The mathematical theory of black holes. Mathematical connections with some sectors of String Theory and Number Theory

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The mathematical theory of black holes. Mathematical connections with some sectors of String Theory and Number Theory

Michele Nardelli1,2

1Dipartimento di Scienze della Terra

Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy

2Dipartimento 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,

₂ ₂ 8 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 e2 ≈

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:

1₂ 29.2 2 / 3 ≅     H G hc ☼ ☼☼ ☼, (1.3)

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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:

4/3

( )

4/3 3 / 1 4 4 1 3

_ρ

_β

_ρ

β

µ

H a C k P =         ₋       = . (1.4)

There is a general theorem (Chandrasekhar, 1936) which states that the pressure, P , at the centre of _c

a star of a mass M in hydrostatic equilibrium in which the density,

ρ

( )

r , at a point at a radial distance, r , from the centre does not exceed the mean density,

ρ

( )

r , interior to the same point r , must satisfy the inequality,

4/3 2/3 3 / 1 3 / 2 3 / 4 3 / 1 3 4 2 1 3 4 2 1 M G P M G

π

ρ

c

π



ρ

c      ≤ ≤       , (1.5)

where

ρ

denotes the mean density of the star and

ρ

c 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,

2/3 3 / 1 3 / 1 4 4 6 1 3 GM a H k c c _      ≤         ₋      

_π

β

µ

, (1.6) or, equivalently, ₃_/₂ 2 / 1 4 4 2 / 1 1 1 3 6 G a H k M c c         ₋             ≥

β

µ

π

, (1.7)

where in the foregoing inequalities,

β

c 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 G H hc H G hc M c c _      =       ≥       −

β

π

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,

5,48 1 2 / 1 4 2 ≥       − c c M

β

µ

☼. (1.10)

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This inequality provides an upper limit to

(

1−

β

_c

)

for a star of a given mass. Thus, 1−

β

c≤1−

β

_∗, (1.11)

where

(

1−

β

_∗

)

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 _      ₋ = ∗ ∗ ββ µ M ☼. (1.12)

We note that the following values:

0,80599 0,80901699 6 3 / 1 ≈ =      π ; 6 1,381976 1,39057647 2 / 1 ≈ =       π ; 2 0,18736 0,183689 135 3 = ≈ π

are well connected with various fractional powers of Phi 1,61803399... 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 p are occupied consistently with the Pauli exclusion-0 principle. If n

( )

p dp denotes the number of electrons, per unit volume, between p and p+dp, then the assumption of complete degeneracy is equivalent to the assertion

( )

8₃ p2 h p

n = π

(

p≤ p₀

)

; n

( )

p =0

(

p> p₀

)

. (1.13)

The value of the threshold momentum, p₀, 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 E_kin 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 p T dp h E π , (1.16)

where v_p and T_p are the velocity and the kinetic energy of an electron having a momentum p . If we set

v_p = p/m and T_p = p2/2m, (1.17)

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5/3 2 3 / 2 5 0 3 3 20 1 15 8 n m h p m h P       = = π π (1.18) and 5/3 2 3 / 2 5 0 3 3 40 3 10 8 n m h p m h E_kin       = = π π . (1.19)

Thence, we obtain the following expressions:

=

∫

0 0 3 3 3 8 p pdp v p h P π 5/3 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 p T dp h E π 5/3 2 3 / 2 5 0 3 3 40 3 10 8 n m h p m h      = = π π . (1.19b)

34345 , 8 37758 , 8 3 8π ₌ _≈ ; 3 0,048486 0,04863 20 1 2/3 ₌ _≅       π ; 0,072729 0,07294 3 40 3 2/3 ₌ _≅       π ;

are well connected with various fractional powers of Phi 1,61803399... 2 1 5 = + = Φ (see Appendix A)

According to the expression for the pressure given by eq. (1.18), we have the relation

P=K1ρ5/3 where

(

)

5/3 2 3 / 2 1 3 20 1 H m h K e

µ

π

     = , (1.20)

where µ_e is the mean molecular weight per electron. We can rewrite the expression also as follows:

(

)

5/3 5/3 2 3 / 2 3 20 1

_ρ

µ

π

m H h P e       = . (1.20b)

Thus, already for a degenerate star of solar mass (with µe=2) the central density is

6 10 19 .

