J. Rogers, SE Ohio
and a Proposal for High-Precision Measurement of 1/G as a Pure
Inertial Constant
Abstract
The gravitational constant G does not describe a physical property of
the universe. It is a conversion factor — a metrological patch — that
exists because humans defined the unit of force (the Newton) in a way
that is misaligned with the natural geometric structure of mass-distance
interactions. This paper demonstrates that by redefining force to carry
units of kg²/m², G vanishes from the law of gravitation entirely and
reappears as k = 1/G inside Newton's Second Law, F = kma. In this
reframing, k is a pure inertial constant with no connection to gravity
as a phenomenon. We then propose a high-precision experimental design to
measure k directly through a clean inertial acceleration experiment
using laser interferometry — bypassing the torsion balance entirely and
achieving precision that Cavendish-style gravity experiments cannot
reach.
1. The Problem with the Newton
The SI unit of force — the Newton — is defined as:
1 N = 1 kg · m/s²
This definition was chosen for convenience. It makes Newton's Second
Law trivially true:
F = ma
Because force is defined as mass times acceleration, F = ma contains
no physical information whatsoever. It is a tautology. It is a statement
about unit definitions, not about nature.
This convenience, however, creates a serious problem. When Newton
wrote down the law of universal gravitation:
F = G · (m₁ m₂) / r²
the constant G had to be inserted to make the units balance. The left
side has units of kg·m/s². The right side, without G, has units of
kg²/m². G carries units of m³/(kg·s²) precisely to bridge this
mismatch.
G is not telling us something about gravity. G is telling us that our
unit system is incoherent relative to the natural geometry of mass
interactions.
2. Redefining Force to Kill G in Gravity Law
Define a new unit of force such that force carries units of kg²/m².
That is, force is defined as the natural product of mass-mass
interaction over distance squared:
F_new ≡ m₁ m₂ / r² [units: kg²/m²]
Under this definition, the law of gravitation becomes exactly:
F_new = m₁ m₂ / r²
G has disappeared. Not because we set G = 1 by fiat, but because the
force unit is now defined in the same geometric terms as the right-hand
side. There is no mismatch to correct.
This is not a new physics claim. No experiment is affected. All
predictions remain identical. We have changed nothing about the
universe. We have only chosen a unit origin that is coherent with the
natural geometry of mass interactions.
3. Where G Goes: The Inertial Constant k
Because we have redefined force, Newton's Second Law can no longer be
a tautology. F_new and ma do not have the same units:
• F_new has units of kg²/m²
• ma has units of kg·m/s²
To write a second law connecting force to motion, we must introduce a
constant k that carries the unit mismatch:
F_new = k · m · a
Dimensional analysis forces the value of k. Since F_new = k ·
F_SI, and F_SI = ma, we get:
k ≈ kg s²/m^3
k is not a new constant. It is G, relocated. Previously G sat inside
the gravity equation as a signal of metrological incoherence. Now k sits
inside the inertial equation for the same reason. The physics is
identical. What has changed is where the constant lives — and that
change has consequences for measurement.
4. Why This Matters for Measurement
G is the least precisely measured fundamental constant in physics.
After centuries of effort, it is known only to approximately 5
significant figures — far worse than constants like c (exact by
definition), h (exact by definition), or e (10 significant figures).
The reason is straightforward: the Cavendish torsion balance is
trying to detect an extraordinarily weak gravitational signal against a
background of seismic noise, thermal drift, and mechanical vibration.
The signal-to-noise ratio is brutal. Every attempt to improve precision
runs into the same physical limitations of the torsion balance
geometry.
In the reframed system, k is a pure inertial constant. It has
nothing to do with gravity as a phenomenon. Measuring k requires:
Acceleration measurement by laser interferometry can resolve
displacements at the picometer scale. This is not the limiting factor.
The question is whether we can realize a force in kg²/m² units with
sufficient precision to make the experiment meaningful — and the answer
is yes, as described in the following section.
5. Experimental Proposal: The Inertial Calibration
Experiment
The goal is to measure k = 1/G by applying a known force in kg²/m²
units to a known mass and measuring the resulting acceleration with
laser interferometry. The experiment is entirely non-gravitational in
character — gravity is used once, at the beginning, to define the unit
of force, and then plays no further role.
5.1 Step One: Define and Realize the Unit Force
The unit force in the new system is defined to be eactly by the kg^2 and m^2 in itse definition. The force is exactly the two masses squared divided by the meter measurements. No uncertainty in the gravity measurement. Just like in the past F = ma had no uncertainty.
