Charged Rotating Black Holes on DGP Brane

0.16 MB·English

by  Daeho Lee

KIAS-P07073
Charged Rotating Black Holes on DGP Brane
Daeho Lee†,1, Ee Chang-Young†,2, and Myungseok Yoon‡,3
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0 †Department of Physics, Sejong University, Seoul 143-747, Korea
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‡Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea
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ABSTRACT

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- We consider charged rotating black holes localized on a three-brane in the DGP model.
p
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h Assuming a Z2-symmetry across the brane and with a stationary and axisymmetric metric
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ansatz on the brane, a particular solution is obtained in the Kerr-Schild form. This so-
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lution belongs to the accelerated branch of the DGP model and has the characteristic of
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the Kerr-Newman-de Sitter type solution in general relativity. Using a modified version of
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Boyer-Lindquist coordinates we examine the structures of the horizon and ergosphere.
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i
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PACS: 04.40.Nr, 04.50.-h, 04.70.-s
[email protected]
[email protected]
[email protected]
1 Introduction
Recent astronomical observations indicate that our universe is in the phase of accelerated
expansion [1]. There has been much recent interest in the idea that our universe may be
a brane embedded in some higher dimensional space. One of the models along this line
proposed by Dvali, Gabadadze and Porrati (DGP) [2] is known to contain a branch of
solutions exhibiting self accelerated expansion of the universe [3].
The brane world black holes in the Randall-Sundrum(RS) model [4] have been studied by
many authors. Firstly, Chamblin et al. [5] presented evidence that a non-rotating uncharged
black hole on the brane is described by a “black cigar” solution in five dimensions. Then,
Dadhich et al. [6] showed that the Reissner-N¨ordstrom metric is an exact solution of the
effective Einstein equations on the brane, and Shiromizu et al. [7] derived the effective
gravitational equations on the brane. A solution for charged brane world black holes in the
RSmodelwasobtainedin[8]andthechargedrotatingcasewasobtainedin[9]. Inparticular,
Aliev et al. [9] found exact solutions of charged rotating black holes in the Kerr-Schild form
[10] using a stationary and axisymmetric metric ansatz on the brane.
In the case of the DGP model, approximate Schwarzschild solutions had been obtained
in [11, 12, 13, 14]. An exact Schwarzschild solution on the brane was obtained in [15]. In the
DGP model, the sources are assumed to be localized on a brane by a certain mechanism not
related to gravity itself. In [16], it was discussed that if the sources were not localized, the
brane with the induced graviton kinetic term has effectively repulsive gravity and it would
push any source off the brane. As a result, ordinary black holes cannot be held on the brane.
However, authors of [15] commented that charged black holes could still be quasilocalized if
the corresponding gauge fields are localized. Recently, motivated by the above suggestion,
an exact solution of charged black holes on the brane in the DGP model was obtained in
[17]. However, up to now no solution of charged rotating black holes on the DGP brane is
obtained.
Here, we intend to improve this situation a bit. We try to obtain an exact solution of
charged rotating black holes on the brane in the DGP model by noting a particular set of
conditions that satisfies the constraint equation. We first obtain the solution in the Kerr-
2
Schild form [10]. Then, by using a modified Boyer-Lindquist coordinate transformation we
find the horizon and ergosphere. In doing this, we use a stationary and axisymmetric metric
ansatz on the brane and the solution exhibits the characteristics of self accelerated expansion
of the brane world universe.
This paper is organized asfollows. In section 2, we set the actionand equations of motion
of the DGP model following the approach of Ref. [18]. In section 3, we get a rotating black
hole solution on the brane in the absence of Maxwell field and examine the properties of the
solution. In section 4, we extend the result of section 3 and find a solution for the charged
rotating case. In section 5, we conclude with discussion. In this last section, we discuss a
possible bulk solution consistent with our on-brane solution.
