AdS waves as exact solutions to quadratic gravity

AdS waves as exact solutions to quadratic gravity

I˙brahim Gu¨llu¨,1,*Metin Gu¨rses,2,†Tahsin C¸ ag˘r
S¸is¸man,1,‡and Bayram Tekin1,x 1

Department of Physics, Middle East Technical University, 06531 Ankara, Turkey

2

Department of Mathematics, Faculty of Sciences Bilkent University, 06800 Ankara, Turkey (Received 18 February 2011; published 8 April 2011)

We give an exact solution of the quadratic gravity in D dimensions. The solution is a plane-fronted wave metric with a cosmological constant. This metric solves not only the full quadratic gravity field equations but also the linearized ones which include the linearized equations of the recently found critical gravity. A subset of the solutions change the asymptotic structure of the anti-de Sitter space due to their logarithmic behavior.

DOI:10.1103/PhysRevD.83.084015 PACS numbers: 04.50.
h, 04.20.Jb, 04.30.
w

I. INTRODUCTION

Quadratic deformations of Einstein’s gravity have al-ways attracted attention since the inception of general relativity for various reasons. Initial motivation was to understand the uniqueness of general relativity, but later on it was realized that perturbative quantum gravity re-quired these terms [1]. Generically, quadratic gravity theo-ries at the linearized level describe massless and massive spin-2, and massive spin-0 modes. Massive spin-2 mode ruins perturbative unitarity due to its ghost nature. Recently, a new interest in quadratic theories arose, since it was shown that in three dimensions the ghost disappears [2], and in D-dimensions in the anti-de Sitter (AdS) back-ground, certain quadratic theories become ‘‘critical’’ with only a massless spin-2 excitation just like the Einstein gravity [3,4]. These observations led us to consider exact solutions to the quadratic gravity models. The existence of a cosmological constant changes the structure of the field equations and the solutions dramatically. For example, the plane wave metric solves all higher order gravity field equations coming from string theory with zero cosmologi-cal constant [5,6]. With a cosmologicosmologi-cal constant this does not work; however, if one starts with a plane-fronted AdS-wave [7,8], one may have exact solutions to quadratic curvature gravity models as we show below. In general, save the Schwarzschild-anti de Sitter solution, no exact solution is known in generic quadratic gravity theories. In some specific theories, such as the Einstein-Gauss-Bonnet theory, some static spherically-symmetric solutions are known [9]; in the new massive gravity of [2], AdS-wave solutions were found in [10], and types D and N solutions were found in [11]. In this paper, in generic D-dimensions, we find the AdS-wave solutions for general quadratic theories including the critical gravity. By construction, our exact solutions also solve the linearized wave equations.

The layout of the paper is as follows: In Sec. II, we briefly review the quadratic gravity theory and specifically the critical gravity in D-dimensions. In Sec.III, we discuss the AdS-wave metric, compute the Riemann and the rele-vant tensors, and derive the field equations. In Sec.IV, we present the solutions of the field equations deferring one rather cumbersome case to the Appendix.

II. CRITICAL POINTS OF QUADRATIC GRAVITIES

This work was inspired by the critical gravity models [3,4] which are a certain subset of quadratic curvature gravities. Therefore, we will briefly recapitulate these models. The action of the quadratic gravity is

I ¼Z dD_xpffiffiffiffiffiffiffi
_g1
ðR
20Þ þ R 2_{þ R}2  þ ðR2 
4R2þ R2Þ  : (1)

The (source-free) field equations were given in [12,13] as 1
 R
1₂gR þ0g  þ2RR
1₄gR  þð2þÞðgh
rrÞR þ2½RR
2RRþR R
2RR
1₄gðR2 
4R2þR2Þ þhR
1 2gR  þ2R
1 4gR  R ¼ 0: (2)

