Charged Black Hole With a Scalar Hair in (2+1) Dimensions

7/30/2019 Charged Black Hole With a Scalar Hair in (2+1) Dimensions

1/23

a r X i v : 1 3 0 5 . 5 4 4 6 v 1 [ g r - q c ] 2 3 M a y 2 0 1 3

Charged black hole with a scalar hairin (2 + 1) dimensions

Wei Xu and Liu ZhaoSchool of Physics, Nankai University, Tianjin 300071, China

email : [email protected] and [email protected]

Abstract

We obtain and analyze an exact solution to Einstein-Maxwell-scalar theoryin (2 + 1) dimensions, in which the scalar eld couples to gravity in a non-minimal way, and it also couples to itself with the self-interacting potential solelydetermined by the metric ansatz. A negative cosmological constant naturallyemerges as a constant term in the scalar potential. The metric is static andcircularly symmetric, and contains a curvature singularity at the origin. Theconditions for the metric to contain 0, 1, 2 horizons are identied, and the effects

of the scalar and electric charges on the size of the black hole radius are discussed.Under proper choices of parameters, the metric degenerates into some previouslyknown solutions in (2 + 1)-dimensional gravity.

Keywords: charged hairy black hole, (2 + 1)-dimensional gravity, non-minimalcoupling, self-interacting potential

PACS: 04.20.Jb, 04.40.Nr, 04.70.-s

1 Introduction

Gravity in (2 + 1)-dimensional spacetime has been a fascinating area of theoreticalinvestigations during the last few decades. Such studies were initiated in the earlyeighties of the last century [14]. It was once believed [4] that there is no black holesolutions in (2 + 1) dimensions in the absence of matter source, because there is nopropagating degrees of freedom. However, since the discovery of BTZ [5] MTZ [6]black holes and the asymptotic conformal symmetry [7, 8], it becomes increasinglyclear that gravity in (2 + 1) dimensions is much more interesting in its own right, notonly because black hole solutions exist, but also because that such theories are idealtheoretical laboratories for studying AdS/CFT(CMT) [718], gravity-uid dual [ 19],

Correspondance author

1
http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1http://lanl.arxiv.org/abs/1305.5446v1mailto:[email protected]:[email protected]:[email protected]:[email protected]://lanl.arxiv.org/abs/1305.5446v1


2/23

(holographic) phase transitions [ 20, 21] etc. Moreover, the study of gravity in (2 + 1)dimensions is also expected to shed some light on the understanding of more realisticor complicated cases of 4- and higher dimensional gravities.

Recently, (2 + 1)-dimensional gravity with matter source has attracted considerableinterests. Besides the standard Maxwell source [ 6, 22, 24], the inclusion of extra scalareld(s) [8, 10, 1215, 20, 23, 24, 4347, 4965], higher rank tensor elds [2833], highercurvature terms [ 3441] and/or gravitational Chern-Simons terms [ 1, 2, 42] are alsointensively studied. Unlike gravities in 4- and higher dimensions, it is possible toinclude a nite number of higher rank tensor elds in (2 + 1) dimensions [2833].The inclusion of gravitational Chern-Simons terms will bring in propagating degrees of freedom in (2 + 1) dimensions [25, 26]. Moreover, it is often much easier to obtain andanalyze black hole solutions in (2 + 1) dimensions than in other dimensions.

In this paper, we are aimed at studying black hole solution in an Einstein-Maxwell-scalar gravity with a non-minimally coupled scalar eld in (2 + 1) dimensions. Gravitycoupled with a scalar eld is not a new idea. Black hole solutions in such theories areknown as hairy black holes, and there is already a huge number of literatures on thissubject [8, 10, 1215, 20, 23, 24, 4371], and the spacetime is not only restricted to be(2 + 1)-dimansional [48, 6671]. The scalar eld may be coupled either minimally[8, 1214, 23, 24, 5359] or non-minimally [911, 6071] to gravity, and it may or maynot couple to itself through a self-interacting potential U (). In the model we shall bedealing with, couples to gravity in a non-minimal way, and it also couples to itself via a self-potential V (). The action reads

I = 12 d3x g R g R 2 2V () 14F F , (1)

where is a constant signifying the coupling strength between gravity and the scalareld, and we have set the gravitational constant equal to unity. A similar action in4-dimensional spacetime was studied in [70].

