Scale Expanding Cosmos Theory III - Gravitation

Scientific Paper
Title	Scale Expanding Cosmos Theory III - Gravitation
Read in full	Link to paper
Author(s)	C Johan Masreliez
Keywords	Locally curved spacetime, Gravitational roll-off, Gravitational field energy, Black hole avoidance
Published	2004
Journal	Apeiron
Volume	11
Number	4
No. of pages	22
Pages	30-51

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Abstract

In the Scale Expanding Cosmos (SEC) the gravitational potential is modified by the cosmological scale expansion. The range of the gravitational field rolls off close to the Hubble distance and the presence of matter modifies the gravitational vacuum field, which evaluated in the cosmological reference frame contains negative energy that should equal the gravitating mass energy mc². A freely falling particle never reaches the event-horizon, which could prevent the formation of black holes. Although the results presented in this paper are tentative, two definite conclusions may be made in the SEC model:

The event-horizon is a true singularity.
Any spherically symmetric solution (other than the cosmological line element) of the Einstein's equations necessarily must modify the vacuum energy-momentum tensor generating negative field energy.

Overview

This is the third paper in Masreliez's Scale Expanding Cosmos series. The first, Scale Expanding Cosmos Theory I – An introduction, set out the model and its cosmological predictions; the second, Scale Expanding Cosmos Theory II – Cosmic Drag, dealt with cosmic drag, galaxy formation and planetary ephemeride discrepancies. Here Masreliez turns to the local gravitational field: what happens to Schwarzschild's exterior solution when the cosmological scale expansion of the SEC is carried into it.

The central claim is that the familiar Newtonian form of the potential is modified by a multiplicative "roll-off" function f(r), so that in the far field P(r) = −Gm f(r)/r. The roll-off diminishes the range of gravitation at distances comparable to the Hubble distance T. Two further consequences follow. First, the presence of matter no longer permits a vacuum energy-momentum tensor that vanishes: the gravitational field acquires negative energy which, on the calculations of section 4, comes out equal to −mc², so that the total energy of the universe is zero. Second, in the near field a freely falling particle never crosses what the standard model would call the event horizon, which suggests that black holes do not form in the SEC.

The contrast with the Standard Cosmological Model (SCM) is structural rather than merely quantitative. In the SCM there is no cosmological reference frame, hence — Masreliez argues — the vacuum energy-momentum tensor must vanish, since energy is not invariant in general relativity and would otherwise depend on the choice of coordinates. The SEC has a preferred line element and a cosmological reference frame, and this is exactly what permits gravitational field energy to be given a definite meaning. Masreliez is explicit that the results are "tentative" and that "there is no exact solution to the GR equations in the SEC"; only the two numbered conclusions in the abstract are offered as definite.

The argument

Schwarzschild's solution in the SEC

Masreliez starts from the standard line element in the two metric functions n(r) and l(r), whose Schwarzschild exterior forms are n = 1 − r₀/r and l⁻¹ = 1 − r₀/r. The SEC version multiplies the whole line element by the scale factor exp(2t/T), where T is the Hubble distance (with c = 1). Crucially, in the SEC the vacuum energy-momentum tensor does not vanish; it equals the Cosmic Energy tensor of the theory, whose components carry the same exponential factor.

Writing n = exp(N), l = exp(L), he sets out the temporal (G⁰₀), radial (G¹₁) and angular (G²₂, G³₃) components of the field equations. The pivotal technical result of the section is negative: differentiating the radial relation and repeatedly substituting yields an angular relation that no pair n, l satisfying the first two can also satisfy (apart from the trivial n = l = 1). "There is no simultaneous solution to all three equations. Therefore, any pair of functions n(r) and l(r) necessarily modifies the energy-momentum tensor." This is the abstract's second definite conclusion: the presence of matter must induce gravitational field energy.

He then makes the choice n(r) = f(r)(1 − r₀/r), l⁻¹ = 1 − r₀/r, which eliminates the large (e^−L − 1)/r² terms and leaves rest terms of order 1/T². In the far field the three equations separately give three different roll-off functions — exp(−1.5r²/T²), exp(−0.5r²/T²) and cos(√2 r/T) — confirming that no single f solves all three.

The SEC action integral

To pick a "best" f(r), Masreliez generalises Hilbert's action. Where Hilbert varies only the metric coefficients, the SEC action I_SEC = ∫(S²G − K T_SEC)√−g d⁴x treats the scale factor S as an independent parameter changing in discrete increments during a "piecewise continuous cosmological expansion." Since S²·T_SEC does not depend on the metric, the variation splits into two independent parts that must each vanish: the Hilbert action, and the new constraint G − K T_SEC = 0. He notes that in the SCM, with constant scale and no vacuum energy, the SEC action reduces to Hilbert's, since G = −R. He also flags forward to the next paper in the series, where the increment ΔS² is proposed to model quantum mechanical wave functions as modulation of the scale.

Applying the constraint in the far field yields a differential equation which reduces to the Helmholtz form ∇²f + 6f/T² = 0, with the closed-form solution

f_a(r) = sin(u)/u, u = √6 r/T.

Masreliez remarks that this "action roll-off function looks like a compromise between" the three single-equation solutions, and Figure 1 plots them together.

