Local Lorentz Invariance

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Local Lorentz invariance says that the outcome of a local nongravitational experiment does not depend on the velocity or orientation of a freely falling laboratory. Modern tests do not search for an arbitrary “failure of relativity.” They choose a coefficient basis, calculate its orientation- and boost-dependent signatures, and ask whether those signatures occur in clock, resonator, or spin-precession records. No compelling violation is established in the evidence considered here through 2026-08-19; every bound remains conditional on the chosen sector, coefficient conventions, and nuisance model.

Required background. Equivalence Principles and Gravitational Measurement supplies the distinction between local Lorentz invariance and the other parts of the equivalence principle. Probability and Statistical Inference supplies likelihood, covariance, and model-comparison language.

Helpful background. Weak-Field Gravity helps distinguish a preferred-frame signal from ordinary weak-field effects.

From a preferred direction to laboratory harmonics

A convenient effective description assigns small background coefficients $c_a$ to operators that would break particle Lorentz symmetry while preserving observer-coordinate covariance. The coefficients are conventionally reported in a Sun-centered celestial-equatorial frame. Earth’s rotation and orbital motion then transform a fixed celestial tensor into time-dependent laboratory components. For a specified transition or resonator mode, linear response gives

h_i(t)=\sum_a R_{ia}(t)c_a,

where $h_i$ is the predicted fractional frequency or precession shift and $R_{ia}$ contains the apparatus latitude, quantization-axis direction, turntable angle, species-dependent matrix elements, and the retained order in the boost. The transformation and clock sensitivities are worked out for minimal and nonminimal matter coefficients in Kostelecký and Vargas 2018, Eqs. (1)–(21) and Appendix B.

Rotation produces harmonics at the sidereal frequency $\omega_\oplus$ and its multiples; an active turntable translates some of them to sidebands around the turntable frequency. Boost coefficients can add annual modulation at $\Omega_\oplus$ and mixed frequencies $m\omega_\oplus+n\Omega_\oplus$ . This frequency list is not universal: spin, photon, electron, proton, neutron, and gravity-sector operators populate different rows of $R$ . The Sun-centered frame and the sector-dependent definitions are tabulated in Kostelecký and Russell 2011, §§II–IV and Tables 2–24.

A simple orientation harmonic

For a toy symmetric spatial tensor $c_{JK}$ , suppose a resonator responds as $h=\hat n^Jc_{JK}\hat n^K$ . Let its axis rotate in the $X$ – $Y$ plane, $\hat{\mathbf n}=(\cos\omega t,\sin\omega t,0)$ . Expanding gives

h(t)=\frac{c_{XX}+c_{YY}}2 +\frac{c_{XX}-c_{YY}}2\cos2\omega t +c_{XY}\sin2\omega t.

The factor of two has a physical meaning: an unoriented cavity axis returns to the same configuration after a half turn. Its constant part can be degenerate with the ordinary frequency offset, while the modulated terms select two anisotropic combinations. A real clock or cavity requires its full material and sector response; this calculation explains the harmonic structure without treating every SME coefficient as one tensor of this form.

Time series, calibration, and likelihood

The measured quantity may be a clock ratio, a difference between orthogonal cavity frequencies, or a comagnetometer precession combination. A reproducible record therefore needs UTC-referenced timestamps; dead-time and data-quality masks; laboratory location; axis and turntable encoders; magnetic, thermal, pressure, tilt, and laser-power monitors; and calibration records connecting readout to fractional frequency. Selection rules must be fixed before inspecting the target harmonics: excluding intervals because they look anomalous can bias the amplitude toward zero or away from it.

One useful analysis model is

y(t_j)=y_0+\sum_k\left[A_k\cos(\omega_k t_j)+B_k\sin(\omega_k t_j)\right] +\sum_q \nu_q X_q(t_j)+n(t_j),

with target frequencies fixed by $R$ , environmental regressors $X_q$ , and correlated noise $n$ . For residual vector $r=y-h(c)-X\nu$ , a Gaussian likelihood has $-2\ln L=r^{\mathsf T}C^{-1}r+\ln\det C+\text{constant}$ . The covariance $C$ must represent colored clock noise and gaps, not merely independent samples. Weakly informative nuisance priors should be reported; priors on $c_a$ must also be explicit when the response matrix is rank deficient. Simultaneous fitting preserves covariance among nearby sidereal, solar-day, annual, and turntable sidebands, whereas fitting each coefficient alone silently assumes all others vanish.

