From Photon–Matter Dynamics to Field Quantization: A Framework for Electromagnetic, Gravitational, and Cosmological Behavior
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This preprint manuscript presents the main theoretical results developed in this work. The framework and presentation will be further refined for the final journal submission. This version has not undergone peer review.
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From Photon–Matter Dynamics to Field Quantization: A Framework for Electromagnetic, Gravitational, and Cosmological Behavior Sourabh Singha,∗ aTata Institute of Fundamental Research Hyderabad, 36/P Gopanpally, Hyderabad, 500046, Telangana, India Abstract We develop a unified, covariant framework for analyzing energy–momentum exchange between photons and massive particles. Starting from a particle-based description of emission and absorption, we obtain a compact derivation of the relativistic Doppler formula in which recoil exchange plays a central role. We then extend the same covariant structure to the electromagnetic field. In the rest frame of a massive charged particle, we construct a quantization scheme by promoting the charge-density contribution to an operator, which yields a harmonic-oscillator representation of the field energy. This approach preserves Lorentz covariance and reproduces familiar Lamb-type radiative corrections without relying on free-field assumptions. Finally, we explore a weak-field correspondence between electric and gravitational acceleration, which allows the quantization method to be applied to the linearized gravitational field. Treating positive and negative operator components leads to harmonicoscillator–type energy, from which a uniform background energy density and simple large-scale expansion behavior emerge. Keywords: Relativistic Doppler Effect, Field Quantization, Quantum Electrodynamics, General Relativity, Quantum Gravity, Lorentz Transformations, Cosmological Expansion, Electromagnetic Mass, Velocity Addition, Cosmology 1. Introduction For over a century, the relativistic Doppler effect and the velocity addition formula have been treated as distinct consequences of special relativity, one derived from wavefront geometry or time-dilation arguments, and the other attributed to kinematic transformations in spacetime [1–5]. In this work, we present a unified, particle-based ∗Corresponding author. Emails: [email protected],[email protected]
derivation of both phenomena using a single energy–momentum conservation framework, without relying on wave arguments or explicit time-dilation reasoning. This common footing reproduces Einstein’s Doppler formula and the standard velocityaddition law, while clarifying the physical meaning of each term and showing that recoil effects, often treated as separate corrections [20–22], emerge directly from the kinematics. To uncover the connection between the Doppler effect, particle energy transformations, and the velocity addition rule, we revisit the Doppler effect from first principles. Our aim is to develop a unified, particle-based formulation in which recoil emerges naturally. We begin by reexamining the foundational structure of velocity and energy transformations, starting from Galilean to Einsteinian formulations. These transformations also underpin the relativistic Doppler effect, where Einstein reinterpreted Doppler’s original wave-based derivation using time dilation to obtain the frequency shift for light [4]. Experimental validation followed in the Ives–Stilwell test for longitudinal shifts [6,7], and later in direct measurements of the transverse Doppler effect [8]. While the wave-based picture has been historically prominent, a complementary particle-based approach was introduced by Schrödinger [9,10], who derived the geometric-mean Doppler formula using energy and momentum conservation. Related work by Jauncey [11], Fermi [12] recovered the relativistic Doppler effect through atomic recoil. Davisson [13] pointed out that the observed frequency in emission processes subtly differs from the rest frame value due to recoil, underscoring the need to carefully disentangle relativistic and recoil effects. A later treatment by Barnett [14] presented the Doppler shift as incorporating recoil effects, with the relevant velocity taken as the average of the absorbing or emitting particle’s velocities before and after the interaction. While these treatments [11–14] successfully captured key features of the phenomenon, they often assumed the emitted photon’s energy to be hν, without fully accounting for the portion of energy transferred to the recoiling atom. While earlier works, including those by Schrödinger, Redžić [15,16], and Giuliani [19], recognized that the energy from atomic transitions is shared between the photon and the recoiling atom [17,18], their treatments left the physical meaning of each term in the Doppler formula largely implicit. Giuliani [19] correctly emphasized that the Doppler contribution originates physically from recoil, although his 2
discussion was at a qualitative level. In standard treatments, however, the connection between frame–dependent energy shifts and recoil effects is still obscured by the mathematical formalism. As noted by Redžić [10], standard derivations do not clearly expose the physical origin of the Doppler components, making intuitive interpretation challenging. Along the way, this formulation resolves longstanding ambiguities associated with the additional recoil correction in light–matter interactions, revealing that these contributions are already implicit in the relativistic structure. Traditional treatments often introduce a recoil correction term, h2ν′2 2mc2, to account for momentum exchange [20– 22]. However, such corrections may obscure the fundamental kinematics, leading to conceptual confusion and quantitative discrepancies, including in high-precision contexts such as Doppler-broadened spectroscopy. However, both the non-relativistic formulation (Eq. D.8) and the relativistic analysis (Eqs. 13 and 16) show that this contribution is already inherent to the Doppler framework. In the relativistic case, the recoil-associated shift is embedded within the longitudinal term γhν(Vcos θ′/c), eliminating the need for any separate correction. This cancellation arises naturally in both formulations, indicating that the recoil contribution is not external to the theory but emerges directly from the transformation structure. The same conclusion applies to two-body decays involving massive particles, where recoil effects are fully accounted for through energy-momentum conservation, with no additional terms required. To build intuition for this structure, consider a classical analogy: when a ball is projected from a moving platform, it appears more energetic in the lab frame if projected in the direction of motion due to two contributions: the kinetic energy already present from the platform’s motion, and the energy imparted during projection. Together, these yield the term γγomc2. Additionally, the energy transfer depends on the ball’s momentum along the platform’s motion, leading to the term γγomV Vocos θ′ with a natural cos θ′factor; this is the only component causes recoil and alters the ball’s energy. Unlike massive particles, a photon alters its energy via frequency shifts, analogous to how massive particles undergo velocity transformations. When emitted by a moving atom, momentum conservation causes atomic recoil, slightly reducing its velocity and producing a blue-shift. This energy comes from internal transitions, with 3
