First Feasible Tabletop Test of Quantum Gravity via Gravitational Aharonov-Bohm Effect

First Feasible Tabletop Test of Quantum Gravity via Gravitational Aharonov-Bohm Effect Shan Gao Research Center for Philosophy of Science and Technology, Shanxi University, Taiyuan 030006, P. R. China E-mail: [email protected] November 12, 2025 Abstract Testing whether gravity is quantum or classical remains a cornerstone challenge in fundamental physics. We introduce a practical tabletop experiment to probe gravity’s quantum nature via the gravitational Aharonov-Bohm (AB) effect in atom interferometry. Our approach hinges on two pivotal insights: First, unlike electromagnetic AB effects where constant charge ensures gauge-invariant phases for closed paths, semiclassical gravity couples to the particle’s total energy, rendering the AB phase gauge-dependent and unobservable when energies differ between paths due to environmental noise or controlled perturbations. Second, in quantum gravity—treating source, particle, and field as quantum—the total phase splits into a gaugedependent particle contribution and a gauge-invariant source backreaction, yielding precisely half the nominal AB phase as observable. This halved phase discriminates quantum from classical gravity without requiring massive superpositions, aligning with existing technology. Remarkably, Overstreet et al.’s (Science 375, 226 (2022)) data already reveal a nonzero measured AB phase, providing the first experimental evidence for gravitational quantization. 1 Introduction The quest to unify quantum mechanics and general relativity into a theory of quantum gravity stands as one of the most profound pursuits in modern physics [1, 2]. While quantum effects dominate at microscopic scales and gravity at cosmic ones, their intersection—quantum gravity—remains elusive, with direct experimental probes hindered by the Planck scale’s inaccessibility (lP∼10−35 m). Recent proposals, such as the Bose-Marletto-Vedral (BMV) schemes [3, 4, 5], seek gravitational entanglement between massive superpositions as a signature of quantum gravity, yet these demand Herculean feats: sustaining delocalized states of microgram-scale masses over micrometers for milliseconds amid relentless decoherence [6, 7]. Such challenges underscore the need for alternative, more feasible tests that sidestep large superpositions while remaining sensitive to gravity’s quantum essence [8, 9]. A promising avenue emerges from quantum interference phenomena, particularly the gravitational analog of the Aharonov-Bohm (AB) effect [10, 11, 12, 13, 14, 15, 16, 17]. In this setup, a quantum particle (e.g., atom or photon) in superposition traverses paths near a gravitational source, accumulating a phase from the potential despite negligible local fields. Unlike electromagnetic AB effects, where gauge invariance holds for closed paths due to constant charge, gravity’s coupling to total energy introduces a vulnerability: any differential energy fluctuations—ubiquitous in real experiments—make the semiclassical phase gauge-dependent and unobservable. 1 Here, we propose a novel test exploiting this distinction: a gravitational AB interferometer with paths exhibiting differing energy changes, either from inherent noise or induced perturbations. Semiclassical gravity predicts a null observable phase, while quantum gravity—via mutual backreaction—yields a gauge-invariant half-AB phase. This approach is tabletop-ready, compatible with current atom interferometers, and circumvents BMV-like hurdles. Furthermore, our framework reinterprets existing data: Overstreet et al.’s landmark measurement of a nonzero observable gravitational AB phase [17], traditionally viewed as semiclassical confirmation, aligns instead with the predictions of quantum gravity, offering first evidence for gravity’s quantization. We detail the theoretical underpinnings, gauge analysis, and experimental pathways, paving the way for definitive tests. 