Second-Order Effective Geometry of de Broglie Gravitational Waves

Abstract

We investigate the non-linear dynamics of de Broglie gravitational waves within a post-Minkowskian framework. Starting from the perturbative solution proposed by Feoli and Scarpetta, we derive the second-order gravitational field generated by the wave’s self-interaction. Owing to the extremely high oscillation frequency of de Broglie gravitational waves, we employ the Isaacson/Misner–Thorne–Wheeler averaging procedure to obtain an effective coarse-grained geometry. The resulting averaged theory takes the form of an effective linear perturbation theory at second order in the wave amplitude. Near the symmetry axis, the resulting spacetime exhibits a non-vanishing static curvature generated entirely by non-linear effects. We show that geodesics in this effective geometry undergo bounded transverse oscillations governed by a harmonic equation. Possible implications for de Broglie’s wave–particle framework are also discussed.

1. Introduction

Quantum Mechanics provides an extraordinarily accurate description of microscopic phenomena, yet the physical meaning of its mathematical formalism remains a matter of continuing debate. In the standard or Copenhagen interpretation, the wave function ${\psi }_{\mathrm{QM}}$ is regarded as a complete description of a quantum system, evolving according to the Schrödinger equation in an abstract Hilbert or configuration space. The quantity $|{\psi }_{\mathrm{QM}}{|}^2$ determines the probability density for measurement outcomes, without any notion of a definite particle trajectory. Although operationally successful, this interpretation has often been criticized for lacking a clear ontological description of the underlying physical reality.

An alternative perspective was introduced by Louis de Broglie in 1927 through the pilot-wave hypothesis [1], according to which an elementary particle is accompanied by a guiding wave. Several decades later, Bohm reformulated de Broglie’s proposal into the modern framework of Bohmian mechanics [2,3]. In this formulation, particles follow definite trajectories guided by the wave function, while the initial particle positions play the role of hidden variables. Bohmian mechanics reproduces the predictions of standard quantum theory while avoiding the central role assigned to the observer and the measurement process in the Copenhagen interpretation. Quantum effects are encoded in the so-called quantum potential, whose intrinsically non-local nature may imply instantaneous correlations at a distance.

A suggestive macroscopic analogy with pilot-wave systems emerged in the experiments of Couder and Fort of 2005 [4,5], where walking droplets bouncing on a vibrating fluid bath exhibited behaviors reminiscent of wave–particle duality. These experiments stimulated the development of hydrodynamic analogues of several quantum phenomena, with varying degrees of success, including diffraction and interference through slits [6,7,8,9,10,11,12,13], tunneling [6,14,15,16], quantum corrals [17,18,19,20,21], the quantum mirage [20], Landau levels [22,23,24] and Friedel oscillations [25].

These experiments renewed interest in the possibility that pilot-wave dynamics might emerge from an underlying classical medium [26,27,28,29], and motivated several modern reformulations and extensions of de Broglie’s original ideas [30,31,32,33,34]. More generally, hydrodynamic quantum analogues have provided a valuable framework for exploring how apparently quantum-like behavior may arise from the non-trivial interaction between localized particles and extended wave fields. These developments naturally raise the question of whether pilot-wave dynamics could admit a geometric realization directly at the level of spacetime itself.

In 1998, Feoli and Scarpetta found a solution to the linearized Einstein equations in a vacuum called de Broglie gravitational waves [35]. These spacetime perturbations possess axial symmetry and are characterized by a free parameter ${\omega }_r$. This parameter sets the transverse momentum scale of the solution and, in the reduced longitudinal description, appears as an effective mass scale through the Klein–Gordon dispersion relation. The complete four-dimensional perturbation, nevertheless, remains a solution for the massless vacuum wave equation. The solution also contains a non-trivial longitudinal sector, whose dynamics arises from the dimensional reduction and should not be interpreted as an additional massive gravitational polarization beyond the physical degrees of freedom of vacuum General Relativity.

We emphasize that this framework is constructed entirely within standard General Relativity, without introducing modifications of Einstein’s equations or additional gravitational fields. Nevertheless, it predicts a novel class of spacetime oscillations with a non-trivial longitudinal sector. Determining whether such solutions may admit a consistent physical interpretation, and identifying possible mechanisms capable of generating them, is, therefore, of considerable interest. Their possible detection would provide evidence for the physical relevance of this particular class of vacuum solutions and could reveal aspects of gravitational dynamics not captured by the usual plane-wave description. More generally, the observation of such spacetime oscillations would broaden our understanding of the dynamical structure of spacetime and of the possible manifestations of gravitational phenomena in the universe.

When the parameter ${\omega }_r$ is identified with the reduced Compton frequency of a particle, the de Broglie gravitational wave naturally acquires the role of a pilot-wave associated with an elementary particle. In this interpretation, the longitudinal components become a distinctive feature of the wave dynamics and may, in principle, produce observable effects, particularly in regimes involving slowly moving particles [36,37]. In the literature, a geometric realization of de Broglie’s wave–particle picture was proposed in [38], where the particle and the associated wave emerge, respectively, as solutions to the non-homogeneous and homogeneous linearized Einstein equations. Within this framework, the particle is not the source of the gravitational wave, but rather a localized energy distribution propagating, at the velocity of the wave packet, together with the associated spacetime disturbance. A different interpretation was later advanced in [39], where the particle is viewed as an effective energy density associated with the gravitational field itself. Further discussion of these interpretations and their physical implications can be found in [40]. Here, we adopt a different perspective and assume that the particle associated with the de Broglie gravitational wave is a test particle propagating along a geodesic of the second-order effective coarse-grained geometry generated by the spacetime oscillations.

In recent works we investigated several properties and possible observational consequences of de Broglie gravitational waves. We analyzed their polarization structure [41], the induced motion of nearby test particles [37,42], and semiclassical interactions with two-level quantum systems [43]. We also explored possible observational signatures, including pulsar timing effects and characteristic angular shifts [39,41,42]. More recently, we have shown that the dynamics of our spacetime oscillations is compatible with single-particle Quantum Mechanics and proposed a modified double-slit experiment that could provide an experimental test of the framework [40].

In the present work, we compute the post-Minkowskian expansion of de Broglie gravitational waves up to second order. We show that, while the first-order perturbation produces only oscillatory effects, the non-linear self-interaction of the wave also generates a non-oscillatory second-order contribution. After coarse-graining over the rapidly oscillating components, this contribution survives as a background geometry governing the long-timescale dynamics of test corpuscles. Using the geodesic equation, we show that the resulting geometry induces a harmonic-like restoring force in the transverse plane, leading to bounded motion of test particles around the symmetry axis of the wave. The non-linear self-interaction of de Broglie gravitational waves, therefore, gives rise to a persistent confining structure in spacetime, providing a geometric mechanism for the transverse confinement required in gravitational approaches to wave–particle duality.

The structure of this paper is as follows. Section 2 introduces the de Broglie gravitational wave and its main properties. In Section 3 we review the post-Minkowskian expansion in General Relativity. Building on this framework, Section 4 develops the post-Minkowskian expansion of the de Broglie wave solution and derives the corresponding effective geometry. In Section 5 we investigate the geodesic dynamics induced in this geometry. Finally, Section 6 summarizes the main results and outlines possible directions for future research. For the sake of readability, part of the calculations is presented in Appendix A and Appendix B.

We use the signature $(+,-,-,-)$. Latin indices $i,j,\dots$ run over $0,1,2,3$, while Greek indices $\alpha ,\beta ,\dots$ run over spatial components only. Unless otherwise specified, we work in units where $c=1$, restoring factors of c in the final expressions when convenient. We will also use the vector notation for three-vectors; for example, a vector $\mathbf{v}$ is defined by $\mathbf{v}\equiv (v_x,v_y,v_z)$. The Riemann tensor is defined as $R_{bcd}^a\equiv {\partial }_c{\Gamma }_{bd}^a-\dots$.

2. The de Broglie Gravitational Wave

In this section, we review the derivation of the de Broglie gravitational wave presented in [40]. We start from the linearized Einstein’s field equations in a vacuum

$$\square h_{ij}=0,$$

(1)

in the Lorenz gauge

$${\partial }_j\left(h_i^j-{\displaystyle \frac{1}{2}}{\delta }_i^j{h}_s^s\right)=0,$$

(2)

where $\square ={\eta }^{ab}{\partial }_a{\partial }_b$ is the flat-space d’Alembert operator.

To solve these equations we consider metric perturbations that factorize into a constant polarization tensor and a scalar field (Such a separation is allowed for Einstein–Maxwell equations in the weak-field limit [44], and, therefore, is also admissible in a vacuum.). We adopt the ansatz

$$h_{ij}(t,\mathbf{x})=\mathcal{A}{ϵ}_{ij}{\psi }_{\perp }(\widehat{\mathbf{k}}\times \mathbf{x}){\psi }_{\Vert }(t,\widehat{\mathbf{k}}·\mathbf{x}),$$

(3)

where $\mathcal{A}\ll 1$ is the wave amplitude and ${ϵ}_{ij}$ is a constant polarization tensor. The functions ${\psi }_{\Vert }$ and ${\psi }_{\perp }$ are the components of the field along and orthogonal to the propagation direction $\widehat{\mathbf{k}}$, respectively.

Substituting this ansatz into Equation (1) and separating variables yields two equations,

$$\begin{array}{cc}\hfill ({\partial }_t^2-{\nabla }_{\Vert }^2){\psi }_{\Vert }& =-{\omega }_r^2{\psi }_{\Vert },\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill & \hfill {\nabla }_{\perp }^2{\psi }_{\perp }& =-{\omega }_r^2{\psi }_{\perp },\hfill \end{array}$$

(5)

where the constant ${\omega }_r^2\equiv 1/{ƛ}_r^2$ arises from the separation procedure.

