---
source_pdf: Falsifiable_Experimental_Signatures_of_Density_Field_Dynamics.pdf
title: "Falsifiable Experimental Signatures of Density Field Dynamics:"
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."
---

Falsifiable Experimental Signatures of Density Field Dynamics:
Phase Velocity Equals One-Way Light Speed in a Nondispersive Vacuum
Gary Alcock1
1

Density Field Dynamics Research Collaboration
(Dated: August 25, 2025)

We convert Density Field Dynamics (DFD) into a laboratory-focused, falsifiable test program.
DFD posits a single scalar field ψ(x) that governs matter dynamics and photon propagation through
a universal, nondispersive vacuum refractive structure. The core operational result is that, when
dispersion is bounded in-band, the electromagnetic phase velocity is the one-way speed of light; hence
precision phase metrology becomes a direct, synchronization-free probe of c1 (x). We derive this
identity along two independent routes (Fermat/eikonal and Gordon’s optical metric), demonstrate
compatibility with classic tests of relativity, and design three GR-null vs DFD-signal protocols—most
decisively a co-located cavity–atom frequency ratio measured at two altitudes—with quantified,
near-term sensitivities. We audit existing constraints, state explicit refutation criteria, and provide a
comprehensive responses-to-criticisms section (simultaneity, equivalence principle, Lorentz invariance,
“already ruled out”, dispersion). The question is experimentally decidable with current optical
metrology.

I.

FROM PRINCIPLE TO PROTOCOL

Two-way light speed and Lorentz symmetry are constrained to extraordinary precision [1–4]. The one-way
speed remains entangled with simultaneity conventions
[5–7]. DFD proposes a dynamical scalar ψ(x) that (i)
fixes a universal vacuum refractive index n = eψ for photons, and (ii) normalizes the Newtonian limit for matter
via a = (c2 /2)∇ψ. In a verified nondispersive band, geometric optics yields vphase = c/n, which, with n = eψ ,
provides the operational bridge c1 = c e−ψ = vphase . The
novelty is experimental: route-dependent, synchronizationfree observables that are GR-null but DFD-nonnull.

II.

DFD DYNAMICS IN ONE PAGE

Adopt a scalar action with a single crossover scale,
 


|∇ψ|
c2
8πG
∇· µ
∇ψ = − 2 (ρm − ρ̄m ),
a = ∇ψ,
a⋆
c
2
(1)
so that ψ is fixed by matter density and yields ψ ≃
−2Φ/c2 in weak fields. Photons propagate by Fermat/optical metric (Sec. III). The weak-field normalization is chosen to reproduce GR’s classic optical tests with
PPN γ = 1 [1, 8]. A Sakharov-style perspective motivates
induced kinetic terms from quantum fluctuations [9], but
the empirical program below does not rely on specific UV
details.

III.
A.

CORE IDENTITY: vphase = c1
Route I: Fermat/eikonal

R
Geometric optics extremizes T [γ] = (1/c) γ n(x) dℓ,
giving vphase = c/n [8, 10, 11]. With n = eψ fixed by

dynamics, c1 = ce−ψ = vphase follows. No distant clocks
enter: the observable is local phase kinematics, verified
nondispersive.

B.

Route II: Gordon’s optical metric

Light in a linear, isotropic, nondispersive medium follows null geodesics of
ds̃2 =

c2
dt2 − dx2
n2 (x)

(2)

[12]. Nullness implies dℓ/dt = c/n; with n = eψ the same
identity follows. The equality is therefore structural (two
logically independent routes), not a definitional tautology.

IV.

EQUIVALENCE PRINCIPLE & LORENTZ
INVARIANCE

Local Lorentz invariance. ψ is a scalar; the light
cone at a point remains isotropic. Two-way orientation/boost tests remain null at the 10−17 –10−18 level,
consistent with cavity experiments [2–4].
Universality for matter. Test bodies obey a =
(c2 /2)∇ψ in the weak field, reproducing free-fall universality and PPN γ = 1 optics [1]. Equivalence principle
tests remain satisfied in this limit.
Where differences appear. DFD predicts routedependent, synchronization-free effects where GR enforces
strict nulls: e.g., a co-located cavity–atom ratio compared
at two altitudes (Sec. VI). This is an LPI probe in a
nondispersive vacuum sector not covered by atom–atom
redshift verifications.

