Page 1
arXiv:0801.0164v1 [hep-th] 30 Dec 2007
OCU-PHYS 286
AP-GR 52
Charged Rotating Kaluza-Klein Black Holes in Five Dimensions
Toshiharu Nakagawa, Hideki Ishihara, Ken Matsuno, and Shinya Tomizawa
Department of Mathematics and Physics,
Graduate School of Science, Osaka City University,
3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan
(Dated: February 2, 2008)
Abstract
We construct a new charged rotating Kaluza-Klein black hole solution in the five-dimensional
Einstein-Maxwell theory with a Chern-Simon term. The features of the solutions are also investi-
gated. The spacetime is asymptotically locally flat, i.e., it asymptotes to a twisted S1bundle over
the four-dimensional Minkowski spacetime. The solution describe a non-BPS black hole rotating
in the direction of the extra dimension. The solutions have the limits to the supersymmetric black
hole solutions, a new extreme non-BPS black hole solutions and a new rotating non-BPS black
hole solution with a constant twisted S1fiber.
PACS numbers: 04.50.+h 04.70.Bw
1
Page 2
I.INTRODUCTION
In the context of string theory, the five-dimensional Einstein-Maxwell theory with a
Chern-Simon term gathers much attention since it is the bosonic sector of the minimal
supergravity. Supersymmetric (BPS) black hole solutions to the five-dimensional Einstein-
Maxwell equations with a Chern-Simon term have been found by various authors. Based
on the classification of the five-dimensional supersymmetric solutions by Gauntlett.et.al. [1],
they have been constructed on hyper-K¨ ahler base spaces, especially, the Gibbons-Hawking
base space. The first asymptotically flat supersymmetric black hole solution, BMPV
(Breckenridge-Myers-Peet-Vafa) solution, was constructed on the four-dimensional Euclid
space [2]. A supersymmetric black hole solution with a compactified extra dimension on the
Euclidean self-dual Taub-NUT base space was constructed by Gaiotto.et.al [3]. It was ex-
tended to multi-black hole solution with the same asymptotic structure [4]. One of the most
interesting properties is that the possible spatial topology of the horizon of each black hole is
the lens space S3/Zniin addition to S3, where niare the natural numbers . Similarly, black
hole solution on the Eguchi-Hanson space [5], which admits horizon topologies of various
lens spaces L(2n;1) = S3/Z2n(n:natural number), was also constructed. Since the latter
black hole spacetimes are asymptotically locally Euclidean, we cannot locally distinguish
these asymptotic structure.
As solutions in five-dimensional Einstein-Maxwell theory with a positive cosmological
constant, black hole solutions on the Euclid base space [6], the Taub-NUT base space [7]
and the Eguchi-Hanson base space [8] were also constructed. In particular, two-black hole
solution on the Eguchi-Hanson space describes a non-trivial coalescence of black holes. In
refes. [8, 9], the authors compared the two-black hole solution on the Eguchi-Hanson space
with the two-black holes solution on the Euclid space [6], and discussed how the coalescence
of five-dimensional black holes depends on the asymptotic structure of spacetime. Two black
holes with the topology of S3coalesce into a single black hole with the topology of the lens
space L(2;1) = S3/Z2in the case of Eguchi-Hanson space, while two black holes with the
topology of S3coalesce into a single black hole with the topology of S3in the Euclid case.
When the action has the Chern-Simon term, black holes can rotate [10, 11].
The most remarkable feature of the five-dimensional black hole solutions is that they
admit non-spherical topology. Emparan and Reall found the first black ring solution to
2
Page 3
the five-dimensional vacuum Einstein equations [12], which describes a stationary, rotating
black hole with the horizon topology S1× S2. Some supersymmetric black ring solutions
have been also found, based on the construction of the solutions by Gauntlett.et.al. [1].
