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Gauge status of the additive constant in the potential two-point function

Establish whether the freedom to add an arbitrary constant to the weak-field potential two-point function G(x,x′) to ensure positive semidefiniteness is purely a gauge choice, and elucidate the resulting physical implications for measurable acceleration variances and spectral densities in tabletop and astrophysical settings.

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Background

In the non-relativistic weak-field limit, the proposed two-point function for the stochastic Newtonian potential is not positive semidefinite. The authors note that adding a constant can make it positive semidefinite over finite ranges without affecting acceleration variances, which are the observables of interest.

They conjecture that this adjustment is a gauge choice but emphasize that the physical implications require fuller understanding, particularly for interpreting experimental constraints across scales.

References

While Eq. eq:freedom is not postive semidefinite, for any finite range of $x,x'$ one can add a large enough constant to insures that~eq:freedom is positive semi-definite. The constant does not effect the variance in acceleration which is the physically meaningful quantity. We conjecture that this is thus a gauge choice, but a fuller understanding of the physical implications is required.

— Anomalous contribution to galactic rotation curves due to stochastic spacetime (2402.19459 - Oppenheim et al., 29 Feb 2024) in Appendix: Section “Comparison with tabletop experiments”