| The Symmetrical Universe: A Cosmological Model by Giulio. C. Cima (Copyrigth © 2004 G. C. Cima Revised 2009 All Rights Reserved) |
Index
1.0 The Cosmological Model
2.0 Mass, Speed of Light, Radius, and Density as Functions of Time
3.0 Time-Space Inversion
4.0 The Current Epoch
5.0 The Case for Equality and Invariance of Energy of Mass and Potential Energy
6.0 The Speed of Light as Function of Time
7.0 The Case for a Universal Gravity Constant
8.0 The Five Variables (M, C, r, d, and A) as Functions of Time
9.0 The Universal Gravitational Acceleration
10.0 Event Horizon, Compton Length, Planck Mass, and Planck Time
11.0 The Radiation Aspect
12.0 Conclusion
Appendix 1: List of Numbered Equations
Appendix 2: The Geometry
Appendix 3: The Age of the Universe
Appendix 4: Red Shift and Galactic Distances
Appendix 5: Avogadro Constant and the Universe
1.0 The Cosmological Model
2.0 Mass, Speed of Light, Radius, and Density as Functions of Time
3.0 Time-Space Inversion
4.0 The Current Epoch
5.0 The Case for Equality and Invariance of Energy of Mass and Potential Energy
6.0 The Speed of Light as Function of Time
7.0 The Case for a Universal Gravity Constant
8.0 The Five Variables (M, C, r, d, and A) as Functions of Time
9.0 The Universal Gravitational Acceleration
10.0 Event Horizon, Compton Length, Planck Mass, and Planck Time
11.0 The Radiation Aspect
12.0 Conclusion
Appendix 1: List of Numbered Equations
Appendix 2: The Geometry
Appendix 3: The Age of the Universe
Appendix 4: Red Shift and Galactic Distances
Appendix 5: Avogadro Constant and the Universe
Table 01 below shows the values of theta, volume A, and hypotenuse “h” at inversion time t:
TABLE 01: TIME-SPACE INVERSION
TABLE 01: TIME-SPACE INVERSION
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4.0 The Current Epoch
The equations for the current epoch are:
Age: to = 1 / Ho
Volume: Ao = (pi/3) ro^3 to
Mass: Mo = (pi/3) ro^3 to do
Radius: ro = (5/4) Co to
Density: do = (3/pi) / [(5/4)^2 G to^3]
For the current age of the universe to = 1 / Ho, see Appendix 3.
Using the equations above, the mass of the universe at the current epoch can also be defined as:
Mo = (5/4) Co^3 to / G (15)
And the values of constants and variables are:
Hubble constant: Ho = 2.29E-18 [sec-1]
Age of universe: to = 4.37E17 [sec] (or 13.85 byrs)
Speed of light: Co = 3E10 [cm sec-1]
Gravitational constant: Go = 6.67E-8 [cm3 g-1 sec-2]
Radius: ro = 1.64E28 [cm]
Volume: Ao = 2.02E102 [cm3 sec]
Density: do = 1.1E-46 [g cm-3 sec-1]
Mass: Mo = 2.22E56 [g]
Rest Energy of Mass: Em = 1.99E77 [g cm2 sec-2]
Potential Energy: Ep = 1.99E77 [g cm2 sec-2]
5.0 The Case for Equality and Invariance of Energy of Mass and Potential Energy
From (1) and (2), solving for M, we have:
Em^2 / Ep = C^4 r / G
Using (4):
r = (5/4) C t
Em^2 / Ep = (5/4) C^5 t / G
Using (12) for C:
Em^2 / Ep = (5/4) Co^5 to / G = 1.99E77 [g cm2 sec-2]
The ratio (Em^2 / Ep) is independent of time and therefore invariant.
The equality of rest energy of mass and potential energy is discussed in "Gravity and Space-Time”).
6.0 The Speed of Light as Function of Time
Equation (12) gives the relation between speed of light and time. At t = 0 the speed of light is mathematically
infinite, or undefined; and at t = infinity, or undefined, the speed of light is zero.
FIGURE 02:
The equations for the current epoch are:
Age: to = 1 / Ho
Volume: Ao = (pi/3) ro^3 to
Mass: Mo = (pi/3) ro^3 to do
Radius: ro = (5/4) Co to
Density: do = (3/pi) / [(5/4)^2 G to^3]
For the current age of the universe to = 1 / Ho, see Appendix 3.
Using the equations above, the mass of the universe at the current epoch can also be defined as:
Mo = (5/4) Co^3 to / G (15)
And the values of constants and variables are:
Hubble constant: Ho = 2.29E-18 [sec-1]
Age of universe: to = 4.37E17 [sec] (or 13.85 byrs)
Speed of light: Co = 3E10 [cm sec-1]
Gravitational constant: Go = 6.67E-8 [cm3 g-1 sec-2]
Radius: ro = 1.64E28 [cm]
Volume: Ao = 2.02E102 [cm3 sec]
Density: do = 1.1E-46 [g cm-3 sec-1]
Mass: Mo = 2.22E56 [g]
Rest Energy of Mass: Em = 1.99E77 [g cm2 sec-2]
Potential Energy: Ep = 1.99E77 [g cm2 sec-2]
5.0 The Case for Equality and Invariance of Energy of Mass and Potential Energy
From (1) and (2), solving for M, we have:
Em^2 / Ep = C^4 r / G
Using (4):
r = (5/4) C t
Em^2 / Ep = (5/4) C^5 t / G
Using (12) for C:
Em^2 / Ep = (5/4) Co^5 to / G = 1.99E77 [g cm2 sec-2]
The ratio (Em^2 / Ep) is independent of time and therefore invariant.
The equality of rest energy of mass and potential energy is discussed in "Gravity and Space-Time”).
6.0 The Speed of Light as Function of Time
Equation (12) gives the relation between speed of light and time. At t = 0 the speed of light is mathematically
infinite, or undefined; and at t = infinity, or undefined, the speed of light is zero.
FIGURE 02:
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This cosmology differs radically from all other cosmologies in that mass, and therefore the speed of light, is
time dependent:
(dM/dt) = (2/5) Mo to^(-2/5) t^(-3/5)
At the current epoch, t = to and
(dM/dt)o = (2/5) Mo / to = 2.03E38 [g sec-1]
This is the rate of mass increase in grams in the whole universe for each second of time.
(Author’s note dated 2007: In “Effect of Time on Ring Structure and Spin – Part A” it is shown that mass is not
created with time as the speed of light decreases but, instead, 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.)
In terms of number of protons, we have:
Mass of proton m = 1.673E-24 [g]
If we assume that all matter is 90% hydrogen and 10% helium, then the total number of protons is:
Po = Mo / [(.9 m) + (.1 2m)] = 1.2E80
This number compares reasonably well with that calculated by the mathematician G. H. Hardy: his number was
1E80.
time dependent:
(dM/dt) = (2/5) Mo to^(-2/5) t^(-3/5)
At the current epoch, t = to and
(dM/dt)o = (2/5) Mo / to = 2.03E38 [g sec-1]
This is the rate of mass increase in grams in the whole universe for each second of time.
(Author’s note dated 2007: In “Effect of Time on Ring Structure and Spin – Part A” it is shown that mass is not
created with time as the speed of light decreases but, instead, 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.)
In terms of number of protons, we have:
Mass of proton m = 1.673E-24 [g]
If we assume that all matter is 90% hydrogen and 10% helium, then the total number of protons is:
Po = Mo / [(.9 m) + (.1 2m)] = 1.2E80
This number compares reasonably well with that calculated by the mathematician G. H. Hardy: his number was
1E80.
10.0 Event Horizon, Compton Length, Planck Mass, and Planck Time
Event horizon: Rs = M G / C^2
Compton length: Lc = h / (M C)
Planck mass: Pm = (h C / G)^(1/2)
Planck length: Pl = (h G / C^3)^(1/2)
Planck time: Pt = Pl / C
Planck constant: h = 6.626E-27 [g cm2/sec]
The event horizon is the radial distance defined by the expansion velocity of the universe
equal to the speed of light. It is named “light event horizon” to distinguish it from the mass event
horizon (see the Author’s “Gravity and Space-Time”).
Compton length Lc is the distance below which quantum effects are dominant.
Planck mass Pm is the mass at which Rs = Lc.
Planck length Pl is the Compton length of a particle with a mass equal to the Planck mass.
Planck time is the Planck length divided by the speed of light.
In this cosmology the five parameters are all time dependent:
a) Planck mass, length, and time are time dependent because of the time dependency of the speed of
light:
C = Co (to / t)^(1/5) (12)
b) Light event horizon and Compton length are time dependent because of the time dependency of both
mass and speed of light:
M = Mo (t / to)^(2/5) (11)
C = Co (to / t)^(1/5) (12)
c) Light event horizon is equal to the radius of the universe at all times.
