A Thermal Basis for Gravitational Quantum Vacuum Polarization Descriptive Account of a Fundamental Process in a Friedman Dust Universe with Einstein's Lambda
| Scientific Paper | |
|---|---|
| Title | A Thermal Basis for Gravitational Quantum Vacuum Polarization Descriptive Account of a Fundamental Process in a Friedman Dust Universe with Einstein's Lambda |
| Read in full | Link to paper |
| Author(s) | James G Gilson |
| Keywords | gravitational, Einstein, universe, Vacuum, General Relativity, dust universe, dark energy, thermal cavity, negative gravity, cosmological Schrödinger equation |
| Published | 2010 |
| No. of pages | 18 |
Read the full paper here
Abstract
A descriptive account is given for the dust universe Friedman lambda model of the universe developed earlier from general relativity by the present author. This description is wrapped around a new and very simple derivation of the model from first principles. The mathematics of this derivation rests on two classical physical equations, the formula for black body radiation and Newton's inverse square law of gravitation, so that without its descriptive wrapping the new derivation which does not involve general relativity directly would occupy about one page of this paper. The descriptive aspect is devoted to showing how the dust universe model can be decomposed into a many subunit form where each galaxy is seen as being a thermal cavity subunit. The time evolution of the whole universe can consequently be seen as being a bundling together of the thermal cavity elements to make up the time evolution structure of the whole universe. Finally, a cosmological Schrödinger equation derived earlier by the present author is significantly generalized to make possible individual quantum state descriptions of the separate galactic thermal cavity elements. Some possible future generalizations of the structure are discussed.
Overview
James G. Gilson, writing from the School of Mathematical Sciences at Queen Mary University of London, here revisits the "dust universe" cosmology he developed in a sequence of earlier papers, beginning with A Dust Universe Solution to the Dark Energy Problem (2005). The model is a Friedman solution with Einstein's Λ in which, as he puts it, "every aspect of this theory depends on Λ to the extent that, if Λ is put equal to zero in the mathematics of the theory, the theory also vanishes." It is related to Lemaître's 1927 structure, but Gilson stresses an irony: "although Lemaître is said to be the father of the big bang, my version of this theory is not of the big bang type", because Lemaître apparently did not know about the negative time branch of his own solution.
The paper's chief novelty is a re-derivation. Where the model previously came out of general relativity and the Friedman equations, Gilson shows it can be obtained instead from just two classical ingredients — the blackbody radiation mass-density formula and Newton's inverse-square law — with one non-classical addition: the postulate of a second kind of mass that is positive in magnitude but negatively gravitating. Stripped of its descriptive wrapping, he says, the derivation "would occupy about one page". The second half decomposes the global model into local units, treating each galaxy as an expanding thermal cavity, and generalizes his earlier cosmological Schrödinger equation so that each such cavity can carry its own quantum state.
The argument
The thermal starting point
Gilson begins from the mass density associated with blackbody radiation,
- ρΓ(t) = aT4(t)/c2, a = π2k4/(15ℏ3c3) = 4σ/c
with σ the Stefan–Boltzmann constant. The physical picture is a bounded cavity in which radiation cannot escape but is reflected back, producing a spatially uniform energy density at rest within the volume. Assuming E = mc2, "rest mass is generated from a thermal system despite the fact that the individual photons of which the system is partially composed have no rest mass. Clearly, the mass production process is a consequence of containment." Everything that follows rests on identifying the right astrophysical realization of that cavity.
Two kinds of positive mass
The extension to Newtonian gravity is the postulate of m+ and m−, both positive quantities, with gravitational strengths G+ = +G and G− = −G. The first is ordinary attracting mass; the second is self-repulsive and repels ordinary mass. Gilson argues qualitatively that the negatively gravitating species must spread out while the ordinary species clumps, and that the two together admit equilibrium configurations: a sphere of total ordinary mass M immersed in a cloud of negative material reaches balance when the negative species forms a uniform static field, with an inward pressure PΔ from the ordinary matter balanced by an outward PΛ from the negative material, PΔ + PΛ = 0. Including the photonic component modifies this to PΔ(t) − PΓ(t) + PΛ = 0; the outward photon pressure shields and weakens the inward gravitational pressure, so "the photons might be described as contributing pseudo negative gravity."
