Unified Electromagnetic and Gravitational Interactions from Phase Harmony and Fermi-Walker Transport

Unified Electromagnetic and Gravitational Interactions from Phase Harmony and Fermi-Walker Transport Donatello Dolce University of Camerino, Piazza Cavour 19F, 62032 Camerino, Italy We show that standard electromagnetism can be obtained from local spacetime geometrodynamics, without postulating gauge invariance, in parallel with standard gravitational geometrodynamics. According to the Hamilton–Jacobi optomechanical analogy (“phase harmony”), interactions admit a dual reading: dynamically, as local changes of four-momentum; geometrically, as local modulations of spacetime recurrences. Spacetime curvature induces redshift and ruler deformation, resulting in gravitational interaction. In parallel, position-dependent local isometries, implemented as Fermi–Walker transport of tetrads, induce precession in particle’s dynamics. The corresponding abelian holonomy defines a U (1) connection and yields electromagnetism, with the Lorentz force appearing as a Coriolis-like effect. This also introduces a S1 (particle’s internal clock) fiber attached to every spacetime point and thus Maxwell kinematics from a purely 4D reinterpretation of the Kaluza-Klein mechanism. Electromagnetic effects, such as Larmor/Thomas precession, Zeeman effect, synchrotron radiation, gravitomagnetism, are consistently described within this geometrodynamical scheme. Since Newton’s principia, interactions are introduced in physics as deviations from uniform motion, producing accelerations on bodies. There is, however, a dual and equally classical description rooted in the Hamilton–Jacobi optomechanical dualism (HJ), [ 1 – 3 ]. Any Hamiltonian system admits an equivalent undulatory description in which the energy and the momentum are proportional to a frequency and wavenumber, hence to the inverse of an instantaneous time periodicity and wavelength, Tand λ, respectively. Its most familiar application is the undulatory mechanics underlying quantum theory, where the proportionality constant h is the Planck constant (we adopt c = ℏ = 1). For a free particle, the four-momentum pµ (tangent covector) and the instantaneous spacetime periodicity τµ = {T, λ} (tangent contravariant vector), form a relativistic invariant pµτµ = 2 π , named here Phase Harmony (PH); e.g. as under Lorentz transformations Λ, where pµ→p′ µ= Λ ν µpνand τµ→τ′µ= Λµ ντν. As a consequence, in the rest frame, the mass m fixes an intrinsic recurrence of Compton period TC in the proper time: p′ µτ′µ = mTC = 2 π . This is known as particle’s Compton (de Broglie) internal clock, [ 4 ]. In this sense, every elementary particle can be regarded as an elementary clock, [ 5 ] and [ 6 ], with (ultrafast) S1 -valued phase θ advancing along its worldline — a further justification is in Rovelli’s statistically motivated observation that generally covariant systems should admit “internal times”, [ 7 , 8 ]. Here, however, we are not interested in quantum effects. The quantum aspects are not relevant for the classical-relativistic derivation of the Electromagnetic (EM) geometrodynamics developed in this work — they are only briefly reviewed in app.(A) for completeness. In this work we remain purely classicalrelativistic and treat ℏ as a conventional constant for most of the paper (set to one for convenience). In general, HJ prescribes that any symplectomorphism (e.g. canonical transformation) makes the local changes of pµ→p′ µ ( x )and τµ→τ′µ ( x )co-modulate so as to preserve PH, [ 9 ]. In fact, writing the canonical 1-forms dθ = pµdxµ with symplectic (closed) 2forms ddθ = dpµ∧dxµ (Darboux’s theorem), Stokes’ lemma implies that the phase accumulated, after any fundamental recurrence interval Tµ ( x )anchored at base point x, is invariant along the symplectic flow: ITµ(x) dθ =p′ µ(x)τ′µ(x) = pµτµ= 2π . (1) The instantaneous spacetime periodicity τµ ( x ) evolves along the symplectic flow, precisely mirroring the local deformation of the manifold chart, and thereby guarantees causality and locality in the undulatory description. For example, in the linear approximation, gravitational interaction can be consistently derived in terms of PH — thus Einstein’s equation recovered by requiring general invariance [ 10 ]. In a weak gravitational field Υ( x ), the local energy shift p′ 0 ( x ) ≃ (1+Υ( x )) p0 implies gravitational redshift