Kerr-Schild method and geodesic structure in codimension-2 brane black holes

Bertha Cuadros-Melgar*

Departamento de Ciencias Fısicas, Universidad Andres Bello, Avenida Republica 252, Santiago, Chile

Susana Aguilar† and Nelson Zamorano‡

Departamento de Fısica, Facultad de Ciencias Fısicas y Matematicas, Universidad de Chile,Avenida Blanco Encalada 2003, Santiago, Chile

(Received 9 December 2009; published 16 June 2010)

We consider the black hole solutions of five-dimensional gravity with a Gauss-Bonnet term in the bulk

and an induced gravity term on a 2-brane of codimension 2. Applying the Kerr-Schild method we derive

additional solutions which include charge and angular momentum. To improve our understanding of this

geometry we study the geodesic structure of such spacetimes.

DOI: 10.1103/PhysRevD.81.126010 PACS numbers: 11.25.�w, 04.20.Fy, 04.50.Gh, 04.70.�s

I. INTRODUCTION

One of the unsolved questions of braneworld cosmologyis the existence of localized black holes on the brane. Thispuzzle has been investigated almost since the appearanceof these alternative-to-classical-gravity models. Incodimension-1 scenarios the natural first proposal was toconsider the Schwarzschild metric and study its blackstring extension into the bulk [1]. Unfortunately, as intuitedby the authors, this string is unstable to classical linearperturbations known as Gregory-Laflamme instability [2].Since then there has been intensive research to find a fullmetric by using numerical techniques [3] or by solving thetrace of the projected Einstein equations on the brane [4]with a special ansatz [5] or making certain assumptions onthe projected Weyl term coming from the bulk [6]. Alower-dimensional version with a Banados-Teitelboim-Zanelli (BTZ) [7] black string was also considered inRef. [8]. This solution was obtained from the so-calledC metric. The thermodynamical analysis showed that thestring remains stable when its transverse size is comparableto the four-dimensional anti–de Sitter (AdS) radius and canbe unbalanced by a Gregory-Laflamme instability abovethat scale, breaking up to a BTZ brane black hole.

In codimension 2 the first attempt was proposed inRef. [9] as a generalization of the four-dimensionalAryal-Ford-Vilenkin black hole [10] pierced by a cosmicstring. The rotating version was also presented in Ref. [11],and a complete study of the gray-body factors was consid-ered in Ref. [12]. Another proposal came by considering afive-dimensional bulk with a Gauss-Bonnet term and in-duced gravity on the brane [13]. The solutions are basicallyof three types. The first one is the familiar BTZ black hole,which can be extended into the bulk with a regular horizon.The second one adds a short distance correction and de-

scribes a BTZ black hole conformally coupled to a scalarfield. There is a third solution family that can accommo-date any brane metric coefficient nðrÞ provided that it canyield a physically acceptable brane energy-momentumtensor. The corresponding generalization to six-dimensional bulks was worked out in Ref. [14], where itis shown that the only possible solution, namely, aSchwarzschild-AdS brane black hole, needs matter in thebulk. For all these solutions the Gauss-Bonnet term plays afundamental role leading to a consistency relation thatdictates the kind of bulk or brane matter necessary tosustain a black hole on the brane.The arbitrariness of nðrÞ in one of the above-mentioned

solutions is an interesting feature that deserves more dis-cussion. This fact motivated us to look for more solutionsthat could exhibit physically acceptable brane energy-momentum tensors. Specifically, we wondered if chargedand rotating metrics can be included in this general nðrÞ, inparticular, taking into account that the rotating BTZ blackhole has nondiagonal elements that were not included inthe original derivation of the solutions found in Ref. [13].With this purpose we recur to the so-called Kerr-Schildmethod.The Kerr-Schild coordinates appeared for the first time

when obtaining the Kerr metric starting from a flat emptyspace. Later on, Taub [15] generalized this method in orderto obtain new solutions by adding to a background metric aterm proportional to a null geodesic vector and a scalarfunction H. The resulting metric is not a coordinate trans-formation but a new spacetime with a different geometry.The selection of H, although arbitrary, must fulfill certaincriteria. Not only does it have to satisfy the Einsteinequations when inserted in the new metric, but it mustalso yield a physically meaningful energy-momentum ten-sor. The advantage of this generalized Kerr-Schild (GKS)method is that the new Einstein equations are linear in Hwhen written in their covariant and contravariant compo-nents. We should also mention that the resulting metric can

*berthaki@gmail.com†saguilar@ing.uchile.cl‡nzamora@dfi.uchile.cl

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have nondiagonal terms that can be cast off by means of acoordinate transformation when no angular momentum isinvolved. The GKS transformation appears as a useful toolin astrophysics where, for example, it can generate asingularity free metric that describes orbits close to aKerr black hole event horizon [16]. Moreover, this kindof metric can be applied in numerical relativity to outlinethe geometry at the horizon and even extend it to theinterior of the black hole, given the absence of singularitiesof these coordinates in this region. The GKS transforma-tion is also used to identify apparent horizons in boostedblack holes due to its invariance under Lorentz boost [17].Other solutions were also worked out in Ref. [18]. Thismethod has also been used to find exact vacuum solutionsin the context of multidimensional gravity [19].

In this paper we applied the GKS method to extradimensions. The background solutions are the black holemetrics described previously. Because of the Gauss-Bonnetcontribution, the Einstein equations are not linear anymoreeven when written in their covariant-contravariant form; infact, they are at most quadratic in the functionH. However,as we will see in this work, the method proved to besuccessful since the equations are still solvable, and itwas possible to obtain more solutions including chargeand angular momentum. It was also viable to pass fromBTZ metrics (pure or charged) to the so-called correctedBTZ one, which includes a brane scalar field.

Given all these solutions it is pertinent to explore themain properties of such spacetimes. In general relativityone possible way to understand the geometrical aspect ofthe gravitational field is to study geodesics in the spacetimepermeated by this field. Over the years, the motion ofmassive and massless test particles in background geome-tries of various higher-dimensional theories of gravity hasbeen investigated [20]. In this work we analyze the time-like and null geodesic motion in the brane black holespacetimes mentioned above. With this aim we solve theEuler-Lagrange equation for the variational problem asso-ciated with the corresponding metrics. The set of orbitsturned out to be very rich.

The paper is organized as follows. In Sec. II we make abrief review of the brane black holes obtained in Ref. [13].Section III is devoted to the application of the GKSmethodto these models displaying the corresponding results andnew solutions. In Sec. IV we present a complete study oftimelike and null geodesic behavior in the whole family ofbackgrounds. Finally, in Sec. V we discuss our results andconclude.

