1
On the analysis of various equations concerning the modified White Hole. New
possible mathematical connections with the Ramanujan Recurring Numbers, the
DN Constant and some sectors of Number Theory and String Theory
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this paper, we analyze various equations concerning the modified White Hole. We
describe the new possible mathematical connections with the Ramanujan Recurring
Numbers, the DN Constant and some sectors of Number Theory and String Theory
1
M.Nardelli studied at Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II,
Largo S. Marcellino, 10 - 80138 Napoli, Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” -
Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via
Cintia (Fuorigrotta), 80126 Napoli, Italy
2
A. Nardelli studies at the Università degli Studi di Napoli Federico II - Dipartimento di Studi Umanistici –
Sezione Filosofia - scholar of Theoretical Philosophy
2
From
Modified White Hole Enthalpy Coupled to Quantum Bose-Einstein Condensate
at Extremely Low Entropy - Kubeka, A.S., Amani, A. and Lekala, M. (2023) -
Journal of Modern Physics , 14, 1587-1599.
I. Analysis of the enthalpy of the modified white hole
The enthalpy of the modified white hole is given by
. (1)
We analyze the number theoretic properties and the Ramanujan recurring numbers
properties of the enthalpy equation as follows:
Input
Exact result
Alternate forms
3
Expanded form
Alternate forms assuming a, b, and S are positive
Expanded logarithmic form
4
Alternate form assuming S>0, sqrt(S)/sqrt(π)>0, and S/π-(b sqrt(S))/sqrt(π)≠0
Derivative
Indefinite integral
1. Indefinite integral
From the above following indefinite integral result
5
we obtain:
-(2 π (-a + 1/2 a^2 b π^(3/2) sqrt(1/S) + a^2 π log(sqrt(S)/sqrt(π))))/(b sqrt(π) sqrt(S) -
S)
Input
Alternate forms
6
Alternate forms assuming a, b, and S are positive
Expanded logarithmic form
Alternate form assuming sqrt(π) b sqrt(S)-S≠0, S>0, and sqrt(S)/sqrt(π)>0
7
Derivative
Indefinite integral
From the indefinite integral result
we obtain:
-(a^2 π (-3 + a b π^(3/2) sqrt(1/S) + 2 a π log(sqrt(S)/sqrt(π))))/(3 (b sqrt(π) sqrt(S) -
S))
8
Input
Alternate forms
9
Expanded form
Alternate form assuming a, b, and S are positive
Expanded logarithmic form
Alternate form assuming sqrt(π) b sqrt(S)-S≠0, S>0, and sqrt(S)/sqrt(π)>0
10
Derivative
Indefinite integral
i.e.
11
2. Volume analysis
Because of supersymmetry of space at extremely low entropy, then it is therefore
possible to consider the vortices of the quantum vacuum schematized as cubes or
octahedrons loops. We also assume that the quantum Van der Waals fluid [4, 6] are
characterized by smooth spheres. In reality, the quantum vacuum will have n-
dimensional hyperspheres in which the compactified dimensions "roll up" and
octahedrons representing the "fluctuations", containing vibrating quantum Van der
Waals fluid particles.
a. Octahedron volume
From the indefinite integral result
for V = 1/3*√2*a^3 (octahedron volume) and V = (4/3*π*r^3) (sphere volume),
where r = (a/2), considering (1/3*√2*a^3) and (4/3*π*(a/2)^3) ,
for V = (1/3*√2*a^3), we obtain:
(-(πa^3 (π^(3/2) ab√(1/S)+2πa log(√S/√π)-4))/(4(3√π b√S-3S))) (1/3*√2*a^3)
Input
12
Exact result
Alternate forms
13
Expanded form
Alternate forms assuming a, b, and S are positive
Expanded logarithmic form
Alternate form assuming sqrt(π) b sqrt(S)-S≠0, S>0, and sqrt(S)/sqrt(π)>0
14
Derivative
Indefinite integral
We obtain the following indefinite integral result
i.e.
15
b. Sphere volume
From the indefinite integral result
For V = (4/3*π*(a/2)^3)
we obtain:
((-(πa^3 (π^(3/2) ab√(1/S)+2πa log(√S/√π)-4))/(4(3√π b√S-3S)))) (4/3*π*(a/2)^3)
Input
Exact result
16
Alternate forms
17
Expanded forms
Alternate forms assuming a, b, and S are positive
Expanded logarithmic form
Alternate form assuming sqrt(π) b sqrt(S)-S≠0, S>0, and sqrt(S)/sqrt(π)>0
18
Derivative
Indefinite integral
We obtain the following indefinite integral result
i.e.
19
c. Number theoretic properties of the volume
i. DN Constant (Del Gaudio-Nardelli Constant)
Now dividing the two indefinite integral results for the octahedron and the sphere
volumes respectively:
and
we obtain:
(-(a^7 π (-32 + 7 a b π^(3/2) sqrt(1/S) + 7 a π log(S/π)))/(336 sqrt(2) (3 b sqrt(π)
sqrt(S) - 3 S)))/(-(a^7 π^2 (-32 + 7 a b π^(3/2) sqrt(1/S) + 7 a π log(S/π)))/(4032 (b
sqrt(π) sqrt(S) - S)))
Input
20
Exact result
Expanded form
Expanded logarithmic form
Alternate form
0.9003163161571…. =
(DN Constant)
Alternate form
Property as a function
Parity
21
Indefinite integral
Global maximum
Global minimum
Limit
22
Alternative representations
23
Series representations
24
Definite integral over a sphere of radius R
Definite integral over a cube of edge length 2 L
With regard the definite integral result over a sphere of radius
and over a cube of edge length
dividing the two results, considering L = R = a :
we obtain:
25
((8√2 a^3)/3) 1/((16√2 a^3)/π)
Input
Result
Decimal approximation
0.523598775598….
