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arXiv:0808.2347v1 [gr-qc] 18 Aug 2008
Rotating Black Branes in Brans-Dicke-Born-Infeld Theory
S. H. Hendi∗
Physics Department, College of Sciences, Yasouj University, Yasouj 75914, Iran
Research Institute for Astrophysics and Astronomy of
Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
In this paper, we present a new class of charged rotating black brane solutions in the higher
dimensional Brans-Dicke-Born-Infeld theory and investigate their properties. Solving the
field equations directly is a non-trivial task because they include the second derivatives of the
scalar field. We remove this difficulty through a conformal transformation. Also, we find that
the suitable Lagrangian of Einstein-Born-Infeld-dilaton gravity is not the same as presented
in [12]. We show that the given solutions can present black brane, with inner and outer
event horizons, an extreme black brane or a naked singularity provided the parameters of the
solutions are chosen suitably. These black brane solutions are neither asymptotically flat nor
(anti)-de Sitter. Then we calculate finite Euclidean action, the conserved and thermodynamic
quantities through the use of counterterm method. Finally, we argue that these quantities
satisfy the first law of thermodynamics, and the entropy does not follow the area law.
I. INTRODUCTION
Lately there have been some renewed interest in the Brans-Dicke (BD) theory of gravitation [1].
On one hand, it is important for cosmological inflation models [2], in which the scalar field allows
the inflationary epoch to end via bubble nucleation without the need for fine-tuning cosmological
parameters (the ”graceful exit” problem). Also, it was found that in the low-energy regime, the
theory of fundamental strings can be reduced to an effective BD one [3].
Because scalar-tensor gravitation can agree with general relativity (GR) in the post-Newtonian
limit, it is important to study strong field examples in which the two theories may give different
predictions. These examples may not only provide further experimental and observational tests
that might distinguish between GR and scalar-tensor gravitation, but they may also illuminate the
structure of both theories.
The BD theory incorporates the Mach principle, which states that the phenomenon of inertia
must arise from accelerations with respect to the general mass distribution of the universe. This
theory is self-consistent, complete and for |ω| ≥ 500 in accord with solar system observations and
∗hendi@mail.yu.ac.ir
2
experiments [4], where ωis an adjustable parameter. In this theory, the matter couples minimally
to the metric and not directly to scalar field. Indeed, the scalar field does not exert any direct
influence on matter, its only role is that of participating in the field equations that determine
the geometry of the spacetime. More recently, many authors have investigated the gravitational
collapse and black hole formation in the BD theory [5, 6].
Till now, nonlinear charged rotating black hole solutions for an arbitrary value of ωhas not
been constructed. In this paper, we want to construct exact rotating black brane solutions in BD-
Born-Infeld (BDBI) theory for an arbitrary value of ωand investigate their properties. One can
find that solving the field equations directly is a non-trivial task, because they include the second
derivatives of the scalar field. We remove this difficulty through a conformal transformation. By
using this transformation, the BDBI action reduce to Einstein-Born-Infeld-dilaton (EBId) action,
and one can solve their field equations analytically.
The idea of the non-linear electrodynamics (BI) was first introduced in 1934 by Born and Infeld
in order to obtain a finite value for the self-energy of point-like charges [7]. Although it become less
popular with the introduction of QED, in recent years, the BI action has been occurring repeatedly
with the development of superstring theory, where the dynamics of D-branes is governed by the
BI action [8, 9]. Lately, black hole solutions in BI gravity with or without a cosmological constant
have been considered by many authors [10, 11]. Both of the Lagrangians of EBId gravity presented
here and in [12] show similar asymptotic behavior but only the one we considered is consistent
with BD theory.
The outline of this paper is organized as follows. In Sec. II, we give a brief review of the
field equations of BDBI theory in Jordan (or string) and Einstein frames. In Sec. III, we obtain
charge rotating solution in (n+ 1)-dimensions with krotation parameters and investigate their
(asymptotic) properties. Sec. IV is devote to calculation of the finite action, the conserved and
thermodynamic quantities of the (n+ 1)-dimensional black brane solutions with a complete set of
rotational parameters. We finish our paper with some concluding remarks.
