---
source_pdf: Completing_Local_Position_Invariance_Tests__A_Cavity_Atom_Frequency_Ratio_Protocol.pdf
title: "Completing Local Position Invariance Tests:"
site: https://densityfielddynamics.com/
author: Gary Alcock
framework: Density Field Dynamics (DFD)
format_note: "Markdown extracted from the source PDF for clean AI ingestion. No editorial changes; mathematical typography in the PDF is authoritative."
---

Completing Local Position Invariance Tests:
A Cavity–Atom Frequency Ratio Protocol
Gary Alcock1
1

Los Angeles, USA

September 2025

Summary. Local Position Invariance (LPI) is a cornerstone of General Relativity, tested via
gravitational redshift with atomic clocks and matter [1, 2, 3, 4, 5]. However, no direct test has
yet compared cavity-stabilized optical frequencies (photon sector) to atomic transitions (matter
sector) across a gravitational potential. We propose a protocol to close this gap: measure
the fractional slope of co-located cavity–atom frequency ratios transported between two fixed
altitudes.

Observable
Define the cavity–atom ratio:
∆R(M,S)
∆Φ
≡ ξ (M,S) 2 ,
(M,S)
c
R

(M )

ξ (M,S) = αw − αL

(S)

− αat .

(1)

Here the coefficients are:
• αw : photon-sector weight, normalized to 1 in GR.
(M )

• αL : cavity length sensitivity for material M (e.g. ULE or Si).
(S)

• αat : atomic transition sensitivity for species S (e.g. Sr or Yb).
• ξ (M,S) : net slope coefficient for cavity–atom ratio with material M and species S.
• GR predicts ξ (M,S) = 0, i.e. a strict null.
• Any reproducible nonzero ξ would indicate sector-dependent deviation from LPI.

Definitions and identifiability
To isolate contributions, define:
Sr
ULE
δtot ≡ αw − αL
− αat
,

Si
ULE
δL ≡ αL
− αL
,

Yb
Sr
δat ≡ αat
− αat
.

The four measured slopes across two cavity materials (ULE, Si) and two atomic species (Sr,
Yb) then map to three independent combinations (Table 1).
1

Table 1: Mapping of measured cavity–atom ratios to sector parameters.
Measured slope

Combination

Identified parameter

ULE/Sr
Si/Sr
ULE/Yb
Si/Yb

ULE − αSr
αw − αL
at
Si
Sr
αw − αL − αat
ULE − αYb
αw − αL
at
Si
Yb
αw − αL − αat

δtot
δtot + δL
δtot + δat
δtot + δL + δat

Numerical scale
For Earth gravity g ≃ 9.8 m/s2 ,
g ∆h
= 1.1 × 10−14 (∆h = 100 m).
c2
Thus the natural scale is at 10−14 per 100 m altitude change, within reach of current 10−16
optical clock precision.

Controls and feasibility
The protocol envisions static comparisons at two fixed altitudes (e.g. basement vs. rooftop
labs, or ground vs. tower). Only stationary data are analyzed, avoiding artifacts from transport
in motion.
Corrections and controls:
• Dispersion/thermo-optic: dual-λ probing within the low-loss band, bounding |εdisp | ≲
10% [6, 7, 8].
• Elastic sag: orientation flips distinguish mechanical artifacts (sign-reversing) from genuine redshift (sign-preserving). In optimized silicon cavities, sag effects can be suppressed
below 10−16 [9, 10].
• Environmental: vibration, temperature, pressure, and magnetic reversals, plus hardware swaps, encode residual offsets in the covariance, suppressing bias [11, 5].
Feasibility. All required components are already demonstrated separately: ultra-stable cavities
at 10−16 [9, 10], optical clocks reaching below 10−18 [11, 5], and long-term LPI clock tests
[2, 4, 3]. Combining these into a cavity–atom slope test is therefore technically feasible with
current infrastructure.

Motivation
Existing LPI tests compare like with like: atom–atom or matter–matter systems [1, 2, 4, 5].
A cavity–atom comparison probes an untested cross-sector combination (photon vs. atomic
transitions). This experiment therefore closes a missing gap in the LPI test suite. Even a null
result would provide the first direct constraint on this sector and complete the phenomenological
mapping of LPI across independent systems.

2

Falsification criterion
• GR: ξ = 0 at all materials/species.
• Experimental discriminator: any reproducible nonzero ξ at or above ∆Φ/c2 would indicate
violation of LPI in this sector.

Acknowledgments
I thank colleagues in precision metrology for discussions of geodesy and cavity–clock systematics.

References
[1] R. F. C. Vessot et al., “Test of relativistic gravitation with a space-borne hydrogen maser,”
Phys. Rev. Lett. 45, 2081 (1980).
[2] N. Huntemann et al., “Improved limit on a temporal variation of mp /me from comparisons
of Yb+ and Cs atomic clocks,” Phys. Rev. Lett. 113, 210802 (2014).
[3] E. Peik et al., “Limit on the present temporal variation of the fine structure constant,”
Phys. Rev. Lett. 93, 170801 (2004).
[4] R. Lange et al., “Atomic clock system for improved tests of the universality of the gravitational redshift,” Phys. Rev. Lett. 126, 011102 (2021).
[5] W. F. McGrew et al., “Atomic clock performance enabling geodesy below the centimetre
level,” Nature 564, 87 (2018).
[6] M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[7] L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
[8] G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010).
[9] T. Kessler et al., “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical
cavity,” Nature Photonics 6, 687 (2012).
[10] S. Häfner et al., “8 × 10−17 fractional laser frequency instability with a long roomtemperature cavity,” Opt. Lett. 40, 2112 (2015).
[11] T. L. Nicholson et al., “Systematic evaluation of an atomic clock at 2 × 10−18 total uncertainty,” Nature Communications 6, 6896 (2015).

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