Implications of the ds Interval Under Lorentz Transformations Jose Oreste Mazzini

International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 5, 2025 DOI: 10.5281/zenodo.17471276 ©2025 RS Publication, rspub[email protected]om 494 Original Article Implications of the ds Interval Under Lorentz Transformations Jose Oreste Mazzini [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: IJASTR6900EAD31D2D6 Received: 2025-09-29 Published: 2025-10-29 DOI: https://dx.doi.org /10.5281/zenodo.17 471276 Page No: 494-504 This paper re-examines the invariance of energy and mass under Lorentz transformations by distinguishing between boosts—active transformations involving actual energy changes—and passive transformations, such as translations or rotations, which involve no physical energy exchange. Building on the author’s previous work, which demonstrated that rest mass remains invariant under active boosts, this study shows that the spacetime interval ds is not invariant in active boosts. In contrast, for passive transformations, ds remains invariant, implying the invariance of proper time and relativistic mass. Additionally, the paper reinterprets electromagnetic waves as entities with non-zero ds corresponding to the wavelength λ = cτ , where τ is the proper time periodicity of energy. Furthermore, the relativistic Lagrangian is redefined, as well, restricting Minkowski’s metric to passive transformations. This interpretation supports the Theory of Space, where the fourth dimension cτ represents energy’s intrinsic wavelength, independent of the observer. Several implications for classical electrodynamics, relativity, quantum mechanics, field theory, and thermodynamics are discussed, suggesting that energy invariance under non-boosted conditions provides a unified four-dimensional (3+1) geometric foundation for modern physics. Keywords: mass at rest, observed mass, proper mass, mass invariance, energy invariance, Lorentz transformation. Cite This Paper: Jose Oreste Mazzini (2025). "Implications of the ds Interval Under Lorentz Transformations". INTERNATIONAL JOURNAL OF ADVANCED SCIENTIFIC AND TECHNICAL RESEARCH (IJASTR), vol. 15, no. 5, 2025, pp. 494-504. DOI: https://dx.doi.org/10.5281/zenodo.17471276 [email protected] This paper re-examines the invariance of energy and mass under Lorentz transformations by distinguishing between boosts—active transformations involving actual energy changes—and passive transformations, such as translations or rotations, which involve no physical energy exchange. Building on the author’s previous work, which demonstrated that rest mass remains invariant under active boosts, this study shows that the spacetime interval ds is not invariant in active boosts. In contrast, for passive transformations, ds remains invariant, implying the invariance of proper time and relativistic mass. Additionally, the paper reinterprets electromagnetic waves as entities with non-zero ds corresponding to the wavelength λ=cτ , where τis the proper time periodicity of energy. Furthermore, the relativistic Lagrangian is redefined, as well, restricting Minkowski’s metric to passive transformations. This interpretation supports the Theory of Space, where the fourth dimension cτ represents energy’s intrinsic wavelength, independent of the observer. Several implications for classical electrodynamics, relativity, quantum mechanics, field theory, and thermodynamics are discussed, suggesting that energy invariance under non-boosted conditions provides a unified four-dimensional (3+1) geometric foundation for modern physics. 1 Introduction. (Declaration note: Because Planck’s interval τis a core interval of nature, the interval ds will be selected to match this quantum interval; therefore, Minkowski’s dτ ≡τproviding a physical meaning in every step of this study). This is an extension of the author’s recent paper [1] where Lorentz invariance and Minkowski contraction must be used for a boosted transformations (meaning active boost which involve energy) since: (Ei/c)2−p2 i=m2 0c2=constant (1) This holds properly for all energies increased in the same particle of rest mass m0. In other words, for energy E2> E1(a boost from 1 to 2), the invariance is the rest mass: (E2/c)2−p2 2= (E1/c)2−p2 1=m2 0c2=constant (2) This invariance is of great importance because it tells how to add types of energy. The rest energy entails a rest frame of reference, which concludes that not everything is relative to the observer’s view. Every physical energy increment (active boost) must be referenced to the domain inherent to the particle’s spatial location (zone’s local