The two AdS vacua of the theory satisfy


0 2
þ f2 ¼ 0; f  ðD þ Þ ðD
4Þ ðD
2Þ2þ  ðD
3ÞðD
4Þ ðD
1ÞðD
2Þ: (3) *e075555@metu.edu.tr †_{gurses@fen.bilkent.edu.tr} ‡_{sisman@metu.edu.tr} x btekin@metu.edu.tr

Around any of these vacua, the linearized equations read [12] cGLþ ð2 þ Þ  gh
rrþ 2 D
2g  RL þ hGL 
2 D
1gR L  ¼ 0; (4)

for the metric perturbation h_ g_
g_. Here,GL is

the linearized Einstein tensor, and c is given by

c  1
þ 4 D D
2 þ 4 D
1 þ 4ðD
3ÞðD
4Þ ðD
1ÞðD
2Þ : (5) The critical theory is obtained as follows: One chooses 4ðD
1Þ þ D ¼ 0, which then kills the massive spin-0 mode, and sets RL_¼
2

D
2h ¼0. Then, in the transverse

gauge r_h

¼ 0, the linearized equations simplify to

ð h þ cÞGL ¼ 0; (6) or more explicitly  h
_{ðD
1ÞðD
2Þ}4
M2h
4 ðD
1ÞðD
2Þ  h ¼ 0; (7) where M2 ¼
1   c þ 4 ðD
1ÞðD
2Þ  ; (8)

and the point M2 ¼ 0 defines the critical point where one is left only with a massless spin-2 excitation.

III. ADS-WAVE METRIC

The quadratic field equations are highly nontrivial, therefore the form of the metric ansatz is important in finding solutions. Here, we take the D-dimensional AdS-wave metric (which is conformally related to the pp-AdS-wave metric) to be in the Kerr-Schild form [14,15] as

g¼ gþ 2V; (9)

where g_is the metric of the AdS space. The vector  _¼

g¼ g is assumed to be null and geodesic with

respect to both gand g, that is [16]

_ _{¼ g}

¼ g¼ 0;

_r

 ¼ r ¼ 0:

(10)

The inverse metric can be found as

g_{¼ g}
_2V_{; (11)}

which is an exact form. Here, note the similarity with a perturbation analysis where the metric perturbation is

defined as h_ g_
g_; and, at the linearized level, the inverse metric becomes g_{¼ g}
_h_{. Now, let us}

choose the coordinates on AdS to be in the conformally flat form

g¼ 
2; (12)

where  ¼ kx, k is a constant vector, and we choose

the flat space coordinates as x _{¼ ðu; v; x}1_{;    ; x}n_{Þ with}

n ¼ D
2. This choice of g_simplifies the construction. Here, u and v are null coordinates, hence more explicitly

dxdx¼ 2dudv þ

Xn m¼1

ðdxm_Þ2_{: (13)}

Then, the vector k ¼ ð0; 0; k1;    ; knÞ is related to the

cosmological constant as R_¼ 2

D
2g;  ¼

ðD
1ÞðD
2Þ 2‘2 ; (14)

where _‘12 Pn_m¼1kmkm>0, hence we will be working

only with  < 0 (note that to conform with the usual notation, we introduced the AdS radius ‘). With this coor-dinate choice, _naturally becomes

dx¼ du ) @ ¼ 2@v: (15)

The function V is assumed not to depend on v, that is

@V ¼0: (16)

This choice is extremely important; since, with it, h_ 2V becomes transverse rh¼ rh¼ 0 and

traceless h  g_h

¼ gh¼ 0. Furthermore, as

we show below by explicit calculations, the parts of the curvature tensors which are quadratic in V drop out with this choice. It is also important to realize that if V ¼cðuÞ2 ,

then g_ corresponds to just a coordinate transformed version of g. This is clear since in this case, one can

simply define ðu;~vÞ in such a way that the two-dimensional subspace metric cðuÞdu2þ 2dudv becomes 2dud~v. This fact will play a role in deciding how our solutions should decay at infinity.