II. BTZ-STRINGLIKE SOLUTIONS ONCO-DIMENSION-2 BRANEWORLDS

We consider the following gravitational action in fivedimensions with a Gauss-Bonnet (GB) term in the bulk andan induced three-dimensional curvature term on the brane[13]:

Sgrav ¼�Z

d5xffiffiffiffiffiffiffiffiffiffiffiffi�gð5Þ

q½Rð5Þ þ �ðRð5Þ2 � 4Rð5Þ

MNRð5ÞMN

þ Rð5ÞMNKLR

ð5ÞMNKLÞ� þ r2cZ

d3xffiffiffiffiffiffiffiffiffiffiffiffi�gð3Þ

qRð3Þ

�M3

5

2

þZ

d5xLbulk þZ

d3xLbrane; (1)

where � ( � 0) is the GB coupling constant, r2c ¼ M3=M35

is the induced gravity ‘‘crossover’’ scale, which marks thetransition from 3D to 5D gravity, and M5 and M3 are thefive- and three-dimensional Planck masses, respectively.The above induced term has been written in the particu-

lar coordinate system in which the metric is

ds25 ¼ g��ðx; �Þdx�dx� þ d�2 þ b2ðx; �Þd�2: (2)

Here g��ðx; 0Þ is the brane metric, whereas x� denotes

three dimensions, � ¼ t; r; �, and � and � denote theradial and angular coordinates, respectively, of the twoextra dimensions (� may or may not be compact, and 0 �� < 2�). Capital M and N indices will take values in thefive-dimensional space.The Einstein equations resulting from the variation of

the action (1) are

Gð5ÞNM þ r2cG

ð3Þ�� g

�Mg

N�

�ð�Þ2�b

� �HNM

¼ 1

M35

�TðBÞNM þ TðbrÞ�

� g�MgN�

�ð�Þ2�b

�; (3)

where

HNM ¼ ½12gNMðRð5Þ2 � 4Rð5Þ2

KL þ Rð5Þ2ABKLÞ � 2Rð5ÞRð5ÞN

M

þ 4Rð5ÞMPR

NPð5Þ þ 4Rð5ÞN

KMPRKPð5Þ � 2Rð5Þ

MKLPRNKLPð5Þ �:

(4)

To obtain the braneworld equations we expand the met-ric around the brane as

bðx; �Þ ¼ ðxÞ�þOð�2Þ: (5)

At the boundary of the internal two-dimensional spacewhere the 2-brane is situated, the function b behaves asb0ðx; 0Þ ¼ ðxÞ, where a prime denotes derivative withrespect to �. In addition, we demand that the space in thevicinity of the conical singularity is regular, i.e., @� ¼ 0

and @�g��ðx; 0Þ ¼ 0 [21]. The extrinsic curvature in the

particular gauge g�� ¼ 1 that we are considering is given

by K�� ¼ g0��. Using the fact that the second derivatives

of the metric contain �-function singularities at the posi-tion of the brane, the nature of the singularity gives thefollowing relations [21]:

b00

b¼ �ð1� b0Þ�ð�Þ

bþ nonsingular terms; (6)

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K0��

b¼ K��

�ð�Þb

þ nonsingular terms: (7)

From the above singularity expressions and using theGauss-Codacci equations, we can match the singular partsof the Einstein equations (3) and get the ‘‘boundary’’Einstein equations

Gð3Þ�� ¼ 1

M23

TðbrÞ�� þ 2�ð1� ÞM

35

M23

g��: (8)

We look for black string solutions of the Einstein equa-tions (3) using the five-dimensional metric (2) in the form

ds25 ¼ f2ð�Þ½�nðrÞ2dt2 þ nðrÞ�2dr2 þ r2d�2�þ d�2 þ b2ð�Þd�2; (9)

where we have supposed the existence of a localized (2þ1) black hole on the brane, whose metric is given by

ds23 ¼ �nðrÞ2dt2 þ nðrÞ�2dr2 þ r2d�2: (10)

In the bulk we consider only a cosmological constant�5. Then, from the bulk Einstein equations

Gð5ÞMN � �HMN ¼ � �5

M35

gMN: (11)

By combining the ðrr;��Þ equations we get�_n2 þ n €n� n _n

r

��1� 4�

b00

b

�¼ 0; (12)

while a combination of the ð��; ��Þ equations gives�f00 � f0b0

b

��3� 4

�

f2

�_n2 þ n €nþ 2

n _n

rþ 3f02

��¼ 0;

(13)

where a dot implies derivatives with respect to r. Thesolutions of Eqs. (12) and (13) are summarized in Table I[13].

In this table, l is the length of three-dimensional AdSspace. The BTZ solution is given by [7]

n2ðrÞ ¼ �Mþ r2

l2: (14)

When its mass is positive, the black hole has a horizon at

r ¼ lffiffiffiffiffiM

p, and the radius of curvature of the AdS3 space

l ¼ ð��3Þ�1=2 provides the necessary length scale to de-fine this horizon. For the mass �1<M< 0, which isdimensionless, the BTZ black hole has a naked conicalsingularity, while for M ¼ �1 the vacuum AdS3 space isrecovered.The corrected BTZ solution corresponds to a BTZ black

hole with a short distance correction term,

nðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Mþ r2

l2�

r

s; (15)

and it describes a BTZ solution conformally coupled to ascalar field [22].To introduce a brane we solve the corresponding junc-

tion conditions given by the boundary Einstein equations(8) using the induced metric shown in (10). For the casewhen nðrÞ corresponds to the BTZ black hole (14), and thebrane cosmological constant is given by �3 ¼ �1=l2, wefound that the energy-momentum tensor in (8) is null.Therefore, the BTZ black hole is localized on the branein vacuum.When nðrÞ is of the form given by (15), the energy-

momentum tensor necessary to sustain such a solution onthe brane is given by

T� ¼ diag

�

2r3;

2r3;�

r3

�; (16)

which is conserved on the brane [23].These solutions extend the brane BTZ black hole into

the bulk. The warp function f2ð�Þ gives the shape of a‘‘throat’’ to the horizon, whose size is defined by the scaleffiffiffiffi�

p, which is fine-tuned to the length scale of the five-

dimensional AdS space.