rest frame). The energetic boost not only produces a variation in ds and τ, but its full interpretation is that it physically changes the time dilation (seen as reduction of proper time), the space contraction and the mass increases of the boosted particle. International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ Implications of the ds Interval Under Lorentz Transformations Jose Oreste Mazzini Abstract Keywords: mass at rest, observed mass, proper mass, mass invariance, energy invariance, Lorentz transformation 495 When a boost is given, the velocity of the particle changes from v1to v2and the corresponding change in ds is as follows: ds2 2=c2τ2 2= (cdt0)2−dx2 2−dy2 2−dz2 2(3) ds2 2=c2dt2 2(1 −1/c2(dx2 2/dt2 2+dy2 2/dt2 2+dz2 2/dt2 2)) (4) ds2 2=c2dt2 0(1 −v2 2/c2) = c2dt2 0/γ2 2=c2τ2 2(5) And, ds2 1=c2dt2 0(1 −v2 1/c2) = c2dt2 0/γ2 1=c2τ2 1(6) Thus, ds2 2−ds2 1=c2dt2 0(1/γ2 2−1/γ2 1) = c2(τ2 2−τ2 1) (7) Therefore, the change in the interval is negative and not invariant under boosted transformations. And, the proper time τgets reduced (∆τ < 0 for v2> v1) due to the kinetic energy increment. This agrees with Planck’s equation that energy increments when τdecrements. Unfortunately, usual deductions on Lorentz invariance assume this rest mass constant with misleading conclusions for non-boosted transformations (passive or no energy change). Maybe in the early days of SR, the rest mass energy (m0) was considered something inherent of the particles that can never change, discarding any other transformation that considers changes of this rest energy. A restriction that can explain why only one type of transformations was developed for boosted and non-boosted. With a closer analysis, this rest energy adds vectorially with the kinetic energy and not as scalars. This issue reveals that rest energy is also a kinetic type, and by that, it can be subject to changes upon observation. If equation (1) is applied for non-boosted transformations, it will consider unchanged rest mass together with the unchanged energy Ei=Ej, concluding in an unchanged momentum. This is a singular case; thus, not a variety of cases as expected for declaring an invariance of non-boosted cases. Equation (1) is useful only when the value of energy is changed Ei6=Ej, which are cases different from non-boosted or passive ones. Literally, Lorentz invariance in all transformations isn’t wrong; it only misleads the non-boosted or passive cases. They also affirm the ds invariance under all transformations, which is specifically for no energy exchange (∆ds =ch∆Eij /(EiEj) = 0), and thus, τinvariance (∆τ= ∆ds/c = 0). Contrary under boosted transformations where ds varies, and so its τ(∆τ= ∆Eτiτj/h). Therefore, the invariance of non-boosted or passive transformations must be guaranteed or sustained under another invariance; i.e., the energy invariance as the name suggests. This non-boosted invariance was developed in the previous paper (1), concluding that referential velocity viand so γiwill vary in such a way that the referential time tiwill vary inversely, giving a reduced contribution of referential mass mi. By this, an overall constant energy is obtained, resulting in the expected ds,τinvariance. Note that τinvariance implies no time dilation change, International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 496 no space contraction change, and no mass change over the particle (proper values conserved, i.e., a non-boosted transformation). A transformation that just changes referential mass and referential velocity keeping ds invariance, —a useful transformation quite different from Lorentz invariance when a boost is applied (explained in the previous paragraph). See in Figure 1 the different “triangles” or possible transformations between inertial frames, all maintaining the same hypotenuse (same energy or non-boosted case). This study solves the twin paradox by clarifying the core difference between boosted and nonboosted transformations. The traveling twin suffers an energetic boost (a time dilation), whereas his brother is seen through a non-boosted or passive transformation (no time dilation). As experienced in our daily life with the precision of GPS systems. With the reasoning, in the barn-ladder paradox, only the ladder will suffers the space contraction. FIGURE 1: Non-boosted transformations, referential mass and momentum from different inertial frames of reference conserving the total energy. Note the importance of energy aside from the conservation law: energy is what produces the gravitational effect and not only the rest mass (Einstein’s contribution versus