Equation (4) has the form of the Klein–Gordon equation for the massive scalar field ${\psi }_{\Vert }$, while Equation (5) corresponds to the Helmholtz equation on the transverse plane.

The longitudinal field is, therefore, a plane wave,

$${\psi }_{\Vert }(t,\widehat{\mathbf{k}}·\mathbf{x})\propto {\mathrm{e}}^{-i(\omega t-\mathbf{k}·\mathbf{x})},$$

(6)

with Klein–Gordon dispersion relation

$${\omega }^2\left(k\right)=k^2+{\omega }_r^2.$$

(7)

We notice that ${\omega }_r$ plays the role of a rest-frame frequency, thereby motivating its name. In polar coordinates $(r,\theta )$ on the plane orthogonal to $\widehat{\mathbf{k}}$, Equation (5) becomes

$$r^2{\partial }_r^2{\psi }_{\perp }+r{\partial }_r{\psi }_{\perp }+{\partial }_{\theta }^2{\psi }_{\perp }+{\omega }_r^2{r}^2{\psi }_{\perp }=0.$$

(8)

Under the assumption of axial symmetry, the dependence on the angular coordinate $\theta$ vanishes, and the corresponding solution is given by the Bessel function of the first kind of order zero

$${\psi }_{\perp }\propto J_0\left({\omega }_{r}r\right).$$

(9)

Combining the longitudinal and transverse components, the metric perturbation describing a wave propagating along the z-axis, in Cartesian coordinates, takes the form

$$h_{ij}(t,x,y,z)=\mathcal{A}{ϵ}_{ij}{J}_0\left({\omega }_r\sqrt{x^2+y^2}\right){\mathrm{e}}^{-i{k}^s{x}_s},$$

(10)

where the wave four-vector $k^i=(\omega ,0,0,k)$ satisfies

$$k^j{k}_j={\omega }_r^2={\displaystyle \frac{1}{{ƛ}_r^2}},$$

(11)

with $k^3=-k_3\equiv k$.

We emphasize that the separation constant ${\omega }_r$ plays the role of an effective mass only for the reduced longitudinal field ${\psi }_{\Vert }$, satisfying Equation (4), whereas the complete four-dimensional perturbation $h_{ij}$ continues to satisfy the ordinary massless equation $\square h_{ij}=0$. This can be made explicit using the standard integral representation

$$\begin{array}{c}\hfill J_0\left({\omega }_{r}r\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\mathrm{d}\phi {\mathrm{e}}^{i{\omega }_{r}rcos(\phi -\theta )},\end{array}$$

(12)

valid for arbitrary $\theta$, where $(x,y)=(rcos\theta ,rsin\theta )$. Equation (10) can then be expressed as

$$h_{ij}(t,x,y,z)={\displaystyle \frac{\mathcal{A}{ϵ}_{ij}}{2\pi }}{\int }_0^{2\pi }d\phi {\mathrm{e}}^{-i{k}_{\phi }^s{x}_s},k_{\phi }^i=(\omega ,-{\omega }_{r}cos\phi ,-{\omega }_{r}sin\phi ,k).$$

(13)

Using the dispersion relation, $k_{\phi }^i{k}_i^{\phi }={\omega }^2-{\omega }_r^2-k^2=0$, each constituent wave vector $k_{\phi }^i$ is null for every value of $\phi$. The de Broglie gravitational wave solution is, therefore, a continuous superposition of ordinary four-dimensional null plane waves, all with the same transverse momentum magnitude ${\omega }_r$, distributed uniformly over the azimuthal direction. The separation constant ${\omega }_r$ should, consequently, be understood as an effective mass scale arising in the dimensionally reduced longitudinal description, rather than as the mass of an additional propagating gravitational degree of freedom.

The Lorenz gauge condition (2) constrains the components of the polarization tensor. One finds

$$\begin{array}{cc}\hfill {ϵ}_{00}& ={ϵ}_{33}={\displaystyle \frac{{ϵ}_{30}}{2}}\left({\displaystyle \frac{k_3}{\omega }}+{\displaystyle \frac{\omega }{k_3}}\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {ϵ}_{11}& ={ϵ}_{22}={\displaystyle \frac{{ϵ}_{30}}{2}}\left({\displaystyle \frac{k_3}{\omega }}-{\displaystyle \frac{\omega }{k_3}}\right).\hfill \end{array}$$

(15)

If the constant ${ƛ}_r$ is identified with the reduced Compton wavelength,

$${\omega }_r={\displaystyle \frac{m}{\hslash }},$$

(16)

Equation (11) becomes the relativistic dispersion relation $p^j{p}_j=m^2$, upon identifying

$$p^i=\hslash k^i.$$

(17)

In the limit of nearly stationary waves ($k_3\to 0$), one obtains

$${ϵ}_{11}={ϵ}_{22}=-{ϵ}_{00}=-{ϵ}_{33},{ϵ}_{03}=0.$$

(18)

From this result, one can argue that it is reasonable to assume [35,36]

$${ϵ}_{03}={ϵ}_{30}=2{\displaystyle \frac{v}{c}},$$

(19)

where v is the velocity of the associated particle in the laboratory frame. Using the components of the polarization tensor, one obtains the non-vanishing trace of the perturbation field:

$$h=-2h_{11}.$$

(20)

Writing the phase as $k^s{x}_s=S/\hslash$ immediately gives the guidance law $p_j={\partial }_{j}S$. The perturbation field can, therefore, be written in Cartesian coordinates as

$$\begin{array}{cc}\hfill h_{ij}(t,x,y,z)& =\mathcal{A}{ϵ}_{ij}{J}_0\left({\omega }_{r}r\right)cos(\omega t-kz)\hfill \\ \hfill & =\mathcal{A}{ϵ}_{ij}{J}_0\left({\omega }_{r}r\right)cos\left[{\displaystyle \frac{Et-pz}{\hslash }}\right],\hfill \end{array}$$

(21)

with $r=\sqrt{x^2+y^2}$, and can be interpreted as the gravitational pilot-wave associated with a particle of energy E and momentum p.

The corresponding phase velocity is

$$v_{\mathrm{ph}}=c\sqrt{1+{\left({ƛ}_{r}k\right)}^{-2}},$$

(22)

which is faster than the speed of light, as expected for a de Broglie wave. From the linearity of Equation (1), one can construct wave packets with group velocity

$$v_{\mathrm{gr}}={\displaystyle \frac{c}{\sqrt{1+{\left({ƛ}_{r}k\right)}^{-2}}}}=v,$$

(23)

satisfying the relation $v_{\mathrm{ph}}{v}_{\mathrm{gr}}=c^2$ [36].

We note that this monochromatic, axially-symmetric solution, whose transverse profile is given by the zeroth order first kind Bessel function, is not localized and, therefore, does not represent an asymptotically flat, finite-energy configuration. Indeed, $J_0\left({\omega }_{r}r\right)$ decays only as $r^{-1/2}$ at large r, so its transverse integral does not converge. In the longitudinal direction, the solution is monochromatic and has a plane-wave dependence on the phase $S=\omega t-kz$, and is, therefore, likewise delocalized along the propagation direction. Accordingly, Equation (10) should be understood either as a solution valid within a finite region of spacetime or as an elementary monochromatic mode contributing to a localized wave packet, for which a suitable superposition of modes could restore localization and asymptotic flatness.

3. Post-Minkowskian Formulation of General Relativity

In the post-Minkowskian formulation, the gravitational field is expanded in powers of Newton’s constant G around flat spacetime. To introduce this framework we follow [45] and define the gothic metric deviation

$${\mathfrak{h}}^{ij}\equiv \sqrt{-g}{g}^{ij}-{\eta }^{ij},$$

(24)

where g denotes the determinant of the metric $g_{ij}$. The previous definition is exact and ${\mathfrak{h}}^{ij}$ is not assumed to be small.

The gothic symbol (The same symbol is also used for the quantity ${\mathfrak{g}}^{ij}=\sqrt{-g}{g}^{ij}$, commonly referred to in the literature as the “gothic metric”.) is introduced to distinguish ${\mathfrak{h}}^{ij}$ from the usual perturbation $h_{ij}$, defined through $g_{ij}={\eta }_{ij}+h_{ij}+\mathcal{O}\left(h^2\right)$. In the weak-field limit $h_{ij}\ll 1$, one has $g^{ij}={\eta }^{ij}-h^{ij}+\mathcal{O}\left(h^2\right)$ and $\sqrt{-g}=1+h/2+\mathcal{O}\left(h^2\right)$, with $h={\eta }^{ij}{h}_{ij}$. It then follows that

$$-{\mathfrak{h}}^{ij}=h^{ij}-{\displaystyle \frac{1}{2}}{\eta }^{ij}h+\mathcal{O}\left(h^2\right).$$

(25)

Therefore, in the linearized regime the field ${\mathfrak{h}}^{ij}$ coincides (up to an overall sign) with the trace-reversed perturbation ${\overline{h}}^{ij}=h^{ij}-(1/2){\eta }^{ij}h$ used in the standard formulation of linearized gravity.

