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

Unified Electromagnetic and Gravitational Geometrodynamics from Phase Harmony and Fermi-Walker Transport Donatello Dolce University of Camerino, Piazza Cavour 19F, 62032 Camerino, Italy Abstract We prove that standard electromagnetism precisely originates from local spacetime geometrodynamics, without postulating gauge invariance, in a way that is complementary and unified with respect to ordinary 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 construction also introduces an S1 fiber attached to every spacetime point and thus the Maxwell kinetic term 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. 1 INTRODUCTION. 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 a wave-number, hence to the inverse of a time and a spatial periodicity, Tand λi, respectively. Its most familiar application is the undulatory mechanics underlying quantum theory, where the proportionality constant ℏ is the reduced Planck (we adopt c = ℏ = 1). For a free particle, the four-momentum pµ (tangent covector) and the instantaneous spacetime periodicity τµ = {T, λi} (tangent contravariant vector), form a relativistic invariant pµτµ = 2 π , named here Phase Harmony (PH); e.g. as under Lorentz transformations Λwhere pµ→p′ µ = Λ ν µpν and τ→τ′µ= Λµ ντν. Thus, in the rest frame, the mass m fixes an intrinsic recurrence of Compton period TC in the proper time: mTC = 2 π . This is 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 — only mentioned in app.(B). We remain purely classical-relativistic 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) 2-forms 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: HTµ(x)dθ = p′ µ ( x ) τ′µ ( x ) = pµτµ = 2 π . 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 2 [ 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 geometric 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 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]. We need to introduce an ad hoc formalism to investigate PH under LLT. In fact, pointparticle Hamiltonian formalism misses PH entirely. 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. An useful formalism is provided by Elementary Cycles Theory (ECT), [ 8 , 9 , 15 – 23 , 41 ], 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 ]. Projected on the plane of the local frame that carries the S1 phase θ , the modulations τ′µ ( x )effectively describe local precessions of an internal gyroscope [ 24 , 25 ], and the internal transformation identifies an U (1) abelian connection which is precisely the EM gauge field Aµ ( x ). Electromagnetism (EM) is thus 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). We will investigate first this mechanics in flat spacetime, par.(II). The generalization to curved spacetime, where EM emerges along with gravitational interaction, is then straightforward, par.(III). Finally, par.(IV), we provide a 4D interpretation of the Kaluza–Klein (KK) mechanism, [ 26 , 27 ], showing that the Maxwell kinetic term emerges directly from geometry, thereby reinforcing the geometrodynamical origin of EM. App.(A) collects examples and ap3 plications (Thomas/Larmor precession, Zeeman, synchrotron, etc.). App.(B) briefly presents quantum implications and phenomenology, with faithful references to the existing literature. I. PHASE HARMONY AND FORMALISM. We consider a scalar charged particle of mass m , free of any interaction in a classicalrelativistic field description — flat spacetime ηµν = diag (+ ,−,−,− ). Then we select the single mode ϕp ( x )of four-momentum pµ by imposing related PBCs, ϕp ( xµ ) = ϕp ( xµ + Tµ ), [ 15 ], such that pµTµ= 2π: SF ree =ITµd4xL(∂µϕp, ϕp;m),(1) where L = 1 2h∂µϕ† p∂µϕp−m2ϕ† pϕpi and HTµ represents the PBCs of the interval Tµ anchored at generic base point x . 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.(B). Eventually, the result of ordinary field is reconstructed by integrating over all the possible fundamental modes ϕp ( x ). 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 , is ϕp ( x ) = Nexp [ −ipµxµ ], where N is a normalization constant — its validity can be restricted to a neighbor of the base point x. It is well known from string or eXtra Dimensional (XD) theories [ 26 – 29 ] 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]. II. GEOMETRODYNAMICAL ORIGIN OF EM, WITHOUT GRAVITY. Now we switch on (EM) interaction by introducing in each point x an infinitesimal LLT x→x′ ( x )with ea µ ( x ) = ∂x′a ∂xµ , [ 15 ]. This is a position dependent isometry, preserving the flat metric ηµν →η′ µν(x) = ea µ(x)eb ν(x)ηab =ηµν. No gravitation is introduced. 