Elvang et.al. found the first supersymmetric black ring solution with asymptotic flatness
on the four-dimensional Euclidean base space, which is specified by three parameters, mass
and two independent angular momentum components [13]. Gauntlett and Gutowski also
constructed a multi-black ring solution on the same base space [14, 15]. The BPS black rings
with three arbitrary charges and three dipole charges on the flat space were also constructed
in [16, 17]. The BPS black ring solutions on the Taub-NUT base space [16, 18, 19, 20] and
the Eguchi-Hanson space [21]were constructed. The black ring solution on Eguchi-Hanson
space has the same two angular momentum components and the asymptotic structure on
time slices is asymptotically locally Euclidean. The S1-direction of the black ring is along
the equator on a S2-bolt on the Eguchi-Hanson space. It has the limit to the BMPV black
hole with the topology of the lens space L(2;1) = S3/Z2.
In addition to the BPS solutions, the non-BPS black hole solutions have also been studied
by several authors. Cvetic et.al. [22] found a non-extremal, charged and rotating black hole
solution with asymptotic flatness. [30] In the specified limits, the solution reduces to the
known solutions: the same angular momenta case of the Myers-Perry black hole solution [23],
Exact solutions of non-BPS Kaluza-Klein black hole solutions are found in neutral case [24,
25] and charged case [26] . These solutions have a non-trivial asymptotic structure, i.e.,
they asymptotically approach a twisted S1bundle over the four-dimensional Minkowski
spacetime. The horizons are deformed due to this non-trivial asymptotic structure and have
a shape of a squashed S3, where S3is regarded as a twisted bundle over a S2base space.
The ratio of the radius S2to that of S1is always larger than one.
Wang proposed that a kind of Kaluza-Klein black hole solutions can be generated by
the ‘squashing transformation’ from black holes with asymptotic flatness [27]. In fact, he
regenerated the five-dimensional Kaluza-Klein black hole solution found by Dobiasch and
Maison [24, 25] from the five-dimensional Myers-Perry black hole solution with two equal
angular momenta.[31]
In this article, using the squashing transformation, we construct a new non-BPS rotating
charged Kaluza-Klein black hole solutions in the five-dimensional Einstein-Maxwell theory
with a Chern-Simon term, which is the generalization of the Kaluza-Klein black hole solution
3
Page 4
in Ref. [24, 25, 26]. Applying the squashing transformation to the Cvetic et.al.’s charged
rotating black hole solution [22] in vanishing cosmological constant case, we obtain the
new Kaluza-Klein black hole solution. We also investigate the features of the solution.
The spacetime is asymptotically locally flat, i.e., it asymptotically approaches a twisted S1
bundle over the four-dimensional Minkowski spacetime. The solution describes a non-BPS
black hole boosted in the direction of the extra dimension. It has various limits, f.g., to the
supersymmetric BMPV black hole solution, to the supersymmetric Kaluza-Klein black hole
solution and to the extreme non-BPS black hole solutions.
The rest of this article is organized as follows. In Sec.II, we present a new Kaluza-Klein
black hole solution in the five-dimensional Einstein-Maxwell theory with a Chern-Simon
term. In Sec.III, we investigate the basic features of the solution. In Sec.IV, we study the
limit of our solution to various known and unknown black hole solutions. In Sec.V, we
summarize the results in this article.
II.SOLUTION
In the metric of squashing Kaluza-Klein black hole in ref. [26], a function of the radial
coordinate k(r), which describes the squashing of the horizons, appeares. Wang pointed
out that the function k(r) would give a transformation from asymptotically flat solutions to
Kaluza-Klein type solutions. He call this squashing transformation.
Applying the squashing transformation to the non-BPS charged rotating black hole so-
lution found by Cvetic et.al. [22], we construct a new charged, rotating Kaluza-Klein black
hole solution to the five-dimensional Einstein-Maxwell theory with a Chern-Simon term.