II. FIELD EQUATION AND CONFORMAL TRANSFORMATION
In n+ 1 dimensions, the action of the BDBI theory with one scalar field Φ and a self-interacting
potential V(Φ) can be written as
IG=−1
16πZM
dn+1x√−gΦR − ω
Φ(∇Φ)2−V(Φ) + L(F),(1)
3
where Ris the Ricci scalar, ωis the coupling constant, Φ denotes the BD scalar field and V(Φ) is
a self-interacting potential for Φ and L(F) is the Lagrangian of BI
L(F) = 4β2 1−s1 + F2
2β2!,(2)
In Eq. (2), the constant βis called BI parameter with dimension of mass, F2=Fµν Fµν where
Fµν =∂µAν−∂νAµis electromagnetic tensor field and Aµis the vector potential. In the limit
β→ ∞,L(F) reduces to the standard Maxwell form L(F) = −F2, while L(F)→0 as β→0.
Varying the action (1) with respect to the metric, scalar and vector fields give the field equations
as
Gµν =ω
Φ2∇µΦ∇νΦ−1
2gµν (∇Φ)2−V(Φ)
2Φ gµν +1
Φ∇µ∇νΦ−gµν ∇2Φ
+2
Φ
1
4gµν L(F) + FµλFλ
ν
q1 + F2
2β2
,(3)
∇2Φ = 1
2 [(n−1) ω+n]h
(n+ 1)L(F) + 4F2
q1 + F2
2β2
+(n−1)ΦdV (Φ)
dΦ−(n+ 1) V(Φ)i,(4)
∂µ
√−gF µν
q1 + F2
2β2
= 0,(5)
where Gµν and ∇µare the Einstein tensor and covariant differentiation corresponding to the metric
gµν respectively. Solving the field equations (3)-(5) directly is a non-trivial task because the right
hand side of Eq. (3) includes the second derivatives of the scalar. We can remove this difficulty by
the conformal transformation
¯gµν = Φ2/(n−1)gµν ,
¯
Φ = n−3
4αln Φ,(6)
where
α= (n−3)/p4(n−1)ω+ 4n(7)
One may note that αgoes to zero as ωgoes to infinity and the BD theory reduces to Einstein
theory. By this transformation, the action (1) transforms to
¯
IG=−1
16πZM
dn+1x√−¯g¯
R − 4
n−1(∇¯
Φ)2−¯
V(¯
Φ) + ¯
L(¯
F , ¯
Φ),(8)
4
where ¯
Rand ¯
∇are the Ricci scalar and covariant differentiation corresponding to the metric ¯gµν ,
and ¯
V(¯
Φ) is
¯
V(¯
Φ) = Φ−(n+1)/(n−1)V(Φ)
The Born-Infeld Lagrangian coupled to a dilaton field, ¯
L(¯
F , ¯
Φ) corresponding to the metric ¯gµν is
given by
¯
L(¯
F , ¯
Φ) = 4β2e−4α(n+1)¯
Φ/[(n−1)(n−3)]
1−s1 + e16α¯
Φ/[(n−1)(n−3)] ¯
F2
2β2
,(9)
In the limit β→ ∞,¯
L(¯
F , ¯
Φ) reduces to the standard Maxwell field coupled to a dilaton field
¯
L(¯
F , ¯
Φ) = −e−4α¯
Φ/(n−1) ¯
F2.(10)
On the other hand, ¯
L(¯
F , ¯
Φ) →0 as β→0. It is convenient to set
¯
L(¯
F , ¯
Φ) = 4β2e−4α(n+1)¯
Φ/[(n−1)(n−3)] ¯
L(¯
Y),(11)
where
¯
L(¯
Y) = 1 −p1 + ¯
Y , (12)
¯
Y=e16α¯
Φ/[(n−1)(n−3)] ¯
F2
2β2.(13)
It is notable that this action is different from the action of EBId that has been presented in [12].
Due to the fact that the action of EBId in [12] is not consistent with conformal transformation,
one can find that Eq. (8) is the suitable action for EBId gravity.
Varying the action (8) with respect to ¯gµν ,¯
Φ and ¯
Fµν , we obtain equations of motion as
¯
Rµν =4
n−1¯
∇µ¯
Φ¯
∇ν¯
Φ + 1
4¯
V¯gµν −1
(n−1) ¯
L(¯
F , ¯
Φ)¯gµν
−2e−4α¯
Φ/(n−1)∂¯
Y¯
L(¯
Y)¯
Fµλ ¯
Fλ
ν λ −2¯
F2
(n−1) ¯gµν ,(14)
¯
∇2¯
Φ = n−1
8
∂¯
V
∂¯
Φ+α
2(n−3) (n+ 1)¯
L(¯
F , ¯
Φ) −8e−4α¯
Φ/(n−1)∂¯
Y¯
L(¯
Y)¯
F2,(15)
∂µh√−¯ge−4α¯
Φ/(n−1)∂¯
Y¯
L(¯
Y)¯
Fµν i= 0.(16)
Therefore, if (¯gµν ,¯
Fµν ,¯
Φ) is the solution of Eqs. (14)-(16) with potential ¯
V(¯
Φ), then
[gµν , Fµν ,Φ] = exp −8α¯
Φ
(n−1) (n−3)¯gµν ,¯
Fµν ,exp 4α¯
Φ
n−3 (17)
is the solution of Eqs. (3)-(5) with potential V(Φ).
