An Energy-Periodicity Representation of Special Relativity: Its implications to the forth dimension and to QM

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.17726060 Original Article An Energy-Periodicity Representation of Special Relativity: Its implications to the forth dimension and to QM 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: IJASTR69260BACC0C8F Received: 2025-10-27 Published: 2025-11-26 DOI: https://dx.doi.org /10.5281/zenodo.17 726060 Page No: 102-109 This paper revisits the geometric and energetic foundations of Special Relativity (SR) and Quantum Mechanics (QM), proposing that SR kinematics can be represented in a Euclidean energy– periodicity geometry without altering empirical predictions. The minus sign in Minkowski’s metric is interpreted as an orientation convention rather than evidence of a fundamentally pseudo-Euclidean space. By introducing energy as a primary invariant, E = h/τ , and the corresponding fourth dimension curl dimension λ = cτ , the formalism unifies relativistic and quantum periodicity. Within this framework, the sequential presence of eigenstates and their phase relations naturally explain measurement single state, uncertainty, spin, the arrow of time, and conservation laws in a unified Euclidean representation of spacetime-energy. Keywords: Special Relativity; Minkowski metric; Quantum Mechanics; Lorentz invariance; Lorentz transformations; covariant; cotravariant; Cauchy inequality. Cite This Paper: Jose Oreste Mazzini (2025). "An Energy-Periodicity Representation of Special Relativity: Its implications to the forth dimension and to QM". INTERNATIONAL JOURNAL OF ADVANCED SCIENTIFIC AND TECHNICAL RESEARCH (IJASTR), vol. 15, no. 6, 2025, pp. 102-109. DOI: https://dx.doi.org/10.5281/zenodo.17726060 ©2025 RS Publication, rspub[email protected]om 102 Jos´e Oreste Mazzini Lima, Peru [email protected] This paper revisits the geometric and energetic foundations of Special Relativity (SR) and Quantum Mechanics (QM), proposing that SR kinematics can be represented in a Euclidean energy–periodicity geometry without altering empirical predictions. The minus sign in Minkowski’s metric is interpreted as an orientation convention rather than evidence of a fundamentally pseudo-Euclidean space. By introducing energy as a primary invariant, E=h/τ, and the corresponding fourth dimension curl dimension λ=cτ, the formalism unifies relativistic and quantum periodicity. Within this framework, the sequential presence of eigenstates and their phase relations naturally explain the measurement of a single state, uncertainty, spin, the arrow of time, and conservation laws in a unified Euclidean representation of space-time-energy. Einstein’s 1905 formulation of Special Relativity provided a revolutionary connection between mass, energy, and motion, while Minkowski’s 1908 geometric interpretation introduced the concept of a four-dimensional continuum combining space and time. The traditional understanding of Minkowski space as “pseudo-Euclidean”—with metric signature (−,+,+,+)—stems from identifying temporal and spatial coordinates as components of a single invariant quantity ds2=−c2dt2+dx2+dy2+dz2. However, a careful geometric and energetic reconstruction reveals that this metric was originally Euclidean in nature; the apparent minus sign arises from the orientation of physical vectors, not from a fundamental property of space. Einstein’s total energy as a vectorial addition of mass energy and kinetic energy can be represented in a triangle (see Figure 1A). Minkowski’s space-time can be presented as a vectorial or orthogonal addition between ds, dt0and ∆x; i.e. Minkowski’s triangle (Figure 1B). Both expressions of the same physical relation viewed from different conceptual domains. Reinterpreting them within a common energetic framework clarifies that the space is indeed Euclidean, and that the so-called pseudo-Euclidean formulation was a historical simplification. Notation. tdenotes coordinate time in a given inertial frame; t0denotes the corresponding interval in the particle’s rest frame; τdenotes the internal period associated with energy via E=h/τ; and λ=cτ is the associated 4th dimension cycle. ©2025 RS Publication, [email protected] International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17726060 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 103 An Energy-Periodicity Representation of Special Relativity: Its implications to the forth dimension and to QM 1 Introduction Abstract Keywords: Special Relativity, Minkowski metric, Quantum Mechanics, Lorentz invariance, Lorentz transformations, covariant, cotravariant, Cauchy inequality. Figure 1: Comparison between Einstein’s and Minkowski’s energy triangles. The total energy (hypotenuse in Einstein’s triangle) corresponds to the side ct0in Minkowski’s construction. Both relations are geometrically Euclidean, differing only by the interpretation of the vector components. Drawing D reveals Feynman’s [4] antimatter opposite internal time −ct0. Einstein’s triangle expresses the total energy as the vector addition of rest and kinetic components: E2= (m0c2)2+ (pc)2.