Testing classical-quantum gravity with geodesic deviation

Abstract: A novel semiclassical gravity model proposed by Oppenheim et al., that consistently describes interactions between quantum systems and a classical gravitational field, has recently attracted considerable attention. However, the limitations and phenomenological viability of this model have not yet been thoroughly investigated. In this work, based on the model, we study quantum fluctuations of geodesic deviation coupled with a classical gravitational field. We analytically derive the strain spectrum expected from the fluctuations and show that the original Oppenheim et al. model can be tested with the current observational sensitivity of gravitational-wave experiments. Furthermore, motivated by the novel semiclassical model, we construct two additional models: a modified Oppenheim et al. model that is manifestly consistent with Einstein equation, and a classical-quantum model with environment-induced noise. We analyze the strain spectra predicted by these two models through comparison with those of the original Oppenheim et al. model and perturbative quantum gravity.

• The paper derives closed-form strain power spectra from decoherence and stochastic metric noise for three classical–quantum gravity models using an Einstein–Langevin treatment of geodesic deviation.
• The analysis finds that white-noise models have theoretical pathologies and are naively excluded by combining LIGO limits with matter-wave interferometry bounds, while an environment-induced model can remain compatible with observations.
• The environmental model can produce a minimum strain near 10⁻⁴² Hz⁻¹ᐟ² at 100 Hz and closely mimic quantum-gravity vacuum fluctuations, showing that detection alone may not prove gravity is fundamentally quantum.

Overview

This paper by Hirotani and Matsumura examines the phenomenology of the postquantum-classical gravity program of Oppenheim et al. (Grudka et al., 2024, Oppenheim et al., 2023) through a specific observable: fluctuations of geodesic deviation between two point masses coupled to a fundamentally classical gravitational field. The authors derive, analytically, the strain power spectral density induced by both the decoherence kernel $D_{\mu\nu\rho\sigma}$ acting on the quantum deviation and the noise kernel $N_{\mu\nu\rho\sigma}$ sourcing stochastic metric fluctuations, and they confront these spectra with current and projected gravitational-wave detector sensitivities. Two new models are constructed along the way: an Einstein-consistent modification of the Oppenheim et al. kernels, and a model in which the gravitational field is treated as quantum but rendered effectively classical by environment-induced noise.

Framework: CQ dynamics and the Langevin equation

The analysis rests on the CPTP master-equation formulation of hybrid classical–quantum (CQ) systems, recast in a path-integral form combining the Schwinger–Keldysh integral for the quantum subsystem with a Fokker–Planck integral for the classical one. The CQ action contains two symmetric, positive-semidefinite kernels whose matrix product obeys the decoherence–diffusion trade-off relation $DN \geq 1/4$, which for gravity becomes $DN \geq (4\pi G_N)^2$. Small $N$ forces near-satisfaction of the linearized Einstein equation in the path integral; large $D$ suppresses the difference of stress tensors across Schwinger–Keldysh branches.

For two masses $M$ and $m$ with separation $\xi^\mu$, expanded about flat spacetime in Fermi normal coordinates, the effective action reduces to a free kinetic term for the transverse deviation plus a coupling to $R^{(1)}_{0a0b}\xi^a\xi^b$. Integrating out the metric perturbation via the Feynman–Vernon influence functional—solving the Einstein–Langevin equation with retarded Green's function and Gaussian-averaging over the stochastic source $N_{\mu\nu\rho\sigma}$0—yields a Langevin equation $N_{\mu\nu\rho\sigma}$1 with vanishing mean force and two-point function

$N_{\mu\nu\rho\sigma}$2

The dissipative kernel $N_{\mu\nu\rho\sigma}$3, arising from gravitational radiation reaction, is neglected on the grounds that $N_{\mu\nu\rho\sigma}$4 is small; this omission has consequences discussed below. The strain spectrum is defined as $N_{\mu\nu\rho\sigma}$5, with each term obtained from the Fourier transform of the corresponding force correlator divided by $N_{\mu\nu\rho\sigma}$6.

Spectra of the three models

Original Oppenheim et al. model. With local white-noise kernels proportional to $N_{\mu\nu\rho\sigma}$7 parameterized by $N_{\mu\nu\rho\sigma}$8 and constants $N_{\mu\nu\rho\sigma}$9, $DN \geq 1/4$0 saturating the trade-off, the spectrum splits into a decoherence piece scaling as $DN \geq 1/4$1 at low frequency and a noise piece scaling as $DN \geq 1/4$2 at high frequency. Taking the geometric mean yields a minimum strain spectrum independent of $DN \geq 1/4$3—a floor below which no parameter choice can push the signal. Two structural problems are identified:

• Divergence in the far-future limit: because the scale-free white noise accumulates over the IR cutoff time $DN \geq 1/4$4 (taken as the age of the universe), the spectrum diverges as $DN \geq 1/4$5. The paper notes this may be an artifact of neglecting dissipation, but leaves the question open.
• Pathological $DN \geq 1/4$6 dependence: for $DN \geq 1/4$7 the noise contribution diverges; for $DN \geq 1/4$8 it vanishes; and for $DN \geq 1/4$9 the noise kernel fails to be positive semidefinite and the strain spectrum becomes complex. Only $DN \geq (4\pi G_N)^2$0 or $DN \geq (4\pi G_N)^2$1 yield well-defined predictions.