4 × g

cm3; and this density corresponds to a threshold momentum p₀ =1.29mc 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 ₂ ₂

)

1/2 / 1 p m c m p v_p + = and T_p =mc2

[

(

1+ p2/m2c2

)

1/2−1

]

, (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

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P= Af

( )

x and ρ=Bx3, (1.22) where ₃ 5 4 3h c m A=π , ₃ 3 3 3 8 h H c m B= π µe (1.23) and f

( )

x =x

(

x2 +1

) (

1/2 2x2 −3

)

+3sinh−1x. (1.24) Thence, P can be rewritten also as follows:

₃ 5 4 3h c m P=π x

(

x2+1

) (

1/2 2x2 −3

)

+3sinh−1x. (1.24b) And similarly E_kin = Ag

( )

x , (1.25) g

( )

x =8x3

[

(

x2 +1

)

1/2 −1

]

− f

( )

x . (1.26) Thence, the eq. (1.25) can be rewritten also as follows:

E_kin = ₃ 5 4 3h c m π

[

₍

₎

]

1 1 8x3 x2+ 1/2− −x

(

x2 +1

) (

1/2 2x2−3

)

+3sinh−1x. (1.26b)

According to eqs. (1.22) and (1.23), the pressure approximates the relation (1.20) for low enough electron concentrations

(

x<<1

)

; but for increasing electron concentrations

(

x>>1

)

, the pressure tends to 4/3 3 / 1 3 8 1 hcn P       = π . (1.27)

This limiting form of relation can be obtained very simply by setting v_p =c 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 p with the aid of eq. (1.14) directly leads to eq. (1.27). The relation between 0

P and ρ corresponding to the limiting form (1.28) is

(

)

4/3 4/3 3 / 1 3 8 1

_ρ

µ

π

H hc P e       = . (1.29)

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=

∫

0 = 0 4 0 3 3 3 3 2 3 8 p p h c dp p h c P π π

(

)

4/3 4/3 3 / 1 3 8 1

ρ

µ

π

H hc e       = . (1.29b)

12538820 , 0 123093 , 0 3 8 1 1/3 ₌ _≅       π and 3 2,094395 2,12461180 2 ≈ = π ,

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, K₂, in the pressure-density relation. We have accordingly,

(

)

(

)

2 2 2 / 3 2 / 3 2 lim 5,76 1 197 . 0 018 . 2 4  = −      =       = e eH G hc G K M

µ

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 1,61803399...

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 M →Mlim.

It is more convenient to express the electron pressure in terms of ρ and β_e defined in the manner

4 3 1 1 aT T H k p e e e e β β ρ µ = − = , (1.31)

where p now denotes the electron pressure. Then, analogous to eq. (1.4), we can write _e

4/3 3 / 1 4 1 3

_ρ

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

µ

(

)

4/3 3 / 1 3 8 1 H hc e

µ

π

     , (1.33)

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the pressure p given by the classical perfect-gas equation of state will be greater than that given by e

the equation if degeneracy were to prevail, not only for the prescribed ρ and T , but for all ρ and

T having the same β_e. Inserting for a its value given in eq. (1.8), we find that the inequality (1.33) reduces to 960₄ 1− >1 e e ββ π , (1.34) or, equivalently 1−β_e>0,0921=1−β_ω. (1.35)

We note that the value 960₄ =9,85534≅9,88854382

π and 0,0921≅0,09184494,

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

4

( ) ( )

2,018 2 / 3     = G C M πβ π (1.36)

where C

( )

β

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 β =β_ω gives

( )

2 2 2 / 3 2 / 3 65 , 6 1 197 . 0 −  = −      =

µ

β

ω 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 1,61803399...