From (4):
Rs = r = (5/4) C t
In addition, from:
Rs = M G / C^2 = (5/4) C t
Solving for G, we get:
G = (5/4) C^3 t / M
Which is equation (18) discussed in Section 10.0.
Finally, there is a convergence time at which:
1) Light event horizon (or radius of the universe) and Compton length are equal;
2) Planck mass and the mass of the universe are equal; and
3) Planck time and the convergence time are equal.
This convergence time is 2.67E-104 [sec] after t = 0 as shown in the tables below.
The proof is as follows:
1) Time at which Rs = Lc:
Rs = M G / C^2
Lc = h / (M C)
Solving for M:
M = (h C / G)^(1/2)
Using (11) for M and (12) for C and simplifying:
Mo = [h Co to / (G t)]^(1/2)
Solving for t:
t = (h Co to / G) / Mo^2
Using (15) for Mo:
t = (4/5)^2 (h G) / (Co^5 to) (i)
2) Time at which Pm = M:
M = (h C / G)^(1/2)
Using (11) for M and (12) for C:
Mo = [h Co to / (G t)]^(1/2)
Solving for t:
t = (h Co to / G) / Mo^2
Using (15) for Mo:
t = (4/5)^2 (h G) / (Co^5 to) (ii)
3) Time at which Pt = t:
Pl = (h G / C^3)^(1/2)
Pt = (4/5) Pl / C = t
Substituting:
t = (4/5) (h G)^(1/2) / C^(5/2)
Using (12) for C and simplifying:
t = (4/5)^2 (h G) / (Co^5 to) (iii)
Equations (i), (ii), and (iii) are identical, which proves that there is a convergence time at which light event
horizon radius and Compton length are equal; Planck mass and the mass of the universe are equal; and
Planck time and the convergence time are equal.
TABLE 05: Radius (r) and Compton Length (Lc) for Age of Universe (t) in Seconds
Event horizon: Rs = M G / C^2
Compton length: Lc = h / (M C)
Planck mass: Pm = (h C / G)^(1/2)
Planck length: Pl = (h G / C^3)^(1/2)
Planck time: Pt = Pl / C
Planck constant: h = 6.626E-27 [g cm2/sec]
The event horizon is the radial distance defined by the expansion velocity of the universe
equal to the speed of light. It is named “light event horizon” to distinguish it from the mass event
horizon (see the Author’s “Gravity and Space-Time”).
Compton length Lc is the distance below which quantum effects are dominant.
Planck mass Pm is the mass at which Rs = Lc.
Planck length Pl is the Compton length of a particle with a mass equal to the Planck mass.
Planck time is the Planck length divided by the speed of light.
In this cosmology the five parameters are all time dependent:
a) Planck mass, length, and time are time dependent because of the time dependency of the speed of
light:
C = Co (to / t)^(1/5) (12)
b) Light event horizon and Compton length are time dependent because of the time dependency of both
mass and speed of light:
M = Mo (t / to)^(2/5) (11)
C = Co (to / t)^(1/5) (12)
c) Light event horizon is equal to the radius of the universe at all times.
From (4):
Rs = r = (5/4) C t
In addition, from:
Rs = M G / C^2 = (5/4) C t
Solving for G, we get:
G = (5/4) C^3 t / M
Which is equation (18) discussed in Section 10.0.
Finally, there is a convergence time at which:
1) Light event horizon (or radius of the universe) and Compton length are equal;
2) Planck mass and the mass of the universe are equal; and
3) Planck time and the convergence time are equal.
This convergence time is 2.67E-104 [sec] after t = 0 as shown in the tables below.
The proof is as follows:
1) Time at which Rs = Lc:
Rs = M G / C^2
Lc = h / (M C)
Solving for M:
M = (h C / G)^(1/2)
Using (11) for M and (12) for C and simplifying:
Mo = [h Co to / (G t)]^(1/2)
Solving for t:
t = (h Co to / G) / Mo^2
Using (15) for Mo:
t = (4/5)^2 (h G) / (Co^5 to) (i)
2) Time at which Pm = M:
M = (h C / G)^(1/2)
Using (11) for M and (12) for C:
Mo = [h Co to / (G t)]^(1/2)
Solving for t:
t = (h Co to / G) / Mo^2
Using (15) for Mo:
t = (4/5)^2 (h G) / (Co^5 to) (ii)
3) Time at which Pt = t:
Pl = (h G / C^3)^(1/2)
Pt = (4/5) Pl / C = t
Substituting:
t = (4/5) (h G)^(1/2) / C^(5/2)
Using (12) for C and simplifying:
t = (4/5)^2 (h G) / (Co^5 to) (iii)
Equations (i), (ii), and (iii) are identical, which proves that there is a convergence time at which light event
horizon radius and Compton length are equal; Planck mass and the mass of the universe are equal; and
Planck time and the convergence time are equal.
TABLE 05: Radius (r) and Compton Length (Lc) for Age of Universe (t) in Seconds
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Time at which r = Rs = Lc is 2.67E-104 [sec]
TABLE 06: Radius (r), Mass (M), Planck Mass (Pm), and Planck Time (Pt) for Age of Universe (t) in Seconds
TABLE 06: Radius (r), Mass (M), Planck Mass (Pm), and Planck Time (Pt) for Age of Universe (t) in Seconds
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Time at which Pm = M and Pt = t is 2.67E-104 [sec]
Before convergence it was a time of total unpredictability. The mass of the universe was smaller than Planck
mass; the age of the universe was smaller than Planck time; and the radius of the universe was smaller than
Compton length; but all was within the light event horizon. All events within this horizon were probabilistic
events. Anything could have happened: at the beginning there was total, absolute, unfettered freedom. And
then, at about 1E-104 seconds, everything changed. Mass got greater than Planck mass, age greater than
Planck time, and radius greater than Compton length, but all still within the event horizon. All events still are
and will always be within the event horizon, including light which is defining it.
11.0 The Radiation Aspect
This is a symmetrical universe, and symmetry is based on dualities: in our case, matter and radiation. Matter
can be defined as unidirectional energy moving at less than the speed of radiation (i.e. light); radiation is omni
directional energy (laser notwithstanding) moving at its own speed. Given the proper conditions, one can be
transformed into the other. Here we are not interested in local phenomena, but rather in the phenomenon
called universe and in the relation between matter and radiation as universal quantities. This relation is based
on the time dependence of mass and speed of light. And so we start with Planck’s wavelength distribution of
blackbody radiation which, after integration over all wavelengths, gives the density of radiation:
u = [8 pi^5 (k T)^4] / [15 (h C)^3] (i)
Where:
u radiation density [erg cm-3]
T temperature [K]
k Boltzmann constant 1.38E-16 [erg K-1)
h Planck constant 6.626E-27 [erg sec]
C speed of light [cm sec-1]
Equation (i) is commonly written as:
u = a T^4
where:
a = (8 pi^5 k^4) / [15 (h C)^3]
1) Relation between Time and Temperature
To define the relation between time and temperature we substitute equation (12) for C in (i) and simplify:
u = b T^4 t^(3/5)
Where t is time and b is:
b = (8 pi^5 k^4) / [15 (h Co)^3 to^(3/5)]
The energy of radiation in ergs is then obtained by multiplying the density of radiation u by the volume
V = (pi/3) r^3 (that is, volume A from equation (5) divided by t):
Eu = (pi/3) r^3 u
Using (10) for r and simplifying:
Eu = b T^4 t^3 (20)
Where b has been redefined as:
b = (8 pi^6 ro^3 k^4) / [45 (h Co to)^3] (20a)
From (20) we define the ratio between the radiation energy at any time to that of the current epoch as:
Eu / Euo = 1 = (T / To)^4 (t / to)^3
which is equal to 1 because of conservation of energy. This gives the relation between temperature and time:
T = To (to / t)^(3/4) (21)
where To is the current background radiation temperature of 2.7 [K].
2) The Relation between Time and Density of Radiation
To define the relation between time and density of radiation we substitute equation (12) for C and equation
(21) for T into (i) and simplify:
u = b1 t^(-12/5) (22)
where b1 is defined as:
b1 = [8 pi^5 to^(12/5) (k To)^4] / [15 (h Co)^3] (22a)
3) The Relation between Mass of Matter and Mass of Radiation
From equation (1)
M = Em / C^2
we define mass of radiation as:
Mu = Eu / C^2
from which:
Mu / M = Eu / Em (23)
From equation (20), the energy of radiation Eu for the current epoch (Euo) is equal to 1.84E72 [erg] and is
invariant due to conservation of energy, and so is for Em. Therefore, the ratio:
Mu / M = Eu / Em = 1.84E72 / 1.99E77 = 9.3E-6
is also invariant.