2 The Gravitational AB Effect: Semiclassical Analysis The AB effect offers a powerful lens for examining quantum phases induced by gauge fields along particle trajectories. In electromagnetism, the electric and magnetic AB effects yield gaugeinvariant phases for closed paths, thanks to the constant coupling charge [11, 12, 13]. However, the gravitational AB analog exhibits a fundamental difference: gravity couples to the particle’s total energy,1which can fluctuate due to environmental interactions or controlled perturbations. This coupling renders the semiclassical gravitational AB phase gauge-dependent—and thus unobservable—for closed paths whenever the particle’s energy changes differently along the two interferometer arms. This distinction underpins our proposal for testing the quantum nature of gravity. 2.1 Electric AB Effect: Gauge Invariance from Constant Charge In the electric AB effect, a charged particle accumulates a phase due to the electromagnetic fourpotential Aµalong its trajectory γ: ϕEM =e ℏIγ Aµdxµ,(1) where eis the particle’s electric charge. Under a gauge transformation Aµ→Aµ+∂µΛ, the phase shift becomes δϕEM =e ℏIγ ∂µΛdxµ=e ℏ[Λ(tf)−Λ(ti)] .(2) For a closed path in spacetime (where ti=tf), the boundary terms cancel, resulting in δϕEM = 0. The key to this invariance is the constancy of the coupling ethroughout the particle’s evolution, ensuring that the phase remains physically meaningful and observable. 2.2 Gravitational AB Effect: From Non-Relativistic Approximation to Fully Relativistic Derivation In semiclassical gravity, the gravitational field is treated as classical, sourced by the expectation value of the quantum matter stress-energy tensor:2 Gµν[g]=8πG⟨ˆ Tµν⟩.(3) 1This point is often ignored in the literature about the gravitational AB effect. See, however, [18, 19, 20]. 2We work in natural units where ℏ=c= 1 for theoretical expressions, except where ℏand care explicitly restored for dimensional clarity in interactions, phases, and experimental contexts. 2 This corresponds to evolving the particle’s wavefunction in an external classical potential Φ(x, t) generated by the source mass. Traditionally, analyses adopt a non-relativistic approximation, where a particle of rest mass m interacts via the Hamiltonian ˆ Hint =mΦ(x, t).(4) The resulting gravitational AB phase along path γis ϕNR grav =−1 ℏZγ mΦ(x(t), t)dt. (5) Since mis constant—analogous to ein electromagnetism—this phase appears gauge-invariant for closed paths, mirroring the electromagnetic case. This non-relativistic view has dominated discussions of the gravitational AB effect. However, a rigorous derivation requires starting from general relativity in the weak-field limit, where the metric is perturbed as gµν =ηµν +hµν,|hµν| ≪ 1.(6) The particle couples to this perturbation through its stress-energy tensor Tµν, with the interaction Lagrangian Lint =−1 2hµνTµν.(7) For a static Newtonian potential, the leading term is h00 =−2Φ/c2, yielding the interaction Hamiltonian ˆ Hint =Zd3xΦ(x, t) c2ˆ T00(x, t),(8) where ˆ T00 is the energy density operator, encompassing rest mass, kinetic energy, and other contributions. For a quantum state |ψ⟩along trajectory γin the AB effect (where the wave packet is welllocalized and WKB approximation is valid), the accumulated phase is ϕgrav =−1 ℏZγ E(t) c2Φ(x(t), t)dt, (9) with E(t) = Rd3x⟨ψ|ˆ T00(x)|ψ⟩being the particle’s instantaneous total energy. Under a gauge transformation in linearized gravity, h00 →h00 + 2 ˙χ(t)/c2, the phase shift is δϕgrav =−1 ℏZγ E(t) c2˙χ(t)dt =−1 ℏc2[E(t)χ(t)]tf ti+1 ℏc2Zγ ˙ E(t)χ(t)dt. (10) Even for closed paths (ti=tf), the integral term persists if ˙ E(t)= 0, making the phase gaugedependent. To illustrate this, consider a two-path interferometer with paths γ1and γ2, along which the particle has constant energies E1and E2(including kinetic and rest energy). Under a gravitational gauge transformation, the phase accumulated along each path is δϕi=−1 ℏZγi Ei c2˙χ(t)dt =−Ei ℏc2χ(tf)−χ(ti), i = 1,2.