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A REGULAR BLACK HOLE IN THE BRANEWORLD

ALEXIS LARRANAGA1,a, CARLOS BENAVIDES1,b, CHRISTIAN RODRIGUEZ2,c

1National Astronomical Observatory. National University of Colombia

E-maila: [email protected]

E-mailb: [email protected] of Physics. National University of Colombia

E-mailc: [email protected]

Received May 13, 2013

We find a new black hole solution in Randall-Sundrum II braneworld scenario inthe presence of an anisotropic perfect fluid in the brane inspired by the noncommutative

geometry, thought to be an effective description of a quantum-gravitational spacetime.

We find that the curvature singularity at the origin is smeared out by the noncommu-

tative fluctuations and study the horizon structure to show that the geometry admits

two, one or no horizon depending on the value of the mass parameter M with respect to

two threshold masses which depend on the noncommutative geometry induced minimal

length. On the thermodynamics side, we show that the black hole temperature, instead

of a divergent behaviour at small scales, admits both a minimum and a maximum value,

before cooling down towards a zero temperature remnant.

Key words: quantum aspects of black holes, thermodynamics, strings and

branes.

PACS : 04.70.Dy, 11.25.-w, 05.70.-a.

1. INTRODUCTION

The braneworld scenario describes our 4-dimensional world as a brane that is

embedded in a higher dimensional bulk and that supports all gauge fields excluding

the gravitational field that lives in the whole spacetime. There are many braneworld

models in the cosmological context as well as descriptions of local self-gravitating

objects. Particularly, black hole solutions on the brane are interesting because they

have considerably richer physical aspects than black holes in general relativity [1–9].

As well as the existence of branes, another of the outcomes of string theory

is that target spacetime coordinates become noncommuting operators on a D-brane

[10, 11]. Thus, string-brane coupling puts in evidence the necessity of spacetime

quantization as well as the noncommutativity of spacetime coordinates, which can

be encoded in a commutator of the form

[xµ, xν ] = iµν , (1)

where µν is an antisymmetric matrix that can be diagonalized as µν = 2diag(ij, ij,. . .) and is a constant with the dimensions of length that determines the fundamental

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58 Alexis Larranaga, Carlos Benavides, Christian Rodriguez 2

cell of spacetime in a similar way as discretizes the phase space [12]. An importantresult obtained by Smailagic and Spallucci [13,14] shows that the effects of noncom-

mutativity can be taken into account by keeping the standard form of the Einstein

tensor in the field equations and introducing a modified energy–momentum tensor.

This fact gives rise to a series of solutions known as noncommutative geometry in-

spired black holes, whose line element singularities are cured by the existence of the

minimal length [15–19]. The strategy to derive these solutions consists in prescrib-

ing an adequate energy-momentum tensor (i.e. one that accounts for the noncommu-

tative fluctuations of the manifold at the origin and that vanishes for distances larger

than the noncommutative geometry typical scale ) and solving the corresponding

Einstein’s field equations with this source.

The main purpose of this paper is to obtain a black hole solution consideringthe effect of spacetime noncommutativity on the braneworld. We will consider the

simplest framework to obtain the solution: the second Randall-Sundrum II model

with a single brane in a Z2-symmetric, 5-dimensional, asymptotically anti-de-Sitter

bulk [20] in the presence of an anisotropic perfect fluid inspired by the noncommuta-

tivity. In the last section we present the temperature associated with the event horizon

of the noncommutative inspired braneworld black hole to show that the evaporation

process admits both a minimum and a maximum value of temperature before cooling

down towards a zero temperature remnant.

