitative and qualitative purposes and Fig, 3 where the same quantities are shown in absolute values to more clearly identify the relative strengths of these energy densities in the shell to core transition regions. Fig. 4 is constructed to complement the data exhibited in

Results
We are considering, in the present undertaking, the gravitational energy realm. Incorporating the symbolism utilized in 2 , we write for the minimum-radius-extremum (the "geometric maximum-energydensity" feature) of a "geometric" gravitational distortion, r0 geo (min) = Rs geo (min) ≡ Schwarzschild radius(min) = 2 (Mc 2 ) grav_min G c −4 (10) and defining the "minimum-geometric-radius", as derived from the geometric characterization 2 of the Fermi-constant GF, as r0W, we produce π 0 3 , of the physical volume of the energy-density 5 distribution of these structures (reference Figures 1 and 2).

18
For a gravitational distortion, R0e and Rse are zero.

19
If, in the absence of a physical structural constraint, one posits a "minimum" curvature, or a "minimum" 20 EM-energy-density condition (which was posited 2 for the "electron-mimic" and which is equivalent to a 21 "maximum" geometric EM core-radius) as the "stability" criterion to produce the maximum-core-radius- By using the associated "geometric-energy-density minimum" as the constraint for a "maximizing 27 gravitational core-radius principle", we produce the more classical "HOLE-like" structure (the "distorted- energy of the structure 5 . Therefore this "distortional-geometry hole-structure", created at the "birth of the 50 universe" (also see reference 13 ), registers as a viable candidate for the structure of "black holes".

51
Mass-energy "black-hole" growth rates 9-13 however range from "~1 solar mass/3000 years (for the Milky 52 Way Galaxy)" up to "~1 solar mass/20 years (for NGC 4594)", therefore the "Milky Way Black-Hole 53 mass-accretion rate" allows for even a "zero-mass black-hole" at creation-time. Accretion rates are in part 54 based on distance, times and the universe-expansion model (see reference 14 ) and would be subject to 55 revision according to the model selected.

67
A "dark-matter" mass-energy ratio, to the (2/5) power, as a distribution-function factor, would produce 68 the desired "experimentally observed" number of galaxies. the magnetic field component, Fd mag 2 , is non-zero at the "radial field zero", Fd 14 2 = 0, or "core radius".

83
This field feature would seem responsible for accretion-disk and galaxy-matter rotational-distribution 84 behavior. Actually, the Fd 14 2 fields contain r -6 elements of a magnitude comparable to the magnetic-field