In the absence of the scalar potential V () and the Maxwell eld, the constant value = 18 will make the coupling between gravity and the free scalar eld conformallyinvariant. In the presence of V (), however, the conformal symmetry will in generalbe broken. Nonetheless, the special value = 18 for the coupling constant will greatlysimplify the solution. So, we will stick to this particular value of the gravity-scalarcoupling. Notice that we did not include explicitly a cosmological constant term inthe action, however, it will turn out that, as far as black hole solution is concerned, anegative cosmological constant will automatically emerge.

The rest of the paper is organized as follows. In Section 2 we describe the exactsolution to the eld equations which follow from the action ( 1) as well as the associatedscalar potential. Some of the basic properties of the scalar potential are discussed.Meanwhile the basic geometric properties of the metric are also outlined. In Section 3we describe some special degenerated cases of the solution, some of which are alreadyknown in the literature. Section 4 is devoted to the analysis of the metric, in particular,the conditions for the metric to behave as an asymptotic AdS 3 spacetime with a naked

2


3/23

singularity, as an extremal charged hairy black hole and as a non-extremal chargedhairy black hole are identied. In Section 5, we discuss the effect of the scalar andelectric charges on the size of the black hole horizons. Finally, in Section 6 we makesome discussions and outline some of the open problems which we intend to solve insubsequent investigations.

2 The scalar potential and the static, circularly sym-metric solution

By a straight forward variational process and discarding all possible boundary terms,we can write down the eld equations associated with the action ( 1) as follows,

G T [] T [A ] + V ()g = 0 , (2) R V = 0 , (3)

gF = 0 , (4)where

V = V (),

g ,T [] =

1

2g

+ g

+ G 2,

T [A ] =12

F F 14

g F F .

As mentioned earlier, we take = 18 throughout this paper. Note that we did notexplicitly specify the scalar potential. Actually, it will be determined uniquely by theform of the metric ansatz to be given below. The same phenomenon also happen inthe study of 4-dimensional hairy black holes [ 6668].

2.1 The circularly symmetric solution

We are interested in static, circularly symmetric solutions. To obtain such a solution,we assume that the metric takes the following form,

ds2 = f (r )d t2 +1

f (r )dr 2 + r 2d2 , (5)

where the coordinate ranges are given by < t < , r 0, . We alsoassume that both the scalar eld and the Maxwell eld A depend only on the radialcoordinate r . Under such assumptions, the Maxwell equation ( 4) gives

A dx = Q lnrr 0 dt, (6)

3


4/23

where Q and r 0 are integration constants, Q R corresponds to the electric charge,

r 0 > 0 corresponds to the radial position of the zero electric potential surface, whichcan (but not necessarily) be set equal to +

.

Eq.( 2) has only 3 non-trivial components, i.e. the ( tt ), (rr ) and ( ) components.The ( tt ) and ( rr ) components together give rise to

3ddr

(r )2

(r )d2

dr 2 (r ) = 0 .

Solving this equation we get

(r ) = 1

k r + b, (7)

even without providing a concrete form for the scalar potential V (). For (r ) notto be singular at nite nonzero r , we require k 0, b 0, and k and b cannot besimultaneously zero. Note that the special choice b = 0, k = 0 corresponds to constant.

In this paper, we are interested in solutions with non-constant scalar eld, so wewill be considering only the k = 0 branch of solutions. Inserting ( 7) into the remainingeld equations ( 2) and (3), we can obtain explicit solutions for f (r ) and V ((r )) as afunction of r . However the result is too much complicated. In particular, f (r ) containsterms which are proportional to the product of two logarithm functions and termsproportional to the special function dilog( r ) dened as

dilog(x) = x

1

ln( t)1 t

dt.

It doesnt make sense to reproduce the complicated result here. Signicant simpli-cations arises if we take the choice k = 18B and b =

18 . In this case, the scalar eld

becomes

(r ) = 8Br + B , (8)and the metric function reads

f (r ) = 3 Q2

4+ 2

Q2

9Br Q2

12

+B3r

ln(r ) +r 2

2, (9)

where and are integration constants. In order that the above f (r ) is a solution,V ((r )) as a function of r must take a very special form. We can invert (r ) for r andinsert the result in V ((r )) to get the scalar potential V ():

V () = 12

+1

51212

+

B 26

118432

Q2

B 2192 2 + 48 4 + 5 6

+13

Q2

B 222

(8 2)2

110246 ln

B (8

2)

2 . (10)