(11) 3 The gravitational AB phase difference between the two paths is therefore δϕAB =δϕ1−δϕ2=−E1−E2 ℏc2hχ(tf)−χ(ti)i.(12) Now we can see that when E1=E2, the AB phase is gauge-invariant, as expected. But when E1=E2, the AB phase depends explicitly on the gauge function χ(t), even for closed paths. This demonstrates that any nonzero energy difference along the interferometer makes the semiclassical gravitational AB phase gauge-dependent, implying it is unobservable within semiclassical gravity. In conclusion, the relativistic treatment reveals that the gravitational AB phase couples to the total energy E(t), not merely the rest mass m. Any energy fluctuations—inevitable due to environmental noise in realistic experiments—render the semiclassical phase unobservable. 2.3 Energy Spread Does Not Restore Gauge Invariance One might argue that quantum energy uncertainty in realistic wavepackets could mitigate this gauge dependence. Consider a state with finite energy spread: |ψ⟩=ZdEa(E)|E⟩,Z|a(E)|2dE = 1.(13) Each component accumulates a phase |ψγ⟩=ZdEa(E) exp −i ℏZγ E(t)Φ(t) c2dt|E⟩.(14) A gauge transformation introduces an E(t)-dependent factor exp[−(i/ℏ)RE(t)(∂tχ/c2)dt] for each component. If E(t) varies, this induces relative phases across the superposition, preventing gauge invariance. The energy spread may reduce interference visibility but does not eliminate the structural dependence on ˙ E(t). Thus, any nonzero ˙ E(t) renders the particle’s gravitational AB phase gauge-dependent and unobservable. As detailed in subsequent sections, the only gauge-invariant contribution arises from the source backreaction in a quantum treatment of gravity, yielding precisely half the total AB phase. 2.4 Impact of Energy Perturbations on Gauge Dependence in the AB Phase Consider an atom moving along two paths in a gravitational AB interferometer. Suppose an external system, such as an EM pulse, interacts with the particle along one path, changing its energy from E0to E0+δE. Under a local gravitational gauge transformation Φ(x, t)→Φ(x, t) + ˙χ(x, t)/c2, the gauge-dependent contribution to the atom’s gravitational AB phase is δϕatom =−1 ℏc2Ztf ti δE ˙χ(xa, t)dt =−δE ℏc2χ(xa, tf)−χ(xa, ti).(15) The EM pulse originates from a macroscopic laser system rigidly fixed near the source mass. The corresponding gravitational phase variation of the EM subsystem is δϕEM =1 ℏc2Ztf ti δE ˙χ(xem, t)dt =δE ℏc2χ(xem, tf)−χ(xem, ti).(16) Now the key point is that the atom and the EM field occupy very different positions in the external gravitational potential of the source. The atom’s trajectory spans a vertical separation 4 such as ∆xa∼10−3m, while the laser optics are stationary at xem with negligible displacement. Thus the net gauge-phase mismatch between atom and EM field induced by the same pulse-energy δE, ∆(δϕ) = −δE ℏc2nχ(xa, tf)−χ(xem, tf)−χ(xa, ti)−χ(xem, ti)o is generically nonzero. Consequently, the EM pulses cannot restore gauge invariance: the atom’s gravitational AB phase remains gauge-dependent and therefore unobservable within a semiclassical framework. 3 Quantum Gravity Framework for the Gravitational AB Effect To probe the quantum nature of gravity, we now extend the analysis beyond the semiclassical approximation by treating the source mass, the interfering particle, and the gravitational field as fully quantum degrees of freedom within linearized (weak-field) quantum gravity. By integrating out the gravitational field, we obtain an effective action that captures the mutual gravitational interactions between the source and the particle.3This action decomposes into two symmetric yet physically distinct terms: Scross[Tp, Ts] = Sp←s+Ss←p,(17) where Sp←s=−4πG ZZ d4y d4x Tαβ p(y)Gαβµν(y−x)Tµν s(x),(18) describes the influence of the source’s stress-energy tensor Tµν son the particle’s Tαβ p, and Ss←p=−4πG ZZ d4y d4x Tαβ s(y)Gαβµν(y−x)Tµν p(x),(19) represents the reciprocal backreaction of the particle on the source. Here, Gαβµν is the symmetric graviton Green’s function. Although these terms are formally symmetric under