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11 A LINK BETWEEN BLACK HOLESAND THE GOLDEN RATIO

J. A. Nieto1

Facultad de Ciencias Fsico-Matematicas de la Universidad Autonomade Sinaloa, 80010, Culiacan Sinaloa, Mexico

and

Mathematical, Computational & Modeling Sciences Center, Arizona StateUniversity, PO Box 871904, Tempe AZ 85287, USA

Abstract

We consider a variational formalism to describe black holes solution inhigher dimensions. Our procedure clarifies the arbitrariness of the radius pa-rameter and, in particular, the meaning of the event horizon of a black hole.Moreover, our formalism enables us to find a surprising link between blackholes and the golden ratio.

Keywords: black holes, constraint Hamiltonian systems, golden ratioPacs numbers: 04.60.-m, 04.65.+e, 11.15.-q, 11.30.LyJune, 2011

1nieto@uas.uasnet.mx, janieto1@asu.edu

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It is well known that the ansatz for a spherically symmetric static blackhole solution in a d-dimensional space-time Md can be written as (see Refs.[1]-[5] and references therein)

=

e

f(r) 0 00 eh(r) 0

0 0 2(r)ij(k)

, (1)

where ij(k) determines a maximally spherically symmetric space in (d 2)

dimensions. From this metric one finds that the only non-vanishing Riccitensor components are (see Ref. [6]-[8] and references therein)

R11 = efh(

f

2+

f 2

4 f h

4+

(d 2)2

f

), (2)

R22 = f2 f

2

4+

f h

4+

(d 2)2

h

(d 2)

, (3)

Rij = eh{(h f)

2 (d 3)2}ij + k(d 3)ij . (4)

Here, we used the notation A dAdr

for any function A = A(r). Moreover, wehave k = {1, 0, 1}. Thus, the Ricci scalar R = R becomes

R = eh{f f22+ f h

2+ (d 2)(h f)

2(d 2)

(d 3)(d 2) 22}+ k(d 3)(d 2)2.

(5)

On the other hand, we have

= e f+h2 (d2), (6)

where and denote the determinant of and ij , respectively. Conse-quently, up to total derivative, the higher dimensional Einstein-Hilbert action

S =1

2

Md

R, (7)

gives

S = (d2)2

Md

[1{((d2)F)((d 3) 2

2+ 2 F

F

)}

+{k(d 3)F(d4)}].(8)

2

Here, we used the notation F e f2 and eh2 . Note that the case d = 2is exceptional. Similar conclusion can be obtained in the case of d = 3. Forour purpose it turns out convenient to assume that d 2 6= 0 and d 3 6= 0.Observe that in (8) acts as auxiliary field.

Performing variations of the action (8) with respect to F , and onederives the following equations

2

+ (d 3)

2

2 2

k(d 3)22 = 0, (9)

(d 3) 2

2+ 2

FF

k(d 3)22 = 0 (10)

and

(d4)(d3)2

2

2 F

F

+ F

F (d 3)((

F

F)

)

k(d4)(d3)2

22 = 0,

(11)

respectively. Using (1)-(4) and considering the gravitational field equations

R 12R = 0,

which can be obtained from the action (7), one can verify that the formulae(9)-(11) are consistent. In particular one can combine (9) and (10) to obtain

FF +

=

. (12)

One recognizes in this formula the typical relation between F , and ob-tained after setting R11 = 0 and R22 = 0 in (2) and (3) and making thecombination ef+hR11 +R22 = 0.

We shall assume that one may associate with (8) the expression (see Refs.[1]-[5] and references therein)

L = 12[1((d2)F){(d 3) 2

2+ 2 F

F

}

+{k(d 3)F(d4)}].(13)

The object L is a kind of Lagrangian, but one should keep in mind that there isnot time dependence in (13). So, rigorously speaking, (13) is not a Lagrangian.For this reason, and since it was derived from the action (8), let us call L in(13) Lagravity, from Lagrangian and gravity relation. We observe that the

3

variable plays the role of Lagrange multiplier. This suggests that we seek apossible analogue of (13) with constraint Hamiltonian formulation. (in orderto be consistent we shall call the kind of Hamiltonian H associated with LHagravity.) For this purpose let us first introduce the redefinition

(d2)F1, (14)of the Lagrange multiplier . In terms of the Lagravity, (13) reads

L = 12[1((d 3)

2

2+ 2

FF

) + (k(d 3)F22(d3))]. (15)