The equivalent mass of all radiation in the universe is approximately equal to one hundred
thousandth of the mass of the universe at any one time. This ratio is invariant.
The above is shown in Table 07 and 08 below: in Table 07 the age of the universe is shown as the log of 1E0
to 1E18 seconds; in Table 08 the age of the universe is at Planck time (1E-43), at time-space inversion (5.81E-
31), and at 1 second.
TABLE 07: TEMPERATURE, DENSITY, AND MASS OF RADIATION
Before convergence it was a time of total unpredictability. The mass of the universe was smaller than Planck
mass; the age of the universe was smaller than Planck time; and the radius of the universe was smaller than
Compton length; but all was within the light event horizon. All events within this horizon were probabilistic
events. Anything could have happened: at the beginning there was total, absolute, unfettered freedom. And
then, at about 1E-104 seconds, everything changed. Mass got greater than Planck mass, age greater than
Planck time, and radius greater than Compton length, but all still within the event horizon. All events still are
and will always be within the event horizon, including light which is defining it.
11.0 The Radiation Aspect
This is a symmetrical universe, and symmetry is based on dualities: in our case, matter and radiation. Matter
can be defined as unidirectional energy moving at less than the speed of radiation (i.e. light); radiation is omni
directional energy (laser notwithstanding) moving at its own speed. Given the proper conditions, one can be
transformed into the other. Here we are not interested in local phenomena, but rather in the phenomenon
called universe and in the relation between matter and radiation as universal quantities. This relation is based
on the time dependence of mass and speed of light. And so we start with Planck’s wavelength distribution of
blackbody radiation which, after integration over all wavelengths, gives the density of radiation:
u = [8 pi^5 (k T)^4] / [15 (h C)^3] (i)
Where:
u radiation density [erg cm-3]
T temperature [K]
k Boltzmann constant 1.38E-16 [erg K-1)
h Planck constant 6.626E-27 [erg sec]
C speed of light [cm sec-1]
Equation (i) is commonly written as:
u = a T^4
where:
a = (8 pi^5 k^4) / [15 (h C)^3]
1) Relation between Time and Temperature
To define the relation between time and temperature we substitute equation (12) for C in (i) and simplify:
u = b T^4 t^(3/5)
Where t is time and b is:
b = (8 pi^5 k^4) / [15 (h Co)^3 to^(3/5)]
The energy of radiation in ergs is then obtained by multiplying the density of radiation u by the volume
V = (pi/3) r^3 (that is, volume A from equation (5) divided by t):
Eu = (pi/3) r^3 u
Using (10) for r and simplifying:
Eu = b T^4 t^3 (20)
Where b has been redefined as:
b = (8 pi^6 ro^3 k^4) / [45 (h Co to)^3] (20a)
From (20) we define the ratio between the radiation energy at any time to that of the current epoch as:
Eu / Euo = 1 = (T / To)^4 (t / to)^3
which is equal to 1 because of conservation of energy. This gives the relation between temperature and time:
T = To (to / t)^(3/4) (21)
where To is the current background radiation temperature of 2.7 [K].
2) The Relation between Time and Density of Radiation
To define the relation between time and density of radiation we substitute equation (12) for C and equation
(21) for T into (i) and simplify:
u = b1 t^(-12/5) (22)
where b1 is defined as:
b1 = [8 pi^5 to^(12/5) (k To)^4] / [15 (h Co)^3] (22a)
3) The Relation between Mass of Matter and Mass of Radiation
From equation (1)
M = Em / C^2
we define mass of radiation as:
Mu = Eu / C^2
from which:
Mu / M = Eu / Em (23)
From equation (20), the energy of radiation Eu for the current epoch (Euo) is equal to 1.84E72 [erg] and is
invariant due to conservation of energy, and so is for Em. Therefore, the ratio:
Mu / M = Eu / Em = 1.84E72 / 1.99E77 = 9.3E-6
is also invariant.
The equivalent mass of all radiation in the universe is approximately equal to one hundred
thousandth of the mass of the universe at any one time. This ratio is invariant.
The above is shown in Table 07 and 08 below: in Table 07 the age of the universe is shown as the log of 1E0
to 1E18 seconds; in Table 08 the age of the universe is at Planck time (1E-43), at time-space inversion (5.81E-
31), and at 1 second.
TABLE 07: TEMPERATURE, DENSITY, AND MASS OF RADIATION
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TABLE 08: SAME AS TABLE 07 FOR TIME t [sec]
|
Column 2 shows the temperature in degrees Kelvin from equation (21)
Column 3 shows the density of energy of radiation in ergs by cubic centimeter from equation (22).
Column 4 shows the density of energy of mass in ergs by cubic centimeter from:
Em = (5/4) C^5 t / G
which is obtained from (1) after substituting M from (15).
Column 5 shows the mass of radiation Mu = Eu / C^2
Column 6 shows mass M = Em / C^2
4) Frequency and Wavelength of Radiation
The Planck distribution reaches a maximum at a wavelength of:
L = .2 (h C) / (k T)
Using frequency f = C / L
f = (5 k / h) T = (1.04E11) T
Table 09 and 10 below show the results for the same ages as in Table 07 and 08 above.
TABLE 09: TEMPERATURE, FREQUENCY, AND WAVELENGTH OF RADIATION
Column 3 shows the density of energy of radiation in ergs by cubic centimeter from equation (22).
Column 4 shows the density of energy of mass in ergs by cubic centimeter from:
Em = (5/4) C^5 t / G
which is obtained from (1) after substituting M from (15).
Column 5 shows the mass of radiation Mu = Eu / C^2
Column 6 shows mass M = Em / C^2
4) Frequency and Wavelength of Radiation
The Planck distribution reaches a maximum at a wavelength of:
L = .2 (h C) / (k T)
Using frequency f = C / L
f = (5 k / h) T = (1.04E11) T
Table 09 and 10 below show the results for the same ages as in Table 07 and 08 above.
TABLE 09: TEMPERATURE, FREQUENCY, AND WAVELENGTH OF RADIATION
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TABLE 10: SAME AS TABLE 09 FOR TIME t [sec]
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12.0 Conclusion
Matter is the defining element of the universe. Time and space are the two ancillary functions: time dominant
in the early stages of the universe before time-space inversion, and space dominant in the later stages. But
matter can only exist in a spatial and temporal environment. Matter without space or time is meaningless. What
this cosmology suggests is that in order for matter to exist it has to create the space and the time in which to
exist, and the creation function is the responsibility of radiation. This cosmology starts with a homogeneous
state of pure potentiality, with no matter, space, or time. At a time infinitely close to zero the first infinitesimally
small quantity of matter was formed in an infinitely small space-time. This initial condition was a time of total
unpredictability; and then, at an incredibly small time everything changed; but all events were, and still are
and will always be within the event horizon of the universe. According to the definition given in the theoretical
base for this cosmology (“Gravity and Space-Time”), mass event horizons have the property of energy
invariance, that is, energy cannot trespass the horizon in either direction. There is however a difference
between a local event horizon and a universal event horizon. A local horizon is defined by its mass while a
universal horizon is defined by the speed of light that creates the spatiality and temporality necessary for the
existence of its mass.
What this cosmology implies is that the universe started from an infinitesimally small seed or perturbation and
developed from that initial condition according to elementary rules applied repetitively. This is a chaotic
process by definition. Chaos implies unpredictability as long as the initial conditions are not known, even if
the rules are known. And so we will never be able to predict what this is all about. All we can reasonably do is
to approximate it by taking into consideration the limitations of our rational thinking and of our sensorial
faculties, aided by technology. Therefore, as all theories, this is an analogy theory.