Property
Series representations
26
Integral representations
From which, we easily obtain:
6(((8√2 a^3)/3) 1/((16√2 a^3)/π))
Input
Result
27
Decimal approximation
3.141592653…. = π (Ramanujan Recurring Number)
Property
Series representations
Integral representations
28
We obtain easily, also:
π(((8√2 a^3)/3) 1/((16√2 a^3)/π))
Input
Result
Decimal approximation
1.64493406684…≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Property
29
Series representations
Integral representations
30
From which, from the following formula
we obtain:
sqrt(1/( π(((8√2 a^3)/3) 1/((16√2 a^3)/π)))*(4/3))
Input
Exact result
Decimal approximation
0.9003163161571…. =
(DN Constant)
Property
31
Series representations
32
From the previous expression
and the following formula concerning the tetrahedron, octahedron and icosahedron
volumes and their ratios with the sphere volume
=
2^(-1/π) (5 (3 + sqrt(5)) π)^(1/(2 π))
1.6180085459… result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
we obtain:
(-(a^7 π (-32 + 7 a b π^(3/2) √(1/S) + 7 a π log(S/π)))/(336 √(2) (3 b √(π) √(S) - 3
S)))/(-(a^7 π^2 (-32 + 7 a b π^(3/2) √(1/S) + 7 a π log(S/π)))/(4032 (b √(π) √(S) - S)))
(2^(-1/π) (5 (3 + √(5)) π)^(1/(2 π)))
33
Input
Exact result
Expanded form
Alternate form
Expanded logarithmic form
Property as a function
Parity
34
Indefinite integral
Global maximum
Global minimum
35
Limit
Alternative representations
36
Series representations
37
38
Definite integral over a sphere of radius R
Definite integral over a cube of edge length 2 L
From the exact result
for S = 4π and b = 2 :
(3 2^(3/2 - 1/π) (5 (3 + sqrt(5)))^(1/(2 π)) π^(1/(2 π) - 1) (sqrt(π) 1/2 sqrt(4π) –
(4π)))/(3 sqrt(π) 1/2 sqrt(4π) - 3 (4π))
Input
39
Exact result
Decimal approximation
1.4567194935555…
Series representations
40
From which, after some calculations:
((3 2^(3/2 - 1/π) (5 (3 + sqrt(5)))^(1/(2 π)) π^(1/(2 π) - 1) (sqrt(π) 1/2 sqrt(4π) –
(4π)))/(3 sqrt(π) 1/2 sqrt(4π) - 3 (4π)))+MRB const
Input
Exact result
Decimal approximation
1.64457913601756…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
41
Alternate form
From
Modular equations and approximations to π – S. Ramanujan -
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
We have also:
We note that in this modular equation there is the Golden Ratio:
Thence, performing the calculation, we obtain:
[(1/2*(1+√5))(1/2*(3+√13))]^(1/4)*sqrt((sqrt(1/8*(9+√65))+sqrt(1/8*(1+√65)))
Input
42
Exact result
Decimal approximation
2.41587194618680936…
Alternate forms
Minimal polynomial
43
And, from the following Ramanujan’s formula for calculate π
we obtain:
24/(sqrt(65))*ln(((2^(1/4)*([(1/2*(1+√5))(1/2*(3+√13))]^(1/4)*sqrt((sqrt(1/8*(9+√65
))+sqrt(1/8*(1+√65))))))))
Input
Exact result
Decimal approximation
3.14159265361956473959…. ≈ π (Ramanujan Recurring Number)
Property
44
Thence, the following new interesting mathematical connection:
III. Analysis of the thermodynamic volume of the modified white hole
The thermodynamic volume of the modified white hole is given by
The analysis gives various number theoretic properties and the Ramanujan recurring
number properties of the thermodynamic volume. Let's consider the numerator of the
expression above:
(((-5ab(π/S)^(1/2)+2ab^2(π/S)+2a-b/((π*S)^(1/2))+(ab(π/S)^(1/2)-2a) ln(S/π)+2/π)))
Input
Alternate forms
45
Alternate form assuming a, b, and S are positive
Derivative
Indefinite integral
Dividing the alternate form
by
46
we obtain:
(a ((2 π b^2)/S - 5 sqrt(π) b sqrt(1/S) + 2) + a (sqrt(π) b sqrt(1/S) - 2) log(S/π) + (2 -
(sqrt(π) b)/sqrt(S))/π)/(((b(S/π)^(1/2)-S/π)))
Input
Exact result
Thus, we obtain the exact result:
.
1. Volume analysis
From the exact result
47
for V = 1/3*√2*a^3 (octahedron volume) and V = (4/3*π*r^3) (sphere volume),
where r = (a/2), considering (1/3*√2*a^3) and (4/3*π*(a/2)^3) , we obtain:
i. Octahedron volume
((a (2 - 5 b sqrt(π) sqrt(1/S) + (2 b^2 π)/S) + (2 - (b sqrt(π))/sqrt(S))/π + a (-2 + b
sqrt(π) sqrt(1/S)) log(S/π))/((b sqrt(S))/sqrt(π) - S/π))(1/3*√2*a^3)
Input
Exact result
The exact result is equal to:
i.e.
48
ii. Sphere volume:
((a (2 - 5 b sqrt(π) sqrt(1/S) + (2 b^2 π)/S) + (2 - (b sqrt(π))/sqrt(S))/π + a (-2 + b
sqrt(π) sqrt(1/S)) log(S/π))/((b sqrt(S))/sqrt(π) - S/π))(4/3*π*(a/2)^3)
Input
Exact result
The exact result is equal to:
i.e.
49
2. Number theoretic properties of the volume
By dividing the two exact results
/
we obtain:
((√2 a^3 (a((2πb^2)/S-5√π b√(1/S)+2)+a(√π b√(1/S)-2) log(S/π)+(2-(√π
b)/√S)/π))/3((b√S)/√π-S/π)) /((πa^3 (a((2πb^2)/S-5√π b√(1/S)+2)+a(√π b√(1/S)-2)
log(S/π)+(2-(√π b)/√S)/π))/6((b√S)/√π-S/π))
Input
50
Exact result
Decimal approximation
0.9003163161571…. =
(DN Constant)
Property
Expanded logarithmic form
51
Alternative representations
52
53
54
Series representations
55
With regard the decimal approximation
this is the
DN Constant, and also
it has property that it is a transcendental number. The
transcendental numbers have important properties in physics and in particular in
astrophysics, particle physics and cosmology because they allow us to reformulate and
resolve unresolved problems, and in our case, the geometry of quantum gravity of the
very early universe.
56
We obtain also:
2√2/(((√2 a^3 (a((2πb^2)/S-5√π b√(1/S)+2)+a(√π b√(1/S)-2) log(S/π)+(2-(√π
b)/√S)/π))/3((b√S)/√π-S/π)) /((πa^3 (a((2πb^2)/S-5√π b√(1/S)+2)+a(√π b√(1/S)-2)
log(S/π)+(2-(√π b)/√S)/π))/6((b√S)/√π-S/π)))
Input
Exact result
Decimal approximation
3.1415926535897…≈ π (Ramanujan Recurring Number)
Property
Expanded logarithmic form
57
Alternative representations
58
Series representations
59
Integral representations
60
Polar plots (figures that can be related to the closed strings)
61
From
Modular equations and approximations to π – S. Ramanujan -
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
From:
We note that in this modular equation there is the Golden Ratio:
we obtain:
(1/2*(1+√5))*(1/2*(3√5+√41))^(1/4)*[sqrt(1/8*(7+√41))+sqrt(1/8*(√41-1))]
Input
Result
Decimal approximation
5.478939713774177863….