III. THE KERR-SCHILD METHOD

In this section we will apply the generalized Kerr-Schildmethod to the metrics analyzed in Ref. [13] to generatenew solutions. This procedure consists of defining a newmetric gMN starting from a known one gMN as follows:

g MN ¼ gMN þ 2Hðr; �Þ‘M‘N; (17)

TABLE I. BTZ-stringlike solutions in five-dimensional braneworlds of codimension 2.

nðrÞ fð�Þ bð�Þ ��5 Constraints

BTZ coshð�=2 ffiffiffiffi�

p Þ 8 bð�Þ 34� l2 ¼ 4�

BTZ coshð�=2 ffiffiffiffi�

p Þ 2ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ 3

4� -

BTZ coshð�=2 ffiffiffiffi�

p Þ 2ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ 3

4� l2 ¼ 4�

BTZ �1 � sinhð�=�Þ 3l2

� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl2 � 4�Þ=2p8 nðrÞ coshð�=2 ffiffiffiffi

�p Þ 2

ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ 3

4� -

Corrected BTZ coshð�=2 ffiffiffiffi�

p Þ 2ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ 3

4� l2 ¼ 4�

Corrected BTZ �1 2ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ 1

4� l2 ¼ 12�

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and checking if this new metric g�� satisfies the Einstein

equations. Here H is an arbitrary function of the coordi-nates r and � in this specific case, and ‘M is a null geodesicvector (in the background metric) described below (18).

Taub [15] introduced this approach in the form justdescribed. There is an extended family of solutions gen-erated by this method. Among others, it reproduces thestandard Einstein solutions: Schwarzschild, Reissner-Nordstrom, and (obviously) the Kerr solutions startingfrom a flat, empty spacetime. We extend this formalismto the BTZ-stringlike solutions described in the previoussection.

Our case is quite different from those just describedsince it includes the Gauss-Bonnet term in the originalaction (1). The resulting equations are at most quadraticin the function Hðr; �Þ due to the index contractions dis-played in the GB term.

A. Null geodesic vector

Let us consider a vector ‘M ¼ ‘Mðr; �Þ, which obeys thefollowing conditions:

‘M;N‘N ¼ 0; ‘M‘

M ¼ 0; (18)

where capital indices run over all the coordinates (t, r, �,�, and �).

The definition of the following operator will show to beuseful in solving these equations:

D ¼�n2‘r

@

@rþ f2‘�

@

@�

�: (19)

Using this operator and the geodesic equation (18), weobtain the following equations for the t, �, and � compo-nents of the null vector ‘A:

1

f2D‘i ¼ 0; for i ¼ t; �; and �: (20)

The r and � components are more involved and read

1

f2D‘r þ 1

f2

�_n

n3‘2t þ n _n‘2r � 1

r3‘2�

�¼ 0;

2

f2D‘� �

�1

f2

�0� 1

n2‘2t � n2‘2r � 1

r2‘2�

�þ

�1

b2

�0‘2� ¼ 0:

(21)

In our convention an overdot means differentiating withrespect to r while a prime corresponds to a derivative withrespect to �.

An additional constraint comes from ‘M being a nullvector (18). This condition is

1

fð�Þ2�� ‘2t

n2þ n2‘2r þ

‘2�

r2

�þ ‘2� þ ‘2�

b2¼ 0: (22)

The Kerr-Schild formalism requires us to know at leastone explicit solution for the null geodesic ‘A before we

begin our search for a new solution of Einstein’s equations.We start considering the following assumptions about thecoordinate dependence of the radial brane and bulk com-ponents of the lightlike vector, ‘r and ‘�:

‘r ¼ ‘rðrÞ and ‘� ¼ ‘�ð�Þ: (23)

With these assumptions a solution for the set of Eqs. (20)is readily found. For the components i ¼ t, �, and � wehave

‘iðr; �Þ ¼ Ci exp

��1‘�

Z dr

n2‘rðrÞ�

� exp

���1‘r

Z d�

f2ð�Þ‘��: (24)

A simple nontrivial solution can be obtained setting�1 ¼ 0. In this case the solutions are

‘t ¼ E; ‘� ¼ L; and ‘� ¼ K; (25)

where E, L, and K are constants related to the energy andthe brane and bulk components of the angular momentumof the particle following the geodesic.There are still two components left to solve: ‘rðrÞ and

‘�ð�Þ. Introducing these expressions into Eqs. (21) and

doing the corresponding simplifications, we arrive at

nðrÞ2ð _‘2rÞ þ��� _1

n2

�E2 þ ð _n2Þ‘2r þ

� _1

r2

�L2

�¼ 0; (26)

ð‘2�Þ0 þ�1

f2

�0�� 1

n2E2 þ n2‘2r þ 1

r2L2

�þ

�1

b2

�0‘2� ¼ 0:

(27)

Let us work with the last equation. Employing the nullvector condition (22) and factorizing conveniently weobtain an expression that can easily be integrated:

1

fð�Þ2�fð�Þ2

�‘�ð�Þ2 þ K2

bð�Þ2��0 ¼ 0;

from here we find the general solution for ‘�ð�Þ:

‘�ð�Þ2 ¼ � K2

bð�Þ2 þ 2

fð�Þ2 : (28)

A new constant has been introduced here. If K � 0,Eq. (28) is valid as long as its right-hand side remainspositive and � > 0.We now solve ‘r from Eq. (26). This equation can be

written as a total derivative and after one integration be-comes �

� E2

nðrÞ2 þ nðrÞ2‘2r þ L2

r2

�¼ �2;

where � is a constant of integration. However, this constantis not a new parameter since this component must fit in the

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null vector restriction Eq. (22). Inserting this expression inEq. (22) we obtain that �2 þ 2 ¼ 0, fixing �.

With this last step we have solved Eqs. (18) for each ofthe components of the null geodesic vector.

The null geodesic vector takes the following generalexpression:

‘M ¼�E;

1

n2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 �

�L2

r2þ 2

�n2

s; L;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

f2� K2

b2

s; K

�:

(29)

B. Solutions generated by the Kerr-Schild method

We choose the following null geodesic vector:

‘M ¼�1;

1

n2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� L2

r2n2

s; L; 0; 0

�; (30)

where L is a constant associated to the angular momentumof the test particle, and a function H of the form Hðr; �Þ ¼h1ðrÞh2ð�Þ. We use this ansatz in (17) and replace back intothe Einstein equations obtaining eight non-null equations.Factorizing ð�tÞ and ð�rÞ components we arrive at thefollowing equation:

½2h2ð�Þf0 � fh02ð�Þ�½h1ðrÞ þ h01ðrÞr� ¼ 0: (31)

Once we solve Eq. (31), the rest of the equations areautomatically fulfilled provided that the correspondingconstraints in Table I are satisfied.