τ′0 ( x ) ≃ (1 − Υ( x )) τ0 through PH. Likewise, the spatial contractions adjust with the local momentum so that p′ µ ( x ) τ′µ ( x ) = 2 π is preserved along the motion. The curved spacetime manifold (e.g. Schwarzschild) directly encodes the local spacetime modulations of ruler and clocks as dual geometrodynamical manifestation of gravitational interaction. The metric tensor, however, doesn’t encode all possible local modulations of τµ ( x ). For instance, Local Lorentz Transformations (LLTs) are position dependent isometries, yet they locally transform τ′µ ( x ) and p′ µ ( x ), producing Fermi–Walker (FW) transport and precessions [ 11 , 12 ] naturally described by the Ehresmann connection on the tangent space, whose holonomy can generate nontrivial phases, [13]. 2 We introduce a specific formalism to investigate PH under LLT. In fact, ordinary point-particle Hamiltonian formalism misses undulatory mechanics. On the other hand, in ordinary field theory — where gauge invariance is postulated rather then inferred from geometry [ 14 ] — a field is a superposition over all momentum modes ϕp ( x )— an “integral” over all global frames. LLTs thus merely reshuffle field modes, hiding the geometrodynamical origin of gauge interactions, as we will see. The useful formalism is provided by Elementary Cycles Theory (ECT), [ 8 , 9 , 15 – 23 ], which in this paper is exclusively used to isolate a single on-shell field mode ϕp by means of Periodic Boundary Conditions (PBCs) and to track its response to LLTs: ϕp→ϕ′ p′ . Implemented kinematically by FW transport of local tetrads, the induced co-modulations of τµ→τ′µ ( x ) and pµ→p′ µ ( x ), forces a compensating internal transformation of the field mode, ϕ′ p′ ( x )  = ϕp ( x ), necessary to preserve PH, [15]. The modulations τ′µ ( x )effectively describe local precessions of an internal gyroscope when projected on the plane of the local frame that carries the S1 phase θ , [ 24 – 26 ]. Then, the internal transformation identifies an U (1) abelian connection which is precisely the EM gauge field Aµ ( x ). EM is therefore the bookkeeping connection that compensates the FW induces precession of the particle’s internal clock, whose U (1) holonomy reproduces the Lorentz force (Coriolis-like). In par.(II), we develop the geometrodynamical construction of EM in flat spacetime and derive the Lorentz force, gauge invariance and Maxwell equations. The generalization to curved spacetime, where EM emerges along with gravitational interaction, is given in par.(III). In par.(IV), we provide a 4D interpretation of the Kaluza–Klein (KK) mechanism [ 33 , 34 ], showing that the Maxwell kinetic term emerges directly from geometry, thereby reinforcing the geometrodynamical origin of EM. Par.(V) collects examples and applications (Thomas/Larmor precession, Zeeman, synchrotron, etc.). App.(A) briefly presents quantum implications and phenomenology, with faithful references to the existing literature, [8, 9, 15–23]. From a conceptual viewpoint, the present framework is close in spirit to geometrodynamical unification attempts such as Rainich–Misner–Wheeler, [ 28 , 29 ], where EM is “already” contained in the spacetime curvature, and to gauge-theoretic approaches such as those of Utiyama and SciamaKibble, [ 30 , 31 ] where gravity itself is understood as a gauge theory of the local Lorentz/Poincaré group, see [ 27 ] for a review. The crucial difference is that here the starting point is internal S1 dynamics associated to the particle, which induces an abelian holonomy in the Lorentz frame bundle. Our framework, also share analogies with “double copy” constructions that relate gauge theories to gravity amplitudes [32, 43]. I. PHASE HARMONY AND FORMALISM. We consider a scalar charged particle of mass m , free of any interaction in a classical-relativistic field description — flat spacetime ηµν = diag (+ ,−,−,− ). Then we select, in every point x , the single mode ϕp ( x )of four-momentum pµ by imposing related PBCs, [15], ϕp(xµ) = ϕp(xµ+Tµ),(2) such that pµTµ= 2π: SF ree =ITµ d4xL(∂µϕp, ϕp;m),(3) where HTµ represents the PBCs eq.(2) relative to the interval Tµ anchored at generic base point x , and the free Klein-Gordon Lagrangian density is L=1 2∂µϕ† p∂µϕp−m2ϕ† pϕp.