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

(

)

4/3 3 / 1 2 3 / 2 3 / 1 3 8 1 3 4 2 1 H hc K M G e

µ

π

      = <       (1.38) or, equivalently

(

)

2 2 2 / 3 74 , 1 1 16 3 ₌ ₋       < e eH G hc M

µ

π

☼. (1.39)

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Thence, we have the following connection between eq. (1.38) and eq. (1.39):

(

)

4/3 3 / 1 2 3 / 2 3 / 1 3 8 1 3 4 2 1 H hc K M G e

µ

π

      = <      

(

)

2 2 2 / 3 74 , 1 1 16 3 _ ₌ −      < ⇒ _e eH G hc M

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 1,61199 1,618 3 4 1/3 ₌ _≅       _π

and 1,74≅1,74535 are well connected with

various fractional powers of Phi 1,61803399... 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, Γ1, 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,

∆P/P=Γ₁∆

ρ

/

ρ

. (1.40) And the Newtonian criterion for stability is

Γ =

∫

MΓ

( ) ( ) ( )

r P r dM r ÷

∫

MP

( ) ( )

r dM r > 0 1 0 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 Γ₁; 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 Γ1 may be, instability will set in provided the radius is less than a certain determinate multiple of the Schwarzschild radius, R_s =2GM/c2. (1.42)

Thus, if for the sake of simplicity, we assume that Γ1 is a constant through the star and equal to 3

/

5 , then the star will become dynamically unstable for radial perturbations, if R₁<2,4R_S. And further, if Γ₁→∞, instability will set in for all R<

( )

9/8R_S. The radius

( )

9/8R_S 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

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(

)

(

)

_      +       + − − =

_∫

1 0 2 2 3 1 4 1 1 ' 1 11 2 18 5 ξ

ξ

θ

ξ

d d d n n n K , (1.44)

where θ is the Lane-Emden function in its standard normalization (θ =1 at ξ =0), ξ is the dimensionless radial coordinate,

ξ

1 defines the boundary of the polytrope (where θ =0) and

θ

'1 is the derivative of θ at

ξ

₁. If we set n=2 in the eq. (1.44), we obtain:

        +       =

_∫

1 0 2 2 3 1 4 1 1 ' 6 6 1 ξ

ξ

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

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 n→5 and the configuration becomes increasingly centrally condensed.

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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 n→5, the asymptotic behaviour of K is given by

K →2.3056/

(

5−n

)

; (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

(

ρc/ρ ≥106

)

. (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.

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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

N,r =aN+bL+cX, (2.1) v L cX r b N v r a Lr r r +      ₋ ₋ +       ₋ ₊ = , , , 1 1 , (2.2) v X 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

( )

₍

₎

(

)

(

)

(

+

)

+

(

+

) (

[

+

)

−

]

(

+

)

+ + + − + + − + + − = + ∗ n r M L X M nr Mn V M nr r M Mnr nr M V M nr r M r M V V M r Zr r rr r r 2 3 3 3 6 4 3 3 2 3 2 ₂ ₂ ₃ 2 2 , , , , , ,

(

r M

)(

nr M

) (

[

r M

) (

L r M

)

X

]

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 L,X, and N will require a further quadrature. Thus,

rewriting the following equation

_            − + − =       − + v X r X L v r L,r ,r ,r ,r 1 2 , (2.5) in the form

(

)

(

re V

)

dr d nr L e r dr d 2 −v =− −v (2.6) and replacing V by ( ) L M r Z Mr M nr V 3 3 3 + + = + , (2.7)

we obtain the equation



(

−

)

=−

[

−

(

+

)

( )+

]

     + e nr M Z 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

(

)

[

(

)

( )

]

∫

− + − ₊ + − = e nr M Z dr dr d M nr nr L e r v v 3 3 2 , (2.9)

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( )

