Galactic Classification Quantum Gravity and Mass Spectra Cosmological Mass Spectrum each Galaxy having a Quantized Black Hole Core Surface Area Described as under the s p d f g h i... Atomic Symmetry

Scientific Paper
Title	Galactic Classification Quantum Gravity and Mass Spectra Cosmological Mass Spectrum each Galaxy having a Quantized Black Hole Core Surface Area Described as under the s p d f g h i... Atomic Symmetry
Read in full	Link to paper
Author(s)	James G Gilson
Keywords	Dust Universe, Dark Energy, Dark Matter, Newton's Gravitation Constant, Einstein's Cosmological Constant, Cosmological Mass Spectra, Quantised Gravity, Black Holes
Published	2013
No. of pages	27
Pages	1-27

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Abstract

There are two types of fundamental quantum gravitational mass amplitude states that are denoted by the subscripts D and P. The D amplitudes lead to Einstein's usual general relativity mass density functions. The P amplitudes lead to Einstein's additional pressure mass densities, 3P/c². Both of these densities appear in the stress energy momentum tensor of general relativity. Here they appear as solutions to a non-linear Schrödinger equation and carry three quantising parameters (l_D,m) and (l_P,m), The l_D,l_P values are subsets of the usual electronic quantum variable l which is here denoted by l' to avoid confusion. The m parameter is exactly the same as the electronic quantum theory m, there the z component of angular momentum. In this paper, these parametric relations are briefly displayed followed by an account of the connection to the spherical harmonic functions symmetry system that is necessarily involved. Taken together, the two types of mass density can be integrated over configuration space to give quantised general relativity galactic masses in the form of cosmological mass spectra as was shown in previous papers. Here this aspect has been extended to ensure that every galaxy component of the spectra has a quantised black hole core with a consequent quantised surface area. This is achieved by replacing the original free core radius parameter r_ε with the appropriate Schwarzschild radius associated with the core mass. Explanations are given for the choices of two further, originally free, parameters, t_b, θ₀. The main result from this paper is a quantum classification scheme for galaxies determined by the form of their dark matter spherical geometry.

Overview

James Gilson, of the School of Mathematical Sciences at Queen Mary University of London, wrote this paper in June 2013 as the latest instalment in a long series developing what he calls the Dust Universe model — a Friedmann cosmology with Einstein's Λ in which the Dark Energy term is treated as a physical medium and galactic Dark Matter haloes are treated as quantised gravitational states. The proposal at the heart of the series is that self-gravitating isothermal gas in equilibrium can be described by a nonlinear Schrödinger-type equation whose solutions are indexed by angular-momentum quantum numbers (l, m) in exactly the way atomic orbitals are, so that galaxies inherit an s, p, d, f, g, h, i... classification.

This paper's specific contribution is narrow and clearly stated. In the earlier papers the mass spectra depended on three free parameters — the galactic formation time t_b, an angular scale θ₀, and a core radius r_ε introduced purely to cut off a divergence at the origin. Gilson here removes r_ε as a free parameter by identifying it with the Schwarzschild radius of the mass it encloses, on the "consensual view that most, if not all, galaxies have a black hole at their centre." Because the core radius is then fixed by the quantum numbers, the horizon area of each galaxy's central Black Hole becomes quantised too, and the resulting scheme classifies galaxies by "the form of their dark matter spherical geometry."