(4) This means to isolate the field mode ϕp associated to the specific global frame denoted by the particle momentum p. Throughout all this work we however retain only the fundamental mode of the harmonic spectrum resulting from the PBCs — the physical interpretation of the higher modes is briefly reviewed in app.(A). Eventually, the result of ordinary fields is reconstructed by integrating over all the possible fundamental modes ϕp ( x ),i.e. an integral over the spatial momentum p. In this free case the spacetime periodicity τµ is constant at any point x of the system evolution, and equal to the spacetime recurrence Tµ = τµ . The fundamental mode, solution of SF ree , eq.(2), is thus: ϕp(x) = Ne−ipµxµ,(5) where Nis a normalization factor. It is well known from string or eXtra Dimensional (XD) theories [ 33 – 36 ] that PBCs, or more in general combinations of Dirichlet or Neumann BCs, is a perfectly allowed choice to minimize the action at the boundary. The adherence of this PBCs to the principle of stationary action is the fundamental property that guarantees PH, as well as locality and causality, under any canonical transformation of the theory, through Stokes’ lemma, [8, 9, 15–23]. 3 II. GEOMETRODYNAMICAL ORIGIN OF ELECTROMAGNETISM, WITHOUT GRAVITY. Now we switch on (EM) interaction by introducing in each point x an infinitesimal LLT x→x′ ( x ), [ 15 ], with ea µ(x) = ∂x′a(x) ∂xµ.(6) This is a position dependent isometry preserving the flat metric ηµν →η′ µν(x) = ea µ(x)eb ν(x)ηab =ηµν .(7) No gravitational interaction is introduced. Fermi-Walker transport. — The resulting flat yet locally twisted spacetime identifies a congruence of particle’s wordlines with unit time-like field of velocity uµ ( x ) = eµ 0 ( x ). We thus assume that the full tetrad ea µ ( x )is FW transported along each worldline with related, not vanishing field of acceleration aµ(x)=(u·∂)uµ,(8) where u·u= 1 and u·a= 0. The FW transportation law (u·∂)eµ a= Ωµ νeν a,(9) is described by the antisymmetric local generator (Lorentz pure boost), [11, 37]: Ωµν=uµaν−aµuν.(10) Gyroscope. — We restrict our analysis to the abelian component of the generator Ω. For instance this can be achieved by projecting onto a precession plane, characterized by a unit bivector nab = 1 2ϵijea ieb j , with i, j = 1 , 2and ϵ12 = +1, orthogonal to the four-velocity ua ( x ), [ 24 , 37 ]. In the following we thus set ¯ Ω = nabΩab .(11) In other words, the only purpose of this hypothesis is to restrict our study to a unique SO (2) ≃U (1) subgroup associated with the precession of the internal S1 clock phase θ induced by the FW transport, in direct analogy with the precession of a gyroscope projected on a plane orthogonal to the motion, [25, 38, 39], see par.(VI). Induced internal transformation. — Solving the evolution law eq.(9) along the path γ : x0→x and considering that the transformation is infinitesimal, at first order we obtain eµ a(x)≃δµ a+Zγ (u·dy)¯ Ωµ a,(12) where we have aligned the local frame with the effective coordinates at point x0:eν a(x0) = δν a. From the substitution of variables eq.(6) in the free action eq.(3), we see that the LLT induces an infinitesimal rotation of the boundary Tµ→T′µ ( x ) while the Lagrangian density remains invariant and locally flat √−g≃ 1in the neighbors of x (within the limited integration region of the action). Up to the projection on the precession plane eq.(11) of the LLT eq.(6), the transformed action is, [15]: SF ree →SEM =IT′µ(x) d4xL(∂µϕ′ p′, ϕ′ p′;m).(13) The key quantity for implementing PH is the locally rotated tangent (instantaneous) four-periodicity Tµ→τ′a(x) = ea µ(x)Tµ ≃δa µ−Zγ (u·dy)¯ Ωa µTµ(14) rather than the recurrence interval T′µ ( x ). Upon projection onto the precession plane, the modulation of τ′a ( x )formally describes the FW induced local axis precession of an internal gyroscope associated to the particle. In ECT, because of the locally transformed PBCs, we see that LLT induces an effective internal transformation in the field mode ϕ′ p′, solution of eq.(13), ϕ′ p′(x)=ϕp(x),(15) in contrast with the ordinary field description, where local distortions of the flat metric do not induce any internal transformation of the field itself, owing to the absence of boundaries. In fact, in our formalism each field mode is a physical standing wave whose instantaneous four-periodicity is directly determined by the local PBCs of the action. Due to the local rotation of boundary Tµ→T′µ ( x )in the transformed action, the transformed solution ϕ′ p′ along the path γis ϕp(x)→ϕ′ p′(x) = ϕ0e−iRγp′(y)·dy ,(16) where ϕ0 = ϕp ( x0 ). The transformed field solution is, in fact, a locally modulated spacetime wave of instantaneous four-periodicity τ′a(x): i∂aϕ′ p′=p′ a(x)ϕ′ p′,(17) where the four-momentum p′ a ( x ), characterizing our interaction scheme, is locally determined by PH, Tµpµ=τ′a(x)p′ a(x)=2π , (18) resulting from the modified PBCs. 4 Thus, the locally transformed four-momentum p′ a(x)can be written as: pµ(x)→p′ a(x) = eµ a(x)pµ := pa−eAa(x).