5
III. CHARGED ROTATING SOLUTIONS IN n+ 1 DIMENSIONS WITH kROTATION
PARAMETERS
Here we construct the (n+ 1)-dimensional solutions of BD theory with n≥4 and the quadratic
potential
V(Φ) = 2ΛΦ2
Applying the conformal transformation (6), the potential ¯
V(¯
Φ) becomes
¯
V(¯
Φ) = 2Λ exp 4α¯
Φ
n−1,(18)
which is a Liouville-type potential. Thus, the problem of solving Eqs. (3)-(5) with quadratic
potential reduces to the problem of solving Eqs. (14)-(16) with Liouville-type potential.
The rotation group in n+ 1 dimensions is SO(n) and therefore the number of independent
rotation parameters for a localized object is equal to the number of Casimir operators, which is
[n/2] ≡k, where [x] is the integer part of x. The solutions of the field equations (14)-(16) with k
rotation parameter ai, and Liouville-type potential is [13]
d¯s2=−f(r) Ξdt −
k
X
i=1
aidϕi!2
+r2
l4R2(r)
k
X
i=1 aidt −Ξl2dϕi2
−r2
l2R2(r)
k
X
i<j
(aidϕj−ajdϕi)2+dr2
f(r)+r2
l2R2(r)dX2,
Ξ2= 1 +
k
X
i=1
a2
i
l2,
¯
Ftr =qΞβe4α¯
Φ/(n−1)
pq2e8α¯
Φ/(n−3) +β2r2n−2R2n−2,¯
Fϕir=−ai
Ξ¯
Ftr.(19)
where lis a constant, called length scale and dX2is the flat Euclidean metric on (n−k−1)-
dimensional submanifold with volume ωn−k−1. Here f(r), R(r) and ¯
Φ(r) are
f(r) = (1 + α2)2r2
(n−1) 2Λ r
c−2γ
(α2−n)+4(n−3)β2r
c2γ(n+1)/(n−3)
λ!−
m
r(n−1)(1−γ)−1−4(1 + α2)2q2r
c2γ(n−2)
λr2(n−2) (η),(20)
(η) = (n−3)√1 + η
(n−1)η−1
Υ2F11
2,(n−3)Υ
2(n−1) ,1 + (n−3)Υ
2(n−1) ,−η,
η=q2r
c2γ(n−1)(n−5)/(n−3)
β2r2(n−1) ,
6
Υ = α2+n−2
2α2+n−3
λ= (3n−1)α2+n(n−3),(21)
R(r) = exp( 2α¯
Φ
n−1) = r
c−γ,(22)
¯
Φ(r) = −(n−1)γ
2αln r
c,(23)
where cis an arbitrary constant and γ=α2/(α2+1). Using the conformal transformation (17), the
(n+ 1)-dimensional rotating solutions of BD theory with krotation parameters can be obtained
as
ds2=−U(r) Ξdt −
k
X
i=1
aidϕi!2
+r2
l4H2(r)
k
X
i=1 aidt −Ξl2dϕi2
−r2
l2H2(r)
k
X
i<j
(aidϕj−ajdϕi)2+dr2
V(r)+r2
l2H2(r)dX2,(24)
where U(r), V(r), H(r) and Φ(r) are
U(r) = r
c4γ/(n−3) f(r),(25)
V(r) = r
c−4γ/(n−3) f(r),(26)
H(r) = r
c−γ(n−5)/(n−3) ,(27)
Φ(r) = r
c−2γ(n−1)/(n−3) .(28)
The electromagnetic field becomes:
Ftr =qΞβc
r−4γ/(n−3)
pq2+β2r2(n−1)[1−γ(n−5)/(n−3)]c2γ(n−1)(n−5)/(n−3) , Fϕir=−ai
ΞFtr.(29)
It is worth to note that the scalar field Φ(r) and electromagnetic field Fµν become zero as rgoes
to infinity. These solutions reduce to the solutions presented in Ref. [6] as βgoes to infinity. In
the absence of a nontrivial dilaton (α=γ= 0 or ω→ ∞), the above solutions reduce to those
of Ref. [14] and in the limit β→ ∞ and ω→ ∞, these solutions reduce to the solutions of Refs.