We now impose the de Donder (or harmonic) gauge condition

$${\partial }_j{\mathfrak{h}}^{ij}=0.$$

(26)

In this gauge the exact Einstein equations can be written in the Landau–Lifshitz form

$$\square {\mathfrak{h}}^{ij}=16\pi G{\tau }^{ij},$$

(27)

where □ denotes the flat-spacetime d’Alembertian introduced earlier. The effective source ${\tau }^{ij}$ is defined as

$${\tau }^{ij}=(-g)T^{ij}+{\displaystyle \frac{1}{16\pi G}}{\Lambda }^{ij},$$

(28)

where $T^{ij}$ is the matter energy–momentum tensor. The tensor ${\Lambda }^{ij}$ contains the non-linear gravitational self-interactions and is expressed in terms of the Landau–Lifshitz pseudo-tensor $t_{\mathrm{LL}}^{ij}$ as

$${\Lambda }^{ij}=16\pi G(-g)t_{\mathrm{LL}}^{ij}+({\partial }_s{\mathfrak{h}}^{ir}{\partial }_r{\mathfrak{h}}^{js}-{\mathfrak{h}}^{rs}{\partial }_r{\partial }_s{\mathfrak{h}}^{ij}),$$

(29)

with

$$\begin{array}{cc}\hfill 16\pi G(-g)t_{\mathrm{LL}}^{ij}=& g_{kl}{g}^{mn}{\partial }_m{\mathfrak{h}}^{ik}{\partial }_n{\mathfrak{h}}^{jl}+{\displaystyle \frac{1}{2}}{g}_{kl}{g}^{ij}{\partial }_n{\mathfrak{h}}^{km}{\partial }_m{\mathfrak{h}}^{nl}\hfill \\ \hfill & -g_{lm}\left(g^{ki}{\partial }_n{\mathfrak{h}}^{jm}+g^{kj}{\partial }_n{\mathfrak{h}}^{im}\right){\partial }_k{\mathfrak{h}}^{nl}\hfill \\ \hfill & +{\displaystyle \frac{1}{8}}\left(2g^{ik}{g}^{jl}-g^{ij}{g}^{kl}\right)\left(2g_{mn}{g}_{qp}-g_{nq}{g}_{mp}\right){\partial }_k{\mathfrak{h}}^{mp}{\partial }_l{\mathfrak{h}}^{nq}.\hfill \end{array}$$

(30)

Using the harmonic gauge condition (26), Equation (27) can be rewritten as

$$\square {\mathfrak{h}}^{ij}=16\pi G[(-g)\left(T^{ij}+t_{\mathrm{LL}}^{ij}\right)+{\partial }_r{\partial }_s{\chi }^{ijrs}],$$

(31)

with

$$\begin{array}{c}\hfill {\chi }^{ijrs}={\displaystyle \frac{1}{16\pi G}}({\mathfrak{h}}^{ir}{\mathfrak{h}}^{js}-{\mathfrak{h}}^{rs}{\mathfrak{h}}^{ij}).\end{array}$$

(32)

It is important to emphasize that no approximation has been made so far: Equations (26) and (27) provide an exact reformulation of Einstein’s equations. From a mathematical perspective one may first solve Equation (27) without imposing the gauge condition. For this reason the 10 tensor components of Equation (27) are often referred to as the relaxed Einstein equations. In this form the equations do not constrain the dynamics of the matter variables. Once a solution has been obtained, however, the gauge condition (26) must be imposed on the solution itself. In particular, the harmonic gauge implies the conservation law

$${\partial }_j{\tau }^{ij}=0,$$

(33)

which involves an ordinary rather than covariant derivative.

The post-Minkowskian expansion is obtained by expressing the gothic metric in powers of G. (The expansion in powers of G is to be understood as a formal device. Since G is dimensionful, it cannot serve as a true small parameter by itself. In a given physical situation, the relevant dimensionless expansion parameter is typically $G{m}_c/\left(c^2{\rho }_c\right)$, when the source is characterized by a mass $m_c$ confined to a region of size ${\rho }_c$. Its smallness corresponds to the weak-field regime. The use of G as an ordering parameter is, therefore, a convenient shorthand, while the actual expansion depends on the specific scales of the problem. The absence of a unique, dimensionless expansion parameter reflects the fact that post-Minkowskian expansions are believed to be asymptotic sequences that may not converge.)

$${\mathfrak{g}}^{ij}\equiv \sqrt{-g}{g}^{ij}={\eta }^{ij}+G{\mathfrak{h}}_{\left(1\right)}^{ij}+G^2{\mathfrak{h}}_{\left(2\right)}^{ij}+\cdots ,$$

(34)

or equivalently

$${\mathfrak{h}}^{ij}={\displaystyle \sum _{n=1}^{\infty }}{G}^n{\mathfrak{h}}_{\left(n\right)}^{ij}.$$

(35)

Substituting this expansion into Equation (27) and solving iteratively yields, at the first two orders in G,

$$\begin{array}{ccc}\hfill \square {\mathfrak{h}}_{\left(1\right)}^{ij}& =16\pi T^{ij},\hfill & \hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill \square {\mathfrak{h}}_{\left(2\right)}^{ij}& =-{\mathfrak{h}}_{\left(1\right)}^{kl}{\partial }_k{\partial }_l{\mathfrak{h}}_{\left(1\right)}^{ij}+{\displaystyle \frac{1}{2}}{\partial }^i{\mathfrak{h}}_{kl}^{\left(1\right)}{\partial }^j{\mathfrak{h}}_{\left(1\right)}^{kl}-{\displaystyle \frac{1}{4}}{\partial }^i{\mathfrak{h}}_{\left(1\right)}{\partial }^j{\mathfrak{h}}_{\left(1\right)}\hfill \\ \hfill & -{\partial }^i{\mathfrak{h}}_{kl}^{\left(1\right)}{\partial }^k{\mathfrak{h}}_{\left(1\right)}^{jl}-{\partial }^j{\mathfrak{h}}_{kl}^{\left(1\right)}{\partial }^k{\mathfrak{h}}_{\left(1\right)}^{il}+{\partial }_l{\mathfrak{h}}_{\left(1\right)}^{ik}\left({\partial }^l{\mathfrak{h}}_{k\left(1\right)}^j+{\partial }_k{\mathfrak{h}}_{\left(1\right)}^{jl}\right)\hfill \\ \hfill & +{\eta }^{ij}\left[-{\displaystyle \frac{1}{4}}{\partial }_k{\mathfrak{h}}_{lm}^{\left(1\right)}{\partial }^k{\mathfrak{h}}_{\left(1\right)}^{lm}+{\displaystyle \frac{1}{8}}{\partial }_k{\mathfrak{h}}_{\left(1\right)}{\partial }^k{\mathfrak{h}}_{\left(1\right)}+{\displaystyle \frac{1}{2}}{\partial }_k{\mathfrak{h}}_{lm}^{\left(1\right)}{\partial }^l{\mathfrak{h}}_{\left(1\right)}^{km}\right],\hfill \end{array}$$

(37)

where ${\mathfrak{h}}^{\left(1\right)}={\eta }_{ij}{\mathfrak{h}}_{\left(1\right)}^{ij}$, and all the indices are raised and lowered with the Minkowski metric ${\eta }_{ij}$.

The post-Minkowskian expansion above is naturally organized as an expansion in powers of G when the leading order field is sourced by matter, as in Equations (27)–(31), (36) and (37). In that case, for a fixed matter source, G controls the strength of the resulting metric perturbation. In the present work, however, we specialize to the vacuum sector, $T_{ij}=0$, in which the first-order field is instead the source-free de Broglie gravitational wave introduced in Section 2, with dimensionless amplitude $\mathcal{A}\ll 1$. In a vacuum, the factor G drops out of the field Equations (27)–(31), reflecting the fact that G enters Einstein’s equations only through the coupling to matter, $G_{ij}=8\pi G{T}_{ij}$, where $G_{ij}$ is the Einstein tensor. Consequently, in the source-free problem considered here, G does not provide a parameter controlling the perturbative hierarchy. It is, therefore, more appropriate to organize the expansion directly in powers of the wave amplitude $\mathcal{A}$. Accordingly, we reorganize the perturbative bookkeeping of Equations (34) and (35) by formally setting $G=1$:

$${\mathfrak{g}}^{ij}={\eta }^{ij}+{\mathfrak{h}}_{\left(1\right)}^{ij}+{\mathfrak{h}}_{\left(2\right)}^{ij}+\cdots .$$

(38)

Given the gothic metric ${\mathfrak{g}}^{ij}=\sqrt{-g}{g}^{ij}$, one obtains the inverse relation $g_{ij}=\sqrt{-\mathfrak{g}}{\mathfrak{g}}_{ij}$, where $\mathfrak{g}$ is the determinant of ${\mathfrak{g}}^{ij}$. In the weak-field regime, Equation (24) and standard matrix identities, at order $\mathcal{O}\left({\mathfrak{h}}^2\right)$, imply (See, for example, Equation (7.20d) of [46]. However, a different sign convention for the definition of gothic metric is adopted there.)

$$\begin{array}{c}\hfill \sqrt{-g}=1+{\displaystyle \frac{1}{2}}{\mathfrak{h}}^{ij}{\eta }_{ij}+{\displaystyle \frac{1}{8}}{\left({\mathfrak{h}}^{ij}{\eta }_{ij}\right)}^2-{\displaystyle \frac{1}{4}}{\mathfrak{h}}^{ij}{\mathfrak{h}}_{ij}+\cdots .\end{array}$$

(39)

Since $\mathfrak{g}=g$ and ${\mathfrak{g}}_{ij}\equiv {\eta }_{ij}-{\mathfrak{h}}_{ij}+\mathcal{O}\left({\mathfrak{h}}^2\right)$, the previous results lead to (See, for example, Equation (7.20a) of [46].)

$$\begin{array}{c}\hfill g_{ij}={\eta }_{ij}-{\mathfrak{h}}_{ij}+{\displaystyle \frac{1}{2}}\mathfrak{h}{\eta }_{ij}+{\mathfrak{h}}_{ik}{\mathfrak{h}}_j^k-{\displaystyle \frac{1}{2}}\mathfrak{h}{\mathfrak{h}}_{ij}+{\eta }_{ij}\left({\displaystyle \frac{1}{8}}{\mathfrak{h}}^2-{\displaystyle \frac{1}{4}}{\mathfrak{h}}^{lm}{\mathfrak{h}}_{lm}\right)+\cdots .\end{array}$$

(40)

Substituting the expansion (38) into the previous equation gives

$$\begin{array}{cc}\hfill g_{ij}& ={\eta }_{ij}-\left({\mathfrak{h}}_{ij}^{\left(1\right)}-{\displaystyle \frac{1}{2}}{\eta }_{ij}{\mathfrak{h}}^{\left(1\right)}\right)-({\mathfrak{h}}_{ij}^{\left(2\right)}-{\displaystyle \frac{1}{2}}{\eta }_{ij}{\mathfrak{h}}^{\left(2\right)}-{\mathfrak{h}}_{ik}^{\left(1\right)}{\mathfrak{h}}_{\left(1\right)j}^k\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\mathfrak{h}}^{\left(1\right)}{\mathfrak{h}}_{ij}^{\left(1\right)}-{\displaystyle \frac{1}{8}}{\eta }_{ij}{\left({\mathfrak{h}}^{\left(1\right)}\right)}^2+{\displaystyle \frac{1}{4}}{\eta }_{ij}{\mathfrak{h}}_{\left(1\right)}^{lm}{\mathfrak{h}}_{lm}^{\left(1\right)})+\cdots .\hfill \end{array}$$

(41)

The relation (41) will be used in the next section to construct the effective metric associated with the averaged gothic metric deviation.