Gravitational field energy

Section 4 is the heart of the paper. Masreliez reviews the long dispute over gravitational field energy in general relativity: because a local Minkowskian frame can always be found in which the energy-momentum tensor disappears, there seems to be no absolute field energy, which motivated the pseudo-tensors of Einstein, Tolman, Landau and Lifshitz, Møller and Weinberg — all now understood to be related by gauge symmetry and nullifiable at any point. Against this he sets the alternative line of Lorenz (1916) and Levi-Civita (1917), quoting the latter: "The nature of ds² is always such as to balance all mechanical actions; in fact the sum of the energy tensor and the inertial (spacetime) one identically vanishes." He likens this to Newton's third law and d'Alembert's principle.

The SEC, having a cosmological reference frame, permits a postulate: gravitational vacuum field energy density is defined by the energy-momentum tensor evaluated in the cosmological reference frame. With that definition the total field energy of a spherically symmetric field is finite, and Masreliez states two observations.

Observation 1: under quite general assumptions the gravitational field energy equals −mc². Assuming only that f(0) = 1, f(∞) = 0, and that rf(r) and r²f_r(r) vanish appropriately in the limits, integration of the radial and angular contributions gives E¹₁ = −mc² and E²₂ = E³₃ = mc²/2, so that the total E_f = −mc². "This would be a pleasing result, since the total energy of the universe then would be zero."

Observation 2: the total gravitational potential from all matter in the universe is finite. Integrating the rolled-off potential over a uniform density ρ and applying an average limiting operation to handle the oscillation of f_a, he obtains

P_a = −GM/(2T),

with M the matter within the Hubble distance T. "Thus, the gravitational potential in the SEC is finite. This would resolve a longstanding puzzle since the time of Newton."

The near field solution

Assuming that the temporal metric approaches zero at the event horizon and that l⁻¹ stays finite there, Masreliez obtains l ≈ n⁻³ and, very close to r₀, the near-field forms n ∝ (r² − r₀²)^1/4 and l ∝ (r² − r₀²)^−3/4.

He then uses the invariance of the momentum inner product p^μp_μ = m² for radial motion. In the SCM the resulting expression for p₀ at the horizon is finite and "a particle may fall through the event horizon and be swallowed by a black hole." In the SEC the corresponding quantity vanishes at r = r₀, so "particles on geodesics will not cross the event horizon as long as relations (2.5) and (2.6) hold. This suggests that black holes might not form in the SEC."

Two supporting results follow. The rest terms give sharply negative energy density near the horizon whose volume integral diverges as r → r₀, which "might prevent gravitational collapse". And in the Riemann curvature tensor the term g_11,00 = 4g₁₁/T², with g₁₁ = exp(2t/T)·n⁻³, becomes infinite at r = r₀ while the other three terms stay finite — hence the abstract's first definite conclusion, that the event horizon is a true singularity in the SEC.

Discussion

Masreliez closes by noting the astrophysical opening this creates: if black hole formation is prevented, "something quite dramatic must happen at gravitational collapse," which he suggests "could account for the AGNs and be the engine of quasars," and might also bear on gamma ray bursts from the sudden collapse of massive stars. These are offered as suggestions, not derivations.

Assessment

What is genuinely attractive here is the internal economy of the proposal. A single feature of the SEC — the cosmological scale factor and the preferred reference frame that comes with it — is made to do several jobs at once: it supplies a well-defined gravitational field energy where general relativity has only pseudo-tensors; it gives that energy the value −mc², so the universe's total energy is zero; it renders the summed potential of all matter finite, addressing a difficulty that genuinely does go back to Newton; and it does all this while the roll-off is negligible at solar-system scales, since it only bites at distances of order the Hubble distance. The no-simultaneous-solution result of section 2 is also a clean piece of work and is stated with appropriate care as a theorem about the equations rather than a claim about nature.

The difficulties are ones the author himself largely concedes. There is no exact solution: the choice of n and l in (2.9) is made because it eliminates awkward terms and because it happens to yield −mc², not because it is forced. The action roll-off f_a = sin(u)/u is selected as a "compromise" between three incompatible far-field solutions; the choice of the generalised action (3.2) that produces it is itself a postulate, as is the definition of field energy in the cosmological frame. Both the potential result (4.15) and the limiting behaviour of f_a require an ad hoc averaging operation (4.14) to tame an oscillatory function that does not otherwise converge — and an oscillatory roll-off means a potential that periodically changes sign, a feature whose observational consequences are not examined. Masreliez notes in passing that the same lowest-order far-field fit is obtained with the Gaussian f = exp[−(r/T)²], which underlines how weakly the data constrain the choice.

The black-hole conclusion is where the paper runs hardest against established observation. It is offered conditionally — "suggests that black holes might not form" — but since 2004 the evidence for compact objects behaving as general relativity's black holes has strengthened considerably: the orbits of stars around Sagittarius A*, the LIGO gravitational-wave detections of binary black hole mergers with waveforms matching general-relativistic templates, and the Event Horizon Telescope images. A theory in which a freely falling particle never reaches the horizon owes an account of these. Masreliez's own suggestion that prevented collapse might power AGNs, quasars and gamma ray bursts is a gesture at where such an account would go, but it is a single sentence with no quantitative development. Finally, the whole result is conditional on the SEC framework of the earlier papers; a reader who does not accept scale expansion as a physical process gets no independent argument for it here.