2
V.

WHAT EXISTING TESTS DO—AND DO
NOT—CONSTRAIN

Two-way
isotropy
&
boost
(Michelson–Morley/Kennedy–Thorndike;
modern rotating
cavities) are exquisitely null [2–4]; DFD predicts the
same nulls for two-way observables along fixed paths.
Atomic clock redshift confirms GR at ∼ 10−16 per meter
and below [13, 14]; spaceborne tests reach 2.5 × 10−5
relative precision [15]. These are atom–atom or remotetransfer comparisons.
Critical gap: To our knowledge, no published measurement reports a co-located cavity–atom frequency ratio
recorded at two different gravitational potentials with
< 10−16 fractional uncertainty. That is the target of
Protocol C.

VI.

LABORATORY PROTOCOLS (GR-NULL VS
DFD-SIGNAL)

All protocols enforce nondispersion via multiwavelength checks which bound |∂n/∂ω| in the measurement band (so phase=group=front) [10, 16].

Protocol

Observable

DFD signal (order)

A: Crossed cavities δf /f on rotate/∆h
10−16 per m (vertical)
B: Fiber loop
∆ϕ⟳ − ∆ϕ⟲
geometry-locked, < 10−16 eqv.
C: Cavity/atom ratio ∆R/R across ∆h 2g∆h/c2 ≈ 2.2 × 10−14 per 100 m

TABLE I. Order-of-magnitude DFD signals in verified nondispersive band. GR predicts strict nulls for A/B and near-null
for C.

upon comparison between distinct potentials, so R is (to
excellent approximation) constant.
DFD (nondispersive): With ψ ≃ −2Φ/c2 , fcav ∝
−ψ
e
≃ 1 + 2Φ/c2 , while the atomic transition is leadingorder ψ-insensitive (matter-sector universality). Thus
∆R
g ∆h
∆Φ
≃ 2 2 ≈ 2 2 ∼ 2.2 × 10−14 per 100 m.
R
c
c
(3)
A variant with small matter-sector coupling yields ∆R
R =
ξ ∆Φ/c2 with 0 < ξ ≤ 2, still at the 10−16 m−1 scale.
Feasibility: present clocks and cavities reach 10−17 –10−16
[2, 4, 13, 14].
Systematics & controls (all protocols). (i) Multi-λ dispersion bound; (ii) temperature/strain control, Allan budgeting; (iii) polarization scrambles and hardware swaps;
(iv) blind orientation/height reversals; (v) environmental
monitors (pressure, tilt, vibration).

Protocol A: Crossed ultra-stable cavities
(orientation/height sweep)
VII.

Two orthogonal high-Q cavities (length L) support
m c
modes fm ≃ 2L
A change δψ imparts δf /f =
n.
−δn/n = −δψ. Orientation reversals and vertical translations by ∆h probe geometry-locked shifts. Target sensitivity: 10−17 –10−16 fractional, routinely achieved [2, 4].

VIII.

PREDICTED SIGNAL SIZES AND
SENSITIVITY TABLE

REFUTATION CRITERIA (CLEAN KILL
CONDITIONS)

Any of the following falsifies this DFD formulation:
Protocol B: Reciprocity-broken fiber loop (two
heights)

A monochromatic tone circulates both ways around an
asymmetric loop with vertical separation ∆h andR a nonreciprocal element. The accumulated phase Hϕ = ωc n(x) dℓ
yields a forward–backward difference ∝ ψ dℓ that vanishes in GR (static loop, Sagnac subtracted) but not
in DFD if ∇ψ · ẑ ̸= 0. Operate near a zero-dispersion
wavelength; multi-λ tracking bounds dispersion [17].

1. Protocol C yields ∆R/R consistent with zero at or
below |∆Φ|/c2 while dispersion and thermal budgets
pass all checks.
2. Protocols A or B yield nulls where DFD predicts nonzero geometry-locked signals after reversals/path swaps.
3. A verified nonzero dispersion (∂n/∂ω) fully accounts
for any residuals across the band.
Conversely, reproducible nonzero signals that (i) scale
with ∆h or orientation as predicted, (ii) survive multi-λ
tests, and (iii) pass swap/blind controls, would be decisive.