If we choose to solve this equation for h1ðrÞ, we obtainh1ðrÞ ¼ �=r, where � is a constant, and h2ðrÞ remainsfree. When putting back into (17) the new metric turns into

ds2 ¼��f2n2 þ 2

�

rh2

�dt2 � 4

�

r

h2n2

dtdr

þ�f2

n2þ 2

�

r

h2n4

�dr2 þ f2r2d�2

þ d�2 þ b2d�2: (32)

This metric represents a five-dimensional gravity solution,which can be diagonalized in few cases depending on theform of h2ð�Þ. Notice that the line element (32) does notinclude a boundary membrane. As our main interest here isto find braneworld solutions, naturally our next step is toembed a brane in this bulk metric. In order to proceed tothis point, the first requirement we find is that the inducedmetric must fulfill the junction conditions, i.e., the three-dimensional Einstein equations (8). As far as we know, theBTZ metric is the only solution describing a black hole in að2þ 1Þ spacetime; thus, h2ð�Þ becomes constrained to amultiple of the warp factor f2ð�Þ to be able to recover aBTZ-like metric on the brane. In particular, if h2ð�Þ ¼f2ð�Þ, we can make a coordinate transformation to endup with the BTZ string coupled to a brane scalar field as itwill be shown below.

Alternatively, if we solve Eq. (31) for h2ð�Þ, we geth2ð�Þ ¼ f2ð�Þ, and h1ðrÞ becomes arbitrary. In this casethe choices for h1ðrÞ are several and give place to thefollowing new solutions. Nevertheless, as the presence ofthe brane imposes junction conditions (8), we should pointout that the right options for h1ðrÞ will be determined bythe requirement of yielding a physically meaningful braneenergy-momentum tensor.

1. Charged BTZ string

Let us first consider the solution fð�Þ ¼ coshð�=2 ffiffiffiffi�

p Þand bð�Þ ¼ 2

ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ. By choosing h1ðrÞ ¼

Q2

2 lnr and L ¼ 0 we arrive at the following metric:

ds2 ¼ f2��ðn2 �Q2 lnrÞdt2 þ 2Q2 lnr

n2dtdr

þ�n2 þQ2 lnr

n4

�dr2 þ r2d�2

�þ d�2 þ b2d�2:

(33)

In order to obtain a more familiar form of the metric wemake the following coordinate transformation:

dt ¼ dtþ Q2 lnrdr

n2ðn2 �Q2 lnrÞ : (34)

This change cancels out the nondiagonal term such that themetric (33) turns into

ds2 ¼ f2��n2dt2 þ dr2

n2þ r2d�2

�þ d�2 þ b2d�2;

(35)

with n2 ¼ �Mþ r2=l2 �Q2 lnr, which describes acharged BTZ string whose charge is confined to the brane.In order to verify this statement we calculate the braneenergy-momentum tensor necessary to hold this solutionand we obtain

T�� ¼ diag

�� Q2

2r2;� Q2

2r2;Q2

2r2

�: (36)

This is precisely the stress energy tensor related to acharged object in ð2þ 1Þ dimensions.

2. BTZ string coupled to a brane scalar field

Working with the same expressions for fð�Þ and bð�Þwenow choose h1ðrÞ ¼

2r and the resulting metric is

ds2 ¼ f2���n2 �

r

�dt2 þ 2

n2rdtdrþ

�b2r2 þ r

n4r2

�dr2

þ r2d�2

�þ d�2 þ b2d�2: (37)

We now make the coordinate transformation

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dt ¼ dtþ

n2rðn2 � =rÞ ; (38)

to arrive at the following metric:

ds2 ¼ f2��n2dt2 þ dr2

n2þ r2d�2

�þ d�2 þ b2d�2;

(39)

where n2 ¼ �Mþ r2=l2 � =r, which corresponds to aBTZ black hole coupled to a scalar field on the brane. Thissolution had already been found in Ref. [13].

3. Charged BTZ string coupled to a brane scalar field

Another possible combination is to choose the originalmetric to be (35) and h1ðrÞ ¼ =2r. Following the sameprocedure as the previous case, we find the same metric as(39), but with n2 ¼ �Mþ r2=l2 �Q2 lnr� =r. If wecompute the energy-momentum tensor on the brane, wefind

T�� ¼ diag

�� Q2

2r2þ

2r3;� Q2

2r2þ

2r3;Q2

2r2�

r3

�; (40)

which corresponds to a charged BTZ black hole coupled toa scalar field on the brane.

4. BTZ string with angular momentum

In order to add angular momentum to the original BTZstring, we pick h1ðrÞ ¼ c, where c is a constant. In this caseL � 0. Thus, the metric takes the following form:

ds2 ¼ f2��ðn2 � 2cÞdt2 þ 4c

n2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� L2

r2n2

sdtdr

þ 4cLdtd�þ n2r2 þ 2cr2 � 2cn2L2

n4r2dr2

þ 4cL

n2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� L2

r2n2

sdrd�þ ðr2 þ 2cL2Þd�2

�

þ d�2 þ b2d�2: (41)

Introducing the following transformations:

dt ¼ dtþ uðrÞdr; (42)

d� ¼ d�þ vðrÞdr; (43)

with

uðrÞ ¼ 2cr2

n2ðn2r2 � 2r2cþ 2cL2n2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� L2

r2n2

s; (44)

vðrÞ ¼ �Ln2

r2uðrÞ: (45)

Moreover, if we define R2 ¼ r2 þ 2cL2, J ¼ �4L, and~M ¼ Mþ 2cðL2=l2 þ 1Þ, we arrive at the metric for a

rotating BTZ string:

ds2 ¼ f2��n2dt2 þ dR2

n2þ R2

��J

2R2dtþ d�

�2�

þ d�2 þ b2d�2; (46)

where n2ðrÞ ¼ � ~Mþ R2=l2 þ J2=4R2. The correspond-ing energy-momentum tensor on the brane can be calcu-lated, and we find that it vanishes, as it should be for arotating BTZ brane black hole.Analogously, when we use the solutions fð�Þ ¼ 1,

bð�Þ¼�sinhð�=�Þ, and fð�Þ ¼ 1, bð�Þ ¼ 2ffiffiffiffi�

p �sinhð�=2 ffiffiffiffi

�p Þ, we also arrive at several solutions involving

angular momentum and scalar fields. We should stress thatin these cases we did not find any charged solution.