exchange of source and particle, they exhibit crucial differences in gauge behavior, especially for interferometer paths where the particle’s total energy may differ between arms due to environmental noise or controlled perturbations. The term Sp←s—the phase acquired by the particle from the source’s field—is gauge-dependent for non-closed paths or closed paths with varying particle energy, as established in the semiclassical analysis. In contrast, Ss←p—the phase acquired by the source from the particle’s backreaction—is gauge-invariant due to stress-energy conservation (∂µTµν s= 0) and standard boundary conditions. This gauge invariance of the backreaction term is key to detecting quantum gravitational effects, even in realistic setups with energy fluctuations. A prior proposal [21] leveraged this distinction for open interferometer paths. Here, we focus on closed paths where the particle’s energy differs between the two arms, enabling a tabletop test compatible with existing atom interferometry. Consider an initial product state for the particle-source system: |Ψ(0)⟩=1 √2(|L⟩p+|R⟩p)⊗|S0⟩s,(20) where |L⟩pand |R⟩pdenote the particle’s left and right paths, and |S0⟩sis the source’s initial state (with the gravitational vacuum implicit). Unitary evolution under the effective cross-action yields 3See [21] for a detailed derivation of the effective action and its gauge properties. The derivation is also given in Appendix for the purpose of peer review. 5 the entangled state: |Ψ(t)⟩=1 √2hei(Sp←s[L]+Ss←p[L])/ℏ|L⟩p⊗|SL⟩s+ei(Sp←s[R]+Ss←p[R])/ℏ|R⟩p⊗|SR⟩si,(21) where |SL⟩sand |SR⟩sare the evolved source states correlated with each particle path. The total relative phase between the branches is: ∆ϕ=1 ℏ[(Sp←s[L]−Sp←s[R]) + (Ss←p[L]−Ss←p[R])] = 1 ℏ(∆Sp←s+ ∆Ss←p).(22) Critically, ∆Sp←sis gauge-dependent for closed paths with differing particle energies in the arms, rendering it unobservable. However, ∆Ss←premains gauge-invariant, representing a physical phase from quantum gravitational backreaction. For a macroscopic source, the branch overlap |⟨SR|SL⟩| ≈ 1, preserving coherence. In the ideal case of a closed interference loop with identical particle energies in both arms, symmetry implies Sp←s=Ss←p, so each contributes half the total AB phase: ϕtot =ϕparticle +ϕsource, ϕsource =1 2ϕtot.(23) With energy differences, only the gauge-invariant ϕsource survives, isolating the quantum backreaction signature. Note that in the usual gravitational AB effect, the system operates in the nonrelativistic domain, where the particle speed satisfies v≪cand its kinetic energy is much smaller than the rest energy mc2. Consequently, the rest-energy term dominates the accumulated phase, and the source phase still contributes approximately half of the total AB phase predicted by quantum mechanics under the nonrelativistic approximation (constant m). 4 Experimental Implications for Probing Quantum Gravity The asymmetry in gauge behavior between Sp←sand Ss←popens a pathway to empirically distinguish classical from quantum gravity. In the fully quantum framework, the total gravitational phase decomposes into two contributions: the gauge-dependent term Sp←s, describing the particle’s response to the source’s field, and the gauge-invariant term Ss←p, capturing the source’s backreaction to the particle’s field. In realistic interferometry experiments, environmental noise inevitably introduces differential energy changes along the particle’s paths, rendering Sp←sgauge-dependent and unobservable—even for closed loops. This isolates the observable phase to Ss←p, which equals half the total AB phase: ϕobs =1 2ϕAB.