| Effect of Time on Ring Structure and Spin Giulio C. Cima |
According to the model discussed in “The Symmetrical Universe: a Cosmological Model” (Reference 1) mass and
speed of light are functions of time and the relations between their current values and those at any epoch of the
universe are:
mt = mo (t / to)^(2/5) (1A)
C = Co (to / t)^(1/5) (2A)
Where: mo is mass in grams at the current epoch
Co is the current speed of light of 2.998E10 [cm sec-1]
to is the current epoch of 4.37E17 [sec] (or 13.85 billion years)
t is any epoch of the universe
mt is mass at time t
C is the speed of light at time t
In the same reference as above, the current value of to has been defined as:
to = 1 / Ho = 4.37E17 [sec]
Where: Ho is the current value of the Hubble constant of 2.29E-18 [sec-1]
Since both mass and speed of light vary with time it is important to know how the structure of elementary particles
is affected by such variation.
As discussed in “The Rings Model of Elementary Particles” (Reference 2), particles are made up of single right-
handed (positive) and single left-handed (negative) helixes of unit charge; and of double right-handed and double
left-handed helixes of zero charge (Z helixes). The helixes (called time-helixes) are the resultants of the circular
motion of minimum quanta of energy rotating at the speed of light around a center at a distance equal to the
Planck length:
Pl = sqr (h G / C^3)
Where: h is the rationalized Planck constant
G is the universal gravitational constant
Using equation (2A) for C, the Planck length as function of time is:
Pl = Plo (t / to)^(3/10)
Plo = sqr (h G / Co^3)
The radius of time-helixes (the fundamental constituents of all matter) is proportional to
the age of the universe.
For instance:
Age of universe: 13.85 billion years
Time-helix radius: 1.616E-33 [cm]
Age of universe: 1.385 billion years
Time-helix radius: 8.1E-34 [cm]
Age of universe: 60 [sec]
Time-helix radius: 2.817E-38 [cm]
Single helixes are coupled to Z helixes of opposite handedness to form singlets of unit charge and null
handedness. A pair of opposite singlets forms a doublet of zero charge and null handedness. Doublets are strung
together positive end to negative end to form rings of zero charge.
The separation between singlets in a ring is assumed to be equal to:
sep = e^2 / (mt C^2)
Where: e is the electron charge of 4.8E-10 [esu]
mt is the rest mass of the singlet at time t
The separation between two adjacent singlets in a ring is also the chord of the arc subtended by the two singlets:
sep = 2 r sin (pi / s) = e^2 / (mt C^2)
Where: r is the radius of the ring
s is the number of singlets in the ring
From the equation above, the radius of a ring is:
r = {e^2 / (mt C^2)} / {2 sin (pi / s)}
Using equations (1A) for mt and (2A) for C:
mt C^2 = mo Co^2
Where: mo is the current rest mass of the singlet which is equal to the rest mass of the electron of 9.11E-28 [g]
And the radius of a ring of an elementary particle as a function of time is:
r = ro
Where: ro is the radius at the current epoch
The separation between two singlets in a ring is independent of time and, therefore, the
radius of a ring of an elementary particle is also independent of time.
Using again equations (1A) and (2A) with Einstein’s equation we have:
E = mt C^2 = mo Co^2 = Eo
Where: Eo is the rest energy of mass mo at the current epoch
E is the rest energy of mass mt at any epoch of the universe
This is a universal law of conservation of energy which gives only two alternatives: either mass and speed of light
are independent of time (which is the basis of all current theories), or mass and speed of light are dependent on
time and inversely proportional to each other (which is the basis of the theory presented in Reference 1).
Using the proton as an example:
Age of universe: 13.85 billion years
Proton mass: 1.673E-24 [gr]
Speed of light: 2.998E10 [cm sec-1]
Proton rest energy: 0.0015 [erg]
Age of universe: 1.385 billion years
Proton mass: 6.66E-25 [gr]
Speed of light: 4.75E10 [cm sec-1]
Proton rest energy: 0.0015 [erg]
Age of universe: 60 [sec]
Proton mass: 7.56E-31 [gr]
Speed of light: 4.46E13 [cm sec-1]
Proton rest energy: 0.0015 [erg]
Invariant ring geometry implies that there could not have been protons until the age of the universe was such that
its size was greater than the sum of the sizes of all protons.
For time-helixes geometry there will be a time when the radius of the time-helix is such that it may interfere with the
separation between singlets in rings thus causing their breakdown and probably their annihilation. This time,
however, is extremely remote.