IV. GEODESIC STRUCTURE

In this section we study the geodesic behavior in thebackground of the solutions displayed in Table I.Let us begin our study by considering the Lagrangian for

the BTZ black hole in codimension-2 branes with thesolution fð�Þ ¼ coshð�=2 ffiffiffiffi

�p Þ and bð�Þ ¼

2ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ,

L ¼ cosh

��

2ffiffiffiffi�

p�2��nðrÞ2 _t2 þ _r2

nðrÞ2 þ r2 _�2

�þ _�2

þ 42� sinh

��

2ffiffiffiffi�

p�2_�2; (47)

where a dot indicates derivative with respect to the affineparameter �.As it is independent of t, �, and �, we can write the

following equations of motion:

@L@ _t

¼ �2 cosh

��

2ffiffiffiffi�

p�2nðrÞ2 _t ¼ �2E; (48)

@L

@ _�¼ 2 cosh

��

2ffiffiffiffi�

p�2r2 _� ¼ 2L; (49)

@L

@ _�¼ 82� sinh

��

2ffiffiffiffi�

p�2_� ¼ 2K; (50)

where L and K are the angular momenta of the particlerelated to � and � coordinates, respectively. Notice thatEq. (48) gives us a relation between the coordinate time tand the affine parameter �.With these equations the Lagrangian (47) can be written

as

L¼ cosh2�

�

2ffiffiffiffi�

p�� �E2

cosh4ð �2ffiffiffi�

p ÞnðrÞ2þ_r2

nðrÞ2þL2

r2cosh4ð �2ffiffiffi�

p Þ�

þ _�2þ K2

42�sinh2ð �2ffiffiffi�

p Þ¼h; (51)

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where h ¼ 0 or �1 is a parameter describing lightlike andtimelike geodesics, respectively.

A. Geodesics on the brane

Let us consider a particle with K ¼ 0. Using Eq. (51) atthe position of the brane (� ¼ 0) we can find the effectivepotential for the geodesic motion:

_r 2 ¼ E2 � n2ðrÞ�L2

r2� h

�) V2

eff ¼ n2ðrÞ�L2

r2� h

�:

(52)

This equation can be integrated to obtain the orbits aswell:

dr

d�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � n2ðrÞ

�L2

r2� h

�s: (53)

1. BTZ case

For radial geodesics (L ¼ 0) the effective potential be-comes

V2eff ¼ n2ðrÞh ¼

��Mþ r2

l2

�h: (54)

In this case lightlike geodesics are just straight lines.For the timelike case, we display the potential in Fig. 1.

The corresponding orbits can be obtained by direct inte-gration of Eq. (53),

rð�Þ ¼ l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 þM

2

s½1þ sinð2�Þ�1=2; (55)

and they are shown in Fig. 1. We see that geodesics have anoscillatory behavior outside the event horizon; however,there are no stable orbits since some of the particles couldcross it and never return. Notice that particles with E< 0are not allowed since Veff ¼ 0 at the event horizon.

For particles with angular momentum the effective po-tential turns out to be

V2eff ¼

��Mþ r2

l2

��L2

r2� h

�: (56)

We plot this potential for lightlike and timelike cases inFig. 2, where we setM ¼ 0; 5, l ¼ 1, L ¼ 2, and L ¼ 5. Inboth cases the potentials intercept themselves at the eventhorizon. Remark that far from the event horizon the light-like potential has an asymptotic behavior that can beinferred from Eq. (56), i.e., Veff ! L=l.In order to obtain the orbits for the lightlike case (h ¼ 0)

with angular momentum we can integrate Eq. (53) to attain

rð�Þ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�E2 � L2

l2

��2 � ML2

E2 � L2=l2

s; (57)

and we should stress that only the plus sign has physicalmeaning.Analogously, we perform the integration of Eq. (53) in

the timelike case and we arrive at

rð�Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2l2 þMl2 � L2

2

s

��1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4ML2

ðE2 þM� L2=l2Þ2l2s

sinð2�Þ�1=2

:

(58)

The corresponding orbits are presented in Fig. 3. From thisfigure we can see that lightlike geodesics with low energiesfall unavoidably into the event horizon. However, particleswith high energies can escape from the black hole showingthat energy extraction is possible but only with masslessparticles. On the other side, the timelike orbits have basi-cally the same shape as those in the radial case [Fig. 1(b)];the only effect of L is to increase the amplitude of theoscillation. Again some of them can cross the event hori-

(a) (b)

FIG. 1 (color online). Effective potential (left) and orbits (right) for radial timelike particles on the brane. These graphs correspond tol ¼ 1,M ¼ 0; 5. Notice that geodesics have an oscillatory behavior outside the event horizon; however, there are no stable orbits sincesome of the particles could cross it and never return. Particles with E< 0 are not allowed since Veff ¼ 0 at the event horizon.

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zon depending on the energy of the oscillation. In bothlightlike and timelike cases a similar qualitative behaviorwas also found in the study of pure BTZ geodesic structure[24].

2. BTZ with electric charge and scalar field

In this case the resulting potential when adding charge ora scalar field to the BTZ solution is again given by Eq. (52)with the corresponding nðrÞ as follows:

nðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Mþ r2

l2�Q2 lnðrÞ

s; chargedBTZ; (59)

nðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Mþ r2

l2�

r

s; BTZþ scalar field: (60)

Both potentials are shown in Fig. 4.We can deduce from these graphics that the charged

BTZ potential admits oscillating orbits that can fall intothe event horizon when Q is small; however, as Q grows,we see that the minimum of the potential is shifted outside

the horizon making possible the existence of stable oscil-lating or bounded geodesics. The potential correspondingto BTZ coupled to a scalar field shows that just unstableoscillating orbits are allowed.

B. Geodesics in the bulk

Here we study the geodesics that explore the extradimensions. Although standard particles are not allowedto travel outside the brane, we perform this analysis as away to acquire a better understanding of the geometry ofthese solutions. For this analysis we will consider the rcoordinate lying outside the black hole horizon, for in-

stance, at r ¼ 2ffiffiffiffiffiM

pl ¼ 2rH. In this case it will be conve-

nient to write Eq. (51) in a different way:

_� 2 ¼ hþ E2

Mcosh2ð �2ffiffiffi�

p Þ �L2

2Ml2cosh2ð �2ffiffiffi�

p Þ

� K2

42�sinh2ð �2ffiffiffi�

p Þ : (61)

Defining a new variable u as

FIG. 3 (color online). Orbits for lightlike (left) and timelike (right) brane geodesics with angular momentum (L ¼ 2). The energiesof the particles are shown in the legend. Notice that in the lightlike case orbits with low energy fall into the event horizon whereasparticles with high energy can escape from the black hole. For timelike geodesics the orbits have basically the same shape as those inthe radial case [Fig. 1(b)]; the only effect of L is to increase the amplitude of the oscillation.

FIG. 2 (color online). Effective potential for lightlike (left) and timelike (right) brane particles with different angular momenta(L ¼ 1; 2; 5). Remark that all the potentials cross themselves at the event horizon. The lightlike potential has an asymptote given byL=l, while the timelike potential grows with no limit.