We are now ready to compute the Riemann tensor. The connection corresponding to gsplits into two parts:

  ¼ 



þ ; (17)

where _is the Levi-Civita connection corresponding to g, which reads as   ¼
1ð  @ þ  @
gg@Þ: (18)

The nontrivial part of the connection is given as 

¼ ð@V þ @V þ2V rÞ
@V:

(19)

The null vector _ with @_¼ 0 satisfies the following equations, which are frequently used in the computations:

r_ ¼ 1 ð@ þ @Þ; h ¼ 1
D ‘2 ; g_{ð r} Þð rÞ ¼ 1 ‘2; (20) where h  gr

r. The Riemann tensor reduces to

R

¼ Rþ r
r: (21)

Note that the ð


Þ part of the Riemann

tensor becomes zero, since the explicit calculation of 

yields a symmetric tensor in  and  indices

 ¼
@V  @_{V þ}2V  @   : (22)

After a lengthy computation, one finds the Riemann tensor as R  ¼ R
2  2 ½@@ðV 2_Þ þ 2 3 ½g@@ _ðV2_Þ þ 2 3 ½@ðV 2_Þ@_þ2 ½  @@V þ 1 @V@ 
1   @@V  ; (23)

where as usual2A_½B_ A_B_
A_B_. Then, the Ricci tensor R R



 and the curvature scalar can be

com-puted as R¼ 2 D
2g
H; R ¼ 2D D
2; (24) where H is defined as H  4 @V@ _{þ hV þ}4V 2@@ _þ 4 D
2V: (25) The following two relations will be used in the field equations: hR¼
 hH þ 4
2D ‘2 H þ 4 @H@   ; (26) hðVÞ ¼  hV þ 4
2D ‘2 V þ 4 @V@   : (27)

With our metric ansatz, the field Eqs. (2) split into two parts in the form Ag_þ B_. Trace of this equation yields a relation between the effective cosmological constant and the parameters of the theory exactly given as (3), where we have used R ¼_D
22D and R2 _
4R2þ

R2¼_{ðD
1ÞðD
2Þ}42DðD
3Þ. Observe that the  part does not

contribute to the trace equation. To obtain the rest of the field equations, the nontrivial computation is the contrac-tion of two Riemann tensors, that is the R R term.

After a lengthy computation, one obtains

RR¼ 8 ðD
1ÞðD
2Þ  R
 D
2g  ; (28) and similarly, R_R__{¼ 4} D
2  R_
 D
2g  ; RR¼ 2 D
1  Rþ 2 ðD
2Þ2g  : (29)

Finally, the remaining field equations become

ð h þ cÞðHÞ ¼ 0; (30)

where H was given in (25). Observe the similarity of this equation to (6). Then, using

hðÞ ¼ 4 D
1; r_ð_Þ ¼ 1 ð2@ þ @ þ @Þ; (31) we get   h þ 4 @ _@ þ 4ðD
3Þ ðD
1ÞðD
2Þ
M2  h þ 4 @ _@ þ 4ðD
3Þ ðD
1ÞðD
2Þ  V ¼0; (32) where M2 is defined as (8). Therefore, the exact equations of the quadratic gravity reduces to a linear fourth-order wave equation. One can show that putting h¼ 2V

in (4) yields (32). When M2¼ 0, the theory reduces to the linearized equations of the critical theory of [3,4]. There is a fine point here: In the critical theory, to get rid off the massive scalar mode, one imposes a relation between  and , 4ðD
1Þ þ D ¼ 0; but here these parameters are arbitrary, since RL R
R vanishes identically for

the AdS-wave metric. In the next section, we discuss the solutions of (32) in detail.

IV. SOLUTIONS OF THE FIELD EQUATIONS The Eq. (32) is of the form

 h þ 4 @ _@ þ b  h þ 4 @ _@ þ a  V ¼0; (33)

with a ¼_{ðD
1ÞðD
2Þ}4ðD
3Þ and b ¼ a
M2, which are obviously equal for M2 ¼ 0. Whether M2¼ 0 or not changes the behavior of solutions dramatically, therefore we will dis-cuss these cases separately.