Newton’s approach). In the same way, boosts of energy are what generate time dilation and space contraction, and not the relative speed seen by an observer. Additionally, energy contains inertia and not only rest mass (Einstein’s contribution versus Galileo’s approach). On each boosted transformation, total inertia grows/diminishes, and so will be the energy requirement for future changes in velocity (in mass terms, increase/decrease of relativistic mass; e.g., verified in particle accelerators). This inertia is involved in the principle of least action, obtaining a zero slope at this least point (not a crossing point). It is also present in quantum mechanics (QM) with a gradual oscillatory realm, i.e., Euler’s equation (not a square wave, on-and-off fluctuation). 2 Massless photons or electromagnetic waves: While in Minkowski’s [2] standard relativity, the lightlike path satisfies ds = 0, here the interval is reinterpreted as ds =cτ =λ, a finite proper time wavelength corresponding to photon energy periodicity. Although the Lorentz gamma factor at the speed of light goes to infinity γ→ ∞, its referential time also goes to infinity t0→ ∞ due to the zero rest mass Emass =m0c2=0=h/∞. Therefore, equation (5) manages an infinity divided by another infinity, but their quotient has a defined value of the proper time τ. A non-zero value for massless interval’s ds (cτ). A value that is International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 497 not observable-dependent since all reference frames will view the same velocity c (Einstein’s axiom of Special Relativity). If a boost or energy increment is given to the photon, the proper time is reduced accordingly to Planck’s equation; a reduction of interval ds as expressed in the previous section (shorter wavelength). The Minkowski interval ds for light is λ, light’s wavelength (cτ); exactly the author’s 4th dimension. A physical longitudinal parameter that doesn’t depend on the observer, nor on the historic time dilations or the initial time compared to Minkowski’s four dimension ct (event time t). The passage of time or event time will be a temporal parameter and not a longitudinal one, where its value will be the accumulation of cycles of value τplus the circumstantial fraction of this oscillation. Reinforcing the concept, quanta’s energy is not transforming across observers; rather, energy is an intrinsic 4D attribute associated with a periodic proper time (τ=h/E). This implies a geometric or cyclic ontology of energy — the quanta don’t “gain” or “lose” energy by relative motion-observation; only when an interaction or boost is applied. The worldline corresponds to a 45-degree triangle with sizes equal to λ. Not a zero λor cτ or ds that gives just a point in the worldline. A non-zero λmeans a line in the worldline, and for two photons with different frequency, the 45-degree line will be described by more or less segments of intervals λ. FIGURE 2: Geometric relation between rest energy, kinetic energy and total energy. The confusion arises from the gamma factor that, at light’s speed, its value goes to infinity and induces a ds equal to zero; a misleading full time dilation (τ= 0) and full space contraction (λ= 0). A common misinterpretation that even the author has been involved in some papers. But if one considers the other infinite value of massless particles, t0→ ∞, the physics returns to the correct value of τand λ. See Figure 2, case A at some relativistic speed v and case B at almost speed c. Note lower part of the triangle going to zero, i.e., going to an infinite value for γand t0. As indicated previously, from equation (5), the interval cτ is the quotient of ct0/γ → ∞/∞=cτ =ds invariant) ; both infinities must be taken into consideration. Another way to understand these massless particles, is that all the inertial frames will observe the same value of τ. Not really an invarinace of τbecause it is a singular transformation since the referential velocity will always be c (Einstein’s SR axiom). Young Einstein’s thoughts about what would be seen when traveling over a ray of light? The answer is the 4th D fluctuation at the same rate as seen from any inertial frame of reference. International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 498 3 Four-dimensional space: Einstein’s and Minkowski’s deal with the same Lorentz γfactor as can be seen geometrically by triangles. In Einstein’s case, the hypotenuse contains the total energy, one side the kinetic component, and in the other the rest energy (kinetic of the 4th dimension as seen by TS). Dividing the three sides by the total