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u ¼ 2ffiffiffiffi�

psinh

��

2ffiffiffiffi�

p�; (62)

and replacing in Eq. (61) it becomes

_u 2 ¼ "2 ��

L2

2Ml2þ

�1þ u2

4�

��K2

2u2� h

��; (63)

where "2 ¼ E2=M. Thus, we can define an effective po-tential given by

V2effðuÞ ¼

�L2

2Ml2þ

�1þ u2

4�

��K2

2u2� h

��: (64)

For radial geodesics (L ¼ K ¼ 0) notice that as theeffective potential vanishes in the lightlike case, the orbitsare just straight lines. On the other side, the orbits fortimelike geodesics can be found by integrating Eq. (63)and replacing u from Eq. (62). Thus, we obtain

�ð�Þ ¼ 2ffiffiffiffi�

parcsinh

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � 1

psin

��

2ffiffiffiffi�

p��

: (65)

Some of the orbits are depicted in Fig. 5. This graph showsoscillating trajectories that cross the brane. This implies

that particles leaving the brane can return in the future.This fact opens up the possibility for the existence ofshortcuts, paths connecting two points which are shorterin the bulk than on the brane [25]. In addition, note that thetrajectory with the energy that corresponds to the minimumof the effective potential is on the brane.Now we turn to the case of timelike and lightlike geo-

desics with K � 0 and/or L � 0.In the lightlike case, if K ¼ 0 and L � 0, we have a

constant potential Veff ¼ L2

2Ml2. The corresponding timelike

case has the same potential as the L ¼ 0 geodesics, onlyshifted by a constant; thus, the shape of the geodesics is thesame as the one shown in Fig. 5(b).When K � 0 and L ¼ 0, we obtain the orbits displayed

in Fig. 6. We can notice that in the lightlike case theparticles seem to be scattered by a barrierlike potentialnear the brane. Regarding the timelike geodesics, we cansee an oscillating behavior around certain position parallelto the brane, which corresponds to the orbit with theminimal permitted energy. Additionally, the higher theenergies of the particles are, the closer to the brane theycan reach, but never cross it.

(a) (b)

FIG. 5 (color online). Effective potential and timelike orbits for radial particles in the bulk. From the oscillating behavior of thetrajectories we can infer the existence of shortcuts since particles leaving the brane can return to it. Note that the trajectory with theenergy that corresponds to the minimum of the effective potential (E ¼ 1) is entirely on the brane.

FIG. 4 (color online). Effective potential for timelike geodesics on the brane for BTZ with charge (left) and with scalar field (right).These graphs correspond to Q ¼ 1; 2; 3 and ¼ 1; 5; 10. In the former case the potential admits oscillating orbits that can fall into theevent horizon when Q is small; however, as Q grows, the minimum of the potential is shifted outside the horizon making possible theexistence of stable oscillating or bounded geodesics. In the latter case only unstable oscillations are allowed.

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Forthwith, let us check the geodesic behavior for thesolution fð�Þ ¼ 1 and bð�Þ ¼ � sinhð�=�Þ (see Table I).By fixing r we can write an analogous equation to (51):

L ¼ � E2

n2ðrÞ þL2

r2þ _�2 þ K2

�2sinh2ð�=�Þ ¼ h; (66)

so that

_� 2 ¼ E2

n2ðrÞ �L2

r2� K2

�2sinh2ð�=�Þ þ h: (67)

Fixing r the effective potential becomes

V2eff ¼

L2

2Ml2þ K2

�2sinh2ð�=�Þ � h: (68)

Notice that this potential is constant when K ¼ 0. Thecorresponding graph can be seen in Fig. 7. Observe thattimelike and lightlike potentials mark off from each otherjust by a constant that shifts the entire curve.

This is a scattering potential where no other behavior ispossible because of the infinite asymptotic barrier caused

by the diverging term. We can infer that gravitationalsignals or particles already in the bulk cannot reach thebrane and those originated on the brane cannot travel farfrom it.

C. BTZ with angular momentum

In this subsection we study the particular case of geo-desics on the brane and in the bulk for the BTZ black holewith angular momentum in a codimension-2 brane. TheLagrangian can be written as follows:

L ¼ f2�� 1

4

��4Mþ 4r2

l2þ J2

r2

�_t2

þ 4 _r2

ð�4Mþ 4r2=l2 þ J2=r2Þ þ r2�� 1

2

J _t

r2þ _�

�2�

þ _�2 þ b2 _�2; (69)

and the equations of motion become

@L

@ _�¼ 2f2r2

�� 1

2

J _t

r2þ _�

�¼ 2L; (70)

@L@ _t

¼ f2�� 1

2

��4Mþ 4r2

l2þ J2

r2

�_t�

�� 1

2

J _t

r2þ _�

�J

�

¼ �2E; (71)

@L

@ _�¼ 2b2 _� ¼ 2K: (72)

With these equations the Lagrangian becomes

h ¼ � 1

f2E2 � LJE=r2 þML2=r2 � L2=l2

�Mþ r2=l2 þ J2=4r2

þ f2 _r2

�Mþ r2=l2 þ J2=4r2þ _�2 þ K2

b2: (73)

First, we look for the geodesics on the brane. Solving thisequation for _r we obtain

FIG. 7 (color online). Bulk effective potential for f ¼ 1, withL ¼ 0; 2 and K ¼ 1. We can see that close to the brane thepotential has a scattering behavior, while far from the brane itbecomes constant, so that any bulk geodesic behaves like a freeparticle.

FIG. 6 (color online). Orbits for lightlike (left) and timelike (right) bulk particles with angular momentum (K ¼ 2, L ¼ 0). Lightlikegeodesics show a scattering behavior. While timelike particles oscillate near the brane, and the higher their energies are, the closer tothe brane they can reach, but never cross it.

