[Paper Review] Black Holes in Elliptical and Spiral Galaxies and in Globular Clusters

Reginald T. Cahill|ArXiv.org|Aug 24, 2005

Relativity and Gravitational Theory3 references16 citations

TL;DR

This paper proposes a velocity-field-based theory of gravity that replaces Newtonian acceleration with a fundamental velocity field, introducing a self-interaction dynamics governed by the fine structure constant α ≈ 1/137. The theory explains black hole masses in galaxies and globular clusters, the dark matter rotation curve anomaly in spiral galaxies, and the borehole g-anomaly on Earth through a single mechanism: vacuum black hole solutions with mass M_BH = (α/2)M, where M is the total system mass.

ABSTRACT

Supermassive black holes have been discovered at the centers of galaxies, and also in globular clusters. The data shows correlations between the black hole mass and the elliptical galaxy mass or globular cluster mass. It is shown that this correlation is accurately predicted by a theory of gravity which includes the new dynamics of self-interacting space. In spiral galaxies this dynamics is shown to explain the so-called `dark matter' rotation-curve anomaly, and also explains the earth based bore-hole g anomaly data. Together these effects imply that the strength of the self-interaction dynamics is determined by the fine structure constant. This has major implications for fundamental physics and cosmology.

Motivation & Objective

• To explain the observed correlation between supermassive black hole mass and host galaxy or globular cluster mass across 15 orders of magnitude of mass.
• To account for the dark matter rotation-curve anomaly in spiral galaxies without invoking unseen matter.
• To interpret the borehole g-anomaly on Earth as evidence of the same non-Newtonian gravitational dynamics.
• To unify the phenomena of galactic black holes, dark matter effects, and terrestrial gravity anomalies under a single theoretical framework based on velocity field dynamics.
• To demonstrate that the strength of space's self-interaction is determined by the fine structure constant α, linking gravity to quantum-scale constants.

Proposed method

• Assumes the velocity field **v**(r,t) as the fundamental dynamical variable instead of gravitational acceleration.
• Derives gravitational acceleration via the Eulerian form: **g** = ∂**v**/∂t + (**v**·∇)**v**, ensuring Galilean covariance.
• Modifies Newtonian gravity by adding second-order tensor invariants of the velocity gradient tensor D_ij: (tr D)^2 and tr(D^2), with coefficients involving α.
• Introduces a self-interaction term C(**v**,t) = (α/8)((tr D)^2 - tr(D^2)) to model non-Newtonian dynamics.
• Derives vacuum black hole solutions: **v**(r) = -μ r^(-α/4) **r̂**, which produce non-inverse-square-law acceleration g ∝ r^(-(1+α/4)).
• Defines an effective dark matter density ρ_DM = (α/32πG)((tr D)^2 - tr(D^2)) to interpret the non-Newtonian acceleration as arising from self-interaction.

Experimental results

Research questions

• RQ1 Why do supermassive black holes in elliptical galaxies and globular clusters exhibit a mass correlation M_BH ∝ M with their host systems?
• RQ2 What explains the flat rotation curves of spiral galaxies, traditionally attributed to dark matter?
• RQ3 Why does the gravitational acceleration g deviate from the inverse-square law in boreholes on Earth?
• RQ4 Can a single theoretical framework explain black holes, dark matter, and the borehole g-anomaly using a unified dynamical mechanism?
• RQ5 Is the strength of space's self-interaction dynamics quantified by the fine structure constant α?

Key findings

• The observed correlation M_BH/M = α/2 is reproduced across 15 orders of magnitude of mass, from Earth to elliptical galaxies like M87 and globular clusters like M15.
• Best-fit analysis of the data yields α = 1/134, consistent with the fine structure constant α = 1/137.036 within experimental error.
• The non-Newtonian acceleration g ∝ r^(-(1+α/4)) from vacuum black hole solutions matches the observed flat rotation curves of spiral galaxies.
• The borehole g-anomaly on Earth is explained by the same velocity field dynamics, with the anomaly's depth dependence determined by α.
• The theory predicts that the 'dark matter' effect and the black hole effect are manifestations of the same phenomenon: self-sustaining in-flow singularities in space.
• The theory implies that primordial black holes with this dynamics would drive rapid quasar and star formation, reconciling early universe observations with the big bang model.