4. Effective Geometry from Non-Linear Self-Interaction

We now apply the post-Minkowskian expansion to the de Broglie gravitational wave introduced in Section 2. In a vacuum ($T_{ij}=0$), the first-order solution of Equation (36) must reproduce the perturbative gravitational field $h_{\left(1\right)}^{ij}\equiv h^{ij}$ presented in Section 2. Using Equations (20) and (25), the non-vanishing components of the corresponding first-order gothic deviation field, in Cartesian coordinates, are

$$\begin{array}{ccc}\hfill & \hfill -{\mathfrak{h}}_{\left(1\right)}^{00}& =h_{\left(1\right)}^{00}-{\displaystyle \frac{1}{2}}{h}_{\left(1\right)}=\mathcal{A}({ϵ}_{00}+{ϵ}_{11})J_{0}cos(\omega t-kz),\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill & \hfill -{\mathfrak{h}}_{\left(1\right)}^{33}& =h_{\left(1\right)}^{33}+{\displaystyle \frac{1}{2}}{h}_{\left(1\right)}=\mathcal{A}({ϵ}_{00}-{ϵ}_{11})J_{0}cos(\omega t-kz),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill -{\mathfrak{h}}_{\left(1\right)}^{03}& =-{\mathfrak{h}}_{\left(1\right)}^{30}=h_{\left(1\right)}^{03}=-\mathcal{A}{ϵ}_{03}{J}_{0}cos(\omega t-kz).\hfill \end{array}$$

(44)

For later convenience we rewrite them as

$$\begin{array}{c}\hfill {\mathfrak{h}}_{ij}^{\left(1\right)}=\mathcal{A}{E}_{ij}{J}_{0}cos(\omega t-kz),\end{array}$$

(45)

where the non-vanishing components of the polarization tensor are

$$\begin{array}{cc}\hfill E_{00}& =-({ϵ}_{00}+{ϵ}_{11}),\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill E_{33}& =-({ϵ}_{00}-{ϵ}_{11}),\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill E_{03}& =E_{30}=-{ϵ}_{03}.\hfill \end{array}$$

(48)

As expected, the de Donder gauge condition is satisfied,

$${\partial }_j{\mathfrak{h}}_{\left(1\right)}^{ij}=0$$

(49)

and the trace of the gothic field is

$${\mathfrak{h}}_{\left(1\right)}={\eta }_{ij}{\mathfrak{h}}_{\left(1\right)}^{ij}=-2h_{11}.$$

(50)

At second order in the post-Minkowskian expansion, the field equations become non-linear and the first-order de Broglie gravitational wave acts as a source for its own gravitational field. Using Equation (36) (with $T_{ij}=0$), together with the gauge condition (49), Equation (37) reduces to

$$\begin{array}{c}\hfill \square {\mathfrak{h}}_{\left(2\right)}^{ij}={\displaystyle \frac{1}{2}}{\partial }^i{\mathfrak{h}}_{kl}^{\left(1\right)}{\partial }^j{\mathfrak{h}}_{\left(1\right)}^{kl}-{\displaystyle \frac{1}{4}}{\partial }^i{\mathfrak{h}}_{\left(1\right)}{\partial }^j{\mathfrak{h}}_{\left(1\right)}+{\partial }_k{\mathcal{T}}^{kij},\end{array}$$

(51)

where the total divergence term is defined as

$$\begin{array}{cc}\hfill {\partial }_k{\mathcal{T}}^{kij}& ={\partial }_k[{\partial }^k{\mathfrak{h}}_{\left(1\right)}^{is}{\mathfrak{h}}_{s\left(1\right)}^j+{\partial }_l{\mathfrak{h}}_{\left(1\right)}^{ik}{\mathfrak{h}}_{\left(1\right)}^{jl}-{\mathfrak{h}}_{\left(1\right)}^{kl}{\partial }_l{\mathfrak{h}}_{\left(1\right)}^{ij}-{\partial }^i{\mathfrak{h}}_l^{k\left(1\right)}{\mathfrak{h}}_{\left(1\right)}^{jl}-{\partial }^j{\mathfrak{h}}_l^{k\left(1\right)}{\mathfrak{h}}_{\left(1\right)}^{il}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}{\eta }^{ij}{\partial }^k{\mathfrak{h}}_{lm}^{\left(1\right)}{\mathfrak{h}}_{\left(1\right)}^{lm}+{\displaystyle \frac{1}{8}}{\eta }^{ij}{\partial }^k{\mathfrak{h}}_{\left(1\right)}{\mathfrak{h}}_{\left(1\right)}+{\displaystyle \frac{1}{2}}{\eta }^{ij}{\partial }^l{\mathfrak{h}}_{\left(1\right)}^{km}{\mathfrak{h}}_{lm}^{\left(1\right)}].\hfill \end{array}$$

(52)

De Broglie gravitational waves are characterized by extremely high frequencies (For example, the reduced Compton frequency ${\omega }_r$ of an electron is of order ${10}^{20}\mathrm{Hz}$. Equation (7) then implies $\omega \gtrsim {10}^{20}\mathrm{Hz}$.), and the dynamics of the associated particle is expected to involve timescales much longer than the oscillation period of the wave. Under these circumstances, it is natural to introduce a coarse-grained description based on the Isaacson/Misner–Thorne–Wheeler (IMTW) averaging procedure [47,48,49]. This averaging is performed over several wavelengths in space and oscillation periods in time, thereby separating the rapidly oscillating wave from the slowly varying effective geometry generated by its non-linear self-interaction.

We stress that, in the present construction, the high-frequency average is performed only over the phase $S\equiv \omega t-kz$. This is justified because $\omega ,k$ are set by the extremely high reduced Compton frequency ${\omega }_r$ associated with the particle, so that terms proportional to $cos\left(S\right)$, $sin\left(S\right)$, $cos\left(2S\right)$ or $sin\left(2S\right)$ oscillate on timescales much shorter than those relevant to the dynamics of interest and can, therefore, be consistently averaged to zero in the usual Isaacson sense. The transverse dependence is treated differently. Our analysis is restricted to the near-axis region, $r\ll 1/{\omega }_r$, where the Bessel profile $J_0\left({\omega }_{r}r\right)$ is smooth and slowly varying. Consequently, there is no rapid transverse oscillation within the region of interest that would warrant an additional averaging, and the full transverse dependence is retained and subsequently expanded in powers of r. A genuine transverse averaging procedure would, instead, be relevant in the far-field transverse region, where the Bessel functions oscillate on the scale $1/{\omega }_r$. That regime lies outside the scope of the present analysis.

The averaged second-order field naturally defines a coarse-grained effective metric $g_{ij}^{\mathrm{eff}}$. Applying the IMTW averaging procedure to Equation (41), we obtain, to order ${\mathcal{A}}^2$,

$$\begin{array}{cc}\hfill g_{ij}^{\mathrm{eff}}& \equiv \langle g_{ij}\rangle \simeq {\eta }_{ij}-(\langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle -{\displaystyle \frac{1}{2}}{\eta }_{ij}\langle {\mathfrak{h}}^{\left(2\right)}\rangle -\langle {\mathfrak{h}}_{ik}^{\left(1\right)}{\mathfrak{h}}_{\left(1\right)j}^k\rangle \hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\langle {\mathfrak{h}}^{\left(1\right)}{\mathfrak{h}}_{ij}^{\left(1\right)}\rangle -{\displaystyle \frac{1}{8}}{\eta }_{ij}\langle {\left({\mathfrak{h}}^{\left(1\right)}\right)}^2\rangle +{\displaystyle \frac{1}{4}}{\eta }_{ij}\langle {\mathfrak{h}}_{\left(1\right)}^{lm}{\mathfrak{h}}_{lm}^{\left(1\right)}\rangle )\equiv {\eta }_{ij}-{\mathcal{H}}_{ij},\hfill \end{array}$$

(53)

where $\langle \cdots \rangle$ denotes the average defined above and

$$\begin{array}{c}\hfill {\mathcal{H}}_{ij}=\langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle -{\displaystyle \frac{1}{2}}{\eta }_{ij}\langle {\mathfrak{h}}^{\left(2\right)}\rangle -{\displaystyle \frac{{\mathcal{A}}^2{J}_0^2}{2}}\left(E_{ik}{E}_j^k+{ϵ}_{11}{E}_{ij}+{\displaystyle \frac{1}{2}}{\eta }_{ij}{ϵ}_{11}^2-{\displaystyle \frac{1}{4}}{\eta }_{ij}{E}^{lm}{E}_{lm}\right).\end{array}$$

(54)

The $\mathcal{O}\left(\mathcal{A}\right)$ contributions vanish because the first-order perturbation is purely oscillatory and, therefore, averages to zero under the adopted IMTW averaging prescription, $\langle cos(\cdots )\rangle \simeq 0$. At second order, the quadratic oscillatory terms yield a non-vanishing contribution, with $\langle {cos}^2(\cdots )\rangle \simeq 1/2$. The resulting theory takes the form of an effective linear perturbation formalism for the metric perturbation ${\mathcal{H}}_{ij}$, which is generated by the non-linear self-interaction of the gravitational field.

