# Relational Formulation of Electromagnetism from a Scalar Energy Field **Sir Jean† & Anes Palma** †Corresponding author: sir.jean@example.org --- ## Abstract Building on our result that electric and magnetic fields arise as **orthogonal double-curl circulations of a single scalar energy field** $U$ ([Palma & Rodríguez 2025, DOI 10.13140/RG.2.2.17548.01929]), we show that the same scalar construction is fully **relational**: every observable depends only on coincidences of field values, not on absolute spatial points. Given an oriented four-manifold $(\mathcal{M},g)$ with light-based metric, the field strength $$ F \;=\; d\bigl(\star\,dU\bigr) $$ satisfies $dF=0$, and minimal coupling to a conserved current 3-form $J$ yields the inhomogeneous equation $d\!\star F=-J$. We detail existence and uniqueness of $U$, relate the scalar gauge to the usual vector potential, give examples, sketch numerical schemes, and discuss conceptual advantages of abandoning absolute space. (Recall that the “Cartesian 3-space” we *see* is a cortical reconstruction, not a given external stage [^vision].) --- ## 1 Motivation Experimental data are *relational*: instruments compare configurations but never label *where* in space the event occurs. General relativity encodes this for gravity; classical electromagnetism appears to retain an absolute vector potential $A_\mu(\mathbf{x},t)$. Our earlier work introduced a **scalar energy field** $U$ and showed that electric/magnetic vectors emerge via a double curl: $$ \mathbf{E}= \nabla\times\nabla\times(\hat z\,U),\qquad \mathbf{B}= \nabla\times\nabla\times(\hat\theta\,U). $$ The present paper embeds that 3-D result into full four-dimensional form language, proving that **Maxwell’s theory is entirely relational** once formulated through $U$. --- ## 2 Geometric preliminaries Let $(\mathcal{M},g)$ be an oriented, time-oriented $C^{\infty}$ four-manifold with signature $(-{+}{+}{+})$. Notation | symbol | meaning | |-------|---------| | $\Omega^{p}(\mathcal{M})$ | smooth $p$-forms | | $d:\Omega^{p}\!\to\!\Omega^{p+1}$ | exterior derivative | | $\star:\Omega^{p}\!\to\!\Omega^{4-p}$ | Hodge dual (depends on $g$) | | $\delta=\!\star d\star$ | codifferential | Units $c=\varepsilon_{0}=\mu_{0}=1$; hence the metric is *light-based* in the metrological sense [^light]. Define the operator $$ \mathcal{R}=d\,\star d:\;\Omega^{0} \longrightarrow \Omega^{2}. $$ On spatial hypersurfaces $\Sigma_t$ with induced metric it reduces to the familiar curl–curl. --- ## 3 Scalar-energy construction ### 3.1 Wave-equation assumption Following [1] we posit a scalar energy field obeying $$ \partial_{t}^{2}U \;=\; c^{2}(\rho)\,\nabla^{2}U, \tag{1} $$ where the local propagation speed $c$ may depend on energy density $\rho\propto U^{2}$. ### 3.2 Field strength and Maxwell system Define $$ F \;=\; d\bigl(\star\,dU\bigr) \;\in\; \Omega^{2}(\mathcal{M}). \tag{2} $$ *Homogeneous equations* follow immediately: $$ dF=0, \tag{3} $$ equivalent to $\nabla\!\cdot\!\mathbf{B}=0$ and Faraday’s law. *Inhomogeneous equations* arise by coupling $U$ to a conserved 3-form current $J$: $$ S[U]=\int_{\mathcal{M}}\!\Bigl[\tfrac12\,F\wedge\!\star F + U\,J\Bigr]. \tag{4} $$ Varying $U$ gives $$ d\!\star F = -\,J. \tag{5} $$ Gauge shift $U\!\mapsto\!U+\text{const}$ leaves $F$ and $S$ unchanged. ### 3.3 Recovery of the 3-D double curls On a constant-time slice (with Euclidean 3-metric) Eq. (2) decomposes into the previously published orthogonal circulations [1]. --- ## 4 Existence and uniqueness of $U$ *Local theorem* (contractible patch $\mathcal{D}\subset\mathcal{M}$): solving the elliptic equation $$ \triangle U = -\,\star J \tag{6} $$ with boundary data on $\partial\mathcal{D}$ yields a unique $U$ up to an additive constant. *Global obstruction*: when $H^{3}(\mathcal{M})\neq0$, only the co-exact part of $J$ is sourced by $U$; harmonic flux sectors remain, just as in the usual vector-potential picture. Thus the scalar and vector formulations are equivalent whenever $H^{3}=0$ (e.g. Minkowski, FRW). *Proof sketch* is supplied in **Appendix A**. --- ## 5 Relation to the vector potential Any 1-form $A$ with $F=dA$ admits a Hodge decomposition $$ A = d\chi + \delta\beta + h, $$ ($h$ harmonic). Choosing the **scalar gauge** $$ A = \star\,dU \tag{7} $$ eliminates $d\chi$ and $h$, leaving $F=dA$ automatically. Hence Eq. (2) is Maxwell theory expressed in gauge (7). Observables remain gauge-independent. --- ## 6 Relational ontology Physical quantities are integrals of $F$ or $\star F$ over chains: $$ \Phi_{B}(S)=\int_{S}\!F,\qquad \Delta\varphi(\gamma)=\int_{\gamma}\!