Foundations of a Quantum Gravity at Large Scales of Length and its Consequences
| Scientific Paper | |
|---|---|
| Title | Foundations of a Quantum Gravity at Large Scales of Length and its Consequences |
| Read in full | Link to paper |
| Author(s) | Cláudio Nassif |
| Keywords | Quantum Gravity, length, cosmological constant, gravitational field, reference frame |
| Published | 2010 |
| No. of pages | 37 |
Read the full paper here
Abstract
We attempt to find new symmetries in the space-time structure, leading to a modified gravitation at large length scales, which provides the foundations of a quantum gravity at very low energies. This search begins by considering a unified model for electrodynamics and gravitation, so that the influence of the gravitational field on the electrodynamics at very large distances leads to a reformulation of our understanding about space-time through the elimination of the classical idea of rest at quantum level. This leads us to a modification of the relativistic theory by introducing the idea of a universal minimum speed related to Planck minimum length. Such a speed, unattainable by the particles, represents a privileged inertial reference frame associated with a universal background field (a vacuum energy), enabling a fundamental understanding of the quantum uncertainties. The structure of space-time becomes extended due to such a vacuum energy density, which leads to a negative pressure at the cosmological length scales as being an anti-gravity, playing the role of the cosmological constant. The tiny values of the vacuum energy density and the cosmological constant are obtained, being in agreement with current observational results. We estimate the very high value of inflationary energy density of vacuum at Planck length scale. After we find the critical radius of the universe, beyond which the accelerated expansion (cosmological anti-gravity) takes place. We show that such a critical radius is Ruc = rg/2, where rg = 2GM/c2, being rg the Shwarzschild radius of a sphere with a mass M representing the total attractive mass contained in our universe. And finally we obtain the radius Ru0 = 3rg/4(> Ruc) where we find the maximum rate of accelerated expansion. For Ru > Ru0, the rate of acceleration decreases to zero at the infinite, avoiding Big Rip.
Overview
Nassif, writing from the Centro Brasileiro de Pesquisas Físicas in Rio de Janeiro, proposes a modification of Special Relativity in which the familiar upper speed limit c is joined by a universal lower limit V, equally unattainable and equally invariant. The resulting framework, which he names Symmetrical Special Relativity (SSR), replaces the open interval 0 ≤ v < c with the doubly bounded interval V < v ≤ c, and with it abolishes the classical notion of rest at the quantum level. The minimum speed is attached to a preferred, non-Galilean frame he calls the ultra-referential SV — a background field whose energy density is identified with the vacuum, and whose negative pressure is then made to do the work of the cosmological constant.
The departure from the standard account is deliberate and specific: SSR breaks Lorentz symmetry. Where special relativity holds all inertial frames equivalent and lets v be exchanged for −v by an inverse transformation, Nassif's transformation matrix Ω is non-orthogonal, Ω−1 ≠ ΩT, and the exchange fails. He presents this as the natural completion of two projects Einstein left unfinished — the search for a relativistic "ether" that does not contradict the relativity principle, and the derivation of quantum uncertainty from a unified field theory — and situates his minimum length alongside the invariant Planck length of doubly special relativity as its low-energy counterpart. The paper's headline claims are numerical: values for the cosmological constant and vacuum energy density said to agree with observation, and a critical radius beyond which cosmic expansion turns accelerating.
The argument
The electromagnetic content of matter
The construction begins from the field amplitude of a single photon. Normalising a plane wave to energy ℏω gives an amplitude e0 = √(8πℏω), and the photon energy can be written E = pc = ℏω ≡ eph2/4π, so that λ ≡ 4πhc/eph2. Nassif calls eph the "scalar support of electric field" — not a mean value of an oscillating classical field, but a corpuscular quantum quantity carrying the photon's wave-particle duality.