a corresponding decrease in rest mass [23], ensuring energy–momentum conservation. During absorption, the atom’s rest mass increases as δm, which in the lab frame corresponds to an energy gain of γδmc2∝γhν. The second term in the relativistic Doppler formula, γhν(Vcos θ′/c), arises from longitudinal momentum transfer during emission, with only the photon’s momentum component along the atom’s motion producing recoil, which leads to the familiar cos θ′dependence. This particle-based analysis thus reveals the physical origin of both terms in the Doppler expression. The transverse Doppler effect, traditionally viewed as a consequence of time dilation, can also be understood using a similar particle-based framework. Unlike earlier explanations based on time dilation or average recoil velocity [14], our approach reveals the true physical origin of Einstein’s Doppler formula. In this work, we develop a particle-based formulation that accounts for energy shifts in the emission of both massive and massless particles. By analyzing these processes through the lens of energy–momentum conservation, we uncover a deeper structural connection between the relativistic Doppler effect, recoil, and velocity transformation laws. This formulation makes explicit a symmetry between light and matter that remains implicit in conventional treatments. In particular, we show that the Doppler shift and the relativistic velocity addition law follow from the same underlying transformation principle, independent of the emitter’s mass. The mass-independence of the recoil contribution clarifies why, in phenomena such as the Mössbauer effect [30], the observed frequency shift remains finite even when recoil energy is suppressed due to the effective mass of the emitting system. Using the same transformation framework, we next examine how electromagnetic mass transforms. This requires revisiting the standard formulation of electromagnetic field energy established by Maxwell [31] and Heaviside [32], in which the total field energy is written as the sum of electric and magnetic energy densities. This decomposition implicitly treats the electric and magnetic fields as independent contributors to the total energy, as if arising from distinct physical sources. However, this assumption becomes questionable when the electric and magnetic fields originate from the same source observed in different inertial frames. A purely electric field in one inertial frame may appear as a combination of electric and magnetic fields in another. As Einstein emphasized, electric and magnetic fields are not fundamentally distinct but represent different manifestations of a single underlying physical entity, depending 4
on the observer’s state of motion [33]. This viewpoint suggests that Eand Bare components of a unified field tensor, prompting the question of whether the standard expression for electromagnetic energy density remains valid in all frames. Since energy contributes to mass via Einstein’s relation E=mc2[4], the energy stored in electromagnetic fields should also contribute to the inertial mass of a system. It is well established that bound systems possess less total energy, and hence less mass, than their unbound constituents. For instance, an ionized atom is slightly more massive than its neutral counterpart, due to the absence of binding energy or the presence of electrostatic self-energy. This relationship between field energy and mass led to the early notion of electromagnetic mass, introduced by J. J. Thomson in 1881 [34], who likened the energy stored in a magnetic field to the kinetic energy of an equivalent mass. Building on this idea, Abraham [35] proposed a purely electromagnetic model of the electron, attributing its entire inertia to field energy. Lorentz [36] extended this by calculating the electron’s inertial mass as M=4U 3c2, where Uis the electromagnetic energy, derived from the rate of change of field momentum. However, this result raised concerns about energy conservation, prompting Poincaré [37] to introduce internal stresses to stabilize the model. Despite these foundational efforts, the energy and momentum of the electromagnetic field of a moving charge remain conceptually problematic [38], reflecting persistent inconsistencies in classical electrodynamics. These difficulties have motivated several modifications to classical electromagnetic theory [39–48]. Supported by the null result of the Trouton–Noble experiment [49]. Nevertheless, challenges remain, as the energy and momentum associated with electromagnetic fields do not transform as a four-vector. Singal [50] demonstrated that a consistent relativistic description emerges only when all energy and momentum contributions, electromagnetic and mechanical, are considered together. Boyer [51] similarly argued that while electromagnetic and mechanical energy–momentum do not transform individually as four-vectors, their combined total does. Furthermore, the null result of the Trouton–Noble experiment can also be reconciled by accounting for constraint forces [52,53]. It will be shown that adopting non-covariant transformation rules for the electromagnetic field energy leads to inconsistencies with either Lorentz invariance or energy–momentum conservation. By contrast, a manifestly covariant transformation 5
law resolves the difficulty and naturally avoids the long-standing 4/3electromagnetic mass problem [54]. These covariant electromagnetic energy transformations also have implications for energy quantization in quantum field theory. The canonical quantization procedure, introduced by Dirac and co-workers, treats field amplitudes and their conjugate momenta analogously to position and momentum in classical mechanics, thereby enforcing appropriate commutation or anti-commutation relations [58,59,105,134]. This procedure successfully quantized the free electromagnetic field, leading to the development of quantum electrodynamics (QED). The Gupta–Bleuler formalism [61,62] extended this by quantizing the electromagnetic field in a covariant gauge and imposing subsidiary conditions to remove unphysical ghost states, thus preserving Lorentz invariance while ensuring positive norms for observable states. Later, more sophisticated methods such as the Faddeev–Popov approach [63] and BRST quantization [64,65] were developed to handle gauge redundancies in non-Abelian theories. Although successful, these frameworks rely on complex constructions involving ghost fields, indefinite metrics, and cohomological techniques, which obscure the physical interpretation of quantization itself. A clear illustration of the physical significance of field quantization comes from the Lamb shift. The Lamb shift, originally measured by Lamb and Retherford [55], was accurately explained using the quantized free electromagnetic field, as first demonstrated by Bethe [56]. In his calculation, Bethe considered only the quantized free electromagnetic field. However, the Coulombic field is typically regarded as nonquantizable, as it corresponds to a zero wavenumber (k= 0) mode. In this work, we also revisit the foundations of field quantization. Adopting the manifestly covariant formulation of electromagnetic energy transformation proposed by Rohrlich [48]. Specifically, we quantize the electric field in the rest frame of a charged particle and ghost-free route to quantization. Interestingly, the Lamb shift calculated using the electric field associated with the quantized Coulomb interaction turns out to be identical to the Lamb shift obtained by Bethe [56] using the transverse field that arises in Dirac’s quantization of the free electromagnetic field [57]. Furthermore, the covariant energy transformation naturally avoids the difficulty associated with the 4/3mass problem [54]. Because the electromagnetic field and the gravitational field share a similar math6
ematical structure, this naturally motivates the broader quest to reconcile general relativity with quantum mechanics, leading to the exploration of quantum gravity. Attempts to quantize gravity date back to the pioneering work of F. J. W. Schwinger [66] and have been pursued in numerous approaches including linearized gravity, canonical quantization, and perturbative quantum general relativity [67–91]. The canonical quantization approach, initiated by Bryce DeWitt in the 1960s, sought to treat the metric of spacetime as a dynamical variable subject to quantization. This led to the formulation of the Wheeler–DeWitt equation, a functional differential equation describing the quantum state of the universe [83,96,97]. However, this approach encountered significant challenges, including the problem of time and the non-renormalizability of the theory [98,99]. Here, we first express gravity in the weak–field limit [100,101,131–133], where the structure of the equations closely parallels electromagnetism: the gravitational acceleration plays a role analogous to the electric field. Similarly to the previous approach, by treating the mass density as an operator, we demonstrate the quantization of the gravitational field in the weak–field limit, yielding harmonic-oscillator–type energy levels ¯sω a† kak+1 2. Within this framework an effective negative gravitational density arises at the classical level from the negative–frequency sector of the mode expansion. We therefore examine the consequences of this contribution, which enters naturally in the linearized theory. Its uniform, non–clumping character leads to an effective repulsive component that drives an accelerated expansion of space. In this picture the vacuum pressure is effectively set to zero, and the usual cosmological, constant problem, namely the 10122-fold discrepancy between theoretical vacuum energy estimates and the observed value [113,121,122,142,145], does not arise. Moreover, as demonstrated in Appendices G and I, the negative-energy obtained here does not exhibit the Bondi runaway instability [137], and instead shows dynamically stable, non-runaway behaviour. This study presents results: 1. We demonstrate that the relativistic Doppler effect and the velocity addition law share a common mathematical structure. This highlights a unified kinematic origin, where the Doppler shift in light mirrors the energy transformation of massive particles, connecting wave and particle descriptions via energy–momentum 7