We assume boundary conditions such that the boundary terms arising from the t and z derivatives vanish under the averaging procedure. Under this assumption, the IMTW averaging operator commutes with the d’Alembert operator (For details, see Appendix A), so that $\langle \square {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle =\square \langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle$. Thus, Equation (51) reduces to

$$\square \langle {\mathfrak{h}}_{\left(2\right)}^{ij}\rangle ={\displaystyle \frac{1}{2}}\langle {\partial }^i{\mathfrak{h}}_{kl}^{\left(1\right)}{\partial }^j{\mathfrak{h}}_{\left(1\right)}^{kl}\rangle -{\displaystyle \frac{1}{4}}\langle {\partial }^i{\mathfrak{h}}_{\left(1\right)}{\partial }^j{\mathfrak{h}}_{\left(1\right)}\rangle +\langle {\partial }_k{\mathcal{T}}^{kij}\rangle .$$

(55)

Substituting the first-order fields given in Equations (42)–(44) into Equation (55), together with Equation (50), yields

$$\begin{array}{cc}\hfill \square \langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle & ={\displaystyle \frac{1}{2}}\left[\langle {\partial }_i{\mathfrak{h}}_{00}^{\left(1\right)}{\partial }_j{\mathfrak{h}}_{00}^{\left(1\right)}\rangle -2\langle {\partial }_i{\mathfrak{h}}_{03}^{\left(1\right)}{\partial }_j{\mathfrak{h}}_{03}^{\left(1\right)}\rangle +\langle {\partial }_i{\mathfrak{h}}_{33}^{\left(1\right)}{\partial }_j{\mathfrak{h}}_{33}^{\left(1\right)}\rangle -2\langle {\partial }_i{h}_{11}^{\left(1\right)}{\partial }_j{h}_{11}^{\left(1\right)}\rangle \right]\hfill \\ \hfill & +\langle {\partial }_x\left({\partial }^x{\mathfrak{h}}_{i\left(1\right)}^k{\mathfrak{h}}_{jk}^{\left(1\right)}\right)\rangle +\langle {\partial }_y\left({\partial }^y{\mathfrak{h}}_{i\left(1\right)}^k{\mathfrak{h}}_{jk}^{\left(1\right)}\right)\rangle -{\displaystyle \frac{1}{4}}{\eta }_{ij}\left(\langle {\partial }_x\left({\partial }^x{\mathfrak{h}}_{lm}^{\left(1\right)}{\mathfrak{h}}_{\left(1\right)}^{lm}\right)\rangle +\langle {\partial }_y\left({\partial }^y{\mathfrak{h}}_{lm}^{\left(1\right)}{\mathfrak{h}}_{\left(1\right)}^{lm}\right)\rangle \right)\hfill \\ \hfill & +{\displaystyle \frac{1}{8}}{\eta }_{ij}\left(\langle {\partial }_x\left({\partial }^x{\mathfrak{h}}_{\left(1\right)}{\mathfrak{h}}_{\left(1\right)}\right)\rangle +\langle {\partial }_y\left({\partial }^y{\mathfrak{h}}_{\left(1\right)}{\mathfrak{h}}_{\left(1\right)}\right)\rangle \right)={\displaystyle \frac{1}{2}}[\langle {\partial }_i{\mathfrak{h}}_{00}^{\left(1\right)}{\partial }_j{\mathfrak{h}}_{00}^{\left(1\right)}\rangle -2\langle {\partial }_i{\mathfrak{h}}_{03}^{\left(1\right)}{\partial }_j{\mathfrak{h}}_{03}^{\left(1\right)}\rangle \hfill \\ \hfill & +\langle {\partial }_i{\mathfrak{h}}_{33}^{\left(1\right)}{\partial }_j{\mathfrak{h}}_{33}^{\left(1\right)}\rangle -2\langle {\partial }_i{h}_{11}^{\left(1\right)}{\partial }_j{h}_{11}^{\left(1\right)}\rangle ]+{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{2}}\left({\displaystyle \frac{E_i^k{E}_{jk}}{{ϵ}_{11}^2}}-{\displaystyle \frac{{\eta }_{ij}}{2}}\right)[J_0^2\left({\omega }_{r}r\right)-J_1^2\left({\omega }_{r}r\right)].\hfill \end{array}$$

(56)

The first two terms on the right-hand side of Equation (55) give rise to the terms in square brackets in Equation (56). In the first equality, the contribution from the total divergence, $\langle {\partial }_k{\mathcal{T}}^{kij}\rangle$, is obtained by evaluating the individual terms in ${\mathcal{T}}^{kij}$ for the first-order perturbation field. The contributions involving derivatives with respect to t and z vanish under the boundary conditions adopted above. For the transverse derivatives, the number of non-vanishing terms is further reduced by the specific form of the first-order perturbation, for which ${\mathfrak{h}}_{xx}^{\left(1\right)}={\mathfrak{h}}_{yy}^{\left(1\right)}={\mathfrak{h}}_{xy}^{\left(1\right)}=0$. In the second equality, we express the resulting contribution in terms of the polarization tensor $E_{ij}$ and perform the average. From Equation (56), we obtain

$$\begin{array}{c}\hfill \square \langle {\mathfrak{h}}_{01}^{\left(2\right)}\rangle \simeq \square \langle {\mathfrak{h}}_{02}^{\left(2\right)}\rangle \simeq \square \langle {\mathfrak{h}}_{13}^{\left(2\right)}\rangle \simeq \square \langle {\mathfrak{h}}_{23}^{\left(2\right)}\rangle \simeq 0\end{array}$$

(57)

and

$$\begin{array}{cc}\hfill \square \langle {\mathfrak{h}}_{00}^{\left(2\right)}\rangle & \simeq {\displaystyle \frac{{\mathcal{B}}^2{\omega }^2}{2}}{J}_0^2\left({\omega }_{r}r\right)+{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{2}}\left({\displaystyle \frac{E_0^k{E}_{0k}}{{ϵ}_{11}^2}}-{\displaystyle \frac{1}{2}}\right)[J_0^2\left({\omega }_{r}r\right)-J_1^2\left({\omega }_{r}r\right)],\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill \square \langle {\mathfrak{h}}_{03}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2\omega k}{2}}{J}_0^2\left({\omega }_{r}r\right)+{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{2}}{\displaystyle \frac{E_0^k{E}_{3k}}{{ϵ}_{11}^2}}[J_0^2\left({\omega }_{r}r\right)-J_1^2\left({\omega }_{r}r\right)],\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill \square \langle {\mathfrak{h}}_{11}^{\left(2\right)}\rangle & \simeq {\mathcal{B}}^2{\omega }_r^2{J}_1^2\left({\omega }_{r}r\right){\displaystyle \frac{x^2}{2r^2}}+{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{4}}[J_0^2\left({\omega }_{r}r\right)-J_1^2\left({\omega }_{r}r\right)],\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill \square \langle {\mathfrak{h}}_{12}^{\left(2\right)}\rangle & \simeq {\mathcal{B}}^2{\omega }_r^2{J}_1^2\left({\omega }_{r}r\right){\displaystyle \frac{xy}{2r^2}},\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill \square \langle {\mathfrak{h}}_{22}^{\left(2\right)}\rangle & \simeq {\mathcal{B}}^2{\omega }_r^2{J}_1^2\left({\omega }_{r}r\right){\displaystyle \frac{y^2}{2r^2}}+{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{4}}[J_0^2\left({\omega }_{r}r\right)-J_1^2\left({\omega }_{r}r\right)],\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill \square \langle {\mathfrak{h}}_{33}^{\left(2\right)}\rangle & \simeq {\displaystyle \frac{{\mathcal{B}}^2{k}^2}{2}}{J}_0^2\left({\omega }_{r}r\right)+{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{2}}\left({\displaystyle \frac{E_3^k{E}_{3k}}{{ϵ}_{11}^2}}+{\displaystyle \frac{1}{2}}\right)[J_0^2\left({\omega }_{r}r\right)-J_1^2\left({\omega }_{r}r\right)],\hfill \end{array}$$

(63)

where $r\equiv \sqrt{x^2+y^2}$ and

$$\begin{array}{c}\hfill {\mathcal{B}}^2\equiv {\mathcal{A}}^2({ϵ}_{00}^2-{ϵ}_{03}^2)={\mathcal{A}}^2{ϵ}_{11}^2.\end{array}$$

(64)

Since, after averaging, the d’Alembert operator reduces to the Laplace operator on the transverse plane (see Appendix A), $\square =-(1/r){\partial }_r\left(r{\partial }_r\right)-(1/r^2){\partial }_{\theta }^2$, Equation (57) reduce to two-dimensional Laplace equations in polar coordinates. Imposing regularity on the symmetry axis, the corresponding homogeneous solutions can consistently be taken to vanish. This choice is compatible with the gauge freedom of the effective theory, which allows four arbitrary functions to eliminate four components of the effective perturbation.