2 Action and field equations
The DGP gravitational action in the presence of sources takes the form [2]
S = M3 d5x√ g (5)R+ d4x√ h M2R+L , (1)
∗ − − P matter
Z Z
(cid:0) (cid:1)
where R and (5)R are the 4D and 5D Ricci scalars, respectively and L is the Lagrangian
matter
of the matter fields trapped on the brane. Here, the (4+1) coordinates are xA = (xµ,y(=
x5)), µ = 0,1,2,3, and g is the determinant of the five-dimensional metric g , while h is
AB
the determinant of the four-dimensional metric h = g (xµ,y = 0). A cross-over scale is
µν µν
defined by r = m−1 = M2/2M3. There is a boundary(a brane) at y = 0 and Z symmetry
c c P ∗ 2
across the boundary is assumed. The field equations derived from the action (1) have the
form
1 h
(5)G =(5) R g (5)R = κ2 (X +T )δ(y), (2)
AB AB − 2 AB 5sg AB AB
where κ2 = M−2 and κ2 = M−3, while X = δµδνG /κ2 and T = δµδνT is the
4 P 5 ∗ AB − A B µν 4 AB A B µν
energy-momentum tensor in the braneworld.
Now, we consider the metric of the following form [18, 19],
ds2 = g dxAdxB = g (x,y)dxµdxν +2N dxµdy +(N2 +g NµNν)dy2. (3)
AB µν µ µν
3
The (µ5), (55) components of the field equations (2) are called as the momentum and
Hamiltonian constraint equations, respectively, and are given by [18, 19]
Kν K = 0, (4)
∇ν µ −∇µ
R K2 +K Kµν = 0, (5)
µν
−
where K is the extrinsic curvature tensor defined by
µν
1
K = (∂ g N N ), (6)
µν y µν µ ν ν µ
2N −∇ −∇
and is the covariant derivative operator associated with the metric g .
µ µν
∇
Integrating both sides of the field equation (2) along the y direction and taking the limit
of y = 0 on the both sides of the brane we arrive at the Israel’s junction condition [20] on
the Z symmetric brane in the relation [17]
2
G = κ2T +m (K h K). (7)
µν 4 µν c µν − µν
In this paper, we take the electro-magnetic field as the matter source on the brane. Using
(7) in the constraint (4) and (5) we find that the momentum constraint equation is satisfied
identically, while the Hamiltonian constraint equation is written as
1
R Rµν R2 +m2R+κ4T Tµν 2κ2R Tµν = 0, (8)
µν − 3 c 4 µν − 4 µν
where we used T = Tµ = 0.
µ
Finally, combining the Einstein Equations in the bulk(y = 0)
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1
(5)G =(5) R g (5)R = 0 (9)
AB AB AB
− 2
with (7) we arrive at the gravitational field equations on the brane [17]
κ4 1
G = E 4 (Tρ T h T Tρσ)
µν − µν − m2 µ ρν − 2 µν ρσ
c
1 2 1 1
(Rρ R RR + h R2 h R Rρσ)
−m2 µ ρν − 3 µν 4 µν − 2 µν ρσ
c
κ2 2
+ 4 (Rρ T +Tρ R RT h R Tρσ), (10)
m2 µ ρν µ ρν − 3 µν − µν ρσ
c
4
where E is the traceless “electric part” of the 5-dimensional Weyl tensor (5)C [7] and
µν ABCD
m−1 = κ2/2κ2. In what follows we shall set κ2 = 8π.
c 5 4 4
In general, the field equations on the brane are not closed and one needs to solve the
evolution equations into the bulk. However, by assuming a special ansatz for the induced
metric on the brane, one can make the system of equations on the brane closed.
3 Rotating black hole solution
Westartwithastationaryandaxisymmetric metricdescribing arotatingblackholelocalized
on a 3-brane in the DGP model. We write it as the Kerr-Schild form [10] in which the metric
is expressed in a linear approximation around the flat metric:
ds2 = (ds2)flat +f(kµdxµ)2, (11)
where f is an arbitrary scalar function and k is a null, geodesic vector field in both the flat
µ
and full metrics with
k kµ = 0,kνD k = 0. (12)
µ ν µ
Introducing the Kerr-Schild coordinates xµ = u,r,θ,ϕ , we write the metric as [9]
{ }
ds2 = h dxµdxν = [ (du+dr)2+dr2 +Σdθ2
µν
−
+(r2 +a2)sin2θdϕ2 +2asin2θdrdϕ]+H(r,θ)(du asin2θdϕ)2, (13)
−
where
Σ(r,θ) = r2 +a2cos2θ, (14)
and a is the angular momentum per unit mass of the black hole.