conservation. 2. We demonstrate that recoil is not an external correction added to the Doppler shift but an intrinsic part of it. By analyzing the emission process within a conservation, law framework, we show that the frequency shift naturally incorporates both the rest mass change and the longitudinal momentum transfer. The conventional recoil correction is thus embedded in the relativistic formulation itself. 3. We obtain the transverse Doppler effect from emission geometry and relativistic momentum transformations, providing a transparent kinematic interpretation that does not rely on invoking time-dilation arguments explicitly. 4. We show that non-covariant energy–momentum transformation laws for electromagnetic fields are not compatible with covariant conservation principles, motivating the use of a manifestly covariant prescription that yields consistent field transformation behavior. 5. By treating the charge density as an operator, we obtain a field whose mode energy takes the familiar form ℏω(n+1 2). The associated electric field leads, through the dipole interaction d·E, to the correct operator structure required to reproduce Lamb–type radiative shifts, in full agreement with Bethe’s treatment using the quantized transverse field. 6. We extend this approach to gravity by treating the gravitational field in the weak-field limit. We demonstrate the quantization of a gravity field by treating the mass density as an operator, obtaining the familiar field energy ¯sω(n+ 1/2). However, the formulation intrinsically generates cosmic expansion and accommodates negative-energy contributions. 7. Building on the intrinsic structure of the linearized quantization scheme, the negative–frequency sector yields an effective homogeneous repulsive contribution. Interpreted phenomenologically as a uniform gravitational density, this component produces large–scale expansion compatible with observed cosmological scales, while avoiding the usual vacuum–energy discrepancy associated with the cosmological–constant problem. In summary, we first present a unified, particle-based derivation of the relativistic Doppler effect. Our analysis identifies two key energy contributions in the lab frame that naturally lead to the velocity addition formula: the emitter’s initial kinetic energy 8
due to platform motion, and the energy delivered during emission, accompanied by a directional momentum exchange via longitudinal momentum conservation. Since the atom’s mass increases upon absorbing light, a parallel structure arises in the photon case, where γhν corresponds to the transformed rest energy due to mass increase, and γhν Vcos θ′ cencodes the recoil shift associated with longitudinal momentum transfer. This energy-based approach offers a physically transparent interpretation of the transverse Doppler effect, attributing it not to time dilation, but to the projection of momentum along the direction of motion. When applied to photon emission, this yields the full relativistic Doppler shift and explains why the commonly cited recoil correction term is naturally accounted. Using the same reasoning framework, it can be demonstrated that non-covariant energy transformation for the electromagnetic field energy–momentum imply are not compatible with conservation principles. Thus, we proceed with a manifestly covariant formulation, to obtain the transformation laws for E-M field energy and momentum associated with a massive charged particle. Furthermore, in contrast to quantization in quantum electrodynamics, arising from free-field quantization, we develop a quantization scheme for the field in the rest frame of a charged particle by treating the charge density as an operator, recovering the familiar harmonic-oscillator form of the field energy. Remarkably, this approach reproduces the Lamb shift without relying on free-field assumptions, while preserving Lorentz covariance and avoiding ghost states. In the weak–field limit, the structural analogy between gravitational acceleration and the electric field allows the same quantization procedure to be applied to the longitudinal gravitational potential. Treating the positiveand negative-frequency parts of the mass–energy density as operators yields a harmonic–oscillator–type mode structure. Within this linearized framework the negative-frequency sector contributes a uniform, non-clumping background term, which acts as an effective repulsive component. Interpreted phenomenologically, this uniform contribution leads to a large-scale expansion consistent with a linear distance–dependence. A fully rigorous mathematical formulation is required to support this framework. We begin by developing a relativistic treatment of Doppler shifts and energy exchange for massless and massive uncharged systems. 9
lustrates how key quantities evolve from classical kinetic energy expressions to the relativistic energy transformation, and finally to the Doppler shift in frequency. This comparison shows that whether dealing with a classical object, a relativistic massive particle, or a quantum photon, the transformation of energy between inertial frames follows a structurally consistent pattern. In each case, the energy observed in the lab frame contains two key components: the intrinsic energy defined in the source frame, including kinetic energy or energy from mass conversion, and a correction term arising from energy exchange due to recoil. Notably, even for photons, the energy in the lab frame includes a term like γhν and an effective recoil correction imposed by momentum conservation. This underscores that the Doppler shift is not merely a manifestation of time dilation but also reflects genuine energy exchange governed by the same fundamental mechanical principles. It is important to recognize that the very notion of a reference frame moving at subluminal speed arises only in the presence of mass. Without mass, energy necessarily propagates at the speed of light, and the concept of a localized rest frame becomes undefined. However, once a massive system defines a frame of reference, one can meaningfully ask how conservation laws apply in that frame. As shown in the above derivations, these laws remain consistent and complete. Crucially, relative velocity is always defined with respect to inertial frames such as Sand S′, which themselves do not recoil. In contrast, when a particle is emitted, the source does recoil, and even in this case, nature preserves a consistent and exact relationship between space-time transformations and energy–momentum conservation. There is no need to introduce additional terms or corrections, the recoil is not an exception, but rather a natural outcome of the same underlying structure. This unifying structure reinforces the view that classical and relativistic are governed by a single coherent framework. 5. Relativistic Energy Transformation For Charged Objects Suppose a force is applied up to a distance bon a particle, and during this process an energy W(o)is supplied to it, so that W(o) = K.E. =Rb 0F dx. This kinetic energy is given by K.E =mc2(γ−1), which gives the mass as m=K.E c2(γ−1). If a fixed amount of kinetic energy is supplied to particles with different masses, then the final velocity alone determines the mass of the particle. Since electric and magnetic fields 16
