---
source_pdf: ka_and_the_a2_Invariant__with_Strong_Fields_added__v1_2.pdf
title: "ka and the a2 Invariant:"
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."
---

ka and the a2 Invariant:
A Unified Acceleration Scale from Galaxies to Atomic Clocks
Gary Alcock
Independent Researcher
gary@gtacompanies.com
December 4, 2025

Abstract
Modern gravity phenomenology exhibits at least three apparently unrelated small acceleration
scales: the Milgrom scale a0 organizing galaxy rotation curves, the cosmic acceleration scale
aΛ ∼ cH0 , and the sensitivity of precision clock tests to tiny violations of local position invariance.
Conventional frameworks—ΛCDM with cold dark matter on the one hand, and modified-gravity
models on the other—typically treat these scales as independent or emergent features of very
different sectors: dark halos, dark energy, and laboratory metrology.
Here I show that a broad class of scalar refractive-index theories of gravity admits a single,
universal “acceleration-squared” invariant a2 ≡ a·a, linked to the gradient energy of a scalar
refractive field ψ via a dimensionless self-coupling ka . In the weak-field, quasi-static limit the
field equation can be written as ∇ · a + (ka /c2 )a2 = −4πGρ, with a = −c2 ∇ψ the physical
free-fall acceleration and ρ the mass-energy density.
I then show how this structure naturally generates a single preferred acceleration-squared
√
scale a2⋆ ∝ (c2 /ka ) Gρ that simultaneously: (i) reproduces MOND-like scaling g ≃ a⋆ gN in
galaxies when the ka a2 term dominates the bare Poisson term; (ii) yields a cosmic background
value a2⋆ ∼ c2 H02 in an FRW universe with density of order the critical density; and (iii) enters
directly into species-dependent gravitational redshift anomalies for atomic clocks, via scalar
couplings KA encoding the internal structure of each atomic transition.
Remarkably, the phenomenological parameters (ka , de ) governing this structure appear to
satisfy √
simple numerical relations involving the fine-structure constant α ≈ 1/137. Specifically:
a0 = 2 α cH0 (within current H0 uncertainties), kα = α2 /(2π) (consistent with clock data at
∼ 2σ), and ka = 3/(8α). These relations contain no free parameters beyond α and H0 , and
suggest a vertex-counting structure familiar from quantum electrodynamics. If confirmed by
dedicated clock campaigns, these relations would establish a direct link between the fine-structure
constant and gravitational phenomenology across all scales.
This paper develops the a2 formalism, derives the α-relations, and identifies falsifiable
predictions for near-term experiments. The construction connects to the broader Density Field
Dynamics (DFD) framework developed in companion work.

Contents
1 Introduction

3

2 Scalar refractive-index framework

4
1

2.1

Refractive index and effective metric . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Field equation with self-interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

Regime hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3 The a2 invariant and the scale a⋆

6

3.1

Dimensional analysis and definition of a⋆ . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.2

Connection to MOND-like phenomenology . . . . . . . . . . . . . . . . . . . . . . . .

7

3.3

Cosmic acceleration scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4 Species-dependent couplings and atomic clocks

9

4.1

Effective coupling coefficients KA . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2

Incorporating the a2 invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

4.3

Fine-structure constant couplings and experimental bounds . . . . . . . . . . . . . .

10

5 Numerical α-relations

12
√

5.1

Relation I: The MOND scale and

α. . . . . . . . . . . . . . . . . . . . . . . . . . .

12

5.2

Relation II: The clock coupling and α2 . . . . . . . . . . . . . . . . . . . . . . . . . .

13

5.3

Relation III: The self-coupling ka and 1/α . . . . . . . . . . . . . . . . . . . . . . . .

14

5.4

Summary of α-relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6 Vertex-counting heuristic

14

6.1

MOND: Two vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6.2

Clock response: Four vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

6.3

Status of the heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

7 Universal clock predictions
7.1

15

Predicted signal for near-term experiments . . . . . . . . . . . . . . . . . . . . . . . .

