Local Position Invariance

Local position invariance says that the result of a local nongravitational experiment is independent of where and when it is performed. Clock comparisons test this by asking whether two unlike transitions acquire different responses to gravitational potential or epoch after the ordinary relativistic redshift, clock corrections, and transfer link have been modeled. The evidence considered through 2026-08-19 contains stringent null results, but any conclusion is specific to the compared transitions and the assumed coupling model.

Required background. Equivalence Principles and Gravitational Measurement supplies the operational meaning of local position invariance. Probability and Statistical Inference supplies likelihood, covariance, and nuisance-parameter reasoning.

Helpful background. Weak-Field Gravity supplies the weak-potential approximation used in clock redshift templates.

Common redshift and differential clock response

Let $g(t)=\Delta\Phi(t)/c^2$ denote a signed potential difference or a potential variation from a stated reference epoch. For two identical clocks of transition type $A$ on different worldlines, a useful parameterization is

$$y_A(t)=g(t)-\frac{\Delta(v^2)}{2c^2}+\beta_Ag(t),$$

before signal-transfer and apparatus corrections. Local position invariance gives $\beta_A=0$. The anomalous coefficient multiplies the potential term; it does not rescale special-relativistic time dilation. For co-located unlike transitions $A$ and $B$, the common proper-time rate cancels from their ratio. A potential-linked residual can then be written

$$\delta\ln\!\left(\frac{\nu_A}{\nu_B}\right) =(\beta_A-\beta_B)g(t).$$

If instead the anomaly is described by varying dimensionless constants $X_p$, define $\delta\ln X_p=k_pg(t)$ and transition sensitivities $K_{Ap}$ in one common frequency-unit convention. The same response becomes

$$\beta_A-\beta_B=\sum_p(K_{Ap}-K_{Bp})k_p.$$

These are alternative descriptions of one potential-dependent effect, not two contributions to add. Adding both without defining a separate residual coupling double counts the signal. Only differences of the transition sensitivities enter a dimensionless ratio. The classic redshift parameterization and the clock-dependent meaning of $\beta$ are derived in Will 2014, §2.1.3, Eqs. (5)–(6), pp. 12–15.

Earth’s eccentric orbit provides a convenient potential template. To leading order it is an annual cosine with phase fixed by perihelion, while the sensitivity difference selects which coupling is measured. In the $^{171}\mathrm{Yb}^{+}$ E3/E2 ratio, for example, the two transitions share the same ion but have very different relativistic electronic sensitivity; Lange et al. 2021, Eqs. (1)–(3), Figs. 2–3, and Table II found both secular-drift and annual-potential amplitudes compatible with zero.

Data, calibration, and inference

An analysis begins with released ratio points or phase records, not a published final bound alone. Required metadata include timestamps and averaging intervals, transition state and interrogation settings, systematic-shift corrections, frequency-comb and flywheel records, transfer-link direction, clock locations and velocities, an ephemeris, and the potential convention. Calibration must propagate blackbody, Zeeman, Stark, quadrupole, density, servo, Doppler, and link corrections with their correlations. The data-quality and uptime selection must be documented because seasonal maintenance or weather-dependent link availability can correlate with the annual template.

For corrected ratio residuals $r$, one may fit

$$r(t)=a+bt+q\,g(t)+\sum_j \eta_j E_j(t)+\epsilon(t),$$

where $b$ is secular drift, $q$ is the anomalous differential response, and $E_j$ are environmental or transfer regressors. The likelihood must use a covariance that includes colored clock noise, shared references, flywheel extrapolation, and correlations among systematic corrections. Priors on drift and nuisance amplitudes should be stated; a prior on $q$ must not conceal a nearly degenerate annual environmental term. Ephemeris uncertainty is usually small but should still be propagated consistently with station coordinates and tidal models. Exact ratio definitions and sensitivity coefficients are necessary to translate $q$ into fundamental-coupling parameters.