4


5/23

The set of equations ( 6)-(10) constitute a full set of exact solution to the systemdened by the action ( 1) which is not seen in the literatures before. The metric contains4 parameters B,, ,Q . Among these, B and Q have already appeared in the solutionfor the scalar and Maxwell elds respectively. 12 appears in V () as a constantterm, which plays the role of a (bare) cosmological constant. In principle, the constant can either be positive, zero or negative. However, if we wish to interpret the solutionas a black hole solution, will be necessarily negative, because in (2 + 1) dimensions,smooth black hole horizons can exist only in the presence of a negative cosmologicalconstant [27]. Thats why we adopted the notation = 12 from the very beginning.The last parameter is related to the black hole mass M via

=13

Q2

4 M , (11)

as will become clear in the degenerate case of a charged BTZ black hole [5] whichcorresponds to the case of B = 0. At this point, we do not seem to have any principleto determine the allowed range for . However, the forthcoming physical analysis willmake it clear that the value has to subject to some constraints, otherwise the solutionwill become physically unacceptable.

2.2 The scalar potential

Since lim0 V () = 12 , we may split V () into a sumV () = 12 + U (),

where U () encodes the true self interaction of the scalar eld . Apart from thebare cosmological constant term and the following 6 term, all the other terms areproportional to Q

2

B 2 , which implies that the scalar self interaction is in a subtle balancewith the Coulomb charge, though does not couple directly with the Maxwell eld.The seemingly complicated form of the potential ensures that as 0, the leadingterm in the power series expansion of U () behaves as O (6), i.e.

U ()

1

512

1

2+

B 2+

1

9

Q2

B 21

3

2ln

8B

26 +

Q2

B 2O (8), (12)

where only even powers of are present and the coefficients of all the O (8) terms arepositive. Some discussions are in due here.

If Q = 0, then U () degenerates into a 6 potential, with coefficient 1512 12 + B 2 .If, in addition, = B

2

2 , then the self-coupling of the scalar eld vanishes. If < B

2

2 , the potential has a single extremum at = 0 which is a maximum,implying that the scalar potential is unbounded from below and the system isunstable under small perturbations in . If > B

2

2 , the single extremumbecomes a minimum, which is stable against small perturbations in . Thus,stability against small perturbations in requires > B

2

2 .

5


6/23

If Q = 0, U () will possess more than one extrema. This is more easily seen inthe expanded form ( 12). It is obvious that = 0 remains an extremum whenQ = 0. Moreover, the term proportional to 6 ln(2) possesses two minima atsome small nonzero . The inclusion of the power series terms may change thelocation of these minima but the qualitative behavior of U () remain unchanged,i.e. it has two minima at = min = 0 and one maximum at = 0.

From either the original potential ( 10) or its expanded form ( 12), it seems that wecannot take B = 0. However, this observation is completely supercial, becausethe scalar eld also depends on B . If we substitute the value of (r ) into V ()and then look at the resulting expression, it will be clear that the potential isperfectly regular at B = 0.

In order to have more intuitive feelings about the scalar potential at Q = 0, wepresent a plot of U () as a function of as well as a function of r . These are givenin Fig.1 and Fig. 2 respectively. If takes its value at the local maxima, i.e. = 0,

Figure 1: Plot of U () versus , withB = 1, = 1, = 1 and Q = 1.

Figure 2: Plot of U ((r )) versus r , withB = 1, = 1, = 1 and Q = 1.

the scalar eld equation is automatically satised, and the scalar potential is exactlyzero. The corresponding solution is an Einstein-Maxwell-AdS black hole. One maytends to think that if takes its value at any of the minima = min , then thecorresponding solution would correspond to the true vacuum of the system, with aneffective cosmological constant eff = 12 + U (min ) which is more negative than12 . However, this is not the case. If we take = min = 0, then the eld equation(3) will force the Ricci scalar R to be constant, which in turn requires Q = 0. Butwhen Q = 0, the shape of the scalar potential U () changes drastically, and the twominima at non-constant totally disappear. Even though, it is still an importantobservation that when Q = 0, the scalar potential V () possesses two minima which issmaller than the cosmological constant 12 . The physical explanation for these minima

6


7/23

remains open. Due to the very complicated form of the potential U (), we are unableto nd the location of the two minima of U () analytically. Nevertheless, it is easy tond the minima of U () numerically if the parameters B,Q,, were given numericvalues. For instance, setting B = Q = = 1 , = 1, we nd that the minima of U ()are located at

= min , min 1.139824with approximate value

U (min )0.000405

2.3 Some geometric properties of the solution

To further characterize the geometry of the solution, we need to calculate some of the associated geometric quantities. First of all, the Ricci scalar contains a curvaturesingularity at r = 0 if Q = 0,

R = 36 r 3 3 rQ 2 2 + 2 B Q 2 2

62r 3.