A. Thea
b case

One can define V ¼ Vaþ Vb in such a way that each

part satisfies the corresponding second order equation  h þ 4 @ _@ þ a  V_a¼ 0;  h þ 4 @ _@ þ b  V_b¼ 0: (34)

Without loss of generality, let us choose only the nth component of the k_{vector to be nonvanishing; and define}

z ¼ xn_{, then  ¼}z

‘. With this choice, the quadratic

equa-tion reduces to  h þ 4 @ _@ þ a  Va ¼ _z2 ‘2@ 2_{þ 6}
D ‘2 z@zþ a  Va ¼ 0; (35)

where @2 @@. Note that the Vbequation is similar.

Using the separation of variables technique as Va

aðu; x1; x2;    ; xn
1Þaðu; zÞ, we can split the quadratic

equation into two parts:

ð ~r2þ_aÞaðu;x1;x2;;xn
1Þ ¼ 0;  z2 d 2 dz2þð6
DÞz d dzþða‘ 2
_ az2Þ  aðu;zÞ ¼ 0; (36)

where ais an arbitrary real number at this stage and ~r2 

P_n
1

i¼1 @ 2

@ðxi_Þ2. Note that the @ 2

@u@vterm does not appear because

of the v-independence of the solution. On the other hand, the solution will have an arbitrary dependence on u. Depending on the boundary conditions, acan be

continu-ous or discrete. Then, a formal solution will be of the form

Vaðu; x1; x2;    ; xn
1; zÞ ¼

Z

daAðaÞaðu; x1; x2;    ; xn
1; aÞaðu; z; aÞ

þX

a

B_a_a;aðu; x1; x2;    ; xn
1Þ_a;aðu; zÞ; (37)

where AðaÞ and Ba are arbitrary functions of u. Over the

entire ðu; v; xiÞ flat space, the first equation in (36) does not have bounded solutions when a<0. Therefore, we will

take a 2a 0. Here, the discussion bifurcates whether

a¼ 0 or not. Concentrating on the continuous case, first

we start with _a
0. Then, the solutions are of the form aðu; x1; x2;    ; xn
1Þ ¼ c1ðuÞ sinð ~a: ~r þ c2ðuÞÞ; (38)

where ~a is an arbitrary vector with magnitude a, and

~r ¼ ðxiÞ. Now, we come to the second equation in (36) which is in the form of the modified Bessel equation for D ¼5, and can be converted to this form for any other D by the following redefinition:

aðu; zÞ  zðD
5Þ=2faðu; zÞ; (39)

which then yields  z2 d 2 dz2þ z d dz
ð 2 aþ 2az2Þ  faðu; zÞ ¼ 0; (40) where a ¼1₂ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD
5Þ2
_4a‘2 p

. For generic D and non-vanishing a, the solution is given as

aðu; zÞ ¼ zðD
5Þ=2½c3ðuÞIaðzaÞ þ c4ðuÞKaðzaÞ; (41)

where I and K are the modified Bessel functions

of the first and second kind, respectively. V_b will have the similar solutions with _b¼1₂pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD
5Þ2
4b‘2¼

1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD
1Þ2_þ4‘2_M2

p

. As we discussed in Sec. III, we re-quire the solution to go like 1

z2at the boundary z ¼0; the

modified Bessel functions approach z !0 as I_ðzÞ  z_,

K_ z
and K₀
lnz. Therefore, we keep both c₃and c₄. It is important to realize that _aand _bare real. This is automatically satisfied for a since a ¼
2ðD
3Þ_‘2 , and

a ¼ 1₂ðD
1Þ. The reality of b puts a constraint on

M2which is

M2 
ðD
1Þ

2

4‘2 : (42)

This bound is exactly equivalent to the Breitenlohner-Freedman (BF) bound on the mass of a scalar excitation in AdS [17]. When the bound is saturated, b¼ 0 and

logarithmic solutions arise.