2. THE REGULAR BLACK HOLE IN THE BRANE

The gravitational field on the brane is described by the Gauss and Codazzi

equations of 5-dimensional gravity [21],

Gµν = −Λgµν + 8πGT µν + κ45Πµν −E µν , (2)where Gµν = Rµν − 12gµν R is the 4-dimensional Einstein tensor, κ5 is the 5-dimensionalgravity coupling constant, κ45 = (8πG5)

2, with G5 the gravitational constant in five

dimensions and Λ is the 4-dimensional cosmological constant that is given in termsof the 5-dimensional cosmological constant Λ5 and the brane tension λ by

Λ = κ25

2

Λ5 +

κ256

λ2

. (3)

T µν is the energy-momentum tensor of matter confined on the brane, Πµν is aquadratic tensor in the energy-momentum tensor given by

Πµν = 1

12T T µν − 1

4T µσT

σν +

1

8gµν

T αβT

αβ− 13

T 2

(4)

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3 A Regular Black Hole in the Braneworld 59

with T = T σσ and E µν is the projection of the 5-dimensional bulk Weyl tensor C ABCDon the brane that can be written as E µν = δ

Aµ δ

Bν C ABCDn

AnB with nA the unit nor-

mal to the brane. E µν encompasses the nonlocal bulk effect and its only known

property is that it is traceless, i.e. E σσ = 0. For simplicity, we will solve the tracedfield equation,

R = 4Λ−8πGT − κ45

4

T αβT

αβ− 13

T 2

. (5)

To obtain the noncommutative inspired black hole metric, we will propose a

solution on the brane with the form

ds2 =−f (r) dt2 + dr2f (r)

+ r2dΩ2 (6)

where dΩ2 = dθ2 +sin2 θdϕ2 is the line element of the two-sphere S 2. Note that weimpose the condition grr = −g−1tt because we want the induced metric to be close toSchwarzschild’s solution in the limit of large r (or small ).

The noncommutative parameter will provide a regular black hole solution in

the brane by introducing an anisotropic perfect fluid such that it replaces point-like

structures with smeared objects [15–19]. This is accomplished by replacing Dirac

delta functions by Gaussian distributions with a minimal width ,

ρ (r) = M

(4π2)3/2 e−

r

2

/4

2

, (7)

which gives a mass distribution

m (r) =

r0

ρ (r) 4πr2dr = 2M √

π γ

3

2;

r2

42

(8)

where γ is the lower incomplete gamma function,

γ a

b; z

=

z0

ua

b−1e−udu. (9)

Note that in the limit r

→∞one obtains m

→M , from which we can identify

M as the total mass of the black hole. In order to completely define the energy-momentum tensor we use the covariant conservation condition ∇ν T µν = 0 whichgives the non-null components

T tt = T rr = −ρ (r) (10)

T θθ = T ϕϕ = −ρ (r)−

r

2∂ rρ (r) . (11)

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Fig. 1 – Characteristic behaviour of f (x) vs. x = r

for various values of the parameter M . Intercepts

on the horizontal axis give the radii of the horizons.

By solving the contracted field equations (5) with this matter source, we find

the line element coefficient

f (r) = 1− 4√ π

GM

r γ

3

2;

r2

42

+

Λ

3r2− 7

√ 2

643G25

M 2

r γ

1

2;

r2

22

+ 1

48π2G25M

2

r2

7γ

1;

r2

22

+ 13γ

2;

r2

22

+ 2γ

3;

r2

22

.(12)

The horizons of this solution are defined by the equation f (r) = 0 which cannot be solved analytically. However, plotting function f (r) as shown in Figure 1,the horizons correspond to the intercepts with the horizontal axis. Note that noncom-

mutativity introduces a new behaviour with respect Schwarzschild or Schwarzschild-

AdS black holes and similar solutions on the brane [22,23]. Instead of a single event

horizon, there are four different possibilities depending on the value of the parameterM with respect to certain critical values M 0 and M ∗,

1. For M > M ∗ there is just one horizon

2. For M 0 < M < M ∗ there are two distinct horizons

3. For M = M 0 there is one degenerate horizon4. For M < M 0 there is no horizon and therefore no black hole.

When there are two horizons, the largest root of f (r) = 0 corresponds to the

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event horizon r = r+.Replacing solution (12) into the field equations (2), we obtain the components

of the projected Weyl tensor,

E tt = E rr =−E θθ =−E ϕϕ

= − G25M

2

48π2r4

7γ

1;

r2

22

+ 13γ

2;

r2

22

+ 2γ

3;

r2

22

+G25

M 2

48π6

−3

2γ

1;

r2

22

+ γ

2;

r2

22

−γ

3;

r2

22

−7G25M

2

48π43

2 π−1 e−r2/22

r2 , (13)

which is clearly traceless. It is important to note that the brane-world black hole

reported in [22] is obtained from our solution (12) by taking the limit rl →∞, i.e. inthe commutative limit.

