---
source_pdf: Optical__Metric_Scalar_Phenomenology_and_a_Decisive_Cavity__Atom_LPI_Test.pdf
title: "Optical-Metric Scalar Phenomenology and a Decisive Cavity-Atom LPI Test:"
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."
---

Optical-Metric Scalar Phenomenology and a Decisive Cavity-Atom LPI Test:
Phase Velocity as Operational One-Way Light Speed in a Verified Nondispersive Band
Gary Alcock1
1

Independent Researcher
(Dated: August 26, 2025)

We develop an optical-metric scalar phenomenology—a minimal, testable framework in which
a scalar field ψ induces a conformal optical metric for electromagnetism while leaving the matter
metric unchanged at leading order. In a verified nondispersive frequency band, geometric optics
implies the measured electromagnetic phase velocity equals the operational one-way propagation
speed along a path segment. This enables synchronization-free measurements that are strictly null
in general relativity (GR) yet potentially non-null here. We (i) state the assumptions explicitly, (ii)
derive the identity via both Fermat/eikonal optics and Gordon’s optical metric, (iii) anchor the
phenomenology to familiar scalar-tensor and SME language (local Lorentz invariance preserved; local
position invariance potentially violated in the photon sector), (iv) identify the clean experimental
discriminator: a co-located cavity-atom frequency ratio recorded at two gravitational potentials, and
(v) provide a quantitative constraints audit and a realistic error budget showing near-term feasibility
at 10−16 fractional uncertainty and below. The proposal is falsifiable: a null cavity-atom ratio shift
across altitude (after dispersion and thermal controls) kills this class of models; a reproducible nonnull that scales with potential would warrant deeper theory. Our aim is not to redefine simultaneity
but to exploit route-dependent, synchronization-free observables that adjudicate between GR (null)
and this optical-metric scalar sector.

I.

MOTIVATION AND ASSUMPTIONS (MADE
EXPLICIT)

A1. Optical–metric scalar. We posit a scalar field
ψ(x) that conformally rescales the photon sector’s effective metric,

II.

CORE IDENTITY FROM TWO
INDEPENDENT ROUTES

Route I: Fermat/eikonal.
Geometric optics extremR
izes T [γ] = (1/c) γ n(x) dℓ with n = eψ , giving
vphase =

−2ψ(x)

g̃µν = e

ηµν ,

(1)

in the lab frame, so that light rays follow g̃µν –null
geodesics (Gordon–type optics [1, 2]). Matter fields minimally couple to ηµν at leading order. This mirrors well–
studied scalar–tensor ideas [3] and photon–sector extensions in the SME [4] while keeping local Lorentz cones
isotropic.
A2. Nondispersive measurement band. Experiments are restricted to a frequency band where dispersion
is bounded by dual–wavelength checks: |∂n/∂ω| small
enough that phase, group, and front velocities coincide
within the error budget [5, 6]. Outside this band no claim
is made.
A3. Weak–field normalization. In the Newtonian
regime we set ψ ≃ −2Φ/c2 , chosen so that standard weak–
field optical tests (deflection, Shapiro delay) recover the
GR value γ = 1 [7]. Test bodies fall universally; LLI in
the matter sector is respected.
Aim. With (1) and A2–A3, the measurable phase
velocity becomes a synchronization–free probe of an operational one–way light speed along a path segment. We
design a clean LPI discriminator where GR is null.

c
= c e−ψ ≡ c1 (x).
n

(2)

In a verified nondispersive band, phase=group=front [5,
6], so c1 is the operational one–way propagation speed
along γ without distant clocks.
Route II: Optical metric. Light rays are g̃µν –null:
ds̃2 = (c2 /n2 )dt2 − dx2 = 0 [1]. Nullness implies dℓ/dt =
c/n, the same identity. The equality is structural, not
definitional.

III.

RELATION TO EQUIVALENCE
PRINCIPLES AND LLI

Local Lorentz invariance (LLI). Because g̃µν is
conformally flat and isotropic, two–way orientation and
boost tests remain null at 10−17 –10−18 as observed [8–10].
Local position invariance (LPI). The matter sector
respects LPI to leading order (atom vs atom redshifts
match GR). The photon sector, however, samples n =
eψ ; thus cavity vs atom ratios can acquire route/height
dependence. Our decisive observable is precisely an LPI
test in a nondispersive photon sector.

2
TABLE I. Constraints audit. “Null in GR” means strict
null after standard subtractions. The last column states this
model’s expectation under A1–A3.

TABLE II. Order–of–magnitude signals under A1–A3 (nondispersive band enforced). GR is strictly null for A/B and effectively null for C.

Observable

Protocol

Observable

A: Crossed cavities
B: Fiber loop

δf /f on
rotate/∆h
∆ϕ⟳ − ∆ϕ⟲

C: Cavity/atom ratio

∆R/R across ∆h

Constrains

Expectation
here
Two–way cavity rotations
LLI anisotropy Null (matches
data)
Atom–atom redshift
LPI (matter
Matches GR
(lab/space)
sector)
Remote transfer links
Path/time
Orthogonal to
transfer
local ratio
Cavity–atom, single height Local
Constant ratio
calibration
at fixed
conditions
Potentially
Cavity–atom ratio at two Photon vs
heights
matter LPI non–null
(decisive)

Signal
scale
∼ 10−16
per m
< 10−16
eqv.
2g∆h/c2 ≈
2.2×10−14
/100 m

∆h probe geometry–locked shifts. Target stability: 10−17 –
10−16 [8, 10]. GR: null (after standard subtractions).
Here: geometry–locked residuals permitted by A1.