[15, 16]. It is also notable to mention that these solutions are valid for all values of ω.
A. Properties of the solutions
In order to study the general structure of these solutions, we first look for the essential singu-
larities. One can show that the Kretschmann scalar Rµνλκ Rµνλκ diverges at r= 0, and therefore
7
there is a curvature singularity located at r= 0. Seeking possible black hole solutions, we turn to
look for the existence of horizons. Because of the presence of the hypergeometric function in the
equation f(r) = 0, the radius of the event horizon cannot be found explicitly. The roots of the
metric function f(r) are located at
−rγ(n−1)−n
+m
4(1 + α2)2+r+
c−2γΛ
2(n−1)(α2−n)+(n−3)β2r+
c2γ(n+1)/(n−3) (1 −√1 + η+)
(n−1)λ
+
r2[γ(n−2)−(n−1)]
+q22F1h1
2,(n−3)Υ
2(n−1) i,h1 + (n−3)Υ
2(n−1) i,−η+
λc2γ(n−2)Υ= 0 (30)
The angular velocities Ωiare [17]
Ωi=ai
Ξl2,(31)
and the temperature may be obtained through the use of definition of surface gravity, κ,
T+=1
β+
=κ
2π=1
2πr−1
2(∇µχν) (∇µχν),(32)
where χis the Killing vector given by
χ=∂t+
k
X
i=1
Ωi∂φi.(33)
One obtains
T+=−(1 + α2)r+
2πΞh(α2−n)rγ(n−1)
+m
2(1 + α2)2rn−8α2β2r+
c2γ(n+1)/(n−3)
λ1−p1 + η+
−2(α2−n)r+
c2γ(n−2) q2
λΥr2(n−1)
+
2F11
2,(n−3)Υ
2(n−1) ,1 + (n−3)Υ
2(n−1) ,−η+i
=−(1 + α2)r+
2πΞ(n−1) r+
c−2γΛ−2β2r+
c2γ(n+1)/(n−3) 1−p1 + η+(34)
which shows that the temperature of the solution is invariant under the conformal transformation
(6). This is due to the fact that the conformal parameter is regular at the horizon.
Asymptotic Behavior:
α2=n: The solution is ill-defined for α2=nwith a quadratic potential (Λ 6= 0).
α2> n: In this case, as rgoes to infinity the dominant term in Eq. (25) is the second term
(mterm), and therefore the spacetime has a cosmological horizon for positive values of the mass
parameter, despite the sign of the cosmological constant Λ.
α2< n: For α2< n, as rgoes to infinity the dominant term is the first term (Λ term), and
therefore there exist a cosmological horizon for Λ >0, while there is no cosmological horizons if
8
r_
r_
r_
r_
r_
r_
r_ r_
r
r+r+
r+
r+
r+r+
r_
++
r = infinity
r = infinity
r = 0
r = 0r = 0
r = 0
r = infinity
r = infinity
r = infinity
r = infinity
FIG. 1: Penrose diagram for negative Λ and α < 1.
Λ<0 . Indeed, in the latter case (α2< n and Λ <0) the spacetimes associated with the solution
(25)-(28) exhibit a variety of possible causal structures depending on the values of the metric
parameters α,m,qand Λ. One can obtain the causal structure by finding the roots of V(r) = 0.
Unfortunately, because of the nature of the exponents and hypergeometric function in (26), it is
not possible to find explicitly the location of horizons. But, we can obtain some information by
considering the temperature of the horizons. Here, we draw the Penrose diagram to show that
the casual structure is asymptotically well behaved. For reason of economy, we draw the Penrose
diagram only for the solution that presents a black brane with inner and outer horizons (negative Λ
and α < √n). The causal structure can be constructed following the general prescriptions indicated
in [18]. The Penrose diagram is shown in Figs. 1 and 2 for α < 1 and 1 ≤α < √nrespectively.