energy, the hypotenuse equals 1, one side equals v/c and the other equals 1/γ =m0/mrelativistic. The same happens to Minkowski’s triangle of equation (3), the hypotenuse as cdt0, one side as ∆x=dt0v, and the other as cτ. Dividing the three sides by cdt0, the hypotenuse equals 1, one side equals v/c, and the other equals 1/γ =τ/dt0. Note the following difference: in Einstein’s approach, the hypotenuse is linked to the total energy, whereas in Minkowski’s approach, energy’s wavelength cτ (or spacetime interval ds) is in one side. This aligns with Planck’s interval inversely to energy, and De Broglie’s wavelength [3] inversely to energy. Therefore, the arrangement (cdt0, i∆x, j∆y, k∆z) = −→ cτ where i=j=k=ijk =√−1, will represent energy at a side. A Pythagoras subtraction (hypotenuse minus the other side) instead of addition of their sides in Einstein’s approach. This sustains the actual Minkowski’s matrix and metric as (+, -, -, -). But care is needed to identify the parameters involved, as it will easily mislead, like not knowing which sign is applicable or by naming a boost when it isn’t. Inverting the signs (-, +, +, +) will refer to antimatter as Feynman [4] described in his diagrams (negative time). The Minkowski matrix (for the side of the triangle) will ensure proper time and spacetime interval invariance (for non-boosted or passive; see equation (8). −→ cτ0= (cdt0 0, ix0, jy0, kz0)≡    cdt0 0 ix0 jy0 kz0     =    γ−γβ 0 0 −γβ γ 0 0 0 0 1 0 0 0 0 1         cdt0 ix jy kz     (8) The modulus of the matrix equals one, something expected since equation (8) is for non-boosted transformations; it maintains quantum energy invariant by τinvariance (E=h/τ =Cte.). This non-observable dependent is crucial for the 4th D (author’s TS proposal). The vector of equation (8) matches with Hamilton’s quaternions [5] (cdt0, i∆x, j∆y, k∆z) = −→ cτ, where i=j=k=ijk =√−1. A physical four-dimensional space where the contraction of 3D values (∆x, ∆y, ∆z) due to the apparent γ, and the growth of cdt0makes a net constant energy’s wavelength modulus. Equation (9) recalls Hamilton’s great equation derived from quaternions, now used for wave equation; appropriate for the TS proposal. −(∂2/∂x2+∂2/∂y2+∂2/∂z2)=(i∂/∂x +j∂/∂y +k∂/∂z)2(9) Equation (10) is for a boosted Lorentz transformation going from rest up to speed v; preserving rest mass invariance. Note the arrangement is a Pythagoras addition, i.e., the classical and not the Minkowski’s metric. −→ ct0 0= (cτ0, x0, y0, z0)≡    cτ0 x0 y0 z0     =    γ−γβ 0 0 −γβ γ 0 0 0 0 1 0 0 0 0 1         cτ x y z     (10) 4 Relativistic Lagrangian: This study will redefine the relativistic Lagrangian for a free particle; where its mass is considered at rest but its integral is done over many boosted transformations with misleading consequences in International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 499 Physics. A free particle do not necessarily will be at rest, the case must considers all moving particles without external interactions; thus, with its invariant relativistic mass and momentum γm0c. Additionally, the integral must consider non-boosted or passive transformations, if not, the particle won’t be anymore in a free condition. Therefore, the relativistic Lagrangian can be deduced assuming the passage of time as equivalent to the different referential times (ti) of non-boosted transformations. Where the relativistic fourmomentum (p=Etotal/c) will be invariant under any inertial frame (i); observing more or less kinetic (due to vi) matching with less and more referential mass (mi). Therefore, the relativistic fourth-momentum (mRc), of its constant speed from rest, is conveniently taken over time (ti). Thus, the action S will be the accumulation (integral) of this fourth-momentum over time (dti). A minus sign must be involved since greater time goes for less energy. Therefore: S=−mRcZds =−m2 Rc2Zdτ =−m2 Rc2Z(1 −v2/c2)dti(11) Since, S=RLdti L=−m2 Rc2p1−v2/c2(12) 5 Transformation key invariances: Transformation Invariant in Invariant in TS Non-invariant Type Standard SR in TS Passive (non-boosted) Stationary ds,τ, total energy, Referential mass, Transformations relativistic mass, referential velocity, (energy is conserved) four-momentum, fourth-momentum photons as singular case (Minkowski’s metric for matter and antimatter) (Lagrangian with mR) Active (Boosts) ds, Rest mass m0, Rest mass, fourthds,τ, Relativistic fourth-momentum, momentum, mass, (energetic changes) photon’s ds = 0 photon’s ∆ds = ∆λfour-momentum (Minkowski’s metric) (Classical metric) (Lagrangian with m0) 6 Suggested implications for physics:. Aside from