The Quantum Mechanical Mechanism Behind The End Results of The GTR: Matter Is Built On The Lorentz Invariant Framework Energy x Mass x Length**2 ~ h**2

Scientific Paper
Title	The Quantum Mechanical Mechanism Behind The End Results of The GTR: Matter Is Built On The Lorentz Invariant Framework Energy x Mass x Length**2 ~ h**2
Read in full	Link to paper
Author(s)	Tolga Yarman
Keywords	Mass Deficiency, Special Theory of Relativity, General of Theory of Relativity, Quantum Mechanics, Metric Change, Muon
Published	2008
No. of pages	30

Read the full paper here

Abstract

In a previous article, we have provided a whole new approach toward the end results of the General Theory of Relativity (GTR), based on just the energy conservation law, in the broader sense of the concept of "energy" embodying the mass & energy equivalence of the Special Theory of Relativity (STR). Thus, our approach was solely based on this latter theory (excluding the necessity of assuming the principle of equivalence of Einstein). According to our approach, the rest mass of an object embedded in a gravitational field (in fact in any field the object interacts with) decreases as much as the binding energy coming into play. Thereby, based on a general quantum mechanical theorem we prove, its internal energy, weakens as much; thus the classical red shift and time dilation. This theorem (we did not have any room to provide a general proof of, previously), basically says that, if in a relativistic or non-relativistic quantum mechanical description, composed properly, the mass of the object in hand is multiplied by an arbitrary number γ, then the total energy of it, is multiplied by γ, and its size is divided by γ. This number however may very well not be arbitrary. For example, it would specify how much the rest mass of the object is altered when this is embedded in a gravitational field, leading via quantum mechanics, strikingly at once, to the end results of the GTR. This manipulation further yields the invariance of the quantity [energy x mass x size²]. We conclude that, it is this quantum mechanical invariance, necessarily strapped to the square of the Planck Constant, which constitutes a given framework regarding the matter architecture, and insures the end results of the GTR.

Not only that our approach is incomparably simple as compared to the GTR, but it also avoids all incompatibilities (such as the breaking of the relativistic relationship E=mc²), or inconsistencies (such as the breaking of the energy conservation law, as well as the breaking of momentum conservation law), or blockades (such as the impossibility of the quantization of the gravitational field), thus opens a whole clean avenue toward a unification of fields, and understanding of the matter and the universe at all levels, with just the same set of tools.

Since we do not have to use the principle of equivalence of the GTR, amongst others, we could show that, just like the gravitational field, the electric field too slows down the internal mechanism of a clock, had this interacted with the field. This result explains substantially, the retardation of the decay of the muon, bound to a nucleus.

Overview

Tolga Yarman here supplies the quantum-mechanical proof that his earlier papers had asserted without room to demonstrate. His programme replaces general relativity's geometrical machinery with a single principle drawn from energy conservation together with the mass–energy equivalence of special relativity: the rest mass of an object bound in any field decreases by exactly the binding energy that comes into play. Lift a hydrogen atom in a gravitational field and you must supply energy to it; by E = mc² that energy raises its rest mass. Lower it, and the mass falls. Because the atom's internal dynamics — its "clock" — is built out of that mass, the clock must run slow when bound, and the light it emits must be redshifted. Yarman derives the gravitational redshift and gravitational time dilation this way, without ever invoking curved spacetime and, he stresses repeatedly, without assuming the principle of equivalence.

The paper's technical core is a theorem he numbers 5: if in a properly composed quantum mechanical description all masses are multiplied by an arbitrary factor γ, then the total energy is multiplied by γ and the characteristic size is divided by γ. The immediate corollary is that the product E₀M₀R₀² is invariant under such a rescaling, and since it is Lorentz invariant and must be tied to a Lorentz invariant universal constant, it is "strapped to" h², the square of the Planck Constant. Yarman calls the relation [energy × clock mass × (clock space size)²] ~ h² the UMA (Universal Matter Architecture) Cast, and treats it as the structural reason why matter delivers the end results of both special and general relativity. Because the derivation nowhere mentions gravity specifically, he extends it to electric and nuclear binding, and predicts that a muon bound in the electric field of a nucleus must have its decay retarded exactly as if it were in a gravitational well.

The argument

Rest mass tracks binding energy

Yarman works in the limit where the bound particle is negligible compared to the host, so the host is unaffected. His parable is a stone dropped from a raised hand and caught by a lower one: since Earth does not move appreciably, only the stone gains kinetic energy, and the energy the catcher removes comes from the stone alone. Raising the stone conversely deposits energy in it. For a single hydrogen atom, that deposited energy must show up as increased rest mass, "and wherever this mass intervenes, we will observe a related change".

