Clock Networks and Relativistic Geodesy

A clock network turns relativistic redshift into a graph of potential differences and a transient-field search into a test of delayed coherence across stations. Neither task is accomplished by precise clocks alone: synchronization, link calibration, clock systematic budgets, station coordinates, and a time-dependent geopotential model are part of the observable. Through 2026-08-19, networks have demonstrated controlled redshift and geodetic measurements and set model-dependent transient limits, but no new propagating field is established.

Required background. Equivalence Principles and Gravitational Measurement supplies the operational redshift statement. Probability and Statistical Inference supplies covariance, likelihood, and detection-efficiency concepts.

Helpful background. Weak-Field Gravity supplies the weak-potential relation between clock rate and gravitational potential.

Clock ratios on a network graph

Let $U_i$ be the effective potential at station $i$, with sign defined by $d\tau/dt=1+U_i/c^2+\cdots$ for the stated reference time and station motion. For clocks fixed to a uniformly rotating Earth, this convention gives $U=\Phi-\lvert\boldsymbol\Omega_\oplus\times\mathbf r\rvert^2/2$, including the centrifugal contribution. Geodesists often use the opposite sign for their positive geopotential $W=-U$; the sign must be translated with the clock-rate equation. A directed comparison from $j$ to $i$ can be modeled as

$$z_{ij}(t)=\delta_i(t)-\delta_j(t) +\frac{U_i(t)-U_j(t)}{c^2} +b_{ij}(t)+\epsilon_{ij}(t),$$

where $\delta_i$ is the residual clock correction, $b_{ij}$ is link bias, and $\epsilon_{ij}$ is noise. Near Earth, a fractional shift of $10^{-18}$ corresponds to roughly one centimetre of height only when local gravity is nearly uniform. The measured object is potential difference, not geometric height; converting it to height requires a geopotential datum and gravity model.

In matrix form the station differences are generated by the graph incidence matrix. Consequently, adding one constant to every $U_i$ changes no ratio: clocks determine relative potential unless an external datum fixes the gauge. For any directed closed cycle, station and potential terms telescope,

$$\sum_{(ij)\in\mathcal C}z_{ij} =\sum_{(ij)\in\mathcal C}b_{ij}+\text{noise}.$$

Loop closure is therefore a link-consistency test only for comparisons referred to the same epochs and averaging windows, with propagation and nonreciprocity corrections included. It cannot detect a station-clock offset: any node contribution, including an anomalous one, telescopes just like the potential. Nor does a zero loop guarantee unbiased links, because biases of the form $b_{ij}=b_i-b_j$ also telescope. The 1,415-km Paris–Braunschweig optical comparison explicitly treated counter synchronization, clock corrections, link instability, and independently determined geopotential in Lisdat et al. 2016, Figs. 1–3, Table 1, and Methods.

Geopotential and transient predictions

For geodesy, $U_i(t)$ contains a static reference field plus solid-Earth and ocean tides, atmospheric and hydrological loading, polar motion, and local mass changes at the required accuracy. Coordinates, epochs, gravity observations, levelling, and the geoid model must share a reference system. A transportable clock provides a powerful closure: compare it locally before deployment, remotely against the fixed clock, and locally again afterward. This before–away–after design separates potential difference from transport-induced clock offset, as demonstrated in Grotti et al. 2018, Figs. 1–3 and Methods.

A propagating scalar disturbance has a different structure. For direction $\hat{\boldsymbol n}$ and speed $v$, a station response may be written

$$s_i(t)=\kappa_i A\,F\!\left(t-t_0-\frac{\hat{\boldsymbol n}\!\cdot\!\boldsymbol r_i}{v}\right),$$

where $F$ is a declared waveform and $\kappa_i$ is the clock sensitivity. The network prediction includes both amplitude scaling and inter-station delays. A global optical-clock search found no signal consistent with its topological-defect and scalar models Wcisło et al. 2018, Figs. 1–4 and Materials and Methods; a later space-time-separated clock, cavity, and GPS analysis constrained another oscillatory scalar model Filzinger et al. 2025, Eqs. (1)–(8) and Figs. 1–2. These are complementary model tests, not a model-independent exclusion of transients.

Releases, calibration, and selection

Each analysis must pin the ratio or phase release, link logs, clock budgets, station coordinates, geopotential and tide products, leap-second and time-scale realization, waveform code, calibration version, DOI or archive identifier, retrieval date, and file checksums. If measurements are available only on request or as plotted summaries, an exact likelihood reconstruction is not possible and should not be claimed.

Clock calibration includes Zeeman, Stark, blackbody, density, servo, Doppler, and transport shifts. Link calibration includes path-delay drift, amplifier and comb offsets, cycle slips, reciprocal-path failure, asynchronous counters, atmospheric delay for free-space links, and relativistic motion. Shared clocks, fibres, geophysical models, and reference oscillators create covariance across edges.

Selections specify station uptime, minimum link signal, cycle-slip veto, averaging interval, clock operating state, environmental bounds, transient duration and frequency range, propagation-speed range, and outlier policy. Network completeness is the probability that a signal from the declared amplitude, duration, direction, and time population survives these masks. Missing stations can destroy directional or delay information even when the remaining data look quiet.