Protocol C (decisive): Co-located cavity–atom ratio
across altitude
IX.

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

COSMOLOGICAL CONTEXT (BRIEF)

With ψ ≃ −2Φ/c2 , Gordon’s metric reproduces classic
weak-field optics [1, 8]. DFD suggests that nearby structure can bias line-of-sight cosmography at low z, providing
a plausible context for directional H0 inferences; earlyuniverse constraints (CMB/BAO) remain intact [18, 19].

3
These motivate but do not underwrite the laboratory
program.

X.

COMPREHENSIVE RESPONSES TO
STANDARD CRITICISMS

(1) “You haven’t solved simultaneity; one-way c
is conventional.” We agree that simultaneity is conventional in SR. DFD makes a different claim: in a verified
nondispersive band, local phase velocity equals the oneway propagation speed. Our observables are local and
synchronization-free; they exploit route dependence where
GR says strict null. This turns a philosophical impasse
into a falsifiable statement [5–7].
(2) “n = eψ makes c1 = c/n definitional (circular).”
The identity vphase = c/n follows from standard optics
(Fermat/eikonal and Gordon’s metric) independently [8,
10, 12]. DFD then supplies dynamics for ψ (Eq. 1), fixed
by classic-test normalization [1]. The bridge is therefore
derived, not stipulated.
(3) “Equivalence principle is violated: photons
vs matter.” Matter test bodies obey a = (c2 /2)∇ψ
(universality preserved in weak field). Photons see the
optical metric which reproduces GR’s lensing/redshift
(PPN γ = 1). Our key test (Protocol C) is an LPI probe
in the nondispersive vacuum sector; either a residual
appears (then LPI is violated in this sector) or it does
not (DFD is falsified).
(4) “Lorentz symmetry constraints already exclude this.” Two-way isotropy/boost tests [2–4] remain
null in DFD. Differences appear only between paths sampling different ψ (height/orientation). This is not what
SME-style cavity rotations constrain [20].
(5) “Existing optical clocks at different elevations would have seen it.” Published redshift verifications are atom–atom or remote-transfer comparisons
[13–15]. They confirm GR and are orthogonal to our
decisive co-located cavity–atom ratio across altitudes. To
our knowledge, such a ratio-vs-altitude measurement at
< 10−16 is not yet published; Protocol C is designed to
fill this gap.
(6) “This is just superluminal phase velocity;
information rides on front velocity.” Correct in
dispersive media [16]. Our tests operate in a verified
nondispersive band where phase=group=front [10]; the
identity is invoked only under those conditions.
(7) “ψ is ad hoc and parameters are tuned.” The
weak-field normalization is fixed by classic tests; induced-

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gravity arguments [9] motivate scalar kinetic terms. However, our claims do not hinge on UV priors: the laboratory
identity and protocols stand on their own as empirical
tests.
XI.

CONCLUSIONS

We have turned DFD into a concrete, near-term experimental program: (i) a structural identity that makes
phase metrology a one-way-c probe in a verified nondispersive vacuum; (ii) three synchronization-free protocols with
GR-null vs DFD-signal contrasts and quantified targets;
(iii) an explicit constraints audit and clean refutation logic.
The decisive experiment (Protocol C) is implementable
now with optical cavities, clocks, and frequency combs.
Either the geometry-locked phase-velocity effects appear
(opening a new sector of physics) or the present DFD is
falsified.

Appendix A: Geometrical optics and nondispersion

Let S be the eikonal: k = ∇S, ω = −∂t S. For 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 [10, 16].

Appendix B: Round-trip nulls, clocks, and GPS

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

Appendix C: Minimal implementation checklist

Cavities: ULE/silicon spacers; PDH locking; cryogenic option; 10−17 stability [2, 4].
Fibers: zero-dispersion operation; Faraday isolators;
dual-λ phase-tracking [17].
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.

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6:8174, 2015.

4
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