The metric and the gauge potential of the solution are given by
ds2= −w(r)
h(r)dt2+ k(r)2dr2
w(r)+r2
4
?
k(r)(σ2
1+ σ2
2) + h(r)(f(r)dt + σ3)2
?
,(1)
and
A =
√3q
2r2
?
dt −a
2σ3
?
,(2)
4
Page 5
respectively, where the metric functions w(r),h(r),f(r) and k(r) are defined as
w(r) =(r2+ q)2− 2(m + q)(r2− a2)
r4
, (3)
h(r) = 1 −a2q2
r6+2a2(m + q)
2a
r2h(r)
k(r) =(r2
r4
,(4)
f(r) = −
?2m + q
r2
−q2
r4
?
∞− a2)
,(5)
∞+ q)2− 2(m + q)(r2
(r2
∞− r2)2
,(6)
and the left-invariant 1-forms on S3are given by
σ1= cosψdθ + sinψ sinθdφ, (7)
σ2= −sinψdθ + cosψ sinθdφ,
σ3= dψ + cosθdφ.
(8)
(9)
The coordinates r,θ,φ and ψ run the ranges of 0 < r < r∞, 0 ≤ θ < π, 0 ≤ φ < 2π,
0 ≤ ψ < 4π, respectively. In the case of k(r) = 1, i.e., r∞→ ∞, the metric coincides with
that of the Cvetic et.al.’s solution without a cosmological constant. In this article we assume
that the parameters a,m,q, and r∞appearing in the solutions satisfy the inequalities
m > 0,(10)
q2+ 2(m + q)a2> 0,(11)
(r2
∞+ q)2− 2(m + q)(r2
(m + q)(m − q − 2a2) > 0,
m + q > 0,.
∞− a2) > 0,(12)
(13)
(14)
As will be explained later, the inequalities (10)-(13) are the conditions for the existence of
two horizons, and the condition (14) is the requirement for the absence of closed timelike
curves outside the outer horizon. FIG.1 shows the region of the parameters.
III.FEATURES OF THE SOLUTIONS
A.Asymptotic form
In the coordinate system (t,r,θ,φ,ψ), the metric diverges at r = r∞but we see that this
is an apparent singularity and corresponds to the spatial infinity. To confirm this, introduce
5
Page 6
m
q
mq
= −
mqa
=+2
2
m
qr
−
ra
q
=
+
2
−
∞
∞
()
()
22
2
2
(,)
−a a
22
(,)
−∞
r
∞
r
22
(,)
rar
∞∞
−
222
2
FIG. 1: Region of the parameters in the (q,m)-plane.
a new coordinate defined by
ρ = ρ0
r2
r2
∞− r2,(15)
where the constant ρ0is given by
ρ2
0=(r2
∞+ q)2− 2(m + q)(r2
∞− a2)
4r2
∞
.(16)
In terms of the coordinate ρ, the metric is written as follows
ds2= −V (ρ)dt2+ 2H(ρ)dtσ3+ U(ρ)dρ2+ ρ(ρ + ρ0)(σ2
1+ σ2
2) + W(ρ)σ2
3,(17)
where the functions V (ρ), H(ρ), U(ρ) and W(ρ) are given by
V (ρ) =((r2
∞+ q)ρ + ρ0q)2− 2(m + q)r2
∞ρ(ρ + ρ0)
r4
∞ρ2
,(18)
H(ρ) =a(ρ + ρ0)((q2− (2m + q)r2
∞))ρ + q2ρ0)
2r4
∞ρ2
4r2
,(19)
U(ρ) =
∞ρ2
0ρ(ρ + ρ0)
(q2+ 2a2(m + q))(ρ + ρ0)2− 2mr2
W(ρ) =r6
∞ρ(ρ + ρ0) + r4
∞)ρ + q2ρ0)
∞ρ2,(20)
∞ρ3− a2(ρ + ρ0)2((q2− 2(m + q)r2
4r4
∞ρ2(ρ + ρ0)
.(21)
This new radial coordinate ρ runs from 0 into ∞. For ρ → ∞, which corresponds to the
6
Page 7
limit of r → r∞, the metric behaves as
ds2≃ −(r2
∞+ q)2− 2(m + q)r2
r4
∞
∞
dt2+a(q2− (2m + q)r2
∞)
r4
∞
dtσ3
(22)
+dρ2+ ρ2(σ2
1+ σ2
2) +r6
∞− a2(q2− 2(m + q)r2
4r4
∞)
∞
σ2
3.(23)
Next in order to transform the asymptotic frame into the rest frame, we define the coordi-
nates (˜t,˜ψ) given by
˜ψ = ψ +
2a(q2− (2m + q)r2
∞− a2(q2− 2(m + q)r2
4r4
∞)
r6
∞)t,(24)
˜t =
?
