---
source_pdf: Two_Numerical_Relations_Linking_the_Fine_Structure_Constant_to_Gravitational_Phenomenology.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 4, 2025

Abstract
We report 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) at the 1σ level. 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 six-month optical clock campaign currently underway should
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
.
(2)
kα =
2π
theoretical framework has explained why these
scales should be related.
This predicts kα ≈ 8.5 × 10−6 , compared to an
We report 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 parame√
a0 = 2 α cH0 ,
(1) ters. Once α and H are specified, a and k are
0
α
√ 0
determined. The appearance of α in the MOND
where α ≈ 1/137 is the fine-structure constant.
relation and α2 in the clock relation suggests a
This relation is satisfied to within the current
vertex-counting structure familiar from quantum
“Hubble tension”—the discrepancy between earlyelectrodynamics. Such a structure arises natuand late-universe determinations of H0 . The aprally in scalar-tensor frameworks where electropearance of α—a purely electromagnetic constant—
magnetically bound matter couples to a cosmologin a gravitational context is unexpected and, if
ical field [13, 14]. A specific realization—Density
not coincidental, suggests a coupling between elecField Dynamics (DFD)—derives both relations
tromagnetism and gravity at cosmological scales.
from a single Lagrangian [15]; here we focus on
1

the numerical predictions independent of that 2.2 Relation II: Clock coupling
framework.
Local Position Invariance (LPI) requires that
atomic frequency ratios be independent of gravita2 The Numerical Coincidences tional potential [8]. Violations are parameterized
as:
∆νA
∆Φ
We first establish the numerical relations as em= KA 2 ,
(13)
ν
c
pirical facts, independent of any theoretical interA
pretation.
where Φ is the gravitational potential. Under
General Relativity with exact LPI, KA = 1 for all
2.1 Relation I: MOND scale
species, so frequency ratios are potential-independent.
If α couples to gravity, different atomic species
The observed MOND acceleration is [2, 3]:
respond proportionally to their α-sensitivity:
−10
aobs
m/s2 .
(3)
0 = (1.20 ± 0.02) × 10
α
KA = kα · SA
,
(14)
The fine-structure constant is [5]:
α ≡ ∂ ln ν /∂ ln α are calculated from
where SA
A
−3
α = 7.2973525693(11) × 10 ≈ 1/137.036. (4) atomic theory [9, 10, 11]. The differential reThe Hubble parameter remains subject to the sponse between species A and B is:
well-known “Hubble tension” [6]:
H0Planck = 67.4 ± 0.5 km/s/Mpc,
H0SH0ES = 73.0 ± 1.0 km/s/Mpc.
From the fine-structure constant:
√
2 α = 0.1708.

α
α
KA − KB = kα (SA
− SB
).

(5)

(15)

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

(6)

(7)

(8)

cH0SH0ES = 7.09 × 10−10 m/s2 .

(9)

(16)

α
= 0.06,
SSr
α

(17)

∆S = 2.77.

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

α
SCs
= 2.83,

(18)

The 2008 Blatt et al. multi-laboratory analysis found [16]:
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
√
2
2
Planck
−10
2 α cH0
= 1.12 × 10
m/s ,
(10) gravitational potential with amplitude ∆Φ/c =
√
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
−5
lies squarely within this range. The prediction KCs −KSr = ∆Φ/c2 = (−1.2±1.8)×10 , (20)
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 2008 measurement is consistent with this
prediction at 1.9σ:
|kαpred − kαobs |
|0.85 − (−0.4)|
≈ 1.9.
=
σkα
0.7

1. EM-bound matter couples to scalar field
√
( α)
2. Scalar field couples to gravitational poten√
tial ( α)

(23)

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

3. Gravitational potential couples to scalar
√
field ( α)
4. Scalar field modifies atomic transition fre√
quency ( α)
√
Combined: ( α)4 = α2 .
Including a standard loop factor of 2π:

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. A formal
derivation within the DFD framework is given in
Ref. [15].
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.