(24) Alternatively, one could deliberately induce such an energy difference, e.g., via a brief electromagnetic pulse in one arm, to control and confirm this effect. Semiclassical gravity, lacking quantum backreaction, predicts no gauge-invariant phase contribution beyond the unobservable Sp←s, yielding ϕobs = 0. Thus, detecting a nonzero phase equal to 1 2ϕAB for closed loops would constitute direct evidence of quantum gravitational mediation, exploiting gravity’s unique coupling to fluctuating energy—a feature absent in electromagnetic AB effects. This prediction holds robustly for realistic quantum states with energy spreads. While a wavepacket superposition of energy eigenstates accumulates distributed phases, any differential energy change ensures the collective Sp←sremains gauge-dependent. Energy uncertainty may modulate interference contrast but does not restore observability to the particle term; only the 6 backreaction Ss←pcontributes a physical, gauge-invariant phase. As detailed in the next section, this framework reinterprets Overstreet et al.’s atom interferometry results [17], where the measured phase aligns quantitatively with 1 2ϕAB. In summary, nonzero differential energy changes make the particle’s AB phase gauge-dependent and unmeasurable, and the surviving gauge-invariant phase originates from quantum backreaction on the source. Observing half the AB phase in experiments like Overstreet et al.’s provides evidence for gravity’s quantization. This approach circumvents the need for massive superpositions, offering a feasible route to test quantum gravity with current technology. 5 Reanalysis of Overstreet et al.’s Experiment: Gauge Issues and Quantum Backreaction 5.1 Experimental Overview and Theoretical Motivation In their 2022 experiment, Overstreet et al. [17] utilized a light-pulse atom interferometer with 87Rb atoms to measure a gravitational AB phase shift induced by a 1.25 kg tungsten source mass (see also [22, 23, 24, 25]). The setup achieved a large spatial separation of approximately 25 cm between the atomic wave packets. By subtracting the deflection-induced midpoint phase (measured via auxiliary gradiometers) from the total phase, they isolated the “beyond-midpoint” phase ϕB, which captures the gravitational contribution. Notable measurements include ϕB=−125 ±24 mrad at Rx= 4 cm and −182 ±28 mrad at Rx= 9 cm (where Rxis the horizontal displacement), both deviating from zero with over 5σsignificance. These results were interpreted as validating semiclassical gravity—quantum matter evolving in a classical gravitational potential—with agreement to ab initio calculations within experimental uncertainties. 5.2 Standard Semiclassical Interpretation Conventionally, ϕBis attributed to the action difference along the interferometer arms: ϕB=m ℏZ[V(x1, t)−V(x2, t)]dt, (25) where Vis the Newtonian potential from the source, and mis the atomic mass. Analogous to the electromagnetic AB effect, this phase arises even without local forces, manifesting as an influence of the classical potential on the quantum wavefunction. For closed paths, the phase is deemed gaugeinvariant, and the experiment is seen as confirming quantum mechanics in a classical spacetime, with no need for gravitational quantization. 5.3 Gauge Dependence in Semiclassical Gravity However, this interpretation overlooks a critical flaw: in Overstreet et al.’s experiment, where each atom cloud is split into two interferometer paths with net energy difference, the gravitational AB phase becomes gauge-dependent and thus unobservable within semiclassical gravity.4 Gravity couples to the total energy of the particle along its trajectory. Under a linearized gravitational gauge transformation Φ(t)→Φ(t) + ˙χ(t) c2, the accumulated phase transforms as δϕ =1 ℏc2Zγ ˙ E(t)χ(t)dt −1 ℏc2E(t)χ(t)tf ti,(26) 4In Overstreet et al.’s interferometer, each Bragg pulse transfers momentum so that the two paths alternate between distinct kinetic energies during free evolution. 7 where E(t) is the particle’s instantaneous energy and χ(t) is an arbitrary gauge function. For closed paths with constant energy ( ˙ E= 0), the boundary term vanishes, ensuring gauge invariance. However, in realistic large-momentum-transfer interferometers, ˙ E(t)= 0 due to the momentum kicks imparted by the Bragg pulses. In this case, the phase explicitly depends on the gauge function χ(t), demonstrating that the semiclassical phase ϕBis unobservable. Hence, purely semiclassical gravity predicts an observable ϕB= 0. 