IV. MINIMAL DYNAMICS
(PHENOMENOLOGY-FIRST)

For concreteness we adopt a scalar field fixed by local
mass density with a single crossover scale,
 


|∇ψ|
c2
8πG
∇· µ
a = ∇ψ,
∇ψ = − 2 (ρm − ρ̄m ),
a⋆
c
2
(3)
so that ψ ≃ −2Φ/c2 in weak fields. This choice reproduces
GR optics at leading order [2, 7] and serves only to map
lab gradients to potentials. Our empirical claims do not
hinge on UV completion (Sakharov–style motivations exist
[11]).

V.

WHAT EXISTING TESTS DO—AND DO
NOT—CONSTRAIN

We summarize the published landscape (abbrev.):
To our knowledge, no peer–reviewed result reports a
co–located cavity–atom frequency ratio recorded at two
distinct gravitational potentials with < 10−16 fractional
uncertainty. This is the precise gap our Protocol C targets.

VI.

LABORATORY PROTOCOLS (GR–NULL VS
SIGNAL HERE)

B. Reciprocity–broken fiber loop (two heights)

A monochromatic tone circulates around an asymmetric
loop with Rvertical separation ∆h and a Faraday element.
ϕ =H (ω/c) n(x) dℓ yields a forward–backward difference
∝ ψ dℓ that vanishes in GR (static loop, Sagnac removed) but not here if ∇ψ · ẑ ̸= 0.

C. Decisive LPI test: co–located cavity–atom ratio
across altitude

Lock a laser to a vacuum cavity (frequency fcav ∝ c/n)
and compare to a co–located optical atomic transition fat
via a comb. Form R ≡ fcav /fat at altitude h1 , repeat at
h2 = h1 + ∆h.
GR: Moving a co–located package changes neither local
physics nor the ratio; R is constant (excellent approximation).

All protocols enforce nondispersion by dual–wavelength
phase tracking to bound |∂n/∂ω| within the budget [6, 12].

Here (A1–A3): With ψ ≃ −2Φ/c2 , the cavity inherits
fcav ∝ e−ψ ≃ 1 + 2Φ/c2 , while the atomic transition is
leading–order ψ–insensitive (matter–sector universality).
Hence

A. Crossed ultra–stable cavities (orientation/height
sweep)

∆R
∆Φ
g ∆h
≈ 2 2 ≈ 2 2 ∼ 2.2×10−14 per 100 m. (4)
R
c
c

Orthogonal high–Q cavities of length L support fm ≃
(m/2L) (c/n); a change δψ imparts δf /f = −δn/n =
−δψ. Orientation reversals and vertical translations by

Allowing a small matter coupling gives ∆R/R = ξ ∆Φ/c2
with 0 < ξ ≤ 2; still at the 10−16 m−1 scale. State–of–the–
art cavities and clocks reach 10−17 –10−16 [8, 10, 13, 14].

3
TABLE III. Illustrative 1σ fractional budget for ∆R/R over
100 m. Values reflect demonstrated performance in the cited
literature; any one item can be tightened.
Source (mitigation)
Cavity thermal drift (ULE/cryo; drift
cancel by differencing)
Vibration/tilt
(seismic
isolation;
feedforward)
Comb ratio transfer (self–referenced;
optical division)
Atomic ref. (Sr/Yb/Al+ ; short–term
avg)
Residual dispersion (dual–λ bound; linear fit)
Air index/pressure (vacuum enclosure;
sensors)
Magnetic/polarization (scrambling;
swaps)
Quadrature total

VII.

σ (fractional)
5 × 10−16
2 × 10−16
1 × 10−16
1 × 10−16
5 × 10−17
5 × 10−17
3 × 10−17
∼ 7 × 10−16

PREDICTED MAGNITUDES (ORDER OF
ESTIMATE)
VIII. PROTOCOL C FEASIBILITY:
QUANTITATIVE ERROR BUDGET

We list dominant systematics and representative fractional contributions for a 100 m potential step (conservative, room–temp cavities; cryo improves margins). Dual–λ
control is assumed to bound dispersion.
The target signal ∼ 2.2 × 10−14 per 100 m exceeds the
above conservative noise by ≳ 30×. Even a suppressed
coupling ξ ∼ 0.1 remains clearly resolvable. Publishing
Allan deviation σy (τ ), blind height reversals, hardware
swaps, and multi–λ linearity fits close the standard loopholes.

IX.

REFUTATION CRITERIA (CLEAN KILL
CONDITIONS)

Any of the following falsifies this class of optical–metric
scalars:
1. Protocol C yields ∆R/R consistent with zero at or
below |∆Φ|/c2 (or ξ inferred ≪ 10−2 ) while dispersion and thermal budgets pass checks.