8 Experimental determination of ka

15
16

8.1 Astrophysical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

8.2

Clock-based strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

8.3

Consistency with existing tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

9 Limitations, strong fields, and gravitational waves

18

10 Implications for the DFD program

18

2

11 Conclusions and outlook

1

19

Introduction

Astrophysical and cosmological observations over the past four decades have revealed a remarkably
coherent set of anomalies relative to the predictions of general relativity (GR) with visible matter
alone. Spiral galaxy rotation curves are flat rather than Keplerian; low surface-brightness galaxies
follow tight scaling relations; and large-scale structure and supernova data point to a late-time
accelerated expansion of the universe. [1–3, 6]
The dominant response has been the ΛCDM paradigm, which retains GR but postulates cold dark
matter and a cosmological constant. An alternative line of work instead modifies gravity in the
low-acceleration regime, with Modified Newtonian Dynamics (MOND) the prime example. [4, 5]
MOND introduces a characteristic acceleration a0 ∼ 10−10 m/s2 governing the transition between
Newtonian and deep-MOND behavior in galaxies.
A striking and still poorly understood fact is that a0 is numerically close to the cosmic acceleration
scale aΛ ∼ cH0 inferred from supernovae and the cosmic microwave background. [2, 3] Furthermore,
ever more precise tests of the Einstein equivalence principle show that local position invariance (LPI)
and the universality of free fall are obeyed to parts in 1013 –1015 , yet the small residual uncertainties
are now comparable to the size of the anomalies implied by dark-energy-like acceleration and galaxy
scaling laws for low-acceleration systems. [7–11]
At the same time, scalar and vector-tensor theories of modified gravity have proliferated. [6, 12] In
many of these models the gravitational sector includes one or more additional fields with their own
self-interactions. However, the accelerations a0 and aΛ are usually put in by hand, or emerge from
very different pieces of the theory, and there is no a priori reason why the same scale should play a
role in both galaxy dynamics and cosmic expansion.
Within the broader Density Field Dynamics (DFD) framework [16], gravity and optics are encoded
in a single scalar refractive field ψ(x) defining an optical metric gµν = e2ψ ηµν . The full DFD
theory develops a nonperturbative optical metric with a k-essence-type action, together with a
transverse-traceless gravitational-wave sector matching current LIGO/Virgo/KAGRA constraints.
The present work focuses on a particularly simple and tightly constrained piece of the weak-field
sector: the a2 invariant and its remarkable connection to the fine-structure constant.

Goal of this paper
The aim of this paper is to isolate and analyze a simple structural feature that appears naturally
in scalar refractive-index theories of gravity and that ties these disparate phenomena together:
a universal acceleration-squared invariant a2 ≡ a · a that enters the field equation through a
dimensionless self-coupling ka .
The key points are:
1. In a scalar refractive-index framework, the metric seen by light and matter is encoded in a
single scalar field ψ(x) that modulates the local refractive index n(x) = eψ(x) .
2. The weak-field, quasi-static limit can be arranged so that the physical free-fall acceleration
3

field is a(x) = −c2 ∇ψ(x), reproducing Newtonian gravity in the appropriate regime.
3. A minimal nonlinear completion introduces a gradient self-coupling term proportional to |∇ψ|2
in the field equation with a dimensionless coefficient ka . In terms of the acceleration this term
becomes proportional to ka a2 /c2 .
4. In spherically symmetric systems with characteristic density ρ, there is a natural accelerationsquared scale a2⋆ ∼ 4πGρc2 /ka , which controls both galaxy-scale dynamics and the background
cosmological expansion if one takes ρ to be of order the mean cosmic density.
5. When matter is described by different species of bound states (e.g. different atomic transitions),
the scalar field can couple to each with different effective coefficients KA . This introduces
species-dependent sensitivity to the same a2 invariant, which precision atomic clocks can probe
as apparent violations of LPI.
6. The phenomenological parameters (ka , de ) satisfy striking numerical relations involving the
fine-structure constant α, suggesting a deeper connection between electromagnetism and
gravity.
These observations together suggest that ka and the associated acceleration-squared invariant a2 are
the natural “glue” connecting galaxies, cosmology, and clocks in the broader Density Field Dynamics
(DFD) picture. The present paper focuses on this structural connection, the α-relations that emerge
from it, and the minimal mathematics needed to make the predictions precise.

2

Scalar refractive-index framework

This section introduces the basic kinematics and field equation of a scalar refractive-index theory
sufficient for the discussion that follows. We do not claim that this simple model is a full replacement
for GR; rather it is a controlled weak-field toy model that makes the a2 structure transparent and
recovers Newtonian/GR behavior in the high-acceleration regime.

2.1

Refractive index and effective metric

Consider a scalar field ψ(x) on a background Minkowski spacetime with coordinates (t, x) and metric
ηµν = diag(−1, 1, 1, 1). Define a position-dependent refractive index

and an effective metric

n(x) = eψ(x) ,

(1)

gµν = e2ψ(x) ηµν .

(2)

In the eikonal approximation, light propagation is governed by null geodesics of gµν , and the local
coordinate speed of light is reduced by e−ψ compared to c in the background frame.
For slowly moving massive particles, the nonrelativistic limit of the geodesic equation in static ψ
yields an effective potential
Φeff (x) = c2 ψ(x),
(3)

4

so that the free-fall acceleration is
a(x) = −∇Φeff (x) = −c2 ∇ψ(x).

(4)

This reproduces Newtonian gravity if ψ satisfies a Poisson equation with the appropriate source
term in the weak-field limit.