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_r ¼ � 1

f2

��Mþ r2

l2þ J2

4r2

��K2

b2þ _�2 � h

�

þ 1

f4

�E2 � LJE

r2þML2

r2� L2

l2

�: (74)

In this case we will go through the integration to find theorbits directly. As the brane is located at � ¼ 0, fð0Þ ¼ 1in any solution. In addition, we set K ¼ 0 to avoid singu-larities. Thus, the integral for the orbit becomes

�� �0 ¼Z drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�Mþ r2=l2 þ J2=4r2Þhþ ðE2 � LJE=r2 þML2=r2 � L2=l2Þp : (75)

For the timelike case (h ¼ �1) and setting �0 ¼ 0 theorbits are given by

rð�Þ ¼ ��sinð2�=lÞ

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðMl2 þL2 þE2l2Þ2 � ð2ELlþ lJÞ2

q

þ l2

2

�MþE2 �L2

l2

��1=2

: (76)

For the lightlike case (h ¼ 0) we have

rð�Þ ¼ ���

E2 � L2

l2

��2 � ðML2 � LJEÞ

E2 � L2=l2

�1=2

: (77)

The possible orbits are displayed in Fig. 8. These graphsshow that geodesics can cross the event horizon in bothlightlike and timelike cases. According to the energy of theparticle, it can fall directly or after a roundabout in theregion exterior to the horizon. Remark that the circularorbits corresponding to E ¼ 0 wind inside the ergospherewhile the other orbits with E> 0 extend farther. In addi-tion, massless particles with high energies can escape from

the gravitational attraction of the black hole allowingenergy extraction, a behavior already noticed in the pureð2þ 1Þ BTZ case [24].Now we turn to the geodesic motion in the bulk.

Accordingly, we write Eq. (73) taking _r ¼ 0 and r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Ml2 þ 2l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2l2 � J2

pp> rE, where rE is the ergosphere

position. We first check the case fð�Þ ¼ coshð�=2 ffiffiffiffi�

p Þ,and bð�Þ ¼ 2

ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ. After integrating, the or-

bits turn out to be

�ð�ÞL ¼ 2ffiffiffiffi�

parcsinh

�1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4C2�þ �2C2

1

C1�

s �; (78)

�ð�ÞT ¼ 2ffiffiffiffi�

parcsinh

�1ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC21 � 4C2

qsin

��ffiffiffiffi�

p�s �

;

(79)

for lightlike and timelike geodesics, respectively. The con-stants C1 and C2 are given by

C1 ¼ h� C2 þ 4

3

�2l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2l2 � J2

pðE2 � L2=l2Þ þ 2l2E2M� L2M� JLE

8MlðMlþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2l2 � J2

p Þ � 5J2

�; (80)

C2 ¼ K2

42�: (81)

FIG. 8 (color online). Lightlike (left) and timelike (right) brane geodesics for BTZ solution with angular momentum. Both positiveand negative branches are plotted. The energies of each geodesic are displayed on the legend. Notice that in both cases geodesics cancross the horizon, some of them after a roundabout and some others directly. Only in the lightlike case particles with high energies canescape from the black hole allowing energy extraction.

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Some trajectories are shown in Fig. 9 (upper graphs). Inboth cases although the geodesics approach the brane, theycannot cross it. Analogously to the nonrotating case, wesee that the brane acts like a repulsive barrier that blocksthe exchange of signals between brane and bulk.Nevertheless, whereas lightlike geodesics are scatteredwhen trying to reach the brane, timelike trajectories oscil-late about a fix distance from the brane.

Similarly, the trajectories corresponding to the solutionsfð�Þ ¼ 1, bð�Þ ¼ � sinhð�=�Þ and fð�Þ ¼ 1, bð�Þ ¼2

ffiffiffiffi�

psinhð�=2 ffiffiffiffi

�p Þ undergo an analogous pattern dis-

played in Fig. 9 (lower graphs). Again we see a repulsivebarrier behavior at the position of the brane that scatterslightlike and timelike particles. This prevents both signalsoriginated on the brane from permeating the bulk andsignals already in the bulk to arrive at the brane.

V. CONCLUSIONS

In this paper we considered the black hole solutions offive-dimensional gravity with a Gauss-Bonnet term in thebulk and an induced gravity term on a 2-brane of codimen-sion 2 [13]. In order to explore the possibility of havingmore solutions to the Einstein equations we applied theKerr-Schild method using as a background the above-mentioned solutions. This method adds to the original

metric a term depending on a null geodesic vector and ascalar function H. Because of the presence of the GB termthe resulting equations are at most quadratic in H, but stillsolvable by choosing the appropriate geodesic vector.Working out the Einstein equations for the function Hwe found additional solutions, which include charge, an-gular momentum, and brane scalar fields coupled to thebrane black hole. The corresponding brane energy-momentum tensors were also computed. These resultslead us to deduce that the solution with free nðrÞ foundin Ref. [13] does not have any constraint of diagonality.Furthermore, we studied the geodesic behavior in the

background of the original and new solutions. For thispurpose we solved the Euler-Lagrange equations for thevariational problem associated with the metric. Among ourresults we can distinguish different cases according to theenergy of the particle, the associated effective potential,and the components of the angular momentum.In the case of geodesics on the branewe can discriminate

two main cases, timelike and lightlike geodesics. Thetimelike orbits, in general, display oscillations that canbecome unstable making a particle cross the event horizonand never return. Nevertheless, the charged BTZ-stringlikesolution shows an additional behavior. When the charge issmall, the paths are the same as those appearing in un-charged or scalar field coupled solutions; however, as the

FIG. 9 (color online). Lightlike (left) and timelike (right) geodesics in the bulk for BTZ with angular momentum case with f ¼coshð�=2 ffiffiffiffi

�p Þ (up) and f ¼ 1 (down). In both cases we observe that particles cannot reach the brane. We can infer that the brane acts as

a repulsive barrier both confining particles already on the brane and preventing the entrance of bulk particles.

CUADROS-MELGAR, AGUILAR, AND ZAMORANO PHYSICAL REVIEW D 81, 126010 (2010)

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charge grows, the minimum of the effective potential isshifted outside the event horizon and makes possible theexistence of stable oscillations or bounded orbits for par-ticles with low or negative energy. The lightlike geodesicsexhibit particles with low energies crossing the event hori-zon. Nonetheless, particles with high energies can escapefrom the black hole and, thus, allow the extraction ofenergy, a result already noticed in pure ð2þ 1Þ BTZ blackholes geodesic structure [24].

