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source_pdf: Two_Numerical_Relations_Linking_the_Fine_Structure_Constant_to_Gravitational_Phenomenology_v1_3.pdf
title: "Two Numerical Relations Linking the Fine-Structure Constant"
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."
---

Two Numerical Relations Linking the Fine-Structure Constant
to Gravitational Phenomenology
Gary Alcock
Independent Researcher
gary@gtacompanies.com
December 3, 2025

Abstract
We highlight two numerical relations connecting the fine-structure constant α ≈
√1/137 to
gravitational phenomenology. First, the MOND acceleration scale satisfies a0 = 2 α cH0 to
within the current uncertainty in H0 , where c is the speed of light and H0 is the Hubble parameter.
Second, if atomic clock responses to gravitational potential variations are parameterized as
α
α
KA = kα SA
, where SA
are tabulated α-sensitivity coefficients, then existing clock data are
consistent with kα = α2 /(2π) within current ∼ 2σ uncertainties. These relations involve no free
parameters: given α and H0 , both a0 and kα are fixed. We present
the numerical evidence, offer
√
a vertex-counting heuristic that motivates the appearance of α and α2 , and identify falsifiable
predictions for near-term clock experiments. A multi-month optical clock campaign building on
recent cavity-referenced work should be able to confirm or exclude the predicted kα at > 10σ
significance.

1

Introduction

We further note that if clock sensitivities
α,
to gravitational potential follow KA = kα SA
−10
2
The MOND acceleration scale a0 ≈ 1.2×10
m/s where S α ≡ ∂ ln νA /∂ ln α are the relativistic αA
demarcates the transition between Newtonian sensitivity coefficients tabulated by Dzuba, Flamand modified gravitational dynamics in galax- baum, and collaborators [9, 10, 11], then existing
ies [1, 2]. Its numerical proximity to cH0 —the clock comparison data are consistent with
speed of light times the Hubble parameter—has
α2
been noted since MOND’s inception [1, 4], but no
kα =
.
(2)
2π
theoretical framework has explained why these
scales should be related.
This predicts kα ≈ 8.5 × 10−6 , compared to an
We show that the relation is more precise inferred value of (−0.4 ± 0.7) × 10−5 from Sr/Cs
than previously recognized:
clock comparisons [16].
√
Equations (1) and (2) contain no free paramea0 = 2 α cH0 ,
(1)
ters. Once α and H0 are specified, a0 and kα are
√
where α ≈ 1/137 is the fine-structure constant. determined. The appearance of α in the MOND
This relation is satisfied to within the current relation and α2 in the clock relation suggests a
“Hubble tension”—the discrepancy between early- vertex-counting structure familiar from quantum
and late-universe determinations of H0 . The ap- electrodynamics. Such a structure arises natupearance of α—a purely electromagnetic constant— rally in scalar-tensor frameworks where electroin a gravitational context is unexpected and, if magnetically bound matter couples to a cosmolognot coincidental, suggests a coupling between elec- ical field [13, 14]. A specific realization—Density
tromagnetism and gravity at cosmological scales. Field Dynamics (DFD)—derives both relations
1

from a single Lagrangian [15]; here we focus on 2.2 Relation II: Clock coupling
the numerical predictions independent of that
Local Position Invariance (LPI) requires that
framework.
atomic frequency ratios be independent of gravitational potential [8]. Violations are parameterized
2 The Numerical Coincidences as:
∆νA
∆Φ
= KA 2 ,
(13)
ν
c
We first establish the numerical relations as emA
pirical facts, independent of any theoretical inter- where Φ is the gravitational potential. Under
pretation.
General Relativity with exact LPI, KA = 1 for all
species, so frequency ratios are potential-independent.
2.1 Relation I: MOND scale
If α couples to gravity, different atomic species
respond proportionally to their α-sensitivity:
The observed MOND acceleration is [2, 3]:
−10
aobs
m/s2 .
0 = (1.20 ± 0.02) × 10

α
KA = kα · SA
,

(3)

(14)

α ≡ ∂ ln ν /∂ ln α are calculated from
where SA
A
atomic theory [9, 10, 11]. The differential re−3
α = 7.2973525693(11) × 10 ≈ 1/137.036. (4)
sponse between species A and B is:
The Hubble parameter remains subject to the
α
α
KA − KB = kα (SA
− SB
).
(15)
well-known “Hubble tension” [6]:

The fine-structure constant is [5]:

H0Planck = 67.4 ± 0.5 km/s/Mpc,

(5)

For 133 Cs (hyperfine) and 87 Sr (optical):

H0SH0ES = 73.0 ± 1.0 km/s/Mpc.