62
Alternate forms
Expanded form
Minimal polynomial
And, from the following Ramanujan’s formula for calculate π
we obtain:
63
24/(sqrt(205))*ln(((2^(1/4)*(
(1/2*(1+√5))*(1/2*(3√5+√41))^(1/4)*[sqrt(1/8*(7+√41))+sqrt(1/8*(√41-1))]))))))
Input
Result
Decimal approximation
3.1415926535897932385…. ≈ π (Ramanujan Recurring Number)
Property
64
Alternate forms
Expanded logarithmic form
Alternative representations
65
Series representations
66
67
Integral representations
Thence, the following interesting new mathematical connection:
68
From the previous exact result
and from the following formula concerning the tetrahedron, octahedron and
icosahedron volumes and their ratios with the sphere volume
=
2^(-1/π) (5 (3 + sqrt(5)) π)^(1/(2 π))
we obtain:
(((a (2 - 5 b sqrt(π) sqrt(1/S) + (2 b^2 π)/S) + (2 - (b sqrt(π))/sqrt(S))/π + a (-2 + b
sqrt(π) sqrt(1/S)) log(S/π))/((b sqrt(S))/sqrt(π) - S/π))) ((2^(-1/π) (5 (3 + sqrt(5))
π)^(1/(2 π))))
For S = 4π and a = 1/2, b = 1/4 :
69
(((1/2 (2-5*1/4 sqrt(π) sqrt(1/(4π))+(2(1/4)^2 π)/(4π))+(2-(1/4
sqrt(π))/sqrt(4π))/π+1/2(-2+1/4sqrt(π) sqrt(1/(4π))) log((4π)/π))/((1/4
sqrt(4π))/sqrt(π)-(4π)/π)))((2^(-1/π)(5(3+sqrt(5)) π)^(1/(2π))))
Input
Exact result
Decimal approximation
-0.0001410316552756….
Alternate forms
70
Expanded forms
Alternative representations
71
72
Series representations
73
Integral representations
From which, we obtain:
1/4(-1/(((((1/2(2-5*1/4 √(π) sqrt(1/(4π))+(2(1/4)^2 π)/(4π))+(2-(1/4
√(π))/√(4π))/π+1/2(-2+1/4√(π) √(1/(4π))) ln((4π)/π))/((1/4 √(4π))/√(π)-(4π)/π)))((2^(-
1/π)(5(3+√(5)) π)^(1/(2π)))))))-47+4-Φ
74
Input
Exact result
Decimal approximation
1729.033641349248…
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve (1728 = 8
2
* 3
3
). The number 1728 is one less than the Hardy–Ramanujan number
1729 (taxicab number, as it can be expressed as the sum of two cubes in two different
ways (10
3
+ 9
3
= 12
3
+ 1
3
= 1729) and Ramanujan's recurring number)
75
Alternate forms
(1/27(1/4(-1/(((((1/2(2-5/4 √(π) √(1/(4π))+(2(1/4)^2 π)/(4π))+(2-
(1/4√(π))/√(4π))/π+1/2(-2+1/4√(π)√(1/(4π))) ln((4π)/π))/((1/4√(4π))/√(π)-
(4π)/π)))((2^(-1/π)(5(3+√(5)) π)^(1/(2π)))))))-43-√3))^2+1/π
Input
76
Exact result
Decimal approximation
4095.937279484…≈ 4096 = 64
2
, (Ramanujan Recurring Number) that multiplied by
2 give 8192, indeed:
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group. The vacuum energy and dilaton tadpole
to lowest non-trivial order for the open bosonic string. While the vacuum energy is
non-zero and independent of the gauge group, the dilaton tadpole is zero for a unique
choice of gauge group, SO(2
13
) i.e. SO(8192). (From: “Dilaton Tadpole for the Open
Bosonic String “ Michael R. Douglas and Benjamin Grinstein - September 2,1986)
(((1/4(-1/(((((1/2(2-5/4 √(π) √(1/(4π))+(2(1/4)^2 π)/(4π))+(2-(1/4√(π))/√(4π))/π+1/2(-
2+1/4√(π)√(1/(4π))) ln((4π)/π))/((1/4√(4π))/√(π)-(4π)/π)))((2^(-1/π)(5(3+√(5))
π)^(1/(2π)))))))-43-√3))+1)^1/15
Input
77
Exact result
Decimal approximation
1.64381013427….
Alternate forms
78
All 15
th
roots of -42-sqrt(3)+(7 2
1/π-3
(5 (3+sqrt(5)) π)
-1/(2 π)
)/(45/64+15/(8 π)-(15
log(4))/16)
79
Alternative representations
80
From the previous exact result
we obtain:
(-42 - sqrt(3) + (7 2^(1/π - 3) (5 (3 + sqrt(5)) π)^(-1/(2 π)))/(45/64 + 15/(8 π) - (15
log(4))/16))^(1/15)+(MRB const)^(1-1/(4π)+π)
Input
81
Exact result
Decimal approximation
1.644932929652249…≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Alternate forms
82
Expanded logarithmic form
83
IV. Analysis of the enthalpy energy density of the modified white hole
The enthalpy energy density of the modified white hole is given by the following
formula:
and dividing the previous exact result
by the previous alternative form
we obtain:
.