In Reference 2 it is shown that each singlet in a ring is subjected to electrostatic forces from all other singlets in
the ring. The resultant Y component of the overall force is always zero while the resultant X component is always
positive (attractive). This attractive force is centripetal and generates a constrictive stress on the ring unless
counteracted by an opposing force which is provided by the rotation of the ring:
Fc = (Rmt V^2) / r = Rmt r W^2 = Fxtot
Where: Fc is the rotational centripetal force
Rmt is the relativistic mass of the singlet at time t
V is the rotation linear velocity of the singlet
r is the radius of the ring
W is the rotation angular velocity equal to V / r
From the equation above we have:
V = sqr (r Fxtot / Rmt)
W = sqr {Fxtot / (Rmt r)}
Because of the reciprocal dependency among number of singlets, centripetal force, ring velocity, and singlet
relativistic mass the ring rotational velocity can only be determined indirectly by using an iterative process that
would find the relativistic mass of a rotating ring that is equal to the rest mass of a target ring. The following
equations are required:
V^2 = r Fc / Rmt
Rmt = mt / R
R = sqr (1 – Vc)
Vc = (V / C)^2
As shown above, the radius of the ring is independent of time while the singlet rest mass mt at any epoch t of the
universe is given by equation (1A) and the speed of light C by equation (2A).
From the equations above we have:
(r Fc)^2 = (Vc mt)^2 C^4 + Vc (r Fc)^2
Setting: x = Vc
a = mt^2 C^4
b = (r Fc)^2
we have a quadratic of the form:
a x^2 + b x – b = 0
Solving it for the real root:
x = [ – b + sqr (b^2 + 4 a b)] / 2 a
V = C sqr (x)
Rmt = mt / sqr (1 – x)
The Table below displays the results for a ring of 218 doublets (which is the structure of the rings presented in
Reference 2) for three epochs of the universe: current epoch of 13.85 billion years, 1.385 billion years, and 60
second. There are two lines of information for each epoch: the first line is for a ring at rest; the second line is for a
ring with spin.
The singlets separations for a ring at rest and for a ring with spin are invariant, and the singlets energy for a ring at
rest and for a ring with spin is also invariant.
The ratio of spin velocity to the speed of light is invariant and equal to 0.702401. The ratio of singlet relativistic
mass to singlet rest mass is invariant and equal to 1.404925. The ratio of singlet relativistic energy to singlet rest
energy is invariant and equal to 1.404925. The ratio of singlets separation for a ring at rest to singlets separation
for a ring with spin is invariant and also equal to 1.404925.
speed of light are functions of time and the relations between their current values and those at any epoch of the
universe are:
mt = mo (t / to)^(2/5) (1A)
C = Co (to / t)^(1/5) (2A)
Where: mo is mass in grams at the current epoch
Co is the current speed of light of 2.998E10 [cm sec-1]
to is the current epoch of 4.37E17 [sec] (or 13.85 billion years)
t is any epoch of the universe
mt is mass at time t
C is the speed of light at time t
In the same reference as above, the current value of to has been defined as:
to = 1 / Ho = 4.37E17 [sec]
Where: Ho is the current value of the Hubble constant of 2.29E-18 [sec-1]
Since both mass and speed of light vary with time it is important to know how the structure of elementary particles
is affected by such variation.
As discussed in “The Rings Model of Elementary Particles” (Reference 2), particles are made up of single right-
handed (positive) and single left-handed (negative) helixes of unit charge; and of double right-handed and double
left-handed helixes of zero charge (Z helixes). The helixes (called time-helixes) are the resultants of the circular
motion of minimum quanta of energy rotating at the speed of light around a center at a distance equal to the
Planck length:
Pl = sqr (h G / C^3)
Where: h is the rationalized Planck constant
G is the universal gravitational constant
Using equation (2A) for C, the Planck length as function of time is:
Pl = Plo (t / to)^(3/10)
Plo = sqr (h G / Co^3)
The radius of time-helixes (the fundamental constituents of all matter) is proportional to
the age of the universe.
For instance:
Age of universe: 13.85 billion years
Time-helix radius: 1.616E-33 [cm]
Age of universe: 1.385 billion years
Time-helix radius: 8.1E-34 [cm]
Age of universe: 60 [sec]
Time-helix radius: 2.817E-38 [cm]
Single helixes are coupled to Z helixes of opposite handedness to form singlets of unit charge and null
handedness. A pair of opposite singlets forms a doublet of zero charge and null handedness. Doublets are strung
together positive end to negative end to form rings of zero charge.