Appendix 1: List of Numbered Equations
Em = M C^2 (1)
Ep = M^2 G / r (2)
Em = Ep (3)
C = dr/dt = (4/5) r / t (4)
A = (pi/3) r^3 t (5)
C^2 = M G / r (6)
(M / Mo) = (Co / C)^2 (7)
(M /Mo) = (r / ro)^(1/2) (8)
(C / Co) = (r / ro)^(-1/4) (9)
(r / ro) = (t / to)^(4/5) (10)
(M / Mo) = (t / to)^(2/5) (11)
(C / Co) = (to / t)^(1/5) (12)
(d / do) = (to / t)^3 (13)
d = (3/pi) / (5/4)^2 G t^3 (14)
Mo = (5/4) Co^3 to / G (15)
g = M G / r^2 (16)
g = (4/5) C / t (17)
G = (5/4) C^3 t / M (18)
(19) Deleted
Eu = b T^4 t^3 (20)
b = (8 pi^6 ro^3 K^4) / [45 (h Co to)^3] (20a)
T = To (to / t)^(3/4) (21)
u = b1 t^(-12/5) (22)
b1 = [8 pi^5 to^(12/5) (K To)^4] / [15 (h Co)^3] (22a)
Mu / M = Eu / Em (23)
Matter is the defining element of the universe. Time and space are the two ancillary functions: time dominant
in the early stages of the universe before time-space inversion, and space dominant in the later stages. But
matter can only exist in a spatial and temporal environment. Matter without space or time is meaningless. What
this cosmology suggests is that in order for matter to exist it has to create the space and the time in which to
exist, and the creation function is the responsibility of radiation. This cosmology starts with a homogeneous
state of pure potentiality, with no matter, space, or time. At a time infinitely close to zero the first infinitesimally
small quantity of matter was formed in an infinitely small space-time. This initial condition was a time of total
unpredictability; and then, at an incredibly small time everything changed; but all events were, and still are
and will always be within the event horizon of the universe. According to the definition given in the theoretical
base for this cosmology (“Gravity and Space-Time”), mass event horizons have the property of energy
invariance, that is, energy cannot trespass the horizon in either direction. There is however a difference
between a local event horizon and a universal event horizon. A local horizon is defined by its mass while a
universal horizon is defined by the speed of light that creates the spatiality and temporality necessary for the
existence of its mass.
What this cosmology implies is that the universe started from an infinitesimally small seed or perturbation and
developed from that initial condition according to elementary rules applied repetitively. This is a chaotic
process by definition. Chaos implies unpredictability as long as the initial conditions are not known, even if
the rules are known. And so we will never be able to predict what this is all about. All we can reasonably do is
to approximate it by taking into consideration the limitations of our rational thinking and of our sensorial
faculties, aided by technology. Therefore, as all theories, this is an analogy theory.
Appendix 1: List of Numbered Equations
Em = M C^2 (1)
Ep = M^2 G / r (2)
Em = Ep (3)
C = dr/dt = (4/5) r / t (4)
A = (pi/3) r^3 t (5)
C^2 = M G / r (6)
(M / Mo) = (Co / C)^2 (7)
(M /Mo) = (r / ro)^(1/2) (8)
(C / Co) = (r / ro)^(-1/4) (9)
(r / ro) = (t / to)^(4/5) (10)
(M / Mo) = (t / to)^(2/5) (11)
(C / Co) = (to / t)^(1/5) (12)
(d / do) = (to / t)^3 (13)
d = (3/pi) / (5/4)^2 G t^3 (14)
Mo = (5/4) Co^3 to / G (15)
g = M G / r^2 (16)
g = (4/5) C / t (17)
G = (5/4) C^3 t / M (18)
(19) Deleted
Eu = b T^4 t^3 (20)
b = (8 pi^6 ro^3 K^4) / [45 (h Co to)^3] (20a)
T = To (to / t)^(3/4) (21)
u = b1 t^(-12/5) (22)
b1 = [8 pi^5 to^(12/5) (K To)^4] / [15 (h Co)^3] (22a)
Mu / M = Eu / Em (23)
K = 4
T1 and T2 are transforms:
(+) transforms move an element from left to right, and from up to down
(–) transforms move an element from right to left, and from down to up
Integration/differentiation raises/lowers the element’s order.
(+) T2 = (+) T1 (+) K
(–) T2 = (–) K (–) T1
(+) T1 = (+) T2 (–) K
(–) T1 = (+) K (–) T2
The above geometry can be summarized by the following series:
n__________________________
0 2 pi 8 pi
1 2 pi r 8 pi r
2 2 pi r^2 / 2 8 pi r^2 / 2
3 2 pi r^3 / 6 8 pi r^3 / 6
4 2 pi r^4 / 24 8 pi r^4 / 24
.
.
n ( 2 pi r^n) / n! ( 8 pi r^n) / n!
For n = 3 in the 8 pi series, integration of volumes (8 pi r^3 / 6) from r = 0 to r gives the four-dimensional
integrated form (8 pi r^4 / 24).
The four-dimensional form can be envisioned as a three dimensional sphere with r = 0 at t = 0 growing to r = r
at t = t. Its “volume” would be:
A = (pi/3) r^3 t
This four-dimensional form, called the “time-cone”, is assumed to be the geometry of the
universe.
Appendix 3: The Age of the Universe
The age of the universe throughout this discussion has been assumed to be equal to 1/Ho, where Ho is the
Hubble constant with the most recent value of 2.29E-18 [sec-1]. This gives the age of the universe of 4.37E17
[sec], or about 13.85 billion years. The following is the mathematical procedure that was used to back up the
assumption.
We shall start by defining the relation between the density of matter and the radius of the universe as defined
by this cosmology. The relation is obtained from substitution in equations (10) and (13):
(r / ro) = (t / to)^(4/5) (10)
(d / do) = (to / t)^3 (13)
Giving:
(d / do) = (r / ro)^(-15/4) (i)
In an Euclidean universe this relation would have been: (d / do) = (r / ro)^(-3)
Using the standard definition for the ratio R = r / ro and setting ro = 1, equation (i) is then rewritten as:
d = do R^(-15/4) (ii)
From equation (6):
C^2 = M G / r (6)
Substituting for M = (pi/3) r^3 t d, and then for r and d:
C = [(pi/3) r^2 t d G]^(1/2)
C = [(pi/3) R^2 t do R^(-15/4) G]^(1/2)
Using equation (4) for t = (4/5) r / C = (4/5) R / C and simplifying:
C = [(pi/3) (4/5) do G]^(1/3) R^(-1/4) = dR / dt (iii)
or:
dt = [(pi/3) (4/5) do G]^(-1/3) R^(1/4) dR
Integration will give:
t = (4/5) [(pi/3) (4/5) do G]^(-1/3) R^(5/4)
and using (iii):
t = (4/5) R / C = (4/5) r / C (iv)
which is again equation (4).
If we redefine the Hubble equation as V = C = (4/5) H r instead of the standard definition of V = H r then
equation (iv) will give:
t = 1 / H
which is the value that has been used in this cosmology instead of t = 4 / 5 H which would have given the age
of the universe as 3.5E17 [sec] or about 11.1 billion years.
Appendix 4: Red Shift and Galactic Distances
An observation about the time dependence of the speed of light which is worthwhile to note at this time is
related to the calculation of galactic distances. The spectrographic red shift “z” of light emitted from a body,
excluding relativistic effects, is given by:
z = (lo – le) / le = V / Co
(where “l” stands for wavelength, “o” for observed, “e” for emitted, “V” for the velocity of the body, and “Co”
for the speed of light at the current epoch); and from Hubble the distance “d” to the body is:
d = V / H
(where H is the Hubble constant = 2.29E-18 [sec-1] at the current epoch)
However, if the speed of light is time dependent, then the original wavelength (shown below as “lc”) of the
light at the time of emission is not the same as its wavelength “le” at the current epoch:
le = Co / f
lc = C / f
Where C is given by equation (12).
If the frequency “f” of emitted light is independent of time, we then have:
lc / le = C / Co
This means that the higher the speed of light, the longer the wavelength of the light emitted from the body: that
is, the length of the wave is stretched by the higher velocity. When the speed of light decreases, the opposite
effect takes place and the wavelength shrinks.
However, what we are now using for comparison is the emitted wavelength “le” of a body that is the current
wavelength of that body because of the current speed of light. What we should be using instead is the emitted
wavelength “lc” of that body computed at the speed of light at the time of emission.
The table below shows the effect of such use of corrected wavelength.
TABLE 11: CORRECTED GALACTIC DISTANCES
T1 and T2 are transforms:
(+) transforms move an element from left to right, and from up to down
(–) transforms move an element from right to left, and from down to up
Integration/differentiation raises/lowers the element’s order.
(+) T2 = (+) T1 (+) K
(–) T2 = (–) K (–) T1
(+) T1 = (+) T2 (–) K
(–) T1 = (+) K (–) T2
The above geometry can be summarized by the following series:
n__________________________
0 2 pi 8 pi
1 2 pi r 8 pi r
2 2 pi r^2 / 2 8 pi r^2 / 2
3 2 pi r^3 / 6 8 pi r^3 / 6
4 2 pi r^4 / 24 8 pi r^4 / 24
.
.
n ( 2 pi r^n) / n! ( 8 pi r^n) / n!
For n = 3 in the 8 pi series, integration of volumes (8 pi r^3 / 6) from r = 0 to r gives the four-dimensional
integrated form (8 pi r^4 / 24).