For the uncharged case we can set T = 0, and the Hamiltonian constraint equation (8)
µν
is reduced to
1
R Rµν R2 +m2R = 0. (15)
µν − 3 c
Note that the above equation is satisfied with the following two sets of conditions,
R = 0, R Rµν = 0, (16)
µν
5
and
R = 12m2, R Rµν = 36m4. (17)
c µν c
The first set (16) is satisfied with the metric function H(r,θ) = 2Mr/Σ in the metric (13),
which is the usual Kerr solution in general relativity. The second set (17) corresponds to a
non-flat(de-Sitter) case, in which the conditions (17) in terms of the metric function H(r,θ)
in (13) are given by the following:
∂2H 4r∂H 2H
12m2 = + + , (18)
c ∂r2 Σ ∂r Σ
4H ∂H ∂2H 4r2 ∂H 2 2r∂H ∂2H
36m4 = r +a2cos2θ + +
c Σ2 ∂r ∂r2 Σ2 ∂r Σ ∂r ∂r2
(cid:18) (cid:19) (cid:18) (cid:19)
1 ∂2H 2 2
+ + (r4 2a2r2cos2θ+5a4cos4θ)H2. (19)
2 ∂r2 Σ4 −
(cid:18) (cid:19)
The metric function H satisfying the above two equations is given by
2Mr +m2(r4 +6r2a2cos2θ 3a4cos4θ)
H = c − , (20)
Σ
where the parameter M is an arbitrary constant of integration. One can easily check that
the metric (13) with (20) satisfies the equation (8) with T = 0. In the limit a 0, the
µν
→
metric (13) with (20) is reduced to the Schwarzschild-de Sitter black hole solution with the
cosmological constant Λ = 3m2 in general relativity. This corresponds to the solution for
c
the U(r) = 2 case in Ref. [15], and belongs to the accelerated branch of Kerr-de Sitter
−
type solution [21, 22].
In order to check the physical properties of the metric given by (13) with (20) we want
to transform the Kerr-Schild form to the Boyer-Lindquist coordinates. However, the equa-
tions (18) and (19) are not preserved under the transformation for the usual Boyer-Lindquist
coordinates. Thus in order to preserve the equations (18) and (19) under coordinate trans-
formation, we use the following modified transformation of Boyer-Lindquist type:
r2 +a2 a
du = dt dr Xdθ, dϕ = dφ dr Ydθ, (21)
− ∆ − − ∆ −
where ∆ = r2 +a2 H(r,θ)Σ(r,θ), X and Y are functions of r and θ only and satisfy the
−
relation
∂ ∂ r2 +a2 ∂ ∂ a
X(r,θ) = , Y(r,θ) = . (22)
∂r ∂θ ∆ ∂r ∂θ ∆
(cid:18) (cid:19)
(cid:16) (cid:17)
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The newly added terms X and Y can be analytically integrated from the transformation
(21) for the function H given in (20). In fact, the modified transformation (21) that satisfy
(22) leaves the Hamiltonian constraint (15) invariant.
Under the transformation (21), the metric (13) takes the form
Σ
ds2 = (1 H)dt2 + dr2 +2[X H(X Yasin2θ)]dtdθ
− − ∆ − −
+[Σ (1 H)X2 2HXYasin2θ+Y2(r2 +a2 +Ha2sin2θ)]dθ2
− − −
2[(r2 +a2)Y aH(X Yasin2θ)]sin2θdθdφ+(r2 +a2 +Ha2sin2θ)sin2θdφ2
− − −
2Hasin2θdtdφ. (23)
−
Using “MATHEMATICA” we check that the equations (18) and (19) remain unchanged
with the metric (23) for any metric function H(r,θ). Hence, we can use (20) as a solution
for the metric in (23).