∫

( )+ − + − − + + − = Z dr M nr e Mn Z nre L e r v v v 3 3 2 . (2.10)

Thence, we have the following expression:

( ) ( )+

[

−

(

)

( )+

]

− + − ₌₋ ₊       + + −       ₊

∫

e nr M Z 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

=− ( )+ +3₂ Φ r M Z r n L . (2.12)

With this solution for L , equation (2.7) gives

=

(

Z( )+ +Φ

)

r

n

X . (2.13)

As a consequence of these solutions for L and X ,

+ =

(

nr+ M

)

Φ r

X

L 1₂ 3 . (2.14)

To obtain the solution for N , we take the following equation

( )

(

)

(

)

(

)

(

nr M

)

(

L X

)

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 V,r

(

= X,r/n

)

on the right-hand side from the following equation

(

)

(

) (

)

r M L n X L M r r r M M r r M M r N M r r M nr Xr 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

(

)

(

)

(

nr M

)

(

L X

)

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 L+X their solutions (2.12) and (2.13), we find:

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( )

(

)

(

)

( ) 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

Z( )± →A( )±

( )

σ e−iσr

(

r_∗ →+∞

)

; ( )± → ( )±

( )

+i r∗

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

[

(

)

]

( )+

(

)

(

)

( )− ( )− ∗ +       + ∆ + + = + + Z MZr M r r M Z M i 2 2 2 ₃ ₂ _, 2 2 12 6 72 2 12 2 µ µ µ σ µ µ ; (2.20a)

[

(

)

]

( )−

(

)

(

)

( )+ ( )+ ∗ −       + ∆ + + = − + Z MZr M r r M Z M i 3 2 , 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

σ

µ

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 exp φ , (2.22)

we find that the equation we have to solve is

_, + 2− 2− ( )− =0

∗ V

i

φ

_r

σ

φ

; (2.23)

and the appropriate boundary conditions are

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Furthermore, we can rewrite the eq. (2.20a) as follows:

(

)

[

]

( )

(

)

(

)

∗ ( )− + ∗ ∗ +       _            + ∆ + + = + +

_∫

r r MZ dr i M r r M Z M i 3 2 , 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 2σ σ2 φ2 µ2 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

( )

2M −1) for value of l=4, we have the following result:

l=4; 2Mσ =1,61835+0,18832i. (2.26)

We note that the value 1,61835 is very near to the 1,61803398... 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 2 5 2 2 1 2 2 3MH Q H W UH r H µ , Λ 1( )+ = ∆5

[

1( )+ +

(

+ _∗ 2( )+ + 1( )+

)

]

2 3 2Q H MH 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

₀₃ 1₂

( )

2 ₂₃ _, ₂₃_, 2B₂₃ r B B r r B = _r = _r + , r4e2vB₀₂ =2Q_∗2

[

2T −l

( )

l+1V

]

−l

( )

l+1r2B₂₃,

(

)

23 2 ₂ 2 2 2 02 2 2 , 03 2 2 2 r L N Q B e r B e r B e r v _r + v +σ − v = _∗ + , and N_,_r =aN +bL+c

(

X −B₂₃

)

, _, 1 _, 1 _,

(

₂₃

)

2B₂₃ r B X c L v r b N v r a L_r _r _r + − −      ₋ ₋ +       ₋ ₊ = , _, 1 _, 1 2 _, 1 v_,

(

X B₂₃

)

B₀₃ r c L v r b N v r a X_r _r _r _r − +      ₊ ₋ −       ₊ ₋ −       ₋ ₊ − = ,

(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

, ( )0 3

(

2

)

4 3 _∗ − ₋ =r e Mr Q L v , X( )0 =nevr−1, B23( )0 2Q2r 3ev − ∗ − = , and B03( )0 =2Q2r 6e v

(

2Q∗2 +r2 −3Mr

)