(6)

α
SCs
= 2.83,

(16)

α
= 0.06,
SSr
α

(17)

From the fine-structure constant:
√
2 α = 0.1708.

∆S = 2.77.

(7)

(18)

The cosmological acceleration scale cH0 deThe 2008 Blatt et al. multi-laboratory analypends on which H0 is used:
sis found [16]:
cH0Planck = 6.55 × 10−10 m/s2 ,

(8)

cH0SH0ES = 7.09 × 10−10 m/s2 .

(9)

ySr = (−1.9 ± 3.0) × 10−15

(19)

for the amplitude of annual variation in Sr/Cs,
The predicted MOND scale therefore spans: where Earth’s elliptical orbit modulates the solar
√
potential with amplitude ∆Φ/c2 =
(10) gravitational
2 α cH0Planck = 1.12 × 10−10 m/s2 ,
1.65 × 10−10 .
√
2 α cH0SH0ES = 1.21 × 10−10 m/s2 .
(11)
This corresponds to:
−10 m/s2
The observed value aobs
ySr
0 = 1.20 × 10
KCs −KSr =
= (−1.2±1.8)×10−5 , (20)
lies squarely within this range. The prediction
∆Φ/c2
brackets the measurement:
(
and thus:
1.07 (H0 = 67.4)
aobs
0
√
=
(12)
KCs − KSr
2 α cH0
0.99 (H0 = 73.0)
kα =
= (−0.4 ± 0.7) × 10−5 . (21)
∆S α
The agreement is within 7% for Planck and
The predicted value from Eq. (2) is:
within 1% for SH0ES. Resolving the Hubble tension will sharpen this test; for now, the parameterα2
(7.297 × 10−3 )2
√
kαpred =
=
= 8.5 × 10−6 .
free prediction a0 = 2 α cH0 is consistent with
2π
2π
observation.
(22)

2

The difference between prediction and central 3.2 Clock response: Four vertices
value is
For clock response to gravitational potential—
pred
obs
requiring coupling between atomic structure, scalar
|kα − kα |
|0.85 − (−0.4)|
≈ 1.8, (23) field, and gravitational potential—we consider a
=
σkα
0.7
four-vertex process:
i.e. the 2008 result is statistically consistent with
1. EM-bound matter couples to scalar field
the prediction within ∼ 2σ but does not consti√
( α)
tute a detection.
The 2008 error bars were large, precluding
detection. However, the central value is in the
predicted direction (Sr/Cs smallest at perihelion),
and the magnitude is consistent with kα ∼ α2 .

3

Vertex-Counting Heuristic

3.1

MOND: Two vertices

2. Scalar field couples to gravitational poten√
tial ( α)
3. Gravitational potential couples to scalar
√
field ( α)

4. Scalar field modifies atomic transition fre√
quency ( α)
√
Why might α appear in the MOND relation
√
and α2 in the clock relation? We offer a heuris- Combined: ( α)4 = α2 .
tic based on QED vertex counting. A formal
Including a standard loop factor of 2π:
derivation within the DFD framework is given in
α2
Ref. [15].
kα =
.
(25)
2π
In quantum electrodynamics, each interaction
√
vertex contributes a factor of α to the ampliWe present this as a heuristic motivating spetude. If electromagnetically bound matter cou- cific powers of α. The essential point is that the
ples to a scalar field through QED-like vertices, observed numerical relations are consistent with
√
the coupling strength scales as ( α)n where n is a vertex-counting structure, and this structure
the number of vertices.
yields falsifiable predictions.

4

Universal Clock Prediction

For the MOND effect—the modification of graviα with k = α2 /(2π), every atomic
tational dynamics at accelerations below a0 —we If KA = kα SA
α
clock has a predicted gravitational coupling:
consider a two-vertex process:
1. EM-bound matter couples to scalar field
√
( α)

Species

Transition

α
SA

pred
KA
(×10−5 )

133

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

Cs
Rb
1
H
87
Sr
171
Yb+
171
Yb+
27
Al+
199
Hg+
87

2. Scalar field modifies gravitational response
√
( α)
√
Combined amplitude: 2 × α.
This gives:
√
(24)
a0 = 2 α · a ⋆ ,

where a⋆ = cH0 is the cosmological acceleration Table 1: Predicted gravitational couplings KA =
α assuming k = α2 /(2π) = 8.5 × 10−6 . Valkα SA
α
scale.
α from Refs. [9, 10, 11, 12].
ues of SA