(19) It is interesting to notice that this formalism provides a kind of holographic description in which interaction, i.e. p′ a ( x ), is encoded in the local transformations of the boundary T′µ ( x )of the theory through PH, eq.(1), [15]. Here we have introduced an effective potential eAa(x)≃ −pµZγ (dy ·u)Ωµ a,(20) at first order in Ω, so that we formally get the minimal substitution of EM, eq.(19). In the rest of the paper we will check whether this effective potential shares properties with the ordinary EM field. The transformation of the field mode eq.(16) can be thus written as ϕp(x)→ϕ′ p′(x) = V(x)ϕp(x),(21) which now includes a path-dependent Wilson line as effective internal transformation, [15], V(x) = eiRxAa(y)dya.(22) Lorentz force and field strength. — From the generator of the FW transport eq.(10) we find pµ¯ Ωµ a=muµ(aµua−uµaa) = −maa.(23) Since the local frame is fixed in the point of reference x0 , we have ˙ua = aa . The integration of eq.(20) along the worldline yields a gauge flow aligned to the velocity field: Aa(x)≃m eua(x).(24) We are not describing a single worldline but a congruence of motions. In the FW frame, each point carries its own four-velocity ua ( x )(in each point there is associated a clock when a particle passes there). Since the vorticity of the field uµ ( x )is in general not vanishing, [ 25 ], its gradient generates a non-vanishing EM field strength Fab =∂aAb−∂bAa=m e(∂aub−∂bua)= 0 ,(25) and the Bianchi identity is automatically satisfied. Along the test worldline x′ ( s )with velocity u′ = ˙x′ , these dynamics yields the ordinary Lorentz force as Coriolis’, mdu′ µ ds =eFµνu′ν,(26) where the physical kinetic momentum is p′ a ( x ) = mu′ a(x). Gauge invariance U (1) and holonomy. — We want now to extend our description to a generic accelerated frame, with generic local Lorentz generator Ω ′′ , by adding a spatial rotational part of angular frequency ωµto the FW transport Ωeq.(10), [11]: Ω→Ω′′ = Ω + Ω(SR).(27) The local spatial rotation has generator Ω(SR)µ ν=ϵµνρσuρωσ.(28) Upon projection on the precession plane, the rotational part, ¯ Ω(SR) , adds a total derivative term to the gauge field eq.(20), Aa→A′′ a=Aa+∂aχ , χ =ξµpµ,(29) where ξµis a Killing vector such that e∂aξµ=−Zγ (dy ·u)¯ Ω(SR)µ a.(30) The field mode thus acquires a local phase ϕ′ p′(x)→ϕ′′ p′(x) = U(x)ϕ′ p′(x),(31) where U(x) = eieχ(x)∈U(1) ,(32) describes a U (1) gauge orbit of the physical gauge Aa(x)generated by FW. In our construction, the FW part corresponds to a position dependent isometry of the local tetrad, but in general it cannot be written in terms of a global Killing vector. As a consequence, its projection on the clock plane carries a nontrivial U (1) curvature responsible for the physical gauge field Aa ( x ). On the other hand, the rotational part is described by a Killing vector and yields a gauge invariant contribution on the orbit of the physical gauge Aa(x). This can be also seen from the fact that purely rotational part is itself a U (1) holonomy. Its contribution to the transformed boundary T′µ ( x ) →T′′ µ ( x )is a total derivative with no effect on the PBCs. It generates a path-dependent phase that can be absorbed by a U(1) gauge transformation. Spin connection. — The result above can be interpreted in the following way. The transport law induced by LLT, together with PH relating precession to local variation of four-momentum, implies atangent bundle in any point, perfectly mirroring the fiber bundle postulated in ordinary gauge theory for the “internal” vector space. In this way we see that the spin-connection ωijµ of the tangent bundle, projected on the precession plane, identifies the 5 abelian gauge field Aµ , providing the local definition, [37, 40, 41]: eAµ=1 2ϵijωij µ, ωabµ=ea ν∂µebν .(33) In terms of this spin connection, the U (1) gauge transformation corresponds to a local rotation on the precession plane by an angle eχ(x): ω12µ→ω12µ+e∂µχ . (34) This introduces an Ehresmann connection, dη =ds +e mAµdxµ,(35) on the S1 fiber, [ 13 ], invariant under gauge transformation for the shift s→s−e mχ , as we will see in more detail in par.(IV). Full electromagnetism in flat spacetime. — We first derive the EM coupling for the single complex scalar mode. Introduce the covariant derivative Dµ=∂µ−ieAµ,(36) such that, ∂µϕp=∂µ[V−1ϕ′ p′] = V−1Dµϕ′ p′.(37) The term V−1ϕ′ p′ has persistent recurrence Tµ as ϕp , even though ϕ′ p′ has locally modulated recurrence T′µ ( x ). In this way, V−1 can locally tune the mode ϕ′ p′ of local recurrence T′µ ( x )as solution of an action with constant boundary Tµ (and PBCs), obtaining a more familiar and practicable formalism, [ 15 ]. As the derivative term is the only relevant term to the PBCs, it is therefore sufficient to replace it with a covariant derivative term to tune the modulated solution to the static PBCs: SEM =ITµ d4x′L(Dµϕ′ p′, ϕ′ p′).(38) The LLT transformed action eq.(13) of modulated boundary T′µ ( x )is equivalent to the gauged version of the free action eq.(3) of fixed boundary Tµ , having the same modulated solution ϕ′ p′ . Thus, by integrating over al possible momenta pto generalize the result from single modes to the whole scalar field, the FW geometrodynamics associated to LLT are described by gauging the ordinary Klein-Gordon free action, obtaining the standard EM gauge of field theory. For the same consistency with the PBCs at fixed boundary Tµ , the dynamics of Aµ must be gauge invariant, i.e. tunable by parallel transport and covariant derivatives to the recurrence imposed by the action boundary — besides requiring locality and Lorentz invariance, [ 15 ]. Essentially, this is precisely the standard justification that leads to the Maxwell kinetic term in ordinary field theory. The unique lowest-dimension term allowed is therefore of the form −κ 4FµνFµν , with Fµν =∂µAν−∂νAµ=DµAν−DνAµ= 0 (39) gauge invariant. In par.(IV) we will give a further, purely 4D geometric derivation of this result directly from the Ricci tensor, in terms of the KK mechanism. Thus, the transformed action, including the kinetic term of Aµ written in a compatible form with respect to the PBCs at fixed Tµ , is just the standard EM action for a single scalar mode: SEM =ITµ d4xh−1 4˜ Fµν ˜ Fµν +L(˜ Dµϕ′ p′, ϕ′ p′;m)i, (40) where we have normalized the gauge sector ˜ Aµ = Aµ/κ and ˜e = eκ .We have obtained the standard Maxwell equations of EM from FW geometrodynamics in flat spacetime: ∂a˜ Fab =Jb,(41) where Ja=ihϕ′† p′(˜ Daϕ′ p′)−(˜ Daϕ′† p′)ϕ′ p′i.(42) By choosing eµ 3 aligned with the propagation of Aµ we find the transverse physical polarization in the precession plane. III. GEOMETRODYNAMICAL ORIGIN OF ELECTROMAGNETISM AND GRAVITY. Let us now consider the complex scalar mode ϕp on a curved background gµν with PBCs imposed at a fixed boundary Tµ: SGR =ITµ d4x√−gL(g, ∇µϕp, ϕp;m),(43) where ∇µis the Levi-Civita derivative and L(g, ∇µϕp, ϕp;m) = 1 2gµν∇µϕ† p∇νϕp−m2ϕ† pϕp. (44) As mentioned in the introduction, this action can be obtained from the free action eq.(3), SF ree →SGR ,(45) by applying local diffeomorphisms from flat to curved spacetime, ηµν →gµν ,(46) 6 which in turn generates local transformations (diffeomorphisms) of the boundary encoding ordinary gravitational redshift and rulers contraction, [10]. However, for a scalar, ∇µϕp = ∂µϕp . The gravitational modulation of clocks/rulers is entirely encoded in the tetrad and the metric. Fixing the boundary globally to Tµ as in eq.(43), rather than to the transformed one, is thus a convenient gravitational gauge choice: this allows us to keep clocks with the same exact period upon the curved spacetime and calculated the induced modulations of EM origin with reference to these clocks. The generalization of the flat result to this curved background is thus straightforward and consists in repeating the same demonstration of par.(II) where, essentially, all derivatives are replaced with Levi–Civita derivatives. This means to introduce local twists of the curved spacetime by means of infinitesimal LLTs eq.