3.1

kα =

α2
.
2π

(25)

We present this as a heuristic motivating specific powers of α. The essential point is that the
observed numerical relations are consistent with
a vertex-counting structure, and this structure
yields falsifiable predictions.

4

Universal Clock Prediction

α with k = α2 /(2π), every atomic
If KA = kα SA
α
clock has a predicted gravitational coupling:

MOND: Two vertices

For the MOND effect—the modification of gravitational dynamics at accelerations below a0 —we
consider a two-vertex process:

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

1. EM-bound matter couples to scalar field
√
( α)

2. Scalar field modifies gravitational response
√
( α)
√
Combined amplitude: 2 × α.
This gives:
Table 1: Predicted gravitational couplings KA =
√
α
2
−6
a0 = 2 α · a ⋆ ,
(24) kα SA assuming kα = α /(2π) = 8.5 × 10 . Valα
ues of SA from Refs. [9, 10, 11, 12].
where a⋆ = cH0 is the cosmological acceleration
scale.
The prediction is falsifiable: any clock comα − S α ) would
parison yielding KA − KB ̸= kα (SA
B
3.2 Clock response: Four vertices
exclude the universal α-coupling hypothesis.
The Cs/Sr channel has ∆S α = 2.77, among
For clock response to gravitational potential—
the largest available, amplifying any signal by
requiring coupling between atomic structure, scalar
nearly a factor of 50 compared to channels with
field, and gravitational potential—we consider a
∆S α ∼ 0.1.
four-vertex process:
3

5

Comparison with Existing Data6.1 Predicted signal

For kα = α2 /(2π), the expected annual amplitude
is:
pred
The three-laboratory Sr clock comparison [16]
|ySr
| = 3.9 × 10−15 .
(28)
found:
Over a six-month baseline spanning periheySr = (−1.9 ± 3.0) × 10−15 .
(26) lion:


νCs
2
∆
≈ 4 × 10−15 .
(29)
Our prediction for kα = α /(2π):
νSr
∆Φ
pred
ySr
= −∆S α · kα · 2
6.2 Expected significance
c
−6
−10
= −2.77 × 8.5 × 10 × 1.65 × 10
Modern optical clock comparisons achieve frac= −3.9 × 10−15 .
(27) tional uncertainties of ∼ 10−17 at one-day avThe predicted amplitude (−3.9 × 10−15 ) and eraging [18, 19]. Over a six-month campaign,
systematic-limited precision of ∼ 3 × 10−16 is
measured central value (−1.9 × 10−15 ) are:
achievable.
• Same sign (Sr/Cs smallest at perihelion)
If the predicted signal is present:

5.1

Blatt et al. (2008)

• Same order of magnitude
Significance =

• Consistent within 0.7σ

4 × 10−15
≈ 13σ.
3 × 10−16

(30)

This would constitute definitive detection or
The 2008 measurement could not detect this
signal due to large uncertainties, but the data are exclusion.
fully consistent with the prediction.

6.3
5.2

Timeline

Sign convention verification

Data collection is expected to conclude in early
2026,
with results potentially available by midWe explicitly verify the sign agreement. In the
2026. The prediction kα = α2 /(2π) is falsifiable
convention of Ref. [16]:
on this timescale.
• ySr < 0 means νSr /νCs is smallest at perihelion.

7

• Our framework predicts KCs > KSr because
α > Sα .
SCs
Sr
7.1

Discussion
Caveats

• At perihelion (∆Φ < 0), Cs frequency shifts We emphasize several limitations:
more than Sr, so Sr/Cs decreases.
1. The vertex-counting argument presented
The signs are consistent. This is a nontrivial
here is a heuristic. A complete derivacheck.
tion from the DFD Lagrangian is given in
Ref. [15].

6

Prediction for Near-Term Experiments

A six-month Sr–Si cavity comparison campaign is
underway at JILA [17]. If cross-referenced to Cs
standards, this dataset will cover approximately
50% of the annual solar potential cycle with precision far exceeding the 2008 measurements.