4 A. 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 — u·u = 1 and u·a = 0. The FW transportation law ( u·∂ ) eµ a = Ω µ νeν a , is described by the antisymmetric local generator (pure boost) Ωµν=uµaν−aµuν, [11, 30]. B. Gyroscope. — We will focus on the precession of the internal S1 clock phase θ induced by FW transport, in direct analogy with a gyroscope. We therefore restrict our analysis to the abelian component of the generator Ω, by projecting onto a precession plane, characterized by a unit bivector nab = 1 2ϵijea ieb j , ϵ12 = +1, orthogonal to ua ( x ), [ 24 , 25 , 30 ]. In the following we thus set Ω = nabΩab. This selects the unique SO(2) ≃U(1) subgroup for θ, [31, 32]. C. Induced internal transformation. — Solving the evolution law along a path γ : x0→x and considering that the transformation is infinitesimal, at first order we obtain eµ a ( x ) ≃δµ a + Rγ ( u·dy )Ω µ a , where we have aliened the local frame with the effective coordinates at point x0:eν a(x0) = δν a. In eq.(1), 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). The transformed action is, [15], SF ree →SEM =IT′µ(x)d4xL(∂µϕ′ p′, ϕ′ p′;m).(2) The key quantity for implementing PH is the locally rotated tangent (instantaneous) fourperiodicity Tµ→τ′a ( x ) = eaµ ( x ) Tµ , rather than the recurrence interval T′µ ( x ). Due to the projection on the precession plane, the modulation of τ′a ( x )formally describes the FW induces local precession of the axis of an internal gyroscope associated to the particle. In ECT ϕ′ p′ ( x )  = ϕp ( x ), contrarily to ordinary field description where (due to the absence of boundaries) local distortion of the flat metric doesn’t induce any internal transformation 5 of the field itself. In fact, in our formalism each field mode is a physical standing wave whose instantaneous four-periodicity is directly determined by the local PBCs. Due to the locally rotation of boundary Tµ→T′µ ( x )in the transformed action, the transformed field solution along the path γ is a modulated spacetime wave ϕ′ p′ ( x ) = Nexp [ −iRγp′ a(y)dya ], where i∂aϕ′ p′ = p′ a ( x ) ϕ′ p′ . The four-momentum p′ a ( x )is locally determined by PH, Tµpµ = τ′a ( x ) p′ a ( x ) = 2 π , resulting from the modified PBCs — in a sort of holographic description in which interaction is encoded in the local transformations of the boundary of the theory, [ 15 ]. Thus: p′ a(x) = eµ a(x)pµ=pa−eAa(x). Here we have defined an effective eAa ( x ) ≃ −pµRγ ( dy ·u )Ω µ a potential at first order in Ω, so that we formally get the minimal substitution of EM. The induced transformation of the field mode ϕp ( x ) →ϕ′ p′ ( x ) = V ( x ) ϕp ( x ), now includes a path dependent Wilson line as effective internal transformation V(x) = exp[iRxAa(y)dya], [15]. D. Lorentz force and field strength. — From the generator of the FW transport we find pµ Ω µ a = muµ ( aµua−uµaa ) = −maa . Since the local frame is fixed in the point of reference x0 , we have ˙ua = aa . The integration along the worldline yields a gauge flow aligned to the velocity field: Aa ( x ) ≃m eua ( x ). 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 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’s, mdu′ µ ds = eFµνu′ν , where the physical kinetic momentum is p′ a ( x ) = mu′ a(x). E. Gauge invariance U(1) and holonomy. — Now we add the rotational part of angular velocity ωµ , Ω(SR)µ ν = ϵµνρσuρωσ , to the FW: Ω → Ω ′ = Ω + Ω (SR) , [ 11 ]. This adds a total derivative term to the gauge field Aa→A′ a = Aa + ∂aχ , with χ = ξµpµ and ξµ is a Killing vector such that e∂aξµ = −Rγ ( dy ·u ) Ω(SR)µ a . The 6 field mode acquires a local phase ϕ′ p′ ( x ) →ϕ′′ p′ ( x ) = U ( x ) ϕ′ p′ ( x ), where U ( x ) = exp [ ieχ(x) ], describing a U(1) gauge orbit of the effective gauge Aa(x)generated by FW. The rotational part in fact is an holonomy. Its contribution to the transformation of the boundary T′µ ( X ) →T′′ µ ( x )is a total derivative with no effect on the PBCs — up to a gauge transformation. F. Spin connection. — The transport law induced by LLT, together with PH relating precession to local variation of four-momentum, implies a tangent 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 abelian gauge field Aµ , providing the local definition eAµ = 1 2ϵijωijµ , where ωabµ = eaν∂µebν , [ 30 , 33 , 34 ]. 