carry energy, and electromagnetic fields transform under Lorentz transformations, the energy associated with these fields must also transform accordingly. We show, if the energy does not transform as the time component of a four-vector, then it do not remain compatible with Lorentz invariance and energy–momentum conservation. Let the coordinate axes of the S,S′and S′′ reference frames be (X, Y, Z),(X′, Y ′, Z′) and (X′′, Y ′′, Z′′). The charged object has an electric field in the Y-direction. We assume that the charged particle moves with velocity Voalong the X-direction with respect to the S′frame, and let the S′′ frame be its rest frame. In the rest frame S′′, the electric and magnetic fields are E′′ =E′′ ˆyand B′′ = 0. The electromagnetic field energy in S′′ is W′′ =ε(o) 2R E′′ 2dτ′′. Observer from S′frame, observers E′=γ(o)E′′ ˆyand B′=−γ(o)V o c2E′′ ˆzand due to length contraction volume transforms as dτ′=dτ′′/γ(o). Energy associated with electromagnetic field as observed from S′frame is given by W′(y) = 1 2R[ε(o) E′2+ B′2 µ(o)]dτ′. And after substituting the value of E′, B′,dτ′and c2= 1/ε(o)µ(o)we get, W′(y) = γ(o)[1 + V o2 c2]ε(o) 2Z E′′2dτ′′ (17) We define Γ(oy) = γ(oy)[1 + V o2 c2].W′(y)is not transforming like the zeroth component of the four-vector. The total amount of energy stored in the charged object, moving with velocity V o with respect to S′frame is W(oy) = γ(o)mc2+ Γ(oy)W′′ (18) When a force Fis applied over a distance b, the work W(oy)imparted to the system is not converted solely into the kinetic energy of the object. Part of this work accelerates the object, while the rest is associated with the build-up or modification of the surrounding electromagnetic field. Thus, the supplied energy naturally partitions into mechanical kinetic energy and electromagnetic field energy. To conserve the total energy of the system, the charged object reaches a smaller velocity after the force is applied up to the same distance b. If this happens then we say the observed mass of the charged particle is larger than that of the same object which has no charge. If an infinitesimal amount of energy dW is supplied to a charged object such that its velocity increases by dV while moving with velocity V, then Eq. (18) gives dW at any velocity by differentiating with respect to V. The total energy required to accelerate the charged object from rest to a final velocity Vois therefore K.E(o) = ZVo 0 dW dV dV. 17
Consequently, the kinetic energy takes the form K.E(oy) = mc2γ(o)−1+W′′Γ(oy)−1. When electric field points perpendicular to the direction of motion then observed the mass of the charged object will be m′(y) = K.E(oy) c2(γ−1) . If the electric field is perpendicular to the direction of motion then electromagnetic field as observed from S frame is given by, E=γ( E′+ V× B′)and B= γ B′− V c2× E′!. Using E′and B′expressed in terms of E′′ and B′′, we get E=γγ(o)E′′ 1 + V Vo c2ˆyand B=−γγ(o)E′′ c2(V+Vo)ˆz. Due to length contraction, the volume transforms as dτ =dτ′′ γγ(o)1 + V Vo c2. The electromagnetic field energy in the Sframe is W(y) = 1 2R"ε(o) E2+ B2 µ(o)#dτ, and substituting the fields and dτ gives W(y) = γγ(o)W′′ "1 + V+Vo c(1+V Vo/c2)2#1 + V Vo c2. To determine the energy transformation for the charged object, we calculate K.E(α). Consider a charged object of mechanical mass mthat also carries electromagnetic field energy. Let this object move with velocity V, and let Jack, who is in the S′frame, throw the object with velocity Vorelative to S′. The kinetic energy added, as observed from the Sframe, is then K.E(α) = Zd(K.E) dV dV =Zd(K.E). Therefore, K.E(α) = ZV+V o 1+V V o/c2 V d(γmc2) + ZV+V′ 1+V V ′/c2 V d(γM′′c2) + ZV+V o 1+V V o/c2 V d(γ(1 + V2 c2)W′′). And after integrating the above expression we obtain: K.E(α)=γγ(o)mc21 + V Vo c2+γγ′M′′c21 + V V ′ c2 +γγ(o)W′′ "1 + V+Vo c(1+V Vo/c2)2#1 + V Vo c2 −γmc2−γM′′c2−γ1 + V2 c2W′′. This expression cannot be cast in the form γ×(frame–independent term); that is, no factorization exists in which the remaining quantity is independent of the boost 18
velocity V. However, when observed from the frame S, this mass transforms to γ δm, as shown in Eq. (12). That would imply that the rest mass mbecomes frame–velocity dependent, meaning the rest mass would be measured to be different in different reference frames. Furthermore, kinetic energy added as observed from S′frame of reference is K.E(α′) = RV o 0d(γmc2+γ(1 + V2 c2)W′′) + RV′ 0d(γM′′c2). After integrating this expression we get, K.E(α′) = mc2[γ(o)−1]+W′′[γ(o)(1 + V o2 c2)−1] + M′′c2[γ′−1] One can always supply the energy K.E(α′)by converting some mass δm into energy. But when observed from the Sframe this mass should be γ δm, and hence the energy observed from Sframe should be K.E(α) = γ K.E(α′)as shown in Eq. (12). However, we do not observe this in the above case. Hence it is concluded that either invariance principle or conservation of energy do not remain compatible. And this was obvious because electromagnetic field energy is not transforming as a zeroth component of the four-vector. Surprisingly one can get energy transformation for charged object and electromagnetic field by the same procedure as mentioned above, even if the invariance principle does not hold. However, it was proposed that energy and momentum transform as four-vector if one takes constraint force into account [52,53]. It was also proposed that one can get manifestly covariant expression if one modifies the energy density of the electromagnetic field [44–47]. We proceed with manifestly covariant expression [44–47] which is given by Pu= 1 cRθuvdσvbecause mechanical rest mass m can not depend on the frame velocity. By using this expression, electromagnetic field energy calculated by the observer when the charge is at rest frame S′′ is W′′. In S′frame, the electromagnetic energy W′and momentum P′will measured to be W′=γ(o)W′′ and P′=γ(o)V oW′′ c2. The amount of energy needed to be supplied such that charged object can move at velocity Vo is K.E(o) = RV o 0 dW dV dV , which gives K.E(o) = [mc2+W′′](γ(o)−1). Due to this, the charged object will reach at lesser velocity after applying force up to some distance b. Therefore we say that the observed mass of a charged object is larger than that of the same object but have no charge. So the observed mass according to our definition of mass is m′=K.E(o) c2(γ(o)−1) . By substituting the value of K.E(o)we get, m′=m+W′′ c2. By this concept, we can understand electromagnetic mass. However, the process by 19
which electromagnetic fields are gaining energy is important. By replacing mwith m′in Eqs. (1)–(6), we obtain: E(ε) = γγ(o)m′c2[1 + V oV c2](19) Therefore, the observed mass of a charged object is greater than the mass of the same object when it carries no charge. Also, a charged particle at rest has only an electric field. However, once the observer begins to move, both the electric and magnetic fields change in the observer’s frame of reference. Thus, during motion, the combined electric and magnetic field energy of the same charged particle must behave as the zeroth component of a fourvector. 6. Coulombic field Quantization A covariant expression for the electromagnetic energy of a charged particle is E′=Zγϵ0 2 E′2− B′2c2dτ′, as emphasized by Rohrlich. In the particle’s rest frame this reduces to E=ϵ0 2Z E2dτ, since the magnetic field vanishes and only the Coulomb electric field contributes. In quantum electrodynamics (QED), the pure Coulomb field, representing a static electrostatic configuration, is generally regarded as non-quantizable, since it lacks dynamical degrees of freedom and does not correspond to a propagating mode of the electromagnetic field [102–106]. Nevertheless, in the theoretical treatment of the Lamb shift, the quantized electric field associated with the free electromagnetic field is typically employed, even though it originates from Coulombic field [55,56]. In contrast, our approach directly quantizes the Coulombic field by promoting the charge density to an operator. Furthermore, since the field energy involves the term ϵ0 2 E2dτ, the same framework can be extended to a moving reference frame using the Lorentz-invariant quantity E2− B2, consistent with the covariant form of the electromagnetic field tensor. Notably, this approach does not require the introduction of unphysical (ghost) fields. 20
Starting from Lorenz gauge condition we obtain: ∇·A+1 c2 ∂ϕ ∂t = 0.(20) Assume the vector and scalar potential takes the form: A(r, t) = A+kei(k·r−ωt)+A−ke−i(k·r−ωt).(21) ϕ(r, t) = ϕ+ei(k·r−ωt)+ϕ−e−i(k·r−ωt).(22) Under Lorenz gauge condition: ∇2ϕ−1 c2 ∂2ϕ ∂t2=−ρ ε0 . ω2 c2−|k|2ϕ=−ρ ε0⇒ϕ=−ρc2 ε0(ω2−|k|2c2).(23) Where, ϕ+ei(k·r−ωt)=−ρ+c2ei(k·r−ωt) ε0(ω2−|k|2c2)(24) ϕ−e−i(k·r−ωt)=−ρ−c2e−i(k·r−ωt) ε0(ω2−|k|2c2)(25) 1 c2 ∂ϕ ∂t +∇·A= 0 ⇒ ∇·A=iω c2ϕ+ei(k·r−ωt)−iω c2ϕ−e−i(k·r−ωt).(26) For positive part of the frequency we have, ikA+k=iω c2ϕ+⇒A+k=ω c2kϕ+(27) ikA−k=iω c2ϕ−⇒A−k=ω c2kϕ−(28) Thus vector potential can also be written as, A(r, t) = ω c2kϕ+e+i(k·r−ωt)+ω c2kϕ−e−i(k·r−ωt)(29) E=−∇ϕ−∂A ∂t .(30) 21
E(r, t) = −ik(ϕ+ei(k·r−ωt)−ϕ−e−i(k·r−ωt)) + iω2 c2k(ϕ+e+i(k·r−ωt)−ϕ−e−i(k·r−ωt))(31) E(r, t) = −ik(1 −ω2 c2k2)(ϕ+ei(k·r−ωt)−ϕ−e−i(k·r−ωt))(32) Substituting the value of positive and negative part of scalar potential in terms of charge density, we get E(r, t) = ik(1 −ω2 c2k2)( ρ+c2ei(k·r−ωt) ε0(ω2−|k|2c2)−ρ−c2e−i(k·r−ωt) ε0(ω2−|k|2c2))(33) After simplifying we get, E(r, t) = −ikρ+ k2ε0 ei(k·r−ωt)+ikρ− k2ε0 e−i(k·r−ωt)(34) Classically for real charge density, we write: E(r, t) = 2ρ ε0k2ksin(k·r−ωt).(35) For small momentum transfer the density modes admit an effective harmonicoscillator description of the Coulomb field [134]. At larger momenta this approximation breaks down: the composite structure of the density operator dominates and the static potential exhibits the usual short-distance (vacuum-polarization) corrections characteristic of full QED [140]. Nevertheless, the relation between the charge-density modes and the electromagnetic potential remains intact. Promoting the charge density to an operator (Appendix L), the electric field takes the form E(r, t) = X k"ikρk k2ε0 e−i(k·r−ωkt)−ikρ† k k2ε0 ei(k·r−ωkt)#.(36) For clarity, the complex-conjugate (negative-frequency) part is E∗(r, t) = X k"−ikρ† k k2ε0 ei(k·r−ωkt)+ikρk k2ε0 e−i(k·r−ωkt)#.(37) Here, the operator ρkmultiplies the positive-frequency mode e−iωktand therefore acts as the annihilation operator of kmode while its Hermitian conjugate ρ† kmultiplies the negative-frequency mode eiωktand acts as the creation operator of kmode. 22