(1) This forms an Einstein’s and Planck’s right triangle, where Eis total energy at the hypotenuse, E0is the rest-energy side m0c2=nh/dt0, and Ek=pc = (nh/τ)(v/c) is the other side. Minkowski’s interval [2] is obtained from Planck-Einstein’s concept [1, 3] 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). The geometry is strictly Euclidean in the energy domain. Minkowski’s construction, by contrast of equation 1, relates temporal and spatial displacements in the rest frame as (c dt0)2= (v dt0)2+ (c dτ)2.(2) Here dτ corresponds to the proper time associated with the energy wavelength λ=cτ, and t0 for observer’s time ds2=c2dτ2=c2dt2 0−v2dt2 0,(3) The apparent minus sign simply reflects the oriented components of a Euclidean vector (Figure 1B). Equivalently, SR kinematics can be reproduced within an energy–periodicity geometry ©2025 RS Publication, [email protected] International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17726060 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 104 2 Einstein–Minkowski Energy Geometry where the metric remains Euclidean and the sign difference denotes orientation. The invariance of ds has been central to relativity, but it applies strictly to systems where intrinsic energy remains constant. As seen in Einstein triangle (Figure 2C) and its equivalent in Minkowski triangle (Figure 1B); any involvement of energy will affect the length of Minkowski bottom side (∆ ds =c∆τ), thus, not a ds invariance. This is the goodness of previous identification of triangle’s sides with physical spatial displacements. For boosted systems, where energy changes as E=h/τ,τmust vary, so: ds0=cτ06=ds =cτ. (4) From the Minkowski triangle, a boosted type will preserve the hypotenuse (rest mass) meanwhile the kinetic side grows, and the energetic side will diminishes, i.e., a shorter τproper of energy increment. The reason why the Cauchy inequality [6] is reversed; energy is contravariant with space. Thus ds2cannot be invariant under active-energy-changing transformations. The assumption of universal invariance led to the pseudo-Euclidean interpretation, unnecessary once energy is recognized. Figure 2: Three ways rest energy could interact with kinetic energy: A) as scalar (different type of energy), B) Same type, adding as vector the extra dimension but relative to the observer’s frame, C) Same kinetic type linked to a local rest frame, D) Same as ”C” but with kinetic components depending from observer frame. Figure 2A shows the way how mass energy will be added to kinetic if those energy where not of the same kind; an scalar addition of terms (e.g. kinetic energy with electromagnetic energy). Figure 2B show a possible full relativistic way of adding vectors (mass and kinetic as the same kind) but without no link to a rest condition. On each pc increment, a new resultant is obtained. ©2025 RS Publication, [email protected] International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17726060 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 105 3 The Invariance of ds Figure 2C shows the correct way deduced by Einstein in 1905. A rest-energy invariance under active (boosted) transformations indicating that velocity is not fully relative to the observer; their is a local rest condition in nature. The increments in kinetic energy is reflected as a greater side of the trianglem, meanwhile, the bottom side is constant (invariant). The hypotenuse, total energy, will grow revealing a greater Lorentz’s γfactor. When the energy involved is infinite, the γwill be also infinitive and time dilation will go to zero. Do not confuse this condition with the ones in electromagnetic waves and photons that goes at speed c. Photons do not contain infinite energy, their energy is the known h/τ, therefore its proper time is not zero. Meanwhile, photons gamma goes to infinite due to its c speed, their t0will also go to infinite because they are massless. Both