F_unit = M² / r² [= 1 unit of force by
definition]
This is not a measurement. It is a definition. The numerical value of
F_unit in new units is exactly M²/r² by construction. The precision of
this step is limited only by the precision of M and r — both of which
are controlled by national metrology standards to better than 1 part in
10⁸.
The physical realization of this force does involve the gravitational
attraction between the spheres. But we are not measuring that
attraction. We are defining it to be our unit. The Cavendish apparatus
fails because it tries to measure an extremely weak signal. We avoid
that entirely by simply declaring the signal to be 1.
5.2 Step Two: Transfer the Force to the Inertial
Track
The gravitational attraction between the tungsten spheres is used to
calibrate a force transducer — a precision electrostatic actuator or a
cryogenic force balance — that can then apply the same magnitude of
force to a test mass in a completely separate, clean environment.
This separation is critical. The inertial measurement environment
is:
Seismically isolated (optical table on active dampers)
Temperature-controlled to millikelvin stability
In high vacuum (< 10⁻⁸ mbar) to eliminate air damping
Shielded from electromagnetic interference
The test mass m is suspended on a frictionless linear guide (magnetic
levitation or superconducting bearing) so that it is free to accelerate
along one axis without mechanical contact.
5.3 Step Three: Measure Acceleration by Laser
Interferometry
Apply the calibrated 1-unit force F_unit to the test mass m. The test
mass begins to accelerate. Track the displacement x(t) of the test mass
over time using a heterodyne laser interferometer locked to an
iodine-stabilized reference laser.
The interferometer resolves displacement to better than 1 pm over
measurement intervals of seconds. From the displacement-time record,
acceleration a is extracted by fitting to the kinematic relation:
x(t) = ½ a t²
The fit is performed over thousands of independent measurement runs.
Statistical averaging suppresses random noise by √N, where N is the
number of runs. Systematic errors — laser frequency drift, test mass
charging, residual gas pressure — are characterized and subtracted.
5.4 Step Four: Extract k
From the known force F_unit, the known mass m, and the measured
acceleration a, k is directly:
k = F_unit / (m · a)
Since F_unit = 1 by definition in the new unit system:
k = 1 / (m · a)
With m known to 1 part in 10⁸ and a measured to a precision limited
by the interferometer and averaging time, the achievable precision for k
— and therefore for 1/G — is expected to significantly exceed the
current 5-significant-figure limit on G.
6. Error Budget
The dominant error sources and their estimated contributions are as
follows:
Mass standard uncertainty: < 1 part in 10⁸ (traceable to BIPM
kilogram definition via Kibble balance)
Length standard uncertainty: < 1 part in 10⁹ (traceable to
iodine-stabilized laser)
Force transducer calibration: estimated 1-5 parts in 10⁷
(dominant term, improvable with cryogenic force balance)
Interferometer displacement noise: < 1 pm/√Hz (suppressed by
averaging)
Residual gas damping: < 1 part in 10⁸ at 10⁻⁸ mbar
Seismic noise: suppressed by active isolation to < 1 nm RMS at
1 Hz, negligible over measurement timescale
The experiment is not limited by quantum noise, thermal noise, or any
fundamental physical barrier at the target precision. It is limited by
engineering — specifically by the precision of the force transducer.
This is an improvable engineering problem, not a fundamental one.
7. What This Experiment Is — and Is Not
This experiment is not a gravity experiment. It does not measure the
strength of gravitational attraction. Gravity appears only in the single
act of defining the unit force by the geometry of two masses — and at
that step, we are not measuring anything. We are defining.
Everything after that is pure inertia. We are measuring how much a
known mass resists a known force. That is a metrological question about
the relationship between our mass scale, our length scale, and our time
scale. The answer is k = 1/G, but G as a concept plays no role in the
measurement itself.
This is why the precision is achievable. The torsion balance fails
because it is trying to detect gravity through noise. This experiment
detects inertia — which is not weak, not noisy, and not buried under
competing signals. It is the most fundamental mechanical property of
matter, and we have instruments precise enough to measure it.
8. Conclusion
G is a metrological artifact. It exists in the gravitational law
because humans defined force in units (kg·m/s²) that do not match the
natural geometric structure of mass interactions (kg²/m²). Redefining
force to carry units of kg²/m² eliminates G from the gravity law
entirely and moves it — as k = 1/G — into the inertial law F = kma.
In this location, k is measurable by a clean acceleration experiment
using laser interferometry. The experiment has no gravitational signal
to detect, no torsion fiber to stabilize, and no seismic noise problem.
It is limited only by the precision of force transducer engineering,
which is an improvable problem.
The result would be the most precise measurement of G ever achieved —
not by doing a better gravity experiment, but by recognizing that G was
never a gravity constant to begin with.
k = 1/G ≈ 10¹⁰ kg s² / m^3