Results and limitations through 2026-08-19

Evidence or position	Result relevant to the claim	Limitation or uncertainty	Discriminator
Clock-comparison and spin-precession records	Searches constrain coefficient combinations through sidereal and annual harmonics; broad compilations report no compelling violation.	Atomic and nuclear structure models determine the mapping to electron and nucleon coefficients.	Change species or transition while retaining the same celestial phase.
Rotating electromagnetic resonators	Null anisotropy searches probe photon-sector combinations with deliberately swept orientation.	Turntable wobble, tilt, thermal cycles, and material response can occur at the rotation frequency.	Reverse or change the rotation schedule and predict how celestial sidebands move.
Astrophysical propagation	Birefringence, dispersion, and threshold observations supply strong negative evidence for selected photon operators.	Source emission physics and propagation assumptions differ from a local laboratory test.	Require consistent energy, polarization, and sky-direction dependence across sources.
Environmental explanation of a candidate harmonic	Solar heating, magnetic drift, timing error, or an imperfect turntable can imitate periodic structure.	A nuisance correlation can be nonlinear, delayed, or incompletely monitored.	Demand the sidereal phase transformation predicted under apparatus rotation and null response at unpredicted frequencies.

The conservative interpretation is therefore a family of null constraints, not a proof that every possible Lorentz-violating theory is absent. A deviation interpretation becomes credible only after the same coefficient vector predicts harmonics across independent orientations, boosts, and species. Treating one isolated peak as new physics is not a comparably supported position: look-elsewhere effects and solar/sidereal leakage provide concrete evidence against it.

Adversarial discrimination and conclusion

The decisive independent check is a blinded signal-injection study using the actual sampling and a realistic correlated-noise model. Inject sidereal and annual coefficient vectors before quality cuts, recover their amplitudes and covariance, rotate the apparatus, and verify the predicted phase transformation. Frequencies excluded by the chosen coefficients provide null channels. The same pipeline should recover an injected solar-day environmental signal without relabeling it sidereal; a multiyear baseline helps separate the two frequencies.

Reproducibility requires the released time series or a lossless sufficient statistic, response-matrix code, frame and sign conventions, time-standard conversion, selection masks, environmental channels, covariance construction, priors, injection seeds, and the full set of searched harmonics. Without these, a quoted single-coefficient interval cannot be independently reconstructed.

At the stated cutoff, the strongest justified conclusion is that many well-defined coefficient combinations are consistent with zero within experiment-specific uncertainties. Coverage of the full theory space remains open, and some limits are model-dependent because nuclear structure or a single-coefficient assumption enters the map. This conclusion should be reviewed after a superseding time-series or calibration release, a corrected response calculation, a changed standard coefficient convention, or a candidate signal that survives rotation, injection, null-channel, and independent-species tests.

Exercise

A laboratory record contains peaks near one solar day and one sidereal day. Explain why separating the two components in a 30-day run is poorly conditioned, and name two stronger checks.

Solution

The frequencies differ by about one cycle per year. For uniform sampling over a centered interval of length $T$ , the slowly varying part of the template overlap is $\sin(\pi\Delta fT)/(\pi\Delta fT)$ , about $0.989$ for $T=30$ days. Their amplitudes are therefore strongly correlated: a solar environmental oscillation can leak into the sidereal fit. This is poor conditioning, not an absolute resolution theorem; sufficiently informative additional phase or response measurements can help. A longer record separates the phases. Independently rotating the apparatus should transform a coefficient-driven phase as predicted, while blinded injections through the real sampling and cuts test recovery of both components.

References

Kostelecký, V. Alan, and Arnaldo J. Vargas. “Lorentz and CPT Tests with Clock-Comparison Experiments.” Physical Review D 98 (2018): 036003. DOI.
Kostelecký, V. Alan, and Neil Russell. “Data Tables for Lorentz and CPT Violation.” Reviews of Modern Physics 83 (2011): 11–31, with updated tabulations. arXiv:0801.0287.