Likelihood and reproducible checks

Stacking all valid edges gives a model $\boldsymbol z=H\boldsymbol U/c^2+H\boldsymbol\delta+H\boldsymbol s(\theta)+B\boldsymbol\eta+\boldsymbol\epsilon$, where the station signal $\boldsymbol s$ must also be differenced by the incidence matrix $H$. A signal identical at every station at the compared epoch is invisible to these ratios. A normalized Gaussian likelihood uses the full edge-time covariance; robust or heavy-tailed alternatives should be tested for cycle slips. Priors on clock drift, link bias, geophysical corrections, signal time, direction, speed, duration, and amplitude must be stated. Searching many transient templates requires a global, not local, false-alarm probability. Posterior-predictive loop residuals and time-scrambled coincidences test the noise model, while injections through the actual masks calibrate detection efficiency and interval coverage.

Results and limitations through 2026-08-19

Evidence or position	Result relevant to the network claim	Limitation or uncertainty	Independent discriminator
Long-haul coherent optical links	Remote optical clocks agree after synchronization, clock, link, and geopotential corrections.	A single edge cannot localize a clock offset versus link bias.	Add a third station and demand loop closure.
Transportable-clock geodesy	Clock-derived potential differences can agree with independent geodetic determination.	Transport hysteresis, datum choice, and time-variable mass loading limit the inference.	Repeat the local comparison after return and reverse the route.
Multiplexed miniature clock network	A blinded five-ensemble experiment recovered the expected redshift gradient and redundant pair relations.	Common apparatus rejects systematics unavailable to geographically separated clocks.	Compare extracted differences with independently measured separations.
Distributed transient searches	Published searches report null or negative evidence for their declared scalar-field templates.	Sparse topology, uptime, sensitivity assumptions, and trials leave model-dependent coverage.	Require waveform, amplitude, and geometrical delays across independent stations.

The conservative interpretation is that calibrated redshift signals and ordinary clock/link noise explain existing network records. A coherent new-field interpretation remains viable for untested couplings and timescales, but a single clock excursion or one bad edge is evidence against network coherence, not evidence for it. The short-baseline blinded result of Zheng et al. 2023, Eqs. (1)–(2), Figs. 3–4, and Methods usefully shows how redundant pair measurements and an independent spatial scale can test the redshift mapping.

The decisive adversarial sequence is to close every available frequency-ratio loop, reproduce a surveyed potential difference, and then inject a blinded propagating transient into realistically sampled records. Recovery must have the correct inter-station delays and clock sensitivities. Loop tests probe link inconsistency; comparisons with independent clocks at a station and delayed coherence between stations are needed to reject a single-clock disturbance. Reproducibility requires inputs, masks, covariance and prior code, time conversions, graph orientation, geophysical products, template bank, injection seeds, and software environment.

Statistical noise, clock and link calibration, geopotential model discrepancy, and selection efficiency should be reported separately. Direction and propagation speed can remain weakly identified in a sparse network. This conclusion should be reviewed after a superseding clock, link, or geopotential release, a corrected synchronization or response model, a materially expanded network, or a candidate that passes loop closure, surveyed-potential, blinded-delay, and independent-station checks.

Exercise

For a directed triangle, the corrected edge ratios in units of $10^{-18}$ are $z_{AB}=2$, $z_{BC}=3$, and $z_{CA}=-4$. What does the loop say, and can the residual be a static gravitational potential?

Solution

The directed sum is $2+3-4=1$ in units of $10^{-18}$, rather than zero. Static station potentials and clock offsets telescope around a closed loop, so they cannot produce this residual in synchronized comparisons. However, a nonzero realization is not automatically a significant inconsistency. If the edge vector has covariance $C$, the loop variance is $\boldsymbol 1^{\mathsf T}C\boldsymbol 1$; compare the observed sum with its square root. A significant excess points to a link correction, synchronization problem, or inadequate noise model. The loop alone does not identify which edge is wrong, and zero closure would not exclude a station-clock glitch.

References

• Filzinger, Melina, Ashlee R. Caddell, Dhruv Jani, Martin Steinel, Leonardo Giani, Nils Huntemann, and Benjamin M. Roberts. “Ultralight Dark Matter Search with Space-Time Separated Atomic Clocks and Cavities.” Physical Review Letters 134 (2025): 031001. DOI.
• Grotti, Jacopo, et al. “Geodesy and Metrology with a Transportable Optical Clock.” Nature Physics 14 (2018): 437–441. DOI.
• Lisdat, Christian, et al. “A Clock Network for Geodesy and Fundamental Science.” Nature Communications 7 (2016): 12443. DOI.
• Wcisło, Piotr, et al. “New Bounds on Dark Matter Coupling from a Global Network of Optical Atomic Clocks.” Science Advances 4 (2018): eaau4869. DOI.
• Zheng, Xin, Jonathan Dolde, Matthew C. Cambria, Hong Ming Lim, and Shimon Kolkowitz. “A Lab-Based Test of the Gravitational Redshift with a Miniature Clock Network.” Nature Communications 14 (2023): 4886. DOI.