|E(r, t)|2=X k,k′ k·k′ ε2 0k2k′2hiρke−i(k·r−ωkt)−iρ† kei(k·r−ωkt) ×−iρ† k′ei(k′·r−ωk′t)+iρk′e−i(k′·r−ωk′t)i. (38) |E(r, t)|2=X k,k′ k·k′ ε2 0k2k′2hρkρ† k′ei(k′−k)·r−(ωk′−ωk)t+ρ† kρk′ei(k−k′)·r−(ωk−ωk′)t −ρkρk′e−i(k+k′)·r−(ωk+ωk′)t−ρ† kρ† k′ei(k+k′)·r−(ωk+ωk′)ti. (39) Time-averaging (or, equivalently, invoking the rotating-wave approximation) amounts to neglecting the rapidly oscillating terms proportional to e±i(ωk+ωk′)tin |E(r, t)|2, since they average to zero over timescales relevant for physical measurements. Retaining only the non–rapidly oscillating contributions, the electric-field energy can be written as ZV d3rε0 2|E(r, t)|2≈X k,k′ k·k′ 2ε0k2k′2hρkρ† k′e−i(ωk′−ωk)tZV d3r ei(k′−k)·r +ρ† kρk′e−i(ωk−ωk′)tZV d3r ei(k−k′)·ri(40) =X k,k′ k·k′ 2ε0k2k′2hρkρ† k′e−i(ωk′−ωk)tV δk,k′ +ρ† kρk′e−i(ωk−ωk′)tV δk,k′i ZV d3rε0 2|E(r, t)|2≈VX k k·k 2ε0k2k2hρkρ† k+ρ† kρki(41) =VX k k2 2ε0k4hρkρ† k+ρ† kρki=X k V 2ε0k2ρkρ† k+ρ† kρk.(42) ρk=αkak, ρ† k=αka† k, with [ak, a† k′]=δk,k′. Then ZV d3rε0 2|E|2=X k V 2ε0k2α2 kaka† k+a† kak.(43) 23
ZV d3rε0 2|E|2=X k V 2ε0k2α2 k2a† kak+ 1.(44) Matching each mode’s contribution to the harmonic-oscillator energy ℏωka† kak+1 2 gives the per-mode coefficient condition V ε0k2α2 k=ℏωk=⇒α2 k=ε0ℏωkk2 V. Therefore the commutator is [ρk, ρ† k′] = α2 k[ak, a† k′]=α2 kδk,k′=ε0ℏωkk2 Vδk,k′.(45) Ek(r, t) = rε0ℏωkk2 V"ik k2ε0ake−i(k·r−ωkt)−ik k2ε0a† kei(k·r−ωkt)#.(46) Ek(r, t) = rℏωk V ε0 ˆ khiake−i(k·r−ωkt)−ia† kei(k·r−ωkt)i.(47) A key observation (derived explicitly in Appendix J) is that the longitudinal Coulomb field, when quantized through the charge–density operator ρk, acquires an electric–field amplitude larger by a factor of √2compared with the transverse field obtained in Dirac’s quantization of the free electromagnetic field. This arises because a transverse photon divides its energy equally between electric and magnetic parts, whereas a longitudinal Coulomb mode carries all of its energy in the electric field. Despite this difference in normalization, the effective dipole coupling relevant for the Lamb shift turns out to be identical to the standard QED result. The longitudinal matrix element involves the projection dmn ·ˆ k, whose angular average gives 1 3|dmn|2. When multiplied by the additional factor of 2coming from the squared √2enhancement of the electric field, one obtains the familiar 2 3|dmn|2that appears in Bethe’s Lamb–shift calculation based on transverse photons. This explains why the Lamb shift, originally measured by Lamb and Retherford [55] and first calculated by Bethe [56] using the dipole Hamiltonian Hint =−d·E [102,123–126], is reproduced equally well using the quantized longitudinal Coulomb field. 24
Thus, if observed from the moving frame, manifestly covariant form of electromagnetic field energy density for the massive charged particle [48] is W=γ 2R[ε(o) E′2− B′2 µ(o)]dτ′. Since ε0 E′2− B′2 µ0 (48) is invariant under Lorentz transformations, we can directly substitute the energy from the rest frame to obtain W=γℏωka† kak+1 2(49) Transforming as 0th vector of four vector. The electric field in the moving frame can be written as E′=γE=γrℏωk V ε0 ˆ khiake−i(k·r−ωkt)−ia† kei(k·r−ωkt)i.(50) where γis the Lorentz factor. Since B= 0 in the rest frame, the transformed magnetic field follows from the Lorentz transformation as B′=γv×E(r, t) c2.(51) Substituting the expression for E(r, t)yields B′=γv× c2rℏωk V ε0 ˆ khiake−i(k·r−ωkt)−ia† kei(k·r−ωkt)i.(52) Furthermore, for each mode k, the field Hamiltonian takes the usual harmonic– oscillator form Hk=ℏωka† kak+1 2,(53) with eigenstates |nk⟩satisfying Hk|nk⟩=ℏωkn+1 2|nk⟩.(54) The ladder operators must act in the standard way, ak|nk⟩=√n|nk−1⟩, a† k|nk⟩=√n+ 1 |nk+ 1⟩.(55) 7. Quantization of the Gravitational Field To study the quantum behavior of gravitational fields, we begin from the Einstein field equations: 25
we obtain ρΛ≈2×5.9×10−27 kg m−3.(85) where 5.9×10−27 kg m−3denotes the commonly adopted value of the dark–energy density in the literature [112,113,117]. From the second Friedmann equation, one obtains a negative value for H2if the dark–energy density is taken to be negative and allowed to vary with the expansion. However, the density of the dark component does not dilute with expansion. As shown in Appendix H, when the density is treated as constant, one obtains Eq. (H.21), which is consistent with a negative energy density [Eq. (H.22)] and yields a positive contribution to H2. Similarly, the matter energy density scales as ρm∝a−3, and substituting this behaviour into the acceleration equation leads to Eq. (H.15), which is consistent with a positive–energy matter component. In the present universe, radiation and other relativistic species contribute negligibly to the expansion rate, so the dominant terms in H2arise from nonrelativistic matter and dark energy. Accordingly, the total expansion rate may be viewed as the combined contribution, H2 tot ≈H2 Λ+H2 m,(86) where each term corresponds to the standard Friedmann contribution of the respective energy component. Substituting the (gravitationally) negative mass density ρΛinto the constant– density Friedmann relation derived in Appendix H [Eq. (H.22)], we obtain H2 Λ=4πG 3ρΛ.(87) Similarly, inserting the positive matter density ρminto the dust Friedmann relation [Eq. (H.15)] gives H2 m=8πG 3ρm.(88) Observationally, the present Hubble constant is related to the critical density by H2 0=8πG 3ρc.(89) This relation sets the value of ρc, and the measured matter fraction Ωm≃0.3then fixes the present matter density ρm= Ωmρc. 32
When these observational inputs are used in our model, the resulting quantities HΛ and Hmsatisfy the same ratios HΛ/H0and Hm/H0as in the standard ΛCDM model. Thus, the relative contributions of dark energy and matter to the total expansion rate are unchanged, and the overall expansion behaviour is identical to that inferred from observations. As shown in the Newtonian interpretation of a uniform negative–mass density (Eqs. H.23–H.28), one arrives at the relation ˙r2=4πG 3ρdE r2,(90) which is identical in form to the constant–density Friedmann result. Furthermore, Eqs. (63), (64), and (71) show that the dynamical consistency of the model requires the presence of a negative gravitational density, thereby providing a clear physical motivation for such a component. This interpretation presents in a consistent manner the physical role normally played by the term Λc2/3in the Einstein field equations. The Einstein equations with a cosmological constant read Gµν + Λgµν =8πG c4Tµν.(91) In the standard formulation, the additional term Λgµν is inserted by hand. In contrast, the present framework shows that the same effective contribution arises naturally if one substitutes the uniform negative gravitational density obtained above into the Einstein equation. In this way, the role ordinarily played by the cosmological constant is reproduced by the negative–density. Within this interpretation, cosmic expansion emerges directly from the gravitational effect of this background density, and also allows the cosmological redshift to be viewed as a gradual loss of photon energy during propagation in an expanding background [107,108,119,127]. Implications for Cosmological Parameters The present Hubble parameter H0inferred from the expansion law ˙a=Ha is unaffected, ensuring consistency with the standard ΛCDM phenomenology. A similar conclusion holds for the age of the universe. The age t0follows from the integral [117,135,143], t0=Z∞ 0 dz (1+z)H(z),(92) 33