Higher order curvature invariants such as R R and R R are also singular atr = 0, even if Q = 0. However, the expressions for these invariants are much morecomplicated and un-illustrative so we do not reproduce them here. The Cotton tensor

C abc = cR ab bR ac +

14 (

bR g ac cR g ab )is nonvanishing if either B > 0 or Q = 0,

C trt = C ttr = 14

f (r )d3

dr 3f (r ) ,

C r = C r = 14

d3

dr 3f (r ) r 2.

In (2 + 1) dimensions, the nonvanishing Cotton tensor signies that the metric is nonconformally at [72]. Thus the hairy ( B > 0) and charged ( Q = 0) solutions aregeometrically quite different from the case of static uncharged BTZ black hole [ 5].

3 Special cases

Before going into detailed analysis of our solution, we would like to point out some of the degenerated cases. Some of the degenerated cases have already been found in theliterature.

7


8/23

3.1 Charged BTZ black hole

When B = 0, the scalar eld vanishes and the system becomes Einstein-Maxwell-AdS theory. The solution degenerates into the already known static charged BTZ blackhole [5]:

f (r ) = M Q2

2ln(r ) +

r 2

2,

A dx = Q lnrr 0

dt,

V () = 12

,

(r ) = 0 ,

where M is the mass of the BTZ black hole. As mentioned earlier, though this solutionmay be stable in the Einstein-Maxwell-AdS theory, it is an unstable solution in the fulltheory ( 1).

3.2 Uncharged hairy AdS black hole

The Maxwell eld can be removed by setting Q = 0. In this case we get an unchargedhairy AdS black hole solution

f (r ) = 3 +2Br

+r 2

2,

A dx = 0 ,

(r ) = 8Br + B ,V () =

12

+1

51212

+

B 26.

Using (11), we can change into M 3 everywhere in the solution. In this case, theRicci scalar becomes constant,

R = 62

= 6 .

However, higher curvature invariants are still singular at r = 0. This solution hasalready appeared in Section 5.1 of [46]. As will become clear later, we need B

2

2 , 0 in order for this solution to be physically well behaving.

8


9/23

3.3 Conformally dressed black hole

In addition to setting Q = 0, we can choose =

B 2

2 in the mean time. Under such

conditions we reproduce the conformally dressed black hole in (2 + 1) dimensions [ 10]:

f (r ) = 3 +2Br

B 2

2+

r 2

2,

A dx = 0 ,

(r ) = 8Br + B ,V () =

12

.

Note that thought the scalar eld is still present, the self interaction potential U ()vanishes, thus making the scalar eld a free massless eld.

3.4 A special charged hairy AdS black hole in three dimension

We may also choose = B2

2 while keeping Q nonvanishing. Then we get a specialcharged hairy AdS black hole

f (r ) =

3B 2

2 +

Q2

4 2B 2

2 +

Q2

9

B

r Q2

1

2+

B

3rln(r ) +

r 2

2 ,

(r ) = 8Br + B ,A dx = Q ln

rr 0

dt,

V () = 12

118432

Q2

B 2192 2 + 48 4 + 5 6

+13

Q2

B 222

(8 2)2 1

10246 ln

B (8 2)2

.

The merit of this special case is that the scalar potential U () is not ne tuned withthe cosmological constant.

4 Horizon structures

Among the 3 elds g , A and , the latter two are easily understandable. A is just the standard Coulomb potential in (2 + 1) dimensions, is a radially distributedscalar which takes its maximum max = 8 at r = 0 and decreases gradually to zeroas r + when B > 0.

9


10/23

The metric g is more involved to understand. This is because of the complicatedform (9) of the metric function f (r ). As usual, the zeros of f (r ) (if any) will correspondto horizons in the metric. So, we need to nd (at least the condition for the existenceof) the zeros of f (r ).

4.1 Q = 0

When Q = 0, the functions f (r ) and f (r ) are simplied drastically,

f (r ) = 3 +2Br

+r 2

2,

f (r) =

2B

r 2 +

2r

2.