The density of this background is fixed at twice Einstein's dark energy density:
- ρ†Λ = Λc2/(4πG), against ρΛ = Λc2/(8πG)
and is described as "just a few such particles per cubic meter — there is probably no more than one such particle within the human body at any moment of time."
From cavity to Friedman solution
The cosmological form of the blackbody relation is
- ρ(t) = ρ†Λ[T(t)/T(tc)]4, tc = (2RΛ/3c) coth−1(31/2)
with tc ≈ 109 yr the epoch of zero radial acceleration, at which ρ(t) equals the background density exactly. Gilson then insists on a particular reading of ρ(t): not a fixed volume with variable mass content, which would not be a closed system, but a fixed mass M in a conceptual expanding volume VM(t) = (4π/3)r3M(t). Within that volume sits the constant ordinary mass M and a time-varying amount of negative mass M†Λ(t) = VM(t)ρ†Λ. The Newtonian field at the boundary is then
- r̈(t) = −M†Λ(t)G−/r2 − MG+/r2 = r(t)c2Λ/3 − C/(2r2), C = 2MG
Multiplying through by ṙ and integrating, with the constant of integration set to zero by requiring ṙ → ∞ as r → 0 at t = 0, gives ṙ2 = (rc)2Λ/3 + C/r — the Friedman equation — whose solution is
- r(t) = b sinh2/3(±3ct/(2RΛ)), RΛ = (3/Λ)1/2, b = (RΛ/c)2/3C1/3
from which ρM(t) = ρΛ sinh−2(±3ct/(2RΛ)) and the Hubble function H(t) = (c/RΛ) coth(±3ct/(2RΛ)). The presence of ± is what carries the negative-time branch that distinguishes his reading from Lemaître's.
Galaxies as thermal cavities
Gilson emphasises that ρ(t) = ρΛsinh−2(·) "contains no mention of any quantity of mass M, there is no such involvement at all" — it depends only on t, Λ, c and G. Because the derivation is Newtonian and local, it applies equally to sub-regions. Setting M = MU gives the density history of the universe; setting M = Mgalaxy gives that of a galaxy. A consequence he draws out is that "the mass densities of galaxies are all equal at the same cosmological time t for this model", differing only in mass and cavity volume. Since the ordinary matter within a cavity clumps while the conceptual boundary keeps expanding, the measured volume VM,clumped(t) is much smaller than VM(t), so observed galactic densities exceed ρM(t). He offers this as an account of how galaxies separated: partition the early uniform universe into n ≈ 1011 contiguous sub-volumes of random size, each seeding a galaxy, each evolving by the same formula.
The cosmological Schrödinger equation
Having previously shown that the dust universe density can be recovered from the Schrödinger equation with ∇2Ψ = 0 and the feedback potential VC(t) = −(3iℏ/2)H(t), Gilson now adds rather than substitutes that term:
- iℏ ∂Ψ/∂t = −(ℏ2/2m)∇2Ψ + V(r,t)Ψ + VC(t)Ψ
Writing Ψ = Ψ1(r,t)Ψnl,ρ(t) and substituting recovers the ordinary Schrödinger equation for the factor Ψ1. The probability density then factorizes as ρS(r,t) = |Ψ1|2Ψ2nl,ρ(t), giving "a cosmological mass density with position variability of great generality" in which individually described galaxies of any known quantum internal structure can be bundled to describe the whole universe.
In conclusion Gilson proposes that time-shifted coordinates could give each unit its own singularity moment, so that statistical assemblies of such bundles would "ameliorate the difficult idea of a singularity at time zero by at least the same order of magnitude that success at searching for one needle in a haystack is improved by searching for 1011 needles in the same haystack." He regards the classical derivation as reinforcing "the validity and correctness of the Λ term apparently about which Einstein had such doubts."
