Orthogonal Energetic Channels from Microscopic to Macroscopic Scales: A QOS Bridge Between Quantum and Gravitational Phenomena

Orthogonal Energetic Channels from Microscopic to Macroscopic Scales: A QOS Bridge Between Quantum and Gravitational Phenomena Stefalo Acha1 1North Carolina A&T State University, Greensboro, NC, USA∗ This work formalizes an orthogonal energetic decomposition that operates consistently from microscopic to macroscopic scales within the Quantum Omni Synthesis framework. The implosive channel stores inward gravity weighted binding while the explosive channel encodes the total outward acting content that includes microscopic kinetic, radiation, degeneracy, turbulent and magnetic energies in addition to any established bulk rotation. The channels are orthogonal in the sense that their principal directions can be misaligned through an extended medium. The misalignment produces an interior torque that organizes a fraction of the explosive content into coherent spin and maintains rotation. At the quantum scale the same structure generates a pseudo spin and an intrinsic clock. At the macroscopic scale a dimensionless index connects centrifugal response to self gravity and a participation factor calibrates the projection from total explosive content to axial spin. The paper states a mapping between the channels and standard observables, derives a minimal two component field model and a set of virial based relations, and implements the macroscopic mapping on Earth and on the Moon. The analysis emphasizes that energetic drains or gains regulate internal clocks so that every object ages according to its energetic trajectory. Prior QOS manuscripts presented the two channel postulate and its consequences. The present paper consolidates those foundations and provides a cross scale demonstration intended to solidify the bridge between quantum mechanics and general relativity [1–6]. ∗sac[email protected] 2 I. INTRODUCTION The Quantum Omni Synthesis approach organizes dynamics into two orthogonal energetic channels. The implosive channel reflects inward binding that is gravity weighted. The explosive channel reflects outward change that is motion weighted and includes the microscopic and macroscopic energy carriers that support expansion or oppose contraction. The channels do not necessarily cancel. In extended media their lines of action can be offset. The offset yields a net torque that drives rotation. After a steady state is established the observed centrifugal effect is a kinematic consequence of rotation. The cause of rotation is the interior torque that arises from orthogonal forcing through structure. The present paper advances three objectives. First, it states a minimal formalism that is compatible with quantum kinematics and classical mechanics while retaining the orthogonal decomposition. Second, it defines a dimensionless index for the total orthogonality balance and a participation factor that connects centrifugal response to self gravity and projects total explosive content into coherent axial spin. Third, it implements the mapping on Earth and on the Moon and outlines falsifiable tests. The analysis complements prior QOS manuscripts where the two channel postulate was introduced and applied to scalar tensor actions, perturbations, and cosmological dynamics [1–6]. Position within the QOS corpus. The microscopic formalism and effective field theory context are developed in a companion PRD manuscript [5]. The compact object and general relativistic implications are addressed in a companion CQG manuscript [3]. The programmatic unification across scales is discussed in a companion EPJ Plus manuscript [4]. II. ORTHOGONAL FORCES PRINCIPLE AND TORQUE MECHANISMS Consider a continuous body with density ρ(r) and characteristic radius R. Let fimpl(r) denote an implosive force density and