(6) and to identify the congruence of worldlines such that the tetrad eµ a is FW transported. Projecting on the precession plane, PH identifies the abelian connection in terms of the spin connection eAµ = 1 2ϵijωij µ . Its curvature measures the holonomy (non-integrability) of the gyroscope/clock precessions. Rewriting the locally transformed dynamics originated by the LLT back on the fixed boundary Tµ , by inserting parallel transport and covariant the derivative ˜ Dµ = ∇µ−i˜e˜ Aµ to tune the modulated solution ϕ′ p′ and the kinematics of Aµ , we get the unified geometrodynamical description of gravitational and EM interactions for a scalar charged particle of mass m in curved spacetime: SGR+EM =ITµ d4x√−gh−1 4˜ Fµν ˜ Fµν + +L(g, ˜ Dµϕ′ p′, ϕ′ p′;m)i+SEH ,(47) after canonical normalization in the gauge sector. We have added the Einstein-Hilbert action SEH whose study of the boundary goes however beyond the scope of this paper, [ 42 ] — relevant for the problem of the quantization of gravity, according ECT where quantization emerges from BCs, [8, 9, 15–23]. IV. ELECTRODYNAMICS FROM THE RICCI TENSOR. ECT, at the base of our formalism, is equivalent to a massless 5D theory with S1 cyclic XD z of period TC= 2π/m, as proven in detail in [16], dS2=gµν dxµdxν−dz2≡0⇔ds2=gµν dxµdxν, (48) as soon as z is identified with the (cyclic) proper time sencoding the particle’s internal clock S1, z≡s . (49) With this identification we say that the XD is Virtual (VXD). From a purely 4D point of view, we are attaching to every spacetime point a S1 fiber representing the internal clock of the particle when it passes there. Again we keep only the fundamental (classical) mode Φ(x, s)=Φm(s)ϕp(x),(50) where the VXD component is the fundamental mass eigenmode Φm(s)∝eims ,(51) of the virtual KK tower of masses. After decompatification, the fundamental mode Φ( x, s )reproduces precisely the fundamental 4D scalar mode ϕp ( x )of mass m , solution of eq.(43) — the higher virtual KK modes describe the quantum excitations of the same particle rather that the independent particles of the ordinary KK theory, see app.(A) for more detail about this correspondence. By switching on LLTs, eq.(6), with FW transport on the 4D part, the resulting modulated VXD theory is obtained by gauging the original theory: SV XD T OT =ITµ d4xITC ds√−g 2gMN DMΦ′†DNΦ′ +SV XD EH .(52) The tuning to constant boundary in the 4D part is in fact obtained by introducing ∂M→DM= (∂M−ieAµ),(53) with M = ( µ, 5), in agreement with the previous analysis. The local isometries introduce a Wilson line V ( x ) to the original field mode, manifestation of generated EM interaction: Φ(x, s)→Φ′(x, s)=Φm(s)ϕ′ p′(x) = Φm(s)V(x)ϕp(x) = Φm+EM (η)ϕp(x)(54) where we have defined Φm+EM (η)∝eimη ,(55) in terms of the parameter η=s+e mZγ Aµ(x)dxµ.(56) 7 We have recovered again eq.(35), i.e. the Ehresmann connection dη associated, in every spacetime point, to the internal clock precession in the S1fiber. From the point of view of the VXD metric, the geometrodynamics generating EM modifies the original metric eq.(48) yielding formally the KK metric dS2=gµν dxµdxν−(ds +e mAµdxµ)2 =gMN dxMdxN≡0,(57) and s is the VXD. Under the assumption of VXD this defines a 4D metric whose additional terms with respect to the curved metric gµν has interesting links to “double copy” construction, [ 43 , 44 ], and other unification attempts of EM and gravitation, see [ 27 ] for a review. Finally, with the KK metric eq.(57), we apply the KK mechanism in the decompactification of the VXD, which is equivalent to describe the effective 4D geometrodynamics of our ECT framework. By retaining only the fundamental mode of the virtual KK tower and normalizing the gauge sector by a factor κ=e2 m2 1 4πG ,(58) we thus obtain directly eq.(47). This is an unified geometrodynamical description of gravitational and EM interactions for a scalar charged particle of mass m in curved spacetime, where now also the kinetic term of the gauge field has been explicitly obtained from geometrodynamical arguments rather than from consistency arguments. V. BASIC PHENOMENOLOGY Precessions are often labeled as “purely kinematic” effects, yet they occur only in accelerated motion generated by external fields, therefore inseparable from dynamics. In our framework, precession and EM/gravito-EM interaction are two faces of the same geometrodynamical mechanism — much like Earth’s rotation and the Coriolis force. Gravity manifests as deformation of clock rates (metric redshift/length), whereas EM/gravito-EM manifest as precessions of clocks (gyroscopes) — rotations of the instantaneous spacetime periodicity τµ ( x )projected on the precession plane. We illustrate this limited to a few classical-relativistic representative examples, see app.