He then extends this to matter via de Broglie reciprocity and pair production γ → e− + e+, writing the mass itself as electromagnetic in origin, m ≡ melectromag ∝ esbs, with m0c2 = cε0es0bs0ve. A consequence he draws out is that the electron cannot be exactly pointlike: setting ve = 0 gives divergent internal fields, whereas taking Re ~ 10−16 m gives a finite internal field es0 ~ 1023 V/m, constant inside Re and falling as 1/r2 outside.
The fine adjustment constant ξ and the minimum speed
In a gravitational potential φ, with √g00 = 1 + φ/c2, the scalar supports shift by ∆es = es0(√√g00 − 1). Nassif posits that the external fields E and B of the moving charge suffer proportional shifts δE = ξ∆es, δB = ξ∆bs, with ξ a dimensionless gravi-electrical coupling. Expanding the total electromagnetic energy density then yields three terms: ρ(0) ∝ 1/r4 (Coulombic), ρ(1) ∝ 1/r2 (a radiation-like mixing term), and — crucially — ρ(2) ∝ 1/r0, a constant, distance-independent, non-local background that "cannot be cancelled by any transformation." Because ρ(2) ∝ (δB)2 never vanishes, the magnetic field of a charged particle can never be transformed away, and therefore no frame exists in which the particle is at rest.
The value of ξ is fixed by analogy. The fine structure constant αF = e2/ℏc ≈ 1/137 governs charge–charge binding; Nassif introduces a "superfine structure constant" βF = Gme2/ℏc ≈ 1.75 × 10−45 for mass–mass binding, noting αF/βF ~ 1042. Since ξ couples mass to charge, he takes the geometric mean
- ξ = √(αFβF) = √(G/4πε0) · meqe/ℏc ≅ 3.57 × 10−24
and identifies ξ = V/c, giving V = ξc ≅ 1.07 × 10−15 m/s. He justifies the choice of electron mass and charge on the grounds that qe is the smallest free charge in nature (quarks being confined), that the electron is the lightest charged elementary particle, and that it is stable — so meqe is minimal. Writing V = (mee√(c3/ℏ3))lP ties V directly to the Planck length; setting G → 0 sends both V → 0 and lP → 0, restoring ordinary relativity.
The modified time relation and "quantum rest"
The kinematic core is a geometric argument on a light-clock in which the emitting particle cannot be located exactly at the origin of its own frame, the non-localisation being ∆x′v = f(v)∆τ. Requiring symmetry between the two limits — f → V as v → c, f → c as v → V — gives the reciprocal internal speed vint = v02/v, and imposing ∆t = ∆τ at some intermediate v0 fixes
- v0 = √(cV) ≅ 5.65 × 10−4 m/s
which Nassif calls the "quantum rest" — the speed at which the ordinary proper mass and proper time are recovered. The resulting relation replaces the Lorentz factor:
- ∆τ √(1 − V2/v2) = ∆t √(1 − v2/c2), ∆t = Ψ∆τ, Ψ = √(1 − V2/v2) / √(1 − v2/c2)
For v >> v0 this reduces to ordinary time dilation; for v << v0 it gives a new effect Nassif names contraction of time, in which proper time runs faster than improper time and ∆τ → ∞ as v → V. The total energy becomes E = m0c2Ψ, which vanishes at v = V, equals m0c2 at v = v0, and diverges at v = c.
The framework carries a geometric surplus: since ∆x′v = (V∆τ)(c∆τ)/(v∆τ), the interval relation acquires a fifth, temporal coordinate ∆x′5 = V∆τ, normally hidden and becoming manifest only as v → V. The invariant is restored as ∆S5 = c∆τ√(1 − V2/v2), and the relation vt2 + v2 = c2 now has maximum temporal speed c√(1 − ξ2) rather than c, so that both the spatial and temporal limits become unattainable. Nassif also notes that because v = 0 is forbidden, motion cannot reverse in one spatial dimension, which he offers as a reason why more than one spatial dimension is required and as a source of temporal irreversibility. The oscillating transverse components he identifies with Schrödinger's zitterbewegung.