For r r , X and Y with (20) can be approximately written as
c
≪
X 6m2a2sin2θ r (r12 +a2)(r12 −a2cos2θ) + 2T1ln(rr1 −1) +(r r ) +η (θ),
≈ − c − (r r )(r r )2 (r r )3 1 ↔ 2 1
(cid:20) − 1 1 − 2 1 − 2 (cid:21)
Y 6m2a3sin2θ (r12 −a2cos2θ) + 2T2ln(rr1 −1) +(r r ) +η (θ), (24)
≈ c (r r )(r r )2 (r r )3 1 ↔ 2 2
(cid:20) − 1 1 − 2 1 − 2 (cid:21)
where T = r4 2r3r r r a2sin2θ + a4cos2θ, T = r r a2cos2θ and r , r are two
1 1 − 1 2 − 1 2 2 1 2 − 1 2
roots of the equation r2 2Mr+a2 = 0. When the crossover scale r is infinite (or m = 0)
c c
−
and both η (θ) and η (θ) are set to be zero, then the equations (18) and (19) are preserved
1 2
with the metric function H = 2Mr. This corresponds to the exact Kerr solution in general
Σ
relativity once we identify the pararmeter M as the mass of the black hole.
The governing equation for horizon radius is given by
∆ = r2 +a2 2Mr m2(r4 +6r2a2cos2θ 3a4cos4θ) = 0. (25)
− − c −
The metric (23) with (20) has three horizons located at r and r , provided the total mass
± CH
M lies in the range M M M where M and M are given by
1e θ=π/2 2e θ=0,π 1e 2e
| ≤ ≤ |
1 1
M = α A3/2, M = α+A3/2 (26)
1e 2e
3√6m − 3√6m
c c
p p
with
α = 1+36m2a2 18m2a2cos2θ(1+12m2a2)+216m4a4cos4θ(1 4m2a2cos2θ) (27)
c − c c c − c
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and
A = 1 12m2a2(1+cos2θ). (28)
− c
Here r , which is smaller than the crossover scale r , is a cosmological horizon, r and r
CH c + −
are outer and inner horizon, respectively.
The horizons can be expressed explicitly as follows:
1 1
1/2 −1/2 1/2 −1/2
r = D D 4Mm D , r = D + D +4Mm D ,
± 2m − ± + − c − CH 2m − − + c −
c (cid:18) q (cid:19) c (cid:18) q (cid:19)
(29)
where
1/3 −1/3
1 2 2
D = C C +A + (30)
±
± 3 " B +√B2 4A3 B +√B2 4A3 #
(cid:18) − (cid:19) (cid:18) − (cid:19)
with C = 1 6m2a2cos2θ and B = 2C[C2 +36m2a2(1+3m2a2cos4θ)] 108m2M2. Note
− c c c − c
that the horizons r and r always have real positive values if the total mass lies between
± CH
the masses M and M .
1e θ=π/2 2e θ=0,π
| |
For r r , the outer and inner horizons can be approximated as
c
≪
M M2 a2(1 3m2a2cos2θ)
r ± − − c . (31)
±
≈ 1 6m2a2cos2θ
p − c
In the limit of r , we get M a, M , r M √M2 a2, and r
c 1e 2e ± CH
→ ∞ → → ∞ → ± − → ∞
independent of the angle θ as one can expect from (25).
For the ergosphere we calculate the condition
g = r2 +a2cos2θ 2Mr m2(r4 +6r2a2cos2θ 3a4cos4θ) = 0, (32)
tt − − c −
which has three boundaries of the ergosphere known as the “static limit” surfaces located at
r± and rCH.
E E
Setting r± and rCH as
E E
1 1
r± = D˜1/2 D˜ 4Mm D˜−1/2 , rCH = D˜1/2 + D˜ +4Mm D˜−1/2 ,
E 2m − ± + − c − E 2m − − + c −
c (cid:18) q (cid:19) c (cid:18) q (cid:19)
(33)
where
1/3 −1/3
1 2 2
D˜ = C C +A˜ + (34)
±
± 3  B˜ + B˜2 4A˜3! B˜ + B˜2 4A˜3! 
− −
 p p 
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with A˜ = 1 24m2a2cos2θ and B˜ = 2C[C2 +36m2a2cos2θ(1+3m2a2cos2θ)] 108m2M2,
− c c c − c
we obtain
r− r < r r+ < rCH r , (35)
E ≤ − + ≤ E E ≤ CH
with the equalities holding at θ = 0,π.