− − ∗ . (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 = 2 1 2 2 , 1 2 2 2 2 0 1 3 2 1 2 e nr M n nrH Q H r H Q nrH e H e n N N _r v v v µ ϖ ϖ ϖ µ ϖ ϖ (2.28) = ( )0Φ− 1₂

(

nrH₂( )+ +Q_∗ H₁( )+

)

r L L µ , (2.29) = ( )0Φ+ H₂( )+ r n X X , (2.30) = ( )Φ− ∗2 1( )+ 0 23 23 H r Q B B µ , (2.31) = ( )Φ− ∗ ( )+ − 4∗

(

2( )+ + _∗ 1( )+

)

2 , 1 2 0 03 03 2 nrH Q H 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:

( )

∫

(

( ) ( )

)

∗ ( )+ − + ∗ + ₊ ₋ = 2 1 2 1 0 23 23 H r Q dr r e H Q nrH B B v µ ϖ µ , (2.34) ( )

(

( ) ( )

)

∗ ( )+ ∗

(

( )+ _∗ ( )+

)

− + ∗ + ₊ ₋ ₋ ₊ =

_∫

4 2 1 2 , 1 2 1 2 0 03 03 2 nrH Q H 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 Vi( )± (i=1,2), 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 Zi( )± . Thus, it can

be verified, the potentials are, in fact, given by

( ) i i i i i i f f dr df V =±β +β +κ ∗ ± 2 2 , (2.36) where κ =µ2

(

µ2 +2

)

, β_i =q_j, and

(

)

j i q r r f + ∆ = ₃ ₂

µ

(

i,j=1,2;i≠ j

)

, (2.37) thence, the eq. (2.36) can be rewritten also as follows

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( )

(

)

(

)

(

) (

)

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

µ

2

µ

. (2.37b)

The solutions, Z_i( )+ and Z_i( )− 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 ₃ ₂ 2 2 2 2 2

µ

σ

µ

(

i,j=1,2;i≠ j

)

. (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 Z_i( )+ and Z_i( )− .

In view of the relation (2.38) between the solutions belonging to axial and polar perturbations, the characteristic frequencies will be the same for Z_i( )+ and Z_i( )− . 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

H₁=Z₁cos

ψ

−Z₂sin

ψ

; H₂ =Z₂cos

ψ

+Z₁sin

ψ

, (2.39)

where the amplitudes H₁ and H₂ of the electromagnetic and gravitational (wave-like) disturbances (of some specified frequencies) are related to the function Z₁ and Z₂.

With regard the complex characteristic frequencies

σ

₁ and

σ

₂ (belonging to Z₁( )± and Z₂( )± ) of the quasi-normal modes for a range of values of Q_∗ and l , we obtain, for Z₁ and Q_∗ =0.9 and

2

=

l , the value 0.61939 that is very near to the 0,61803398... 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 Z_i( )± remain formally unaltered, the potential barriers, V_i( )± , 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 Z_i( )− 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

(

i,j=1,2;i≠ j

)

and

(

r₋<r<r₊,+∞>r_∗ >−∞

)

, (2.40) where ₋ − + + ∗ =r+ r −r − r−r r lg 2 1 lg 2 1

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

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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:

σ

=±i

σ

_j

[

j=1,2;n=1,2...,m

]

. (2.43) The boundary conditions we must now impose are

( )

− ∗

( )

+ ∗ ∗ →A e i r +B e i r r Z

σ

(

r→r₋+0;r_∗ →+∞

)

→ −i r∗ e σ

(

r→r₊−0;r_∗ →−∞

)

. (2.44)

The coefficients A

( )

σ

and B

( )

σ

in equation (2.44) are related to the reflexion and the transmission amplitudes,

( )

σ

( )

σ

− = = _∗ 1 1 1 T T A ,

( )

σ

( )

σ

− − = = _∗∗ 1 1 T R T R B , (2.45) so that A

( )

σ 2− B

( )

σ 2 =1. (2.46)

With regard the amplification factors, A

( )

σ 2, appropriate for the potential V₁( )± and V₂( )± for 2

2

75 .