We may divide the solution in three sub cases:

If > 0, f (r ) will always be positive, implying that the metric correspondsto an asymptotically AdS 3 spacetime containing a naked singularity. This is aphysically uninteresting case;

If = 0, then the metric degenerates into the (2 + 1)-dimensional empty topo-logical AdS spacetime;

If, instead, < 0, then f (r ) will remain positive for r [0, + ). Meanwhile,f (r ) as r 0, f (r ) + as r + . This implies f (r ) increasesmonotonically and hence contains exactly one zero. The zero of f (r ) correspondsto the event horizon of a neutral AdS black hole with a scalar hair.

Combining with the analysis made in subsection 2.2, we see that the physicallyacceptable range for the parameter is B

2

2 , 0 at Q = 0.

4.2 Q = 0

The form of f (r ) with nonvanishing Q is much more complicated than the Q = 0 case.To nd whether f (r ) has some zeros, we need to know the asymptotic behavior andthe number of extrema of f (r ).

From ( 9) it can be seen that f (r ) is dominated by the term r2

2 in the far region, so,f (r ) + as r + . On the other hand, as r 0, f (r ) is dominated by the termQ

2 B3

ln( r )r if B > 0, or by the term Q

2

2 ln(r ) if B = 0. So, f (r ) always approaches + as r 0. Remember that we have excluded the possibility of choosing B < 0 in orderthat (r ) is not singular at nite nonzero r . Combining the asymptotic behaviors atboth ends, we see that for Q = 0, f (r ) will approach +

at both ends. Therefore,

f (r ) has to have some extrema, and the total number of extrema must be odd. Let us

10


11/23

denote the location of the extrema of f (r ) by r X . If f (r ) has more than one extrema, anextra index may be adopted to distinguish these different extrema if necessary. Clearly,the position r X of every extremum of f (r ) obeys f (r X) = 0, or more conveniently

(r X)2f (r X) = 0 , (13)

where r 2f (r ) is given by the following expression

r 2f (r ) = 2 B B2

9+ +

13

Q2B ln(r ) 12

Q2r +2r 3

2. (14)

However, the converse needs not to be true: if r 2f (r ) happens to be zero at some of its extremum r i (not to be confused with r X), then r i will correspond to an inectionpoint of f (r ), rather than an extremum.

In order to determine the number of roots for f (r ), we need to consider two distinctcases, i.e. B = 0 and B > 0. For B = 0, it is easy to get the location of the extremumof f (r ) using (13) and (14). The only real positive extremum of f (r ) in this case islocated at r = r X = 12 |Q|. Inserting the value of r X and B = 0 into the solution ( 9),we get the following:

If < Q2

6 ln(r X), we have f (r X) < 0. So f (r ) will have two zeros, each correspondsto a black hole horizon. Among these, the outer horizon is the event horizon.This case corresponds to a charged non-extremal AdS black hole without the

scalar hair;

If = Q2

6 ln(r X), we have f (r X) = 0. It is evident that r ex is the only root of f (r ),which corresponds to the horizon of a charged extremal AdS black hole withoutthe scalar hair. In this particular case, it may be better to replace r X with r ex ,which stands for the radius of the extremal black hole;

If > Q2

6 ln(r X), then f (r X) > 0, there will be no zeros for the function f (r ), sothe metric becomes an asymptotically AdS spacetime with a naked singularityat the origin, which is a physically uninteresting case.

The case with B > 0 is much more complicated comparing to the B = 0 case. Itis impossible to solve (13) analytically to get r ex , so we turn to look at the extrema of r 2f (r ). We introduce the following polynomial function:

g(r ) 62 rddr

r 2f (r ) = 36 r 3 3Q22r + 2 BQ 22 . (15)Every real positive root of g(r ) will correspond to an extremum or an inection point of r 2f (r ), and the collection of signs of r 2f (r ) at its extrema will determine the numberof roots thereof. Fortunately, the function g(r ) is simple enough so that its roots canbe found analytically. In Appendix A, we shall present the details about the roots of g(r ).