Assessment
The paper's genuinely attractive feature is the derivation itself. It is a real result, and an old and well-known one in a new dress: the Friedman equation for a matter-plus-Λ universe can be obtained by Newtonian reasoning about a uniform sphere, because Birkhoff's theorem guarantees that the interior dynamics of a homogeneous ball are decoupled from the exterior. Gilson is right that this makes the structure "more classical intuitively acceptable" than its general relativistic form, right that the sinh2/3 scale factor is the exact ΛCDM matter-plus-Λ solution, and right that the same local reasoning can be applied to a sub-region. His observation that the resulting ρ(t) carries no reference to M at all is correct and is the reason the argument scales. The presentation is careful about what it is doing — he repeatedly flags VM(t) as a conceptual defined volume rather than a measurable one — and the acknowledgement to Kilmister and Rindler suggests the general-relativistic side had competent scrutiny.
The difficulties begin with what the classical derivation actually buys. Because the standard model's expansion history is the same sinh2/3 solution, nothing observational distinguishes Gilson's cosmology from ΛCDM at the level of the scale factor; what is offered is a re-interpretation of Λ as a real population of negatively gravitating particles at density Λc2/(4πG) rather than as a vacuum energy. That reinterpretation is asserted, not derived, and it is the load-bearing step: the whole Newtonian route works only because M†Λ(t) has been assigned exactly the density that reproduces the Λ term. The factor of two between ρ†Λ and ρΛ is introduced without argument, and it is precisely the factor needed to convert the relativistic active gravitational mass of a Λ fluid (ρ + 3p/c2 = −2ρΛ) into a Newtonian source. In other words the general relativity Gilson says the derivation does not require has been silently encoded in the constant.
The thermal foundation is the weakest element. The blackbody relation ρΓ = aT4/c2 is used to set ρ(t) ∝ T4, but the model is explicitly a dust universe, in which the matter density scales as a−3; a genuine radiation density scales as a−4, and these cannot be the same quantity. Combining ρ ∝ T4 with ρ ∝ V−1 yields T ∝ a−3/4, which conflicts with the T ∝ a−1 scaling of the cosmic microwave background — the one temperature history in cosmology that is directly measured, through the excitation of fine-structure levels in absorbers along quasar sight lines out to z ≈ 3. The paper does not identify T(t) with the CMB temperature, but nor does it say what else it is, and since the entire derivation enters through the blackbody formula this is not a detail. Relatedly, the claim that "the mass densities of galaxies are all equal at the same cosmological time" is either vacuous, if VM is a defined rather than physical volume, or false, since observed galaxies differ in mean density by orders of magnitude; Gilson notices the problem and answers it by clumping, but clumping is then doing all the work the formula was supposed to do.
The cosmological Schrödinger section is the least substantiated. Adding a purely time-dependent term VC(t) to the Hamiltonian is mathematically harmless — as Gilson's own factorization shows, it simply multiplies the wavefunction by a common factor and changes no measurable relative phase or expectation value — so the "strong unification of quantum mechanics and cosmology" claimed in the conclusion is, by the paper's own algebra, a decomposition that leaves ordinary quantum mechanics untouched. That VC is imaginary means the resulting evolution is non-unitary, which the paper does not discuss. And no observational consequence is drawn from any of it: there are no predicted numbers, no fits to the supernova or CMB data whose references appear in the bibliography, and no test that would distinguish the negatively-gravitating-particle picture from a cosmological constant. The negative-time branch, presented as the feature that makes the model "not of the big bang type", is mentioned only through the ± inside the sinh and never developed.
Read as what it says it is — a descriptive account wrapped around a one-page derivation — the paper succeeds at its stated task and is honest about being a restatement rather than a new prediction. Its interest lies in the pedagogical clarity of the Newtonian route and in the specific proposal that Λ be read as a dilute real substance, a proposal that would become testable only if the negative species were given properties beyond its density.