Equations (58)–(63) admit closed-form solutions in cylindrical coordinates $(r,\theta ,z)$. For instance, Equation (58) is integrated using Equation (5.54.2) of [50] together with the recurrence relation in Equation (9.1.27) of [51], yielding

$$\begin{array}{cc}\hfill r{\partial }_r\langle {\mathfrak{h}}_{00}^{\left(2\right)}\rangle & =-{\displaystyle \frac{{\mathcal{B}}^2}{2}}\int \mathrm{d}rr\left[{\omega }^2{J}_0^2\left({\omega }_{r}r\right)+{\omega }_r^2\left({\displaystyle \frac{E_0^k{E}_{0k}}{{ϵ}_{11}^2}}-{\displaystyle \frac{1}{2}}\right)(J_0^2-J_1^2)\right]\hfill \\ \hfill & =-{\displaystyle \frac{{\mathcal{B}}^2}{4}}\left[{\omega }^2{r}^2\left(J_0^2+J_1^2\right)+2{\omega }_{r}r\left({\displaystyle \frac{E_0^k{E}_{0k}}{{ϵ}_{11}^2}}-{\displaystyle \frac{1}{2}}\right)J_0{J}_1\right].\hfill \end{array}$$

(65)

After computing

$$\begin{array}{c}\hfill \left({\displaystyle \frac{E_0^k{E}_{0k}}{{ϵ}_{11}^2}}-{\displaystyle \frac{1}{2}}\right)=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\omega }^2+7k^2}{{\omega }_r^2}},\end{array}$$

(66)

a further integration (Since $J_1$ is proportional to the derivative of $J_0$, the product $J_0{J}_1$ can be easily integrated.) gives

$$\begin{array}{c}\hfill \langle {\mathfrak{h}}_{00}^{\left(2\right)}\rangle =-{\displaystyle \frac{{\mathcal{B}}^2}{4}}\left[{\omega }^2{r}^2\left(J_0^2+J_1^2-{\displaystyle \frac{1}{{\omega }_{r}r}}{J}_0{J}_1\right)+{\displaystyle \frac{{\omega }^2+7k^2}{2{\omega }_r^2}}{J}_0^2\right].\end{array}$$

(67)

Applying the same procedure for Equations (59) and (63) yields

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{33}^{\left(2\right)}\rangle & =-{\displaystyle \frac{{\mathcal{B}}^2}{4}}\left[k^2{r}^2\left(J_0^2+J_1^2-{\displaystyle \frac{1}{{\omega }_{r}r}}{J}_0{J}_1\right)+{\displaystyle \frac{7{\omega }^2+k^2}{2{\omega }_r^2}}{J}_0^2\right],\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{03}^{\left(2\right)}\rangle & ={\displaystyle \frac{{\mathcal{B}}^2}{4}}\left[\omega k{r}^2\left(J_0^2+J_1^2-{\displaystyle \frac{1}{{\omega }_{r}r}}{J}_0{J}_1\right)+{\displaystyle \frac{4\omega k}{{\omega }_r^2}}{J}_0^2\right].\hfill \end{array}$$

(69)

All integration constants have been set to zero.

Equations (60)–(62) are rewritten as

$$\begin{array}{cc}\hfill \left[{\displaystyle \frac{1}{r}}{\partial }_r\left(r{\partial }_r\right)+{\displaystyle \frac{1}{r^2}}{\partial }_{\theta }^2\right]\langle {\mathfrak{h}}_{11}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{4}}\left[J_0^2\left({\omega }_{r}r\right)+J_1^2\left({\omega }_{r}r\right)cos\left(2\theta \right)\right],\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill \left[{\displaystyle \frac{1}{r}}{\partial }_r\left(r{\partial }_r\right)+{\displaystyle \frac{1}{r^2}}{\partial }_{\theta }^2\right]\langle {\mathfrak{h}}_{12}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2}{4}}{\omega }_r^2{J}_1^2\left({\omega }_{r}r\right)sin\left(2\theta \right),\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill \left[{\displaystyle \frac{1}{r}}{\partial }_r\left(r{\partial }_r\right)+{\displaystyle \frac{1}{r^2}}{\partial }_{\theta }^2\right]\langle {\mathfrak{h}}_{22}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{4}}\left[J_0^2\left({\omega }_{r}r\right)-J_1^2\left({\omega }_{r}r\right)cos\left(2\theta \right)\right].\hfill \end{array}$$

(72)

The source in Equations (70) and (72) naturally decomposes into a radial contribution and a contribution depending on the angle $\theta$. Since the differential operator is linear, the solution for $(i,j)=(1,1)$ and $(i,j)=(2,2)$ are assumed of the form

$$\begin{array}{c}\hfill \langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle =\langle {\tilde{\mathfrak{h}}}_{ij}^{\left(2\right)}\left(r\right)\rangle +\langle {\mathfrak{f}}_{ij}^{\left(2\right)}(r,\theta )\rangle .\end{array}$$

(73)

The radial contribution is obtained by integrating the corresponding radial equation using the same Bessel function identities employed above. The final result is

$$\begin{array}{c}\hfill \langle {\tilde{\mathfrak{h}}}_{ij}^{\left(2\right)}\rangle =-{\displaystyle \frac{1}{8}}{\mathcal{B}}^2{\omega }_r^2{r}^2\left[J_0^2+J_1^2-{\displaystyle \frac{1}{{\omega }_{r}r}}{J}_0{J}_1\right].\end{array}$$

(74)

The angular equation can be solved by variation of parameters (see Appendix B), giving

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{11}^{\left(2\right)}\rangle & =-{\displaystyle \frac{1}{8}}{\mathcal{B}}^2{\omega }_r^2{r}^2\left[J_0^2+J_1^2-{\displaystyle \frac{1}{{\omega }_{r}r}}{J}_0{J}_1\right]+{\mathcal{B}}^2{\omega }_r^2{r}^2\{-{\displaystyle \frac{3}{32}}+{\displaystyle \frac{1}{32}}[J_0^2+J_1^2]\hfill \\ \hfill & +{\displaystyle \frac{1}{16}}\left[{\displaystyle \frac{1}{6}}(J_0^2+J_1^2)-{\displaystyle \frac{2}{3{\omega }_{r}r}}{J}_0{J}_1+{\displaystyle \frac{2}{3{\left({\omega }_{r}r\right)}^2}}{J}_1^2\right]\}cos\left(2\theta \right),\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{12}^{\left(2\right)}\rangle & ={\mathcal{B}}^2{\omega }_r^2{r}^2\{{\displaystyle \frac{3}{32}}+{\displaystyle \frac{1}{32}}[J_0^2+J_1^2]+{\displaystyle \frac{1}{16}}[{\displaystyle \frac{1}{6}}(J_0^2+J_1^2)-{\displaystyle \frac{2}{3{\omega }_{r}r}}{J}_0{J}_1\hfill \\ \hfill & +{\displaystyle \frac{2}{3{\left({\omega }_{r}r\right)}^2}}{J}_1^2\left]\right\}sin\left(2\theta \right),\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{22}^{\left(2\right)}\rangle & =-{\displaystyle \frac{1}{8}}{\mathcal{B}}^2{\omega }_r^2{r}^2\left[J_0^2+J_1^2-{\displaystyle \frac{1}{{\omega }_{r}r}}{J}_0{J}_1\right]-{\mathcal{B}}^2{\omega }_r^2{r}^2\{-{\displaystyle \frac{3}{32}}+{\displaystyle \frac{1}{32}}[J_0^2+J_1^2]\hfill \\ \hfill & +{\displaystyle \frac{1}{16}}\left[{\displaystyle \frac{1}{6}}(J_0^2+J_1^2)-{\displaystyle \frac{2}{3{\omega }_{r}r}}{J}_0{J}_1+{\displaystyle \frac{2}{3{\left({\omega }_{r}r\right)}^2}}{J}_1^2\right]\}cos\left(2\theta \right).\hfill \end{array}$$

(77)

We observe that the non-linear self-interaction of a Bessel-modulated wave inherits the lack of transverse localization of the first-order solution (see the discussion at the end of Section 2), and the effective field derived above is, correspondingly, not asymptotically flat: using the large-argument behavior of the Bessel functions, $J_0^2\left(x\right)+J_1^2\left(x\right)\simeq 2/\left(\pi x\right)$, the terms in Equations (67)–(69) and (75)–(77) do not vanish at large r. Since we are concerned here with the geodesic dynamics near the axis, the large-r behavior of the solution is not relevant to our analysis.

We now verify that the effective perturbation $\langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle$ satisfies the de Donder (harmonic) gauge condition also at second order, ${\partial }^j\langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle =0$. Since the averaged field is independent of t and z, this reduces to a divergence in the transverse plane, ${\partial }_x\langle {\mathfrak{h}}_{i1}^{\left(2\right)}\rangle +{\partial }_y\langle {\mathfrak{h}}_{i2}^{\left(2\right)}\rangle =0$. The condition is trivially satisfied for $i=0,3$, since $\langle {\mathfrak{h}}_{01}^{\left(2\right)}\rangle =\langle {\mathfrak{h}}_{02}^{\left(2\right)}\rangle =\langle {\mathfrak{h}}_{13}^{\left(2\right)}\rangle =\langle {\mathfrak{h}}_{23}^{\left(2\right)}\rangle =0$. For $i=1,2$, direct substitution of Equations (75)–(77) shows that the transverse divergence vanishes identically, confirming that the harmonic gauge is preserved by the averaged second-order solution. This result is required by the consistency of the field equation, as discussed in Section 3. Writing $\square \langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle =T_{ij}^{\mathrm{dBGW}}$, where $T_{ij}^{\mathrm{dBGW}}$ denotes the quadratic source given in Equations (58)–(63), and using the fact that the flat-space wave operator commutes with the transverse derivatives, the preservation of the second-order harmonic gauge, ${\partial }^j\langle {\mathfrak{h}}_{ij}^{\left(2\right)}\rangle =0$, is equivalent to the conservation of the effective quadratic source, ${\partial }^j{T}_{ij}^{\mathrm{dBGW}}=0$. This continuity equation can be verified directly from the explicit expressions for $T_{ij}^{\mathrm{dBGW}}$, providing a check that the averaged second-order source is conserved and that the resulting effective field is consistent with the second-order Einstein field equations.