Einstein-consistent modification. The original noise kernel violates the Bianchi-consistency condition $DN \geq (4\pi G_N)^2$2 required of any stochastic source entering the Einstein–Langevin equation. The authors replace it with a transverse projector structure $DN \geq (4\pi G_N)^2$3 built from $DN \geq (4\pi G_N)^2$4, which manifestly satisfies the constraint. The resulting spectrum is numerically close to that of the original model—the modification makes little practical difference to geodesic-deviation fluctuations—but it removes the $DN \geq (4\pi G_N)^2$5 pathology entirely, so constraints derived from it apply independently of $DN \geq (4\pi G_N)^2$6.

Environment-induced noise model. Here the gravitational field is assumed fundamentally quantum but coupled to environmental degrees of freedom with energy scale $DN \geq (4\pi G_N)^2$7; tracing out the environment produces colored noise with support only above threshold, encoded in $DN \geq (4\pi G_N)^2$8. This kernel is also Bianchi-consistent and requires no artificial UV cutoff at the separation scale $DN \geq (4\pi G_N)^2$9. The spectrum is monotonically increasing in frequency, scales as $N$0 at low frequency and $N$1 at high frequency, and—opposite to the original model—is noise-dominated at low frequency and decoherence-dominated at high frequency. A minimum strain $N$2, derived via the arithmetic–geometric mean inequality applied to the trade-off relation $N$3, holds for arbitrary kernel profiles.

A notable qualitative finding: for small $N$4, the minimal environmental spectrum closely tracks the perturbative quantum-gravity prediction $N$5 from graviton vacuum fluctuations [PhysRevD.103.044017]. The authors state plainly that this implies CQ models may be experimentally difficult to distinguish from genuine quantized gravity, and that even experimental confirmation of such a model would not constitute direct evidence for intrinsic quantum behavior of spacetime.

Experimental constraints

Combining the derived spectra with detector sensitivities yields bounds on the model parameters:

Model	Constraint from LIGO ($N$6 at 100 Hz, $N$7 km)
Original Oppenheim et al.	$N$8
Einstein-consistent	$N$9
Environmental ($D$0 Hz)	$D$1

The central quantitative claim of the paper is that these LIGO-derived windows, when naively intersected with the pre-existing bounds from matter-wave interferometry and LISA Pathfinder (Grudka et al., 2024), namely $D$2, leave no overlap—so the original Oppenheim et al. model with white noise would be observationally excluded. The same conclusion would extend to the other two models if the interferometric bound transfers, though the authors flag this transfer as unverified for $D$3. Importantly, the minimal environmental strain $D$4 at 100 Hz lies far below LIGO sensitivity, so a tuned kernel choice evades exclusion entirely. Constraints from future detectors (DECIGO, LISA, KAGRA, superconducting levitated detectors) are also mapped, with their relative strength depending strongly on arm length and frequency band.

Two caveats are stated explicitly: the constraints are "expected" rather than rigorous, since a full optical-readout analysis for a LIGO-type experiment in the CQ framework was not performed; and the environmental bound weakens as $D$5 increases, since fewer frequency modes then contribute to the noise.

Limitations and open questions

Several assumptions bound the validity of the results. The Langevin equation neglects the dissipative kernel $D$6 from gravitational-wave radiation, justified only for sufficiently small $D$7; whether the far-future divergence of the scale-free-noise spectra persists once dissipation is included is left unresolved. The cosmic expansion contribution to the accumulated noise is ignored, and extending the CQ formalism to curved backgrounds is deferred. For the original model, the ill-defined derivatives of white noise and the non-positive-semidefiniteness of its kernel in part of the $D$8 range remain unaddressed beyond restricting the allowed parameter region. Finally, whether the interferometric bounds of Grudka et al. apply to the environmental model's parameters is an open question bearing directly on whether that model is excluded.

Conclusion

The paper provides closed-form strain spectra for geodesic-deviation fluctuations in three variants of postquantum-classical gravity and demonstrates that white-noise-type CQ models fall within reach of current gravitational-wave detectors, with a naive combination of bounds excluding the original Oppenheim et al. construction. It identifies concrete theoretical deficiencies of the original kernels—Bianchi inconsistency, far-future divergence, and a pathological $D$9 window—and shows that a transverse-projector modification cures the latter two without materially changing the spectrum. The environment-induced model offers a more self-consistent effective description with colored noise, but its minimal spectrum mimics perturbative quantum gravity, sharpening the interpretive question of what any positive detection would actually establish about the quantum nature of gravity.