(A) for outlooks about quantum phenomenology. If we apply the FW transport to a uniform circular motion of radius R and angular velocity ω , in the approximation Rω ≪ 1, the effective magnetic field generated by the vorticity of uµ turns out to be B ≃ ∇ × A= −m e∇ × u. For a rigid rotation, we have Larmor frequency ω≃ − e 2m B, consistently describing the relation between the precession ω of the magnetic moment of a particle of charge e and mass m and the associated magnetic field Bnecessary to preserve PH. For a particle with spin, viewed as an internal clock, the appropriate generalization of the transport law is the Bargmann–Michel–Telegdi equation. This ensures PH of the spin, consistently describing the effective EM interaction. Similarly to Larmor, for a particle of spin vector S, the Thomas precession angular velocity ωT = γ2 γ+1 v × a, follows precisely from the spin transport law, describing the non-commutativity of boosts, [45, 46]. Furthermore, any loss of synchronization of the spin/clock precession, such as in the time-varying four-velocity uµ along a synchrotron orbit, is corrected by the emission of EM radiation as bookkeeper, restoring phase alignment and resulting precisely in the observed synchrotron radiation. Gravito-EM, [ 47 ], is another example that shows the relationship between precessions associated to FW transport and the mathematical laws of EM, the difference is that in Gravito-EM precession is of gravitational origin, originated by the curvature of spacetime, whereas in pure EM geometrodynamics the precession originates from LLT, i.e. local twists of the spacetime. VI. COMMENTS AND OUTLOOKS The basic formal chain from PBCs (internal clock S1 ), local isometries, U (1) holonomy of the boundary, and finally to the EM laws was already fully developed in [ 15 ] and further reported in subsequent papers [ 8 , 9 , 15 – 17 , 19 – 23 ]. However, the specific FW geometrodynamics that constitute the original content of the present work were only sketched in the past work. Paper [ 15 ] also presents the full generalization to QED, which can be directly applied to the present results; see also [8] and app.(A). The projection onto the precession plane has the purpose of restricting the gauge group relevant to our analysis to the Abelian U (1). More generally, PH points to a (not necessarily identical) relationship between the Lorentz group and induced internal gauge symmetries on the tangent bundle, which is the natural fiber bundle singled out by PH. In the simple Abelian case considered here, the unbroken U (1) EM holonomy is linked to the compact diagonal U (1) subgroup of the Lorentz group. Extending the present geometrodynamical mechanism to larger gauge groups is nontrivial, it goes beyond the scope of this paper and requires a careful analysis of the interplay between noncompact spacetime symmetries and 8 (possibly broken) gauge symmetries on the tangent bundle. This analysis must also take into account that, in our framework, particle masses have a clear geometrical meaning as the duration of proper-time recurrences (internal clock), with interesting analogies to XD extensions of the Standard Model of the ElectroWeak interactions, in particular by means of the VXD formalism, [15, 16, 33–36]. CONCLUSIONS. We have identified geometrodynamics underlying EM. Local twists of spacetime encoded by FW transport of locally accelerated frames drive the precession of each particle’s phase recurrence, which behaves as an internal gyroscope or clock. If restricted to the projection onto a spatial plane perpendicular particle’s velocity, PH defines an Abelian subgroup U (1), with associated holonomy, yielding EM without postulating gauge invariance, alongside the standard gravitational geometrodynamics based on curved spacetime. In particular, PH implies an internal S1 fiber (internal clock), whose Ricci geometry reproduces the Maxwell kinetic term, via a purely 4D generalization of the KK mechanics. Classical EM phenomena (Thomas and Larmor precession, Zeeman effect, synchrotron radiation, gravitomagnetism) are consistently reinterpreted in this framework. The present analysis is entirely classical–relativistic, but it is based on the framework of ECT which provides a natural extension of its validity to quantum mechanics and to a variety of quantum phenomena in condensed matter and high-energy physics, [ 8 , 9 , 15 – 23 ]. In brief, light acts as the phase connection that keeps nature’s elementary clocks synchronized. [1] V. Arnol’d, Mathematical Methods of Classical Mechanics (Springer New York, 2013). [2] A. Fasano, S. Marmi, S. Marmi, et al.,Analytical mechanics: an introduction (Oxford University Press on Demand, 2006). [3] J. Masoliver and A. Ros, European journal of physics 31, 171 (2009). 