For the sake of completeness, let us write the solution

Vðu; ~r; zÞ ¼ zðD
5Þ=2½c_a;1ðuÞI_aðz_aÞ þ c_a;2ðuÞK_aðz_aÞ sinð ~_a: ~r þ c_a;3ðuÞÞ þ zðD
5Þ=2_½c

b;1ðuÞIbðzbÞ þ cb;2ðuÞKbðzbÞ sinð ~b: ~r þ cb;3ðuÞÞ: (43)

The full solution can be obtained by integrating the first part of (43) with respect to ~a, and the second part with

respect to ~b as in (37). Here, ca;1ðuÞ and ca;2ðuÞ depend

on ~a, and cb;1ðuÞ and cb;2ðuÞ depend on ~b. In odd

dimensions for D 5, c_2;a should vanish, otherwise V is unbounded at the boundary of AdS at z ¼0. (Note that we do not worry about the disconnected component of the boundary, which is just the single point z ¼ 1, since this point just compactifies the boundary of AdS from RD
1_{! S}D
1_{.) For generic values of M}2_{, }

b is not an

integer or an odd integer, therefore in general c_2;bmust be kept.

Let us now consider the _a¼ _b¼ 0 case. [The case when only one of these parameters vanish follows from the discussion above and the discussion below.] In this case, aðu; x1; x2;    ; xn
1Þ ¼ cðuÞ þ ~qðuÞ:~r, but we take

~

qðuÞ ¼0 to have a bounded solution at j~rj ! 1. The other equation reduces to  z2 d 2 dz2þ z d dz
 2 a  f_aðu; zÞ ¼ 0; (44) whose solution is faðzÞ ¼ ca;1ðuÞzjajþ ca;1ðuÞz
jaj.

Adding also the solution of the b equation, one gets

Vðu; zÞ ¼ c_a;1ðuÞzD
3þ c_a;2ðuÞ 1 z2 þ zðD
5Þ=2_ðc

b;1ðuÞzjbjþ c

b;2ðuÞz
jbjÞ: (45)

When M2>0, because of the last term the spacetime is not asymptotically AdS as one approaches the boundary z ¼0: Namely, Vðu; zÞ  c_b;2ðuÞz
ð2þÞ where  >0, hence c_b;2ðuÞ ¼ 0. On the other hand, when 0 > M2>
ðD
1Þ2

4‘2 , all the terms are allowed. When the BF bound

(42) is saturated, one has (b¼ 0)

Vðu; zÞ ¼ c_a;1ðuÞzD
3þ c_a;2ðuÞ 1 z2 þ zðD
5Þ=2_c b;1ðuÞ þ cb;2ðuÞ ln  z ‘  ; (46)

which yields an asymptotically AdS metric.

Up to now, we have implicitly assumed D >3, but in fact our expressions are also valid for D ¼3 for which (45) and (46) reduce to the results given in [10].

B. Thea ¼ b case which includes the critical theory In this case, a ¼
2ðD
3Þ_‘2 and Va, as found above (43), is

a solution, but this is not the only solution: One should consider the full quadratic theory;

ðz2_@2_{þ ð6
DÞz@}

z
2ðD
3ÞÞ2V ¼0: (47)

Defining W  ðz2@2þ ð6
DÞz@z
2ðD
3ÞÞV; so that,

ðz2_@2_{þ ð6
DÞz@}

z
2ðD
3ÞÞW ¼ 0, since we know

the solution of the latter equation from the above discus-sion, we can simply consider the nonhomogeneous equa-tion, where W is a source term. The 
0 case is somewhat cumbersome and not particularly illuminating, therefore we defer it to the Appendix, and here study the  ¼0 case. Then, the solution to the quadratic equation is Wðu; zÞ ¼ c₁ðuÞzD
3þ c₂ðuÞz
2: (48) The nonhomogeneous equation becomes