The argument

Two families of gravitational states

Gilson works with two mass-amplitude types. The D amplitudes yield the ordinary general-relativistic mass density ρ; the P amplitudes yield the pressure contribution 3P/c², the second mass-like term that appears in the stress–energy–momentum tensor of General Relativity. Both are written as solutions of the nonlinear Schrödinger equation, and each carries its own relation between the gravitational quantum number l and the ordinary atomic angular-momentum number, which Gilson writes l′ to avoid confusion:

D: l′ = 2l − 1, so l_D = (l′ + 1)/2

P: l′ = 2(2l − 1), so l_P = (l′ + 2)/4

with −l′ ≤ m ≤ l′ as usual. The paper tabulates these from l′ = 0 to 20. Gilson flags an unresolved problem in his own table immediately: the derived l values come out as multiples of 1/2 and 1/4, whereas "only integral values have been found in the isothermal l-state theory." Half-integers might conceivably come from spin, which the isothermal theory does not yet contain, but "there is no existing explanation for those which reduce to multiples of 1/4." He states plainly that he cannot resolve this and proceeds "accepting the existence of theoretical gaps that possible may be filled some day."

Spherical harmonics and the mass spectra

A long section rehearses the standard machinery of spherical harmonics — the Laplacian in polar coordinates, associated Legendre functions, tesseral harmonics, and the normalisation integral

∫∫ Y_m,nY*_m,n sin θ dθ dφ = (4π/(2n+1))·(n+m)!/(n−m)!

Gilson draws out one methodological contrast that is central to his scheme. In atomic quantum theory these integrals are set to unity because the object of interest is a probability. Here the object of interest is a mass, so the gravitational states "make great use of the same normalisation factors in the very inverse way" — the un-normalised integrals, expressed in l and m, are precisely what generates the mass spectrum. The resulting generating functions are, for the two families,

M_l,m,D = (c²Λs(t_b)/G) · [(2l−1)^4l 2l θ₀^2l r_ε³ / 3(4l−3)] · A(2l−1, m)

M_l,m,P = (c²Λs(t_b)/G) · [(2l−1)^8l−2(4l−1)θ₀^4l−1r_ε³ / (8l−5)] · A(4l−2, m)

where A is a ratio of gamma functions and s(t) = sinh⁻²((3Λ)^1/2ct/2) comes from the dust-universe solution. Each spectrum is thus a product of a purely radial, time-dependent factor and a purely angular factor S_l,m(θ₀).

Quantising the core

The densities ρ_D,l,m(r) and ρ_P,l,m(r) diverge at r = 0. Gilson's original device was to freeze the density at its value ρ(r_ε) inside a sphere of radius r_ε, giving a finite core mass M_C = 4πρ(r_ε)r_ε³/3 with r_ε arbitrary. The new step is to demand self-consistency: set M_C = c²r_ε/2G, that is, require r_ε to be the Schwarzschild radius of the mass it contains. Solving gives closed forms

r_ε,D³ = [3c²ε_m(4l−1)Γ(2l−m) / 8πGρ(t_b)θ₀^2l(2l−1)^4lΓ(2l+m)]^3/2

and its P-family analogue, so that the horizon area a = 4πr_ε² is fixed by (l, m). A third case is treated as well: a core made of dark-energy mass alone would require r_ε = R_Λ/√2 with area 2πR_Λ², which Gilson himself calls "a very unlikely size for the core of any galaxy though still a possibility of theoretical interest."

Fixing the remaining parameters

Two free parameters survive. Gilson argues t_b should simply be left arbitrary, since s(t_b) multiplies every line equally and only rescales the whole spectrum — spectra are therefore "possible sets of values associated with some arbitrary but definite evolutionary time," and "no mass member of such a set would have necessarily certainly occurred physically." θ₀ is different: it sets the relative spacing of the lines and is therefore intrinsic. He can find no theoretical way to determine it, and no observational way either, because "at this time, the (l, m) structure is not known for any galaxy and the actual galactic masses generally are only known very approximately." He therefore fixes it by hand, tuning θ₀ = 2.97845 so that the l = 1, m = 0 line takes the value

M_G = R_Λc²/G = (3c⁴/ΛG²)^1/2 ≈ 2.00789 × 10⁵³ kg

a quantity that, he remarks, "for unknown reasons give a number of kilograms which could be the actual mass of our universe."