((2/(-S/π+b(S/π)^(1/2)) [1-2πa ln((S/π)^(1/2))-abπ(π/S)^(1/2)]))/(((a((2πb^2)/S-5√π
b√(1/S)+2)+a(√π b√(1/S)-2) log(S/π)+(2-(√π b)/√S)/π)/(b√(S/π)-S/π)))
84
Input
Exact result
Expanded form
Alternate forms assuming a, b, and S are positive
85
Alternate forms
86
87
Indefinite integral
Then the analysis gives the following number theoretic properties and the Ramanujan
recurring number properties of the enthalpy energy density. The exact result is:
,
and for , we obtain:
(2 (1 - a b π^(3/2) sqrt(1/(4π)) - 2 a π log(sqrt(4π)/sqrt(π))))/((2 - (b
sqrt(π))/sqrt(4π))/π + a(2 - 5 b sqrt(π) sqrt(1/(4π)) + (2 b^2 π)/(4π)) + a(-2 + b sqrt(π)
sqrt(1/(4π))) log((4π)/π))
88
Input
Exact result
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
89
Contour plot
Alternate forms
Expanded forms
90
Expanded logarithmic form
Alternate form assuming b≠4 and π a b
2
-5 π a b+π a b log(4)+4 π a-4 π a
log(4)-b≠-4
Root
Root for the variable b
91
Series expansion at a=0
Series expansion at a=∞
Derivative
Indefinite integral
92
Limit
From the following alternate form
i.e.
we obtain:
2π(2-abπ-4aπ log(2))/(4-b+2πa(1/2 (-4+b)(-1+b))+2πa(1/2 (-4+b)) log(4))
Input
Exact result
93
Alternate forms
Expanded form
Expanded logarithmic form
Root for the variable b
94
Indefinite integral
From the indefinite integral result
i.e., the indefinite integral for the previous alternative form
,
we obtain:
-(2 π (-2 a + 1/2 a^2 π (b + log(16))))/(4 - b + 2 π a(1/2 (-4 + b) (-1 + b)) + π a(1/2 (-4
+ b)) log(16))
Input
95
Result
Alternate forms
Expanded form
Alternate form assuming a and b are positive
96
Expanded logarithmic form
Alternate form assuming b≠4 and 2 π a b
2
-10 π a b+π a b log(16)+8 π a-4 π a
log(16)-2 b≠-8
Roots
Root for the variable b
97
Series expansion at a=0
Series expansion at a=∞
Derivative
98
Indefinite integral
Limit
Taking the limit of the previous alternative form, as , we get
,
for V = 1/3*√2*a^3 (octahedron volume) and V = (4/3*π*r^3) (sphere volume),
where r = (a/2), considering (1/3*√2*a^3) and (4/3*π*(a/2)^3) , we obtain:
i. Octahedron volume
(-(2 π (-2 a + 1/2 a^2 π (b + log(16))))/(4 - b + a (-4 + b) (-1 + b) π + 1/2 a (-4 + b) π
log(16)))(1/3*√2*a^3)
Input
99
Result
3D plot (figure that can be related to a D-brane/Instanton)
Contour plot
100
The key observation from the above figures, is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear apart. Continuing
further, we obtained the following properties, that the alternate forms are:
Alternate forms
Expanded forms
101
Alternate form assuming a and b are positive
Expanded logarithmic form
Alternate form assuming b≠4 and 2 π a b
2
-10 π a b+π a b log(16)+8 π a-4 π a
log(16)-2 b≠-8
Roots
Root for the variable b
102
Series expansion at a=0
Series expansion at a=∞
Derivative
103
Indefinite integral
Limit
For the octahedron volume we have
i.e.
,
104
ii. Sphere volume
(-(2 π (-2 a + 1/2 a^2 π (b + log(16))))/(4 - b + a (-4 + b) (-1 + b) π + 1/2 a (-4 + b) π
log(16)))(4/3*π*(a/2)^3)
Input
Result
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
105
Contour plot
The observation in the above Figures, is the same as in the previous Figures.
Continuing further, we obtained the following properties, that the alternate forms are:
Alternate forms
106
Expanded forms
Alternate form assuming a and b are positive
Expanded logarithmic form
Alternate form assuming b≠4 and 2 π a b
2
-10 π a b+π a b log(16)+8 π a-4 π a
log(16)-2 b≠-8
107
Roots
Root for the variable b
Series expansion at a=0
Series expansion at a=∞
108
Derivative
Indefinite integral
Limit
109
For the sphere volume we have
i.e.
,
With regard the ratio of the two expressions/results concerning the Octahedron
volume to the sphere volume and its number theoretic properties, we have that:
(-(2 sqrt(2) πa^3 (1/2 πa^2 (b+log(16))-2a))/(3 (πa (b-1)(b-4)+1/2 πa (b-4) log(16)-
b+4)))/(-(π^2 a^3 (1/2 πa^2 (b+log(16))-2a))/(3 (πa (b-1)(b-4)+1/2 πa (b-4) log(16)-
b+4)))
Input
Result
Decimal approximation
0.9003163161571…. =
(DN Constant)
110
Property
Reduced logarithmic form
Alternative representations
111
Series representations
112
where
. It has the
following series representations.
We obtain also:
2√2/((-(2 sqrt(2) πa^3 (1/2 πa^2 (b+log(16))-2a))/(3 (πa (b-1)(b-4)+1/2 πa (b-4)
log(16)-b+4)))/(-(π^2 a^3 (1/2 πa^2 (b+log(16))-2a))/(3 (πa (b-1)(b-4)+1/2 πa (b-4)
log(16)-b+4))))
Input
Result
Decimal approximation
3.141592653589…≈ π (Ramanujan Recurring Number)
Property
113
Reduced logarithmic form
Alternative representations
114
Series representations
Integral representations
115
From
Modular equations and approximations to π – S. Ramanujan -
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
We note that in this modular equation there is the Golden Ratio:
116
(1/2*(1+√5))*(6+√35)^(1/4)*(1/2*(7^(1/4)+sqrt(4+√7)))^(3/2)*[sqrt(1/8*(43+15√7+
(8+3√7) sqrt(10√7)))+sqrt(1/8*(35+15√7+(8+3√7) sqrt(10√7)))]
Input
Exact result
Decimal approximation
82.121716259777959592….
Minimal polynomial
117
And, from the following Ramanujan’s formula for calculate π
we obtain:
24/(sqrt(1225))*ln(((2^(1/4)*(
(1/2*(1+√5))*(6+√35)^(1/4)*(1/2*(7^(1/4)+sqrt(4+√7)))^(3/2)*[sqrt(1/8*(43+15√7+
(8+3√7) sqrt(10√7)))+sqrt(1/8*(35+15√7+(8+3√7) sqrt(10√7)))])))
Input
Exact result
Decimal approximation
3.14159265358979323…. ≈ π (Ramanujan Recurring Number)
118
Property
Thence, the following new interesting mathematical connection:
119
Analysis of the equation of state of the modified white hole
The equation of state of the modified white hole is given by
The analysis gives the following number theoretic properties and the Ramanujan
recurring number properties of the equations of state:
-(1-1/(4π))*1/(2πr^2)*[V/(2πr^2-r^2+br) (1-(a ln r)/2-ab/(4r))+(V-(2πr^2)/(1-1/(4π))
(r-r/(2π)+b/(4π)))P]+a/(8πr)-ab/(16πr)
Input
120
Exact result
The exact result is
and considering , we obtain to be, from
that is equal to
i.e.