The four-dimensional form can be envisioned as a three dimensional sphere with r = 0 at t = 0 growing to r = r
at t = t. Its “volume” would be:
A = (pi/3) r^3 t
This four-dimensional form, called the “time-cone”, is assumed to be the geometry of the
universe.
Appendix 3: The Age of the Universe
The age of the universe throughout this discussion has been assumed to be equal to 1/Ho, where Ho is the
Hubble constant with the most recent value of 2.29E-18 [sec-1]. This gives the age of the universe of 4.37E17
[sec], or about 13.85 billion years. The following is the mathematical procedure that was used to back up the
assumption.
We shall start by defining the relation between the density of matter and the radius of the universe as defined
by this cosmology. The relation is obtained from substitution in equations (10) and (13):
(r / ro) = (t / to)^(4/5) (10)
(d / do) = (to / t)^3 (13)
Giving:
(d / do) = (r / ro)^(-15/4) (i)
In an Euclidean universe this relation would have been: (d / do) = (r / ro)^(-3)
Using the standard definition for the ratio R = r / ro and setting ro = 1, equation (i) is then rewritten as:
d = do R^(-15/4) (ii)
From equation (6):
C^2 = M G / r (6)
Substituting for M = (pi/3) r^3 t d, and then for r and d:
C = [(pi/3) r^2 t d G]^(1/2)
C = [(pi/3) R^2 t do R^(-15/4) G]^(1/2)
Using equation (4) for t = (4/5) r / C = (4/5) R / C and simplifying:
C = [(pi/3) (4/5) do G]^(1/3) R^(-1/4) = dR / dt (iii)
or:
dt = [(pi/3) (4/5) do G]^(-1/3) R^(1/4) dR
Integration will give:
t = (4/5) [(pi/3) (4/5) do G]^(-1/3) R^(5/4)
and using (iii):
t = (4/5) R / C = (4/5) r / C (iv)
which is again equation (4).
If we redefine the Hubble equation as V = C = (4/5) H r instead of the standard definition of V = H r then
equation (iv) will give:
t = 1 / H
which is the value that has been used in this cosmology instead of t = 4 / 5 H which would have given the age
of the universe as 3.5E17 [sec] or about 11.1 billion years.
Appendix 4: Red Shift and Galactic Distances
An observation about the time dependence of the speed of light which is worthwhile to note at this time is
related to the calculation of galactic distances. The spectrographic red shift “z” of light emitted from a body,
excluding relativistic effects, is given by:
z = (lo – le) / le = V / Co
(where “l” stands for wavelength, “o” for observed, “e” for emitted, “V” for the velocity of the body, and “Co”
for the speed of light at the current epoch); and from Hubble the distance “d” to the body is:
d = V / H
(where H is the Hubble constant = 2.29E-18 [sec-1] at the current epoch)
However, if the speed of light is time dependent, then the original wavelength (shown below as “lc”) of the
light at the time of emission is not the same as its wavelength “le” at the current epoch:
le = Co / f
lc = C / f
Where C is given by equation (12).
If the frequency “f” of emitted light is independent of time, we then have:
lc / le = C / Co
This means that the higher the speed of light, the longer the wavelength of the light emitted from the body: that
is, the length of the wave is stretched by the higher velocity. When the speed of light decreases, the opposite
effect takes place and the wavelength shrinks.
However, what we are now using for comparison is the emitted wavelength “le” of a body that is the current
wavelength of that body because of the current speed of light. What we should be using instead is the emitted
wavelength “lc” of that body computed at the speed of light at the time of emission.
The table below shows the effect of such use of corrected wavelength.
TABLE 11: CORRECTED GALACTIC DISTANCES
|
(d): distance (l): wavelength (o): observed (e): emitted (c): corrected
The first column shows the distance “do” of the body in billion light years based on the current speed of light.
The second column shows the ratio of wavelength observed “lo” to wavelength emitted “le” based on the
current speed of light:
(lo /le) = (H d / Co) + 1
The third column shows the ratio of wavelength emitted “lc” for the speed of light at the time of emission to
wavelength emitted “le”, or:
lc / le = C / Co
The fourth column shows the ratio of observed wavelength “lo” to corrected wavelength “lc” (the ratio that
should be used to estimate distances):
lo / lc = (lo/ le) / (lc / le)
And the fifth column show the resultant corrected distance “dc” of the body in billion light years:
(lo / lc) – 1 = V / C = H dc / C
From which:
dc = [(lo / lc) – 1] C / H
There is another way of looking at the same problem, giving approximately the same results. Equation (12)
relates the speed of light to the age, or time “t”, of the universe:
C = Co (to / t)^(1/5) = dd / dt
(where “dd” is the distance differential and “dt” the time differential)
Integrating C with respect to time, from a time “t” to the current epoch “to”, we get a distance “d1”:
d1 = (5/4) Co to - (5/4) Co (to)^(1/5) (t)^(4/5) = (ro – r)
This is the integrated distance that the light would have traveled if it had started at time “t”, and it is greater
than the observed distance “do” because the aggregated speed of light C is greater than the constant Co. So
the light couldn’t have started at time “t”: it had to have started at a time later than “t”. To find that time we must
use an iterative method that increments “t” until “d1” is equal to the observed distance “do”, and then
calculate the corrected distance “dc” from that time. Table 13 below shows the results.
TABLE 12: CORRECTED GALACTIC DISTANCES
The first column shows the distance “do” of the body in billion light years based on the current speed of light.
The second column shows the ratio of wavelength observed “lo” to wavelength emitted “le” based on the
current speed of light:
(lo /le) = (H d / Co) + 1
The third column shows the ratio of wavelength emitted “lc” for the speed of light at the time of emission to
wavelength emitted “le”, or:
lc / le = C / Co
The fourth column shows the ratio of observed wavelength “lo” to corrected wavelength “lc” (the ratio that
should be used to estimate distances):
lo / lc = (lo/ le) / (lc / le)
And the fifth column show the resultant corrected distance “dc” of the body in billion light years:
(lo / lc) – 1 = V / C = H dc / C
From which:
dc = [(lo / lc) – 1] C / H
There is another way of looking at the same problem, giving approximately the same results. Equation (12)
relates the speed of light to the age, or time “t”, of the universe:
C = Co (to / t)^(1/5) = dd / dt
(where “dd” is the distance differential and “dt” the time differential)
Integrating C with respect to time, from a time “t” to the current epoch “to”, we get a distance “d1”:
d1 = (5/4) Co to - (5/4) Co (to)^(1/5) (t)^(4/5) = (ro – r)
This is the integrated distance that the light would have traveled if it had started at time “t”, and it is greater
than the observed distance “do” because the aggregated speed of light C is greater than the constant Co. So
the light couldn’t have started at time “t”: it had to have started at a time later than “t”. To find that time we must
use an iterative method that increments “t” until “d1” is equal to the observed distance “do”, and then
calculate the corrected distance “dc” from that time. Table 13 below shows the results.
TABLE 12: CORRECTED GALACTIC DISTANCES
|
(do): observed distance (dc): corrected distance
(C): speed of light at corrected distance
The corrected distances are about 13% to 0% less than those estimated using a constant speed of light.
If, however, we use the redefined Hubble equation shown in Appendix 3: d = (5/4) V / H instead of : d = V / H
then the corrected distances are even smaller than those shown in Tables 12 and 13 above, and range from
about 23% to 40% less than the distances based on the current speed of light, as show in the table below.
TABLE 13: CORRECTED GALACTIC DISTANCES USING REDEFINED HUBBLE EQUATION
(C): speed of light at corrected distance
The corrected distances are about 13% to 0% less than those estimated using a constant speed of light.
If, however, we use the redefined Hubble equation shown in Appendix 3: d = (5/4) V / H instead of : d = V / H
then the corrected distances are even smaller than those shown in Tables 12 and 13 above, and range from
about 23% to 40% less than the distances based on the current speed of light, as show in the table below.
TABLE 13: CORRECTED GALACTIC DISTANCES USING REDEFINED HUBBLE EQUATION
|
(d): distance (l): wavelength (o): observed (e): emitted (c): corrected
Appendix 5: Avogadro Constant and the Universe
Avogadro constant is the number of atoms or molecules in one mole of substance. It has a value of
6.0221367E23. If we assume that the Sun is an average star, then the number of such stars in the universe is:
Mu / Ms = 2.22E56 / 1.99E33 = 1.12E23
where Mu is the mass of the universe and Ms is the mass of the Sun.
If the mass of the average star were 5.4 times less (3.69E32) than that of the Sun, then the number of stars in
the universe would be equal to the Avogadro constant.