Transformations and the twin paradox
The speed composition law becomes
- vRel = (v′ ∓ v ± V) / (1 ∓ v′v/c2 ± v′V/c2)
which reduces to the Lorentz law when V → 0. It gives "V − V" = V and "V + V" = V — so V behaves as an "absolute zero of movement" — and preserves the invariance of c. Since the transformation matrix satisfies Ω−1 = ΩT/detΩ with detΩ ≠ ±1, v cannot be exchanged for −v; Nassif argues this asymmetry eliminates the twin paradox, since only the non-Galilean frame moves with respect to the covariant SV. He verifies that the Maxwell wave equation remains covariant under the new transformations, the extra factor detΩ = Ψ2[1 − β2(1 − α)2] being strictly positive because v > V always.
Cosmological consequences
Writing the four-velocity with the modified factor and inserting it into the perfect-fluid energy-momentum tensor Tμν = (p + ε)UμUν − pgμν, the limit v → V gives T00vacuum = −p. Since T00 must be positive, p < 0: a negative vacuum pressure emerges from the limit rather than being postulated, with ε = −p and equation of state w = −1.
Identifying the potential from E = m0c2(1 + φ/c2) gives φ = c2(Ψ − 1), which is attractive for v0 ≤ v < c and repulsive for V < v ≤ v0, with the strongest repulsion φΛ = −c2 at v = V. Modelling the universe as a sphere of radius Ru filled with uniform vacuum energy, φΛ = −ΛRu2/6, and equating this to −c2 gives
- Λ = 6c2/Ru2, ρ(Λ) = 3c4/4πGRu2
For the Hubble radius RH0 = cT0 ≈ 1.3 × 1026 m (T0 = 13.7 Gyr) this yields Λ0 ≈ 3 × 10−35 s−2 and ρ(Λ0) ≈ 2 × 10−29 g/cm3; at the Planck length it yields ΛP ~ 1087 s−2 and ρ(ΛP) ~ 10113 J/m3, "122 orders of magnitude beyond" the present value, which he offers as an inflationary vacuum field.
Finally he sets an effective curvature against the two contributions,
- Reff = R + Λ = −6GM/Ru3 + 6c2/Ru2
Setting Reff = 0 gives the critical radius Ruc = rg/2 with rg = 2GM/c2, and with M ~ 1052 kg this is Ruc ~ 1025 m — an order of magnitude below the present Hubble radius, so the universe is currently accelerating. Setting dReff/dRu = 0 gives Ru0 = 3rg/4 = 1.5Ruc, where acceleration peaks at Reff.max ~ 10−33 s−2, after which it declines toward zero at infinity — so the model, he notes, avoids a Big Rip. A correction term +6GMlP/Ru4 is added so that gravity vanishes at the Planck scale, producing a five-stage expansion history: initial inflation, deceleration, maximum deceleration, transition, and the present accelerating phase.
Assessment
There is real architectural ambition here, and several features are genuinely attractive. The most striking is that the negative pressure is not inserted: it falls out of the v → V limit of an otherwise ordinary perfect-fluid tensor, and the equation of state w = −1 emerges rather than being chosen. That is a better structural motivation for a cosmological constant than simply postulating one. The symmetry of the construction is also elegant — the reciprocal pairing of v and vint = v02/v, the way the uncertainty ∆x′v grows precisely as momentum falls, and the recovery of both Lorentz kinematics and Galilean rest as the single limit V → 0, which is the right kind of correspondence to demand. The paper is honest about which results are conjectural, and its identification of V with lP gives it a stated point of contact with the doubly-special-relativity literature rather than leaving it isolated.