We decompose the Hubble rate into contributions from each component, H2(z) = H2 m(z)+H2 r(z) + H2 Λ+H2 k(z),(93) where, starting from the Friedmann equation, H2(z) = ˙a a2 =8πG 3ρtot(z)−k a2(z),(94) where we know, H2 m(z) = 8πG 3ρm0(1+z)3,(95) H2 r(z) = 8πG 3ρr0(1+z)4,(96) H2 Λ=4πG 3ρΛ,(97) H2 k(z) = −k a2(z)=−k(1+z)2.(98) Because the rescaling leaves all Hiunchanged, the functional form of H(z)is identical, and therefore the predicted age of the universe remains the same as in the standard ΛCDM model. As shown in Appendix K, G and H the interaction takes the form of a 1/r2force, which, when averaged over large scales owing to the uniformity of matter and dark energy, produces an effectively produces a linear relation between recessional velocity and distance, consistent with the Hubble law. It is important to emphasize that the analysis presented above does not constitute a complete cosmological model. We have not imposed the full hierarchy of Friedmann–Lemaître equations, nor have we considered the evolution of perturbations, structure formation, or the detailed thermal history of the universe. The present work is therefore restricted to demonstrating that the quantization scheme, by construction, introduces a uniform negative–energy density whose classical limit reproduces a repulsive, cosmological–constant–like acceleration. How it interacts with matter (partially discussed in Appendix I) and perturbations, are questions that lie beyond the scope of the present analysis and must be addressed in future work. In summary, the negative-mass interpretation is consistent with Eqs. (60), (63), and (64) in general relativity, mirroring the structure of Eqs. (24) and (25) in the quantized electromagnetic field. 34
Absence of the Cosmological–Constant Problem In the standard approach, the cosmological constant is identified with the vacuum energy of quantum fields, leading to an expected energy density ρvac that is larger than the observed dark–energy density ρobs Λby a factor of order 10122 (see, e.g., [113, 117,145]). In contrast, the present framework does not introduce vacuum energy into the Einstein equations. The only contribution that gravitates is the uniform negative gravitational density ρ−. Thus, in this picture the dark–energy density is not the residual of a huge vacuum contribution, but a direct consequence of the quantized gravitational background. The usual 10122-fold discrepancy does not arise, because the large vacuum energy terms of ordinary quantum fields do not appear as independent gravitational sources in the first place. 9. Conclusion In summary, we present a unified, particle-based derivation of the relativistic Doppler effect that explicitly separates contributions from rest-frame motion and directional recoil. It shows that the commonly applied recoil correction term cancels within the full relativistic formulation rather than requiring an additive fix. By applying energy–momentum conservation, we identify the physical origin of each term in the Doppler formula. Notably, it provides a purely kinematic explanation of the transverse Doppler effect. This framework naturally recovers the longitudinal and transverse Doppler shifts and also draws a conceptual parallel to the gravitational case, where a photon’s energy varies with potential in a manner analogous to inertial frame transformations [27–29]. Our results reveal a shared kinematic basis for both massive and massless particles, enabling an intuitive understanding of relativistic motion and energy exchange. Furthermore, by using energy conservation for a charged particle, we conclude that the observer measures a larger mass due to the transformed electromagnetic field. For consistency with the invariance principle, the total energy and momentum of a charged object must transform as a four-vector, as suggested by Rohrlich. Since mass is the time component of the four-momentum, the total electromagnetic energy must transform in the same way; otherwise, invariance and energy conservation will not consistent. 35
Thus we proceed with a covariant quantization scheme for the electromagnetic field in the rest frame of a charged particle. Within this framework, the quantized charge density acts as the creation and annihilation operator for k mode. This approach aims to provide a consistent Lamb shift without relying on the free-field assumption, while preserving Lorentz covariance and avoiding ghost states. Thus we proceed with a covariant quantization scheme for the electromagnetic field in the rest frame of a charged particle. In this formulation the operator-valued charge density ρkplays the role of the creation and annihilation operator for each k–mode, providing a source-driven quantization without invoking a free-field decomposition. This framework remains Lorentz covariant and avoids the introduction of ghost degrees of freedom, while allowing the Lamb shift to be obtained directly from the sourced interaction. In the weak–field limit the analogy with the longitudinal electric field allows the same operator structure to be applied to linearized gravity. The mode expansion then produces positiveand negative-frequency components of the energy–density operator. This structure naturally yields a uniform background energy density and produces a repulsive contribution at large distances. This provides a simple phenomenological route to a cosmological–constant–like acceleration within the linearized theory. Both the static electric and gravitational cases lead to an energy spectrum of the form E=ℏωa†a+1 2. Acknowledgments I acknowledges the support of the Department of Atomic Energy, Government of India. Appendix A. Derivation of the relativistic Velocity Addition Rule As shown in Eq.(4) of the main text, this follows from the conservation of energy, we have γ(ε)mc2+γ(δ)M′′c2=γ(M′′ +m)c2+γK.E(α′). By substituting Eq. (2) from the main text into the above expression, we obtain: γ(ε)mc2+γ(δ)M′′c2=γ(M′′ +m)c2+γ(mc2(γ(o)−1)+ M′′c2(γ′−1)) (A.1) 36
Using the conservation of momentum shown in Eq. (5) of the main text and by substituting Eq. (2) from the main text, we have the following, γ(δ)M′V(δ)+γ(ε)mV (ε)=γ(M′′ +m)V+γ[m(γ(o)−1)+ M′′(γ′−1)]V(A.2) After canceling common terms in Eq. (A.1), γ(δ)M′′ +γ(ε)m=γγ(o)m+γγ′M′′ (A.3) The above equation can be rearranged as, ϕ=γ(δ)M′′ =γγ(o)m+γγ′M′′ −γ(ε)m(A.4) V(δ)=cr1−M′′2 Φ2(A.5) The conservation of momentum in the S′frame, given by γ(o)mVo=−γ′M′′V′, can also be written as: M′′2(γ′2−1)=m2(γ(o)2−1) (A.6) By rearranging Eq. (A.2), we get γ(δ)M′′V(δ)=γ[mγ(o) + M′′γ′]V−γ(ε)mV (ε) Using the expressions for γ(δ)M′′ and V(δ)from Eqs. (A.4) and (A.5), respectively, we obtain, ϕcp(1 −M′′2 ϕ2)=γ[mγ(o) + M′′γ′]V−γ(ε)mV (ε) Squaring both sides of the above equation and rearranging terms, we obtain: ϕ2−M′′2= (γ[mγ(o) + M′′γ′]V c−γ(ε)mV(ε) c)2 Substituting the expression for ϕfrom Eq. (A.4), we obtain: (γγ(o)m+γγ′M′′ −γ(ε)m)2−M′′2= (γ[mγ(o)+M′′γ′]V c−γ(ε)mV(ε) c)2 37
After rearranging the terms, we arrive at: γ(o)m+γ′M′′]2−2γγ(ε)m[γ(o)m+γ′M′′][1 −V V (ε) c2]+ m2−M′′2= 0 Expanding and rearranging the above equation, we obtain: γ(o)2m2+m2+M′′2(γ′2−1) −2γγ(ε)γ(o)m2−2γγ(ε)γ′mM′′+ 2γ(o)γ′mM′′ + 2γγ(ε)γ(o)m2V V (ε) c2+ 2γγ(ε)γ′mM′′ V V (ε) c2= 0 Substituting Eq. (A.6), canceling common terms, and dividing the equation by 2m, we obtain: γ(o)2m−γγ(ε)γ(o)m−γγ(ε)γ′M′′ +γ(o)γ′M′′+ γγ(ε)γ(o)mV V (ε) c2+γγ(ε)γ′M′′ V V (ε) c2= 0 By factoring out mγ(o)and −M′′γ′, the above equation can be rewritten as: γ(o)m[γ(o)−γγ(ε)+γγ(ε)V V (ε) c2] = −M′′γ′[γ(o)−γγ(ε)+γγ(ε)V V (ε) c2] Above equation holds for any mass of the bus and ball. Thus [γ(o)−γγ(ε) + γγ(ε)VV(ε) c2] = 0. γ(o)−γγ(ε)+γγ(ε)V V (ε) c2= 0 (A.7) Squaring the Eq. (A.7), substituting the expression for γ(ε), and then expanding and rearranging the terms, we obtain a quadratic equation in V(ε): V(ε)2 c2[V2 c2+γ(o)2 γ2]−V(ε)[2V c2]−[γ(o)2 γ2−1] = 0 (A.8) The solution of Eq. (A.8) yields the standard velocity addition formula: V(ε) = V±Vo 1±V Vo c2 . 38
Appendix B. Relativistic Energy Dynamics: Recoil and Transfer One can notice, how energy transformation is govern by energy conservation. Thus to derive this result, one can rearrange Eq. (A.1) to get transformed energy for ball as, γ(ε)mc2=γ(M′′ +m)c2+γ(mc2(γ(o)−1)+M′′c2(γ′−1)) −γ(δ)M′′c2(B.1) We substitute the value of V(δ) = V±Vo 1±V Vo c2 in γ(δ)and then by rearranging Eq. (A.1), we obtain: γ(ε)mc2=γ(M′′ +m)c2+γ(mc2(γ(o)−1)+M′′c2(γ′−1))− γγ′M′′c2[1 + V V ′ c2](B.2) By rearranging the above term, we obtain: γ(ε)mc2=γγ(o)mc2−γγ′M′′V′V(B.3) Using the conservation of momentum in the S′frame and referring to Eq. (1) from the main text, given by γ(o)mVo=−γ′M′′V′, we obtain: γ(ε)mc2=γγ(o)mc2[1+V V o/c2](B.4) It can be observed that the second term, γγomV Vo, arises specifically due to recoil, as it appears only after applying momentum conservation in the S′frame. This term determines whether the energy of the particle increases or decreases. The magnitude arises due to the conservation of momentum in the S′frame and depends on the direction in which the object is moving. Appendix C. Non-Relativistic Energy Dynamics: Recoil and Transfer To better interpret the physical meaning of each term, we provide a corresponding derivation within the framework of classical (non-relativistic) mechanics. Suppose S be the rest frame of the observer. Consider that the bus and the frame S′are moving with same velocity V with respect to S frame. Let the coordinates axes of S and S′ frame is (X, Y, Z) and (X′,Y′,Z′). Let the mass of the bus be M′which includes the mass of the Jack. Jack has a ball of mass m on his hand. And he throws a 39