There is another way of looking at the same question. Noting that mass times gravity constant equals volume
acceleration:
[g] [cm3g-1sec-2] ----> [cm3sec-2]
then, if we take the second derivative of the volume V = (pi/3) r^3 (that is, time-cone volume A divided by t)
using equation (10) for r, and evaluate it at t = to, we have:
(d2V/dt2)o = (84/25) (pi/3) (ro^3) [to^(2/5)] = 8.13E49 [cm3sec-2]
and:
(d2V/dt2)o / (Ms G) = 6.11E23
which is again very close to the Avogadro constant.
This does not mean that stars are universal atoms dispersed in a universal mole. What it implies, however, is
that there is symmetry between the very small and the very large and that the concept of size
is relative.
==========================================================================================
Copyright © 2004 Giulio. C. Cima Revised 2009 All Rights Reserved
==========================================================================================
Back to Home Page
Appendix 5: Avogadro Constant and the Universe
Avogadro constant is the number of atoms or molecules in one mole of substance. It has a value of
6.0221367E23. If we assume that the Sun is an average star, then the number of such stars in the universe is:
Mu / Ms = 2.22E56 / 1.99E33 = 1.12E23
where Mu is the mass of the universe and Ms is the mass of the Sun.
If the mass of the average star were 5.4 times less (3.69E32) than that of the Sun, then the number of stars in
the universe would be equal to the Avogadro constant.
There is another way of looking at the same question. Noting that mass times gravity constant equals volume
acceleration:
[g] [cm3g-1sec-2] ----> [cm3sec-2]
then, if we take the second derivative of the volume V = (pi/3) r^3 (that is, time-cone volume A divided by t)
using equation (10) for r, and evaluate it at t = to, we have:
(d2V/dt2)o = (84/25) (pi/3) (ro^3) [to^(2/5)] = 8.13E49 [cm3sec-2]
and:
(d2V/dt2)o / (Ms G) = 6.11E23
which is again very close to the Avogadro constant.
This does not mean that stars are universal atoms dispersed in a universal mole. What it implies, however, is
that there is symmetry between the very small and the very large and that the concept of size
is relative.
==========================================================================================
Copyright © 2004 Giulio. C. Cima Revised 2009 All Rights Reserved
==========================================================================================
Back to Home Page
1.0 The Cosmological Model
There is no complex mathematics in this cosmology: the mathematics that is used can be understood by
anyone with a scientific background; and the descriptive part is also kept as simple and as clear as possible.
All values are expressed in the exponential or scientific format, and their units are in the CGS system
(centimeter, gram, second). Whenever used, the subscript “o” indicates current epoch or observed values.
The theoretical foundation of this cosmological model is presented in the Author’s “Gravity and Space-
Time”. The model is based on the following five equations:
Em = M C^2 (1)
Ep = M^2 G / r (2)
Em = Ep (3)
C = dr / dt = (4/5) r / t (4)
A = (pi/3) r^3 t (5)
From (1), (2), and (3):
C^2 = M G / r (6)
Where:
Em is the invariant rest energy of mass of the universe.
Ep is the invariant potential energy of the universe.
C is the speed of light.
A is the time-cone volume of the universe.
G is the gravitational constant equal to 6.67E-8 [cm3 g-1 sec-2].
M is the mass of the universe
r is the radius of the universe.
t is time or age of the universe.
Equation (1) Em = M C^2 relates energy to mass using the square of the speed of light C as the constant of
proportionality.
Equation (2) Ep =M^2 G / r relates potential energy to mass and to position using the gravitational constant
G as the constant of proportionality.
Equation (3) Em = Ep (see below)
Equation (4) C = dr / dt is the differential equation that relates speed to distance and time.
Equation (4) C = (4/5) r / t is discussed in Section 3.0.
Equation (5) A = (pi/3) r^3 t is the equation for the volume of the time-cone (See Appendix 2). Equation (5)
can be expressed as function of radius or as function of time using equation (4) as follows:
A = (pi/3) r^4 (4/5) / C
A = (pi/3) t^4 [(5/4) C]^3
In this model, the first two equations describe relations among quantities available in the universe as a
whole. Thus, Em is the total rest energy of mass available in the universe; Ep is the total potential energy
available; and M is the total mass of the universe.
The ratio between the universal rest energy of mass Em and the universal potential energy
Ep is equal to 1 and the magnitudes of the two energies are invariant (see the Author’s “Gravity
and Space-Time” and Paragraph 5.0).
There is no complex mathematics in this cosmology: the mathematics that is used can be understood by
anyone with a scientific background; and the descriptive part is also kept as simple and as clear as possible.
All values are expressed in the exponential or scientific format, and their units are in the CGS system
(centimeter, gram, second). Whenever used, the subscript “o” indicates current epoch or observed values.
The theoretical foundation of this cosmological model is presented in the Author’s “Gravity and Space-
Time”. The model is based on the following five equations:
Em = M C^2 (1)
Ep = M^2 G / r (2)
Em = Ep (3)
C = dr / dt = (4/5) r / t (4)
A = (pi/3) r^3 t (5)
From (1), (2), and (3):
C^2 = M G / r (6)
Where:
Em is the invariant rest energy of mass of the universe.
Ep is the invariant potential energy of the universe.
C is the speed of light.
A is the time-cone volume of the universe.
G is the gravitational constant equal to 6.67E-8 [cm3 g-1 sec-2].
M is the mass of the universe
r is the radius of the universe.
t is time or age of the universe.
Equation (1) Em = M C^2 relates energy to mass using the square of the speed of light C as the constant of
proportionality.
Equation (2) Ep =M^2 G / r relates potential energy to mass and to position using the gravitational constant
G as the constant of proportionality.
Equation (3) Em = Ep (see below)
Equation (4) C = dr / dt is the differential equation that relates speed to distance and time.
Equation (4) C = (4/5) r / t is discussed in Section 3.0.
Equation (5) A = (pi/3) r^3 t is the equation for the volume of the time-cone (See Appendix 2). Equation (5)
can be expressed as function of radius or as function of time using equation (4) as follows:
A = (pi/3) r^4 (4/5) / C
A = (pi/3) t^4 [(5/4) C]^3
In this model, the first two equations describe relations among quantities available in the universe as a
whole. Thus, Em is the total rest energy of mass available in the universe; Ep is the total potential energy
available; and M is the total mass of the universe.
The ratio between the universal rest energy of mass Em and the universal potential energy
Ep is equal to 1 and the magnitudes of the two energies are invariant (see the Author’s “Gravity
and Space-Time” and Paragraph 5.0).
2.0 Mass, Speed of Light, Radius, and Density as Functions of Time
The equations are:
From (1): (M / Mo) = (Co / C)^2 (7)
From (2): (M / Mo) = sqr (r / ro) (8)
From (7), (8): (C / Co) = (r / ro)^(-1/4) (9)
(r / ro) = (t / to)^(4/5) (10)
From (8), (10): (M / Mo) = (t / to)^(2/5) (11)
From (7), (11): (C / Co) = (to / t)^(1/5) (12)
(d / do) = (to / t)^3 (13)
All of the equations above are obtained directly from the indicated sources, with the exception of equations
(10) and (13) which are obtained as follows:
Equation (10): radius as function of time:
From (4) and (9):
C = dr/dt = Co (ro / r)^(1/4)
dt = (r / ro)^(1/4) dr / Co
t = (4/5) r^(5/4) / [Co ro^(1/4)]
r = [(5/4) Co ro^(1/4) t]^(4/5)
At this point we have to express Co in terms of ro and to; so we start with Co = ro / to, although we know
that this is incorrect because it assumes that the speed of light has been invariant from t = 0 to t = to. The
result is:
(r / ro) = [(5/4) t / to]^(4/5)
Which is incorrect, since at t = to, r should be equal to ro and not to (5/4)^(4/5) ro.
If we instead use Co = (4/5) ro / to then:
(r / ro) = (t / to)^(4/5) (10)
Therefore:
C = dr / dt = (4/5) r / t (4)
For a more rigorous method, see Appendix 3.