5.4 Gauge-Invariant Backreaction in Quantum Gravity The observed nonzero ϕBthus requires a gauge-invariant mechanism beyond semiclassics. We posit that it stems from the quantum backreaction of the gravitational field, encoded in the gaugeinvariant term Ss←p=−4πG ZZ d4y d4x Tαβ s(y)Gαβµν(y−x)Tµν p(x),(27) where Tαβ sand Tµν pare the stress-energy tensors of the source and probe atom, respectively, and Gis the graviton propagator. This term mediates virtual graviton exchange, yielding a phase that remains gauge-invariant despite energy fluctuations. In the weak-field regime, this backreaction contributes half the full AB phase (for constant m): ϕ(quantum) B=1 2ϕAB.(28) This prediction aligns quantitatively with the measured magnitudes, supporting a quantum gravitational origin. 5.5 Quantitative Comparison and Best-Fit Analysis of AB Phases Here we analyze approximate data points extracted from the experiment’s figure (Fig.2C), at horizontal positions Rx≈ −0.30,−0.20,−0.10,0.00,0.04,0.10 m: ϕexp ≈ −0.07,0.04,−0.01,−0.07,−0.13,−0.18rad,(29) with theoretical full AB phases ϕAB ≈0,0.02,0.03,−0.07,−0.23,−0.22rad,(30) and half AB phases ϕ1/2 AB =1 2ϕAB. The best-fit scaling cminimizes the discrepancy: c=Piϕexp,iϕtheory,i Piϕ2 theory,i .(31) Results yield cfull ≈0.696 for the full AB (fit ϕfull fit =cfullϕAB) and c1/2≈1.392 for the half AB (fit ϕ1/2 fit =c1/2ϕ1/2 AB ≈0.696ϕAB). The relative mismatch before scaling is δtheory =(1−cfull for full AB, (c1/2−1)/c1/2for half AB.(32) Numerically: δfull ≈0.304 (30%) and δ1/2≈0.282 (28%), indicating the half AB prediction is marginally closer to the data. Theory Best-fit scale cRelative mismatch δ Full AB 0.696 30% Half AB 1.392 28% 8 5.6 Implications and Experimental Consequences If ϕBoriginates from the gauge-invariant backreaction, Overstreet et al.’s results already furnish first evidence for gravitational quantization: 1. Semiclassics predict ϕB= 0 due to gauge dependence. 2. Quantum gravity predicts ϕB=1 2ϕAB, matching the observed scale and sign. 3. The data thus disfavor purely classical gravity. While the original study claims consistency with semiclassics, our gauge analysis reveals that the phase’s observability point to quantum backreaction. Enhanced precision in follow-up experiments could solidify this interpretation. 6 Proposed Experiments and Future Prospects Building on our reinterpretation of the gravitational AB effect and its implications for quantum gravity, we outline several enhanced experimental proposals to rigorously test the predicted half-AB phase shift arising from gauge-invariant quantum backreaction. These tests leverage existing and emerging technologies in atom and photon interferometry. We emphasize feasibility with current setups, potential challenges such as environmental decoherence and noise mitigation, and expected outcomes under semiclassical versus quantum gravity models. Additionally, we suggest broader future directions inspired by recent advancements in the field, including space-based and entangled interferometry. 1. Enhanced Precision in Overstreet-Type Experiments: Replicate and refine the Overstreet et al. setup [17] with improved measurement accuracy to directly compare the observed beyond-midpoint phase ϕBagainst the half-AB prediction. Current atom interferometers using 87Rb or similar atoms achieve phase sensitivities on the order of milliradians; enhancements could involve longer free-fall times (e.g., via drop towers or fountains exceeding 1 s) or advanced laser cooling to reduce velocity spreads below 1 mm/s. By minimizing systematic errors—such as vibrations through active isolation platforms—and increasing statistics (e.g., 106atoms per shot), the experiment could resolve whether ϕBprecisely matches 1 2ϕAB (quantum prediction) or vanishes (semiclassical, due to gauge dependence from unavoidable energy fluctuations). Recent developments in portable atom interferometers [27] could enable field-deployable versions for varied source masses, testing mass-dependence of the backreaction. 