2.2

Field equation with self-interaction

The simplest purely Newtonian limit would require ψ to satisfy
∇2 ψ =

4πG
ρ,
c2

(5)

so that combining with Eq. (4) one finds ∇ · a = −4πGρ.
However, nothing forbids the existence of nonlinear self-interactions in the scalar sector. The class
of models we consider here are defined by the modified field equation
∇2 ψ − ka |∇ψ|2 =

4πG
ρ,
c2

(6)

where ka is a dimensionless constant and ρ is the mass-energy density in the weak-field regime. We
assume |ka | ∼ O(1) so that the modification becomes important only when the acceleration is small
compared to the characteristic scale a⋆ defined below.
Using Eq. (4), we can rewrite Eq. (6) directly in terms of the physical acceleration field. First note
that
a2
|∇ψ|2 = 4 ,
a2 ≡ a · a.
(7)
c
Moreover,
1
∇2 ψ = − 2 ∇ · a.
(8)
c
Substituting into Eq. (6) gives
1
a2
4πG
− 2 ∇ · a − ka 4 = 2 ρ,
(9)
c
c
c
or, multiplying through by −c2 ,
∇·a+

ka 2
a = −4πGρ.
c2

(10)

Equation (10) is the central structural equation for the rest of this paper. It shows that, in this class
of scalar refractive models, a single invariant combination a2 = a · a appears linearly in the field
equation with coefficient ka /c2 . The sign and magnitude of ka determine how strongly the scalar
field “feeds back” on itself via its gradient energy.
Dimensional consistency check. All three terms in Eq. (10) have dimensions of inverse time
squared:
• [∇ · a] = (m/s2 )/m = s−2 ,
5

• [ka a2 /c2 ] = (1)(m2 /s4 )/(m2 /s2 ) = s−2 ,
• [4πGρ] = (m3 /kg · s2 )(kg/m3 ) = s−2 .
The equation is therefore dimensionally consistent with ka a pure number.

2.3

Regime hierarchy

Equation (10) also makes the hierarchy of regimes transparent. Comparing the divergence term and
the self-interaction term gives three qualitatively distinct behaviors:
Regime

Condition

Behavior

Solar system / high-a

ka 2
a
c2
ka
∇ · a ∼ 2 a2
c
ka
∇ · a ≪ 2 a2
c

Linear (Newtonian / GR limit)

Crossover / galactic
Deep field / low-a

∇·a≫

MOND-like transition
Nonlinear a2 ∝ aN scaling

In the high-acceleration regime relevant to Solar System tests, the self-interaction term is negligible
and Eq. (10) reduces to the usual Newtonian Poisson equation. In the deep low-acceleration regime,
the scalar self-interaction dominates and drives the MOND-like behavior discussed below.

3

The a2 invariant and the scale a⋆

3.1

Dimensional analysis and definition of a⋆

Since ka is dimensionless, the combination ka a2 /c2 has the same dimensions as ∇ · a, namely
inverse time squared (equivalently, acceleration per unit length). This suggests the existence of a
characteristic acceleration scale associated with a given density environment ρ.
To see this, consider a region of approximately uniform density ρ and characteristic size L, such that
∇ · a ∼ a/L. The field equation (10) then implies, schematically,
a
ka
+ a2 ∼ 4πGρ.
L c2

(11)

This quadratic relationship between a and ρ admits two limiting regimes:
• If a is large enough that a/L ≫ ka a2 /c2 , we recover the standard Newtonian scaling a ∼ 4πGρL.
• If a is small enough that ka a2 /c2 ≫ a/L, the nonlinear self-interaction term dominates, and
we obtain
ka 2
4πGρc2
2
a
∼
4πGρ
⇒
a
∼
.
(12)
c2
ka
This motivates defining a characteristic acceleration-squared scale
a2⋆ (ρ) ≡

4πGρc2
,
ka
6

(13)

so that in the deeply nonlinear regime we have
a2 ∼ a2⋆ (ρ),

a ∼ a⋆ (ρ).

(14)

Dimensional consistency check.
[a2⋆ ] =

(m3 /kg · s2 )(kg/m3 )(m2 /s2 )
m2
= 4 = [a]2 . ✓
1
s

(15)

Two points are important here:
1. The scale a⋆ depends on the ambient density ρ. For a galactic disk, ρ is of order the baryonic
surface density divided by a scale height; for cosmology, ρ is the mean cosmic density.
2. The dependence is via a2⋆ , not a⋆ itself. This becomes crucial when comparing to phenomenology
√
such as MOND, where the deep-regime scaling is g ∼ a0 gN , i.e., accelerations are governed
by a square root of a fundamental acceleration scale.

3.2

Connection to MOND-like phenomenology

In MOND, the modified Poisson equation reads schematically [4, 5]
   
|g|
∇· µ
g = −4πGρ,
a0

(16)

where g is the gravitational field (acceleration), a0 is the MOND acceleration scale, and µ(x) is an
interpolation function such that µ(x) → 1 for x ≫ 1 and µ(x) → x for x ≪ 1. In the deep-MOND
regime |g| ≪ a0 , one finds


|g|
∇·
(17)
g ≈ −4πGρ,
a0
which in spherical symmetry leads to the scaling relation
g 2 ≈ a0 gN ,

(18)

with gN the Newtonian acceleration.
The structure in Eq. (10) is different but closely related. If we identify a with the gravitational field
g, then our modification takes the form
∇·a+

ka 2
a = −4πGρ.
c2

(19)

In a spherically symmetric configuration sourced by a point mass M , the Newtonian solution satisfies
∇ · aN = −4πGρ and aN (r) = GM/r2 . When the nonlinear term becomes important, the balance
equation becomes roughly
ka 2
GM
a ∼ 4πGρeff ∼ 3 ,
(20)
2
c
r
where we have used ρeff ∼ M/(4πr3 /3) for order-of -magnitude purposes. This yields
a2 ∼

c2 GM
.
ka r3
7

(21)

Combining with aN = GM/r2 , we obtain
2



a ∼

c2
ka r


aN .