X.

ADDRESSING STANDARD CRITICISMS
DIRECTLY

(1) “One–way c is conventional; you cannot measure it.” We do not alter simultaneity conventions. We
identify local, synchronization–free, route–dependent observables that are null in GR but not necessarily in the
photon sector of an optical–metric scalar. The equality
“phase=one–way speed” is invoked only in a band where
phase=group=front is verified [5, 6, 15–17].
(2) “Vacuum cannot have a refractive index.”
We never posit a material medium. We posit an effective
optical metric (a standard construct since Gordon [1, 2])
in which photons see n = eψ . This is squarely within
scalar–tensor/SME phenomenology [3, 4]. Two–way LLI
tests remain null and satisfied.
(3) “Equivalence principle is broken.” Matter test
bodies obey universal free fall; atomic redshift tests match
GR [13, 14, 18]. The proposed discriminator is LPI in
the photon sector: cavity (photon) vs atom (matter). If
nature is GR in both sectors, Protocol C is null and the
model is ruled out.
(4) “Existing experiments would already have
seen this.” Published demonstrations involve atom–atom
redshifts or remote transfers; none report the co–located
cavity–atom ratio at two potentials with < 10−16 sensitivity (Table I). Our error budget shows clear headroom.
(5) “Phase velocity is not signal velocity.” Correct
in dispersive media. We operate only in a verified nondispersive band where phase, group, and front velocities
coincide within budget [5, 6].

XI.

CONCLUSIONS

We have recast “DFD” as an optical–metric scalar phenomenology that is conservative (LLI preserved; GR optics
recovered in the weak field) yet falsifiable by a single decisive, synchronization–free test: the co–located cavity–atom
ratio across a potential difference. The identity linking
phase velocity to operational one–way speed is established by two independent routes and invoked only under
a verified nondispersion assumption. With current optical
metrology, the proposal is executable; either the ratio
is null (model killed) or a controllable, potential–scaling
residual appears (then the photon sector merits renewed
scrutiny).

2. Protocols A/B give nulls after reversals/path swaps
where a geometry–locked residual was predicted
under A1–A3.
3. A verified nonzero dispersion fully accounts for any
residuals across the band.
Conversely, a reproducible, potential–scaling non–null
that survives the above controls would motivate a fuller
theory (or sharpen SME bounds).

Appendix A: Geometrical optics and nondispersion

Let S be the eikonal: k = ∇S, ω = −∂t S. With ω =
(c/n)|k|, vphase = ω/|k| = c/n and vg = ∂ω/∂|k| = c/n;
the Sommerfeld–Brillouin front velocity coincides in the
nondispersive limit [5, 6].

4
Appendix B: Round–trip nulls, clocks, and GPS

Appendix C: Minimal implementation checklist

R
For a fixed path γ, T2w = (2/c) γ n dℓ; at fixed geometry, orientation rotations preserve two–way times
(Michelson–Morley nulls). Atom–atom redshift verifications rely on matter clocks and remote transfers consistent
with GR [13, 14, 18]; our decisive observable is a local
cavity–atom ratio across altitude.

Cavities: ULE/silicon spacers; PDH locking; cryogenic option; 10−17 stability [8, 10].
Fibers: zero–dispersion operation; Faraday isolators;
dual–λ phase tracking [12].
Clocks: Sr/Yb lattice or Al+ logic; comb–based ratio
readout [13, 14].
Analysis: publish σy (τ ); blinded reversals; multi–λ linearity fits; environmental logs.

[1] W. Gordon, Annalen der Physik 377, 421 (1923).
[2] V. Perlick, Ray Optics, Fermat’s Principle, and Applications to General Relativity, Lecture Notes in Physics
Monographs, Vol. 61 (Springer, 2000).
[3] C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).
[4] V. A. Kostelecký and N. Russell, Rev. Mod. Phys. 83, 11
(2011), updated annually; see arXiv:0801.0287.
[5] L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
[6] M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
[7] C. M. Will, Living Reviews in Relativity 17, 4 (2014).
[8] C. Eisele, A. Y. Nevsky, and S. Schiller, Phys. Rev. Lett.
103, 090401 (2009).
[9] S. Herrmann et al., Phys. Rev. D 80, 105011 (2009).

[10] M. Nagel et al., Nature Communications 6, 8174 (2015).
[11] A. D. Sakharov, Sov. Phys. Dokl. 12, 1040 (1968).
[12] G. P. Agrawal, Fiber-Optic Communication Systems, 4th
ed. (Wiley, 2010).
[13] C. W. Chou, D. B. Hume, T. Rosenband, and D. J.
Wineland, Science 329, 1630 (2010).
[14] W. F. McGrew et al., Nature 564, 87 (2018).
[15] H. Reichenbach, The Philosophy of Space and Time
(Dover, New York, 1958).
[16] W. F. Edwards, American Journal of Physics 31, 482
(1963).
[17] D. Malament, Noûs 11, 293 (1977).
[18] P. Delva et al., Phys. Rev. Lett. 121, 10.1103/PhysRevLett.121.231101 (2018).