The separation between singlets in a ring is assumed to be equal to:
sep = e^2 / (mt C^2)
Where: e is the electron charge of 4.8E-10 [esu]
mt is the rest mass of the singlet at time t
The separation between two adjacent singlets in a ring is also the chord of the arc subtended by the two singlets:
sep = 2 r sin (pi / s) = e^2 / (mt C^2)
Where: r is the radius of the ring
s is the number of singlets in the ring
From the equation above, the radius of a ring is:
r = {e^2 / (mt C^2)} / {2 sin (pi / s)}
Using equations (1A) for mt and (2A) for C:
mt C^2 = mo Co^2
Where: mo is the current rest mass of the singlet which is equal to the rest mass of the electron of 9.11E-28 [g]
And the radius of a ring of an elementary particle as a function of time is:
r = ro
Where: ro is the radius at the current epoch
The separation between two singlets in a ring is independent of time and, therefore, the
radius of a ring of an elementary particle is also independent of time.
Using again equations (1A) and (2A) with Einstein’s equation we have:
E = mt C^2 = mo Co^2 = Eo
Where: Eo is the rest energy of mass mo at the current epoch
E is the rest energy of mass mt at any epoch of the universe
This is a universal law of conservation of energy which gives only two alternatives: either mass and speed of light
are independent of time (which is the basis of all current theories), or mass and speed of light are dependent on
time and inversely proportional to each other (which is the basis of the theory presented in Reference 1).
Using the proton as an example:
Age of universe: 13.85 billion years
Proton mass: 1.673E-24 [gr]
Speed of light: 2.998E10 [cm sec-1]
Proton rest energy: 0.0015 [erg]
Age of universe: 1.385 billion years
Proton mass: 6.66E-25 [gr]
Speed of light: 4.75E10 [cm sec-1]
Proton rest energy: 0.0015 [erg]
Age of universe: 60 [sec]
Proton mass: 7.56E-31 [gr]
Speed of light: 4.46E13 [cm sec-1]
Proton rest energy: 0.0015 [erg]
Invariant ring geometry implies that there could not have been protons until the age of the universe was such that
its size was greater than the sum of the sizes of all protons.
For time-helixes geometry there will be a time when the radius of the time-helix is such that it may interfere with the
separation between singlets in rings thus causing their breakdown and probably their annihilation. This time,
however, is extremely remote.
In Reference 2 it is shown that each singlet in a ring is subjected to electrostatic forces from all other singlets in
the ring. The resultant Y component of the overall force is always zero while the resultant X component is always
positive (attractive). This attractive force is centripetal and generates a constrictive stress on the ring unless
counteracted by an opposing force which is provided by the rotation of the ring:
Fc = (Rmt V^2) / r = Rmt r W^2 = Fxtot
Where: Fc is the rotational centripetal force
Rmt is the relativistic mass of the singlet at time t
V is the rotation linear velocity of the singlet
r is the radius of the ring
W is the rotation angular velocity equal to V / r
From the equation above we have:
V = sqr (r Fxtot / Rmt)
W = sqr {Fxtot / (Rmt r)}
Because of the reciprocal dependency among number of singlets, centripetal force, ring velocity, and singlet
relativistic mass the ring rotational velocity can only be determined indirectly by using an iterative process that
would find the relativistic mass of a rotating ring that is equal to the rest mass of a target ring. The following
equations are required:
V^2 = r Fc / Rmt
Rmt = mt / R
R = sqr (1 – Vc)
Vc = (V / C)^2
As shown above, the radius of the ring is independent of time while the singlet rest mass mt at any epoch t of the
universe is given by equation (1A) and the speed of light C by equation (2A).
From the equations above we have:
(r Fc)^2 = (Vc mt)^2 C^4 + Vc (r Fc)^2
Setting: x = Vc
a = mt^2 C^4
b = (r Fc)^2
we have a quadratic of the form:
a x^2 + b x – b = 0
Solving it for the real root:
x = [ – b + sqr (b^2 + 4 a b)] / 2 a
V = C sqr (x)
Rmt = mt / sqr (1 – x)
The Table below displays the results for a ring of 218 doublets (which is the structure of the rings presented in
Reference 2) for three epochs of the universe: current epoch of 13.85 billion years, 1.385 billion years, and 60
second. There are two lines of information for each epoch: the first line is for a ring at rest; the second line is for a
ring with spin.
The singlets separations for a ring at rest and for a ring with spin are invariant, and the singlets energy for a ring at
rest and for a ring with spin is also invariant.