121
We have already analyzed the thermodynamic volume of the modified white hole,
that is given by:
Considering the numerator of the expression above:
(((-5ab(π/S)^(1/2)+2ab^2(π/S)+2a-b/((π*S)^(1/2))+(ab(π/S)^(1/2)-2a) ln(S/π)+2/π)))
for S = 4π
,
From
the denominator is:
,
Thence:
122
-5ab√(π/(4π))+(2ab^2) π/(4π)+2a-b/√(π(4π))+(ab√(π/(4π))-2a) log(((4π)/π))+2/π
Input
Exact result
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
Geometric figure
123
Contour plot
From the above Figures, we note that the gravitational potential is almost zero as the
self-vacuum perturbations have not started to take effects resulting in the flat quantum
vacuum geometry.
Alternate forms
Expanded logarithmic form
124
Alternate form assuming True
Roots
Polynomial discriminant
Integer root
Root for the variable b
Derivative
125
Indefinite integral
From the indefinite integral result
we obtain:
((-4 + b) (-a + 1/2 a^2 π (-1 + b + log(4))))/(2 π)
Input
126
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
Geometric figure
Contour plot
127
From the plots, we also observe that the vacuum quantum geometry starts to be
unevent, meaning the seeds for the gravitational potential are starting to take effects
due to the self-perturbations starting to taking effect.
Alternate forms
Alternate form assuming a and b are positive
Expanded logarithmic form
Alternate form assuming True
128
Root
Roots
Polynomial discriminant
Root for the variable b
Derivative
Indefinite integral
129
Again, from the indefinite integral result
we obtain:
((-4 + b) (-a^2 + 1/3 a^3 π (-1 + b + log(4))))/(4 π)
Input
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
Geometric figure
130
Contour plot
From the plots, we also observe that the vacuum quantum geometry starts to be more
unevent that in the above Figures. That is, the gravitational potential of the quantum
vacuum geometry is growing, as a results of the growth of the vacuum self-
perturbations
Alternate forms
131
Alternate form assuming a and b are positive
Expanded logarithmic form
Alternate form assuming True
Root
Roots
Polynomial discriminant
132
Root for the variable b
Derivative
Indefinite integral
Local minimum
From the indefinite integral result
for V = 1/3*√2*a^3 (octahedron volume) and V = (4/3*π*r^3) (sphere volume),
where r = (a/2), considering (1/3*√2*a^3) and (4/3*π*(a/2)^3) , we obtain:
133
i. Octahedron volume
((((-4 + b) (-a^3 + 1/4 a^4 π (-1 + b + log(4))))/(12 π))) (1/3*√2*a^3)
Input
Result
134
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
Geometric figure
135
Contour plot
From the above Figures, we now observe that the vacuum gravitational potential has
grown exponentially and infinitely high as a result of the exponentially grown and
infinitely high growth vacuum self-perturbations near i.e. the energy density
of the universe at the big bang.
Alternate forms
136
Expanded form
Expanded logarithmic form
Alternate form assuming True
Root
Roots
Polynomial discriminant
137
Root for the variable b
Derivative
Indefinite integral
Local minimum
For the octahedron we have
i.e.
138
we obtain:
ii. Sphere volume
((((-4 + b) (-a^3 + 1/4 a^4 π (-1 + b + log(4))))/(12 π))) (4/3*π*(a/2)^3)
Input
Result
139
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
Geometric figure
140
Contour plot
As in the previous Figures in case of the octahedron, here, we also observe that the
vacuum gravitational potential has grown exponentially and infinitely high as a result
of the exponentially grown and infinitely high growth vacuum self-perturbations near
i.e. the energy density of the universe at the big bang.
Alternate forms
141
Alternate form assuming a and b are positive
Expanded logarithmic form
Alternate form assuming True
Root
Roots
Polynomial discriminant
142
Root for the variable b
Derivative
Indefinite integral
Local minimum
For the sphere we have
i.e.
143
Dividing the two previous expressions
and
we obtain:
(((a^3 (-4 + b) (-a^3 + 1/4 a^4 π (-1 + b + log(4))))/(18 sqrt(2) π)))/((1/72 a^3 (-4 + b)
(-a^3 + 1/4 a^4 π (-1 + b + log(4)))))
Input
Result
Decimal approximation
0.9003163161571…. =
(DN Constant)
144
Property
Reduced logarithmic form
Alternative representations
145
Series representations
146
With regard the decimal approximation
, it
is the DN Constant and also a transcendental number). It also the reduced logarithmic
form.
We obtain also:
2√2/((((a^3 (-4 + b) (-a^3 + 1/4 a^4 π (-1 + b + log(4))))/(18 sqrt(2) π)))/((1/72 a^3 (-
4 + b) (-a^3 + 1/4 a^4 π (-1 + b + log(4))))))
Input
Result
Decimal approximation
3.14159265358979…≈ π (Ramanujan Recurring Number)
Property
Reduced logarithmic form
147
Alternative representations
Series representations
148
Integral representations
149
From
Modular equations and approximations to π – S. Ramanujan -
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
From the following approximation
we consider:
24/(sqrt(142)) ln[sqrt(1/4*(10+11√2))+sqrt(1/4*(10+7√2))]
Input
Result
Decimal approximation
3.141592653589…. ≈ π (Ramanujan Recurring Number)
150
Property
Alternate forms
Expanded logarithmic form
Reduced logarithmic form
151
Alternative representations
152
Series representations
153
Integral representations
Thence, the following new interesting mathematical connection:
154
On the application of the formulas of the volumes of an octahedron and a sphere
to quantum gravity
In this section we apply the number theoretic properties and the Ramanujan recurring
number properties to the quantum geometry of the white hole. With regard to a sphere
inscribed in an octahedron, we have the following formulas.
Figure: sphere inscribed in an octahedron
V
0
=
, V
s
=
where
We take the ratio between the two above formulas for the octahedron and sphere in
the above equation:
, (for ) ,
with the decimal approximation,
(which is the DN Constant, and a transcendental number)
The series representations are:
for (not (
and
))
155
for ( and )
from which we obtain
with the decimal approximation
(which is the trace of the instanton shape and
Ramanujan Recurring Number, and it is also a transcendental number).