Equation (13): density of matter as function of time:
From (5) and (4):
M = (pi/3) r^3 t d
r = (5/4) C t
Substituting in (6):
C^2 = (pi/3) [(5/4) C]^2 t^3 d G
Solving for d:
d = (3/pi) / [(5/4)^2 G t^3] (14)
And:
(d / do) = (to / t)^3 (13)
The equations are:
From (1): (M / Mo) = (Co / C)^2 (7)
From (2): (M / Mo) = sqr (r / ro) (8)
From (7), (8): (C / Co) = (r / ro)^(-1/4) (9)
(r / ro) = (t / to)^(4/5) (10)
From (8), (10): (M / Mo) = (t / to)^(2/5) (11)
From (7), (11): (C / Co) = (to / t)^(1/5) (12)
(d / do) = (to / t)^3 (13)
All of the equations above are obtained directly from the indicated sources, with the exception of equations
(10) and (13) which are obtained as follows:
Equation (10): radius as function of time:
From (4) and (9):
C = dr/dt = Co (ro / r)^(1/4)
dt = (r / ro)^(1/4) dr / Co
t = (4/5) r^(5/4) / [Co ro^(1/4)]
r = [(5/4) Co ro^(1/4) t]^(4/5)
At this point we have to express Co in terms of ro and to; so we start with Co = ro / to, although we know
that this is incorrect because it assumes that the speed of light has been invariant from t = 0 to t = to. The
result is:
(r / ro) = [(5/4) t / to]^(4/5)
Which is incorrect, since at t = to, r should be equal to ro and not to (5/4)^(4/5) ro.
If we instead use Co = (4/5) ro / to then:
(r / ro) = (t / to)^(4/5) (10)
Therefore:
C = dr / dt = (4/5) r / t (4)
For a more rigorous method, see Appendix 3.
Equation (13): density of matter as function of time:
From (5) and (4):
M = (pi/3) r^3 t d
r = (5/4) C t
Substituting in (6):
C^2 = (pi/3) [(5/4) C]^2 t^3 d G
Solving for d:
d = (3/pi) / [(5/4)^2 G t^3] (14)
And:
(d / do) = (to / t)^3 (13)
3.0 Time-Space Inversion
We could consider the volume of the time-cone as being equal to a base volume V times its height t, or A =
V t = (pi/3) r^3 t. We could then plot the base volume V on the horizontal axis versus time t on the vertical
axis of a coordinate system and obtain a world line for the time-cone. We can furthermore define an angle
theta as equal to tan (theta) = t / V where V = (pi/3) r^3, and an hypotenuse h = sqr (t^2 + V^2). If we then
solve for theta using (12) for the speed of light as function of time C = Co (to /t)^(1/5), we obtain an inversion
of time-cone volume from 90 degrees to 0 degrees at a time between approximately 1E-33 to 1E-28
seconds, as shown below.
At theta = 45 degrees, tan (theta) = 1, or t = V = (pi/3) r^3
Using (4) and (12) we have:
1 / t = (pi/3)^(5/7) [(5/4) Co]^(15/7) to^(3/7)
t = 5.85E-31 [sec]
h = sqr (2) t = 8.27E-31
C = Co (to / t)^(1/5) = 1.13E20 [cm/sec]
What this means is that earlier than 5.85E-31 seconds the expansion of the universe is time dominated,
while after 5.85E-31 seconds the expansion is volume, or space, dominated. The time-cone world line goes
straight up parallel to the vertical axis from t = 0 to about t = 1E-33 seconds during which period time was
dominating all events, and then between 1E-33 and 1E-28 seconds takes a 90 degrees turn and becomes
almost parallel to the horizontal axis, and since then space (volume) has been dominating all events. There
are no factors affecting the time-space inversion other than the geometry of the time-cone itself and the
time dependency of the speed of light. The inversion happens at almost the same time as the inflation of
the current inflation theory and is consistent with that theory.
Figure 01 below shows the time-cone angle (theta) as function of time (in red) and the logarithm of the value
of “h” also as function of time (in blue). The inversion point is at theta = 45 degrees at t = 5.85E-31 seconds
where h = sqr (2) t = 8.27E-31
FIGURE 01:
We could consider the volume of the time-cone as being equal to a base volume V times its height t, or A =
V t = (pi/3) r^3 t. We could then plot the base volume V on the horizontal axis versus time t on the vertical
axis of a coordinate system and obtain a world line for the time-cone. We can furthermore define an angle
theta as equal to tan (theta) = t / V where V = (pi/3) r^3, and an hypotenuse h = sqr (t^2 + V^2). If we then
solve for theta using (12) for the speed of light as function of time C = Co (to /t)^(1/5), we obtain an inversion
of time-cone volume from 90 degrees to 0 degrees at a time between approximately 1E-33 to 1E-28
seconds, as shown below.
At theta = 45 degrees, tan (theta) = 1, or t = V = (pi/3) r^3
Using (4) and (12) we have:
1 / t = (pi/3)^(5/7) [(5/4) Co]^(15/7) to^(3/7)
t = 5.85E-31 [sec]
h = sqr (2) t = 8.27E-31
C = Co (to / t)^(1/5) = 1.13E20 [cm/sec]
What this means is that earlier than 5.85E-31 seconds the expansion of the universe is time dominated,
while after 5.85E-31 seconds the expansion is volume, or space, dominated. The time-cone world line goes
straight up parallel to the vertical axis from t = 0 to about t = 1E-33 seconds during which period time was
dominating all events, and then between 1E-33 and 1E-28 seconds takes a 90 degrees turn and becomes
almost parallel to the horizontal axis, and since then space (volume) has been dominating all events. There
are no factors affecting the time-space inversion other than the geometry of the time-cone itself and the
time dependency of the speed of light. The inversion happens at almost the same time as the inflation of
the current inflation theory and is consistent with that theory.
Figure 01 below shows the time-cone angle (theta) as function of time (in red) and the logarithm of the value
of “h” also as function of time (in blue). The inversion point is at theta = 45 degrees at t = 5.85E-31 seconds
where h = sqr (2) t = 8.27E-31
FIGURE 01:
7.0 The Case for a Universal Gravity Constant
Presented below is a proof for the invariance of G.
Let’s assume that G is time dependent. From (6):
G = C^2 r / M
Differentiating the four time-dependent variables G, C, r, and M with respect to time we have:
dG/dt = [M (2 C dC/dt r + C^2 dr/dt) – dM/dt ( C^2 r)] / M^2
Multiplying both sides by M^2 and dividing by (M C^2 r) we have:
dG/dt / G = 2 dC/dt / C + dr/dt / r – dM/dt / M (i)
Differentiating (10):
dr/dt = 4/5 ro to^(4/5) t^(-1/5)
dr/dt / r = 4 / (5 t)
Differentiating (11):
dM/dt = 2/5 Mo to^(-2/5) t^(-3/5)
dM/dt / M = 2 / (5 t)
Differentiating (12):
dC/dt = – 1/5 Co to^(1/5) t^(-6/5)
dC/dt / C = – 1/ (5 t)
Substituting into (i), we have:
dG/dt / G = – 2 / (5 t) + 4/ (5 t) – 2 / (5 t)
dG/dt / G = 0
The rate of change of the gravity parameter G is zero, which means that G is NOT time dependent and,
therefore, is an invariant universal constant.
Note that the rate of change of G (i.e. dG/dt) is zero because the rate of change of M (i.e. dM/dt) is NOT zero.
Had dM/dt been zero, G would have been time dependent, and dG/dt / G = 2 / (5 t).
Presented below is a proof for the invariance of G.
Let’s assume that G is time dependent. From (6):
G = C^2 r / M
Differentiating the four time-dependent variables G, C, r, and M with respect to time we have:
dG/dt = [M (2 C dC/dt r + C^2 dr/dt) – dM/dt ( C^2 r)] / M^2
Multiplying both sides by M^2 and dividing by (M C^2 r) we have:
dG/dt / G = 2 dC/dt / C + dr/dt / r – dM/dt / M (i)
Differentiating (10):
dr/dt = 4/5 ro to^(4/5) t^(-1/5)
dr/dt / r = 4 / (5 t)
Differentiating (11):
dM/dt = 2/5 Mo to^(-2/5) t^(-3/5)
dM/dt / M = 2 / (5 t)
Differentiating (12):
dC/dt = – 1/5 Co to^(1/5) t^(-6/5)
dC/dt / C = – 1/ (5 t)
Substituting into (i), we have:
dG/dt / G = – 2 / (5 t) + 4/ (5 t) – 2 / (5 t)
dG/dt / G = 0
The rate of change of the gravity parameter G is zero, which means that G is NOT time dependent and,
therefore, is an invariant universal constant.
Note that the rate of change of G (i.e. dG/dt) is zero because the rate of change of M (i.e. dM/dt) is NOT zero.
Had dM/dt been zero, G would have been time dependent, and dG/dt / G = 2 / (5 t).
8.0 The Five Variables (M, C, r, d, and A) as Functions of Time
Figure 03 below shows the behavior of the five main variables as ratios to their current epoch’s values.
The speed of light C (red), the mass M (black), and the radius r (blue) have their major changes in the first
two to four billion years of the life of the universe, after which they settle pretty much into an almost linear
behavior. That, however, is not the case for volume A (magenta) and density d (cyan) which are going
through extensive exponential changes as the universe grows older.