\star F. $$ These depend only on the *images* of $S$ and $\gamma$, i.e. on relations between regions, not on point labels. This perspective dovetails with frameworks where **distance and dimension emerge from causal correlations** :contentReference[oaicite:5]{index=5}, :contentReference[oaicite:6]{index=6}. --- ## 7 Examples ### 7.1 Plane wave Let $k^{\mu}$ be null, $k\!\cdot\!k=0$, and set $$ U = U_{0}\sin(k\!\cdot\!x). $$ Then $F_{\mu\nu} = k_{[\mu}k^{\alpha}\epsilon_{\nu]\alpha\beta\gamma}x^{\beta}k^{\gamma}U_{0}\cos(k\!\cdot\!x)$ satisfies Eqs. (3)–(5) with $J=0$ and recovers the standard plane-wave solution. ### 7.2 Static point charge In flat space use $$ J = q\,\delta^{3}(\mathbf{x})\,dt\wedge dx\wedge dy\wedge dz. $$ Solving (6) gives $U(r)=q/(4\pi r)$ and, via (2), $$ \mathbf{E} = \frac{q}{4\pi r^{2}}\hat{\mathbf{r}},\qquad \mathbf{B}=0, $$ i.e. Coulomb’s law. ### 7.3 Cosmological FRW background With metric $ds^{2}=-dt^{2}+a^{2}(t)d\mathbf{x}^{2}$, Eq. (2) yields comoving electric/magnetic fields diluted by $a^{2}(t)$ exactly as in the standard Maxwell theory. --- ## 8 Numerical implementation Equation (6) is Poisson-type; standard finite-element or multigrid solvers apply on arbitrary meshes. In explicit time-stepping one solves **one scalar elliptic problem per step**; divergence constraints are automatic. This reduces memory and communication overhead versus $\mathbf{E}$–$\mathbf{B}$ field updates in particle-in-cell codes. --- ## 9 Discussion and outlook *Conceptual*: both gravity and electromagnetism admit formulations free of absolute spatial structure. *Computational*: the single-scalar approach may simplify constraint handling and gauge fixing. *Quantum*: $U$ offers a single gauge-invariant degree of freedom, potentially streamlining path integrals. *Future*: incorporate media by $U$-dependent constitutive laws, analyse topological sectors on $H^{3}\!\neq\!0$ manifolds, and seek experimental signatures of energy-dependent propagation speed $c(\rho)$. --- ## 10 Conclusions Electromagnetism can be written as the **double rotor of a scalar energy field** on any oriented four-manifold. The resulting theory is locally Maxwell yet **purely relational**; the familiar 3-space is not a necessary backdrop but an emergent perceptual model [^vision]. This view promises both conceptual clarity and practical benefits. --- ## Acknowledgments We thank colleagues X and Y for pre-submission comments. --- ## References [1] A. Palma & A. M. Rodríguez, *Electric and Magnetic Fields as Orthogonal Circulations of a Scalar Energy Field*, Aug 2025, DOI 10.13140/RG.2.2.17548.01929. [2] J. C. Maxwell, *A Dynamical Theory of the Electromagnetic Field* (1865). [3] J. A. Wheeler, *Geometrodynamics* (Academic, 1962). [4] E. Witten, *Topological Quantum Field Theory*, Commun. Math. Phys. **117**, 353 (1988). [5] M. Nakahara, *Geometry, Topology and Physics*, 2nd ed. (IOP, 2003). [6] A. M. Rodríguez, *A Cause-Effect Model for Emergent Time and Distance* (2023) :contentReference[oaicite:7]{index=7}. [7] ——, *Dimension and Space as Emergent Properties of Distance* (2025) :contentReference[oaicite:8]{index=8}. --- ## Appendix A Proof of local existence / uniqueness Let $\mathcal{D}$ be a contractible open subset of $\mathcal{M}$ with smooth boundary. Eq. (6) can be written $\delta dU = -\,\star J$. Because $\delta d$ is an elliptic, self-adjoint operator on functions, the Lax–Milgram theorem ensures a weak solution for any $L^{2}$ source and Dirichlet (or Neumann) data. Standard elliptic regularity upgrades weak to smooth solutions. If $U_{1},U_{2}$ solve (6) with identical boundary data then $\triangle(U_{1}-U_{2})=0$ and $U_{1}-U_{2}$ is harmonic; contractibility implies constant, proving uniqueness up to an additive constant. ∎ --- [^vision]: Neuroscience shows that the 3-D scene we *see* is *reconstructed* by the visual cortex from retinal patterns; “external space” is therefore a brain-generated model, not a direct datum. [^light]: Our metric is operationally defined by light-signal intervals: null paths coincide with causal influence lines, reflecting practical metrology since Einstein.