∞ρ2
0
r6
∞− a2(q2− 2(m + q)r2
∞)t.(25)
Then the metric takes the following asymptotic form
ds2≃ −d˜t2+ dρ2+ ρ2(σ2
1+ σ2
2) + L2˜ σ2
3,(26)
where ˜ σ3= d˜ψ + cosθdφ and the size of extra dimension L is given by
L2=r6
∞− a2(q2− 2(m + q)r2
4r4
∞)
∞
.(27)
It should be noted that the coefficient of ˜ σ2
3approaches a constant. Hence the spacetime
is asymptotically locally flat, i.e., the asymptotic form of the metric is a twisted S1bundle
over four-dimensional Minkowski spacetime.
The Komar mass at the infinity, the angular momenta associated with the Killing vector
fields ∂φand ∂ψat the infinity, and the electric charge can be obtained as
MKomar= π2r6
∞(mr2
∞− q2) − 2a4(m + q)q2− a2(q4− 4mq2r2
2r2
∞+ (4m2+ 4mq + 3q2)r4
∞)
∞(r6
∞− a2(q2− 2(m + q)r2
∞))ρ0
L,
Jφ= 0,
Jψ= −πa(a2q3+ 3q2r4
4r4
∞
√3
2πq.
(28)
∞− 2(2m + q)r6
∞− a2(q2− 2(m + q)r2
∞)
∞)L,
?r6
Q = −
(29)
B.Near-Horizon geometry and regularity
As is seen later, the solution within the region of the parameters in FIG.1 has two horizons,
an event horizon at r = r+> 0 and an inner horizon at r = r−> 0, which are determined
7
Page 8
by the equation w(r) = 0, where r±are given by
r2
±= m ±
?
(m + q)(m − q − 2a2).(30)
In the coordinate system (t,r,θ,φ,ψ), the metric diverges at r = r+and r = r−but these
are apparent. In order to remove this apparent divergence, we introduce the coordinates
(t′,ψ′) such that
dt = dt′+
?h(r)k(r)
w(r)
?h(r)f(r)k(r)
w(r)
dr,(31)
dψ = dψ′−
dr.(32)
Then the metric takes the form of
ds2= −w(r)
h(r)dt′2−2k(r)
?h(r)dt′dr +r2
4
?
k(r)(σ2
1+ σ2
2) + h(r)(f(r)dt′+ σ′
3)2?
.(33)
This metric well behaves at r = r±, i.e., it is smooth at this place.
Moreover, define the coordinates (v,ψ′′) given by
v = t,(34)
ψ′′= ψ′+ f(r+)t, (35)
and then the metric can be rewritten as follows
ds2= −
?w(r)
+r2
2h(r)(f(r) − f(r+))σ′′
h(r)+r2
4h(r)(f(r) − f(r+))2
?
dv2−
2k(r)
?h(r)dvdr
1+ σ2
3dt +r2
4
?
k(r)(σ2
2) + h(r)σ
′′2
3
?
.(36)
The Killing vector field V = ∂vbecomes null at r = r+and furthermore, V is hypersurface
orthogonal from Vµ∝ dr there.