Concerning the geodesics in the bulk, we fixed r > rHand studied two main cases, namely, f ¼ coshð�=2 ffiffiffiffi

�p Þ

and f ¼ 1. In the first case, we observed that timeliketrajectories corresponding to particles without angular mo-mentum are oscillating about the position of the brane.This implies that a particle leaving the brane can return to itopening up the possibility of shortcuts, i.e., paths that areshorter in the bulk than on the brane [25]. The samebehavior is verified for particles having an angular mo-mentum component along the brane (L � 0). When K (theparticle’s angular momentum component along the bulk) isswitched on, lightlike geodesics encounter a barrierlikepotential around the brane. Moreover, although timeliketrajectories are still oscillating and approach the branemore and more as their energy increases, they never crossit. This result prevents the exchange of bulk and branesignals; i.e., particles living in the bulk do not enter thebrane, and particles residing on the brane cannot go far intothe bulk. In the case f ¼ 1, both timelike and lightlikeparticles undergo scattering near the brane, but move in

straight lines far from it because the effective potentialbecomes constant at this far region.The case of a BTZ black stringlike object with angular

momentum was treated separately due to the nondiagonalterms appearing in the metric. We found that both lightlikeand timelike geodesics on the brane can cross the horizondirectly or after a roundabout in the ergosphere (particleswith the lowest allowed energy) or even outer regions.However, massless particles with high energies are ableto escape allowing energy extraction from the black hole.With respect to bulk geodesics, we determined that regard-less of the expression for f, the effective potential has abarrierlike structure around the brane, which both yieldsscattering bulk geodesics and confines particles already onthe brane. In particular, timelike geodesics in the back-ground of f ¼ coshð�=2 ffiffiffiffi

�p Þ oscillate around a fixed dis-

tance from the brane but never make contact with it.It would be interesting to investigate if these features

remain valid in codimension-2 (3þ 1) brane scenarios,and if these results may help uncover other interactionsbetween the brane and the bulk. However, as this is out ofthe scope of this paper, we expect to address these ques-tions in a future work.

ACKNOWLEDGMENTS

B.C.-M. and S. A. thank the hospitality of Facultad deCiencias Fısicas y Matematicas of Universidad de Chile,where this work was accomplished.

[1] A. Chamblin, S.W. Hawking, and H. S. Reall, Phys. Rev.

D 61, 065007 (2000).[2] R. Gregory, Classical Quantum Gravity 17, L125 (2000);

R. Gregory and R. Laflamme, Phys. Rev. Lett. 70, 2837(1993); G. Gibbons and S.A. Hartnoll, Phys. Rev. D 66,064024 (2002).

[3] T. Shiromizu and M. Shibata, Phys. Rev. D 62, 127502(2000); A. Chamblin, H. S. Reall, H. A. Shinkai, and T.

Shiromizu, Phys. Rev. D 63, 064015 (2001); T. Wiseman,

Phys. Rev. D 65, 124007 (2002).[4] T. Shiromizu, K. I. Maeda, and M. Sasaki, Phys. Rev. D

62, 024012 (2000); A.N. Aliev and A. E. Gumrukcuoglu,

Classical Quantum Gravity 21, 5081 (2004).[5] R. Casadio, A. Fabbri, and L. Mazzacurati, Phys. Rev. D

65, 084040 (2002); K.A. Bronnikov, V.N. Melnikov, and

H. Dehnen, Phys. Rev. D 68, 024025 (2003); P. Kanti and

K. Tamvakis, Phys. Rev. D 65, 084010 (2002).[6] N. Dadhich, R. Maartens, P. Papadopoulos, and V.

Rezania, Phys. Lett. B 487, 1 (2000); M. Bruni, C.

Germani, and R. Maartens, Phys. Rev. Lett. 87, 231302(2001); G. Kofinas, E. Papantonopoulos, and I. Pappa,

Phys. Rev. D 66, 104014 (2002); G. Kofinas, E.

Papantonopoulos, and V. Zamarias, Phys. Rev. D 66,

104028 (2002); A.N. Aliev and A. E. Gumrukcuoglu,Phys. Rev. D 71, 104027 (2005).

[7] M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett.69, 1849 (1992).

[8] R. Emparan, G. T. Horowitz, and R. C. Myers, J. HighEnergy Phys. 01 (2000) 007; 01 (2000) 021.

[9] N. Kaloper and D. Kiley, J. High Energy Phys. 03 (2006)077.

[10] M. Aryal, L. H. Ford, and A. Vilenkin, Phys. Rev. D 34,2263 (1986).

[11] D. Kiley, Phys. Rev. D 76, 126002 (2007).[12] D. Dai, N. Kaloper, G.D. Starkman, and D. Stojkovic,

Phys. Rev. D 75, 024043 (2007).[13] B. Cuadros-Melgar, E. Papantonopoulos, M. Tsoukalas,

and V. Zamarias, Phys. Rev. Lett. 100, 221601 (2008).[14] B. Cuadros-Melgar, E. Papantonopoulos, M. Tsoukalas,

and V. Zamarias, Nucl. Phys. B810, 246 (2009).[15] A. H. Taub, Ann. Phys. (N.Y.) 134, 326 (1981).[16] S. S. Komissarov, Mon. Not. R. Astron. Soc. 326, L41

(2001).[17] M. F. Huq, M.W. Choptuik, and R. Matzner, Phys. Rev. D

66, 084024 (2002).[18] R. Alonso and N. Zamorano, Phys.Rev. D 35, 1798

KERR-SCHILD METHOD AND GEODESIC STRUCTURE IN . . . PHYSICAL REVIEW D 81, 126010 (2010)

126010-13

(1987); L. Pena and N. Zamorano, report, 2005 (unpub-lished).

[19] A. Anabalon, N. Deruelle, Y. Morisawa, J. Oliva, M.Sasaki, D. Tempo, and R. Troncoso, Classical QuantumGravity 26, 065002 (2009).

[20] J. Ponce de Leon, Int. J. Mod. Phys. D 12, 757 (2003); W.Mueck, K. S. Viswanathan, and I. V. Volovich, Phys. Rev.D 62, 105019 (2000); D. Youm, Mod. Phys. Lett. A 16,2371 (2001); F. Dahia, C. Romero, L. F. P. Silva, and R.Tavakol, J. Math. Phys. (N.Y.) 48, 072501 (2007); S. Das,S. Ghosh, J.-W. van Holten, and S. Pal, J. High Energy

Phys. 04 (2009) 115.[21] P. Bostock, R. Gregory, I. Navarro, and J. Santiago, Phys.

Rev. Lett. 92, 221601 (2004).[22] C. Martinez and J. Zanelli, Phys. Rev. D 54, 3830 (1996).[23] G. Kofinas, Classical Quantum Gravity 22, L47 (2005).[24] N. Cruz, C. Martinez, and L. Pena, Classical Quantum

Gravity 11, 2731 (1994).[25] H. Ishihara, Phys. Rev. Lett. 86, 381 (2001); R. Caldwell

and D. Langlois, Phys. Lett. B 511, 129 (2001); E.Abdalla, A. G. Casali, and B. Cuadros-Melgar, Int. J.Theor. Phys. 43, 801 (2004).

CUADROS-MELGAR, AGUILAR, AND ZAMORANO PHYSICAL REVIEW D 81, 126010 (2010)

126010-14