The series representations are
,
156
,
with the integral representations
We note that, from the sum of the first nine numbers excluding 0, i.e.,
(these are the fundamental numbers, from which, through
infinite combinations, all the other numbers are obtained), we obtain the following
interesting formula:
157
where is the golden ratio,
is the constant. The exact result of the above
equation is then given by
,
with the decimal approximation
(which is a trace of the instanton shape and
Ramanujan Recurring Number)
The alternate forms are
,
,
,
From which the expanded forms are
,
,
158
and making input
,
then we get exact results
with the decimal approximation
(which is a Ramanujan Recurring Number)
The alternate form are:
,
,
from which the expanded forms are
,
159
.
All
roots of
are
(real, principal root)
(real root).
Furthermore, from the input:
= 8,
where value is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
The series representations are:
for (not (
and
)),
160
,
for ( and )
And by the input
The value 24 is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 24 "modes" corresponding to the
physical vibrations of a bosonic string representing a bosons. From the analysis, we
observe that the is no number theoretic connection with physical vibrations of
fermionic strings at extremally low entropy. This fact is confirmed by the fact that the
Higgs bosons at the moment of the big bang and infinitesimally shortly thereafter,
facilitated the creation of fermions (matter and antimatter particles). Thus, we note that
the ingredients for the formation of electromagnetic radiation from photons (a Boson),
and the formation of matter from the Higgs boson after the big bang, are intrinsic
properties of the vacuum energy in pre big bang.
The series representations are
for (not (
and
)),
161
for ( and ) ,
By the input
The number 4096 = 64
2
, is the Ramanujan Recurring Number, that when multiplied by
2 give 8192. The total amplitude vanishes for gauge group SO(8192) for bosonic string
SO(8192), while the vacuum energy is negative and independent of the gauge group.
The vacuum energy and dilaton tadpole to lowest non-trivial order for the open bosonic
string. While the vacuum energy is non-zero and independent of the gauge group, the
dilaton tadpole is zero for a unique choice of gauge group, SO(2
13
) i.e. SO(8192). This
could be the implications for a pre-big bang scenario where only self-perturbative
bosonic strings lived when the enthalpy was extremely low as discussed above. This
regime contains all the intrinsic properties of superstrings inherent in the bosonic
strings that as observed, would at the big bang give effect to the properties of matter
(fermions) as Higgs Boson. This number theoretic connection to the gauge group
SO(8192), gives a much more compelling relevance of the bosonic string theory
SO(8192), to quantum gravity and places this string theory where it should
appropriately be in the evolution of the universe from a quantum gravity perspective
rather than it be neglected because it doesn’t include fermionic strings to confirm to
post big-bang reality. The vanishing of the bosonic string’s amplitude could be
explained by the effect of extreme low entropy on the quantum vacuum geometry.
Thus, as the entropy increases infinitesimally as a result of the vacuum self-
perturbation then also is the amplitude of the vibrating bosonic strings from zero.
162
Is right to indicate that the “vanishing of the amplitude of the bosonic string could be
the results of string theory itself”, but here, we give a much more elaborate explanation
of what could be happening.
We further proceed and make the input
This result is very near to the mass of candidate glueball
scalar meson.
Furthermore, occurs in the algebraic formula for the j-invariant of an elliptic
curve
. The number is one less than the Hardy–Ramanujan
number (taxicab number, as it can be expressed as the sum of two cubes in two
different ways
and Ramanujan's recurring number).
Since bosons are made of gauge bosons and scalar bosons (meson), then this number
theoretic analysis perhaps confirm that the number , confirm the fact that both the
gauge and scalar bosons are actually different states of a single bosonic string, and that
these states are isomorphic or that the states vibrations are synchronised with the state
of the bosonic string. This also imply that each state lives inside a cubic or octahedron
as a spherical cloud, and that the total sum of these two states is the state of the bosonic
string. Taking the cross section of the bosonic string, we realise that it must be a
rectangular, or a two shaped octahedron. As the string vibrates in difference
frequencies, so is the two spherical cloud states inside the string. That is, the string
vibrations simply excites the gauge bosons i.e Photon, gluon, W and Z inside one
cube/octahedron, and the scalar boson i.e. Higgs inside the other cube/octahedron.
Furthermore, if we bring the picture of loop quantum gravity (LQG) with the property
of a discontinues quantum geometry, we can therefore, think of the graviton living on
the vertices of the rectangles or the octahedrons. This graviton then acts a glue binding
the bosonic strings lattice together forming a complete cross section of alternating
states of between the gauge bosons and scalar bosons. This arrangement of states then
gives a precise supersymmetric quantum picture of the vacuum geometry at low
entropy.
But the geometry further reveals very important fact, that since the vacuum geometry
is discontinues, then we observe that there is no relation whatsoever between the
quantum vibrational frequencies of the strings, and that of the vertices of the vacuum
geometry where the graviton lives.
163
Ashtekar et al., (2021) asserted that gravity is simply a manifestation of spacetime
geometry. Thus, the graviton cannot be a string boson, however, there is a duality
between gravity and strings. Also, gauge bosons have spin-1, while the graviton has
spin-2. Then lastly, because of the thermodynamic constraints we were able to arrive
at the results we have, now this bring us to this fundamental question; that string theory
and LQG theory are two intrinsic aspects of a complete quantum gravity theory we are
after? That is, without the other no complete and compelling quantum geometry can be
attained, as it is done here? This need to be investigated further.
The series representations are
,
,
.
We input
With a decimal approximation
,
164
result that is a very good approximation to the value of the golden ratio
1.618033988749… (which is a Ramanujan Recurring Number). The is the
repeating decimal.
The series representations
for (not (
and
)),
for ( and )
165
From inputting the transcendental number, we obtain, from the following formula :
,
with the decimal approximation
All
roots of
are
(real, principal roo),
(real
root). Thus, the series representations of Eq. (254) are
,
,
166
The integral representations are:
,
It is plausible to hypothesize that π and φ, in addition to being important mathematical
constants, are constants that also have a fundamental relevance in the various sectors
of Theoretical Physics and Cosmology
From
, we obtain:
With the decimal approximation
(which is the DN Constant, and a transcendental number).
All
roots of
are
(real, principal root), and
(real
root). The series representations are
167
for (not (
and
)),
for ( and )
Now, we have the following convex regular polyhedrons volumes:
Tetrahedron Volume
Octahedron Volume
Icosahedron Volume
Regarding the Sphere the volume is
168
With regard the sphere inscribed in a Tetrahedron, we have, for
, the following
ratio:
Thus:
(((√2 d^3)/12))*1/(4/3*π*(d/2)^3)
Input
Result
Decimal approximation
0.225079079….