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Danehkar, Int. J. Mod. Phys. D 29, 2043001 (2020). [48] L. d. Broglie, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47, 446 (1924). [49] A. Einstein, Arch. Sci. Phys. Natur. 29 (1910). [50] R. Penrose, “Cycles of Time. An Extraordinary View of The Universe,” (Knopf, New York, 2011). [51] D. Dolce, Quant. Stud. Math. Found. 12, 26 (2025). ———————— APPENDIX ———————— Appendix A: Further comments The reader not interested in foundations of QM can skip this part, which is not relevant to the purely classical-relativistic analysis developed in this paper The analysis of more advanced phenomenology, beyond the examples discussed above, quickly involves Quantum Mechanics (QM). The quantization properties and quantum implications of ECT have been studied in detail in [ 8 , 9 , 15 – 23 ], from which we faithfully extract only the essential aspects that are directly relevant to the present work. Second quantization — [ 8 ] — The most general solution of the free action, eq.(3), is a superposition of all the harmonic modes (indicated by the “tilde” symbol) allowed by the PBCs ϕp ( x ) →˜ ϕp ( x ), which directly implies a description in Hilbert spaces and the introduction of ladder operators ˆap and ˆa† p , for the field modes ϕp ( x ). In [ 8 ] we have proven that the constraint of PBCs as in eq.(3) formally play the same role of the ordinary commutator of second quantization for the (fundamental) field mode ϕp ( x ): (˜ ϕp(xµ+Tµ)PBCs ≡˜ ϕp(xµ) Tµpµ= 2π⇔[ˆap,ˆa† p]=1. This represents the bridge from our geometrodynamical description of EM interaction to Quantum Field Theory. The PBCs naturally reproduce the ordinary normal ordered quantized energy spectrum En (p) = nω (p) prescribed by second quantization for the classical mode ϕp ( x ). It is also true that the field mode of zero momentum (rest particle), second quantization defines a mass spectrum (rest energy spectrum) mn = nω (0) = nm where m = 2 π/TC , which is naturally reproduced in ECT by the internal clock S1 of Compton recurrence, when all the harmonic modes allowed by the PBCs are considered. This heuristic argument shows that that ECT is actually dual to an XD theory where the XD is virtual,i.e. a free KK theory where the higher KK modes mn are not independent classical particles but quantum excitations of the same particle of mass m ,i.e. the virtual KK tower is the quantized energy spectrum prescribed by second quantization for the zero momentum mode, see virtual KK tower in par.(IV). The rigorous demonstration of this correspondence is given in [16]. Canonical quantization — [ 9 ] — More in general, as proven in [ 9 ], under the very general hypothesis of symplectic manifolds equipped with a non-degenerate, closed 2-form, the result of PH imposed as constraint (PBCs) is a quantization formally equivalent to the canonical quantization of the system. In particular the PBCs formally plays a role equivalent to the canonical commutation relations of canonical QM (Dirac rule): ˜ ϕ(x, t)PBCs ≡˜ ϕ(x, t +T(x)) ⇔[x,ˆ p′] = i{x,p′}=iI, where ˆpi=−i∂ ∂xi, and {,}is the Poisson bracket. The general validity of this results includes Riemannian geometry, opening an novel prospective to the problem of the quantization of gravity. It is interesting to notice the fundamental role played by gauge invariance — whose holonomy originates from canonical transformations in a generalization of the analysis of the present paper — in defining the correct pre-quantum operators of geometric quantization which eventually originate the equivalence between the Poisson bracket of the cyclic classical dynamics and the canonical commutators of QM — see also [51]. Feynman Path Integral and QED — [ 15 ] — In agreement with the previous correspondences, ECT has also a correspondence, at a classical level, to the ordinary Feynman path integral [ 15 – 18 , 21 – 23 ]. In particular, the formal generalization to scalar QED