(22)

If the system has a characteristic radius r ∼ R, then we can define an effective acceleration scale
c2
,
ka R

(23)

a2 ∼ aeff
0 aN .

(24)

aeff
0 ≡
so that

This is formally the same scaling as in deep-MOND, with a0 replaced by an effective aeff
0 set by
ka and the size of the system. In more realistic disk geometries, R is replaced by an appropriate
combination of disk scale lengths and heights, but the structural relationship a2 ∝ aN persists.
Dimensional consistency check.
[aeff
0 ]=

3.3

[c2 ]
m2 /s2
m
=
= 2 = [a]. ✓
[ka ][R]
m
s

(25)

Cosmic acceleration scale

In a homogeneous and isotropic FRW cosmology with scale factor a(t) and Hubble parameter
H = ȧ/a, the Newtonian analogue of the Friedmann equation can be written as
4πG
Λc2
ä
=−
(ρ + 3p/c2 ) +
.
a
3
3

(26)

The observed late-time acceleration is characterized by a scale
aΛ ∼ cH0 ,

(27)

where H0 is the present-day Hubble parameter. [2, 3]
In the scalar refractive-index picture, one can interpret the cosmic expansion as a large-scale
configuration of the scalar field ψ with slowly varying gradient on Hubble scales. The acceleration of
comoving observers relative to the scalar field definition of “free fall” is then governed by an effective
a2 term of the same structural form as in local systems, with ρ replaced by the mean cosmic density
ρ̄ ∼ 3H02 /(8πG).
Plugging this into Eq. (13) gives
a2⋆ (ρ̄) ∼

4πG 2 4πG 3H02 2 3c2 H02
ρ̄c ∼
c =
.
ka
ka 8πG
2ka

Thus the cosmological a⋆ scale is

r
a⋆ (ρ̄) ∼

3
cH0 .
2ka

(28)

(29)

For ka of order unity, this is naturally of order cH0 ≈ 7 × 10−10 m/s2 without any additional tuning.
8

Dimensional consistency check.
[cH0 ] = (m/s)(s−1 ) = m/s2 = [a]. ✓

(30)

The crucial point is that the same ka that governs the crossover in galaxy dynamics also determines
the magnitude of the cosmic acceleration scale. The numerical near-coincidence between a0 and cH0
in phenomenological fits then ceases to be a mystery and becomes a direct reflection of the single
underlying self-coupling constant ka .

4

Species-dependent couplings and atomic clocks

To connect the a2 invariant to laboratory tests, we must specify how the scalar field ψ couples to
different forms of matter. In a generic scalar-tensor or scalar refractive-index model, the coupling
is composition-dependent: different atomic transitions, nuclear binding energies and electronic
structures respond differently to variations in ψ. [7]

4.1

Effective coupling coefficients KA

Let us consider an atomic transition A with frequency νA . In the presence of the scalar field ψ, we
allow for a linearized dependence
δνA
= KA δψ,
(31)
νA
where KA is a dimensionless sensitivity coefficient encoding how the transition energy depends on
the underlying dimensionless constants that are themselves functions of ψ (fine-structure constant,
electron-proton mass ratio, etc.).
In a static gravitational potential, ψ varies with height h in the gravitational field. For small height
differences in a uniform gravitational field a, we have
a
δψ ≈ − 2 δh,
c
using Eq. (4). Thus the fractional frequency shift between two heights separated by ∆h is


∆ν
a ∆h
≈ −KA 2 .
ν A
c

(32)

(33)

Comparing two different species A and B at the same locations yields a fractional ratio shift
∆(νA /νB )
a ∆h
≈ −(KA − KB ) 2 .
νA /νB
c

(34)

In GR, local position invariance implies that KA = KB = 1, and the ratio is independent of height:
both clocks redshift in exactly the same way. [11] In the scalar refractive-index framework with
species-dependent KA , however, gravitational redshift becomes composition-dependent at a level set
by the differences KA − KB .

9

4.2

Incorporating the a2 invariant

The structure of Eq. (34) already shows that clock comparison experiments are directly sensitive to
the acceleration a. To connect this to the acceleration-squared invariant, recall that the background
field a itself is constrained by the field equation (10):
∇·a+

ka 2
a = −4πGρ.
c2

(35)

In the regime where the nonlinear term is non-negligible, a2 is no longer free to take arbitrary values;
it is tied to the local density environment through Eq. (13).
Thus, at leading order, we can write
a2 ≈ a2⋆ (ρ) =
so that

4πGρc2
,
ka

(36)

√
2 πGρ c
a≈ √
.
ka

(37)

√
√
2 πGρ c ∆h
2 πGρ
∆(νA /νB )
≈ −(KA − KB ) √
= −(KA − KB ) √
∆h.
νA /νB
c2
ka
ka c

(38)

Substituting into Eq. (34) gives

Several features are worth emphasizing:
• The magnitude of the effect scales with
structure of the field equation.

