Gravitational Redshift Tests

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A gravitational-redshift test compares accumulated clock phase after removing relative motion, signal-transfer delay, trajectory error, and clock-specific shifts. The observable is a frequency ratio tied to two worldlines and one reference system; a quoted “height sensitivity” is already a model-dependent conversion through a geopotential. The test isolates local position invariance only when those layers are explicit.

Required background. Equivalence Principles and Gravitational Measurement supplies proper time, clock comparison, and LPI. Probability and Statistical Inference supplies likelihood, covariance, and interval concepts; the Gaussian clock-residual model below is the bounded bridge used here.

Helpful background. Weak-Field Gravity supplies the metric and potential expansion behind the redshift prediction.

Proper time and the measured ratio

Use a weak, slowly varying field with $g_{00}=-(1+2\Phi/c^2)$ , where $\Phi$ is the Newtonian potential and $\Phi\to0$ far from an isolated source. To first post-Newtonian order,

\frac{d\tau}{dt}=1+\frac{\Phi}{c^2}-\frac{v^2}{2c^2}+O(c^{-4}).

This follows by inserting $d\mathbf x=\mathbf v\,dt$ into $c^2d\tau^2=-ds^2$ and expanding the square root; products of $\Phi/c^2$ and $v^2/c^2$ enter only at the omitted order. For identical stationary clocks $A$ and $B$ , define $\nu_A$ and $\nu_B$ as their tick rates referred to the same coordinate time after ideal transfer correction. Then

y_{BA}\equiv\frac{\nu_B-\nu_A}{\nu_A} =\frac{\Phi_B-\Phi_A}{c^2}.

Because $\Phi=-GM/r$ , the higher clock has larger $\Phi$ and runs faster. This rate ratio should not be confused with the frequency of one photon received upstairs: for stationary endpoints that received-to-emitted local photon-frequency ratio is $1+(\Phi_A-\Phi_B)/c^2$ , the opposite signed difference. A redshift test must say which comparison it reports.

An LPI-violation parameter is often introduced as $y_{BA}=(1+\alpha)\Delta\Phi/c^2$ . This $\alpha$ is transition- and framework-dependent unless universality has been assumed. It multiplies the potential contribution, not the special-relativistic velocity term. It must not absorb Doppler, orbit, or link errors by definition. Near Earth’s surface, $\Delta\Phi\simeq gh$ gives $y\simeq1.09\times10^{-16}(h/{\rm m})$ for $g=9.81\ {\rm m\,s^{-2}}$ ; a centimetre corresponds to about $1.1\times10^{-18}$ .

Real data are phase or frequency comparisons at finite times. One useful model is

y^{\rm obs}(t) =y_{\rm GR}(t)+\alpha\frac{\Delta\Phi(t)}{c^2} +y_{\rm Dop}(t)+y_{\rm link}(t) +\sum_k\nu_k X_k(t)+n(t),

where $y_{\rm GR}=[\Delta\Phi-\Delta(v^2)/2]/c^2$ at this order, $y_{\rm Dop}$ denotes remaining first-order signal-transfer Doppler terms, and $X_k$ describe clock temperature, magnetic field, blackbody, density, and other systematic channels. Each velocity contribution is included once. The sign of every term follows the declared emitter/receiver order and coordinate time scale.

Data and calibration layers

A reproducible comparison pins frequency-ratio or phase records, clock trajectories, velocity and potential products, time-transfer observables, and evaluated clock systematics. The BIPM distinguishes realization of the SI second from the relativistic comparison of distant standards; relativistic corrections are part of interpreting the measurement, not a change in the atomic definition BIPM 2026, §2.3.1, p. 126, and §2.3.6, p. 137, PDF.