let fexp(r) denote an explosive force density. The net torque about a chosen axis is τ=ZV r×fimpl +fexpdV. (1) The spin evolution follows I˙ ω=τ−τdiss,(2) with Ithe axial moment of inertia, ωthe angular velocity, and τdiss the net dissipative torque that includes tidal, viscous, magnetic, and other sinks [7–11]. In a steady state ˙ ω= 0 and the driving torque balances dissipation. Once rotation is present the centrifugal acceleration at cylindrical radius ris ac(r) = ω2r=∂ ∂r 1 2ω2r2,(3) which is the radial gradient of the rotational part of the explosive energy per unit mass. In extended geophysical and astrophysical media, baroclinic misalignment, Maxwell stresses from large scale fields, and Reynolds stresses from organized turbulence are representative sources for τ. III. TOTAL EXPLOSIVE AND IMPLOSIVE ENERGIES ACROSS SCALES Work in a center of mass and co rotating frame. The explosive channel is the entire outward acting content Etot exp =Krot +ZVhuth +urad +udeg +uturb +u(+) magidV, (4) where Krot =1 2Iω2,uth is microscopic thermal kinetic energy density, urad =aT 4is radiation energy density when relevant, udeg is degeneracy energy density, uturb is macroturbulent kinetic energy density, and u(+) mag denotes the outward supporting component of magnetic energy. The implosive channel is the inward binding Etot impl =|Ugrav|+Eint bind,(5) where Ugrav <0 is the gravitational binding energy and Eint bind collects other inward bindings appropriate to scale. 3 Define a total orthogonality index σ2≡2Etot exp Escale impl , Escale impl ≡GM2 R.(6) Only a fraction of Etot exp needs to organize into coherent axial rotation. Model this by a participation factor χ∈[0,1], Krot =χ Etot exp.(7) With I=kMR2one obtains the spin law 1 2kMR2ω2=χσ2 2 GM2 R⇒ω=rχ kσrGM R3.(8) In the limit where the rotational term dominates Etot exp and χ=k, Eq. (8) reduces to the classical balance used in rotational geodesy. A. Virial correspondence and pressure form The scalar virial relation for a self gravitating object in a long lived state reads [26–28] 2Tmicro + 2 Krot +Wmag + 3 ZV P dV +Epress rad =|Ugrav|,(9) where Tmicro =Ruth dV ,Wmag is the net magnetic stress contribution, and Epress rad is the radiation pressure term. For an ideal gas one has Tmicro =3 2RP dV . In that case the pressure form gives 3ZV P dV ≃ |Ugrav| − 2Krot −Wmag −Epress rad ,(10) which exhibits the same separation into outward support that includes microscopic kinetic and rotation versus inward gravitational binding. IV. CLASSICAL ROTATIONAL BALANCE PARAMETER AND QOS INDEX Define the rotational explosive scale per unit mass and the gravitational implosive scale per unit mass by Erot exp m=1 2ω2R2,Egrav impl m=GM R.(11) The standard rotational balance parameter is q≡ω2R3 GM =2Erot exp Egrav impl .(12) Introduce the QOS orthogonality index for the rotational mapping ς2≡q, (13) so that ω=ςrGM R3,Erot exp m=ς2 2 GM R.(14) With I=kMR2, Krot =1 2Iω2=kς2 2 GM2 R.(15) Equation (12) is standard in geodesy and planetary rotation studies in another notation and can be measured from ω,R, and GM [17–21]. In the present framework qis the special case of the total mapping where Etot exp ≈Krot and where χ=k. 4 V. MICROSCOPIC MODEL AND PSEUDO SPIN CLOCK Consider a two component field Ψ = (ψimpl, ψexp)⊤with a quadratic Lagrangian density L=1 2h(1 −ς2)∂µψimpl∂µψimpl +ς2∂µψexp∂µψexpi−V(ψimpl, ψexp),(16) where Vcouples the channels. For a uniform background and a symmetric quadratic coupling the Euler Lagrange equations produce two normal modes with frequencies Ω±= Ω0g±(ς, λ) for positive functions g±. The relative phase advances at a constant rate and defines a pseudo spin clock. This construction preserves stability and hyperbolicity while keeping the orthogonal–channel coupling explicit, and it maps to quantum expectation values via the quantum virial statement when the potential has a power law form [12–15]. VI. MACROSCOPIC IMPLEMENTATION ON EARTH AND ON THE MOON Unless stated otherwise each body is treated as a rigid or quasi rigid rotator with an effective moment of inertia I=kMR2held constant over the interval of interest. A. Earth Adopt M≃5.97 ×1024 kg, R≃6.371 ×106m, GM ≃3.986 ×1014 m3s−2, sidereal ω≃7.292 ×10−5s−1, and k≃0.3308. Then q=ω2R3 GM ≈3.45 ×10−3, ac=ω2R≈3.39 ×10−2m s−2,(17) and Krot ≈1 2kMR2ω2≈2.13 ×1029 J.