2.1. THE ESSENTIAL SINGULARITY

Now, we would like to clarify what happens if the starting object has mass

smaller than M 0. Figure 1 shows that in this case there is no event horizon and thus

the object may be a naked singularity. However, we are going to show that this is not

the case by studying the curvature scalar near r = 0. Using equation (5) we obtain

R|r=0 = 4Λ + 32πGM

(4π2)3/2 +

κ453

M 2

(4π2)3 (14)

so the curvature is actually constant. An eventual naked singularity is replaced by a

de Sitter or Anti-de Sitter regular geometry around the origin. As is well known, other

attempts to avoid curvature singularity at the origin of Schwarzschild metric in ge-

neral relativity have been made by matching de Sitter and Schwarzschild geometries

along matter shells [24–27] or by imposing a regular black hole geometry to derive

the corresponding energy-momentum tensor from the field equations [28,29]. In this

paper we start from an energy-momentum tensor originating from non-commutative

geometry and with this source we solve Einstein equation to obtain the regular solu-tion.

3. THERMODYNAMICS

Black hole’s temperature is defined in terms of the surface gravity at the event

horizon by

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Fig. 2 – Characteristic behaviour of the temperature vs. x+ = r+

for different values of the parameter

M . The dotted curve is the temperature for the commutative black hole with cosmological constant

[30].

T = κ

2π =

1

4π ∂ rf (r)|r=r+ . (15)

By using the property of the gamma function,

∂

∂uγ a

b, u

= e−uu−1+a

b , (16)

the temperature associated to the black hole gives

T = 1

4π

4GM √

πr2+γ

3

2; r2+42

− 2√

π

GM

e−

r2+

42 + 2Λ

3 r+ +

7√

2

643G25

M 2

r2+γ

1

2; r2+22

− 7

322G2

5

M 2

r2+e−

r2+

22 − 1

24π2G2

5

M 2

r3+

7γ

1; r2+22

+ 13γ

2;

r2+22

+2γ

3;

r2+22

+

1

48π2G25M

2

r2+e−

r2+

22

7 +

13

2

r2+2

+ r4+24

.

(17)

The plot of this temperature in Figure 2 shows a new property of our black

hole. The commutative black hole with cosmological constant [30] is represented by

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the dotted line while continuous lines correspond to our solution for different valuesof mass. Note that for large values of r+ the temperatures coincide but for small radii

our solution admits both a minimum T min (just as in the noncommutative case found

in [31]) and a maximum temperature T max. From this point, the solution cools down

to the zero temperature extremal black hole remnant configuration. This behaviour

shows clearly that noncommutativity plays the same role in braneworld models as in

Quantum Field Theory: it removes short distance divergences.

4. CONCLUSION

We have obtained a black hole solution in Randall-Sundrum II braneworld sce-

nario in the presence of an anisotropic perfect fluid inspired by noncommutative ge-

ometry such that it replaces point-like structures with smeared objects. In general,

the bulk spacetime is obtained by solving the full five-dimensional field equations,

and the geometry of the embedded brane is deduced. However, due to the complexity

of the five-dimensional equations, we have solved the effective field equations for the

induced metric on the brane to obtain the black hole solution, and therefore we have

not studied fully the effect of the braneworld black hole on the bulk geometry.

