Electromagnetic Energy Invariance via Time-Phase Separation of Maxwell's Equations

International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17882397 ©2025 RS Publication, rspub[email protected]om 198 Original Article Electromagnetic Energy Invariance via Time-Phase Separation of Maxwell’s Equations Jose Oreste Mazzini Lima, Peru [email protected] International Journal of Advanced Scientific and Technical Research Available online on http://www.rspublication.com/ijst/index.html ISSN 2249-9954 ARTICLE INFO ABSTRACT ©2025 RS Publication Paper ID: IJASTR69389001A94DF Published: 2025-12-10 DOI: https://dx.doi.org/ 10.5281/zenodo.1788 2397 Page No: 198-206 In classical treatments of electromagnetic wave propagation, combining Faraday’s and Amp`ere–Maxwell’s equations typically yields an oscillatory instantaneous Poynting vector, even though the time-averaged energy flux remains constant. Conventional tensor and complex-formalism approaches resolve this issue using phase considerations implicitly. This work proposes an alternative interpretation that introduces an explicit time-phase difference of π/2 between Faraday’s and Amp`ere–Maxwell’s equations, yielding a combined field representation in which the energy density remains invariant at all times during propagation. The formulation reproduces electromagnetic invariants within a real (non-complex) framework and emphasizes energy consistency under both passive Lorentz transformations (moving observers) and active transformations (moving charges). This approach offers a conceptually accessible complement to tensor and complexvector methods, enabling a broader understanding of classical electromagnetism without replacing existing formalism. Keywords: Electromagnetism; Riemann–Silberstein vector; signed-Euclidean geometry; Lorentz invariance; electromagnetic invariance; charge radiation. Cite This Paper: Jose Oreste Mazzini (2025). "Electromagnetic Energy Invariance via Time-Phase Separation of Maxwell's Equations". INTERNATIONAL JOURNAL OF ADVANCED SCIENTIFIC AND TECHNICAL RESEARCH (IJASTR), vol. 15, no. 6, 2025, pp. 198-206. DOI: https://dx.doi.org/10. 5281/zenodo.17882397 Lima - Peru oremazz[at]gmail.com In classical treatments of electromagnetic wave propagation, combining Faraday’s and Amp`ere–Maxwell’s equations typically yields an oscillatory instantaneous Poynting vector, even though the time-averaged energy flux remains constant. Conventional tensor and complex-formalism approaches resolve this issue using phase considerations implicitly. This work proposes an alternative interpretation that introduces an explicit time-phase difference of π/2 between Faraday’s and Amp`ere–Maxwell’s equations, yielding a combined field representation in which the energy density remains invariant at all times during propagation. The formulation reproduces electromagnetic invariants within a real (non-complex) framework and emphasizes energy consistency under both passive Lorentz transformations (moving observers) and active transformations (moving charges). This approach offers a conceptually accessible complement to tensor and complex-vector methods, enabling a broader understanding of classical electromagnetism without replacing existing formalism. In standard classical electromagnetism, a monochromatic plane wave in vacuum is typically expressed as E= ˆyE0cos φ, cB= ˆz cB0cos φ, φ =kx −ωt, (1) The corresponding Poynting vector oscillates as S=1 µ0 E×(cB) = ˆxE0B0 µ0 cos2φ, (2) implying an instantaneous energy flux that varies over time despite a constant average value. Though this behavior is normally resolved using complex representation or tensor methods, although classical interpretations often describe electric and magnetic fields as oscillating in phase. Both fields oscillate in phase, with instantaneous energy peaking and vanishing at the same time. In this work, an explicit time-phase shift of π/2 is introduced between the electric and magnetic field contributions when combining Faraday’s and Amp`ere–Maxwell’s equations. By doing so, a unified physical field is constructed in which the energy density of electromagnetic waves remains constant even instantaneously, mirroring the treatment in quantum and tensor approaches but without invoking complex amplitudes or Minkowski formalism. Jose Oreste Mazzini Abstract Keywords: Electromagnetism; Riemann–Silberstein vector; signed-Euclidean geometry; Lorentz invariance; electromagnetic invariance; charge radiation. 