3

The prediction is falsifiable: any clock com• At perihelion (∆Φ < 0), Cs frequency shifts
α
α
parison yielding KA − KB ̸= kα (SA − SB ) would
more than Sr, so Sr/Cs decreases.
exclude the universal α-coupling hypothesis.
The signs are consistent. This is a nontrivial
The Cs/Sr channel has ∆S α = 2.77, among
the largest available, amplifying any signal by check.
nearly a factor of 50 compared to channels with
∆S α ∼ 0.1.
6 Prediction for Near-Term Ex-

periments
5

Comparison with Existing Data

A multi-month Sr–Si cavity comparison campaign, extending the work of Ref. [17], would
cover a substantial fraction of the annual solar
The three-laboratory Sr clock comparison [16]
potential cycle with precision far exceeding the
found:
2008 measurements. If cross-referenced to Cs
ySr = (−1.9 ± 3.0) × 10−15 .
(26) standards, such a dataset could decisively test
the kα relation.
2
Our prediction for kα = α /(2π):

5.1

Blatt et al. (2008)

6.1

∆Φ
pred
ySr
= −∆S α · kα · 2

For kα = α2 /(2π), the expected annual amplitude
in Cs/Sr is:

c
= −2.77 × 8.5 × 10−6 × 1.65 × 10−10
= −3.9 × 10−15 .

Predicted signal

(27)

pred
|ySr
| = 3.9 × 10−15 .

(28)

The predicted amplitude (−3.9 × 10−15 ) and
Over a six-month baseline spanning perihemeasured central value (−1.9 × 10−15 ) are:
lion:


νCs
∆
≈ 4 × 10−15 .
(29)
• Same sign (Sr/Cs smallest at perihelion)
νSr
• Same order of magnitude

6.2

Expected significance

• Statistically consistent within measurement
achieve fracuncertainty: the ySr amplitudes differ by Modern optical clock comparisons
−17 at one-day avtional
uncertainties
of
∼
10
only 0.7σ, and the corresponding kα values
eraging [18, 19]. Over a six-month campaign,
differ by ≈ 1.8σ
systematic-limited precision of ∼ 3 × 10−16 is
The 2008 measurement could not detect this achievable.
signal due to large uncertainties, but the data are
If the predicted signal is present:
fully consistent with the prediction.
4 × 10−15
Significance =
≈ 13σ.
(30)
3 × 10−16
5.2 Sign convention verification
This would constitute a definitive detection or
We explicitly verify the sign agreement. In the
exclusion
of the specific kα = α2 /(2π) hypothesis.
convention of Ref. [16]:
• ySr < 0 means νSr /νCs is smallest at perihelion.

7

Discussion

• Our framework predicts KCs > KSr because 7.1 Caveats
α > Sα .
SCs
We emphasize several limitations:
Sr

4

1. The vertex-counting argument presented
here is a heuristic. A complete derivation from the DFD Lagrangian is given in
Ref. [15].

3. The numerical coincidence would remain
unexplained.

8

Conclusion

2. The 2008 measurement has large uncertainties. While consistent with our prediction, We have presented two numerical relations:
√
it is also consistent with zero.
a = 2 α cH (within H uncertainty), (31)
0

α2
kα =
2π

3. The factor of 2π in Eq. (2) arises from loop
integration in the formal derivation [15].

0

0

(consistent with data at ∼ 2σ).

(32)
4. The MOND prediction depends on H0 , which
These relations contain no free parameters. A
is currently uncertain at the ∼ 8% level due
vertex-counting heuristic motivates the appearto the Hubble tension [6, 7].
√
ance of α (two vertices) and α2 (four vertices),
5. Alternative explanations for a0 ≈ cH0 ex- connecting MOND phenomenology to atomic
ist [20, 21], though none predict the specific clock physics through the fine-structure constant.
√
factor of 2 α.
The formal derivation within the DFD framework
is given in Ref. [15].
7.2 If confirmed
The prediction kα = α2 /(2π) ≈ 8.5 × 10−6
can be tested at > 10σ precision by ongoing
If a future campaign measures kα consistent with
and planned optical clock campaigns. If conα2 /(2π), the implications include:
firmed, this would establish a direct link be1. First detection of LPI violation. This tween the fine-structure constant and gravitawould be the first confirmed departure from tional phenomenology—a connection uniquely
the Einstein Equivalence Principle in clock suggested by DFD.
comparisons.
2. α–gravity coupling. The fine-structure Acknowledgments
constant would be directly implicated in We thank J. Ye and the JILA optical frequency
gravitational physics.
metrology group for valuable discussions.
3. Parameter-free prediction. Both a0 and
kα would be determined by α and H0 alone.

References

4. Unification hint. The same constant
(α) appearing in MOND and clock physics
would suggest a common origin, as realized
in the DFD framework [15].

7.3

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√
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5

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6