The paper closes with the Mathematica source that prints the spectrum, and sample pages of output. The listing runs from the l = 1 head, where masses are of order 10⁵³ kg, down to l = 18, where the tabulated entries fall to 10⁻³⁷ kg.

Assessment

The mathematics in this paper is competent and the presentation unusually honest about its own gaps — Gilson repeatedly marks which results are established and which are not. The reduction of a free parameter is a real methodological gain: turning r_ε from an arbitrary regulator into a quantity fixed by the Schwarzschild condition is exactly the kind of move that makes a phenomenological model more constrained rather than less. Tying the black-hole horizon area to a discrete index is also, in isolation, a respectable idea; horizon-area quantisation appears independently in loop-quantum-gravity and Bekenstein-type arguments, so the ambition is not eccentric. And the organising analogy — that galactic haloes might, like atoms, carry a spherical-harmonic label — is at least a testable-sounding proposal about morphology rather than a purely verbal one.

The difficulties, however, run deep, and several are visible in the paper itself. The most serious is that the l values derived from the D and P relations come out in quarters, which the author concedes has "no existing explanation" and which he sets aside rather than resolves. A quantum number that the theory's own algebra produces in a form the theory cannot interpret is not a small blemish; it means the correspondence between the gravitational states and the atomic ones — the paper's whole organising claim — is not actually established. Nothing in the paper derives the nonlinear Schrödinger equation for self-gravitating dust from general relativity or from any variational principle; it is imported from earlier work in the series and its status as an approximation is never characterised. There is no ħ anywhere, and no argument for why a classical isothermal gas sphere should have discrete states at all.

The parameter fixing is the second problem, and it undercuts the paper's stated aim. Having removed one free parameter, Gilson disposes of t_b by declaring the spectrum arbitrary up to overall scale, and disposes of θ₀ by tuning it to reproduce a chosen number. The chosen number is R_Λc²/G, identified with the mass of the universe "for unknown reasons." This is a coincidence of the Dirac large-number family — with R_Λ ~ 10²⁶ m, c²R/G is automatically of order the mass within the Hubble radius, because that is what the Schwarzschild relation at cosmological scale gives — so it is a dimensional identity rather than a prediction. Fixing the one parameter that controls line spacing by matching it to such a coincidence means the spectrum's structure is set by fiat, and the paper is candid that the alternative, calibrating on a real galaxy, is unavailable because no galaxy's (l, m) is known.

That last admission points to the decisive gap: nothing here is compared with data. The paper contains no galaxy, no measured mass, no rotation curve, and no observational test. The abstract of the earlier work claims the quantised halo velocity curves are "decisively flat," but flat rotation curves follow from any roughly isothermal halo profile and do not discriminate this model from a standard NFW or pseudo-isothermal halo. Meanwhile the printed spectrum runs down through 10⁻³⁷ kg — sub-electron masses presented as galactic spectral lines — with no selection rule offered to say which lines are physical, and Gilson explicitly disclaims that any given member need have occurred. A spectrum that ranges over ninety orders of magnitude and predicts nothing about which values are occupied cannot be falsified by a galaxy survey.

Two further conflicts with established measurement deserve naming. The scheme ties every galaxy's central black hole mass rigidly to its halo quantum numbers, whereas the observed correlation is with the bulge — the M–σ relation between black hole mass and stellar velocity dispersion is tight, roughly a factor of two in scatter, while black hole mass correlates only weakly and indirectly with total halo mass. And the identification of the halo with a coherent quantum state indexed by a single (l, m) is hard to reconcile with the observed lumpy, triaxial, merger-built structure that both weak lensing maps and the substructure inferred from strong-lensing flux-ratio anomalies reveal.

Judged as what it announces itself to be — one step in a long private research programme, tightening a parameter and setting up a classification scheme — the paper does what it says. Judged as cosmology, it remains a formal exercise: mathematically self-consistent within its own assumptions, unconnected at every point where it might have been connected to the sky.