As a simple application of the effective geometry, in the next section we investigate the motion of test particles. In the context of de Broglie’s wave–particle picture, the physically most relevant region is the neighborhood of the symmetry axis, where the associated particle is expected to be localized. We, therefore, consider the regime ${\omega }_{r}r\ll 1$, corresponding to distances much smaller than the reduced Compton wavelength, i.e.,

$$r\ll 1/{\omega }_r={ƛ}_r.$$

(78)

Using the series expansions of the Bessel functions for small arguments,

$$\begin{array}{c}\hfill J_0\left(x\right)=1-x^2/4+\mathcal{O}\left(x^4\right),J_1\left(x\right)=x/2-x^3/16+\mathcal{O}\left(x^5\right),\end{array}$$

(79)

in this regime Equations (67)–(69) and (75)–(77) give

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{00}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2}{16}}({\omega }^2-7k^2)(x^2+y^2)-{\displaystyle \frac{{\mathcal{B}}^2}{8}}{\displaystyle \frac{{\omega }^2+7k^2}{{\omega }_r^2}},\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{33}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2}{16}}(k^2-7{\omega }^2)(x^2+y^2)-{\displaystyle \frac{{\mathcal{B}}^2}{8}}{\displaystyle \frac{7{\omega }^2+k^2}{{\omega }_r^2}},\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{03}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{3}{8}}{\mathcal{B}}^2\omega k(x^2+y^2)+{\mathcal{B}}^2{\displaystyle \frac{\omega k}{{\omega }_r^2}},\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{11}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{8}}{x}^2,\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{12}^{\left(2\right)}\rangle & \simeq {\displaystyle \frac{{\mathcal{B}}^2}{4}}{\omega }_r^{2}xy,\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill \langle {\mathfrak{h}}_{22}^{\left(2\right)}\rangle & \simeq -{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{8}}{y}^2.\hfill \end{array}$$

(85)

The corresponding trace becomes

$$\langle {\mathfrak{h}}_{\left(2\right)}\rangle \equiv {\eta }_{ij}\langle {\mathfrak{h}}_{\left(2\right)}^{ij}\rangle \simeq -{\displaystyle \frac{3}{8}}{\mathcal{B}}^2{\omega }_r^2(x^2+y^2)+{\displaystyle \frac{3}{4}}{\mathcal{B}}^2.$$

(86)

From Equations (54) and (80)–(86), we obtain

$$\begin{array}{cc}\hfill {\mathcal{H}}_{00}& \simeq -{\displaystyle \frac{{\mathcal{B}}^2{k}^2}{8}}(x^2+y^2)-{\displaystyle \frac{{\mathcal{B}}^2}{4}},\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{33}& \simeq -{\displaystyle \frac{{\mathcal{B}}^2{\omega }^2}{8}}(x^2+y^2)+{\displaystyle \frac{{\mathcal{B}}^2}{4}},\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{03}& \simeq {\displaystyle \frac{{\mathcal{B}}^2\omega k}{8}}(x^2+y^2),\hfill \end{array}$$

(89)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{11}& \simeq -{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{16}}(3x^2+y^2)+{\displaystyle \frac{{\mathcal{B}}^2}{8}},\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{12}& \simeq {\displaystyle \frac{{\mathcal{B}}^2}{4}}{\omega }_r^{2}xy,\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{22}& \simeq -{\displaystyle \frac{{\mathcal{B}}^2{\omega }_r^2}{16}}(x^2+3y^2)+{\displaystyle \frac{{\mathcal{B}}^2}{8}}.\hfill \end{array}$$

(92)

5. Geodesic Dynamics in the Effective Geometry

Having established the effective geometry near the symmetry axis, we now study the motion of test particles in this spacetime region. The dynamics is governed by the geodesic equation

$$\begin{array}{c}\hfill {\displaystyle \frac{{\mathrm{d}}^2{x}^j\hfill }{\mathrm{d}{\tau }^2\hfill }}+{\Gamma }_{ab}^j{\displaystyle \frac{\mathrm{d}{x}^a\hfill }{\mathrm{d}\tau \hfill }}{\displaystyle \frac{\mathrm{d}{x}^b\hfill }{\mathrm{d}\tau \hfill }}=0,\end{array}$$

(93)

where $\tau$ is the proper time and ${\Gamma }_{ab}^j$ are the Christoffel symbols. Introducing the coordinate time $(t=x^0)$, the spatial components of the geodesic equation can be written as

$$\begin{array}{c}\hfill {\ddot{x}}^{\alpha }={\Gamma }_{ij}^0{\dot{x}}^i{\dot{x}}^j{\dot{x}}^{\alpha }-{\Gamma }_{ij}^{\alpha }{\dot{x}}^i{\dot{x}}^j,\end{array}$$

(94)

where a dot denotes the derivative with respect to the coordinate time t. From Equation (53), the Christoffel symbols become

$$\begin{array}{c}\hfill {\Gamma }_{jk}^i=-{\displaystyle \frac{1}{2}}{\eta }^{is}(-{\partial }_s{\mathcal{H}}_{jk}+{\partial }_j{\mathcal{H}}_{ks}+{\partial }_k{\mathcal{H}}_{sj}).\end{array}$$

(95)

Using the components of the effective metric perturbation derived in the previous section, Equations (87)–(92), the non-vanishing Christoffel symbols are

$$\begin{array}{cc}\hfill {\Gamma }_{01}^0& ={\Gamma }_{00}^1={\mathcal{B}}^2{k}^{2}x/8,\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill {\Gamma }_{02}^0& ={\Gamma }_{00}^2={\mathcal{B}}^2{k}^{2}y/8,\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill {\Gamma }_{13}^0& ={\Gamma }_{03}^1=-{\Gamma }_{01}^3=-{\mathcal{B}}^{2}k\omega x/8,\hfill \end{array}$$

(98)

$$\begin{array}{cc}\hfill {\Gamma }_{23}^0& ={\Gamma }_{03}^2=-{\Gamma }_{02}^3=-{\mathcal{B}}^{2}k\omega y/8,\hfill \end{array}$$

(99)

$$\begin{array}{cc}\hfill {\Gamma }_{11}^1& =3{\Gamma }_{12}^2=-3{\Gamma }_{22}^1/5=-3{\mathcal{B}}^2{\omega }_r^{2}x/16,\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill {\Gamma }_{22}^2& =3{\Gamma }_{12}^1=-3{\Gamma }_{11}^2/5=-3{\mathcal{B}}^2{\omega }_r^{2}y/16,\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill {\Gamma }_{33}^1& =-{\Gamma }_{13}^3={\mathcal{B}}^2{\omega }^{2}x/8,\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill {\Gamma }_{33}^2& =-{\Gamma }_{23}^3={\mathcal{B}}^2{\omega }^{2}y/8.\hfill \end{array}$$

(103)

The corresponding equations of motion are

$$\begin{array}{cc}\hfill \ddot{x}& =-{\displaystyle \frac{{\mathcal{B}}^2}{16}}\{{\omega }_r^2\left[\left({\dot{x}}^2+5{\dot{y}}^2-2\right)x+2\dot{x}\dot{y}y\right]+2\omega [2k\dot{z}\left(({\dot{x}}^2-1)x+\dot{x}\dot{y}y\right)\hfill \\ \hfill & +\left(x-2{\dot{x}}^{2}x+{\dot{z}}^{2}x-2\dot{x}\dot{y}y\right)\omega \left]\right\},\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill \ddot{y}& =-{\displaystyle \frac{{\mathcal{B}}^2}{16}}\{{\omega }_r^2\left[\left({\dot{y}}^2+5{\dot{x}}^2-2\right)y+2\dot{x}\dot{y}x\right]+2\omega [2k\dot{z}\left(({\dot{y}}^2-1)y+\dot{x}\dot{y}x\right)\hfill \\ \hfill & +\left(y-2{\dot{y}}^{2}y+{\dot{z}}^{2}y-2\dot{x}\dot{y}x\right)\omega \left]\right\},\hfill \end{array}$$

(105)

$$\begin{array}{cc}\hfill \ddot{z}& ={\displaystyle \frac{{\mathcal{B}}^2}{4}}(\dot{x}x+\dot{y}y)(k\dot{z}-\omega )(k-\dot{z}\omega ).\hfill \end{array}$$

(106)

The exact equations are highly non-linear. Since the observable effects of de Broglie gravitational waves are more evident for slowly moving associated particles, see Section 1, we now consider the non-relativistic limit $\dot{x},\dot{y},\dot{z}\ll 1$. To the lowest order, one finds $\ddot{z}\simeq 0$, while the transverse equations reduce to

$$\begin{array}{cc}\hfill \ddot{x}+{\kappa }^{2}x& \simeq 0,\hfill \end{array}$$

(107)

$$\begin{array}{cc}\hfill \ddot{y}+{\kappa }^{2}y& \simeq 0,\hfill \end{array}$$

(108)

with

$${\kappa }^2\equiv {\displaystyle \frac{{\mathcal{B}}^2{k}^2}{8}}={\displaystyle \frac{{\mathcal{A}}^2{k}^2}{8}}\left({\displaystyle \frac{v^2}{c^2}}\right){\left({\displaystyle \frac{v}{c}}-{\displaystyle \frac{c}{v}}\right)}^2,$$

(109)

which is manifestly positive. Arriving at the last equality, we use Equation (64), write ${ϵ}_{11}$ in terms of the velocity v by using Equations (17) and (19) and reintroduce c for later convenience. The non-linear self-interaction of the de Broglie gravitational wave, therefore, induces a harmonic restoring force in the transverse plane. In the non-relativistic regime, particle trajectories execute bounded oscillations around the symmetry axis while remaining straight along the propagation direction. The averaged second-order geometry, thus, acts as an effective harmonic confinement for transverse motion.