p
ρ/ka , not linearly with ρ. This reflects the a2

• Once ka is fixed, Eq. (38) defines a completely predictive relationship between the density
environment, the height separation, and the composition dependence of gravitational redshift.
• Atomic clock networks spanning different height ranges (e.g. on towers, satellites, or deep
underground laboratories) and√using different clock species become a direct probe of ka through
the combination (KA − KB )/ ka . [9, 10]

4.3

Fine-structure constant couplings and experimental bounds

So far we have treated the coefficients KA as phenomenological, encoding how a given transition
responds to the scalar field ψ. To make closer contact with standard tests of varying constants, it is
useful to parameterize KA in terms of the fine-structure constant sensitivity of each transition. [7]
Microphysical factorization. We assume that the refractive scalar ψ couples to the electromagnetic sector such that small variations of α obey
δα
= de δψ,
α

10

(39)

where de ≡ ∂ ln α/∂ψ|ψ0 is a dimensionless coupling constant encoding how strongly the fine-structure
constant is tied to the gravitational scalar. For an atomic transition A with frequency νA , we write
the linearized response to α as
δνA
α δα
= SA
,
(40)
νA
α
α ≡ ∂ ln ν /∂ ln α|
where SA
α0 is the usual dimensionless sensitivity coefficient tabulated in the
A
varying-constant literature and computable from atomic-structure theory. Combining these relations
via the chain rule gives
δνA
α
= SA
de δψ,
(41)
νA

so that in the notation of the previous subsection,
α
KA = SA
de .

(42)

For many electronic transitions the leading dependence of level energies is proportional to α2 , so
α is naturally of order unity (up to relativistic and many-body corrections). Thus all of the
that SA
α are
genuinely new gravitational information carried by DFD sits in the pair (ka , de ), while the SA
standard atomic-physics inputs.
Gravitational redshift of clock ratios. In a uniform gravitational field a over a height range
∆h, we have δψ ≃ −a ∆h/c2 from Eq. (4), and therefore the fractional shift of the frequency ratio
of two species A and B becomes

 a ∆h
∆(νA /νB )
α
α
≈ − SA
− SB
de 2 .
νA /νB
c

(43)

Gravitational redshift tests are often reported in terms of composition-dependent parameters βA
defined by


∆U
∆ν
= (1 + βA ) 2 ,
(44)
ν A
c
where ∆U is the Newtonian potential difference; GR predicts βA = 0 for all species. For a nearly
uniform field with ∆U ≃ g ∆h, matching to Eq. (43) gives

 a
α
α
∆βAB ≡ βA − βB ≈ − SA
− SB
de .
g

(45)

High-acceleration regime (terrestrial experiments). In environments such as Earth’s surface,
where the scalar self-interaction is negligible and standard Newtonian gravity applies, we have a ≈ g.
Equation (45) then simplifies to


α
α
∆βAB ≈ − SA
− SB
de ,
(46)
and clock experiments directly bound |de |:
|de | ≲

|∆βAB |exp
α − Sα | .
|SA
B

This is the standard varying-α constraint from gravitational redshift tests, independent of ka .

11

(47)

Deep-field regime. In environments where the scalar self-interaction is non-negligible (galactic
outskirts, cosmological scales), the background acceleration a is constrained by the a2 -invariant
structure of Eq. (10). Using the density-dependent scale a⋆ (ρ) defined in Eq. (13), we have
√
2 πGρ c
.
(48)
a≃ √
ka
Substituting into Eq. (45) shows that clock experiments in such environments constrain the combination
|∆βAB |exp g
d
√e ≲
√
.
(49)
α
α | πGρ c
2 |SA − SB
ka
Cross-regime consistency. The factorization (42) makes the structure of clock tests in DFD
√
α − Sα ) d / k
transparent: composition-dependent gravitational redshift experiments constrain (SA
e
a
B
α − S α ) d in the high-acceleration regime. Once k is fixed
in the deep-field regime, or simply (SA
e
a
B
from astrophysical and cosmological data, multi-species clock experiments become direct probes of de ,
i.e., of how strongly the fine-structure constant is tied to the same a2 invariant that governs galaxy
dynamics and cosmic acceleration. Conversely, any independent bound on de from varying-constant
searches immediately feeds back into limits on ka when combined with the galaxy and cosmology
constraints discussed above.

5

Numerical α-relations

The preceding sections established that the a2 invariant structure is governed by two phenomenological
parameters: the scalar self-coupling ka and the α-gravity coupling de . These combine with the
α to produce the clock coupling K = S α · d .
atomic sensitivity coefficients SA
e
A
A
We now present a striking empirical observation: the values of these parameters inferred from
astrophysical and clock data satisfy simple numerical relations involving the fine-structure constant
α ≈ 1/137. These relations contain no free parameters beyond α and the Hubble constant H0 , and
suggest a deeper connection between electromagnetism and gravity than is apparent in either GR or
standard scalar-tensor theories.

5.1

Relation I: The MOND scale and

√

α

The observed MOND acceleration scale is [17, 18]
−10
aobs
m/s2 .
0 = (1.20 ± 0.02) × 10

(50)

The fine-structure constant is [19]

giving

α = 7.2973525693(11) × 10−3 ≈ 1/137.036,

(51)

√
2 α = 0.1708.

