The unification of the fundamental forces remains a central goal in theoretical physics, with electromagnetism providing the paradigmatic example of a successful quantum field theory. In this work, we develop a quantum electrogravitics (QEG) framework in which gravity emerges as a U ( 1 ) gauge theory directly analogous to electromagnetism, thereby realizing a U ( 1 ) × U ( 1 ) quantum field theory (QFT) that couples the two interactions at the level of gauge potentials and currents. The construction begins from a relativistic velocity dependence of gravitational mass and proceeds systematically to a Lorentz-like gravitational force law, an implied four-potential, a gravitomagnetic version of Maxwell’s equations, and finally a fully quantized theory whose tree-level amplitudes recover both the Coulomb and Newtonian potentials in the appropriate limits.

The key insight motivating the approach is the observation that centripetal accelerations in gravitational orbits can be reinterpreted through the relativistic reduction of gravitational mass.

The remainder of the paper is organized as follows. Section 2 derives the gravitational Lorentz force and the associated gravitomagnetic field. Section 3 constructs the implied four-potential and field-strength tensor, emphasizing its epistemological status. Section 4 obtains the gravitomagnetic Maxwell equations from the Bianchi identity and the action principle. Section 5 quantizes the uncoupled theory and verifies the classical limits. Section 6 introduces field and charge couplings, derives the coupled equations of motion (Section 6.1), and computes the mixed propagators (Section 6.2). In Section 7 we compute the potentials in the classical limit. We conclude with remarks on possible experimental signatures and extensions.

By treating gravity as an induced U ( 1 ) gauge symmetry parallel to electromagnetism, the present framework offers a concrete, calculable QEG theory whose predictions can be confronted directly with laboratory-scale tests of gravitomagnetism and electrogravitational coupling.

We derive a gravitational analog of the Lorentz force equation by considering the velocity dependence of gravitational mass, as proposed in alternative gravitational theories [1]-[3]. Starting from the relativistic form of gravitational mass, we approximate the force on a test mass in a weak gravitational field and employ vector identities to obtain a form resembling the electromagnetic Lorentz force, introducing a gravitomagnetic field. This derivation highlights the illusory nature of gravitomagnetism arising from mass-velocity dependence.

In electromagnetism, the Lorentz force on a charged particle is given by

where q is the charge, E is the electric field, B is the magnetic field, and v is the velocity. Analogous effects in gravity, termed gravitomagnetism, have been explored [1], but may stem from the velocity dependence of gravitational mass rather than a true magnetic-like field. Here, we derive the gravitational Lorentz force from the assumption that gravitational mass m g decreases with velocity as.

Consider a test body with gravitational mass m g in a gravitational field E g . The force is

Substituting the velocity-dependent mass and expanding for v ≪ c

assuming a stationary frame relative to the field source.

Using the vector identity

and neglecting the second term for near-perpendicular motion (e.g., circular orbits), we obtain

Define the gravitomagnetic field

satisfying

and (when v is re-interpreted as the source velocity)

where ρ g is the mass density and κ is the gravitational constant from Gauss’ law

The force then takes the Lorentz-like form

This holds for small velocities and perpendicular motion, mimicking gravitomagnetism but originating purely from mass-velocity dependence.

The model is constructed under the following starting assumptions and scope: it applies in the weak-field, low-velocity regime with motion that is near-perpendicular to the gravitational field lines, and it incorporates the nonstandard postulate of a velocity-dependent gravitational mass. These restrictions define the domain in which the subsequent field-theoretic construction is intended to operate.

It is clear that E g and B g take exactly the same form as the electric and magnetic fields E e and B e . Carrying on this analogy we introduce a gravitational 4-potential B μ with components.

where A g is the gravitomagnetic vector potential (distinct from the field B g ).

The gravitational fields are recovered in the usual manner:

We can define the gravitational field-strength tensor

where B μ = ( ϕ , A g ) . Working out the components yields

The structural similarity of the gravitational fields to their electromagnetic counterparts motivates an analogous definition of the field-strength tensor G μ ν and the derivation of the corresponding Maxwell-like equations.

Using the framework developed above, we construct an uncoupled U ( 1 ) × U ( 1 ) quantum field theory in which the electromagnetic and gravitomagnetic sectors couple to the same matter field but remain independent of each other. Electromagnetism uses potential A μ with charge q e and gravity uses B μ with charge q g = g m 0 , where g ~ κ . Here g is the dimensionless gravitational gauge coupling, and the square-root factor ensures that the tree-level potential reproduces Newton’s law once the attractive sign is inserted

The Lagrangian is

with

In the gravitational sector the overall minus sign is imposed by hand on the interaction term − q g J g μ B μ (and correspondingly on the tree-level potential) to produce attraction between like gravitational charges, as required phenomenologically for positive masses. This choice overrides the generic repulsive sign expected for massless spin-1 gauge-boson exchange between identical charges in a U ( 1 ) theory.