Writing the reduced mass of the electron–proton system as μ_0∞ in empty space, he sets the binding energy E_B(R) = [μ_0∞ − μ₀(R)]c², and equates it to first order with the Newtonian GMμ_0∞/R. Treating the mass change as continuous during quasistatic lowering gives the rigorous form

μ₀(R) = μ_0∞ e^−α(R),   α(R) = GM/Rc²

He insists that the electron charge and h remain untouched in any field, both being Lorentz invariant, but argues that the gravitational constant G is not Lorentz invariant — "so it is not as 'universal' as one may think it is" — since only the product GM'm has the invariant character of a squared charge.

Redshift and clock retardation

Since the Bohr total energy E_n∞ = −2π²e⁴μ_0∞/h²n² is proportional to the reduced mass, a fractional mass change gives an identical fractional change in every transition frequency:

Δν_n→m/ν_n→m = Δμ₀/μ_0∞ = −E_B(R)/μ_0∞c²

which is the gravitational redshift. Turning to periods, he invokes de Broglie's hν₀ = m₀c² for the intrinsic periodic phenomenon of a mass at rest, so that the associated period T₀ stretches by 1/[1 − E_B/m_0∞c²]. He checks this against the Bohr orbital period T_e∞ = h³/(4π²e⁴m_e∞), which is inversely proportional to the electron mass and so stretches by the same factor. And since c is unchanged while frequency drops, the corresponding wavelength — the "clock space size" — must stretch equally. A footnote flags that this is "the opposite of what the general theory of relativity establishes; in this latter theory indeed, when embedded in a gravitational, masses increase, and lengths contract."

He then applies the same reasoning to radioactivity. For alpha decay, T_α∞ = 2 ln2 m_α∞R_∞²e^γ/h, and he argues in a footnote that the barrier transmission coefficient e^γ is unaltered, because the masses M_∞ and m_α∞ change together, the lengths R_∞ and r_0∞ change together, and since lengths and periods both stretch, velocities are unchanged. The half-life therefore stretches by the same universal factor.

Why Coulomb and Newton must go as 1/r²

Appendix A argues that the inverse-square form is not empirical but imposed by special relativity. Yarman notes that H = force × mass × length³ has the dimensions of h² and is therefore Lorentz invariant, and that for a dipole in motion along the line joining its poles, the product (mass × length) is invariant because mass dilates and length contracts by the same factor. Assuming Coulomb's force goes as qQ/r₀ⁿ and demanding invariance of H then forces n = 2. The same argument transfers to Newton's law once one notes that GMm, not G alone, is the invariant. He takes this to mean both force laws are as universal as the special theory itself, and that any other force law — Yukawa's, or the weak force governing beta decay — must be built the same way.

The field, and the electron's internal dynamics

Yarman revises the field concept along the way. Force is the measurable primitive; field is "only an extended concept" and cannot be measured. The total relativistic energy of two masses or charges is "not anyway materialized by the surrounding space, but only by the 'internal dynamics' of the charges of concern". He also reports (citing his own earlier work) that Coulomb's force on a moving test charge is reduced by the factor (1 − v₀²/c₀²)^1/2, contrary to the usual assumption.

This leads to his sharpest structural claim. A point cannot be a material being, so the electron cannot be a point: it must have an internal dynamics, and perhaps "its 'mass' is simply the 'internal energy' of the 'electric property' which we call 'electric charge'". A bound electron is therefore not the same object as a free one — "one cannot make an omelette, and keep the eggs as they are, prior to cooking". The muon makes this testable, because its internal dynamics is directly legible in its decay rate.

Theorem 5 and the UMA Cast

Appendix B proves the theorem on the time-independent Schrödinger equation for J nuclei and I electrons with Coulomb potentials, and then on the Dirac equation. The essential restriction is that the description must exclude "synthetic potential energies": the potential must carry a 1/r₀ background dependence, as Coulomb and Yukawa do, these being the two forms yielded by the Klein–Gordon equation built on p²c² + m_0∞²c⁴ = E². Multiplying all masses by γ then multiplies the eigenvalue by γ and divides the characteristic length by γ, whence E₀M₀R₀² is invariant. Yarman stresses that this is not dimensional analysis: the quantity would fail to be invariant under mass rescaling if the potentials were not of the STR-compatible form, even though the dimensions would be unobjectionable either way.

He then offers an empirical illustration, Figure 1: for ten alkali and hydrogen diatomic molecules (H₂, Li₂, LiNa, Na₂, NaK, K₂, KRb, Rb₂, RbCs, Cs₂) the lowest classical vibrational period is plotted against M₀^1/2r₀²/(r₀/r₀₀)^1/2 and found to lie on a line, T₀ ~ M₀^1/2r₀². He reads this as the matter architecture showing itself directly, so that "one can test the validity of the STR, already at rest".