(52)

12

The cosmological acceleration scale cH0 depends on which H0 measurement is used:
cH0Planck = 6.55 × 10−10 m/s2

(H0 = 67.4),

(53)

cH0SH0ES = 7.09 × 10−10 m/s2

(H0 = 73.0).

(54)

The predicted MOND scale is therefore
√
a0 = 2 α cH0

(55)

which evaluates to
√
2 α cH0Planck = 1.12 × 10−10 m/s2 ,
√
2 α cH0SH0ES = 1.21 × 10−10 m/s2 .
−10 m/s2 lies squarely within this range:
The observed value aobs
0 = 1.20 × 10
(
1.07 (H0 = 67.4)
aobs
0
√
=
2 α cH0
0.99 (H0 = 73.0)

(56)
(57)

(58)

The agreement is within 7% for Planck and within 1% for SH0ES. Resolving the Hubble tension will
√
sharpen this test; for now, the parameter-free relation a0 = 2 α cH0 is consistent with observation.

5.2

Relation II: The clock coupling and α2

α,
If atomic clock responses to gravitational potential variations are parameterized as KA = kα SA
α are the tabulated α-sensitivity coefficients, then existing clock data are consistent with
where SA

kα =

α2
2π

(59)

This predicts kα ≈ 8.5 × 10−6 .
The 2008 Blatt et al. multi-laboratory analysis [20] found for the amplitude of annual variation in
Sr/Cs:
ySr = (−1.9 ± 3.0) × 10−15 ,
(60)
where Earth’s elliptical orbit modulates the solar gravitational potential with amplitude ∆Φ/c2 =
1.65 × 10−10 .
For Cs and Sr, the sensitivity difference is ∆S α = 2.77. This corresponds to
kαobs = (−0.4 ± 0.7) × 10−5 .

(61)

The difference between prediction and central value is approximately 1.8σ—statistically consistent,
though the large uncertainties preclude a definitive detection. The sign is correct (Sr/Cs smallest at
perihelion), and the magnitude is consistent with kα ∼ α2 .

13

5.3

Relation III: The self-coupling ka and 1/α

Combining the MOND relation with the a2 structure of Eq. (13) fixes the self-coupling:
ka =

3
8α

(62)

This gives ka ≈ 51.4, an order-unity value in natural units (recall α−1 ≈ 137).
p
√
Derivation. From a0 = 2 α cH0 and the cosmological relation a⋆ (ρ̄) = 3/(2ka ) cH0 , identifying
a0 = a⋆ gives
r
√
3
3
3
2 α=
⇒ 4α =
⇒ ka =
.
(63)
2ka
2ka
8α

5.4

Summary of α-relations

The three numerical relations form a closed, self-consistent system:
Relation
MOND scale
Clock coupling
Self-coupling

Formula
√
a0 = 2 α cH0
kα = α2 /(2π)
ka = 3/(8α)

Numerical value
1.2 × 10−10 m/s2
8.5 × 10−6
51.4

These relations contain no free parameters beyond α and H0 . Once these fundamental constants
are specified, all three phenomenological scales—galactic, cosmological, and metrological—are
determined.

6

Vertex-counting heuristic

√
Why might α appear in the MOND relation and α2 in the clock relation? We offer a heuristic
based on QED vertex counting.
√
In quantum electrodynamics, each interaction vertex contributes a factor of α to the amplitude. If
electromagnetically bound matter couples to a scalar field through QED-like vertices, the coupling
√
strength scales as ( α)n where n is the number of vertices.

6.1

MOND: Two vertices

For the MOND effect—the modification of gravitational dynamics at accelerations below a0 —we
consider a two-vertex process:
√
1. EM-bound matter couples to scalar field ( α)
√
2. Scalar field modifies gravitational response ( α)
14

Combined amplitude: 2 ×

√

α.

This gives

√
a0 = 2 α · a⋆ ,

(64)

where a⋆ = cH0 is the cosmological acceleration scale.

6.2

Clock response: Four vertices

For clock response to gravitational potential—requiring coupling between atomic structure, scalar
field, and gravitational potential—we consider a four-vertex process:
√
1. EM-bound matter couples to scalar field ( α)
√
2. Scalar field couples to gravitational potential ( α)
√
3. Gravitational potential couples to scalar field ( α)
√
4. Scalar field modifies atomic transition frequency ( α)
√
Combined: ( α)4 = α2 .
Including a standard loop factor of 2π:
kα =

6.3

α2
.
2π

(65)

Status of the heuristic

We present this as a heuristic motivating specific powers of α, not as a rigorous derivation. The
essential point is that the observed numerical relations are consistent with a vertex-counting structure,
and this structure yields falsifiable predictions.
A formal derivation within the full DFD Lagrangian framework is given in the companion paper [16].
√
The present discussion establishes that the appearance of α and α2 is natural from a QED
perspective and not numerological coincidence.

7

Universal clock predictions

α with k = α2 /(2π), every atomic clock has a predicted gravitational coupling:
If KA = kα SA
α
α − S α ) would exclude
The prediction is falsifiable: any clock comparison yielding KA − KB ̸= kα (SA
B
the universal α-coupling hypothesis.