ðz2_@2_{þ ð6
DÞz@}

z
2ðD
3ÞÞV

¼ c₁ðuÞzD
3_{þ c}

2ðuÞz
2; (49)

which after rescaling Vðu; zÞ ¼ zðD
5Þ=2fðu; zÞ can be transformed to  z2 d 2 dz2þ z d dz
ðD
1Þ2 4  fðzÞ ¼ c₁ðuÞzðD
1Þ=2_{þ c} 2ðuÞzð1
DÞ=2: (50)

The general solution of this equation is

Vðu; zÞ ¼ d₁ðuÞzD
3þd2ðuÞ z2 þ 1 D
1  c₁ðuÞzD
3
c2ðuÞ z2  lnz ‘  ; (51)

which was also obtained recently in [18], in the context of D-dimensional Log gravity. For generic D, the solution is not asymptotically AdS, unless c₂ðuÞ vanishes. For D ¼ 3, this equation again reduces to the corresponding expres-sion given in [10].

V. CONCLUSIONS

We have found exact AdS-wave solutions in the generic quadratic gravity theory with a cosmological constant. The metrics we have found also solve the linearized field equations of the same theory. When we restrict the qua-dratic theory by choosing M2¼ 0, which boils down to eliminating one of the parameters of the quadratic theory, the solutions we found in this case also solve the critical gravity theory defined recently. Depending on the value of M2, asymptotic behavior of the solution changes dramati-cally. Energy and some other physical properties of our solutions, and their conformal field theory duals need to be investigated. It would also be interesting to take the

solutions presented here as background and study the spin-2 fluctuations.

Finally, with adjusted  and M2, the metrics we have found will also solve any higher (cubic or more) curvature gravity models and constitute an example of spacetimes studied in [19]. This can be seen as follows: the linearized version of a generic gravity theory constructed from the contractions of the Riemann tensor around AdS will be exactly of the form (4) which is solved by the AdS-wave metrics obtained above. Since by construction the exact field equations reduce to the linearized equations for these solutions, AdS-wave solves the full nonlinear theory at any order in the curvature.

ACKNOWLEDGMENTS

M. G. is partially supported by the Scientific and Technological Research Council of Turkey (TU¨ BI˙TAK) and Turkish Academy of Sciences (TU¨ BA). The work of I. G., T. C. S., B. T. is supported by the TU¨ BI˙TAK Grant No. 110T339, and METU Grant No. BAP-07-02-2010-00-02.

APPENDIX: SOLUTION FOR THE CRITICAL THEORY WITH

0

Let us consider the a ¼ b theory for the 
0 case. The corresponding fourth-order equation reduces to the qua-dratic nonhomogeneous equation;

ðz2_@2_{þ ð6
DÞz@} z
2ðD
3ÞÞV ¼ zðD
5Þ=2_½c 1ðuÞIðzÞ þ c2ðuÞKðzÞ  ½c₃ðuÞei ~: ~r_{þ c} 4ðuÞe
i ~:~r; (A1)

where  ¼ 1₂ðD
1Þ, but we shall restrict to the positive  case; and instead of the sines and cosines we choose the exponentials. This equation can be solved with the help of the Green’s function technique. First, we would like to take care of the ~r dependence using the Fourier transform (for the sake of simplicity, here we choose  to be continuous, but the discrete case follows similarly)

Vðu; z; ~rÞ ¼ 1 ð2ÞðD
3Þ=2

Z

dD
3p ~Vðu; z; ~pÞei ~p: ~r_{; (A2)}

then (A1) after defining ~Vðu; z; ~pÞ ¼ zðD
5Þ=2fðu; z; ~pÞ, re-duces to  z2 d 2 dz2þ z d dz
ð 2_{þ p}2_z2_Þ_{fðu; z; ~pÞ} ¼ ð2ÞðD
3Þ=2_½c 1ðuÞIðzÞ þ c2ðuÞKðzÞ