terms goes to infinite and their quotient holds proper time τ. A common misconception that proper time, in electromagnetic waves and photons, is zero (ds = 0); an error that the author was involuntarily involved in its first papers. Since mass energy and kinetic energy are the same type of energy, the mass energy can change upon the observation of any reference frame in the same way that the apparent velocity changes. But the combined energy is unchanged or invariant as shown in Figure 2D for passive (nonbooted) transformations. The apparent velocity (vertical side) and the apparent mass-energy (bottom side) will change obtaining a constant hypotenuse length (total energy, ds invariance). No time dilation or space contraction since energy is unchanged as seen in the quadrant of the circle (Figure 2D). Note that previous work [5] has identified mass energy as kinetic energy at the 4th dimension and subject to variations depending the observer frame. Energy, expressed as E=h/τ, is an intrinsic property of a system, not an observer-dependent one. The apparent relativity of total energy, under non-booested transformation, arises from considering rest energy invariant (from boosted transformations) and vary only the kinetic energy due to the apparent velocity. These kinetic component being observed-dependent must be considered together with the observer-dependent contribution to mass-energy; both are kinetic energies and observer-dependent (vector addition; thus, the same type of energy but from different coordinate). From the knowledge of Planck’s periodicity and the existence of quantum spin, this paper analyzes a 4th dimension rotational model of circumference given by curled λcorresponding to proper’s view. A moving observer will considered it as a spiral of length c2τ2=v2τ2+ reducedlambda2. Since the spiral length λmust be constant from SR axiom, speed c is constant from all observers and τis also constant (non-boosted); this will reduce the lambda of the apparent mass-energy from observers view. This changes our actual understanding by combining both energies; concluding in that total energy is conserved under non-boosted transformations, i.e., invariant or observer-independent(Figure 2D). Recognizing this resolves the paradox between quantum invariance and relativistic relativity, clarifying that there are two types of Lorentz transformations; the active (boosted) and passive (non-boosted) ones. This last type maintains physical values of space contraction, time dilation and energy invariance; the opposite of boosted transformations (energetic interactions over the system). ©2025 RS Publication, [email protected] International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17726060 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 106 4 Energy as Observer-Independent Applying a boosted transformation to the 4th dimension rotational model, where, the curl at rest is λ0=c τ0. When boosted to a new speed v, the spiral seen from the rest frame satisfies: λ2 1=c2t2 1=v2t2 1+c2t2 0,(5) yielding t1=γt0,(6) the standard Lorentz time-dilation relation. This result follows from the invariance of rest energy and is consistent with muon-lifetime and GPS measurements. The observer-independence of τand its corresponding longitudinal value cτ, can redefine the 4th dimension as λ, energies wavelength. Energy as a core parameter not only involved in the gravitational effect, in the relativistic inertia, and in Lorentz’s boosted transformations. Reserving the passage of time (also event’s time) to the accumulation of τcycles plus a fractional part. This fraction of the curled 4th-D cycle appears in the quantum phase: Ψ(r, τ, t) = eiE(t−t0)/~Ψ(r, τ, t0),∆t≡t−t0 τ.(7) Here ∆tis the number (possibly fractional) of completed internal cycles since t0; when ∆tis an integer the system returns to its original eigenstate. The dimensionless phase angle 2π∆t ensures continuous evolution of the wavefunction over many cycles. From the quantum evidence upon measurement reveling a discrete eigenstate plus the arrow of time, emerges that cycles are random sequential progression of eigenstates. Each eigenstate defines a 3D configuration, while transitions occur along the invariant 4th dimension λ= cτ. The total energy remains conserved through these alternations, fulfilling conservation laws naturally in Euclidean 4D geometry. The sequential progression provides a geometric basis for the wavefunction: the “probability amplitude” is a 4D oscillation connecting eigenstates through cτ. Each