Calibration includes the local oscillator and transition response, transmitter/receiver delays, cable and fibre asymmetry, atmospheric or ionospheric propagation, station coordinates, orbit solution, and reference-frame transformations. First-order Doppler terms are usually much larger than the gravitational signal; two-way links cancel them only to the accuracy allowed by reciprocity and synchronization. Second-order Doppler, tidal potentials, and centrifugal contributions remain in the relativistic model.

Selection must list clock switches, phase discontinuities, invalid orbit arcs, tracking gaps, environmental excursions, and any daily detrending. Completeness means time and orbital-phase coverage sufficient to separate the redshift template from drift and periodic systematics. If detrending projects onto the signal, its transfer function must enter the forward model.

With residual vector $\mathbf r(\alpha,\boldsymbol\nu)$ and covariance $C$ , use a normalized likelihood including $\det C$ . Clock noise is often colored, and trajectory or magnetic-field errors can be coherent at orbital frequency. Shared calibration parameters should therefore be fitted or propagated jointly rather than converted into independent error bars.

Two discriminating examples

The eccentric Galileo-satellite test used 1008 days of orbit and passive-hydrogen-maser clock solutions. Orbital eccentricity modulates potential and velocity at a known period. The analysis removed daily offset and drift, modeled colored clock noise, and estimated orbit, magnetic, and temperature systematics independently from the clock signal Delva et al. 2018, Eqs. (1)–(6), Figs. 2–4, and Table II, PDF. Satellite laser ranging supplied an external diagnostic of orbit error. The example shows why a periodic potential history, not merely a large altitude difference, is experimentally valuable.

At a radically smaller scale, Bothwell et al. spatially resolved the frequency gradient across one millimetre-scale strontium ensemble. Pixel-level density and second-order Zeeman corrections preceded a linear spatial fit; independent synchronous regions provided a cross-check Bothwell et al. 2022, Figs. 3–4, Methods “Analysis” and “Systematics,” and Table 1. This does not remove apparatus physics—it makes its spatial structure measurable.

An independent redshift check should recover the common prediction from an up/down interchange or common-clock closure, vary trajectory and Doppler models, and inject a blinded nonzero $\alpha$ . If the injection is absorbed by clock drift or link parameters, the experiment cannot support the intended conclusion.

What the results establish

Evidence assessed through 2026-08-19 supports the metric redshift across laboratory and satellite scales within the tested clock, trajectory, and transfer models. It does not show that all transitions have identical anomalous couplings, and it does not by itself select Einstein’s field equation. A result is conditional on the named data product, calibration, selections, likelihood, priors, covariance, and potential convention.

The retained reproduction should specify software and ephemeris versions, clock and orbit file identities, reference time scale, all cuts, systematic templates, and one command that reconstructs $y^{\rm obs}-y_{\rm GR}$ and the injection test. A corrected clock solution, orbit product, transfer calibration, or geopotential convention triggers substantive reanalysis.

Exercise

Clock A is stationary in the chosen coordinates. Clock B is at a potential higher by $\Delta\Phi$ and moves with speed $v$ . What leading fractional rate difference should be modeled?

Solution

Subtract the proper-time rates using the same coordinate time. The potential contribution is $\Delta\Phi/c^2$ , while the moving upper clock contributes $-v^2/(2c^2)$ . Therefore $y=\Delta\Phi/c^2-v^2/(2c^2)$ at the displayed order, before link and clock-systematic terms. Fitting only the potential term would bias an anomalous-redshift parameter because special-relativistic time dilation can share the same trajectory modulation.

References

Bothwell, Tobias, et al. “Resolving the Gravitational Redshift across a Millimetre-Scale Atomic Sample.” Nature 602 (2022): 420–424. DOI.
Bureau International des Poids et Mesures. The International System of Units (SI), 9th ed., version 4.01. Sèvres: BIPM, 2019, updated June 2026. DOI; Open PDF.
Delva, Pacôme, et al. “A Gravitational Redshift Test Using Eccentric Galileo Satellites.” Physical Review Letters 121 (2018): 231101. DOI; Open PDF.