The analysis of the solution shows that the geometry admits two, one or no

horizon depending on the value of the mass parameter M with respect to two thresh-

old masses M 0 and M ∗ which depend on the noncommutative geometry induced

minimal length . In agreement with the expected behavior from quantum gravity,

our effective geometry in the brane is singularity free. Specifically, we have shown

that there is a minimal mass M 0 to which a black hole can decay through Hawking

radiation, so it does not end up into a naked singularity due to the finiteness of the

curvature at the origin. The regular geometry and the residual mass M 0 are both

manifestations of the Gaussian delocalization of the source inspired by the noncom-

mutative spacetime.

From the thermodynamical analysis, the regularization takes place eliminating

the divergent behaviour of the temperature. As a consequence, the possibility of

horizon extremization gives a temperature that instead of a divergent behaviour at

short scales, admits both a minimum and a maximum values before cooling down

towards a zero temperature remnant configuration.

Acknowledgements. This work was supported by the Universidad Nacional de Colombia. Her-

mes Project Code 17318.

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REFERENCES

1. A. Chamblin, S.W. Hawking and H.S. Reall, Phys. Rev. D 61, 065007 (2000).

2. A. Chamblin, H.S. Reall, H. Shinkai and T. Shiromizu, Phys. Rev. D 63, 064015 (2001).

3. S. Nojiri, O. Obregon, S.D. Odintsov and S. Ogushi, Phys. Rev. D 62, 064017 (2000).

4. P. Kanti and K. Tamvakis, Phys. Rev. D 65, 084010 (2002).

5. R. Casadio, A. Fabbri and L. Mazzacurati, Phys. Rev. D 65, 084040 (2001).

6. C. Germani and R. Maartens, Phys. Rev. D 64, 124010 (2001).

7. R. Casadio and L. Mazzacurati, Mod. Phys. Lett. A 18, 651-660 (2003).

8. R. Casadio and B. Harms, Phys. Lett. B 487, 209-214 (2000).

9. R. Casadio, Annals Phys. 307, 195-208 (2003).

10. E. Witten, Nucl. Phys. B 460, 335 (1996).

11. N. Seiberg, E. Witten, JHEP 9909, 032 (1999).

12. A. Smailagic, E. Spallucci, J. Phys. A 37, 7169 (2004).

13. A. Smailagic, E. Spallucci, J. Phys. A 36, L467 (2003).

14. A. Smailagic, E. Spallucci, J. Phys. A 36, L517 (2003).

15. A. Smailagic and E. Spallucci, Phys. Rev. D 65, 107701 (2002).

16. A. Smailagic and E. Spallucci, J. Phys. A 35, L363 (2002).

17. P. Nicolini, A. Smailagic, E. Spallucci, Phys. Lett. B 632, 547 (2006).

18. P. Nicolini, Int. J. Mod. Phys A 24, 7, 1229 (2009).

19. S. Ansoldi, P. Nicolini, A. Smailagic, E. Spallucci, Phys. Lett. B 645, 261 (2007).

20. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999).

21. T. Shiromizu, K. Maeda and M. Sasaki, Phys. Rev. D 62, 024012 (2000).

22. N. Dadhich, R. Maartens, P. Papadopoulos and V. Rezania. Phys. Lett. B 487, 1 (2000).

23. A. Sheykhi and B. Wang, Mod. Phys. Lett. A24, 2531-2538 (2009).

24. A. Aurilia, G. Denardo, F. Legovini, E. Spallucci, Phys. Lett. B 147, 258 (1984).25. A. Aurilia, G. Denardo, F. Legovini, E. Spallucci, Nucl. Phys. B 252, 523 (1985).

26. A. Aurilia, R.S. Kissack, R. Mann, Phys. Rev. D 35, 2961 (1987).

27. V.P. Frolov, M.A. Markov, V.F. Mukhanov, Phys. Rev. D 41, 383 (1990).

28. D.A. Easson, JHEP 0302, 037 (2003).

29. S.A. Hayward, Phys. Rev. Lett. 96, 031103 (2006).

30. S. W. Hawking and D.N. Page, Commun. Math. Phys. 87, 577 (1983).

31. P. Nicolini and G. Torrieri. JHEP 1108, 097 (2011).

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