Results and limitations through 2026-08-19

Evidence or position	Result relevant to local position invariance	Limitation or uncertainty	Independent discriminator
Identical-transition clocks at different potentials	Experiments recover the common metric redshift and provide negative evidence for a clock-specific anomaly.	Geopotential, motion, and frequency transfer enter the same measured difference.	Close the comparison with reciprocal links and an independently surveyed potential.
Dissimilar optical transitions in one ion	Multiyear E3/E2 ratios show no established secular or annual anomalous response.	Sparse epochs, transition-sensitivity calculations, and annual systematics limit interpretation.	Blind an annual coupling and require recovery at the ephemeris-fixed phase.
Optical-to-microwave comparisons	Different electronic and hyperfine sensitivities break some coupling degeneracies; reported drifts remain compatible with zero.	Shared caesium standards and flywheel extrapolation correlate multiple points.	Repeat with an independent microwave realization and a direct optical ratio.
Environmental or transfer explanation of a candidate	Temperature, magnetic field, uptime, and link delay can produce annual structure without new physics.	An unmonitored lagged response may survive linear decorrelation.	Move or reverse the link while retaining the same clock pair and potential template.

Within the declared models, dissimilar clocks have not shown a reproducible position-dependent nongravitational outcome. A deviation interpretation requires more than an annual peak. Its amplitude must scale with calculated transition sensitivities, its phase must follow the potential rather than local seasons, and independent links and standards must reproduce it. Agreement with the common metric redshift does not alone constrain every differential clock coupling; the two claims concern different observables. Any model comparison must state its priors and test sensitivity to the allowed seasonal nuisance responses.

A two-stage adversarial test

First compare identical transitions over a known height or trajectory and recover $g(t)$ with the expected sign and scale. This checks potential modeling, relativistic motion, synchronization, and transfer without relying on differential atomic sensitivity. Galileo satellite eccentric-orbit data illustrate how orbit, clock correction, and redshift parameters enter a controlled test in Delva et al. 2018, Eqs. (1)–(6), Figs. 2–4, and Table II.

Second, analyze unlike transitions with a blinded injected value of $q$. The pipeline must recover the injected coupling under real sampling, distinguish it from a free-phase seasonal sinusoid, and return zero in link-reversal and same-transition null channels. Rotating the sensitivity basis among several ratios should give invariant constraints on the same underlying coupling vector.

Reproducibility requires ratio and clock-correction data, masks, ephemeris and geopotential inputs, transfer calibration, sensitivity-coefficient sources, covariance code, priors, and injection settings. At the cutoff, no universal scalar-coupling exclusion follows: unmeasured directions in sensitivity space remain open, and the mapping is model-dependent. This conclusion should be reviewed after a superseding clock or transfer release, a corrected sensitivity calculation, improved potential reconstruction, or a candidate that survives same-transition calibration, blinded recovery, seasonal alternatives, and an independent standard.

Exercise

Two co-located transitions have $K_{A\alpha}=6$ and $K_{B\alpha}=1$. Their ratio has a fitted annual amplitude $q_g=(5\pm3)\times10^{-18}$ for a potential template of amplitude $g_0=1.65\times10^{-10}$. Estimate the inferred coupling $k_\alpha=d\ln\alpha/dg$ and explain why this is not evidence for a violation.

Solution

The ratio amplitude is $(K_{A\alpha}-K_{B\alpha})k_\alpha g_0$, so $k_\alpha=q_g/(5g_0)=(6.1\pm3.6)\times10^{-9}$. The estimate differs from zero by only about $1.7$ standard deviations before accounting for alternative annual systematics or multiple searches. Therefore it is statistically compatible with zero and cannot identify local-position violation without the fixed phase, calibration, covariance, and independent-transition checks described above.

References

• Delva, Pacôme, et al. “A Gravitational Redshift Test Using Eccentric Galileo Satellites.” Physical Review Letters 121 (2018): 231101. DOI.
• Lange, R., N. Huntemann, J. M. Rahm, C. Sanner, H. Shao, B. Lipphardt, Chr. Tamm, S. Weyers, and E. Peik. “Improved Limits for Violations of Local Position Invariance from Atomic Clock Comparisons.” Physical Review Letters 126 (2021): 011102. DOI.
• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17 (2014): 4. DOI.

Further reading

• Safronova, M. S., D. Budker, D. DeMille, Derek F. Jackson Kimball, A. Derevianko, and Charles W. Clark. “Search for New Physics with Atoms and Molecules.” Reviews of Modern Physics 90 (2018): 025008. DOI.