Property
Series representations
169
With regard the sphere inscribed in a Octahedron, we have, for
, the following
ratio:
Thus:
170
((1/3*√2*d^3)/(4/3*π*(d/2)^3))
Input
Result
Decimal approximation
0.9003163161571…. =
(DN Constant)
Property
Series representations
171
With regard the sphere inscribed in a Icosahedron, we have, for
, the following
ratio:
Thus:
(5/12*(3+√5)*d^3)/(4/3*π*(d/2)^3)
Input
172
Result
Decimal approximation
4.166730504921….
Property
Expanded form
Alternate forms
Series representations
173
Now, we note that from the following interesting formula
we obtain:
(((((5/12*(3+√5)*d^3)/(4/3*π*(d/2)^3))*1/((1/3*√2*a^3)/(4/3*π*(a/2)^3)) *1/((((√2
d^3)/12))*1/(4/3*π*(d/2)^3)))))^(1/(2π))
174
Input
Exact result
Decimal approximation
1.6180085459…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
Alternate form
Series representations
175
176
Integral representation
Number connections to the Planck multipole spectrum frequency and to the
hypothetical Gluino mass
We note that, from the number 8, we obtain as follows:
,
, (True)
,
(True)
,
From Figure 16 below, we notice how from the numbers 8 and 2 we get 64, 1024, 4096
and 8192, and that 8 is the fundamental number. In fact 8
2
= 64, 8
3
= 512, 8
4
= 4096.
We define it "fundamental number", since 8 is a Fibonacci number, which by rule,
divided by the previous one, which is 5, gives 1.6 , a value that tends to the golden
ratio, as for all numbers in the Fibonacci sequence
Figure 16: “Golden” Range number scale
177
Finally we note how 8
2
= 64, multiplied by 27, to which we add 1, is equal to 1729, the
so-called "Hardy-Ramanujan number". Then taking the 15th root of 1729, we obtain a
value close to ζ(2) that 1.6438 ..., which, in turn, is included in the range of what we
call "golden numbers"
Furthermore for all the results very near to 1728 or 1729, adding 64 = 8
2
, one obtain
values about equal to 1792 or 1793. These are values almost equal to the Planck
multipole spectrum frequency (Black Body Radiation) 1792.35 and to the hypothetical
Gluino mass.
178
Appendix
From: A. Sagnotti – AstronomiAmo, 23.04.2020
In the above figure, it is said that: “why a given shape of the extra dimensions? Crucial,
it determines the predictions for α”.
We propose that whatever shape the compactified dimensions are, their geometry must
be based on the values of the golden ratio and ζ(2), (the latter connected to 1728 or
1729, whose fifteenth root provides an excellent approximation to the above mentioned
value) which are recurrent as solutions of the equations that we are going to develop.
It is important to specify that the initial conditions are always values belonging to a
fundamental chapter of the work of S. Ramanujan "Modular equations and
Approximations to Pi" (see references). These values are some multiples of 8 (64 and
4096), 276, which added to 4096, is equal to 4372, and finally e
π√22
179
We have, in certain cases, the following connections:
Fig. 1
Fig. 2
180
Fig. 3
Stringscape - a small part of the string-theory landscape showing the new de Sitter solution as a local
minimum of the energy (vertical axis). The global minimum occurs at the infinite size of the extra
dimensions on the extreme right of the figure.
Fig. 4
182
Where ζ(2+it) :
Input
Plots
Roots
183
Series expansion at t=0
Alternative representations
Series representations
184
Integral representations
Functional equations
With regard the Fig. 4 the points of arrival and departure on the right-hand side of the
picture are equally spaced and given by the following equation:
185
we obtain:
2Pi/(ln(2))
Input:
Exact result:
Decimal approximation:
9.06472028365….
Alternative representations:
186
Series representations:
Integral representations:
187
From which:
(2Pi/(ln(2)))*(1/12 π log(2))
Input:
Exact result:
Decimal approximation:
1.6449340668…. = ζ(2) =
188
From:
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
189
We note that, with regard 4372, we can to obtain the following results:
27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
Input
Result
Decimal approximation
1729.0526944….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan number
1729 (taxicab number)
Alternate forms
190
Minimal polynomial
Expanded forms
Series representations
191
192
Or:
27((4096+276)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
Input
Result
Decimal approximation
1729.0526944…. as above
Alternate forms
193
Minimal polynomial
Expanded forms
Series representations
194
195
From which:
(27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ)^1/15
Input
Exact result
Decimal approximation
1.64381856858…. ≈ ζ(2) =
Alternate forms
196
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
197
Series representations
198
Integral representation
199
From:
An Update on Brane Supersymmetry Breaking - J. Mourad and A. Sagnotti -
arXiv:1711.11494v1 [hep-th] 30 Nov 2017
From the following vacuum equations:
we have obtained from the results almost equals of the equations, putting
instead of
a new possible mathematical connection between the two exponentials. Thence, also
the values concerning p, C, β
E
and correspond to the exponents of e (i.e. of exp).
Thence we obtain for p = 5 and β
E
= 1/2:
200
Therefore, with respect to the exponentials of the vacuum equations, the Ramanujan’s
exponential has a coefficient of 4096 which is equal to 64
2
, while -6C+ is equal to -
. From this it follows that it is possible to establish mathematically, the dilaton
value.
For
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
201
Now, we have the following calculations:
= 1.6272016… * 10^-6
from which:
= 1.6272016… * 10^-6
0.000244140625
=
= 1.6272016… * 10^-6
Now:
202
And:
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
Thence:
0.000244140625
=
Dividing both sides by 0.000244140625, we obtain:
=
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
203
Result:
0.00666501785…
Series representations:
Now:
= 0.0066650177536
=
= 0.00666501785…
204
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
205
Integral representation:
In conclusion:
and for C = 1, we obtain:
=
Note that the values of n
s
(spectral index) 0.965, of the average of the Omega mesons
Regge slope 0.987428571 and of the dilaton , are also connected to
the following two Rogers-Ramanujan continued fractions:
206
(hp://www.bitman.name/math/arcle/102/109/)
Also performing the 512
th
root of the inverse value of the Pion meson rest mass
139.57, we obtain:
((1/(139.57)))^1/512
Input interpretation:
Result:
0.99040073.... result very near to the dilaton value = and to the
value of the following Rogers-Ramanujan continued fraction:
207
From
Properties of Nilpotent Supergravity - E. Dudas, S. Ferrara, A. Kehagias and A.
Sagnotti - arXiv:1507.07842v2 [hep-th] 14 Sep 2015
We have that:
We analyzing the following equation:
We have:
208
(M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ- sqrt6/k))]^2
i.e.
V = (M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ-
sqrt6/k))]^2
For k = 2 and φ = 0.9991104684, that is the value of the scalar field that is equal to
the value of the following Rogers-Ramanujan continued fraction:
we obtain:
V = (M^2)/3*[1-(b/euler number * 2/sqrt6) * (0.9991104684- sqrt6/2) * exp(-
(2/sqrt6)(0.9991104684- sqrt6/2))]^2
Input interpretation:
Result:
209
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
210
Derivative:
Implicit derivatives
Global minimum:
211
Global minima:
From:
we obtain:
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
212
Result:
Plots:
Alternate form assuming M is real:
-12.2723 result very near to the black hole entropy value 12.1904 = ln(196884)
Alternate forms:
213
Expanded form:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
214
Indefinite integral:
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
215
From b that is equal to
From:
we obtain:
1/3 (0.0814845 ((225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2 ) +
1)^2 M^2
Input interpretation:
Result:
216
Plots: (possible mathematical connection with an open string)
M = -0.5; M = 0.2
(possible mathematical connection with an open string)
M = 2 ; M = 3
Root:
Property as a function:
Parity
Series expansion at M = 0:
217
Series expansion at M = ∞:
Definite integral after subtraction of diverging parts:
For M = - 0.5 , we obtain:
1/3 (0.0814845 ((225.913 (-0.054323 (-0.5)^2 + 6.58545×10^-10 sqrt((-0.5)^4)))/(-
0.5)^2 ) + 1)^2 * (-0.5^2)
Input interpretation:
218
Result:
-4.38851344947*10
-16
For M = 0.2:
1/3 (0.0814845 ((225.913 (-0.054323 0.2^2 + 6.58545×10^-10 sqrt(0.2^4)))/0.2^2 ) +
1)^2 0.2^2
Input interpretation:
Result:
7.021621519159*10
-17
219
For M = 3:
1/3 (0.0814845 ((225.913 (-0.054323 3^2 + 6.58545×10^-10 sqrt(3^4)))/3^2 ) + 1)^2
3^2
Input interpretation:
Result:
1.57986484181*10
-14
For M = 2:
1/3 (0.0814845 ((225.913 (-0.054323 2^2 + 6.58545×10^-10 sqrt(2^4)))/2^2 ) + 1)^2
2^2
220
Input interpretation:
Result:
7.021621519*10
-15
From the four results
7.021621519*10^-15 ; 1.57986484181*10^-14 ; 7.021621519159*10^-17 ;
-4.38851344947*10^-16
we obtain, after some calculations:
sqrt[1/(2Pi)(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17 -
4.38851344947*10^-16)]
Input interpretation:
221
Result:
5.9776991059*10
-8
result very near to the Planck's electric flow 5.975498 × 10
−8
that
is equal to the following formula:
We note that:
1/55*(([(((1/[(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17
-4.38851344947*10^-16)])))^1/7]-((log^(5/8)(2))/(2 2^(1/8) 3^(1/4) e log^(3/2)(3)))))
Input interpretation:
Result:
1.6181818182… result that is a very good approximation to the value of the golden
ratio 1.618033988749...
222
From the Planck units:
Planck Length
5.729475 * 10
-35
Lorentz-Heaviside value
Planck’s Electric field strength
1.820306 * 10
61
V*m Lorentz-Heaviside value
Planck’s Electric flux
5.975498*10
-8
V*m Lorentz-Heaviside value
Planck’s Electric potential
1.042940*10
27
V Lorentz-Heaviside value
223
Relationship between Planck’s Electric Flux and Planck’s Electric Potential
E
P
* l
P
= (1.820306 * 10
61
) * 5.729475 * 10
-35
Input interpretation:
Result:
Scientific notation:
1.042939771935*10
27
≈ 1.042940*10
27
Or:
E
P
* l
P
2
/ l
P
= (5.975498*10
-8
)*1/(5.729475 * 10
-35
)
Input interpretation:
Result:
1.042939885417*10
27
≈ 1.042940*10
27
224
Fig. 1
It is therefore possible to consider the vortices of the "quantum vacuum" schematized
as cubes or octahedrons (the + sign inside a given vortex indicates its counterclockwise
rotation, while the - sign indicates its clockwise rotation). Between vortex and vortex
there is a layer of "bubbles"-universes (or universes-spheres), which flows, as in the
simplified two-dimensional drawing, from A to B
225
Fig. 2
Proposal
Image of space-time at quantum scale: the circles in red represent the points
corresponding to the compactified dimensions and the hexagons in blue, represent the
"fluctuations" (potential universes - green circles) of the quantum vacuum (2D). In
reality, we will have n-dimensional hyperspheres in which the compactified
dimensions "roll up" and octahedrons representing the "fluctuations", containing
spheres (bubbles of potential universes), of the quantum vacuum
226
Acknowledgments
We would like to thank Professor Augusto Sagnotti theoretical physicist at Scuola
Normale Superiore (Pisa – Italy) for his very useful explanations and his kindness and
availability. We would like to thank also Professor Amos Kubeka Financial
Economist, Mathematician and Professional Physicist, Professor of Applied
Mathematics at University of South Africa for his availability and kindness.
227
References
Thermodynamics of a modified Schwarzschild white hole in the presence of a
cosmological constant - Kubeka, A.S, and Amani A. (2022) - International Journal of
Modern Physics A, 37(9), 2250039.
Modified White Hole Enthalpy Coupled to Quantum Bose-Einstein Condensate
at Extremely Low Entropy - Kubeka, A.S., Amani, A. and Lekala, M. (2023) -
Journal of Modern Physics , 14, 1587-1599.
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
An Update on Brane Supersymmetry Breaking - Jihad Mourad and Augusto
Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017
Properties of Nilpotent Supergravity - Emilian Dudas, Sergio Ferrara, Alex
Kehagias and Augusto Sagnotti - arXiv:1507.07842v2 [hep-th] 14 Sep 2015
See also:
The Geometry of the MRB constant by Marvin Ray Burns
https://www.academia.edu/22271085/The_Geometry_of_the_MRB_constant
(See also Page 29 the applications of the CMRB in various sectors of Theoretical
Physics (String Theory) and Cosmology )
http://xoom.virgilio.it/source_filemanager/na/ar/nardelli/michele%20and%20antonio
%20papers/Try%20to%20beat%20these%20MRB%20constant%20records!%20-
%20Online%20Technical%20Discussion%20Groups%E2%80%94Wolfram%20Com
munity%20b.pdf