1 Introduction Electromagnetic Energy Invariance via Time-Phase Separation of Maxwell's Equations International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 199 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ This framework also reinforces energy invariance under passive Lorentz transformations and provides an interpretation of energy redistribution under active transformations (charge dynamics). Rather than replacing conventional tensor treatments, this model offers an educational and conceptual alternative that emphasizes real-time energy coherence within classical field theory. Figure 1: Comparison between conceptual A) Einstein’s and B) Minkowski’s energy triangles. The total energy (hypotenuse in Einstein’s triangle) corresponds to the side ct0in Minkowski’s construction. C) Antimatter opposite internal time −ct0with metric (-1, 1, 1, 1). D) EM, boosted transformation (square of energy). E) EM passive (non-boosted) transformation ( 1, -1, -1, -1). F) Conducting current density with metric (-1, 1, 1, 1). In previous work [1], the Lorentz interval ds and its implications for energy invariance under different classes of transformations were examined. Where Minkowski’s interval [2] is obtained from Planck-Einstein’s concept [3, 4] when its triangle’s three sides are divided by nh/(τcdt0) (see Figure 1A and B); thus, an Euclidean space with signs ( 1, -1, -1, -1) due to a subtraction between vectors. The inverse relation of energy and momentum with time and space (E∝1/t and p∝1/λ), explains why the relation between the vectors of ds,dt0and ∆xare subtracted in the Minkowski interval, and why they cannot be treated separately in spacetime; a covariant (e.g. space and time) and contravariant relation between physical parameters (e.g. energy and time). Extending that analysis to classical electromagnetic fields using a time phase alternative to the RS field. The main idea is to introduce a time phase between Faraday’s and Amp`ereMinkowski’s equations. This reproduces the electromagnetic invariant inside a Euclidean framework and provides a direct geometric interpretation; a “subtraction between vectors” suggested by Minkowski-type energy diagrams. Maxwell’s equations can be written in a compact form and the energy density during electromagnetic wave propagation remains strictly constant when considering this phase. The International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 200 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ formalism also clarifies the behavior of electromagnetic fields and energy under passive and active Lorentz transformations of charged particles. Since 1900, Planck [3] revealed that the energy appears discretely in nature, later, Einstein [4] clarified it by the concept of package or quanta attached to proper time τ. A concept not known by Maxwell in 1865 [5] nor Poynting in 1884 [6] where the only oscillation discuss was the generated in purpose and the electromagnetic wave (massless and chargeless condition). From this intrinsic cyclic presence in nature, all electric and magnetic fields are fluctuating, and classical Maxwell equations must update this realm (τ=h/Energy). But care is needed to identify what phase difference is present between these two equations before combining them. Actual assumption of equal phase conducts to the discussed oscillatory energy propagation. The introduction of a phase time to the future π/2 over φof equation 1 solves this issue, i.e. φ0=φ−π/2 and double phase effect φ00 will be equivalent to changing signs. φ=kx −ωt, (cos φ)0= cos(φ−π/2) = sin φ, (sin φ)0= sin(φ−π/2) = −cos φ(3) (cos φ)00 = ((cos φ)0)0= (sin φ)0=−cos φ, (cos2φ)00 = (cos φ)0(cos φ)0== −cos2φ(4) The phase change over time, during the fluctuation (0 ≤t<τ), will be express as t0=t−τ/4, φ0=kx −ωt0, φ0=φ+π/2,(5) ∂(cos φ)00 ∂ct =−∂(cos φ)0 ∂ct0=−∂sin φ ∂ct0,∂(cos φ)00 ∂ct =∂(sin φ)0 ∂ct =−∂cos φ ∂ct ,(6) This phase changer gives the perfect tool to combine Faraday’s with Amp`ere-Maxwell’s equations with a phase π/2 between them, yielding to the united field F. Being the conducting current density Jkparallel to the charge density ρmovement. Fcos φ=Ecos φ+cB(cos φ)0(7) 2.1 Combining electric and magnetic fields Here is a step-by-step development using an oscillatory Maxwell’s equation with considerations of equations 3, 4 and 6. From Gauss electrics equation ∇ · (Ecos φ) = ρcos φ ε0 =cρ cos φ c ε0 Multiplying cboth sides of Gauss magnetic equation, and including a phase change ∇ · (cB(cos φ)0)=0 Combining previous equations 2 The appropriate time phase International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 201 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ ∇ · (Ecos φ+cB(cos φ)0) = cρ cos φ c ε0 (8) ∇ · (Fcos φ) = cµ0(cρ) cos φ(9) Multiplying c/c on right side of Faraday’s equation, and changing sign by a double phase change (note no phase change affecting both sides) ∇ × (Ecos φ) = −∂(Bcos φ) ∂t = +∂(cB(cos φ)00 ) ∂ct From equation 6 ∇ × (Ecos φ) = −∂(cB(cos φ)0) ∂ct0(10) Multiplying both sides by cof Amp`ere-Maxwell’s equation; including in both sides a time phase (note this will differ one phase π/2 from Faraday’s equation) ∇ × (cB(cos φ)0) = cµ0(Jk(cos φ)0) + c µ0ε0 ∂(E(cos φ)0) ∂t From equation 6 ∇ × (cB(cos φ)0) = cµ0(Jk(cos φ)0)−∂(Ecos φ) ∂ct0(11) Combining equation 10 with 11 ∇ × (Ecos φ+cB(cos φ)0) = cµ0(Jk(cos φ)0)−(∂Ecos φ+cB(cos φ)0) ∂ct0, ∇ × (Fcos φ) = cµ0(Jk(cos φ)0)−∂(Fcos φ) ∂ct0, Regrouping terms (∂ ∂ct0+∇×)(Fcos φ)=+cµ0(Jk(cos φ)0) (12) Defining the phase-shifted operator: ∂µ= ( ∂ ∂ct0,∇) and construct the combined field: F=Ecos φ+cBcos(φ−π/2), F =Ecos φ+cBsin φ(13) This new presentation of combined