previous distinction between active and passive transformations with different fourdimensional vectors and invariances, metrics, photon’s non-zero ds, redefined relativistic Lagrangian, etc.; this paper give the following list of further implications for physics that deserves analysis: International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 500 6.1 Special Relativity (SR): •Einstein’s great contribution in SR was to give physical meaning to Lorentz’s gamma factor and to include rest energy in the total energy. Unfortunately, the fact that this total energy is relative (observer-dependent) has caused some confusion over time. •The author’s Theory of Space (TS) reveals that the energy of the quantum oscillation [6] is what is considered as rest energy by SR. An oscillation between 3D and the proposed fourth dimension cτ. TS four-dimensional oscillatory realm of nature. •Lorentz γexplains what happens to physical values of space and time, where TS contributes in pointing out that the passage of time (universal reference) remains unalterable. Energy will change the scale of this time and the bodies will behave physically in this new scale. This makes possible that particles with different τand different accumulative values for their event time, can meet (future common events) many times through time. For example, in the twin paradox, at the end of the journey, the brothers can hug each other. The same interpretation applies to the geometrical distance (space occupied) which remains unalterable. Energy will change the scale of proper distance and the body’s will behave physically in this new scale. This makes possible that particles with different τand different accumulative values for their position, can meet (future common events) many times in a common place. For example, in µdecay over the surface of Earth, at the end of the journey, the contracted distance (µ0sperspective) will match the Earth’s observed distance. An analogy for the case Lorentz γ→ ∞: the handle of the proper clock continues moving, indicating the flow of time, and pointing the ”now” instance. The new interpretation is that the scale of the dial changes in such a way that all the circumference indicates the same value; thus, a complete time dilation with ∆t→0 while the handle continuous moving. The same thought for space analogy; engineer’s scale rule, the length of the ruler doesn’t change while the different scales indicate different length values. This will guide for a correct understanding of black holes avoiding singularities. •Rest energy as a kinetic energy of the oscillation in the 4th D which is at speed c. This quantum perspective matches with Einstein’s relativistic axiom of common speed c from any reference frame. Non-boosted transformations change this referential oscillatory speed, but with the contribution of the referential speed, the total speed c remains constant for all observers (energy conservation and proper values invariance). 6.2 Classical Electrodynamics: •The invariance of proper energy and space implies that electromagnetic energy density and field intensities are not merely observer-dependent projections but reflect a 3+1D realm with intrinsic oscillations in 3D space. This explains why both fields (Eand icB) [7] grow and diminish together; not one growing at the expense of the other (as pendulums do by sharing kinetic and potential energy). •The Lorentz force law could then be reinterpreted as describing how the projection of invariant field energy varies along a 3D trajectory, rather than the field itself transforming. •Non-boosted transformations will change the referential displacement current and referential fields, maintaining invariant its combined field (E2−c2B2), accomplishing conservation laws). Maxwell’s equations would preserve their form, but the Poynting vector would explain 3D fluctuation while the energy distribution is invariant in the 3+1D manifold. 