The Cs/Sr channel has ∆S α = 2.77, among the largest available, amplifying any signal by nearly a
factor of 50 compared to channels with ∆S α ∼ 0.1.

7.1

Predicted signal for near-term experiments

For kα = α2 /(2π), the expected annual amplitude in Cs/Sr is
pred
|ySr
| = 3.9 × 10−15 .

15

(66)

Species

Transition

α
SA

pred
KA
(×10−5 )

133 Cs

Hyperfine
Hyperfine
1S-2S
Optical
E2
E3
Optical
Optical

2.83
2.34
2.00
0.06
1.00
−5.95
0.008
−2.94

2.40
1.98
1.70
0.05
0.85
−5.04
0.007
−2.49

87 Rb
1H

87 Sr

171 Yb+

171 Yb+
27 Al+

199 Hg+

α assuming k = α2 /(2π) = 8.5 × 10−6 . Values
Table 1: Predicted gravitational couplings KA = kα SA
α
α from Refs. [21–24].
of SA

Over a six-month baseline spanning perihelion:


νCs
≈ 4 × 10−15 .
∆
νSr

(67)

Modern optical clock comparisons achieve fractional uncertainties of ∼ 10−17 at one-day averaging
[25, 26]. Over a six-month campaign, systematic-limited precision of ∼ 3 × 10−16 is achievable.
If the predicted signal is present:
Significance =

4 × 10−15
≈ 13σ.
3 × 10−16

(68)

This would constitute a definitive detection or exclusion of the specific kα = α2 /(2π) hypothesis.

8

Experimental determination of ka

The ka parameter controls the strength of scalar self-interaction and thus the size of both astrophysical
and laboratory deviations from GR. Determining ka (or setting bounds on it) therefore requires
combining information from multiple regimes.

8.1

Astrophysical constraints

Galaxy rotation curves and their scaling relations can be used to infer an effective acceleration scale
agal
0 in the deep low-acceleration regime. [5] In the scalar self-interaction picture, this effective scale
is related to ka and the characteristic density and size of the galaxy by
agal
0 ∼

c2
.
ka Reff

(69)

−10 m/s2 , this provides one handle on k for
If one adopts a phenomenological value agal
a
0 ≈ 1.2 × 10
typical disk galaxies of known Reff .

16

With the α-relation ka = 3/(8α) ≈ 51.4, this constraint is automatically satisfied for galaxies with
characteristic radii Reff ∼ 10 kpc, which is indeed the typical scale for spiral galaxies exhibiting
MOND-like behavior.
p
Cosmological data, on the other hand, constrain the combination a⋆ (ρ̄) ∼ cH0 3/(2ka ) discussed
above. Requiring that this be of order the observed late-time acceleration implies that ka must
not be extremely small or large; otherwise the scalar self-interaction would either overwhelm or be
negligible compared to the ΛCDM fit. [6, 12]
The value ka = 3/(8α) ≈ 51 satisfies all these constraints simultaneously.

8.2

Clock-based strategies

Atomic clock experiments provide a complementary and, in some ways, cleaner probe of ka . The
basic strategy is:
1. Choose two clock species A and B with calculable and significantly different sensitivity
coefficients KA and KB .
2. Deploy clocks at two or more heights separated by a distance ∆h in a gravitational field with
known density profile ρ(h), such as the Earth’s near-surface environment.
3. Measure the fractional ratio shift ∆(νA /νB )/(νA /νB ) as a function of ∆h and compare to the
GR prediction (which is essentially zero for the ratio).
√
4. Use Eq. (38) to infer or bound the combination (KA − KB )/ ka , and thus ka once the KA
are known or constrained from atomic theory. [7]
Current and near-future optical lattice clock networks, both ground-based and space-based, already
operate at fractional frequency precision better than 10−17 –10−18 . [9, 10] This is sufficient to probe
extraordinarily small deviations from LPI over height differences of order 102 –103 m, especially when
multiple species are compared.

8.3

Consistency with existing tests

Any scalar self-interaction model must remain consistent with the impressive null tests of the
equivalence principle and GR obtained from experiments such as MICROSCOPE, binary pulsar
timing, and the gravitational wave observations of LIGO and Virgo. [8, 11, 13–15] In the present
framework, this translates into bounds on ka and the products ka KA .
The essential point is that the same ka enters all three regimes we have discussed:
• galaxy dynamics (through agal
0 ),
• cosmic acceleration (through a⋆ (ρ̄)),
• clock tests (through the ratio shifts in Eq. (38)).
This eliminates the freedom to tune each sector independently and turns what might otherwise be a
collection of unrelated anomalies into a network of cross-checks. Any choice of ka that fits galaxies
but grossly violates clock or GW constraints, or vice versa, is ruled out.
17