Several steps are nevertheless asserted rather than derived. The proportionality δE = ξ∆es linking external field shifts to internal ones is introduced by analogy, with no derivation from the field equations; everything downstream depends on it. The value of ξ is then fixed by a geometric mean, ξ = √(αFβF), which is motivated by the observation that the coupling is of "mass–charge" type but is not obtained from any calculation — many other combinations of αF and βF would be equally "of the type meqe". The choice of the electron in particular is defended on plausibility grounds (smallest free charge, lightest stable charged particle) rather than by a mechanism, which leaves V — and hence Λ — resting on a species selection.
The most serious internal difficulty is that the cosmological result appears to be a consequence of the model's geometry rather than of its physics. The identification φΛ = −c2, combined with the Newtonian potential of a uniform sphere, gives Λ = 6c2/Ru2 directly. But this is essentially the statement that the vacuum energy inside the Hubble volume is of order the Hubble-scale gravitational binding energy — a coincidence relation of the same family as the well-known ρΛ ~ c4/GRH2, which reproduces the observed magnitude for any theory that makes the vacuum density scale as Ru−2. Note that ξ, the quantity built from me and qe and carrying all the model's specific content, does not appear in the final expressions for Λ or ρ(Λ) at all. The "agreement with observation" therefore tests the scaling assumption, not the minimum-speed hypothesis. Worse, the relation Λ = 6c2/Ru2 makes Λ time-dependent, decreasing as Ru−2, whereas the quantity constrained by observation is a constant: the Type Ia supernova Hubble diagram, the baryon acoustic oscillation scale, and the CMB acoustic peaks jointly bound the dark-energy equation of state to w = −1.03 ± 0.03 with no detected evolution, and a Λ falling as Ru−2 corresponds to w = −1/3, which is excluded at high significance and would not produce accelerated expansion at all. The paper does not reconcile its derived scaling with the constant Λ it inserts into the field equations at eq. (99).
The kinematic claims face a different problem. A minimum speed V ≅ 10−15 m/s is not a remote regime: it corresponds to a kinetic energy for an electron of order 10−60 J, far below anything a laboratory probes, but the transformation law that contains V applies at all speeds. Since Ω is non-orthogonal and Lorentz symmetry is explicitly broken, the model predicts frame-dependent effects suppressed by powers of ξ ~ 10−24. Modern Lorentz-invariance tests — the Michelson–Morley-type rotating optical cavity experiments, which bound anisotropy in the speed of light at the level of parts in 1018, and clock-comparison and Hughes–Drever experiments bounding preferred-frame effects at parts in 1027 for some coefficients — are in the range where such a violation would need to be shown to hide. The paper does not compute what SSR predicts for any of these, so the compatibility is unestablished rather than demonstrated. The claim that "contraction of time" occurs for v << v0 = 5.65 × 10−4 m/s is more troubling still, because that is an ordinary laboratory speed — roughly half a millimetre per second. Atomic clocks have been compared at relative velocities and height differences far smaller than this, most sharply in the optical-clock experiments that resolved gravitational redshift over a 33 cm height difference; the model's prediction of an anomalous ∆τ > ∆t regime below v0 is not confronted with that data, and it is not obvious how the effect would evade it given that v0 is macroscopic.
Two smaller points. The identification of M ~ 1052 kg as "the total attractive mass contained in our universe", used to get Ruc, is taken from the same Ωm ≈ 0.3 concordance fit that the model is meant to explain, so the critical-radius result is partly circular. And the treatment of the electron's internal structure, giving Re ~ 10−16 m and a constant interior field, sits uneasily with the electron's measured pointlike behaviour: the anomalous magnetic moment agrees with QED point-particle calculation to better than a part in 1012, and high-energy scattering bounds any substructure well below 10−18 m, an order of magnitude tighter than the figure Nassif uses.
What the paper does achieve is a self-consistent kinematics with two invariant speeds, a coherent geometric account of why quantum non-localisation should grow as momentum falls, and a mechanism by which negative vacuum pressure appears without being assumed. Those are not small things. What it does not yet do is derive its central coupling, or bring its cosmological scaling and its Lorentz violation into contact with the measurements that most directly bear on them.