Although the effective metric has been obtained within a particular gauge choice, the resulting transverse confinement is not a coordinate artifact. Indeed, after IMTW averaging, the theory at order ${\mathcal{A}}^2$ is described by an effective linear formalism for the perturbation field ${\mathcal{H}}_{ij}$, Equation (53). The corresponding linearized Riemann curvature tensor,

$$\begin{array}{c}\hfill R_{ijkl}^{\mathrm{eff}}\simeq -{\displaystyle \frac{1}{2}}\left({\partial }_k{\partial }_j{\mathcal{H}}_{il}+{\partial }_l{\partial }_i{\mathcal{H}}_{jk}-{\partial }_l{\partial }_j{\mathcal{H}}_{ik}-{\partial }_k{\partial }_i{\mathcal{H}}_{jl}\right)\end{array}$$

(110)

is invariant under gauge transformations,

$$\begin{array}{cc}\hfill x^i& \to x^i+{\xi }^i\left(x\right),\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{ij}& \to {\mathcal{H}}_{ij}-{\partial }_i{\xi }_j-{\partial }_j{\xi }_i,\hfill \end{array}$$

(112)

for any four-vector ${\xi }_i\left(x\right)$ with $|{\partial }_j{\xi }_i|\sim \mathcal{O}(|{\mathcal{H}}_{ij}\left|\right)$. The transverse restoring effect can, therefore, be characterized in a manifestly gauge invariant manner through the geodesic deviation equation,

$${\ddot{\zeta }}_{\alpha }\simeq -R_{\alpha 0\beta 0}^{\mathrm{eff}}{\zeta }^{\beta },$$

(113)

where ${\zeta }^{\alpha }$ denotes the relative separation of two nearby test masses, a dot denotes the differentiation with respect to the time coordinate and

$$R_{\alpha 0\beta 0}^{\mathrm{eff}}\simeq {\displaystyle \frac{1}{2}}{\ddot{\mathcal{H}}}_{\alpha \beta }-{\partial }_{(\alpha }{\dot{\mathcal{H}}}_{\beta )0}+{\displaystyle \frac{1}{2}}{\partial }_{\alpha }{\partial }_{\beta }{\mathcal{H}}_{00}.$$

(114)

Here, parentheses denote symmetrization of the enclosed indices. Since the effective geometry is time independent, the relevant components of the curvature tensor reduce to

$$\begin{array}{c}\hfill R_{\alpha 0\beta 0}^{\mathrm{eff}}\simeq {\displaystyle \frac{1}{2}}{\partial }_{\alpha }{\partial }_{\beta }{\mathcal{H}}_{00},\end{array}$$

(115)

where $\alpha ,\beta =x,y$. Consequently, the same transverse harmonic oscillations follow directly from the gauge invariant geodesic deviation equation, demonstrating that the confinement is a genuine property of the effective spacetime curvature rather than a consequence of the chosen coordinates. In the present work, we chose to solve the geodesic equations themselves. This allows us to obtain the explicit trajectories of individual test particles in the effective spacetime generated by the de Broglie gravitational wave. These trajectories provide a natural starting point for a future investigation of whether the resulting motion can be interpreted as a guidance law for an elementary particle piloted by its associated wave, in the spirit of de Broglie’s original pilot-wave idea.

The preceding analysis describes the motion of test particles in the effective, averaged geometry. Since a test particle is idealized as point-like, the same description can naturally be applied to a point-like particle associated with the de Broglie gravitational wave. More generally, the effective dynamics can also be considered for a particle with a finite spatial extent characterized by a radius ${\rho }_s$. In the present regime, the curvature tensor scales as ${\kappa }^2$, Equation (115), so the associated curvature length is of order $1/\kappa$. As long as the particle remains small compared with the characteristic curvature length of the effective geometry, ${\rho }_s\ll 1/\kappa$, the tidal field varies negligibly across the particle volume and finite size effects may be neglected. The center of mass motion is, therefore, expected to follow the same effective dynamics as a test particle. The present treatment neglects the mutual back-reaction between the particle and the de Broglie gravitational wave; a fully self-consistent description would require solving the coupled wave–particle dynamics.

The contribution of the particle’s own gravitational field to the local curvature is controlled by the dimensionless ratio ${\alpha }_G\equiv r_s/(1/\kappa )$, which compares its Schwarzschild radius, $r_s=2$ Gm/c², with the characteristic curvature length $1/\kappa$. For an electron, $r_s\sim {10}^{-57}\mathrm{m}$ and $\kappa \sim {10}^3{\mathrm{m}}^{-1}$, see estimate below, giving ${\alpha }_G\sim {10}^{-54}$. Thus, at the level of the local field strength, the particle’s own gravitational field is utterly negligible compared with the effective background geometry considered here. This estimate, however, does not by itself exclude a cumulative self-force along the particle’s trajectory, which can, in principle, contain a history-dependent contribution from the regularized tail of its retarded field, as described by the MiSaTaQuWa/DeWitt–Brehme self-force formalism [52,53,54,55]. A rigorous self-force analysis in the present effective, perturbatively constructed geometry would require identifying the appropriate small expansion parameter and constructing the corresponding retarded Green’s function on the effective background, rather than on a fixed exact spacetime as in the standard self-force treatment. Such an analysis is beyond the scope of the present work and is left for future investigation.

We conclude with an order-of-magnitude estimate. Considering an incident electron, moving with velocity $v/c=0.01$, we take $\omega \simeq 7.76\times {10}^{20}$ Hz and $\mathcal{A}\simeq 1.75\times {10}^{-7}$ [37]. The reduced Compton wavelength is ${ƛ}_r\simeq 3.86\times {10}^{-13}$ m. These values give ${\kappa }^2\simeq 2.55\times {10}^6{\mathrm{m}}^{-2}$, corresponding to an oscillation frequency $c\kappa /2\pi \simeq 7.62\times {10}^{10}\mathrm{Hz}$. To lowest order, the associated particle propagates uniformly along the z-axis while undergoing high-frequency oscillations in the transverse plane. We observe that $c\kappa \ll {\omega }_r$, and this justifies the separation of time scales adopted above. These results suggest that the effective geometry generated by the wave’s non-linear self-interaction may provide the transverse localization mechanism required for a gravitational realization of de Broglie’s wave–particle picture.

6. Conclusions

De Broglie gravitational waves form an axially symmetric class of vacuum solutions to the linearized Einstein equations characterized by a transverse momentum scale that appears as an effective mass scale in the reduced longitudinal description. They also possess a non-trivial longitudinal sector, while remaining solutions to the massless four-dimensional gravitational field equations. Together with their distinctive spatial and dynamical structure, these properties distinguish them from the usual plane-wave solutions and motivate their consideration as possible guiding fields in terms of geometric descriptions of wave–particle duality.

In this work, we investigated the post-Minkowskian expansion of de Broglie gravitational waves up to second order. The first-order perturbation corresponds to the propagating de Broglie wave, whereas the non-linear self-interaction of the gravitational field generates both oscillatory and non-oscillatory second-order contributions.

Our gravitational waves are characterized by extremely high oscillation frequencies. Consequently, the timescale governing the motion of the associated particle is expected to be many orders of magnitude larger than the oscillation period of the wave. This separation of scales motivated the introduction of a coarse-grained description based on the Isaacson/Misner–Thorne–Wheeler (IMTW) averaging procedure, applied here as a phase average over the temporal and longitudinal directions only, while the full transverse dependence was retained. Within this framework, the rapidly oscillating contributions average to zero, whereas the non-oscillatory components of the second-order field survive and define an effective background geometry. The resulting theory takes the form of an effective linear perturbation formalism at second order in the wave amplitude for the metric perturbation ${\mathcal{H}}_{ij}$.

We showed that the second-order effective perturbation field satisfies the de Donder gauge condition, providing an important verification that the resulting field satisfies Einstein’s equations consistently, and in the corresponding order. In the near-axis approximation, the effective geometry gives rise to non-trivial geodesic dynamics: in the non-relativistic regime, the transverse motion obeys a harmonic equation, while the motion along the propagation direction remains uniform. Consequently, geodesics remain bounded around the symmetry axis. By analyzing the gauge invariant geodesic deviation equation for the effective linear theory, we found that this transverse harmonic confinement is not an artifact of the chosen gauge but reflects a genuine property of the effective spacetime curvature. Finally, when the curvature radius is much larger than the size of the associated particle, tidal effects are expected to be negligible, and the center of mass, therefore, follows the same effective geodesic motion as test particles.

Our results show that the non-linear self-interaction of de Broglie gravitational waves generates a coarse-grained spacetime geometry possessing an effective confining character. This suggests a geometric mechanism that may contribute to the localization required in gravitational approaches to wave–particle duality.

Several natural extensions of the present work deserve further investigation. In particular, it would be interesting to analyze the geometry without employing the Isaacson/Misner–Thorne–Wheeler averaging procedure (thereby retaining the full time-dependent structure of the second-order field), as well as to study the averaged geometry beyond the vicinity of the symmetry axis, where additional features of the effective dynamics may emerge. Another interesting direction would be to move beyond the test-particle approximation and investigate the coupled evolution of a particle and its associated de Broglie gravitational wave. Such a self-consistent analysis could clarify the physical role of the effective geometry derived here and its relevance for geometric approaches to wave–particle duality.