 ½c₃ðuÞ ð ~
~pÞ þ c₄ðuÞ ð ~ þ ~pÞ: (A3) From the solutions of the homogeneous part, we can con-struct the Green’s function (OG ¼
1) as

Gðz; z0; pÞ ¼1 z0 

IðzpÞKðz0pÞ 0 < z < z0;

I_ðz0pÞK_ðzpÞ z0< z < 1; (A4) where we have used the Wronskian WfIðpzÞKðpzÞg ¼

1

z. Therefore, the solution of (A3) becomes

fðu; z; ~pÞ ¼ ½d₁ðuÞIðzpÞ þ d2ðuÞKðzpÞ þ ð2ÞðD
3Þ=2½c3ðuÞ ð ~
~pÞ

þ c₄ðuÞ ð ~ þ ~pÞZ1

0 dz

0_{Gðz; z}0_{; pÞ½c}

1ðuÞIðz0Þ þ c2ðuÞKðz0Þ: (A5)

We can carry out the p integrals using Z dD
3pfðpÞei ~p: ~r _{¼ ð2Þ}ðD
3Þ=2_rð5
DÞ=2 Z1 0 dpfðpÞp ðD
3Þ=2_J ðD
5Þ=2ðprÞ; (A6) where Jn is the Bessel function of the first kind. Since the

Fourier transform of IðzpÞ diverges, we must choose

d₁ðuÞ ¼ 0. For D ¼ 4 and D ¼ 5, d₂ðuÞ must also be zero, since the integral involving KðzpÞ diverges.

Then, the integral involving KðzpÞ for D > 5 gives

z
₂~F₁ð₂
1;3₂
1; 
1;
r_z22Þ where the second factor

is the regularized hypergeometric function. As z !0, this expression diverges. Therefore, for all D, there is no con-tribution from the homogeneous part, and c₂ðuÞ should vanish in the nonhomogeneous part since that term di-verges in the ½0; z integral, yielding finally

Vðu; z; ~rÞ ¼ zðD
5Þ=2½c₁ðuÞei ~: ~rþ c₂ðuÞe
i ~:~rZ

1 0 dz 0_{Gðz; z}0_{; ÞI} ðz0Þ ¼ zðD
5Þ=2_½c 1ðuÞei ~: ~rþ c2ðuÞe
i ~:~r  K_ðzÞZz 0 1 z0Iðz 0_ÞI ðz0Þdz0þ IðzÞ Z1 z 1 z0Kðz 0_ÞI ðz0Þdz0  : (A7)

This solution is valid for a given ~. The general solution can be obtained by integrating this solution over the (D
3)-dimensional ~-space. Here, c₁ðuÞ and c₂ðuÞ also depend on ~. Using Mathematica, one can find the integrals in terms of the hypergeometric functionpFqða; b; zÞ and the digamma functionc

Zz 0 dz 01 z0Iðz 0_ÞI ðz0Þ ¼ ðzÞ2 22þ1_{½ð þ 1Þ}2 2F3  ;  þ1 2;  þ 1;  þ 1; 2 þ 1; 2z2  ; Z1 z 1 z0Kðz 0_ÞI ðz0Þdz0¼ 2z2 8ð2
_1Þ3F4  1; 1;3₂; 2; 2; 2
;  þ 2; 2_z2
ð
ÞðzÞ2 4þ1_{ð þ 1Þ}2F3  ;  þ1 2;  þ 1;  þ 1; 2 þ 1; 2z2 
1 2  lnz₂
cðÞ
1 2  : (A8)

Specifically, for D ¼4, that is  ¼3₂, around z ¼0, one has

Vðu; z; ~rÞ  ½c₁ðuÞei ~: ~r_{þ c}

2ðuÞe
i ~:~r3=2z½lnðzÞ
1:3963; (A9)

which gives an asymptotically AdS metric.

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