state corresponds to a stage on a continuous but periodic 4D trajectory. This progression affirms that under interactions, e.g., measurements, the system will reveal its current eigenstate, avoiding conflicts with the collapse of the 4D superposition of eigenstates. The uncertainty principle arises from phase differences between conjugate variables, not stochasticity. Position–time describing the eigenstate at 3D, and momentum–energy preserving conservation laws at the 4th D; a complementary projections of one 3 + 1 D oscillatory process. Their phase lag π/2 enforces: ∆E∆t&~/2,(8) a deterministic phase constraint in 4D Euclidean space. Conservation laws are preserved at the 4th D even if the system is composed by split spaces (entangled particles). This passage through the 4th D makes it possible to overcome a 3D barrier (tunneling effect). ©2025 RS Publication, [email protected] International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17726060 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 107 5 Time dilation in boosted transtormations 6 Sequential quantum eigenstates in the 4th dimension 7 The phase nature of uncertainty and locality at the 4th D Figure 3: Sequential quantum eigenstates represented as oscillations between 3D configurations through the 4th dimension λ=cτ. The 4D oscillation ensures conservation of energy, momentum, and charge, meanwhile at 3D, other physical parameters depends on its eigenstate. Figure 4: Helical propagation in 4D: four helices represent the full symmetry of matter and antimatter. The dynamics of the curled 4th D explains rest energy or energy at the 4th D, and its angular momentum as spin. ω= 2π/τ (9) With a random CW and CCW behavior per cycle through time. Matter and antimatter arise from upward CW/CCW and downward CW/CCW respectively. Antimatter downward helices shares the same global arrow of time as matter but an opposite internal phase. The unidirectional arrow of time emerges from the impossibility to reverse an random sequential structure. ©2025 RS Publication, [email protected] International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17726060 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 108 8 Helical Propagation and Spin Restoring Minkowski geometry as Euclidean dissolves the conceptual divide between SR and QM. The 4th dimension cτ defines the system’s intrinsic energy and periodicity. Lorentz active (boosted) transformations act as energetic interaction over the quantum system with scaled physical values, not geometrical deformations. Passive (non-boosted) transformations unifies view and preserves Einstein’s and Minkowski’s insights while refining their interpretation. The framework reconciles relativistic geometry and quantum periodicity through the same invariant: the energy–wavelength relation. Recasting Minkowski geometry in energetic terms shows that SR and QM share the same underlying periodic structure. Under passive (non-boosted) transformations, the invariant quantity is the internal period τ, this energy–periodicity representation preserves all verified Lorentzinvariant predictions yet provides a Euclidean geometric interpretation linking relativity and quantum mechanics through the same invariant relation λ=cτ. [1] A. Einstein, “Zur Elektrodynamik bewegter K¨orper,” Annalen der Physik, 17 (1905). [2] H. Minkowski, “Raum und Zeit,” Address delivered at the 80th Assembly of German Natural Scientists and Physicians, Cologne (1908). [3] M. Planck, “¨ Uber das Gesetz der Energieverteilung im Normalspektrum,” Annalen der Physik, 4 (1900). [4] 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. [5] J. O. Mazzini, “Implications of the ds Interval Under Lorentz Transformations,” International Journal of Applied Science and Technology Research, Vol. 15, No. 5, 2025. [6] Agustin-Luis Cauchy, ”Sur les formules qui r´esultent de l’emploie du signe et sur ¿ ou ¡, et sur les moyennes entre plusieurs quantit´es”, Cours d’Analyse, 1er Partie: Analyse Alg´ebrique 1821; OEuvres Ser.2 III 373-377, 1821. ©2025 RS Publication, [email protected] International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17726060 WEB: http://www.rspublication.com/ Issue 15 volume 6 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 109 9 Discussion 10 Conclusion Acknowledgment. This work builds upon the foundational contributions of Einstein, Minkowski, and Planck, whose century-old insights into the unity of space, time, and energy continue to inspire deeper clarification of modern physics. References