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 . This introduces an Ehresmann connection dη = ds + e mAµdxµ 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). G. Full electromagnetism in flat spacetime. — We first derive the electromagnetic coupling for the single complex scalar mode. Introduce the covariant derivative Dµ = ∂µ−ieAµ , such that, ∂µϕp = ∂µ [ V−1ϕ′ p′ ] = V−1Dµϕ′ p′ . The term V−1ϕ′ has persistent recurrence Tµ as ϕp , even though ϕ′ p′ ( x )has locally modulated recurrence T′µ . In this way, V−1 can locally tune the mode ϕ′ p′ of local recurrence T′µ as solution of an action with constant boundary Tµ, obtaining o more familiar and practicable formalism, [15]. As the derivative term is the only relevant term to the PBCs, it is therefore sufficient to replace this derivative term with a covariant derivative term to tune the modulated solution to the static PBCs SEM =ITµ d4x′L(Dµϕ′ p′, ϕ′ p′).(3) The LLT transformed action eq.(2) of modulated boundary T′µ ( x )is equivalent to the gauged free action written at fixed Tµ, having the same solution. 7 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 — in par.(IV) will give a further geometric justification in terms of KK mechanism. The unique lowest-dimension term allowed is therefore of the form −κ 4FµνFµν with Fµν =∂µAν−∂νAµ=DµAν−DνAµ= 0, gauge invariant. 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-mode scalar: SEM =ITµd4x−1 4˜ Fµν ˜ Fµν +L(˜ Dµϕ′ p′, ϕ′ p′;m),(4) where we have normalized the gauge sector ˜ Aµ = Aµ/κ and ˜e = eκ .We have obtained the standard Maxwell equations of EM from the FW geometrodynamics: ∂a˜ Fab = Jb , where Ja = ihϕ′† p′(˜ Daϕ′ p′)−(˜ Daϕ′† p′)ϕ′ p′i . By choosing eµ 3 aligned with the direction of the EM field we find that the physical polarization is transverse, and lives in the precession plane. III. GEOMETRODYNAMICAL ORIGIN OF EM AND GRAVITY. Let us now consider a complex scalar mode ϕp on a curved background gµν with PBCs imposed at a fixed boundary Tµ: SGR =ITµd4x√−gL(g, ∇µϕp, ϕp;m)(5) where ∇µ is the Levi-Civita derivative and L ( g, ∇µϕp, ϕp ; m ) = 1 2hgµν∇µϕ† p∇νϕp−m2ϕ† pϕpi . As mentioned in the introduction, this action can be obtained from the free action eq.(1) by applying local diffeomorphisms from flat to curved spacetime ηµν →gµν , which in turn generates local transformations of the boundary encoding gravitational redshift and rulers contractions. 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.(5) is thus a convenient gravitational gauge choice: we 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 straightforward and consists in repeating the same demonstration where, essentially, all derivatives are replaced 8 with Levi–Civita derivatives. This means to introduce local twists of the curved spacetime by means of infinitesimal LLTs 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 2ϵijωijµ . Its curvature measure the holonomy (nonintegrability) of the gyroscope/clock precessions. Rewriting the locally transformed dynamics back on the fixed boundary Tµ and inserting parallel transport 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: SGR+EM =ITµd4x√−g−1 4˜ Fµν ˜ Fµν + +L(g, ˜ Dµϕ′ p′, ϕ′ p′;M)+SEH ,(6) after canonical normalization in the gauge sector, where ˜ Dµ = ∇µ−i˜e˜ Aµ . We have added the Einstein-Hilbert action SEH whose study of the boundary goes however beyond the scope of this paper, [35]. IV. ELECTROMAGNETODYNAMICS FROM RICCI TENSOR. ECT, at the base of our formalism, is equivalent to a massless 5D theory with cyclic XD z of period TC = 2 π/m , as soon as z is identified with the (cyclic) proper time s encoding the particle’s internal clock S1 , [ 16 ]: dS2 = gµνdxµdxν−dz2≡ 0 →ds2 = gµνdxµdxν if the z≡s. With this identification we say that the XD is virtual (VXD). Again we keep only the fundamental (classical) mode Φ( x, s ) = Φ m ( s ) ϕp ( x ), with Φ m ( s ) = exp [ ims ], of the virtual KK tower of masses, reproducing precisely, after decompatification, the fundamental 4D scalar mode ϕp ( x )of mass m , solution of eq.(5) — the higher virtual KK modes describe the quantum excitations of the same particle rather that the independent particles of the ordinary KK theory, app.(B). By switching on LLTs with FW transport on the 4D part of the resulting modulated VXD theory is obtained by gauging the original theory: SV XD T OT =ITµ d4xITC ds√−g 2hgMN DMΦ′†DNΦ′i +1 16πG Zd4xZds√−gRV XD.(7) 9 particle.” L. de Broglie (1924), [38]. [5] “By a clock we understand anything characterized by a phenomenon passing periodically through identical phases so that we must assume, by the principle of sufficient reason, that all that happens in a given period is identical with all that happens in an arbitrary period”, A. Einstein (1910), [39]. 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