Also it is worth to write down the asymptotic behavior of the Ricci scalar. Indeed, the form of
the Ricci scalar for large values of ris:
R=−n2
(n−3)2l2
(2α2+n−3)[4α2+ (n+ 1)(n−3)]
n−α2c
r2(n−1)γ/(n−3) (35)
9
r_
r_
r_
r_
r_
r_
r_ r_
r
r+r+
r+
r+
r+r+
r_
++
r = 0
r = 0r = 0
r = 0
i0
i0
J+
J
J
J
JJ++
+
_
_
_
J
J_
FIG. 2: Penrose diagram for negative Λ and 1 ≤α < √n.
which does not approach a nonzero constant as in the case of asymptotically AdS spacetimes. It
is worth to mention that the Ricci scalar of the solution (24)-(27) goes to zero as r goes to infinity,
but with a slower rate than that of an asymptotically flat spacetimes in the absence of the scalar
field.
Equation (34) shows that the temperature is negative for the two cases of (i)α > √ndespite
the sign of Λ, and (ii ) positive Λ despite the value of α. As we argued above in these two cases
we encounter with cosmological horizons, and therefore the cosmological horizons have negative
temperature. Indeed, the metric of Eqs. (24)-(28) has two inner and outer horizons located at r−
and r+, provided the mass parameter mis greater than mext, an extreme black brane in the case
of m=mext, and a naked singularity if m < mext only for negative Λ and α < √nwhere
mext =4(1 + α2)2
rγ(n−1)−n
ext rext
c−2γΛ
2(n−1)(α2−n)+(n−3)β2rext
c2γ(n+1)/(n−3) (1 −√1 + ηext)
(n−1)λ
+
r2[γ(n−2)−(n−1)]
ext q22F1h1
2,(n−3)Υ
2(n−1) i,h1 + (n−3)Υ
2(n−1) i,−ηext
λc2γ(n−2)Υ
(36)
in Eq. (36), rext is the root of temperature relation (34) such that
1−Λ
2β2rext
c−8γ/(n−3)2
−1 = ηext,(37)
10
where
ηext =q2rext
c2γ(n−1)(n−5)/(n−3)
β2r2(n−1)
ext
.
Note that in the absence of scalar field (α=γ= 0) mext reduces to that obtained in [16].
Next, we calculate the electric charge of the solutions. According to the Gauss theorem, the
electric charge is the projections of the electromagnetic field tensor on special hypersurfaces. De-
noting the volume of the hypersurface boundary at constant tand rby Vn−1= (2π)kωn−k−1, the
electric charge per unit volume Vn−1can be found by calculating the flux of the electric field at
infinity, yielding
Q=Ξq
4πln−2(38)
Comparing the above charge with the charge of black brane solutions of Einstein-Born–Infeld-
dilaton gravity, one finds that charge is invariant under the conformal transformation (6). The
electric potential U, measured at infinity with respect to the horizon, is defined by [19]
U=Aµχµ|r→∞ −Aµχµ|r=r+,(39)
where χis the null generators of the event horizon (33). One can easily show that the vector
potential Aµcorresponding to electromagnetic tensor (29) can be written as
Aµ=qc(3−n)γ
ΓrΓ2F11
2,(n−3)Υ
2(n−1) ,1 + (n−3)Υ
2(n−1) ,−ηΞδt
µ−aiδi
µ(no sum on i),(40)
where Γ = (n−3)(1 −γ) + 1. Therefore the electric potential is
U=qc(3−n)γ
ΞΓr+Γ2F11
2,(n−3)Υ
2(n−1) ,1 + (n−3)Υ
2(n−1) ,−η+(41)
IV. ACTION AND CONSERVED QUANTITIES
The action (1) does not have a well-defined variational principle, we should add the boundary
action to it for ensuring well-defined Euler-Lagrange equations. The suitable boundary action is
Ib=−1
8πZ∂M
dnx√−γK Φ,(42)
where γand Kare the determinant of the induced metric and the trace of extrinsic curvature
of boundary. In general the action IG+Ib, is divergent when evaluated on the solutions, as is
the Hamiltonian and other associated conserved quantities. For asymptotically (A)dS solutions of
11
Einstein gravity, the way that one deals with these divergences is through the use of counterterm
method inspired by (A)dS/CFT correspondence [20]. Although, in the presence of a non-trivial
BD scalar field with potential V(Φ) = 2ΛΦ2, the spacetime may not behave as either dS (Λ >0)
or AdS (Λ <0). In fact, it has been shown that with the exception of a pure cosmological constant
potential, where α= 0, no AdS or dS static spherically symmetric solution exist for Liouville-type
potential [21]. But, as in the case of asymptotically AdS spacetimes, according to the domain-
wall/QFT (quantum field theory) correspondence [22], there may be a suitable counterterm for the