2. Controlled Energy Perturbations in Modified Configurations: Introduce deliberate differential energy changes mid-path in one interferometer arm to explicitly break energy constancy and probe gauge dependence. For instance, apply a short optical pulse (e.g., 10 ns duration, resonant with an atomic transition) to induce a controlled excitation or momentum kick in one arm, shifting the total energy while preserving coherence. This could be implemented in a Mach-Zehnder geometry with a 1 kg source mass, similar to Overstreet, but with pulse timing synchronized to the Raman beams. Under semiclassical gravity, the particle phase becomes fully gauge-dependent, predicting ϕobs = 0; quantum gravity yields ϕobs =1 2ϕAB. Challenges include compensating for recoil-induced path deflections via counter-pulses; feasibility is high with current optical control in cold-atom labs. 9 For a point particle with proper time τ, the energy-momentum density takes the form: Pµν(τ) = mdxµ p dτ dxν p dτ ,(59) where mis the particle’s mass. Substituting this expression: δSp←s=−1 2Zτf τi dτ mdxµ p dτ dxν p dτ ∂µξν(xp(τ)).(60) Recognizing that the directional derivative along the worldline satisfies: dxµ p dτ ∂µξν=dξν dτ ,(61) we obtain: δSp←s=−1 2Zτf τi dτ mdxν p dτ dξν dτ .(62) Integrating by parts: δSp←s=−1 2mdxν p dτ ξν(xp(τ))τf τi +1 2mZτf τi dτ d2xν p dτ2ξν(xp(τ)).(63) Since ξµ(x) is an arbitrary gauge function, the boundary term and the integral term are independent. This means that only when both terms are zero, can the probe phase Sp←sbe gauge-invariant. This requires that the motion of the particle is geodesic motion: d2xν p dτ2= 0, and its path is closed and its four-velocities in the two paths of the interferometer are the same. In other words, the probe phase Sp←sis gauge-invariant only for particles with constant four-velocities and closed paths. For open paths with distinct endpoints, the boundary term is generally non-zero, and thus Sp←s is gauge-dependent. For closed paths, when the four-velocities (such as the total energy) differ between the two paths or when there are four-velocity fluctuations (such as energy fluctuations) along the paths, Sp←sis also gauge-dependent. In the former case, the boundary term is generally non-zero, and in the latter case, the integral term is generally non-zero, both making the probe phase gauge-dependent. (2). Gauge Invariance of the Source Phase Ss←p Under an infinitesimal linearized diffeomorphism: hµν →hµν +∂µξν+∂νξµ,(64) the variation of the source phase functional Ss←pis given by: δSs←p=−1 2Zd4x Tµν s(x)∂µξν(x).(65) We now integrate by parts: δSs←p=−1 2Zd4x[∂µ(Tµν s(x)ξν(x)) −(∂µTµν s(x)) ξν(x)] .(66) The second term vanishes due to stress-energy conservation of the source: ∂µTµν s(x) = 0.(67) 16 The first term is a total derivative that can be converted to a boundary term via the divergence theorem: Zd4x ∂µ(Tµν s(x)ξν(x)) = I∂V dΣµTµν s(x)ξν(x).(68) The boundary term can be decomposed as Z∂V dΣµTµν sξν=ZΣtf d3x nµTµν sξν−ZΣti d3x nµTµν sξν+ZS∞ dΣµTµν sξν,(69) where Σtiand Σtfare the initial and final time slices, and S∞is the spatial boundary at infinity. For a localized source, the stress-energy tensor vanishes at spatial infinity: Tµν s→0 as |x| → ∞, so the integral over S∞is zero. In the gravitational AB setup, the measured phase is the difference between two particle paths. Since Tµν sis defined as the flat-spacetime stress-energy tensor of the source—unaffected by the probe—it is identical for both paths. Consequently, the contributions from the initial and final time slices, Σtiand Σtf, both cancel when computing the phase difference. Therefore, the gauge variation of the source-probe phase vanishes exactly: δSs←p= 0,(70) regardless of whether the particle paths are open or closed. This shows that the source phase Ss←p is fully gauge-invariant in the gravitational AB effect. 17