This can be simplified further by introducing the two coordinates q1 and q2 inthe following form:

eq1 (16)and

F eq2 . (17)In fact, in terms q1 and q2, one sees that (15) can be written as

L = 12[1((d 3)(q1)2 + 2(q1)(q2)) + m20], (18)

where

m20 = k(d 3)e2(d3)q1

e2q2

). (19)

Before proceeding further, it is convenient to verify that we are in the rightroute, by first writing the Euler-Lagrange equations associated with (18). Forthe coordinate q1 we have the equation

d

dr(1((d 3)q1 + q2) (d 3)m20 = 0, (20)

and for q2 we get

d

dr(1q1) m20 = 0. (21)

While for we find

2((d 3)(q1)2 + 2q1q2)m20 = 0. (22)

The idea now is to show that from these equations one can derive (9)-(11).For this purpose, one first note that from (20) and (21) one finds

4

ddr(1q2) = 0. (23)

Thus, by writing (21), (22) and (23) in terms F , and we obtain

(d 3) 2

2+

+FF

k(d 3)22 = 0, (24)

(d 3) 2

2+

2FF

k(d 3)22 = 0 (25)

and

(d 2)FF

FF

+FF = 0, (26)

respectively. We first note that (25) is just (10). Now, multiplying (24) by(d 3) and combining the resultant formula with (25) and (26) one sees that(11) follows. Finally, it is not difficult to obtain (9) from (24) and (25).

For our goal it turns out convenient to solve directly the equations (21)-(23). Combining (21) and (22) one formally get

d

dr(1q1) = (d 3)(q1)2 + 2q1q2 (27)

or

q1 + q1 = (d 3)(q1)2 + 2q1q2. (28)

This expression suggests that we define the quantity

e(d2)q1eq2, (29)which is, of course, consistent with (14). In terms of , (28) gives

(d 2)(q1)2 + q1q2 q1 + q1 = (d 3)(q1)2 + 2q1q2. (30)

Simplifying this equation, one sees that (30) is reduced to

(q2 +

)q1 = q1 + (q1)2 (31)

or

5

(q2 +

) =

q1

q1+ q1. (32)

But since q1 = ln and q2 = lnF one discovers that (32) can be rewritten as(12). The solution of the formula (32) is

(q2 + ln) = ln q1 + q1 + ln, (33)

where is a constant. Notice that (33) can also be written as

F = . (34)On the other that from (23) we must have

= q2, (35)

where is another constant. In terms of , F and the equation (35) becomes

= (d2)F . (36)Therefore, using (36) the equation (34) leads to

FF = (d2)

, (37)

where = . The expression (37) implies the equation

d

dr(F2) = d

dr( 2

(d 3)(d3) ), (38)

whose solution is

F2 = a b(d3)

. (39)

Here, a is a constant and b = 2(d3)

. Moreover, one can verify that (39) satisfies

(25) if a = k2. In the Newtonian limit one can set a = k and b = GdM2(d3)c2

,where Gd is the d-dimensional Newton gravitational constant, M is the massassociated with the black hole and c is the light velocity.

Considering (39) one can also determine . In fact, from (34) we see that

2 =22

F2 =22

a b(d3)

. (40)

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Writing F = e f2 and = eh2 one recognizes in (39) and (40) the traditionalsolutions of a black hole. What it is interesting about our formalism is that weobtained these solutions by using the Lagravity (18) rather than the Einstein-Hilbert field equations.

Another advantage of our formalism is that we can now shed some light onthe meaning of the event horizon. Suppose we set = r. From (40) one seesthat is singular when a b

r(d3)= 0. This surface defines the so called event

horizon. It has been always argued that this is not true singularity becauseone can find another set of coordinates (in particular the Kruskal-Szekerescoordinates) that avoids such a singularity. From our formalism, however, (or ) plays a role of a Lagrange multiplier and therefore from (40) one seesthat setting = r gives 2 = 1 and this is equivalent to fix the gauge associatedwith . So, from our perspective the event horizon does not determine a truesingularity because, in relation with the parameter r, such a surface is gaugedependent.

Now, let us compute the analogue of the canonical momenta p1 and p2associated with the coordinates q1 and q2, respectively. From (18) one gets

p1 =Lq1

= 1((d 3)q1 + q2) (41)and

p2 =Lq2

= 1q1. (42)

Considering these results one can now show that (18) can be obtained fromthe first order Lagravity

L = q1p1 + q2p2 2((d 3)(p2)2 + 2p1p2 m20). (43)

The Lagravity (18) can also be written as

L = 12[1(qaqbab) + m

20], (44)

where the metric ab is given by

ab =

((d 3) 1

1 0

). (45)

Similarly, we can write (43) in the following form:

L = qapa 2H. (46)

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Here,

H = abpapb m20 (47)with

ab =

(0 11 (d 3)

). (48)

Note that (48) is the inverse matrix of (45). Of course, H can be interpretedas constraint Hagravity. In fact, it can be shown in straightforward way thatthis constraint satisfies the analogue condition of a first class constraint andtherefore it can be interpreted as the gauge generator of the variable r (seeRef. [9] and references therein).