ball in X′or X direction. Let ball velocity is Voin S′frame. After throwing the ball, bus reduces its velocity and velocity of the bus is measured to be V(δ)from the rest frame of reference. Let V′and Vobe the velocities of the bus and the ball, respectively, measured in the S′frame of reference after the ball is thrown. According to the conservation of momentum in the S′frame, we have: mVo=−M′V′(C.1) In non-relativistic mechanics, the mass of the bus (M′) remains constant. Let the velocity of the bus in the Sframe be denoted by V(δ). Using the Galilean velocity addition rule along with Equation (C.1), we obtain: V(δ)=V+V′=V−mVo/M′(C.2) Thus, using Eq. (C.2), the kinetic energy (K.E(δ)) of the bus after the ball is thrown, as measured in the Sframe, becomes: K.E(δ) = 1 2M′[V+V′]2(C.3) In the S′frame of reference, prior to the act of throwing, both the ball and the bus are at rest, and hence possess zero kinetic energy. Upon throwing the ball, both the ball and the bus acquire nonzero velocities, indicating that kinetic energy has been added to the system. This energy is supplied by Jack: he imparts kinetic energy to the ball via his hand, while an equal and opposite reaction, exerted through his legs, sets the bus in motion in the opposite direction, in accordance with momentum conservation in the S′frame. Consequently, the total kinetic energy added to the system by Jack is distributed between the ball and the bus, and is quantitatively given by: K.E(α′) = 1 2mV 2 o+1 2M′V′2(C.4) where mand M′are the masses of the ball and the bus respectively, V0is the velocity of the ball, and V′is the recoil velocity of the bus, all measured in the S′frame. For non-relativistic case, K.E(α) = K.E(α′). Where is the kinetic energy K.E(α)is observed by the observer in S frame of reference is, K.E(α) = 1 2mV 2 o+1 2M′V′2(C.5) Before the ball is thrown, the kinetic energy of the combined bus-and-ball system, as observed from the inertial frame in S, is given by K.E(β) = 1 2(m+M′)V2(C.6) 40
We assume that the mass of the bus remains unchanged. However, some kinetic energy is added to the system. Therefore, we can write: K.E(β) + K.E(α)=K.E(ε)+K.E(δ)(C.7) where K.E(ε)is the kinetic energy of the ball in S frame of reference. To obtain the total energy of the ball, we substitute Eqs. (C.3), (C.5), and (C.6) into Eq. (C.7), we obtain: K.E(ε) = 1 2(m+M′)V2+ (1 2mV 2 o+1 2M′V′2)−1 2M′[V+V′]2(C.8) Using Eq. (C.1), and after canceling terms and simplifying, we obtain: K.E(ε) = mV 2 2+mV 2 o 2+mV Vo(C.9) One can observe that the first term, 1 2mV 2, arises because the massive ball has already gained energy due to being carried with the bus at velocity V; energy must be supplied to accelerate the ball to this velocity according to its mass. The second term, 1 2mV 2 o, represents the additional energy imparted by Jack during the act of throwing the ball relative to the bus. The third term, mV Vo, arises due to the recoil effect, as captured in Eq. (C.2). This term reflects the fact that, in the Sframe, the bus slows down slightly as a result of the throw, leading to a corresponding increase in the ball’s energy due to momentum conservation. Let θ′be the angle measured in the S′frame between the direction of motion of the bus and the direction in which the ball is thrown. Considering angles in the range 0< θ′<90◦, only the X′-component of the ball’s velocity acts to reduce the velocity of the bus. This X′-component is responsible for conserving momentum. Therefore, by applying Eq. (C.1), we have: V′(x) = −mVocosθ′ M′(C.10) Alternatively, replacing Eq. (C.1) with Eq. (C.10) in the above derivation yields: K.E(ε) = mV o2 2+mV 2 2+mV Vocosθ′(C.11) The above equation illustrates how the observed energy of the ball changes after the throw and clarifies why the transformed energy depends on cos θ, where θis the angle between the direction of motion of the bus and the direction in which the ball 41
Integration Multiply (Eq. H.5) by ˙aand integrate over time: Z˙a¨a dt =−Z4πG 3ρ(t)a˙a dt. (H.7) The left-hand side integrates immediately: Z˙a¨a dt =1 2˙a2.(H.8) Thus, 1 2˙a2=−4πG 3Zρ(t)a˙a dt +C. (H.9) Using ρ(a)=ρ0a−3, the integrand becomes ρ(a)a˙a=ρ0a−2˙a=ρ0 d dt−a−1.(H.10) Therefore, Zρ(a)a˙a dt =ρ0−a−1+const.(H.11) Substituting this result back into Eq. H.9, gives 1 2˙a2=4πG 3ρ0a−1+C. (H.12) Since ρ(a)=ρ0a−3, we rewrite ρ0a−1=ρ(a)a2.(H.13) Thus, 1 2˙a2=4πG 3ρ(a)a2+C. (H.14) Multiply by 2and divide by a2: ˙a a2 =8πG 3ρ(a) + 2C a2.(H.15) (ii) Constant density: ρ(t) = ρconst If we instead assume a constant density ρ(t) = ρconst, then Eq. (H.5) becomes d dt˙a2 2=−4πG 3ρconst d dta2 2,(H.16) leading upon integration to ˙a2 2=−4πG 3ρconst a2 2+C, (H.17) 48
or ˙a2=−4πG 3ρconst a2+ 2C. (H.18) Dividing by a2, we obtain H2=−4πG 3ρconst +2C a2.(H.19) Identifying 2C=−k, Eq. (H.19) becomes H2=−4πG 3ρconst −k a2.(H.20) For a spatially flat model (k= 0), a positive constant density ρconst >0would therefore imply H2<0, which is unphysical. Arepulsive component, on the other hand, satisfies ¨r= +4πG 3ρrep r, (H.21) which leads, for ρrep = const, to H2= +4πG 3ρrep −k a2,(H.22) so that a non–diluting density contributes positively to H2, in analogy with the cosmological–constant (dark energy) term in the Friedmann equation. Newtonian Interpretation of a Uniform Negative–Mass Density For additional intuition, consider a Newtonian model in which the Universe is filled with a uniform negative mass density ρ−<0. The enclosed (gravitational) mass within radius ris then M−(r) = 4π 3ρ−r3.(H.23) The corresponding radial acceleration of a test particle is ¨r=−GM−(r) r2=−4πG 3ρ−r, (H.24) which is directed outward when ρ−<0, as expected for a repulsive (dark–energy–like) component. Multiplying by ˙rgives ˙r¨r=−4πG 3ρ−r˙r=−4πG 3ρ− d dtr2 2.(H.25) 49
For a constant (non–diluting) density ρ−, this integrates to ˙r2=−4πG 3ρ−r2+C. (H.26) At the center r= 0, the enclosed mass vanishes. Thus at ˙r(0) = 0, implies that the integration constant must be C= 0. Equation (H.26) then becomes ˙r2=−4πG 3ρ−r2=4πG 3ρdE r2,(H.27) where we defined ρdE ≡ −ρ−>0as the effective dark–energy density. Introducing the scale factor through r(t) = a(t)ximmediately yields ˙a2=4πG 3ρdE a2,(H.28) which is identical to the result obtained in Eq (H.22) for a constant, repulsive energy density. This Newtonian picture admits two equivalent interpretations. One may regard the test mass as accelerating away from the center due to the repulsive gravitational effect of the surrounding negative mass density. Alternatively, the same equation may be interpreted geometrically as a uniform expansion. Either viewpoint reproduces the same expansion law once r=a(t)xis introduced. Appendix I. Interaction Between Positive and Negative Gravitational Mass The idea of a hidden or mirrored sector has appeared in a variety of contexts, including mirror–matter models and CPT–symmetric cosmologies [138,139,141]. These works do not introduce negative gravitational mass; rather, they provide examples in which an additional sector is coupled to ordinary gravity. In contrast, the present framework involves a genuinely negative gravitational mass and therefore requires a different dynamical structure. Earlier negative–mass proposals suffered from severe instabilities, most notably the runaway behaviour identified by Bondi [137], which arises when the sign of the gravitational mass is reversed while the inertial law F=ma is left unchanged. In the present construction, the inertial response is reversed in a matched way, F=−m a, 50
As the gravitational force between two negative masses is repulsive, it follows that: F= + GM1M2 r2. The force still acts along the radial unit vector, so for two particles at r1and r2, F12 = +GM1M2 |r12|2ˆ r12,F21 =−GM1M2 |r12|2ˆ r12, and Newton’s third law remains intact. Consequently, momentum is conserved, even with the reversed force law. Energy exchange. In this sector the work relation becomes dW =−mv dv, yielding a kinetic energy T−=−1 2mv2. The negative sector therefore absorbs energy while keeping the total energy conserved, without exhibiting the unbounded growth characteristic of the runaway solutions in ordinary negative–mass models. Appendix J. Coulomb–field quantization and the Lamb shift In our formulation the electric field associated with the Coulomb interaction is expressed, in Fourier space, in terms of the charge-density operator ρkas EL(r, t) = X k"−ikρk ε0k2e−i(k·r−ωkt)+ikρ† k ε0k2ei(k·r−ωkt)#,(J.1) Promoting the charge density to an operator and writing ρk=αkak, ρ† k=αka† k,[ak, a† k′]=δk,k′,(J.2) we determine the coefficient αkby requiring that the field energy HCoul =ZV d3rε0 2EL(r, t) 2,(J.3) reproduce the usual harmonic–oscillator spectrum HCoul =X k ℏωka† kak+1 2.(J.4) And a straightforward calculation gives the normalization: α2 k=ε0ℏωkk2 V.(J.5) Substituting Eq.(J.5) into Eq. (J.1), the Coulomb field takes the compact form EL(r, t) = X krℏωk V ε0 ˆ khi ake−i(k·r−ωkt)−i a† kei(k·r−ωkt)i,(J.6) 51