(18) These characterize the rotational mapping. The total explosive content Etot exp today is dominated by the integrated thermal reservoir with a subdominant rotational part. The participation factor χ= 2Krot/Etot exp can be inferred once an interior model provides Etot exp. The sign of the secular trend in day length is dominated by tidal torques at present which modify the coherent fraction rather than the total energy. B. Moon Adopt M≃7.342 ×1022 kg, R≃1.7371 ×106m, GM ≃4.905 ×1012 m3s−2, sidereal T≃27.3217 days with ω= 2π/T, and k≃0.393. Then q≈7.58 ×10−6, ac≈1.23 ×10−5m s−2,(19) and Krot ≈3.09 ×1023 J.(20) The small value of qreflects the weak centrifugal response of a locked rotator. The total explosive content is set by the thermal inventory with rotational energy negligible in the present budget. VII. CLOCK FACTOR AND AGING Aging is defined as the accumulated count of internal cycles along a trajectory through a two dimensional state plane. The horizontal axis tracks change of shape or position that is associated with kinetic processes and the explosive channel. The vertical axis tracks change of energy configuration that is associated with gravity weighted binding and the implosive channel. 5 A. State plane, orthogonality index, and local clock Let the state evolve along Γ : t7→ (x(t), y(t)).(21) For the rotational mapping define ς2(t) = q(t) = 2Erot exp(t) Egrav impl(t),(22) and define the local clock factor f(t) = 1−ς2(t)γeff (t),(23) where γeff collects kinematic contributions. B. Age as cycle count Let Ω0be the natural internal angular frequency in a reference configuration. The accumulated phase is ϕ(t) = Zt t0 Ω0f(τ)dτ, (24) and the age measured as cycles is A(t) = ϕ(t) 2π=Ω0 2πZt t01−ς2(τ)γeff (τ)dτ. (25) C. Energetic modulation and sensitivities From f(t) one has df dt =−γeff dς2 dt +1−ς2dγeff dt .(26) Since ς2= 2Erot exp/Egrav impl, dς2 dt =2 Egrav impl dErot exp dt −2Erot exp Egrav impl2 dEgrav impl dt .(27) If GM and Rvary slowly then dς2 dt ≃2R3 GM ω˙ω. (28) For the total mapping one may differentiate Eq. (8) and obtain the fractional sensitivity ˙ω ω=1 2˙χ χ+˙σ σ+1 2 ˙ G G+˙ M M−3˙ R R!.(29) Equation (29) shows that a drain of total explosive content decreases σand therefore decreases ωin the absence of strong changes in Mor R. Tidal torques act primarily through ˙χ < 0 at the present epoch for Earth. D. Trajectory formulation Let sbe an arc length like parameter with speed vs=ds/dt. Introduce a positive weight w(x, y). Then A[Γ] = Ω0 2πZΓ1−ς2(x, y)γeff (x, y)ds vs .(30) 6 VIII. TESTABLE IDENTIFICATION IN THE TOTAL ENERGY MAPPING Two statements can be tested. H1 (participation). A constant participation ratio χwithin a class maps total explosive content into coherent rotation through Eq. (7). H2 (rotational proxy). In bodies where rotational energy dominates Etot exp the rotational mapping with q=ς2 suffices and Krot tracks a class dependent proxy for Etot exp. A population that spans locked, slow, and fast rotators can be used to infer χand to check whether it is approximately constant within a class. Observations of day length trends and tidal power provide independent constraints on the exchange between coherent spin and other channels [22–25]. IX. DISCUSSION The orthogonal channels framework is consistent with standard mechanics once torque is recognized as the origin of spin. The centrifugal effect is an outcome of established