It is also interesting to write the second order Lagravity

L = 12m0(q

aqbab)1/2, (49)

which, by using the corresponding Euler-Lagrange equation for , can beobtained from (44). It is straightforward to see that the associated action

S =12

drm0(q

aqbab)1/2, (50)

is invariant under reparamitrazation r = r(r). This explain the reason forexistence of the arbitrary function and the Hagravity constraint H.

If the cosmological constant is included via the usual extended actionS = 1

2

Md

(R 2) one can show that the Lagravity (49) also followsbut with the mass m20 now given by

m20 = k(d 3)e2(d3)q1

e2q2 2

d 2e2(d2)q1e2q

2

. (51)

It is worth mentioning that, in four dimensions, that is in d = 4, from ourformalism arises an intriguing and fascinating result. In such a case the metric(45) is reduced to

ab =

(1 11 0

). (52)

Of course, the matrix (52) can be diagonalized. We first find the eigenvaluesby computing the determinant

1 11 = 2 1 = 0. (53)

8

Surprisingly, one recognizes in (53) the famous formula which determines thegolden ratio. In fact, the two roots of (53) are

1 =1 +

5

2(54)

and

2 =15

2. (55)

The formula (54) is the so called golden ratio or golden number (see Refs.[10]-[13] and references therein).

In terms of 1 and 2 the Lagravity (49) becomes

L = 12(5)1/2

m0(abab)

1/2, (56)

where

1 = 1q1 + q2,

2 = 2q1 + q2.

(57)

Here, ab = diag(1,1). Thus, the expression (56) formally establishes aconnection between the golden ratio and black holes. (It is worth mentioningthat in Ref. [14] a different relation between the golden ratio and black holeshas been found. In fact, in such a reference it is shown that rotating blackholes make a phase transition when the ratio of the square of its mass to thesquare of its angular momentum is equal to the golden ratio.)

Summarizing we have developed a variational method that can help toclarify the meaning of the black holes solution in any dimension. Our procedureis based in an analogy with the constraint Hamiltonian formalism. As a rewardfor our efforts we found an interesting link between two fascinating concepts:black holes and the golden ratio.

Acknowledgments: I would like to thank to E. R. Estrada and L. A. Beltranfor helpful comments. This work was partially supported by PROFAPI-UAS,2011.

.

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References

[1] L. Andrianopoli, R. DAuria, S. Ferrara and M. Trigiante, Lecture Notesin Phys. 737 (2008) 661; arXiv: hep-th/0611345.

[2] A. Ceresole and G. DallAgata, JHEP 0703 (2007) 110; arXiv:hep-th/0702088.

[3] L. Andrianopoli, R. DAuria, E. Orazi and M. Trigiante, JHEP 0711(2007) 032; arXiv: 0706.0712.

[4] G. Lopes Cardoso, A. Ceresole, G. DallAgata, J. M. Oberreuter and J.Perz, JHEP 0710 (2007) 063; arXiv: 0706.3373.

[5] B. Vercnocke, Hidden Structures of Black Holes, PhD. Thesis, Depart-ment of Physics and Astronomy, Katholieke Universiteit Leuven Facultyof Science (2010).

[6] C. Castro, J. A. Nieto Int. J. Mod. Phys. A 22 (2007) 2021.

[7] C. Castro, J. A. Nieto, L. Ruiz and J. Silvas, Int. J. Mod. Phys. A 24(2009) 1383.

[8] I. S. Sokolnikoff, Tensor Analysis: Theory and Applications to Geometryand Mechanics of Continua, (Lincoln, LIN, United Kingdom 1976).

[9] V. M. Villanueva, J. A. Nieto, L. Ruiz and J. Silvas, J. Phys. A 38 (2005)7183; hep-th/0503093.

[10] H. D. Ebbinghaus and J. H. Ewing Ed., Numbers (Springer, 1991).

[11] M. Livio, Golden Ratio, (Broaway Books, 2002).

[12] G. E. Runion, Golden Section, (Dale Seymour Publications, 1990).

[13] S. Olsen, The Golden Section: Natures Greatest Secret, (BloomsburyPublishing USA, 2006)

[14] P. C. W. Davies, Class. Quan. Grav. 6 (1989) 1909.

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