FIGURE 03:
Figure 03 below shows the behavior of the five main variables as ratios to their current epoch’s values.
The speed of light C (red), the mass M (black), and the radius r (blue) have their major changes in the first
two to four billion years of the life of the universe, after which they settle pretty much into an almost linear
behavior. That, however, is not the case for volume A (magenta) and density d (cyan) which are going
through extensive exponential changes as the universe grows older.
FIGURE 03:
Table 02 below gives numerical values for the five main variables for twice the current epoch; the
current epoch; half the current epoch; 1/10 the current epoch; 1/100 the current epoch; at 1 sec; at
time-space inversion; and at Planck time.
TABLE 02: THE FIVE VARIABLES AT SELECTED TIMES
current epoch; half the current epoch; 1/10 the current epoch; 1/100 the current epoch; at 1 sec; at
time-space inversion; and at Planck time.
TABLE 02: THE FIVE VARIABLES AT SELECTED TIMES
9.0 The Universal Gravitational Acceleration
The universal gravitational acceleration is given by:
g = M G / r^2 (16)
From (2): Ep = M^2 G / r = M g r
From (1), (3) and (4):
Ep = Em = M C^2 = M g r
g = C^2 / r = (4/5) C / t (17)
From (16) and (17) or from (4) and (6):
G = (5/4) C^3 t / M (18)
Table 03 below shows the values of:
the speed of light C = Co (to / t)^(1/5)
its deceleration dC/dt = (-1/5) Co to^(1/5) t^(-6/5)
the mass of the universe M = Mo (t / to)^(2/5)
its rate of change dM/dt = (2/5) Mo to^(-2/5) t^(-3/5)
its deceleration d2M/dt2 = (-6/25) Mo to^(-2/5) t^(-8/5)
the gravitational acceleration g = (4/5) C / t = -4 dC/dt
the gravity constant G = (5/4) C^3 t / M
for the age of the universe in billion years.
The universal gravitational acceleration is given by:
g = M G / r^2 (16)
From (2): Ep = M^2 G / r = M g r
From (1), (3) and (4):
Ep = Em = M C^2 = M g r
g = C^2 / r = (4/5) C / t (17)
From (16) and (17) or from (4) and (6):
G = (5/4) C^3 t / M (18)
Table 03 below shows the values of:
the speed of light C = Co (to / t)^(1/5)
its deceleration dC/dt = (-1/5) Co to^(1/5) t^(-6/5)
the mass of the universe M = Mo (t / to)^(2/5)
its rate of change dM/dt = (2/5) Mo to^(-2/5) t^(-3/5)
its deceleration d2M/dt2 = (-6/25) Mo to^(-2/5) t^(-8/5)
the gravitational acceleration g = (4/5) C / t = -4 dC/dt
the gravity constant G = (5/4) C^3 t / M
for the age of the universe in billion years.
TABLE 03: SPEED OF LIGHT DECELERATION, SPEED OF MASS INCREASE AND ITS DECELERATION,
AND UNIVERSAL GRAVITATIONAL ACCELERATION AT SELECTED AGES
Age [byrs] C [cm/sec] dC [cm/sec2] M [g] dM [g/sec] d2M [g/sec2] g [cm/sec2]
1 4.49E+10 -1.54E-07 9.89E+55 6.79E+38 -6.99E+21 6.16E-07
2 4.12E+10 -9.17E-08 1.18E+56 5.24E+38 -3.50E+21 3.67E-07
4 3.70E+10 -4.84E-08 1.45E+56 3.81E+38 -1.49E+21 1.94E-07
6 3.45E+10 -3.20E-08 1.67E+56 3.09E+38 -8.59E+20 1.28E-07
8 3.28E+10 -2.35E-08 1.85E+56 2.65E+38 -5.70E+20 9.41E-08
10 3.15E+10 -1.84E-08 2.01E+56 2.35E+38 -4.12E+20 7.37E-08
12 3.05E+10 -1.50E-08 2.15E+56 2.12E+38 -3.14E+20 6.01E-08
13 3.00E+10 -1.37E-08 2.21E+56 2.03E+38 -2.79E+20 5.50E-08
The Gravity Constant G is equal to 6.67E-08 [cm3/g sec2] at all times
The next table shows the same values at Planck time, at time-space inversion, and at 1 second:
TABLE 04: SAME AS TABLE 03 FOR TIME t [sec]
t [sec] C [cm/sec] dC [cm/sec2] M [g] dM [g/sec] d2M [g/sec2] g [cm/sec2]
1.00E-43 4.03E+22 -8.06E+64 1.23E+32 4.91E+74 -2.95E+117 3.22E+65
5.85E-31 1.13E+20 -3.85E+49 1.57E+37 1.07E+67 -1.10E+97 1.54E+50
1.00E+00 1.01E+14 -2.02E+13 1.94E+49 7.78E+48 -4.67E+48 8.10E+13
The Gravity Constant G is equal to 6.67E-08 [cm3/g sec2] at all times
The universal gravitational acceleration is directly proportional to the speed of light and inversely
proportional to time, resulting in a decreasing but positive acceleration equal to four times the speed of
light deceleration. The gravity constant G is directly proportional to the cube of the speed of light and to
time and inversely proportional to mass, in such a way as the balance among the three variables results
in the invariance of G.
Figure 04 below shows the universal gravitational acceleration and the gravitational constant G against
the age of the universe in billion years, up to 30 billion years (about twice the current epoch). It is
coincidental that the value of the acceleration is almost identical to the value of G for the current epoch.
They were identical when the age of the universe was about 11.8 billion years (see Appendix 3).
FIGURE 04:
AND UNIVERSAL GRAVITATIONAL ACCELERATION AT SELECTED AGES
Age [byrs] C [cm/sec] dC [cm/sec2] M [g] dM [g/sec] d2M [g/sec2] g [cm/sec2]
1 4.49E+10 -1.54E-07 9.89E+55 6.79E+38 -6.99E+21 6.16E-07
2 4.12E+10 -9.17E-08 1.18E+56 5.24E+38 -3.50E+21 3.67E-07
4 3.70E+10 -4.84E-08 1.45E+56 3.81E+38 -1.49E+21 1.94E-07
6 3.45E+10 -3.20E-08 1.67E+56 3.09E+38 -8.59E+20 1.28E-07
8 3.28E+10 -2.35E-08 1.85E+56 2.65E+38 -5.70E+20 9.41E-08
10 3.15E+10 -1.84E-08 2.01E+56 2.35E+38 -4.12E+20 7.37E-08
12 3.05E+10 -1.50E-08 2.15E+56 2.12E+38 -3.14E+20 6.01E-08
13 3.00E+10 -1.37E-08 2.21E+56 2.03E+38 -2.79E+20 5.50E-08
The Gravity Constant G is equal to 6.67E-08 [cm3/g sec2] at all times
The next table shows the same values at Planck time, at time-space inversion, and at 1 second:
TABLE 04: SAME AS TABLE 03 FOR TIME t [sec]
t [sec] C [cm/sec] dC [cm/sec2] M [g] dM [g/sec] d2M [g/sec2] g [cm/sec2]
1.00E-43 4.03E+22 -8.06E+64 1.23E+32 4.91E+74 -2.95E+117 3.22E+65
5.85E-31 1.13E+20 -3.85E+49 1.57E+37 1.07E+67 -1.10E+97 1.54E+50
1.00E+00 1.01E+14 -2.02E+13 1.94E+49 7.78E+48 -4.67E+48 8.10E+13
The Gravity Constant G is equal to 6.67E-08 [cm3/g sec2] at all times
The universal gravitational acceleration is directly proportional to the speed of light and inversely
proportional to time, resulting in a decreasing but positive acceleration equal to four times the speed of
light deceleration. The gravity constant G is directly proportional to the cube of the speed of light and to
time and inversely proportional to mass, in such a way as the balance among the three variables results
in the invariance of G.
Figure 04 below shows the universal gravitational acceleration and the gravitational constant G against
the age of the universe in billion years, up to 30 billion years (about twice the current epoch). It is
coincidental that the value of the acceleration is almost identical to the value of G for the current epoch.
They were identical when the age of the universe was about 11.8 billion years (see Appendix 3).
FIGURE 04:
Appendix 2: The Geometry
Let us consider on the left the formulas for the circumference of a circle, the area of a circle, and the volume of
a cone; and on the right the surface area of a sphere, and the volume of a sphere.
FIGURE 05:
Let us consider on the left the formulas for the circumference of a circle, the area of a circle, and the volume of
a cone; and on the right the surface area of a sphere, and the volume of a sphere.
FIGURE 05: