Mathematical Connections Between Black Holes, String Theory and Number Theory
1. 1
The mathematical theory of black holes. Mathematical connections with some sectors of
String Theory and Number Theory
Michele Nardelli 2
,
1
1
Dipartimento di Scienze della Terra
Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10
80138 Napoli, Italy
2
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
Abstract
In this paper we have described in the Section 1, some equations concerning the stellar evolution
and their stability. In the Section 2, we have described some equations concerning the perturbations
of Schwarschild black-hole, the Reissner-Nordstrom solution and the Schwarzschild geometry in D
= d + 1 dimensions. Furthermore, in these sections, we have showed the mathematical connections
with some sectors of Number Theory, principally with the Ramanujan’s modular equations and the
aurea ratio (or golden ratio)
1. On some equations concerning the stellar evolution and their stability. [1]
The success of the quantum theory may be traced to two basic facts: first, the Bohr radius of the
ground state of the hydrogen atom, namely,
8
2
2
2
10
5
.
0
4
−
×
≈
me
h
π
cm, (1.1)
where h is Planck’s constant, m is the mass of the electron and e is its charge, provides a correct
measure of atomic dimensions; and second, the reciprocal of Sommerfeld’s fine-structure constant,
137
2 2
≈
e
hc
π
, (1.2)
gives the maximum positive charge of the central nucleus that will allow a stable electron-orbit
around it.
With regard the stellar structure and stellar evolution, the following combination of the dimensions
of a mass provides a correct measure of stellar masses:
2
.
29
1
2
2
/
3
≅
H
G
hc
☼
☼
☼
☼, (1.3)
2. 2
where G is the constant of gravitation and H is the mass of the hydrogen atom.
The radiation pressure in the equilibrium of a star, is given from the following equation:
( ) 3
/
4
3
/
4
3
/
1
4
4
1
3
ρ
β
ρ
β
β
µ
C
a
H
k
P =
−
= . (1.4)
There is a general theorem (Chandrasekhar, 1936) which states that the pressure, c
P , at the centre of
a star of a mass M in hydrostatic equilibrium in which the density, ( )
r
ρ , at a point at a radial
distance, r , from the centre does not exceed the mean density, ( )
r
ρ , interior to the same point r ,
must satisfy the inequality,
3
/
2
3
/
4
3
/
1
3
/
2
3
/
4
3
/
1
3
4
2
1
3
4
2
1
M
G
P
M
G c
c ρ
π
ρ
π
≤
≤
, (1.5)
where ρ denotes the mean density of the star and c
ρ its density at the centre.
The right-hand side of inequality (1.5) together with P given by eq. (1.4), yields, for the stable
existence of stars, the condition,
3
/
2
3
/
1
3
/
1
4
4
6
1
3
GM
a
H
k
c
c
≤
−
π
β
β
µ
, (1.6)
or, equivalently,
2
/
3
2
/
1
4
4
2
/
1
1
1
3
6
G
a
H
k
M
c
c
−
≥
β
β
µ
π
, (1.7)
where in the foregoing inequalities, c
β is a value of β at the centre of the star. Now Stefan’s
constant, a , by virtue of Planck’s law, has the value
3
3
4
5
15
8
c
h
k
a
π
= . (1.8)
Inserting this value a in the equality (1.7) we obtain
( )
2
2
/
3
2
2
/
3
3
2
/
1
2
/
1
4
2 1
1873
,
0
1
2
135
1 H
G
hc
H
G
hc
M
c
c
=
≥
− π
β
β
µ . (1.9)
We observe that the inequality (1.9) has isolated the combination (1.3) of natural constants of the
dimensions of a mass; by inserting its numerical value given in eq. (1.3), we obtain the inequality,
48
,
5
1
2
/
1
4
2
≥
− c
c
M
β
β
µ ☼. (1.10)
3. 3
This inequality provides an upper limit to ( )
c
β
−
1 for a star of a given mass. Thus,
∗
−
≤
− β
β 1
1 c , (1.11)
where ( )
∗
− β
1 is uniquely determined by the mass M of the star and the mean molecular weight,
µ , by the quartic equation,
2
/
1
4
2 1
48
,
5
−
=
∗
∗
β
β
µ M ☼. (1.12)
We note that the following values:
80901699
,
0
80599
,
0
6
3
/
1
≈
=
π
; 39057647
,
1
381976
,
1
6
2
/
1
≈
=
π
; 183689
,
0
18736
,
0
2
135
3
≈
=
π
are well connected with various fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ (see
Appendix A)
In a completely degenerate electron gas all the available parts of the phase space, with momenta
less than a certain “threshold” value 0
p are occupied consistently with the Pauli exclusion-
principle. If ( )dp
p
n denotes the number of electrons, per unit volume, between p and dp
p + ,
then the assumption of complete degeneracy is equivalent to the assertion
( ) 2
3
8
p
h
p
n
π
= ( )
0
p
p ≤ ; ( ) 0
=
p
n ( )
0
p
p > . (1.13)
The value of the threshold momentum, 0
p , is determined by the normalization condition
( )
∫ =
=
0
0
3
0
3
3
8
p
p
h
dp
p
n
n
π
, (1.14)
where n denotes the total number of electrons per unit volume. For the distribution given by (1.13),
the pressure p and the kinetic energy kin
E of the electrons (per unit volume), are given by
∫
=
0
0
3
3
3
8 p
pdp
v
p
h
P
π
(1.15)
and
∫
=
0
0
2
3
8 p
p
kin dp
T
p
h
E
π
, (1.16)
where p
v and p
T are the velocity and the kinetic energy of an electron having a momentum p . If
we set
m
p
vp /
= and m
p
Tp 2
/
2
= , (1.17)
appropriate for non-relativistic mechanics, in eqs. (1.15) and (1.16), we find
4. 4
3
/
5
2
3
/
2
5
0
3
3
20
1
15
8
n
m
h
p
m
h
P
=
=
π
π
(1.18)
and
3
/
5
2
3
/
2
5
0
3
3
40
3
10
8
n
m
h
p
m
h
Ekin
=
=
π
π
. (1.19)
Thence, we obtain the following expressions:
∫
=
0
0
3
3
3
8 p
pdp
v
p
h
P
π 3
/
5
2
3
/
2
5
0
3
3
20
1
15
8
n
m
h
p
m
h
=
=
π
π
; (1.18b)
∫
=
0
0
2
3
8 p
p
kin dp
T
p
h
E
π 3
/
5
2
3
/
2
5
0
3
3
40
3
10
8
n
m
h
p
m
h
=
=
π
π
. (1.19b)
We note that the following values:
34345
,
8
37758
,
8
3
8
≈
=
π
; 04863
,
0
048486
,
0
3
20
1
3
/
2
≅
=
π
; 07294
,
0
072729
,
0
3
40
3
3
/
2
≅
=
π
;
are well connected with various fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ (see
Appendix A)
According to the expression for the pressure given by eq. (1.18), we have the relation
3
/
5
1ρ
K
P = where
( ) 3
/
5
2
3
/
2
1
3
20
1
H
m
h
K
e
µ
π
= , (1.20)
where e
µ is the mean molecular weight per electron. We can rewrite the expression also as follows:
( )
3
/
5
3
/
5
2
3
/
2
3
20
1
ρ
µ
π H
m
h
P
e
= . (1.20b)
Thus, already for a degenerate star of solar mass (with 2
=
e
µ ) the central density is 6
10
19
.
4 × g
cm3
; and this density corresponds to a threshold momentum mc
p 29
.
1
0 = and a velocity which is
c
63
.
0 . Consequently, the equation of state must be modified to take into account the effects of
special relativity. And this is easily done by inserting in eqs. (1.15) and (1.16) the relations,
( ) 2
/
1
2
2
2
/
1 c
m
p
m
p
vp
+
= and ( )
[ ]
1
/
1
2
/
1
2
2
2
2
−
+
= c
m
p
mc
Tp , (1.21)
in place of the non-relativistic relations (1.17). We find that the resulting equation of state can be
expressed, parametrically, in the form
5. 5
( )
x
Af
P = and 3
Bx
=
ρ , (1.22)
where
3
5
4
3h
c
m
A
π
= , 3
3
3
3
8
h
H
c
m
B e
µ
π
= (1.23)
and
( ) ( ) ( ) x
x
x
x
x
f 1
2
2
/
1
2
sinh
3
3
2
1 −
+
−
+
= . (1.24)
Thence, P can be rewritten also as follows:
3
5
4
3h
c
m
P
π
= ( ) ( ) x
x
x
x 1
2
2
/
1
2
sinh
3
3
2
1 −
+
−
+ . (1.24b)
And similarly
( )
x
Ag
Ekin = , (1.25)
( ) ( )
[ ] ( )
x
f
x
x
x
g −
−
+
= 1
1
8
2
/
1
2
3
. (1.26)
Thence, the eq. (1.25) can be rewritten also as follows:
=
kin
E 3
5
4
3h
c
m
π
( )
[ ]
1
1
8
2
/
1
2
3
−
+
x
x ( ) ( ) x
x
x
x 1
2
2
/
1
2
sinh
3
3
2
1 −
+
−
+
− . (1.26b)
According to eqs. (1.22) and (1.23), the pressure approximates the relation (1.20) for low enough
electron concentrations ( )
1
<<
x ; but for increasing electron concentrations ( )
1
>>
x , the pressure
tends to
3
/
4
3
/
1
3
8
1
hcn
P
=
π
. (1.27)
This limiting form of relation can be obtained very simply by setting c
vp = in eq. (1.15); then
∫ =
=
0
0
4
0
3
3
3
3
2
3
8 p
p
h
c
dp
p
h
c
P
π
π
; (1.28)
and the elimination of 0
p with the aid of eq. (1.14) directly leads to eq. (1.27). The relation between
P and ρ corresponding to the limiting form (1.28) is
( )
3
/
4
3
/
4
3
/
1
3
8
1
ρ
µ
π H
hc
P
e
= . (1.29)
Thence, the eq. (1.28) can be rewritten also as follows:
6. 6
∫ =
=
0
0
4
0
3
3
3
3
2
3
8 p
p
h
c
dp
p
h
c
P
π
π
( )
3
/
4
3
/
4
3
/
1
3
8
1
ρ
µ
π H
hc
e
= . (1.29b)
We note that the following values:
12538820
,
0
123093
,
0
3
8
1
3
/
1
≅
=
π
and 12461180
,
2
094395
,
2
3
2
≈
=
π
,
are well connected with various fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ (see
Appendix A)
In this limit, the configuration is an Emden polytrope of index 3. And it is well known that when the
polytropic index is 3, the mass of the resulting equilibrium configuration is uniquely determined by
the constant of proportionality, 2
K , in the pressure-density relation. We have accordingly,
( )
( )
2
2
2
/
3
2
/
3
2
lim 76
,
5
1
197
.
0
018
.
2
4 −
=
=
= e
eH
G
hc
G
K
M µ
µ
π
π ☼. (1.30)
We note that the value 5,76 is well connected with the mean of the following values: 5,562
and 5,890 that is equal to 5,726 that is a fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ
(see Appendix A)
It is clear from general considerations that the exact mass-radius relation for the degenerate
configurations must provide an upper limit to the mass of such configurations given by eq. (1.30);
and further, that the mean density of the configuration must tend to infinity, while the radius tends
to zero, and lim
M
M → .
It is more convenient to express the electron pressure in terms of ρ and e
β defined in the manner
4
3
1
1
aT
T
H
k
p
e
e
e
e
β
β
ρ
µ −
=
= , (1.31)
where e
p now denotes the electron pressure. Then, analogous to eq. (1.4), we can write
3
/
4
3
/
1
4
1
3
ρ
β
β
µ
−
=
e
e
e
e
a
H
k
p . (1.32)
Comparing this with eq. (1.29), we conclude that if
=
>
−
2
3
/
4
3
/
1
4
1
3
K
a
H
k
e
e
e
ρ
β
β
µ ( ) 3
/
4
3
/
1
3
8
1
H
hc
e
µ
π
, (1.33)
7. 7
the pressure e
p given by the classical perfect-gas equation of state will be greater than that given by
the equation if degeneracy were to prevail, not only for the prescribed ρ and T , but for all ρ and
T having the same e
β . Inserting for a its value given in eq. (1.8), we find that the inequality (1.33)
reduces to
1
1
960
4
>
−
e
e
β
β
π
, (1.34)
or, equivalently
ω
β
β −
=
>
− 1
0921
,
0
1 e . (1.35)
We note that the value 88854382
,
9
85534
,
9
960
4
≅
=
π
and 09184494
,
0
0921
,
0 ≅ ,
are well connected with various fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ (see
Appendix A)
On the standard model, the fraction β (=gas pressure/total pressure) is a constant through a star. On
this assumption, the star is a polytrope of index 3 as is apparent from eq. (1.4); and, in consequence,
we have the relation
( ) ( )
018
,
2
4
2
/
3
=
G
C
M
π
β
π (1.36)
where ( )
β
C is defined in eq. (1.4). Equation (1.36) provides a quartic equation for β analogous to
eq. (1.12) for ∗
β . Equation (1.36) for ω
β
β = gives
( )
2
2
2
/
3
2
/
3
65
,
6
1
197
.
0 −
−
=
= µ
µ
βω
H
G
hc
M ☼ = R . (1.37)
We note that the value 0,197 is well connected with 0,19513485 and that 6,65 is well
connected with the mean of the following values: 6,47213 and 6,87538. It is equal to 6,6737
that is a fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ (see Appendix A)
By virtue of the inequality (1.5), the maximum central pressure attainable in a star must be less than
that provided by the degenerate equation of state, so long as
( ) 3
/
4
3
/
1
2
3
/
2
3
/
1
3
8
1
3
4
2
1
H
hc
K
M
G
e
µ
π
π
=
<
(1.38)
or, equivalently
( )
2
2
2
/
3
74
,
1
1
16
3 −
=
< e
eH
G
hc
M µ
µ
π
☼. (1.39)
8. 8
Thence, we have the following connection between eq. (1.38) and eq. (1.39):
( ) 3
/
4
3
/
1
2
3
/
2
3
/
1
3
8
1
3
4
2
1
H
hc
K
M
G
e
µ
π
π
=
<
( )
2
2
2
/
3
74
,
1
1
16
3 −
=
<
⇒ e
eH
G
hc
M µ
µ
π
☼. (1.39b)
We conclude that there can be no surprises in the evolution of stars of mass less than 0,43☼ (if
2
=
e
µ ). The end stage in the evolution of such stars can only be that of the white dwarfs.
We note that the values 618
,
1
61199
,
1
3
4
3
/
1
≅
=
π and 74535
,
1
74
,
1 ≅ are well connected with
various fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ (see Appendix A)
In the framework of the Newtonian theory of gravitation, the stability for radial perturbations
depends only on an average value of the adiabatic exponent, 1
Γ , which is the ratio of the fractional
Lagrangian changes in the pressure and in the density experienced by a fluid element following the
motion; thus,
ρ
ρ /
/ 1∆
Γ
=
∆ P
P . (1.40)
And the Newtonian criterion for stability is
( ) ( ) ( ) ( ) ( )
∫ ∫ >
÷
Γ
=
Γ
M M
r
dM
r
P
r
dM
r
P
r
0 0
1
1
3
4
. (1.41)
If
3
4
1 <
Γ , dynamical instability of a global character will ensue with an e-folding time measured
by the time taken by a sound wave to travel from the centre to the surface. When one examines the
same problem in the framework of the general theory of relativity, one finds that, again, the stability
depends on an average value of 1
Γ ; but contrary to the Newtonian result, the stability now depends
on the radius of the star as well. Thus, one finds that no matter how high 1
Γ may be, instability will
set in provided the radius is less than a certain determinate multiple of the Schwarzschild radius,
2
/
2 c
GM
Rs = . (1.42)
Thus, if for the sake of simplicity, we assume that 1
Γ is a constant through the star and equal to
3
/
5 , then the star will become dynamically unstable for radial perturbations, if S
R
R 4
,
2
1 < . And
further, if ∞
→
Γ1 , instability will set in for all ( ) S
R
R 8
/
9
< . The radius ( ) S
R
8
/
9 defines, in fact,
the minimum radius which any gravitating mass, in hydrostatic equilibrium, can have in the
framework of general relativity. It follows from the equations governing radial oscillations of a star,
in a first post-Newtonian approximation to the general theory of relativity, that instability fro radial
perturbations will set in for all
2
1
2
3
/
4 c
GM
K
R
−
Γ
< , (1.43)
where K is a constant which depends on the entire march of density and pressure in the equilibrium
configuration in the Newtonian framework. It is for this reason that we describe the instability as
global. Thus, for a polytrope of index n , the value of the constant is given by
9. 9
( )
( )
+
+
−
−
= ∫
1
0
2
2
3
1
4
1
1
'
1
11
2
18
5 ξ
ξ
ξ
ξ
θ
θ
θ
ξ
d
d
d
n
n
n
K , (1.44)
where θ is the Lane-Emden function in its standard normalization ( 1
=
θ at 0
=
ξ ), ξ is the
dimensionless radial coordinate, 1
ξ defines the boundary of the polytrope (where 0
=
θ ) and 1
'
θ is
the derivative of θ at 1
ξ . If we set 2
=
n in the eq. (1.44), we obtain:
+
= ∫
1
0
2
2
3
1
4
1
1
'
6
6
1 ξ
ξ
ξ
ξ
θ
θ
θ
ξ
d
d
d
K . (1.44b)
We note that 6 is 24
4
1
× and 24 is related to the physical vibrations of the superstring by the
following Ramanujan’s modular equation:
( )
+
+
+
⋅
=
−
∞
−
∫
4
2
7
10
4
2
11
10
log
'
142
'
cosh
'
cos
log
4
24
2
'
'
4
0
'
2
2
w
t
itw
e
dx
e
x
txw
anti
w
w
t
w
x
φ
π
π
π
π
.
Thence, we have the following possible mathematical connection:
( )
⇒
+
+
+
⋅
×
−
∞
−
∫
4
2
7
10
4
2
11
10
log
'
142
'
cosh
'
cos
log
4
4
1
2
'
'
4
0
'
2
2
w
t
itw
e
dx
e
x
txw
anti
w
w
t
w
x
φ
π
π
π
π
+
∫
1
0
2
2
3
1
4
1
1
'
6
6
1 ξ
ξ
ξ
ξ
θ
θ
θ
ξ
d
d
d
. (1.44c)
In the following Table, we have the values of K for different polytropic indices. It should be
particularly noted that K increases without limit for 5
→
n and the configuration becomes
increasingly centrally condensed.
10. 10
n K
0 0.452381
1.0 0.565382
1.5 0.645063
2.0 0.751296
2.5 0.900302
3.0 1.12447
3.25 1.28503
3.5 1.49953
4.0 2.25338
4.5 4.5303
4.9 22.906
4.95 45.94
It has been possible to show that for 5
→
n , the asymptotic behaviour of K is given by
( )
n
K −
→ 5
/
3056
.
2 ; (1.45)
and, further, that along the polytropic sequence, the criterion for instability (1.43) can be expressed
alternatively in the form
3
/
4
1
2
2264
.
0
1
2
3
/
1
−
Γ
<
c
GM
R c
ρ
ρ
( )
6
10
/ ≥
ρ
ρc . (1.46)
We note that the value 0,2264 is well connected with 0,229 that is a fractional powers of Phi
...
61803399
,
1
2
1
5
=
+
=
Φ
Furthermore, the values of the precedent table, are well connected with the following
fractional powers of Phi (see Appendix A):
0,445824; 0,572949; 0,629514; 0,752329; 0,891649; 1,128493; 1,259029; 1,500000;
2,250000; 4,500000; 22,180339; 41,1246 + 4,8541.
In this section we have considered only the restrictions on the last stages of stellar evolution that
follow from the existence of an upper limit to the mass of completely degenerate configurations and
from the instabilities of relativistic origin. From these and related considerations, the conclusion is
inescapable that black holes will form as one of the natural end products of stellar evolution of
massive stars; and further that they must exist in large numbers in the present astronomical
universe.
11. 11
2. On some equations concerning the perturbations of Schwarzschild black-hole and the
Reissner-Nordstrom solution. [2]
With the reduction of the third-order system of the following equations
cX
bL
aN
N r +
+
=
, , (2.1) cX
L
v
r
b
N
v
r
a
L r
r
r +
−
−
+
+
−
= ,
,
,
1
1
, (2.2)
X
v
r
c
L
v
r
b
N
v
r
a
X r
r
r
r
−
+
−
−
+
−
+
−
−
= ,
,
,
,
1
2
1
1
, (2.3)
to the single second-order the following equation
( )
( ) ( )
( ) ( ) ( )
( )
[ ]( )+
+
−
+
+
+
+
+
+
−
+
+
−
+
+
−
=
+
∗
X
L
M
r
n
M
nr
Mn
V
M
nr
r
M
Mnr
nr
M
V
M
nr
r
M
r
M
V
V
M
r
Z r
r
r
r
r
r 2
3
3
3
6
4
3
3
2
3
2 3
2
2
2
2
,
,
,
,
,
,
( )( )
( ) ( )
[ ]
X
M
r
L
M
r
M
nr
M
r
r
M
Mnr
nr
3
5
2
3
2
3
3
2
2
2
−
+
−
+
−
−
−
− , (2.4)
for ( )
+
Z , it is clear that the solution for ,
, X
L and N will require a further quadrature. Thus,
rewriting the following equation
−
+
−
=
−
+ X
v
r
X
L
v
r
L r
r
r
r ,
,
,
,
1
2
, (2.5)
in the form
( ) ( )
V
re
dr
d
nr
L
e
r
dr
d v
v −
−
−
=
2
(2.6)
and replacing V by
( )
L
M
r
Z
Mr
M
nr
V
3
3
3
+
+
= +
, (2.7)
we obtain the equation
( ) ( ) ( )
[ ]
+
−
−
+
−
=
+ Z
M
nr
e
dr
d
M
nr
L
e
r
dr
d
M
nr v
v
3
3
3
1 2
. (2.8)
This equation yields the integral relation
( )
( ) ( )
[ ]
∫
+
−
−
+
+
−
= dr
Z
M
nr
e
dr
d
M
nr
nr
L
e
r v
v
3
3
2
, (2.9)
or, after an integration by parts
12. 12
( ) ( )
∫
+
−
+
−
−
+
+
−
= dr
Z
M
nr
e
Mn
Z
nre
L
e
r
v
v
v
3
3
2
. (2.10)
Thence, we have the following expression:
( ) ( )
( ) ( )
[ ]
+
−
+
−
+
−
+
−
=
+
+
−
+ ∫ Z
M
nr
e
dr
d
M
nr
dr
Z
M
nr
e
Mn
Z
nre
dr
d
M
nr v
v
v
3
3
3
3
3
1 . (2.10b)
Defining
( )
dr
M
nr
Z
e
ne
v
v
∫ +
=
Φ
+
−
3
, (2.11)
we have the solution
( )
Φ
+
−
= +
2
3
r
M
Z
r
n
L . (2.12)
With this solution for L , equation (2.7) gives
( )
( )
Φ
+
= +
Z
r
n
X . (2.13)
As a consequence of these solutions for L and X ,
( )Φ
+
=
+ M
nr
r
X
L 3
1
2
. (2.14)
To obtain the solution for N , we take the following equation
( )
( ) ( )
( ) ( )
( )
X
L
M
nr
M
nMr
nr
V
M
nr
r
M
r
M
V
M
r
Z
r
M
Z r
r
r +
+
−
−
+
+
−
+
−
=
−
=
+
∗ 2
2
2
,
,
,
3
3
3
3
2
3
2
2
1 , (2.15)
and substitute for ( )
n
X
V r
r /
,
, = on the right-hand side from the following equation
( ) ( ) ( )
( ) L
M
r
n
X
L
M
r
r
r
M
M
r
r
M
M
r
N
M
r
r
M
nr
X r
2
1
2
2
2
1
2
3
2
4
2
2
,
−
+
+
+
−
+
+
−
−
−
−
−
+
−
=
σ
. (2.16)
In this manner, we obtain
( )
( ) ( ) ( )
( ) +
−
+
+
+
×
−
+
+
−
−
−
−
−
+
−
−
=
∗ L
M
r
n
X
L
M
r
r
r
M
M
r
r
M
M
r
N
M
r
r
M
nr
M
r
n
Z r
2
1
2
2
2
1
2
3
2
1
2
4
2
2
,
σ
( )
( ) ( )
( )
X
L
M
nr
M
Mnr
nr
X
M
nr
nr
M
r
M
+
+
−
−
+
+
−
+ 2
2
2
3
3
3
3
2
3
. (2.17)
Simplifying this last equation and substituting for L and X
L + their solutions (2.12) and (2.13),
we find:
13. 13
( )
( )
( ) ( )
2
4
2
2
2
2
,
2
1
3
6
3
3 r
M
r
r
M
M
Z
r
n
n
Mn
r
M
M
nr
n
Z
M
nr
nr
N r
Φ
−
+
−
+
+
+
+
+
−
+
−
= +
+
∗
σ
. (2.18)
For the eq. (2.11), we obtain the following expression:
( )
( )
( ) ( )
( )
+
−
+
−
+
+
+
+
+
−
+
−
= ∫
+
−
+
+
∗ dr
M
nr
Z
e
ne
r
M
r
r
M
M
Z
r
n
n
Mn
r
M
M
nr
n
Z
M
nr
nr
N
v
v
r
3
1
2
1
3
6
3
3 2
4
2
2
2
2
,
σ
(2.18b)
This completes the formal solution of the basic equations.
Quasi-normal modes are defined as solutions of the perturbation equations, belonging to complex
characteristics-frequencies and satisfying the boundary conditions appropriate for purely outgoing
waves at infinity and purely ingoing waves at the horizon. The problem, then, is to seek solutions of
the equations governing ( )
±
Z which will satisfy the boundary conditions
( ) ( )
( ) r
i
e
A
Z σ
σ −
±
±
→ ( )
+∞
→
∗
r ; ( ) ( )
( ) ∗
+
±
±
→ r
i
e
A
Z σ
σ ( )
−∞
→
∗
r . (2.19)
We observe that the characteristic frequencies σ are the same for ( )
−
Z and ( )
+
Z ; for, if σ is a
characteristic frequency and ( )
−
Z ( )
σ is the solution belonging to it, then the solution ( )
+
Z ( )
σ
derived from ( )
−
Z ( )
σ in accordance with the following relations
( )
[ ] ( )
( ) ( )
( ) ( )
−
−
+
∗
+
+
∆
+
+
=
+
+ r
MZ
Z
M
r
r
M
Z
M
i ,
2
3
2
2
2
2
2
12
6
72
2
12
2
µ
µ
µ
σ
µ
µ ; (2.20a)
( )
[ ] ( )
( ) ( )
( ) ( )
+
+
−
∗
−
+
∆
+
+
=
−
+ r
MZ
Z
M
r
r
M
Z
M
i ,
2
3
2
2
2
2
2
12
6
72
2
12
2
µ
µ
µ
σ
µ
µ ; (2.20b)
will satisfy the boundary conditions (2.19) with
( )
( ) ( )
( ) ( )
( ) M
i
M
i
A
A
σ
µ
µ
σ
µ
µ
σ
σ
12
2
12
2
2
2
2
2
+
+
−
+
= −
+
. (2.21)
It will suffice, then, to consider only the equation governing ( )
−
Z . Letting
( )
= ∫
∗
∗
−
r
dr
i
Z φ
exp , (2.22)
we find that the equation we have to solve is
( )
0
2
2
, =
−
−
+ −
∗
V
i r φ
σ
φ ; (2.23)
and the appropriate boundary conditions are
σ
φ −
→ as +∞
→
∗
r and σ
φ +
→ as −∞
→
∗
r . (2.24)
14. 14
Furthermore, we can rewrite the eq. (2.20a) as follows:
( )
[ ] ( )
( ) ( )
( )
−
∗
+
∗
∗
+
+
∆
+
+
=
+
+ ∫ r
r
MZ
dr
i
M
r
r
M
Z
M
i ,
2
3
2
2
2
2
2
12
exp
6
72
2
12
2 φ
µ
µ
µ
σ
µ
µ . (2.24b)
Solutions of eq. (2.23), satisfying the boundary conditions (2.24), exist only when σ assumes one
of a discrete set of values. A useful identity, which follows from integrating equation (2.23) and
making use of the boundary conditions (2.24), is
( ) ( )
∫ ∫
∞
+
∞
−
∞
+
∞
−
∗
−
∗
+
=
=
−
+
−
2
1
2
1
2 2
2
2
µ
φ
σ
σ
M
dr
V
dr
i . (2.25)
With regard the complex characteristic-frequency σ belonging to the quasi-normal modes of
the Schwarzschild black-hole (σ is expressed in the unit ( ) 1
2
−
M ) for value of 4
=
l , we have
the following result:
4
=
l ; i
M 18832
,
0
61835
,
1
2 +
=
σ . (2.26)
We note that the value 1,61835 is very near to the ...
61803398
,
1
2
1
5
=
+
=
Φ , thence to the
value of the golden ratio.
The Reissner-Nordstrom solution
With regard the Reissner-Nordstrom solution, the reduction of the equations governing the polar
perturbations of the Schwarzschild black-hole to Zerilli’s equation, it is clear that the reducibility of
a system of equations of order five to the pair of the following equations
( ) ( ) ( ) ( )
( )
[ ]
+
∗
+
+
+
+
−
+
∆
=
Λ 1
2
2
5
2
2
2
3 H
Q
MH
W
UH
r
H µ , ( ) ( ) ( ) ( )
( )
[ ]
+
+
∗
+
+
+
+
+
∆
=
Λ 1
2
1
5
1
2
3
2 MH
H
Q
W
UH
r
H µ ,
must be the result of the system allowing a special solution. Xanthopoulos has discovered that the
present system of equations
( ) 23
,
23
,
23
2
2
03
2
1
B
r
B
B
r
r
B r
r
+
=
= , ( )
[ ] ( ) 23
2
2
02
2
4
1
1
2
2 B
r
l
l
V
l
l
T
Q
B
e
r v
+
−
+
−
= ∗ ,
( ) 2
2
23
2
2
2
02
2
2
,
03
2
2
2
r
L
N
Q
B
e
r
B
e
r
B
e
r v
v
r
v +
=
+
+ ∗
−
σ ,
and
( )
23
, B
X
c
bL
aN
N r −
+
+
= , ( ) 23
23
,
,
,
2
1
1
B
r
B
X
c
L
v
r
b
N
v
r
a
L r
r
r −
−
+
−
−
+
+
−
= ,
( ) 03
23
,
,
,
,
1
2
1
1
B
B
X
v
r
c
L
v
r
b
N
v
r
a
X r
r
r
r +
−
−
+
−
−
+
−
+
−
−
= ,
15. 15
allows the special solution
( )
( )
−
+
−
∆
−
= ∗
∗
−
r
Q
r
Q
M
r
M
e
r
N v
2
4
2
2
2
2
0
2
σ , ( )
( )
2
3
0
4
3 ∗
−
−
= Q
Mr
e
r
L v
, ( ) 1
0 −
= r
ne
X v
,
( ) v
e
r
Q
B 3
2
0
23 2 −
∗
−
= , and ( )
( )
Mr
r
Q
e
r
Q
B v
3
2
2 2
2
6
2
0
03 −
+
= ∗
−
−
∗ . (2.27)
The completion of the solution for the remaining radial functions with the aid of the special solution
(2.27) is relatively straightforward. Xanthopoulos finds
( ) ( ) ( ) ( )
( ) ( ) ( )
[ ] ( ) ( )
( )
+
∗
+
+
∗
+
+
+
+
−
−
−
+
+
−
+
Φ
= 1
2
2
2
,
1
2
2
2
2
0
1
3
2
1
2 H
Q
nrH
n
M
nr
e
r
H
Q
nrH
e
H
e
n
N
N v
r
v
v
µ
ϖ
ϖ
ϖ
µ
ϖ
ϖ
(2.28)
( ) ( ) ( )
( )
+
∗
+
+
−
Φ
= 1
2
2
0 1
H
Q
nrH
r
L
L µ , (2.29) ( ) ( )
+
+
Φ
= 2
0
H
r
n
X
X , (2.30)
( ) ( )
+
∗
−
Φ
= 1
2
0
23
23 H
r
Q
B
B
µ
, (2.31) ( ) ( ) ( ) ( )
( )
+
∗
+
∗
+
∗
+
−
−
Φ
= 1
2
4
2
,
1
2
0
03
03 2 H
Q
nrH
r
Q
H
r
Q
B
B r µ
ϖ
µ
, (2.32)
where
( ) ( )
( )
∫
−
+
∗
+
+
=
Φ dr
r
e
H
Q
nrH
v
ϖ
µ 1
2 . (2.33)
Thence, the eqs. (2.31) and (2.32) can be written also as follows:
( ) ( ) ( )
( ) ( )
∫
+
∗
−
+
∗
+
−
+
= 1
2
1
2
0
23
23 H
r
Q
dr
r
e
H
Q
nrH
B
B
v
µ
ϖ
µ , (2.34)
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
+
∗
+
∗
+
∗
−
+
∗
+
+
−
−
+
= ∫ 1
2
4
2
,
1
2
1
2
0
03
03 2 H
Q
nrH
r
Q
H
r
Q
dr
r
e
H
Q
nrH
B
B r
v
µ
ϖ
µ
ϖ
µ . (2.35)
As in the case of the Schwarzschild perturbations, the potentials ( )
±
i
V ( 2
,
1
=
i ), associated with the
polar and the axial perturbations, are related in a manner which guarantees the equality of the
reflexion and the transmission coefficients determined by the equations governing ( )
±
i
Z . Thus, it can
be verified, the potentials are, in fact, given by
( )
i
i
i
i
i
i f
f
dr
df
V κ
β
β +
+
±
=
∗
± 2
2
, (2.36)
where
( )
2
2
2
+
= µ
µ
κ , j
i q
=
β , and
( )
j
i
q
r
r
f
+
∆
= 2
3
µ
( )
j
i
j
i ≠
= ;
2
,
1
, , (2.37)
thence, the eq. (2.36) can be rewritten also as follows
16. 16
( )
( ) ( ) ( ) ( )
j
j
j
j
j
i
q
r
r
q
r
r
q
q
r
r
dr
d
q
V
+
∆
+
+
+
∆
+
+
∆
±
=
∗
±
2
3
2
2
2
2
3
2
2
3
2
µ
µ
µ
µ
µ
. (2.37b)
The solutions, ( )
+
i
Z and ( )
−
i
Z of the respective equations are, therefore, related in the manner
( )
[ ] ( )
( ) ( )
( )
( )
∗
±
±
+
∆
+
+
=
±
+
dr
dZ
q
Z
q
r
r
q
Z
q
i i
j
i
j
j
i
j
m
m
2
2
2
2
2 2
3
2
2
2
2
2
µ
µ
µ
σ
µ
µ ( )
j
i
j
i ≠
= ;
2
,
1
, . (2.38)
It is the existence of this relation which guarantees the equality of the reflexion and the transmission
coefficients determined by the wave equations governing ( )
+
i
Z and ( )
−
i
Z .
In view of the relation (2.38) between the solutions belonging to axial and polar perturbations, the
characteristic frequencies will be the same for ( )
+
i
Z and ( )
−
i
Z . It should also be noticed that there is
no quasi-normal mode which is purely electromagnetic or purely gravitational: any quasi-normal
mode of oscillation will be accompanied by the emission of both electromagnetic and gravitational
radiation in accordance with the following equation
ψ
ψ sin
cos 2
1
1 Z
Z
H −
= ; ψ
ψ sin
cos 1
2
2 Z
Z
H +
= , (2.39)
where the amplitudes 1
H and 2
H of the electromagnetic and gravitational (wave-like) disturbances
(of some specified frequencies) are related to the function 1
Z and 2
Z .
With regard the complex characteristic frequencies 1
σ and 2
σ (belonging to ( )
±
1
Z and ( )
±
2
Z ) of
the quasi-normal modes for a range of values of ∗
Q and l , we obtain, for 1
Z and 9
.
0
=
∗
Q and
2
=
l , the value 0.61939 that is very near to the ...
61803398
,
0
2
1
5
=
−
=
φ , thence to the value
of the golden section.
The considerations, relative to the stability of the Schwarzschild black-hole to external
perturbations apply, quite literally, to the Reissner-Nordstrom black-hole since the only fact
relevant to those considerations was that the potential barriers, external to the event horizon, are real
and positive; and stability follows from this fact.
While the equations governing ( )
±
i
Z remain formally unaltered, the potential barriers, ( )
±
i
V , are
negative in the interval, +
− <
< r
r
r , and in the associated range of ∗
r , namely −∞
>
>
∞
+ ∗
r ; they
are in fact potential wells rather than potential barriers. Thus, the equation now governing ( )
−
i
Z is,
for example,
( )
( )
( ) ( )
−
∗
−
∗
−
+
−
+
∆
−
=
+ i
j
i
i
Z
r
Q
q
r
r
Z
dr
Z
d 2
2
5
2
2
2
4
2
µ
σ ( )
j
i
j
i ≠
= ;
2
,
1
, and ( )
−∞
>
>
+∞
<
< ∗
+
− r
r
r
r , ,
(2.40)
where
−
−
+
+
∗ −
−
−
+
= r
r
r
r
r
r lg
2
1
lg
2
1
κ
κ
, (2.41) 2
2 +
−
+
+
−
=
r
r
r
κ , and 2
2 −
−
+
−
−
=
r
r
r
κ . (2.42)
In view of the relation (2.38) between the solutions, belonging to axial and polar perturbations, it
will, again, suffice to restrict our consideration to equation (2.40); and for convenience, we shall
suppress the distinguishing superscript. An important consequence of the fact that we are now
17. 17
concerned with a short-range one-dimensional potential-well, is that equation (2.40) will allow a
finite number of discrete, non-degenerate, bound states:
j
iσ
σ ±
= [ ]
m
n
j ...,
2
,
1
;
2
,
1 =
= . (2.43)
The boundary conditions we must now impose are
( ) ( ) ( ) ∗
∗ +
−
∗ +
→ r
i
r
i
e
B
e
A
r
Z σ
σ
σ
σ ( )
+∞
→
+
→ ∗
− r
r
r ;
0
∗
−
→ r
i
e σ
( )
−∞
→
−
→ ∗
+ r
r
r ;
0 . (2.44)
The coefficients ( )
σ
A and ( )
σ
B in equation (2.44) are related to the reflexion and the transmission
amplitudes,
( )
( ) ( )
σ
σ
σ
−
=
= ∗
1
1
1
T
T
A , ( ) ( )
( )
( )
( )
σ
σ
σ
σ
σ
−
−
=
= ∗
∗
1
1
T
R
T
R
B , (2.45)
so that
( ) ( ) 1
2
2
=
− σ
σ B
A . (2.46)
With regard the amplification factors, ( )2
σ
A , appropriate for the potential ( )
±
1
V and ( )
±
2
V for
2
2
75
.
0 M
Q =
∗ , we observe that ( )2
σ
A , and, therefore, also ( )2
σ
B , tend to finite limits as 0
→
σ .
This fact has its origin in the existence of bound states of zero energy in the potential wells, 1
V and
2
V . Furthermore, we note that for 30
.
0
=
σ , for 1
V we have the value 1.6168, while for 2
V we
have the value 1.6286. It is easy note that these values are very near to the
...
61803398
,
1
2
1
5
=
+
=
Φ , thence to the value of the golden ratio.
In analyzing the radiation arriving at the Cauchy horizon at −
r , we must distinguish the edges '
EC
and EF in the Penrose diagram. For this reason, we restore the time-dependence, t
i
e σ
, of the
solutions; and remembering that in the interval, +
− <
< r
r
r ,
t
r
u +
= ∗ and t
r
v −
= ∗ , (2.47)
we write, in place of equation (2.44),
( ) ( )
[ ] ( ) u
i
v
i
v
i
e
B
e
A
e
t
r
Z σ
σ
σ
σ
σ +
−
−
∗ +
−
+
→ 1
, . (2.48)
If we now suppose that the flux of radiation emerging from '
'C
D is ( )
v
Ẑ , then
( ) ( )
∫
∞
+
∞
−
= dv
e
v
Z
Z v
iσ
π
σ ˆ
2
1
. (2.49)
This flux disperses in the domain between the two horizons and at the Cauchy horizon it is
determined by
( ) ( ) ( )
u
Y
v
X
t
r
Z +
→
∗, ( )
∞
→
∞
→ u
v ; , (2.50)
where
18. 18
( ) ( ) ( )
[ ] σ
σ
σ σ
d
e
A
Z
v
X v
i
−
+∞
∞
−
∫ −
= 1 , (2.51)
and
( ) ( ) ( )
∫
+∞
∞
−
= σ
σ
σ σ
d
e
B
Z
u
Y u
i
. (2.52)
Thence, the eq. (2.50) can be rewritten also as follows:
( )→
∗ t
r
Z , ( ) ( )
[ ] σ
σ
σ σ
d
e
A
Z v
i
−
+∞
∞
−
∫ −1 ( ) ( )
∫
+∞
∞
−
+ σ
σ
σ σ
d
e
B
Z u
i
. (2.52b)
However, our interest is not in ( )
v
X or ( )
u
Y , per se, but rather in quantities related to them. We are
primarily interested in the radiation an observer receives at the instant of his (or her) crossing the
Cauchy horizon. To evaluate this quantity, we consider a freely falling observer following a radial
geodesic. The four-velocity, U, of the observer is given by the following equations
2
2
2
2
2
1 E
r
L
r
d
dr
=
+
∆
+
τ
;
∆
=
2
r
E
d
dt
τ
; and 2
r
L
d
d
=
τ
ϕ
, where 2
2
2 ∗
+
−
=
∆ Q
Mr
r ,
for 0
=
L ; thus,
E
r
U t
∆
=
2
,
2
/
1
2
2
2
∆
−
∆
=
∗
r
E
r
U r
, and 0
=
= ϕ
θ
U
U , (2.53)
where, consistently with the time-like character of the coordinate r in the interval +
− <
< r
r
r , we
have chosen the positive square-root in the expression for ∗
r
U . Also, it should be noted that we are
allowed to assign negative values for E since the coordinate t is space-like in the same interval.
With the prevalent radiation-field expressed in terms of ( )
t
r
Z ,
∗ , a measure of the flux of radiation,
F , received by the freely falling observer is given by
F =
∆
−
+
∆
= ∗
r
t
j
j
Z
r
E
EZ
r
Z
U ,
2
2
,
2
, . (2.54)
We have seen that as we approach the Cauchy horizon (cf.equations (2.50) – (2.52)),
( ) ( ) ( )
∗
∗
∗ +
+
−
→ r
t
Y
r
t
X
t
r
Z , . (2.55)
Accordingly,
u
v
t Y
X
Z ,
,
, +
→ − and u
v
r Y
X
Z ,
,
, +
−
→ −
∗
; (2.56)
and the expression (2.54) for F becomes
F
∆
−
+
+
∆
−
−
∆
→ − 2
2
,
2
2
,
2
r
E
E
Y
r
E
E
X
r
u
v . (2.57)
19. 19
On EF , v remains finite while ∞
→
u ; therefore,
t
r +
→
∗ , −
−
∗ −
−
→
→ r
r
r
u lg
1
2
κ
as −
→ r
r on EF . (2.58)
Also for 0
>
E , the term in v
X −
, remains finite while the term in u
Y, has a divergent factor (namely,
∆
/
1 ). Hence,
u
u
EF e
EY
r
r
r −
−
+
−
−
−
→ κ
,
2
2
F ( ∞
→
u on EF ). (2.59)
On '
EC , u remains finite while ∞
→
v ; therefore,
t
r −
→
∗ , −
−
∗ −
−
→
→ r
r
r
v lg
1
2
κ
as −
→ r
r on '
EC . (2.60)
And for 0
<
E , the term in u
Y, remains finite while the term in v
X −
, has the divergent factor.
Hence,
v
v
EC e
X
E
r
r
r −
−
−
+
−
−
+
→ κ
,
2
'
2
F ( ∞
→
v on '
EC ). (2.61)
We conclude from the equations (2.59) and (2.61) that the divergence, or otherwise, of the received
fluxes on the Cauchy horizon, at EF and '
EC , depend on
( )
( )
( )
∫
∞
+
∞
− −
−
= σ
σ
σ
σ
σ σ
d
e
Z
T
R
i
Y u
i
u
1
1
, ; (2.62) and
( )
( )
∫
∞
+
∞
−
−
−
−
−
= σ
σ
σ
σ σ
d
e
Z
T
i
X v
i
v 1
1
1
, , (2.63)
where we have substituted for ( )
σ
A and ( )
σ
B from equation (2.45). Furthermore, we can rewrite
the eq. (2.57) as follows:
( )
( ) ( )
( )
( )
∆
−
+
−
−
+
∆
−
−
−
−
∆
→ ∫
∫
∞
+
∞
−
−
∞
+
∞
− 2
2
1
1
2
2
1
2
1
1
r
E
E
d
e
Z
T
R
i
r
E
E
d
e
Z
T
i
r u
i
v
i
σ
σ
σ
σ
σ
σ
σ
σ
σ σ
σ
F
(2.63b)
In particular, if we wish to evaluate the infinite integrals, as is naturally suggested, by contour
integration, closing the contour appropriately in the upper half-plane and in the lower half-plane,
then we need to specify the domains of analyticity of ( )
σ
A and ( )
σ
B , as defined in equations
(2.45). Returning then to the definitions of ( )
σ
A and ( )
σ
B , we can write
( ) ( )
( )
( ) ( )
[ ]
σ
σ
σ
σ
σ
σ +
−
=
−
−
= ,
,
,
2
1
1
2
1
1
x
f
x
f
i
T
R
B (2.64)
and
( )
( )
( ) ( )
[ ]
σ
σ
σ
σ
σ −
−
=
−
= ,
,
,
2
1
1
2
1
1
x
f
x
f
i
T
A , (2.65)
20. 20
where for convenience, we have written x in place of ∗
r and ( )
σ
±
,
1 x
f , and ( )
σ
+
,
2 x
f are solutions
of the one-dimensional wave equations which satisfy the boundary conditions
( ) x
i
e
x
f σ
σ m
→
±
,
1 +∞
→
x ; and ( ) x
i
e
x
f σ
σ ±
→
±
,
2 −∞
→
x . (2.66)
Also ( )
σ
−
,
2 x
f satisfies the integral equation
( ) ( ) ( ) ( )
∫ ∞
−
−
−
−
+
=
−
x
x
i
dx
x
f
x
V
x
x
e
x
f '
,
'
'
'
sin
, 2
2 σ
σ
σ
σ σ
. (2.67)
The corresponding integral equation satisfied by ( )
σ
±
,
1 x
f is
( ) ( ) ( ) ( )
∫
∞
± −
−
=
x
x
i
dx
x
f
x
V
x
x
e
x
f '
,
'
'
'
sin
, 1
1 σ
σ
σ
σ σ
m
m . (2.68)
Adapting a more general investigation of Hartle and Wilkins to the simpler circumstances of our
present problem, we can determine the domains of analyticity of the functions, ( )
σ
−
,
2 x
f and
( )
σ
±
,
1 x
f , in the complex σ -plane, by solving the Volterra integral-equations (2.67) and (2.68) by
successive iterations. Thus, considering equation (2.67), we may express its solution as a series in
the form
( ) ( )
( )
∑
∞
=
−
−
+
=
−
1
2
2 ,
,
n
n
x
i
x
f
e
x
f σ
σ σ
, (2.69)
where
( )
( ) ( ) ( ) ( )
( )
∫ ∞
−
−
−
−
=
−
x
n
n
x
f
x
V
x
x
dx
x
f σ
σ
σ
σ ,
sin
, 1
1
2
1
1
1
2 . (2.70)
By this last recurrence relation,
( )
( ) ( ) ( )
∫ ∫ ∏
∫ ∞
− ∞
−
=
−
−
∞
−
− −
=
−
1 1
0
1
1
2
1
2
sin
...
,
x x n
i
x
i
i
i
i
n
x
n n
n
e
x
V
x
x
dx
dx
dx
x
f σ
σ
σ
σ (2.71)
where x
x =
0 ; or, after some rearrangement,
( )
( )
( )
( )
[ ] ( )
{ }
∫ ∫ ∏
∞
− ∞
−
=
−
−
−
−
−
=
−
0 1
1
1
2
1
2 1
...
2
,
x x n
i
i
x
x
i
n
n
x
i
n n
i
i
x
V
e
dx
dx
i
e
x
f σ
σ
σ
σ . (2.72)
Thence, the eq. (2.69) can be rewritten also as follows:
( ) ∑
∞
=
−
+
=
−
1
2 ,
n
x
i
e
x
f σ
σ
( )
( )
[ ] ( )
{ }
∫ ∫ ∏
∞
− ∞
−
=
−
−
−
−
−
0 1
1
1
2
1 1
...
2
x x n
i
i
x
x
i
n
n
x
i
n
i
i
x
V
e
dx
dx
i
e σ
σ
σ
. (2.72b)
Since
( )→
x
V constant x
e +
κ
2
( )
−∞
→
x , (2.73)
21. 21
it is manifest that each of the multiplicands in (2.72) tends to zero, exponentially, for −∞
→
x for
all
+
−
> κ
σ
Im . (2.74)
In view of the asymptotic behaviour (2.73) for ( )
x
V , we may, compatible with this behaviour,
expect a representation of ( )
x
V for 0
<
x , in the manner of a Laplace transform, by
( ) ( )
∫
∞
+
=
κ
µ
µ
µ
2
x
e
d
x
V V , (2.75)
where ( )
µ
V includes δ -functions at various locations, i.e., ( )
µ
V is a distribution in the technical
sense. With the foregoing representation for ( )
x
V , the first iterate, ( )
( )
σ
−
,
1
2 x
f , of the solution for
( )
σ
−
,
2 x
f , becomes
( )
( ) ( )
[ ] ( )
∫
∫
∞
∞
−
−
−
+
−
=
−
κ
µ
σ
σ
µ
µ
σ
σ
2
2
1
1
2
1
1
1
2
, x
x x
x
i
x
i
e
d
e
dx
i
e
x
f V ; (2.76)
or, inverting the order of the integrations, we have
( )
( ) ( ) ( )
[ ] ( )
∫ ∫
∞
∞
−
−
−
−
+
−
=
−
κ
µ
σ
µ
σ
µ
µ
σ
σ
2
2
1
1
2
1
1
1
2
,
x x
x
x
x
i
x
x
i
e
e
dx
e
d
i
e
x
f V . (2.77)
After effecting the integration over 1
x , we are left with
( )
( ) ( )
( )
∫
∞
−
+ −
=
−
κ
µ
σ
σ
µ
µ
µ
µ
σ
2
1
2
2
, x
x
i
e
i
d
e
x
f
V
. (2.78)
From this last expression, it is evident that ( )
( )
σ
−
,
1
2 x
f has singularities along the negative imaginary
axis beginning at +
−
= κ
σ
Im . Thence, the eq. (2.76) can be rewritten also as follows:
( )
( ) ( )
[ ] ( )
∫
∫
∞
∞
−
−
−
+
−
=
−
κ
µ
σ
σ
µ
µ
σ
σ
2
2
1
1
2
1
1
1
2
, x
x x
x
i
x
i
e
d
e
dx
i
e
x
f V
( )
( )
∫
∞
−
+ −
=
κ
µ
σ
σ
µ
µ
µ
µ
2 2
x
x
i
e
i
d
e
V
. (2.78b)
Entropy of strings and black holes: Schwarzschild geometry in D = d + 1 dimensions [3]
The black hole metric found by solving Einstein’s equation in D dimensions is given by
2
2
2
1
3
3
2
3
3
2
1
1 −
−
−
−
−
−
−
−
−
−
= D
D
D
S
D
D
S
d
r
dr
r
R
dt
r
R
d ω
τ . (2.79)
The horizon is defined by
( )
( )
3
1
2 2
3
16 −
−
−
Ω
−
=
D
D
S
D
GM
D
R
π
(2.80)
22. 22
and its D-2 dimensional “area” is given by
2
2
2
2
−
−
−
−
Ω
=
Ω
= ∫ D
D
S
D
D
S R
d
R
A . (2.81)
Furthermore, the entropy is given by
( )
G
GM
G
A
S D
D
D
4
2
4
2
3
2
−
−
−
Ω
=
= . (2.82)
The entropy in equation (2.82) is what is required by black hole thermodynamics.
To extend a static spherically symmetric geometry to D = d + 1 dimensions, the metric can be
assumed to be of the form
( )
2
1
2
2
1
2
2
2
1
2
2
1
2
2
2
2
2
2
sin
...
sin
...
sin −
−
∆
Φ
+
+
+
+
+
−
= d
d d
d
d
r
dr
e
dt
e
ds θ
θ
θ
θ
θ
θ . (2.83)
Using orthonormal coordinates, the t
t
G
ˆ
ˆ component of the Einstein tensor can be directly calculated
to be of the form
( ) ( )( )( )
−
−
−
+
∆
−
−
= ∆
−
∆
−
2
2
2
ˆ
ˆ 1
2
3
2
'
2 e
r
D
D
r
e
D
Gt
t
. (2.84)
Form Einstein’s equation for ideal pressureless matter, κρ
=
t
t
G ˆ
ˆ . This means
( ) ( )
[ ] κρ
−
=
−
−
−
= ∆
−
−
−
2
3
2
ˆ
ˆ 1
2
2
e
r
dr
d
r
D
G D
D
t
t
(2.85)
which can be solved to give
( ) ( )
( )
∫ −
−
−
∆
−
Ω
−
=
−
=
−
r
D
D
D M
D
dr
r
r
D
r
e
0
2
2
3
2
2
2
'
'
'
2
2
1
κ
ρ
κ
(2.86)
where the solid angle is given by
( )
( )
( )
2
/
1
2 2
/
1
2
−
Γ
=
Ω
−
−
D
D
D
π
. The r
r
Gˆ
ˆ component of the Einstein tensor
satisfies
( ) ( )( )( ) ( )( )
+
∆
+
Φ
−
−
−
=
−
−
−
+
Φ
−
−
−
=
∆
−
∆
−
∆
−
κρ
r
e
D
e
r
D
D
r
e
D
Gr
r
2
2
2
2
ˆ
ˆ '
'
2
1
2
3
2
'
2 . (2.87)
For pressureless matter in the exterior region ( )
P
=
= 0
ρ , we can immediately conclude that
∆
−
=
Φ . Defining the Schwarzschild radius
( ) 2
3
2
2
−
−
Ω
−
=
D
D
S
D
M
R
κ
we obtain the form of the
metric
Φ
−
∆
−
=
−
= 2
3
2
1 e
r
R
e
D
S
. (2.88)
23. 23
If we write ( ) Φ
≡ 2
e
r
F , a useful shortcut for calculating the solution to Einstein’s equation (2.85) is
to note its equivalence to the Newtonian Poisson equation in the exterior region
( ) κρ
−
=
∇ r
F
2
, ( ) Newton
r
F φ
2
1+
= . (2.89)
The Hawking temperature can be calculated by determining the dimensional factor between the
Rindler time and Schwarzschild time. Near the horizon, the proper distance to the horizon is given
by
( )
1
3
2
3
−
−
=
−
D
S
S
R
r
D
R
ρ (2.90)
which gives the relation between Rindler time/temperature units and Schwarzschild
time/temperature units
( )dt
R
D
d
S
2
3
−
=
ω . (2.91)
Thus, the Hawking temperature of the black hole is given by
( )
S
H
R
D
T
2
3
2
1 −
=
π
. (2.92)
Using the first law of thermodynamics, the entropy can be directly calculated to be of the form
( )
κ
π A
D
S
3
2 −
= . (2.93)
Substituting the form ( )G
D 3
8 −
= π
κ for the gravitational coupling gives the previous results in
D-dimensions (see eq. (2.82)). Furthermore, if we substitute ( )G
D 3
8 −
= π
κ in the eq. (2.86), we
obtain the following expression:
( ) ( ) ( )
( )
∫ −
−
−
∆
−
Ω
−
−
⋅
=
−
=
−
r
D
D
D M
D
G
D
dr
r
r
D
r
e
0
2
2
3
2
2
3
8
2
'
'
'
2
2
1
π
ρ
κ
. (2.93b)
We note that this expression can be related by the number 8, with the “modes” that
correspond to the physical vibrations of a superstring by the following Ramanujan function:
( )
+
+
+
⋅
=
−
∞
−
∫
4
2
7
10
4
2
11
10
log
'
142
'
cosh
'
cos
log
4
3
1
8
2
'
'
4
0
'
2
2
w
t
itw
e
dx
e
x
txw
anti
w
w
t
w
x
φ
π
π
π
π
.
24. 24
Thence, we obtain the following mathematical connection:
( ) ( ) ( )
( )
⇒
Ω
−
−
⋅
=
−
=
− ∫ −
−
−
∆
−
r
D
D
D M
D
G
D
dr
r
r
D
r
e
0
2
2
3
2
2
3
8
2
'
'
'
2
2
1
π
ρ
κ
( )
+
+
+
⋅
⇒
−
∞
−
∫
4
2
7
10
4
2
11
10
log
'
142
'
cosh
'
cos
log
4
3
1
2
'
'
4
0
'
2
2
w
t
itw
e
dx
e
x
txw
anti
w
w
t
w
x
φ
π
π
π
π
. (2.93c)
The quantization of the string defines a 1 + 1 dimensional quantum field theory in which the (D – 2)
transverse coordinates ( )
σ
i
X play the role of free scalar fields. The spatial coordinate of this field
theory is 1
σ , and it runs from 0 to π
2 .
The entropy and energy of such a quantum field theory can be calculated by standard means. The
leading contribution for large energy is (setting the string length 1
=
S
l )
( )
2
2
−
= D
T
E π , ( )
2
2 −
= D
T
S π . (2.94)
Using
2
2
m
E = and eliminating the temperature yields ( ) m
D
S π
2
2 −
= or, restoring the units,
i.e. the string length S
l :
( ) S
m
D
S l
π
2
2 −
= . (2.95)
Subleading corrections can also be calculated to give
( ) ( )
S
S m
c
m
D
S l
l log
2
2 −
−
= π (2.96)
where c is a positive constant. The entropy is the log of the density of states. Therefore the number
of states with mass m is
( )
( )
S
c
S
m m
D
m
N l
l
2
2
exp
1
−
= π . (2.97)
The formula (2.97) is correct for the simplest bosonic string, but similar formulae exist for the
various versions of superstring theory.
Now let us compare the entropy of the single string with that of n strings, each carrying mass
n
m
.
Call this entropy ( )
m
Sn . Then
( ) ( )
n
m
nS
m
Sn /
= (2.98)
or
25. 25
( ) ( )
−
−
=
n
m
nc
m
D
m
S S
S
n
l
l log
2
2 π . (2.99)
Obviously for large n the single string is favored. For a given total mass, the statistically most
likely state in free string theory is a single excited string. Thus it is expected that when the string
coupled goes to zero, most of the black hole states will evolve into a single excited string.
These observations allow us to estimate the entropy of a black hole. The assumptions are the
following:
- A black hole evolves into a single string in the limit 0
→
g
- Adiabatically sending g to zero is an isentropic process; the entropy of the final string is the
same as that of the black hole
- The entropy of a highly excited string of mass m is of order S
m
S l
≈ (2.100)
- At some points as 0
→
g the black hole will male a transition to a string. The point at which
this happens is when the horizon radius is of the order of the string scale.
The string and the Planck length scales are related by
2
2
2 −
−
= D
p
D
S
g l
l . (2.101)
At some value of the coupling that depends on the mass of the black hole, the string length will
exceed the Schwarzschild radius of the black hole. This is the point at which the transition from
black hole to string occurs.
Let us begin with a black hole of mass 0
M in a string theory with coupling constant 0
g . The
Schwarzschild radius is of order
( ) 3
1
0
−
≈ D
S G
M
R , (2.102)
and using
2
2 −
≈ D
S
g
G l (2.103)
we find
( ) 3
1
2
0
0
−
≈ D
S
S
S
g
M
R
l
l
. (2.104)
Thus for fixed 0
g if the mass is large enough, the horizon radius will be much bigger than S
l . Now
start to decrease g . In general the mass will vary during an adiabatic process. Let us call the g -
dependent mass ( )
g
M . Note
( ) 0
0 M
g
M = . (2.105)
The entropy of a Schwarzschild black hole (in any dimension) is a function of the dimensionless
variable P
Ml . Thus, as long as the system remains a black hole,
( ) =
P
g
M l constant. (2.106)
Since 2
2
−
≈ D
S
P g
l
l we can write equation (2.106) as
26. 26
( )
2
1
2
2
0
0
−
=
D
g
g
M
g
M . (2.107)
Now as 0
→
g the ratio of the g -dependent horizon radius to the string scale decreases. From
equation (2.80) it becomes of order unity at
( ) 3
2 −
−
≈ D
S
D
P
g
M l
l (2.108)
which can be written
( ) 2
1
g
g
M S ≈
l . (2.109)
Combining equations (2.107) and (2.109) we find
( ) 3
1
0
3
2
0
−
−
−
≈ D
D
D
S G
M
g
M l . (2.110)
As we continue to decrease the coupling, the weakly coupled string mass will not change
significantly. Thus we see that a black hole of mass 0
M will evolve into a free string satisfying
equation (2.110). But now we can compute the entropy of the free string. From equation (2.100) we
find
3
1
0
3
2
0
−
−
−
≈ D
D
D
G
M
S . (2.111)
This is a very pleasing result in that it agrees with the Bekenstein-Hawking entropy in equation
(2.82). In this calculation the entropy is calculated as the microscopic entropy of fundamental
strings.
Appendix A
FINAL TABLES
In this tables we have the various fractional powers of Phi ...
61803399
,
1
2
1
5
=
+
=
Φ that we
have obtained by the following expression: ( )×
Φ
∑ 7
/
n
fractions or numbers of the first line; for n
included in the following numerical interval: [– 113; + 38] . For example:
( ) ( )
[ ] 6666667
,
2
3
4
2
3
4
)
38196601
,
0
61803399
,
1
(
3
4
6666667
,
2
7
/
14
7
/
7
=
×
=
×
+
=
×
Φ
+
Φ
=
−
30. 30
8,62951461 9,70820393 12,94427191 19,41640786 10,47213595
9,13880262 10,28115295 13,70820393 20,56230590 11,09016994
9,88854382 11,12461180 14,83281573 22,24922359 12,00000000
10,47213595 11,78115295 15,70820393 23,56230590 12,70820393
11,29618127 12,70820393 16,94427191 25,41640786 13,70820393
12,00000000 13,50000000 18,00000000 27,00000000 14,56230590
12,94427191 14,56230590 19,41640786 29,12461180 15,70820393
13,96284794 15,70820393 20,94427191 31,41640786 16,94427191
14,78689326 16,63525492 22,18033989 33,27050983 17,94427191
16,00000000 18,00000000 24,00000000 36,00000000 19,41640786
16,94427191 19,06230590 25,41640786 38,12461180 20,56230590
18,27760524 20,56230590 27,41640786 41,12461180 22,18033989
Acknowledgments
I would like to thank Prof. Branko Dragovich of Institute of Physics of Belgrade (Serbia) for his
availability and friendship.
References
[1] S. Chandrasekhar – “On stars, their evolution and their stability” – Reviews of Modern
Physics, Vol. 56, No. 2, Part I, April 1984.
[2] S. Chandrasekhar – “The mathematical Theory of Black Holes” – Oxford University Press, Inc.
1992.
[3] L. Susskind, James Lindesay – “An Introduction to Black Holes, Information and the String
Theory Revolution – The Holographic Universe” – World Scientific Publishing Co. Pte. Ltd.
2005.
![1
The mathematical theory of black holes. Mathematical connections with some sectors of
String Theory and Number Theory
Michele Nardelli 2
,
1
1
Dipartimento di Scienze della Terra
Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10
80138 Napoli, Italy
2
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
Abstract
In this paper we have described in the Section 1, some equations concerning the stellar evolution
and their stability. In the Section 2, we have described some equations concerning the perturbations
of Schwarschild black-hole, the Reissner-Nordstrom solution and the Schwarzschild geometry in D
= d + 1 dimensions. Furthermore, in these sections, we have showed the mathematical connections
with some sectors of Number Theory, principally with the Ramanujan’s modular equations and the
aurea ratio (or golden ratio)
1. On some equations concerning the stellar evolution and their stability. [1]
The success of the quantum theory may be traced to two basic facts: first, the Bohr radius of the
ground state of the hydrogen atom, namely,
8
2
2
2
10
5
.
0
4
−
×
≈
me
h
π
cm, (1.1)
where h is Planck’s constant, m is the mass of the electron and e is its charge, provides a correct
measure of atomic dimensions; and second, the reciprocal of Sommerfeld’s fine-structure constant,
137
2 2
≈
e
hc
π
, (1.2)
gives the maximum positive charge of the central nucleus that will allow a stable electron-orbit
around it.
With regard the stellar structure and stellar evolution, the following combination of the dimensions
of a mass provides a correct measure of stellar masses:
2
.
29
1
2
2
/
3
≅
H
G
hc
☼
☼
☼
☼, (1.3)](https://image.slidesharecdn.com/2003-220918110915-dd53554d/85/20030515v1pdf-1-320.jpg)