The paper closes by claiming its approach avoids the breaking of E = mc², of energy conservation and of momentum conservation, permits quantization of the gravitational field, and generalizes to electric and magnetic dipoles in their respective fields — and by remarking that it "must take quite a captivation" that neither Dirac nor the founders of quantum mechanics ever thought to alter the rest mass of a bound electron.

Assessment

The paper's central identity is genuine and rather pretty. For the hydrogen atom, E_n ∝ μ and r_n ∝ 1/μ, so E_nμr_n² is independent of the reduced mass, and Yarman's own footnote gives the exact statement −8π²μ_0∞r_n∞²E_n∞ = n²h². That scaling is correct, it is not widely displayed in this form, and packaging it as a "matter architecture" is a legitimate pedagogical move. Theorem 5 is likewise a correct statement of a scaling property of the Coulombic Schrödinger problem: multiplying every mass by γ with fixed e and h does scale energies by γ and lengths by 1/γ, and the corresponding empirical regularity for the alkali diatomics is a real check rather than a rhetorical one. The insistence on tracking where binding energy physically resides is also a fair criticism of loose textbook language, and the observation that GMm rather than G carries the invariant character is worth stating.

The difficulties begin where the scaling theorem is asked to carry physical rather than mathematical weight. The premise that a bound object's rest mass decreases by its binding energy is asserted, not derived. What energy conservation requires is that the system — atom plus source — has a mass deficit equal to the binding energy; how that deficit is distributed between constituents and the field is precisely the question, and Yarman simply assigns all of it to the small body and none to the field or the interaction, having declared that fields carry no energy. That declaration is not defended; it is stipulated in the section on the field concept and then used throughout. Nor is the mechanism by which a uniform rescaling of the electron and proton masses could be produced by an external gravitational potential ever exhibited. The theorem says if masses are rescaled, then the clock slows; the physics is entirely in the "if".

The consequence is a prediction the paper does not confront: the mass shift is a local property of the atom, so two different kinds of clock — one whose rate depends on the reduced mass in one way, one in another — need not shift alike, and more importantly, an observer deep in a potential well would find his own atomic constants altered. This is a violation of local position invariance, which is directly constrained by null gravitational redshift experiments — comparisons of clocks based on different transitions (hyperfine versus optical, or different atomic species) as the solar gravitational potential varies over Earth's orbit — which bound any such species-dependent shift at levels many orders of magnitude below what a wholesale rescaling of rest masses would produce. Yarman's footnote asserting that GR has "masses increase and lengths contract" in a gravitational field also misdescribes the theory he is displacing: general relativity assigns no change to a locally measured rest mass at all, and the redshift there is a statement about comparing clocks at different potentials, not about altered local structure. Because the paper never derives the light deflection or perihelion precession here — it refers those to an earlier article — its claim to reproduce "all of the measurable end results of the GTR" cannot be assessed from this text.

Several supporting arguments are weaker than presented. The Appendix A derivation that n must equal 2 is dimensional analysis, whatever the disclaimer: force × mass × length³ has the dimensions of h² identically, and the "invariance" step uses the frame-dependent statement that mass dilates and longitudinal length contracts by reciprocal factors, which cannot by itself select a dynamical exponent. The claim that Coulomb's force on a moving test charge is simply reduced by (1 − v²/c²)^1/2 is not what the Lorentz transformation of the field gives; the force on a moving charge acquires a magnetic contribution and is direction-dependent, not scaled by a single scalar. And the barrier-penetration argument for alpha decay asserts that e^γ is unchanged from a cancellation of ratios, but γ contains Z'z'c/V and R/r₀ terms whose separate invariance is stated rather than shown.

Finally, the muon result — the paper's one claimed experimental confirmation — is not the untouched territory Yarman describes. The decay rate of a negative muon bound in an atomic orbit is indeed reduced relative to the free rate, but this is a long-standing and quantitatively computed effect of standard theory (the Huff correction), arising from the bound muon's relativistic orbital motion and the modified phase space, and it is routinely accounted for in muon capture experiments. That the effect exists is therefore not evidence for the mass-deficiency mechanism specifically, since an established calculation already reproduces it; distinguishing the two would require showing where the predictions differ numerically, which the paper does not attempt. It also promises the muon and beta-decay treatment to a later article, so the reader is left with the alpha-decay case alone. Within its own terms the paper is careful and internally consistent, and its scaling identity deserves to be better known; as a replacement for general relativity it rests on a stipulation about where binding energy lives that it never argues for.

See also

• Tolga Yarman
• General Relativity
• Special Relativity
• Equivalence Principle
• Planck Constant
• Coulomb's Law
• Muon
• Redshift
• Time Dilation
• Mass