Maxwell’s equation contains a time phase to the variables cB, J and ct that is different from Eand the electric charge ρ. Since energy density is proportional to the squared of the fields, this phase difference is crucial for validating energy invariance. International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 202 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 2.2 The norm of the combined field F Since the combined field Fcontains a π/2 (90o) phase difference between its fields (orthogonal between them), the square of the norm is the square of each field. This orthogonality is in time and conducts to |F|2=|E|2cos2φ+c2|B|2sin2φ(14) Applying the plane wave condition |E|2=c2|B|2, with peak amplitudes of E0and cB0 u=ε0 2|F|2=ε0 2E2 0(15) This understanding of classical electromagnetism reveals the invariance of the propagation flux of energy density, inclusive, during the fluctuation; confirming that Faraday’s equation must not be treated in the same time phase as Amp`ere-Maxwell’s equation. An effect already accounted for in tensor analysis, complex vectors, and to photons by quantum mechanics (QM), but still remaining in classical electromagnetism. Consider a point charge q0at rest in a frame S. The electric field is E0(r) = 1 4πε0 q0 r2ˆ r, B0= 0. The associated electromagnetic energy (neglecting self-energy regularization issues) is U0=Zε0 2E2 0dV Now consider a second inertial frame S0moving with constant velocity vrelative to S. The transformed fields (E0, B0) are given by the usual Lorentz transformation rules. In general, a non-zero magnetic field B0appears in S0, as well, an incremented electric field as seen in Maxwell’s triangle (Figure 1E). Observe that the bottom side is invariant due to charge conservation and the only changes are in the hypotenuse and vertical side (subtracting). The square of both sections will be constant; i.e. charge and energy invariance of electromagnetism (negative sign correspond to the vector subtraction). u0=ε0 2E2 0, u0 obs =ε0 2E02−1 2µ0 B02=ε0 2(E02−c2B02) (16) The increment of the electric field due to the observed velocity is (E0=γE0), and the apparent magnetic energy (B0=γE0v/c) (where γis the Lorentz factor). u0 obs =ε0 2(γ2E2 0−γ2E2 0(v/c)2) = ε0 2γ2E2 0(1 −(v/c)2) = ε0 2E2 0=u0(17) Revealing that any increment in the energy of the observed electric field vanishes by the appearance of the energy of the magnetic field yielding to the Lorentz invariance under passive (non-boosted) transformations ( u0=u0 obs). u0 obs =ε0 2E2 0+ε0 2(∆E2−c2∆B2),∆E2−c2∆B2= 0 (18) consistent with standard electromagnetic invariant I1=E2−c2B2,with E⊥B, |E|=c|B| 3 Passive (Non-Boosted) Lorentz Transformations International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 203 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ Similar to the rest mass invariance under active transformations between boosted frames in previous work [1], but in electromagnetism under passive transformations. The presence of the magnetic energy mirrors the observed kinetic energy (but with negative sign) and the increment of the electric energy mirrors the total energy. The charge energy invariance mirrors the rest energy invariance. In contrast to passive transformations between inertial frames, active or boosted transformations correspond to actual accelerations of charged particles and their fields for acquiring a determined speed. When a charge is accelerated from rest to velocity v, work must be done, and the electromagnetic field configuration is physically altered. The resulting radiation and field changes are not mere reparameterizations of the same field. In this context, the charge is not incremented as happens in to “relativist mass” of boosted transformations. What happens is that the boosted energy received is the energy radiated by the charge acquiring the speed v. The electric and magnetic fields already at speed vwill be equivalent to the case of passive (non-boosted) transformations. Therefore, in electromagnetism, the charge does not relate with its local frame at rest as kinetic energy has due to the, sometimes named, relativistic mass increment (really a inertial increment). 5.1 Classical electrodynamics The signed-Euclidean formalism clarifies several aspects of classical electromagnetism:  The apparent oscillation of instantaneous energy density in a propagating wave is seen as an harmonic oscillator (cos φ⇔sin φ) sharing energy between electric and magnetic fields, obtaining a constant energy propagation (energy invariance).  Passive Lorentz transformations introduce an apparent magnetic fields in moving frames, but the signed-Euclidean invariant shows explicitly that the energy incremented by the electric field is vanished by this appearance of the magnetic field ensuring the energy invariance and charge conservation.  A 90oforward in time phase must be considered when combining Faraday’s and Amp`ereMaxwell’s equations. By doing this, electromagnetic energy is non observable-dependent.  