where ˆ k=k/k. For comparison, recall that in the standard quantization of the free electromagnetic field, a single transverse mode with wave vector kand polarization λis written as ˆ E⊥(r, t) = irℏωk 2ε0Vekλhˆakλei(k·r−ωkt)−ˆa† kλe−i(k·r−ωkt)i,(J.7) with ekλ⊥k. The atom–field interaction in the dipole approximation is ˆ Hint =−ˆ d·ˆ EL(r),(J.8) and for an initial atomic state |n, 0⟩the second–order Lamb–shift contribution is ∆En=X m,k⟨m, 1k|ˆ Hint|n, 0⟩ 2 E(0) n−E(0) m−ℏωk ,(J.9) where |m, 1k⟩denotes |m⟩with one quantum. Using Eq. (J.6), the matrix element becomes ⟨m, 1k|ˆ Hint|n, 0⟩=−irℏωk V ε0dmn ·ˆ k,dmn ≡ ⟨m|ˆ d|n⟩,(J.10) and therefore ⟨m, 1k|ˆ Hint|n, 0⟩ 2=ℏωk V ε0dmn ·ˆ k 2.(J.11) For an isotropic distribution of directions, dmn ·ˆ k 2ˆ k=1 3|dmn|2.(J.12) yields an effective factor 1 3|dmn|2from geometry. Using the transverse field (J.7), the dipole interaction Hamiltonian reads ˆ H(⊥) int =−ˆ d·ˆ E⊥(r).(J.13) The relevant matrix element between the initial state |n, 0⟩and the intermediate state |m, 1kλ⟩is ⟨m, 1kλ|ˆ H(⊥) int |n, 0⟩=−irℏωk 2ε0Vdmn ·ekλ,(J.14) so that ⟨m, 1kλ|ˆ H(⊥) int |n, 0⟩ 2=ℏωk 2ε0Vdmn ·ekλ 2.(J.15) The sum over the two transverse polarizations can be written as X λdmn ·ekλ 2=X λ dmn,idmn,j ekλ,iekλ,j =dmn,idmn,jδij −ˆ kiˆ kj,(J.16) 52
where we used the completeness relation for the transverse polarization vectors, X λ ekλ,iekλ,j =δij −ˆ kiˆ kj. Contracting with dmn,idmn,j gives X λdmn ·ekλ 2=|dmn|2−dmn ·ˆ k 2.(J.17) For an isotropic distribution of directions ˆ k, the angular average of the longitudinal projection is Ddmn ·ˆ k 2Eˆ k=1 3|dmn|2,(J.18) so that DX λdmn ·ekλ 2Eˆ k=|dmn|2−1 3|dmn|2=2 3|dmn|2.(J.19) This is the familiar 2 3|dmn|2factor entering Bethe’s Lamb–shift calculation when using two transverse photon polarizations. At first sight the matrix element dmn·ˆ k 2together with the angular average (J.12) appears to give only the geometric factor 1 3|dmn|2, which differs from the familiar 2 3|dmn|2occurring in the standard transverse (QED) calculation. The resolution is simple. A free transverse photon carries its energy equally in the electric and magnetic fields, so only half of the mode energy resides in the electric field that couples to the atomic dipole. By contrast, the Coulomb mode in Eq. (J.4 and J.6) carries all of its energy in the electric field. Normalizing both descriptions to the same total mode energy therefore makes the longitudinal electric field amplitude larger by a factor √2 relative to a transverse photon. Since the Lamb shift depends on the square of the dipole matrix element, the longitudinal contribution is enhanced by an overall factor of 2. Combining this with the angular average (Eq. (J.12)) gives 2×1 3|dmn|2=2 3|dmn|2,(J.20) which is exactly the standard factor obtained in Bethe’s transverse–field treatment. Hence the longitudinal Coulomb–field quantization reproduces the same effective dipole coupling that appears in the conventional QED calculation of the Lamb shift, even though the underlying field modes are longitudinal rather than transverse. 53
Appendix K. Fourier–space derivation of the gravitational law From Eq. (71) we can write the gravitational field in Fourier space as g(k) = 4πiG kρ+ei(k·r−ωt) |k|2−ρ−e−i(k·r−ωt) |k|2.(K.1) Thus the gravitational field in Fourier space becomes g(k) = 4πiG kρ+e−iωt |k|2−ρ−e−iωt |k|2eik·r.(K.2) The corresponding real–space field is obtained from g(r, t) = Zd3k (2π)3g(k)eik·r.(K.3) Using the standard identity Zd3k (2π)3 ik |k|2eik·(r−r0)=−r−r0 4π|r−r0|3, we obtain g(r, t)=−Gρ+ r r3e−iωt +Gρ− r r3e−iωt.(K.4) In the static limit (ω→0) this reduces to the familiar attractive and repulsive 1/r2gravitational laws. Appendix L. Potentials, canonical structure, and mode expansion We adopt the gauge-fixed longitudinal-sector Lagrangian (with Lorenz gauge made dynamical), Lgf =ε0 2(∂tϕ)2−c2(∇ϕ)2+ε0 2(∂tA)2−c2(∇×A)2−ρ ϕ +J·A.(L.1) The current density Jis kept general at this stage. After imposing ∇×A= 0 only the longitudinal component JLcouples to A, and charge conservation ∂tρ+∇·J= 0 fixes JLin terms of ρ, so no additional independent degrees of freedom enter. Canonical momenta follow directly: πϕ=ε0∂tϕ, πA=ε0∂tA.(L.2) The Hamiltonian density is H=1 2ε0 π2 ϕ+ε0c2 2(∇ϕ)2+1 2ε0|πA|2+ε0c2 2|∇×A|2+ρϕ −J·A.(L.3) 54
Hamilton’s equations give ∂tϕ=πϕ ε0 , ∂tA=πA ε0 ,(L.4) and the electric field is E=−∇ϕ−∂tA.(L.5) We now restrict to the longitudinal sector by imposing ∇×A= 0 (hence B= 0) while retaining Aas a longitudinal vector potential. The Lorenz gauge reads 1 c2∂tϕ+∇·A= 0,(L.6) which in Fourier components yields −iωk c2ϕk+ik·Ak= 0 =⇒k·Ak=ωk c2ϕk.(L.7) ϕwave equation ε0(∂2 t−c2∇2)ϕ=ρ, (L.8) Equation (L.8) is an inhomogeneous wave equation, and its solution may be written in terms of the retarded Green’s function of the operator (∂2 t−c2∇2). Convolving this Green’s function with the source ρyields the usual potential. and in Fourier space ε0(−ω2 k+c2k2)ϕk=ρk, D ≡ω2 k−c2k2.(L.9) Because Ais longitudinal we write Ak=ˆ kA∥(k)with ˆ k=k/k. From Eq.(L.7), A∥(k) = ωk c2kϕk.(L.10) Using E=−∇ϕ−∂tA, the positive-frequency longitudinal electric-field amplitude is ˆ E+(k, t) = −ik1−ω2 k c2k2ϕke−iωkt,(L.11) with the Hermitian-conjugate negative-frequency part accordingly. Canonical quantization is imposed by the equal-time commutator [ˆ ϕ(r, t),ˆπϕ(r′, t)] = iℏδ(3)(r−r′),ˆπϕ=ε0∂tˆ ϕ. (L.12) We adopt the discrete box-normalized mode ansatz ˆ ϕ(r, t) = X kβkˆakei(k·r−ωkt)+β∗ kˆa† ke−i(k·r−ωkt),(L.13) ˆπϕ(r, t) = ε0∂tˆ ϕ=X k ε0(−iωk)βkˆakei(k·r−ωkt)−β∗ kˆa† ke−i(k·r−ωkt).(L.14) 55
Substituting these into the canonical commutator and using plane-wave orthogonality fixes the oscillator algebra [ˆak,ˆa† k′] = δkk′.(L.15) We define the vacuum state |0⟩by ˆak|0⟩= 0 ∀k.(L.16) The longitudinal-sector Hilbert space is then the bosonic Fock space generated by repeated action of the creation operators ˆa† kon |0⟩. The spin of the corresponding quanta is determined by the Lorentz transformation properties of the underlying field rather than by the presence of electric charge. Since the longitudinal sector is constructed from a scalar degree of freedom, its excitations carry spin zero. The longitudinal part of the field energy is obtained from ε0 2Rd3rˆ E2. Retaining only the non-oscillatory (energy-conserving) contribution and performing the spatial integral over volume Vgives the per-mode longitudinal energy H∥(k)=ε0V k21−ω2 k c2k22|βk|2ˆa† kˆak+1 2.(L.17) Since ωk>0and ˆa† kˆak≥0, the longitudinal-sector Hamiltonian is bounded from below. Matching mode by mode with the canonical oscillator energy ℏωk(ˆa† kˆak+1 2)determines |βk|2=ℏωk ε0V k21−ω2 k c2k2−2.(L.18) From the Fourier wave equation (L.9) we obtain the operator identity ˆρk=ε0(ω2 k−c2k2)ˆ ϕk=ε0D βkˆak.(L.19) Defining ˆρk=αkˆakand using Eq. (L.18) together with the algebraic identity 1−ω2 k c2k2=−D c2k2(L.20) yields complete cancellation of Dand the final normalization |αk|2=ε0ℏωkk2 V,ˆρk=rε0ℏωkk2 Vˆak.(L.21) Consequently [ˆρk,ˆρ† k′] = ε0ℏωkk2 Vδkk′.(L.22) 56
Gauss-law constraint In the operator formalism, Gauss’s law is not a dynamical equation but a constraint on the physical states. At the operator level it reads ∇· ˆ E(x, t)−ˆρ(x, t) ε0 = 0 (L.23) to be imposed as a constraint on the physical Hilbert space, ∇· ˆ E(x, t)−ˆρ(x, t) ε0|phys⟩= 0.(L.24) This relation is preserved in time by the Heisenberg equations of motion, so it acts as a primary constraint rather than an independent equation of motion. In covariant Gupta–Bleuler quantization one imposes the subsidiary condition ∂µˆ Aµ(+) |phys⟩= 0,(L.25) which removes negative-norm states from the covariantly quantized photon Fock space. In the present construction, however, the Lorenz condition ∂µˆ Aµ= 0 (L.26) holds as an operator identity once the longitudinal relations between ˆ ϕ,ˆ A, and the charge-density operator ˆρare imposed. As a consequence, ∂µˆ Aµ(+) |phys⟩= 0 (L.27) is automatically satisfied for all states and does not impose any further restriction on the physical Hilbert space. We decompose the electric field into longitudinal and transverse parts, ˆ E(x, t) = ˆ EL(x, t) + ˆ ET(x, t),(L.28) with ∇× ˆ EL= 0,∇· ˆ ET= 0.(L.29) Since ˆ ETis divergence–free, Gauss’s law (Eq. L.23) constrains only the longitudinal component: ∇· ˆ EL(x, t) = ˆρ(x, t) ε0 .(L.30) 57
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