In the Feynman gauge the propagators are identical:

In this U ( 1 ) formulation the gravitational gauge field B μ is quantized as a massless vector boson. Consequently, its quantum, the graviton, carries spin 1 (helicity ±1), exactly as the photon does in electromagnetism.

The tree-level amplitudes and their non-relativistic limits yield the position-space potentials:

Thereby recovering Coulumb’s law and Newton’s law, respectively.

Building on the uncoupled framework, we introduce both kinetic mixing β 4 F μ ν G μ ν and charge mixing q g = q g 0 + α q e E , where E is the local electric-field strength. The complete Lagrangian is

For two particles with electric charges q e 1 , 2 and effective gravitational charges q g 1 , 2 = q g 1 , 2 0 + α q e 1 , 2 E (where E is the common external electric-field strength), the tree-level amplitude is

In the non-relativistic limit, the position-space potential becomes

Substituting the charge-mixing expression immediately yields

where a 0 , a 1 , a 2 are explicit combinations of the bare charges, α , and β . The calibration q g 0 ~ m 4 π κ recovers the usual Coulomb and Newtonian terms in a 0 , while the linear and quadratic pieces in E are pure electrogravitic predictions.

The quadratic dependence a 2 E 2 precisely matches the voltage-squared force law observed in the Biefeld-Brown effect for asymmetric high-voltage capacitors. Reports in the literature, including vacuum chamber tests at pressures ~10⁻⁶ Torr [4]-[7], confirm that this E 2 dependence persists where the absence of free ions and atmospheric gases rules out conventional fluid-dynamic or ionized-air-current explanations, though other experiments have demonstrated the contrary [8][9].

Velocity-dependent effects can be incorporated via the relativistic factor E ≈ γ m 0 c 2 , thereby re-deriving the apparent gravitomagnetism discussed earlier

In this work we have constructed a complete QEG theory by treating gravity as a U ( 1 ) gauge interaction fully analogous to electromagnetism. The journey began with the observation that centripetal effects in gravitational orbits, when combined with the relativistic velocity dependence of gravitational mass, naturally yield a Lorentz-like force law. Extrapolating this structure produced a gravitational field-strength tensor G μ ν that is formally identical to the electromagnetic tensor F μ ν , thereby permitting the derivation of a full set of gravitomagnetic Maxwell equations and the subsequent quantization of the theory.

A central philosophical distinction runs through the entire framework: because the gravitomagnetic component arises solely from the velocity dependence of the test mass, the gravitational four-potential B μ and its associated tensor are implied rather than fundamental. The construction is therefore epistemological, an extremely useful mathematical artifact that reproduces observed dynamics, rather than an ontological postulate of an independent gravitomagnetic field.

Nevertheless, the formalism is sufficiently robust to support a full quantum field theory. We presented both the uncoupled U ( 1 ) × U ( 1 ) limit and the coupled version that includes a bilinear mixing term β 4 F μ ν G μ ν together with charge-mixing operators. In the uncoupled case the tree-level propagators are identical and the position-space potentials recover Coulomb’s law for electric charges q e and Newton’s law for gravitational charges q g ~ m 4 π κ . When coupling is introduced, the quadratic action becomes matrix-valued and the mixed propagator 〈 A μ B ν 〉 mediates genuine electrogravitational interference while preserving the long-range inverse-square behavior.

Particularly noteworthy is that the coupled theory naturally generates interaction terms quadratic in the electric-field strength E , a feature that aligns directly with laboratory observations of asymmetric high-voltage capacitors. The framework therefore offers not only conceptual unification at the gauge level but also concrete, calculable predictions that can be confronted with tabletop experiments.

Furthermore, the mixed propagator opens a novel annihilation channel:

with γ g representing the graviton. Although the branching ratio is suppressed by factors of β 2 and ( q g / q e ) 2 ≪ 1 under ordinary conditions, in the extreme environments of quasars, with enormous pair-production rates and relativistic luminosities, cumulative energy leakage into escaping gravitons could produce observable spectral dimming or continuum suppression. This offers a distinctive, testable signature unique to the coupled theory.

This U ( 1 ) × U ( 1 ) QEG theory constitutes a radical departure from conventional approaches to gravitation, yet it remains fully calculable and renormalizable in the same sense as QED. By recasting gravity as an induced yet fully quantized gauge interaction, the present theory opens a new avenue for unifying the long-range forces within a single, experimentally testable quantum framework.