action which removes the divergences. Since our solutions have flat boundary [Rabcd(h) = 0], there
exists only one boundary counterterm
Ict =−1
8πZ∂M
dnx√−γ(n−1)
leff
,(43)
where leff is given by
l2
eff =(n−1)(α2−n)
2ΛΦ3.(44)
One may note that as αgoes to zero, the effective l2
eff of Eq. (44) reduces to l2=−n(n−1)/2Λ of
the (A)dS spacetimes. The total action, I, can be written as
I=IG+Ib+Ict.(45)
The Euclidean actions (45) per unit volume Vn−1can be obtained as
I=β+
c(n−1)γ
4πln−2h(4γ+n−3)q2r−Γ
+
λΓ(γ−1)c2γ(n−2) 2F11
2,(n−3)Υ
2(n−1) ,1 + (n−3)Υ
2(n−1) ,−η+
+(1 + α2)rn−γ(n−1)
+
2(n−1) (α2−1)
(α2−n)hr+
ci−2γΛ−
2 [λ+ (n−1)(n−3)] β2
λr+
c−2γ(n+1)/(n−3) h1−p1 + η+i!i(46)
It is easy to show that the mass Mand the angular momentum Jicalculated in Jordan (or
string) frame is the same as Einstein frame and they are remain unchanged under conformal
transformations, that is
Ji=c(n−1)γ
16πln−2n−α2
1 + α2Ξmai,
M=c(n−1)γ
16πln−2(n−α2)Ξ2+α2−1
1 + α2m
For ai= 0 (Ξ = 1), the angular momentum per unit length vanishes, and therefore aiis the ith
rotational parameter of the spacetime.
12
We calculate the entropy through the use of Gibbs-Duhem relation
S=1
T(M − ΓiCi)−I, (47)
where Iis the finite total action (46) evaluated on the classical solution, and Ciand Γiare the
conserved charges and their associate chemical potentials respectively. It is straightforward to show
that
S=Ξc(n−1)γ
4l(n−2) r(n−1)(1−γ)
+,(48)
for the entropy per unit volume Vn−1. It is worth to note that the area law is no longer valid in
Brans-Dicke theory [23]. Nevertheless, the entropy remains unchanged under conformal transfor-
mations.
Comparing the conserved and thermodynamic quantities calculated in this section with those
obtained in EBId gravity, one finds that they are invariant under the conformal transformation
(17). It is easy to show that these quantities calculated satisfy the first law of thermodynamics,
dM =T dS +
k
X
i=1
ΩidJi+UdQ (49)
V. CLOSING REMARKS
The main goal of this paper is solving the field equations of BD theory in the presence of
non-linear electromagnetic field for an arbitrary value of ω. As one can find, solving the field
equations directly is a non-trivial task because they include the second derivatives of the scalar
field. We could remove this difficulty through a conformal transformation. Indeed, after conformal
transformation, the BDBI action is reduced to EBId action. We found that the suitable Lagrangian
of EBId gravity is not the same as the one presented in [12], because it is not consistent with
conformal transformation. We also found analytical solutions of BDBI theory, using conformal
transformation (6) and investigated their properties. Then we found that these solutions which
exist only for α26=n, have a cosmological horizon for (i)α2> n despite(regardless of) the sign
of Λ, and (ii) positive values of Λ, despite(disregarding) the magnitude of α. For α2< n, the
solutions present black branes with inner and outer horizons if m > mext, an extreme black brane
if m=mext, and a naked singularity otherwise. Also we presented Penrose diagrams and showed
that the black brane solutions are neither asymptotically flat nor (anti)-de Sitter. We computed the
finite Euclidean action through the use of counterterm method and obtained the thermodynamic
13
and conserved quantities of the solutions. We found that the entropy does not follow the area law
in BDBI theory but one can show that it trace the area law in EBId gravity. One can find that
the conserved and thermodynamic quantities are invariant under the conformal transformation
and satisfy the first law of thermodynamics. The study of spherical symmetric solutions of BDBI
theory with non-zero curvature boundary remains to be carried out in the future.
Acknowledgments
I am indebted to S. Bahaadini for her encouragements. This work has been supported financially
by Research Institute for Astronomy and Astrophysics of Maragha.
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