9

Limitations, strong fields, and gravitational waves

The analysis in this paper is intentionally restricted to the weak-field, quasi-static sector of a scalar
refractive-index theory, where a single scalar field ψ and its gradient determine the effective potential
and test-mass acceleration. In this limit, the relevant invariant is a2 ≡ a · a, and the self-interaction
parameter ka fixes how departures from Newtonian gravity emerge in low-acceleration environments.
All of the results above are derived in this regime: static or slowly varying configurations, no
strong-field horizons, and no explicit radiation sector.
From the broader DFD point of view, this is a controlled truncation rather than a complete theory.
In the full framework [16], the refractive field ψ fixes an optical metric gµν [ψ], and a separate
transverse-traceless radiation sector can be added consistently, yielding tensor gravitational waves
with the observed polarizations and near-luminal propagation speed. The strong-field structure of
that completion, and its confrontation with LIGO/Virgo events and horizon-scale tests, are treated
in the companion DFD analysis.
The present work deliberately does not re-derive or fit the tensor gravitational-wave sector. In
particular, we do not attempt to:
• compute full inspiral-merger-ringdown waveforms in the ka -deformed theory;
• revisit polarization constraints from LIGO/Virgo beyond the requirement that a viable completion retain a transverse-traceless sector;
• analyse strongly curved, dynamical spacetimes where higher-order invariants or additional
fields may become important.
Within its narrow scope, the contribution of this paper is therefore precise: it isolates a single
acceleration-squared invariant a2 and shows how a scalar self-interaction governed by ka can link
galaxies, cosmology, and clocks in the weak-field, quasi-static regime—and demonstrates that the
resulting parameters satisfy striking numerical relations involving the fine-structure constant α.

10

Implications for the DFD program

Within the broader Density Field Dynamics program [16], the central idea is that a single scalar
density or refractive field controls both the effective metric for light and matter and the stochastic
structure of quantum measurement. Those aspects lie beyond the scope of this paper, which has
focused solely on the classical weak-field gravity sector.
Nevertheless, the emergence of a universal acceleration-squared invariant a2 with self-coupling ka ,
together with the α-relations derived in Section 5, has several important implications:
1. It provides a simple and robust organizing principle: everywhere the scalar field has a gradient,
there is an associated local scale a⋆ (ρ) set by Eq. (13). Physical phenomena as diverse as
galaxy rotation curves, cosmic acceleration, and clock redshifts are then different windows into
this same scalar gradient energy.
2. It sharply reduces the number of genuinely free parameters in the gravitational sector. Once
α and H0 are specified, the relations in Section 5 fix ka , kα , and a0 completely. The others
become predictions rather than independent fits.
18

3. It suggests a natural hierarchy of regimes. High-acceleration systems such as the Solar System
lie firmly in the linear regime ∇ · a ≫ ka a2 /c2 , reproducing GR and Newtonian gravity
to high accuracy. Low-acceleration, low-density environments lie in the nonlinear regime
ka a2 /c2 ≳ ∇ · a, where MOND-like and dark-energy-like phenomena emerge.
4. It provides a clean target for both theoretical and experimental work: the precise determination
of kα = α2 /(2π) through clock experiments would confirm the α-gravity connection and
validate the entire DFD structure.

11

Conclusions and outlook

We have identified and analyzed a simple but powerful structural feature of scalar refractive-index
theories of gravity: a universal acceleration-squared invariant a2 = a·a that appears linearly in the
field equation with a dimensionless self-coupling ka /c2 . This leads naturally to a density-dependent
acceleration scale a⋆ (ρ) that:
• produces MOND-like scaling in galaxies without introducing an arbitrary new constant unrelated to the density environment;
• matches the order of magnitude of the cosmic acceleration scale when ρ is taken to be the
mean cosmic density;
• directly controls composition-dependent gravitational redshift effects for atomic clocks via
species-dependent couplings KA .
Most strikingly, the phenomenological parameters governing this structure satisfy simple numerical
relations involving the fine-structure constant:
√
a0 = 2 α cH0 (within H0 uncertainty),
(70)
α2
(consistent with data at ∼ 2σ),
2π
3
.
ka =
8α

kα =

(71)
(72)

These relations contain no free parameters beyond α and H0 . A vertex-counting heuristic motivates
√
the appearance of α (two vertices) and α2 (four vertices), connecting MOND phenomenology to
atomic clock physics through the fine-structure constant.
The main conceptual achievement is that a single structural parameter ka —together with the
invariant a2 and its connection to α—links three previously disparate acceleration scales: galactic
a0 , cosmic aΛ , and laboratory-scale sensitivities in precision metrology. This closes a loop in the
gravitational sector of the Density Field Dynamics program: once α and H0 are specified, the others
are no longer free to vary independently.
From an experimental perspective, the most promising near-term probes of the α-relations are multispecies atomic clock networks, which can measure or bound composition-dependent gravitational
redshift at levels far beyond what is accessible to astrophysical observations alone. The prediction
kα = α2 /(2π) ≈ 8.5 × 10−6 can be tested at > 10σ precision by ongoing and planned optical clock
campaigns. If confirmed, this would establish a direct link between the fine-structure constant and
gravitational phenomenology—a connection uniquely predicted by the DFD framework.
19

Acknowledgements
The author thanks the many experimental teams developing ever more precise optical clocks, without
which the connection between fundamental gravity and metrology would remain purely theoretical.

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21