In active transformations, the charge and electromagnetic energy is conserved due to the released energy radiated during the acceleration (boost). 5.2 Special relativity From a relativistic standpoint, the signed-Euclidean picture seen in Maxwell’s conceptual triangle (Figure 1E) offers an intuitive way to understand classical electromagnetism invariants without explicitly invoking the Minkowski metric on spacetime coordinates. Instead, the minus sign of the electromagnetic invariant appears in the combined field F, while the underlying space remains Euclidean. This can be seen as complementary to standard treatments based on the field tensor Fµν and its invariants I1and I2. 4 Active (boosted) Transformations of Charged Particles 5 Implications International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 204 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 5.3 Photon wavefunctions and the Riemann–Silberstein field The Riemann–Silberstein [8] vector F=E+icB plays a central role in photon wavefunction approaches and in quantum optics [9, 10]. In that complex formalism, helicity, polarization, and other quantum properties are naturally encoded. The present signed-Euclidean formalism can be viewed as a time phase analogue: the role of iis played by φ0, but no complex conjugation is introduced, and the focus is on classical invariants rather than on Hilbert-space structures. The cycle τcontain two stages of π/2 or 180otime phase, evoking the spin 1 in photons. This suggests a possible bridge between classical field-based descriptions and quantum interpretations of photons, where the time phase ∓π/2 could be related to internal degrees of freedom such as helicity or duality, while still maintaining the phase framework. This work proposes a reformulation of classical electromagnetism based on explicitly introducing a time-phase difference of π/2 between Faraday’s and Amp`ere–Maxwell’s equations. Through this phase-aware unification, the oscillatory interpretation of instantaneous electromagnetic energy density is eliminated, revealing that energy propagation is invariant even during wave fluctuation. When expressed using the combined field F=Ecos φ+cBcos(φ−π/2), F =Ecos φ+cBsin φ the energy density remains constant and matches the electromagnetic invariant conventionally derived using complex-vector or tensor formalism. This offers a conceptually accessible complementary approach that preserves consistency with Lorentz transformations:  Under passive (coordinate) transformations, the electromagnetic energy remains invariant due to compensation between electric and magnetic field contributions.  Under active transformations, invariance is maintained through radiation exchange during acceleration. Rather than replacing established tensor-based methods, this formulation provides an intuitive, real-time energy consistent representation of Maxwell’s equations. It may serve as a useful pedagogical tool or conceptual bridge between classical and quantum interpretations, particularly in explaining electromagnetic invariance without requiring full complex formalism. Further work may explore potential applications in educational frameworks and extensions toward photon field descriptions. Conflicts of Interest The author declares no conflicts of interest regarding the publication of this work. [1] J. Oreste Mazzini, “Implications of the ds Interval Under Lorentz Transformations,” Int. J. Adv. Sci. Tech. Res. 15(5), 494–504 (2025). DOI: 10.5281/zenodo.17471276. [2] Hermann Minkowski, Raum und Zeit, Address at the 80th Assembly of German Natural Scientists and Physicians, Cologne (1908). 6 Conclusion References International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 205 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ [3] Max Planck, ¨ Uber das Gesetz der Energieverteilung im Normalspektrum, Annalen der Physik, 4 (1900). [4] Albert Einstein, Zur Elektrodynamik bewegter K¨orper, Annalen der Physik, 17 (1905). [5] James Clerk Maxwell “A dynamical theory of the electromagnetic field” (PDF). Philosophical Transactions of the Royal Society of London. 155: 459–512. DOI:10.1098/rstl.1865.0008 (1865). [6] John Henry Poynting “On the Transfer of Energy in the Electromagnetic Field”. Philosophical Transactions of the Royal Society of London. 175: 343–361. DOI:10.1098/rstl.1884.0016 (1884). [7] Richard P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20(2), 367–387 (1948). DOI: 10.1103/RevModPhys.20.367. [8] Ludwick Silberstein, “Elektromagnetische Grundgleichungen in bivectorieller Behandlung,” Ann. Phys. 327(3), 579–586 (1907). DOI: 10.1002/andp.19073270313. [9] Iwo Bialynicki-Birula, “Photon wave function,” Prog. Opt. 36, 245–294 (1996). DOI: 10.1016/S0079-6638(08)70316-0. [10] Iwo Bialynicki-Birula and Z. Bialynicka-Birula, “The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism, J. Phys. A: Math. Theor. 46(5), 053001 (2013). DOI: 10.1088/1751-8113/46/5/053001. International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17882397 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ 206 ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________