rotation. The central rotational observable is q=ω2R3/(GM) which encodes the relative weight of rotational motion and self gravity. The total orthogonality index σand the participation factor χextend this mapping to the full explosive content through Eq. (8). The microscopic Lagrangian (16) displays the same orthogonality as a pseudo spin clock. The clock factor (23) ties energy loss to cycle accumulation and therefore to aging. The present derivations rely on body intrinsic quantities and do not invoke external orbital forcing beyond their role in dissipation (τdiss). Micro to macro equivalence and origin of spin. In the QOS view the pseudo spin at the quantum scale and the axial spin at the planetary scale are regulated by the same total energetic ratio. The total explosive channel comprises microscopic kinetic, radiation, degeneracy, turbulent and magnetic energies in addition to rotation, while the total implosive channel comprises gravitational and other bindings. Misalignment of these channels through an extended medium generates a torque that projects a fraction of the explosive content into coherent rotation. The relation Krot =χ Etot exp with ω=pχ/k σpGM/R3defines a direct mapping from the total energy ratio to the observed spin, which is the macroscopic echo of the pseudo spin rule at the quantum scale. Predictions and observational tests. For Earth the present day drain of total explosive content by internal heat loss is small compared with tidal torques, hence the sign and magnitude of the secular day length trend are dominated by the reduction of participation of energy in coherent spin. For the Moon the present state is tidally locked with very small rotational response and the integrated thermal content sets the explosive budget. Over geological spans a slow decrease of σincreases the clock factor and therefore increases the accumulated cycles at fixed coordinate time. The following quantitative rotational mapping holds for the present epoch: •Earth: q≈3.45 ×10−3,ac≈3.39 ×10−2m s−2,Krot ≈2.13 ×1029 J. •Moon: q≈7.58 ×10−6,ac≈1.23 ×10−5m s−2,Krot ≈3.09 ×1023 J. Falsification. Within a given class the participation ratio inferred from χ= 2Krot/Etot exp should be approximately constant. Large non systematic scatter of χwhen Etot exp is evaluated from independent interior models would falsify H1. When rotational energy dominates the total explosive budget the proportional mapping in H2 predicts that Krot tracks Etot exp up to a class dependent constant; failure of monotonicity at fixed structure would falsify this reduced mapping. X. CONCLUSION A two channel orthogonal energetic description has been developed and applied from microscopic pseudo spin to macroscopic axial rotation. A total orthogonality index and a participation factor connect the full explosive content to the observed spin. The classical rotational balance appears as a special case. A population based test has been proposed that can affirm or reject a constant participation ratio within a class and can evaluate when rotational energy is a sufficient proxy. The motivation is to consolidate a bridge between quantum mechanics and general relativity and to demonstrate that microscopic structure replicates at macroscopic scale through orthogonal energetic channels. 