2. The 2008 measurement has large uncertainties. While consistent with our prediction,
it is also consistent with zero.
3. The factor of 2π in Eq. (2) arises from loop
integration in the formal derivation [15].

4

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 appear√
to the Hubble tension [6].
ance of α (two vertices) and α2 (four vertices),
connecting MOND phenomenology to atomic
5. Alternative explanations for a0 ≈ cH0 ex- clock physics through the fine-structure constant.
ist [20, 21], though none predict the factor The formal derivation within the DFD framework
√
of 2 α.
is given in Ref. [15].
The prediction kα = α2 /(2π) ≈ 8.5 × 10−6
7.2 If confirmed
will be tested at > 10σ precision by ongoing
If a future campaign measures kα consistent with optical clock campaigns. If confirmed, this would
establish a direct link between the fine-structure
α2 /(2π), the implications include:
constant and gravitational phenomenology—a
1. First detection of LPI violation. This connection uniquely predicted by DFD.
would be the first confirmed departure from
the Einstein Equivalence Principle.

Acknowledgments

2. α–gravity coupling. The fine-structure
We thank J. Ye and the JILA optical frequency
constant would be directly implicated in
metrology group for valuable discussions.
gravitational physics.
3. Parameter-free prediction. Both a0 and
kα would be determined by α and H0 alone.

References
[1] M. Milgrom, Astrophys. J. 270, 365 (1983).

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].

[2] S. S. McGaugh,
F. Lelli,
and
J. M. Schombert, Phys. Rev. Lett. 117,
201101 (2016).

[3] F. Lelli, S. S. McGaugh, J. M. Schombert,
and M. S. Pawlowski, Astrophys. J. 836, 152
2
(2017).
If measurements show kα inconsistent with α /(2π)
at high significance:
[4] R. H. Sanders and S. S. McGaugh, Annu.
Rev. Astron. Astrophys. 40, 263 (2002).
1. The universal α-coupling hypothesis would
be ruled out.
[5] E. Tiesinga et al., Rev. Mod. Phys. 93,
√
025010 (2021).
2. The a0 = 2 α cH0 relation would not extend to clock physics.
[6] L. Verde, T. Treu, and A. G. Riess, Nat.
Astron. 3, 891 (2019).
3. The numerical coincidence would remain

7.3

If excluded

unexplained.

8

[7] Planck Collaboration, Astron. Astrophys.
641, A6 (2020).

Conclusion

[8] C. M. Will, Living Rev. Relativ. 17, 4
(2014).

We have presented two numerical relations:
√
a0 = 2 α cH0 (within H0 uncertainty), (31)
kα =

α2
2π

(consistent with data at 1σ).

[9] V. A. Dzuba, V. V. Flambaum, and
J. K. Webb, Phys. Rev. A 59, 230 (1999).

(32) [10] V. V. Flambaum and A. F. Tedesco, Phys.
Rev. C 73, 055501 (2006).
5

[11] E. J. Angstmann, V. A. Dzuba, and
V. V. Flambaum, Phys. Rev. A 70, 014102
(2004).
[12] V. A. Dzuba and V. V. Flambaum, Hyperfine Interact. 236, 79 (2015).
[13] T. Damour and J. F. Donoghue, Phys. Rev.
D 82, 084033 (2010).
[14] T. Damour, Class. Quantum Grav. 29,
184001 (2012).
[15] G.
Alcock,
Zenodo
doi:10.5281/zenodo.XXXXXXX.

(2025),

[16] S. Blatt et al., Phys. Rev. Lett. 100, 140801
(2008).
[17] W. R. Milner et al., Phys. Rev. Lett. 123,
173201 (2019).
[18] T. Bothwell et al., Metrologia 56, 065004
(2019).
[19] S. M. Brewer et al., Phys. Rev. Lett. 123,
033201 (2019).
[20] M. Milgrom, Phys. Lett. A 253, 273 (1999).
[21] E. Verlinde, SciPost Phys. 2, 016 (2017).

6