11


12/23

According to Appendix A, the number of real positive roots of g(r ) will changewhen the value of the parameter B crosses |Q |6 . The signicance of this change in thenumber of real positive roots will be best illustrated if we look at the extremum of g(r ). Taking the rst derivative of g(r ) with respect to r and nding the root of theresulting expression, we nd that g(r ) has only one real positive extremum located atr = |Q |6 . Clearly this is a minimum. Substituting this value of r into g(r ) itself, wend the minimum value of g(r ), which reads

gmin = 2 BQ 22 13

Q33. (16)

If B > |Q |6 , then gmin is positive, which implies that g(r ) has no real positive root,which in turn implies that r 2f (r ) has no extrema for r > 0, so that f (r ) has only oneroot, i.e. f (r ) has only one extremum. If B = |Q |6 , then g(r ) is zero at its extremum.This means that g(r ) has only one root which is located at its minimum. This impliesthat the minimum of g(r ) corresponds to an inection point rather than an extremumof r 2f (r ). So, in the end, r 2f (r ) still has no extremum for r > 0, resulting in theconclusion that f (r ) has only a single extremum for r > 0.

The problem becomes more complicated when 0 < B < |Q |6 . In this case, g(r ) hasthree roots, two of which are positive. Therefore, r 2f (r ) will also have two extrema forr > 0. Among these, the extremum at r 1 is a minimum, and that at r 2 is a maximum,and we have r 2 < r 1.

Now let us assume 0 < B < |Q |6 . Since r 1 and r 2 are both real positive roots of g(r ), we have

r 3i =Q22r i

12 BQ 22

18, i = 1 , 2.

Inserting this into ( 14), we get

r 2i f (r i ) = 2 B w(r i ) , (17)where

w(r i )

Q2

6B(B ln(r i )

r i

B ) .

Eq.( 17) gives the value that r 2f (r ) must take at any of its extrema. In particular, theanalysis made in Appendix A implies w(r 1) < w (r 2).

Depending on the value of the parameter , r 2i f (r i ) will take different signs:

If = w(r 1) or = w(r 2), we have r 21 f (r 1) = 0 or r 22f (r 2) = 0, i.e. r 2f (r ) iszero at one of its extrema. Besides this accidental root, r 2f (r ) has another rootwhich is not at its extrema. So, totally r 2f (r ) will have two roots, one of these(the normal root) corresponds to the extremum of f (r ), the other (the accidentalroot at the extremum) corresponds to an inection point of f (r ). So, in the end,f (r ) will have only one extremum in this case;

12


13/23

If < w (r 1) or > w (r 2), we have either r 22 f (r 2) > r 21f (r 1) > 0 or 0 >r 22f (r 2) > r 21f (r 1). In both cases all the extrema of r 2f (r ) have the same signs,so there is only one positive root of r 2f (r ). Consequently f (r ) will have only asingle extremum for r > 0;

If w(r 1) < < w (r 2), we nd r 22f (r 2) > 0 > r 21f (r 1). The two positive extremaof r 2f (r ) have different signs, indicating that the curve for r 2f (r ) will cross thehorizontal axes three times. Therefore, there are three positive roots for r 2f (r ),each corresponds to an extremum of f (r ).

Summarizing the above discussions, we make the following conclusion on the numberof extrema for the metric function f (r ):

If B = 0 or B |Q |6 , f (r ) has only a single extremum; If 0 < B < |

Q |6 , the number of extrema for f (r ) depends on the value of theparameter . Explicitly,

If w(r 1) or w(r 2), f (r ) still has only a single extremum; If w(r 1) < < w (r 2) will have three extrema.

Therefore, the horizon structure of our solution at Q = 0 will depend cruciallyon the range of parameters.

Since at any extremum r X of f (r ) we have f (r X) = 0, an arbitrary multiple of f (r X) can be added to f (r X) to yield a simplied expression for f (r X). Specically,we take the following combination:

p(r X) B

r X + Bf (r X) + f (r X)

r X(9r X + 6 B )6B

=B

r X + Bf (r X) = 36 ( r X)3 9 Q22 r X 4 BQ 22. (18)

Clearly, p(r X) and f (r X) always have the same sign, so the collection of signs of p(r X)

at all the extrema r X of f (r ) will determine the number of roots of f (r ).Let us consider the extremal case dened via f (r ex) = p(r ex) = 0 and f (r ex) = 0.

In this case, f (r ) happens to be zero at one of its extremum. Since the zeros of f (r ex )and p(r ex) always coincide, we can try to nd the zeros of p(r ex) to get the value of r ex .

Since we have B 0, we may assumeB =

a4|Q|, rex = |Q|, (19)

so that the equation p(r ex) = 0 becomes

363 9 a = 0 . (20)13


14/23

The condition a 0 implies 12 , where the lower bound for correspond to a = 0,i.e. B = 0, as discussed previously. Generically, eq.( 20) can have 3 zeros. However,only one of these is real positive and lies in the range

12 , +

, which is given in

terms of a as

=z 6

+1

2z , z =

3 3 a + 3 a2 3, a 0. (21)The detailed analysis on the solution to ( 20) can be carried out in exactly the sameway as is done in Appendix A for the similar equation ( 29).