0 M

Q_∗ = , we observe that A

( )

σ 2, and, therefore, also B

( )

σ 2, tend to finite limits as σ →0. This fact has its origin in the existence of bound states of zero energy in the potential wells, V1 and

2

V . Furthermore, we note that for σ =0.30, for V₁ we have the value 1.6168, while for V₂ 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, eiσt, of the solutions; and remembering that in the interval, r₋<r<r₊,

u=r_∗+t and v=r_∗−t, (2.47) we write, in place of equation (2.44),

Z

( )

r_∗,t →e−iσv+

[

A

( )

σ −1

]

e−iσv+B

( )

σ e+iσu. (2.48) If we now suppose that the flux of radiation emerging from D'C' is Zˆ

( )

v , then

( )

∫

+∞

( )

∞ − = Z ve dv Z iσv 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

Z

( )

r_∗,t →X

( ) ( )

v +Y u

(

v→∞;u→∞

)

, (2.50) where

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X

( )

v +∞Z

( ) ( )

σ

[

A

σ

]

e−iσvd

σ

∞ −

∫

− = 1 , (2.51) and

( )

∫

+∞

( ) ( )

∞ − =

σ

d e B Z u Y i u . (2.52)

Thence, the eq. (2.50) can be rewritten also as follows: Z

( )

r_∗,t → +∞Z

( ) ( )

σ

[

A

σ

]

e−iσvd

σ

∞ −

∫

−1

∫

+∞

( ) ( )

∞ − +

σ

d e B Z i u . (2.52b)

However, our interest is not in X

( )

v or Y

( )

u , 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 ∆=r2−2Mr+Q_∗2, for L=0; thus, Ut r E ∆ = 2 , 2 / 1 2 2 2       ₋ ∆ ∆ = ∗ r E r Ur , and Uθ =Uϕ =0, (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 Z

( )

r_∗,t , a measure of the flux of radiation,

F , received by the freely falling observer is given by

F =               ₋ ∆ + ∆ = t r_∗ 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)), Z

( )

r_∗,t →X

(

t−r_∗

) (

+Y t+r_∗

)

. (2.55)

Accordingly,

Z_,_t →X_,₋_v+Y_,_u and Z_,_r →−X_,₋_v+Y_,_u

∗ ; (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)

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On EF , v remains finite while u→∞; therefore, r_∗→+t , ₋ − ∗→− − → r r r u 2 1 lg κ as r→r− on EF . (2.58)

Also for E >0, the term in X,−v remains finite while the term in Y,u has a divergent factor (namely,

∆ / 1 ). Hence, _EF EY_ue u r r r ₋ − + − − − → κ , 2 2 F ₍_u→∞_{on EF ). (2.59)}

On EC , ' u remains finite while v→∞; therefore,

r_∗→−t , ₋ − ∗→− − → r r r v 2 1 lg κ as r→r− on EC . (2.60) '

And for E<0, the term in Y,u remains finite while the term in X,−v has the divergent factor.

Hence, EC EX ve v 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 σσ σ σ d e Z T R i Yu i u 1 1 , ; (2.62) and

∫

( )

∞ + ∞ − − −       − − =

σ

_σ

σ

d e Z T i X v 1 i v 1 1 , , (2.63)

where we have substituted for A

( )

σ

and B

( )

σ

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 i v

σ

i u

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 A

( )

σ

and B

( )

σ

, as defined in equations (2.45). Returning then to the definitions of A

( )

σ

and B

( )

σ

, we can write

( )

σσ 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)

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where for convenience, we have written x in place of r_∗ and f₁

(

x,±

σ

)

, and f₂

(

x,+

σ

)

are solutions of the one-dimensional wave equations which satisfy the boundary conditions

(

)

i x e x f ±σ → mσ , 1 x→+∞; and

(

)

x i e x f₂ ,±σ → ±σ x→−∞. (2.66) Also f2

(

x,−

σ

)

satisfies the integral equation

(

)

