of 1 6
Theory for Inertia Predicting the Radius of a Proton that Satisfies Quantum Mechanics and
Classical Gravity!
By!
Ian Beardsey © 2025"
of 2 6
It is the goal here to provide a theory for inertia, that quality of a mass to resist change in
motion. We want the the theory to include not just the quantum mechanics constant for energy
over time Planck’s constant, but to include the universal constant of gravitation G, the
constant the speed of light from relativity, and the fine structure constant for theories of
electric fields so as to bring together the things that have been pitted against one another:
quantum mechanics, relativity, classical physics, electric fields, and gravitational fields. To do
this we will suggest a proton is a 3D cross-section of a 4D hypersphere held in place
countering its motion through time by a normal force that produces its inertia (measured in
mass in kilograms) much the same way we model a block on an inclined plain countered by
friction from the normal force to its motion. The following is the illustration of such a proton as
a cross-sectional bubble in space:!
!
To get the ball rolling, I had found a wave solution to the Earth/Moon/Sun system where the
Earth orbiting the Sun is like an electron orbiting a proton with a quantum mechanical solution.
I found this solution had a characteristic time of one second (Beardsley, A Theory for the
Proton and the Solar System with a Characteristic time of One Second, 2025). But, I found as
well, I could describe the proton as having a characteristic time of one second, and that this
yielded the radius of a proton very close to that obtained by modern experiments. So, it is now
before me to come up with a theory for the proton in terms of these characteristic times.!
The expressions for the characteristic times of 1-second for the proton that I found, were:!
1.
2.
h
c
α
(
1
6 α
2
4πh
G c
)
⋅
r
p
m
p
= 1secon d
ϕ ⋅
π r
p
α
4
G m
3
p
1
3
⋅
h
c
= 1secon d
of 3 6
Where is the golden ratio, is the radius of a proton, and is the mass of a proton. We find
these produce close to the most recent measurements of the radius of a proton, if you equate the left sides
of each, to one another:
3.
4.
To derive this equation for the radius of a proton from first principles I had set out to do it with the Planck
energy, , given by frequency of a particle, and from mass-energy equivalence, :
We take the rest energy of the mass of a proton :
The frequency of a proton is
We see at this point we have to set the expression equal to . So we need to come up with a theory for
inertia that explains it:
The radius of a proton is then
In order to prove our theory for the radius of a proton as incorporating , we will apply our model
outlined involving a normal force, to a 3d cross-section of a 4d hypersphere countering its direction
through time, t. We begin by writing equation 1 as:
5.
Where , the constant of gravitation measures the pliability of space, and the granularity of space, and c
the speed of propagation. measures the inertia endowed in a proton. We write equation 2 as:
r
p
m
p
r
p
= ϕ
h
cm
p
r
p
= 0.816632E − 15m
E = h f
E = m c
2
E = h f
m
p
E = m
p
c
2
f
p
=
m
p
c
2
h
ϕ
m
p
c
2
h
⋅
r
p
c
= ϕ =
m
p
c
h
r
p
m
p
r
p
= ϕ ⋅
h
c
r
p
= ϕ ⋅
h
cm
p
ϕ
F
n
m
p
=
1
6 α
2
4πh
G c
⋅
r
p
1secon d
G
h
m
p
of 4 6
6.
We now say that and that the normal force is
7.
This gives us:
8.
=
Since , we have
9.
This gives
10.
is the cross-sectional area of the proton countering the normal force, , against its motion through
time, this is measured by the constant of gravitation. It is to say that
11.
And, the coupling constant is
12.
Let us see if this is accurate:
1 =
ϕ
9
⋅
π r
p
α
4
G m
3
p
⋅
h
c(1secon d )
2
⋅
h
c
t
1
= 1secon d
F
n
=
h
ct
2
1
1 =
ϕ
9
⋅
π r
p
α
4
G m
3
p
⋅
h
c
⋅ F
n
π
9α
4
⋅
F
n
G
⋅
r
p
m
2
p
(
ϕ
h
cm
p
)
r
p
= ϕ
h
cm
p
1 =
π
9α
2
⋅
F
n
G
⋅
r
2
p
m
2
p
m
p
=
1
3α
2
⋅
π r
2
p
F
n
G
π r
2
p
F
n
G
m
p
∝
Ar eaCr ossSect i on Pr oton ⋅ F
n
G
C =
1
3α
2
F
n
=
h
ct
2
1
=
6.62607E − 34J ⋅ s
(299,792,458m /s)(1s
2
)
= 2.21022E − 42N
of 5 6
We used the experimental value of a proton . And we have demonstrated that our
model of a proton as a 3D cross-section of a 4D hypersphere countering the normal force against its
motion through time gives its inertia that can counter a force at right angles to its motion through time and
the normal force.
It is thought that the proton does not have an exact radius, but that it is a fuzzy cloud of subatomic
particles. As such depending on what is going on can determine its state, or effective radius. It could be
that the proton radius is as large as
Which it was nearly measured to be before 2010 in two separate experiments. Or as small as
Which is closer to current measurements, which have decreased by 4% since 2010, and could get smaller.
In which case the characteristic time, , could be as large as
Using 2/3 as a fibonacci approximation to . Or, it could be as small as
=0.995 seconds
Or perhaps more often it is in the area of:
But, what this tells us is that the unit of a second might be a natural constant. And, since the second comes
from dividing the Earth rotation period into 24 hours, and each hour into 60 minutes, and each minute
m
p
=
18769
3
⋅
π (2.21022E − 42N )
6.674E − 11N
m
2
kg
2
(0.833E − 15m) = 1.68E − 27kg
r
p
= 0.833E − 15m
r
p
=
2
3
⋅
h
cm
p
r
p
=
2
3
⋅
6.62607E − 34
(299,792,458)(1.67262E − 27)
= 0.88094E − 15m
r
p
= ϕ ⋅
h
cm
p
= 0.816632E − 15m
t
1
2
3
⋅
π r
p
α
4
G m
3
p
1
3
⋅
h
c
= 1.03351secon d s
ϕ
ϕ ⋅
π r
p
α
4
G m
3
p
1
3
⋅
h
c
= (0.618)
(352275361)π (0.833E − 15m)
(6.674E − 11)(1.67262E − 27kg)
3
⋅
1
3
⋅
6.62607E − 34
299792458
1
6 α
2
m
p
h 4π r
2
p
G c
= 1.004996352secon d s
of 6 6
into 60 seconds, which ultimately comes to us from the ancient Sumerians who first settled down from
hunting, wandering, and gathering and flaking stones into spearpoints to invent agriculture, writing, and
mathematics, that this might be related to the mechanics of our Solar System. We find if we take the
second as natural we have a wave mechanics solution to our Solar System with a characteristic time of
one second that is connected to the characteristic time of the proton, thus connecting macro scales (the
solar system) to micro scales (the atom). I presented such a theory in my paper Presentation: How a
Characteristic Time of One Second May Describe Physical and Biological Systems in General (Beardsley,
2025).