Now, substituting ( 11), (19) into the equation f (r ex) = 0, we get,

=Q2

6ln(r ex) =

Q2

6ln (|Q|) .

This is the same condition which appeared in the B = 0 case. The only difference liesin that is xed at the value 12 when B = 0, while for B = 0, its value is given by ( 21).

We have already made it clear that when B = 0 or B |Q |6 , or when 0 < B < |Q |6with w(r 1) or w(r 2), f (r ) always has only a single minimum. So, for theseparameter ranges, the zero r ex of f (r ) as described in (19) and( 21) is the only zeroand minimum of f (r ). So, these cases correspond to extremal black holes with horizonradius

r ex = exp6 Q2

.

If at the extremum r X of f (r ), p(r X) fails to be zero, then the corresponding solutionwill not correspond to an extremal black hole. Let us now consider such cases in moredetail.

For all r X r ex = |Q|, we have p(r X) = 9(12 r 2X Q22) p(r ex) = 9 3Q22 Q 22 = 18Q22 > 0,

i.e. p(r X) increases monotonically for r X r ex . So, if p(r X) < 0, then r X must belocated to the left of r ex , i.e. r X < r ex . However, if p(r X) > 0, we cannot deduce fromabove that r X > r ex .

Consider the following identity:

1 B

r X + Bf (r X) = 3

Q2

2ln (r X) .

This is equivalent to

=Q2

6ln (r X) +

r X3(r X + B )

f (r X). (22)

If f (r X) < 0, then

< Q2

6 ln (r X) < Q

2

6 ln (r ex) . (23)

14


15/23

In (23), r X may be extremely close to r ex , while still keeping r X < r ex . So, we maythink of

< Q2

6ln (r ex) (24)

to be the condition which must be imposed on the parameter in order for the metricto have two disjoint horizons, to behave as a non-extremal black hole. If f (r X) > 0,then

>Q2

6ln (r X) . (25)

Under this condition, the metric contains no horizons and corresponds to an asymp-totically AdS 3 spacetime with a naked singularity.

What remains untouched is the real troublesome case with 0 < B < |Q |6 andw(r 1) < < w (r 2). In this parameter range, we have to be careful about how manyzeros there are for f (r ), because f (r ) has three extrema. Let us denote the locationof the three extrema by r X1 , r X2 and r X3 respectively. Let r X1 , r X2 and r X3 be orderedsuch that r X1 < r X2 < r X3 . At these points, r 2f (r ) vanishes, its two positive extremalocate between these zeros, i.e.

r X1 < r 2 < r X2 < r 1 < r X3 .

From Appendix A, we have |Q |6

< r 1 < 3|Q |

6< |Q |

2, so,

r X2 < |Q|2 .Therefore, according to ( 18), we have

p(r X2) =B

r X2 + Bf (r X2 ) = 36 ( r X2)3 9 Q22 r X2 4 BQ 22

= 36 r X2 r 2X2 Q22

4 4 BQ22 < 0,

i.e. f (r X2 ) < 0. Since r X2 lies in between r X1 and r X3 , it corresponds to the local maximum of f (r ), so the above result implies that f (r ) is negative at all its threeextrema. Combining with the asymptotic behavior, we deduce that f (r ) have preciselytwo zeros in this case, so, whenever f (r ) has three extrema, the solution correspondsto a non-extremal charged hairy black hole with AdS asymptotics. Please note thatfor this case, we have

< w (r 2) =Q 2

6B(B ln(r 2) r 2 B ) 1

6, and in general the

radius r + for the outer horizon non-extremal black hole is bigger yet than r ex , we see

16


17/23

that the effect of inclusion of the f B (r ) term in f (r ) is to make f (r ) more negative.Therefore, to compensate for this extra negative contribution, the f (r )|B =0 term mustbecome more positive in order to make a zero for f (r ). In other words, the size of theblack hole must increase as B increases.