∫

(

) ( ) ( )

∞ − − ₊ − ₋ = − _i _x x dx x f x V x x e x f₂ , sin ' ' ₂ ', σ ' σ σ σ σ . (2.67)

The corresponding integral equation satisfied by f₁

(

x,±

σ

)

is

(

)

= ± −

∫

∞

(

−

) ( ) ( )

x x i dx x f x V x x e x f1 , sin ' ' 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, f₂

(

x,−

σ

)

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,

( )

(

)

∫

∏

(

) ( )

∞ − −∞ ₌ − − ∞ − − − = − 0 1 1 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 x₀ =x; 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 _V _x 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 _V _x e dx dx i e σ σ σ . (2.72b) Since V

( )

x → constant e2κ+x

(

x→−∞

)

, (2.73)

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it is manifest that each of the multiplicands in (2.72) tends to zero, exponentially, for x→−∞ for all

Im

σ

>−

κ

₊. (2.74)

In view of the asymptotic behaviour (2.73) for V

( )

x , we may, compatible with this behaviour, expect a representation of V

( )

x for x<0, in the manner of a Laplace transform, by

( )

∫

∞

( )

+ = κ µ µ µ 2 x e d x V V _{, (2.75)}

where V

( )

µ

_includesδ _{-functions at various locations, i.e.,}V

( )

µ

_{is a distribution in the technical} sense. With the foregoing representation for V

( )

x , the first iterate, f2( )1

(

x,−σ

)

, 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 i x x x 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 i x x x x x i e e dx e d i e x f V _{. (2.77)}

After effecting the integration over x₁, we are left with

( )

(

)

( )

(

)

∫

∞ − + − = − κ µ σ σ µ µ µ µ σ 2 1 2 2 , i x e x i d e x f V . (2.78)

From this last expression, it is evident that f₂( )1

(

x,−σ

)

has singularities along the negative imaginary axis beginning at Im

σ

=−

κ

₊. 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 i x x x 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 _dr _r _d r R dt r R dτ ω . (2.79)

The horizon is defined by

(

)

(

)

3 1 2 2 3 16 − −      − Ω − = D D S D GM D R π (2.80)

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and its D-2 dimensional “area” is given by

A=R_SD−2

∫

dΩ_D₋₂ =R_SD−2Ω_D₋₂. (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

ds2 =−e2Φdt2+e2∆dr2 +r2

(

dθ₁2+sin2θ₁dθ₂2+...+sin2θ₁...sin2θ_d₋₂dθ_d2₋₁

)

. (2.83)

Using orthonormal coordinates, the G_ttˆˆ 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 G_tt . (2.84)

Form Einstein’s equation for ideal pressureless matter, G_tˆ_tˆ =κρ. This means

ˆˆ =−

(

−₋₂

)

[

−3

(

1− −2∆

)

]

=−κρ 2 2 e r dr d r D G_tt _D D (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

( )

(

)

(

1 /2

)

2 1/2 2 − Γ = Ω − − D D D

π

. The G_rˆrˆ component of the Einstein tensor satisfies

(

)

(

)(

)

(

)

(

)(

)

_      + ∆ + Φ − − − =       − − − + Φ − − − = − ∆ − ∆ −∆

_κρ

r e D e r D D r e D Grr 2 2 2 2 ˆ ˆ 1 2 ' ' 2 3 2 ' 2 . (2.87)

For pressureless matter in the exterior region

(

ρ

=0=P

)

, 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)

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If we write

( )

≡ 2Φ

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

∇2F

( )

r =−κρ, F

( )

r =1+2φNewton. (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 D A

S= 2 −3 . (2.93)

Substituting the form

κ

=8

π

(

D−3

)

G for the gravitational coupling gives the previous results in D-dimensions (see eq. (2.82)). Furthermore, if we substitute

κ

=8

π

(

D−3

)

G 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 κ ρ π . (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 .