Comparing this with the event horizons (29), we see that the ergosphere lies in the region
r < r < r+ and rCH < r < r , which coincide with the horizon at θ = 0,π.
+ E E CH
4 Charged rotating black hole solution
In this section, we consider the case in which the brane contains a Maxwell field with an
electric charge. Outside the black hole, the Maxwell field can be described by a source-free
Maxwell equations. Thus, we have to solve simultaneously the constraint equation (8) and
the Maxwell equations:
gµνD F = 0, (36)
µ νσ
D F = 0, (37)
[µ νσ]
whereD isthecovariant derivative operatorassociated withthebranemetrich . However,
µ µν
we only need to solve Eqs. (8) and (36), since Eq. (37) is satisfied identically.
Hinted from the characteristic of the Kerr-Newman solution in general relativity which
yieldsR Rµν = 4Q4/Σ4,R Tµν = Q4/2πΣ4,T Tµν = Q4/16π2Σ4,wenotethattheHamil-
µν µν µν
tonian constraint (8) on the brane is satisfied with the following two set of conditions:
4Q4 Q4 Q4
R = 0, R Rµν = , R Tµν = , T Tµν = (38)
µν Σ4 µν 2πΣ4 µν 16π2Σ4
and
4Q4 Q4 Q4
R = 12m2, R Rµν = 36m4 + ,R Tµν = , T Tµν = . (39)
c µν c Σ4 µν 2πΣ4 µν 16π2Σ4
The first set (38) is satisfied with the conventional Kerr-Newman solution which is given by
the following potential one-form A and the metric function H:
µ
Qr
A dxµ = (du asin2θdϕ), (40)
µ
− Σ −
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2Mr Q2
H = − , (41)
Σ
where theparameterQistheelectric charge oftheblack hole. The second set (39)is satisfied
with the following solution:
Qr
A dxµ = (du asin2θdϕ), (42)
µ
− Σ −
2Mr Q2 +m2(r4 +6r2a2cos2θ 3a4cos4θ)
H = − c − . (43)
Σ
In the non-rotating limit, a 0, the second solution, (42) and (43), reduces to the conven-
→
tional charged de Sitter solution with the cosmological constant Λ = 3m2 [25]. This case
c
corresponds to the U(r) = 2 case of Ref. [17], in which it was shown to belong to the
−
accelerated branch.
In order to check the physical properties of the solution, we again make a transforma-
tion of the Boyer-Lindquist type (21). Since the Maxwell equation (36) should transform
covariantly under (21), the potential one-form (42) should also transform covariantly:
Qr Qr
A dxµ = dt asin2θdφ (X Yasin2θ)dθ + dr. (44)
µ
− Σ − − − ∆
(cid:2) (cid:3)
The nonvanishing components of the electromagnetic field tensor F are given by
µν
Q(r2 a2cos2θ)(X Yasin2θ) Q(r2 a2cos2θ)
F = − − , F = − ,
rθ − Σ2 rt Σ2
Qrasin2θ Qa(r2 a2cos2θ)sin2θ Qar(r2 +a2)sin2θ
F = , F = − , F = . (45)
tθ Σ2 φr Σ2 θφ Σ2
Since the Hamiltonian constraint (8) is invariant under (21) with the transformed potential
one-form (44), we only need to check the Maxwell equation (36). Indeed, the above potential
one-form (44) satisfies the Maxwell equation (36) with the metric (23) and (43).
Now, we would like to examine the gravitational effect on the brane due to the extra
dimension. To do that we will calculate the projected Weyl tensor E in (10) using our
µν
potential one-form (44) and the metric (23) with (43). In the charged rotating case the
tensor E is quite complicated to tell anything definite. For instance, Er component is
µν r
given by
m2
Er = c [ 6a2( r4(1 3cos2θ) 2r2a2cos2θsin2θ +a4cos4θ(21 19cos2θ) )
r −Σ2∆ − − −
Q2
+ Q2(r4 +6r2a2cos2θ 3a4cos4θ)]+ (r2 2Mr +Q2). (46)
− Σ2∆ −
10