The ratio of spin velocity to the speed of light is invariant and equal to 0.702401. The ratio of singlet relativistic
mass to singlet rest mass is invariant and equal to 1.404925. The ratio of singlet relativistic energy to singlet rest
energy is invariant and equal to 1.404925. The ratio of singlets separation for a ring at rest to singlets separation
for a ring with spin is invariant and also equal to 1.404925.
As mentioned above, the theory presented in Reference 1 is based on the universal law of conservation of
energy where mass and speed of light are dependent on time and inversely proportional to each other. The
inference is not that elementary particles are created with time as the speed of light
decreases but that the mass of all existing particles increases with time as the speed of
light decreases. This means that the constituents of all elementary particles in the universe (which,
according to the Rings Model, are quanta of energy rotating at the speed of light in single and double time
helixes) are created at time zero with infinite small mass and infinite rotational velocity; and that their mass, not
number, increases with time as the rotational velocity decreases.
energy where mass and speed of light are dependent on time and inversely proportional to each other. The
inference is not that elementary particles are created with time as the speed of light
decreases but that the mass of all existing particles increases with time as the speed of
light decreases. This means that the constituents of all elementary particles in the universe (which,
according to the Rings Model, are quanta of energy rotating at the speed of light in single and double time
helixes) are created at time zero with infinite small mass and infinite rotational velocity; and that their mass, not
number, increases with time as the rotational velocity decreases.
| Part A: Singlets Separation Distance Independent of Spin (September 2007) |
The rings model as presented in Reference 2 is based on the condition that the distance separating adjacent
singlets in a ring is independent of the velocity of rotation of the ring. The effect of that condition on the structure
of a ring is examined in Part 1 where it is shown that the separation between two adjacent singlets in a ring, and
therefore the radius of the ring, is invariant and independent of time.
In “Effect of Charge Interaction on Baryons Rings Configuration and Stability” (Reference 3), the condition that
the separation distance is independent from velocity of rotation of a ring is removed with the following results.
The separation distance at time to and at time t is given by:
Rsep0 = {e^2 / (Rm0 Co^2)} (1B)
Rsept = {e^2 / (Rmt C^2)} (2B)
Where: Rsep0 is the separation at the current epoch to when the relativistic effect of spin velocity is
taken into account.
Rm0 is the relativistic mass of a singlet at the current epoch.
Co is the current speed of light of 2.998E10 [cm sec-1].
Rsept is the separation at any epoch t when the relativistic effect of spin velocity is taken
into account.
Rmt is the relativistic mass of a singlet at any epoch.
C is the speed of light at any epoch.
e is the invariant electron charge.
According to the model discussed in Reference 1 mass and speed of light are functions of time and the relations
between their current values and those at any epoch of the universe are:
mt = mo (t / to)^(2/5) (1A)
C = Co (to / t)^(1/5) (2A)
Where: mo is mass in grams at the current epoch
to is the current epoch of 4.37E17 [sec] (or 13.85 billion years)
t is any epoch of the universe
mt is mass at time t
Using equation (1A) for Rmt and (2A) for C:
Rmt = Rm0 (t / to)^(2/5)
C = Co (to / t)^(1/5)
And substituting in equation (2B) gives:
Rmt C^2 = Rm0 Co^2
Rsept = Rsep0
Even when the condition of independence of separation distance from spin velocity is removed, the
separation between two adjacent singlets is still independent of time and, therefore, the
radius of a ring is also independent of time.
In both Reference 2 and Reference 3 is shown that each singlet in a ring is subjected to electrostatic forces from
all other singlets in the ring. The resultant Y component of the overall force is always zero while the resultant X
component is always positive (attractive). This attractive force is centripetal and generates a constrictive stress
on the ring unless counteracted by an opposing force which is provided by the rotation of the ring:
Fc = (Rmt V^2) / r = Rmt r W^2 = Fxtot
Where: Fc is the rotational centripetal force
V is the rotation linear velocity of the singlet
r is the radius of the ring
W is the rotation angular velocity equal to V / r
From the equation above we have:
V = sqr (r Fxtot / Rmt)
W = sqr {Fxtot / (Rmt r)}
Because of the reciprocal dependency among number of singlets, centripetal force, ring velocity, and singlet
relativistic mass the ring rotational velocity can only be determined indirectly by using an iterative process that
would find the relativistic mass of a rotating ring that is equal to the rest mass of a target ring. The required
equations are identical to those shown in Part A.