6.3 Quantum Mechanics (QM): •Energy quantization (E=h/τ) becomes a geometric identity, not a statistical one. The invariant “fourth dimension” can hold infinite 3D scenerios. •The proper time (τ) — being invariant — defines the quanta’s intrinsic periodicity, linking directly to author’s fourth’s dimension (cτ =λE). Planck’s and Einstein’s contributions reveal discreteness of energy in 3D. A constant action, where fastest presence in 3D implies a more energetic presence and vice versa. •Lorentz transformations would no longer “mix” temporal time and space physically but would express how 3D observers project the same invariant oscillation differently (multiple eigenstates in International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 501 superposition at 4D). •A 3+1 realm (4D) where 3D holds some physical parameters, and the 4th D holds others (embracing conservation laws) resulting in noncommutativity between them due to their phase difference. A 3+1 realm like what ADM [8] formalism holds, a 3D randomly arranged foliation containing each one an eigenstate ϕ(r, τ, t) (multi-determinism, not in-determinism). •TS 3+1 realm (4D) makes possible that entangled particles manages their conserved parameters in the 4th D, even when they are spatially separated in 3D. A concept of locality at the 4th D. •TS 3+1 realm (4D) makes possible to overcome 3D barriers when the particles are in the phase at the 4th dimension, i.e., out of the 3D barrier impediment(tunneling effect). •Coherent particle have the same oscillatory rate and phase. •Wavefunctions (Ψ(r, τ, t)) would then be seen as 3D projections of a 4D cyclic process. Quantum superposition represents multiple 3D projections of a single 4D cyclic state, each realized sequentially in λ-space. Probability thus reflects geometric projection weights, not indeterminacy [9]. In Hilbert’s 4D orthonormal vector space, each coordinate hosts an independent eigenstate, with its corresponding 3D space, and with a probabilistic 4D projection over 3D followed up by Ψ∗Ψ. TS also proposes an aleatory sequential projection supported by a) Heisenberg’s and Schr¨odinger’s contributions of noncommutativity between some physical parameters and the use of complex numbers, b) evidence of individual states when experimental measurement-observation is done, c) the arrow of time since a probabilistic sequential realm is irreversible, and d) the spin as the two angular momenta (cw and ccw) of the quantum oscillation between the 3D and the 4th D. •Born’s rule for superposition of states changes to Kolmogorov’s [10] additive axiom for mutually exclusive events (sequential states). •Energy’s inertia reflects in a gradual projection of 4D over 3D, embracing the importance of phases. Euler’s equation, and complex numbers as part of quantum math. The double-slit experiment [11] reveals this realm through the interference pattern. •Physical parameters of space and particles are two entities that coexist in nature. Where the space oscillates between 3D and the 4th D (3+1 realm), the particle will follow this behavior, i.e., a wavy particle. Not one entity with two antagonist roles as understood by the wave-particle duality [11]. In this quantum system, the space can be split and still behave as one, meanwhile, it doesn’t suffer an interaction-observation. Its particle will be located randomly in any of the subspace at the rate of its oscillation. 6.4 Quantum Field Theory (QFT): •The Lagrangian must be invariant under this 4D periodic structure — meaning fields are inherently periodic in τ. •This introduces a discrete energy spectrum geometrically (through τquantization), rather than via canonical quantization. •Propagators could then reflect phase progression along τ, not propagation through continuous Minkowski time. •The vacuum (zero-energy state) becomes a 4D standing wave — a stable configuration of proper time cycles — removing the need for “vacuum fluctuations” as probabilistic artifacts. 6.5 General Relativity (GR): •Being energy, in closed systems, non-dependable to observers, the energy of the universe is also non-observable dependent. A fix value that astrophysicists can take into consideration. A crucial way to see all the energy (e.g., masses) in the universe, even for dark matter (maybe pure energy) and dark energy (maybe not fluctuating with 3D); being all of those energies not observable-dependent. Furthermore, the gravitational effect is not observable-dependent. •Proper time and mass are invariant when no external energy is applied to the system (non-boosted case). When no external intervention or boost is present, the gravitational potential field is shared with observed kinetic energy; a net zero scenario (e.g., pendulums). •Gravitational redshift and time dilation continue showing gravitational potential contributions International Journal of Advanced Scientific and Technical Research, ISSN 2249-9954 DOI: 10.5281/zenodo.17471276 WEB: http://www.rspublication.com/ Issue 15 volume 5 2025 ____________________________________________________________________________________________________________________ ©2025 RS Publication, [email protected] _________________________________________________________________________________________________________________________________________________ 502