5.2 Effect of the electric charge

We can also separate f (r ) into its Q2 independent and Q2-dependent parts, i.e.

f (r ) = f (r )|Q =0 + f Q 2 (r ), (27)where

f Q 2 (r ) = Q2df (r )d(Q2)

= Q2 14 B9r

12

+ B3r

ln(r ) . (28)

The horizon condition is a proper balance between the f (r )|Q =0 and the f Q 2 (r ) terms.In Subsection 4.1, it was shown that f (r )|Q =0 as r 0 and increases monoton-ically with r . On the other hand, from the above expression, we see that f Q 2 (r ) + as r 0 and decreases monotonically with r . Moreover, in the near end ( r 0),the term 1r ln(r ) in f Q 2 (r ) takes over the term 1r in f (r )|Q =0 , and in the far region(r + ), the term r

2

2 in f (r )|Q =0 dominants, while f Q 2 (r ) becomes increasinglynegative. In effect, the inclusion of the f Q 2 (r ) term in f (r ) results in two different con-sequences: in the near region, f (r ) develops a novel zero which is the inner horizon forthe charged black hole; in the far region, the original zero of f (r )|Q =0 (now being theouter horizon of the charged black hole) gets increased with the inclusion of the f Q 2 (r )term, and as Q2 increases, the radius of the outer horizon also increases monotonically.

6 Discussions

In this paper, we obtained an exact solution for the Einstein-Maxwell-scalar theory in(2 + 1) dimensions, in which the scalar eld couples to gravity non-minimally and isalso self-coupling to itself in a peculiar way. The solution is static, circularly symmetricand the scalar self potential is completely determined by static-ness and the circularsymmetry of the solution. In particular, a negative cosmological constant naturallyemerges as a constant term in the scalar potential, if we require that for certain ranges of parameters the solution represents a charged hairy black hole. Under proper choices of parameter values, our solution degenerates into some already known (2+1)-dimensionalblack hole solutions.

When the electric charge Q is nonzero, the scalar potential possesses three extrema,one maximum at = 0 and two minima at = 0. The scalar eld cannot stay at

the constant value = min = 0, otherwise the eld equations will not be satised.17


18/23

We also identied the conditions for the metric to behave as a charged extremalblack hole, as an asymptotically AdS 3 spacetime with a naked singularity at the originand as a charged non-extremal black hole. When black hole horizons exist, it is shownthat the size of the (outer) horizon increases monotonically with both the scalar chargeand the electric charge.

Some of the related properties and duality relations will become instantly interestingfurther tasks to be worked out. These are

The thermodynamic quantities and the associated laws of thermodynamics; The properties of the boundary CFT, if any;

These days the uid dual of AdS gravity is an active area of study. It will beinteresting to ask whether there is uid dual of the black hole solution given inthis paper;

The solution considered in this paper is only static and circularly symmetric.It would be interesting to ask whether one can nd more complicated solutionsto the same theory. For instance, it is interesting to ask whether one can ndrotationally symmetric solutions an determine the scalar self interaction solelyby the form of the metric ansatz;

The scalar eld in model of this paper is neutral and does not couple to theelectromagnetic eld. It would also be interesting to allow the scalar eld tobecome complex and thus couples directly to the electromagnetic eld.

We leave the answer to all these problems for future works.

Appendix A: roots of the function g(r )

In this appendix we shall solve the root of the function g(r ) given by (15), i.e. the rootof the equation

36r 3 3Q22r + 2 BQ 22 = 0 . (29)With the aid of a computer algebra system like Maple , it is easy to nd that the rootsof the above equation are given by

r 1 =16

z +Q22

z ,

r 2 =1

12 z Q22

z + i 3 z

Q22

z ,

r 3 =1

12 z Q22

z i

3 z Q22

z ,

18


19/23

where

z =3

6BQ 22 +

36BQ 44

Q66. (30)

Among the three roots of g(r ), only the real positive roots are relevant to our problem.So we need to identify which and how many of the roots are real positive.

It is evident that if B |Q |6 , then z is real. It follows that if B > |

Q |6 , then r 1 isreal but negative, while both r 2 and r 3 are complex. If B = |Q |6 , then r 1 and r 2 willbecome degenerate and both are real positive. In this case r 3 is negative. If B < |Q |6 ,z becomes complex. In this case, it is not too difficult to see that the modulus of z isequal to |Q|, so we can denote z as

z =|Q

| (cos + i sin ).

Comparing this expression with the original denition ( 30), we see that must takevalue in the range

6

Charged Black Hole With a Scalar Hair in (2+1) Dimensions

Documents

Transcript of Charged Black Hole With a Scalar Hair in (2+1) Dimensions