7 Body q=ω2R3 GM ac=ω2R[m s−2]Krot [J] Earth 3.45 ×10−33.39 ×10−22.13 ×1029 Moon 7.58 ×10−61.23 ×10−53.09 ×1023 TABLE I. Rotational mapping parameters for Earth and the Moon at the present epoch. Values use standard geodetic constants and representative shape factors k. APPENDIX A: GRAVITATIONAL BINDING ENERGY AND INTEGRALS The exact gravitational binding energy is Ugrav =−ZR 0 G m(r) rdm(r) = −4πG ZR 0 m(r)ρ(r) rr2dr, (31) with m(r) = 4πRr 0ρ(r′)r′2dr′. For a uniform sphere Ugrav =−3 5GM2/R. For a layered body the integral is evaluated layer by layer with ρ(r) taken from an interior model. The scale Escale impl =GM2/R used in Eq. (6) is an order of magnitude proxy that preserves the correct scaling with Mand Rand that is sufficient for population level tests. APPENDIX B: TOTAL EXPLOSIVE CONTENT IN COMMON REGIMES B1. Ideal gas dominated layers For an ideal monoatomic like layer with adiabatic index γ≃5/3, uth =3 2P, Eth =Zuth dV =3 2ZP dV. (32) Hydrostatic equilibrium gives dP/dr =−ρGM(r)/r2and P(R) = 0. Then ZR 0 P4πr2dr =ZR 0"ZR r ρ(r′)GM(r′) r′2dr′#4πr2dr, (33) which evaluates to a fraction of |Ugrav|for simple profiles and equals 1 3|Ugrav| − 2Krot −. . . by the virial relation. B2. Degenerate layers For a nonrelativistic fully degenerate Fermi gas udeg =3 5nEF, EF=ℏ2 2m(3π2n)2/3,(34) with number density n. For a relativistic degenerate gas udeg =3 4nEFwith EF∝n1/3. These are integrated over the layer volume to obtain the degeneracy contribution to Etot exp. B3. Magnetic and radiation support The magnetic energy density is umag =B2/(2µ0). Only the outward supporting component enters u(+) mag. Radiation energy density is urad =aT4with a= 4σSB/c and radiation pressure contributes in the virial equation as Epress rad = R3Prad dV =Rurad dV . 8 APPENDIX C: SENSITIVITY, UNITS, AND CONSISTENCY CHECKS C1. Units The index σ2is dimensionless. Since Etot exp is in joules and Escale impl =GM2/R is in joules, σ2is unitless. The spin law (8) has dimensions of s−1because pGM/R3has units s−1. C2. Differential checks From Eq. (8), ∂ln ω ∂ln χ=1 2,∂ln ω ∂ln σ=1 2,∂ln ω ∂ln M=1 2,∂ln ω ∂ln R=−3 2.(35) A small fractional drain δE of total explosive content with fixed Mand Rgives δω ω≈1 2 δσ σ≈1 4 δEtot exp Etot exp .(36) C3. Earth order of magnitude Let Etot exp be dominated by a thermal reservoir of order 1031 J with Krot ≈2×1029 J. Then χ∼2Krot/Etot exp ∼0.04. A net annual drain of 1.5×1021 J changes σby order 10−10 per year and gives a microsecond level annual change in day length. This is smaller than the observed tidal contribution, consistent with the present interpretation. APPENDIX D: INFERENCE WORKFLOW FOR χAND σ 1. Specify a layered model {ρ(r), T(r), B(r),EOS}. 2. Compute Etot exp from Eq. (4). 3. Compute Escale impl =GM2/R and σfrom Eq. (6). 4. With observed (ω, M, R) and estimated k, compute Krot =1 2kMR2ω2and infer χ= 2Krot/Etot exp. 5. Test H1 within a class by examining the dispersion of χversus structural indicators. NOMENCLATURE GGravitational constant. MMass of the body. RReference radius of the body. GM Standard gravitational parameter. ρ(r) Mass density field. fimpl Implosive force density. fexp Explosive force density. τNet torque about the spin axis. τdiss Net dissipative torque. IAxial moment of inertia with I=kMR2. kShape factor. ωAngular velocity vector. ωMagnitude of angular velocity. ac(r) Centrifugal acceleration at radius r. Egrav impl/m Gravitational implosive scale per unit mass. 9 Erot exp/m Rotational explosive scale per unit mass. qRotational balance parameter ω2R3/(GM). ςRotational orthogonality index with ς2=q. Etot exp Total explosive content. Etot impl Total implosive content. σTotal orthogonality index. χParticipation factor. Krot Coherent rotational energy. Uint Candidate internal energy. cpSpecific heat at constant pressure. TRotation period or temperature by context. γeff Effective kinematic factor. fLocal clock factor. ACKNOWLEDGMENTS The author acknowledges discussions in prior QOS manuscripts that introduced the orthogonal channels and motivated the present consolidation. REFERENCES [1] S. Acha, Superposition of Implosive and Explosive Energetics in the Quantum Omni Synthesis Framework, Zenodo, DOI 10.5281/zenodo.17195560. [2] S. Acha, Quantum Omni Synthesis: A Field Theoretical Framework with a Quantized Gravity Coupling Parameter, Zenodo, DOI 10.5281/zenodo.17179911. [3] S. 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