The Table below displays the results for a ring of 169 doublets (which is the structure of the rings presented in
Reference 3) for three epochs of the universe: current epoch of 13.85 billion years, 1.385 billion years, and 60
second. There are two lines of information for each epoch: the first line is for a ring at rest; the second line is for
a ring with spin.
The singlets separations for a ring at rest and for a ring with spin are invariant, and the singlets energy for a ring
at rest and for a ring with spin is also invariant.
The ratio of spin velocity to the speed of light is invariant and equal to 0.832551. The ratio of singlet relativistic
mass to singlet rest mass is invariant and equal to 1.805221. The ratio of singlet relativistic energy to singlet rest
energy is invariant and equal to 1.805221. The ratio of singlets separation for a ring at rest to singlets separation
for a ring with spin is invariant and also equal to 1.805221.
singlets in a ring is independent of the velocity of rotation of the ring. The effect of that condition on the structure
of a ring is examined in Part 1 where it is shown that the separation between two adjacent singlets in a ring, and
therefore the radius of the ring, is invariant and independent of time.
In “Effect of Charge Interaction on Baryons Rings Configuration and Stability” (Reference 3), the condition that
the separation distance is independent from velocity of rotation of a ring is removed with the following results.
The separation distance at time to and at time t is given by:
Rsep0 = {e^2 / (Rm0 Co^2)} (1B)
Rsept = {e^2 / (Rmt C^2)} (2B)
Where: Rsep0 is the separation at the current epoch to when the relativistic effect of spin velocity is
taken into account.
Rm0 is the relativistic mass of a singlet at the current epoch.
Co is the current speed of light of 2.998E10 [cm sec-1].
Rsept is the separation at any epoch t when the relativistic effect of spin velocity is taken
into account.
Rmt is the relativistic mass of a singlet at any epoch.
C is the speed of light at any epoch.
e is the invariant electron charge.
According to the model discussed in Reference 1 mass and speed of light are functions of time and the relations
between their current values and those at any epoch of the universe are:
mt = mo (t / to)^(2/5) (1A)
C = Co (to / t)^(1/5) (2A)
Where: mo is mass in grams at the current epoch
to is the current epoch of 4.37E17 [sec] (or 13.85 billion years)
t is any epoch of the universe
mt is mass at time t
Using equation (1A) for Rmt and (2A) for C:
Rmt = Rm0 (t / to)^(2/5)
C = Co (to / t)^(1/5)
And substituting in equation (2B) gives:
Rmt C^2 = Rm0 Co^2
Rsept = Rsep0
Even when the condition of independence of separation distance from spin velocity is removed, the
separation between two adjacent singlets is still independent of time and, therefore, the
radius of a ring is also independent of time.
In both Reference 2 and Reference 3 is shown that each singlet in a ring is subjected to electrostatic forces from
all other singlets in the ring. The resultant Y component of the overall force is always zero while the resultant X
component is always positive (attractive). This attractive force is centripetal and generates a constrictive stress
on the ring unless counteracted by an opposing force which is provided by the rotation of the ring:
Fc = (Rmt V^2) / r = Rmt r W^2 = Fxtot
Where: Fc is the rotational centripetal force
V is the rotation linear velocity of the singlet
r is the radius of the ring
W is the rotation angular velocity equal to V / r
From the equation above we have:
V = sqr (r Fxtot / Rmt)
W = sqr {Fxtot / (Rmt r)}
Because of the reciprocal dependency among number of singlets, centripetal force, ring velocity, and singlet
relativistic mass the ring rotational velocity can only be determined indirectly by using an iterative process that
would find the relativistic mass of a rotating ring that is equal to the rest mass of a target ring. The required
equations are identical to those shown in Part A.
The Table below displays the results for a ring of 169 doublets (which is the structure of the rings presented in
Reference 3) for three epochs of the universe: current epoch of 13.85 billion years, 1.385 billion years, and 60
second. There are two lines of information for each epoch: the first line is for a ring at rest; the second line is for
a ring with spin.
The singlets separations for a ring at rest and for a ring with spin are invariant, and the singlets energy for a ring
at rest and for a ring with spin is also invariant.
The ratio of spin velocity to the speed of light is invariant and equal to 0.832551. The ratio of singlet relativistic
mass to singlet rest mass is invariant and equal to 1.805221. The ratio of singlet relativistic energy to singlet rest
energy is invariant and equal to 1.805221. The ratio of singlets separation for a ring at rest to singlets separation
for a ring with spin is invariant and also equal to 1.805221.
| Part B: Singlets Separation Distance Dependent on Spin (October 2009) |