These mean that the hypersurfaces r = r+is a Killing horizon. Similarly, r = r−is also
a Killing horizon. We should also note that in the coordinate system (v,φ,ψ′′,r,θ), each
component of the metric form (36) is analytic on and outside the black hole horizon. Hence
the spacetime has no curvature singularity on and outside the black hole horizon.
The surface gravity κ of the black hole is obtained as
κ =
r2
+− r2
∞− r2
−
r∞(r2
−)
?
(r2
(r2
∞− r2
∞− r2
+)
−)
?
r6
r6
∞− a2(q2− 2(m + q)r2
+− a2(q2− 2(m + q)r2
∞)
+).(37)
8
Page 9
Next, we investigate the shape of the horizon, especially, the aspect ratio of S2base space to
the S1fiber, which characterize the squashing of S3and is denoted by k(r+)/h(r+). In the
case of k(r+)/h(r+) > 1, the event horizon is called oblate, where the radius of S2larger than
that of S1. In the case of k(r+)/h(r+) < 1, the event horizon is called prolate, where the
radius of S2smaller than that of S1. FIG.2 shows the oblate region and the prolate region
in the (m,q)-plane. The shaded region and unshaded region are the oblate region and the
prolate region, respectively. At the boundary of two regions, the ratio is k(r+)/h(r+) = 1,
where the horizon becomes a round S3. Thus unlike the static solution [26] obtained by two
of the present authors, the horizon admits a prolate shape in addition to a round S3.
(,)
−a a
22
oblate
prolate
m
q
(,)
−∞
r
∞
r
22
(,)
rar
∞∞
−
222
2
(()/,/)213323
22
aa
−
@
FIG. 2: The aspect ratio of the outer horizon.
C.Parameters region and absence of CTCs
We consider the condition that the spacetime has non-degenerate horizons and no CTCs
outside the horizons.
FIG.1 shows the region of the parameters in a (q,m)-plane. The horizons appear at r > 0
satisfying w(r) = 0, i.e., the quadratic equation with respect to r2
(r2+ q)2− 2(m + q)(r2− a2) = 0.(38)
9
Page 10
The necessary and sufficient condition that this quadratic equation has two different roots
within the range of 0 < r2< r2
∞is
m > 0,(39)
q2+ 2(m + q)a2> 0,(40)
(r2
∞+ q)2− 2(m + q)(r2
(m + q)(m − q − 2a2) > 0.
∞− a2) > 0,(41)
(42)
As shown in FIG.3, the region satisfying the inequalities (39)-(42) consist of two regions:
Σ = {(q,m)| m − q > 2a2, m + q > 0, m < −q + (q + r2
{(q,m)| m−q < 2a2, m+q < 0, m > −(q2+2a2q)/(2a2)}. However, as will be seen below,
one of the regions Σ′is excluded from the requirement for the absence of CTCs outside the
∞)2/(2(r2
∞− a2))} and Σ′=
black hole horizon.
The positivity of the components gφφand gψψoutside the horizons assures the absence
of CTCs there. Since gφφis always positive definite outside the horizons, it is sufficient to
consider the parameter region such that
gψψ= h(r) =r6+ 2a2(m + q)r2− a2q2
r6
> 0(43)
outside the horizons. It is noted that the function h(r) is a monotony increasing function
of r. Hence, as a result, the condition for the absence of CTC outside the outer horizons
is h(r+) > 0. The curve in FIG.3 which enter only the region Σ′and has endpoints at the
points (0,0) and (−a2,a2) denotes h(r+) = 0. Since the function h(r+) takes a maximum
value of a zero in the region Σ′, it becomes non positive throughout the region Σ′. In
contrast, it takes positive values in the region Σ. This is why in addition to the inequalities
(39)-(42), we impose the inequality m + q > 0 on the parameters.
D.Ergo region
In our solutions, an ergo surface is located at r = re(r+< re< r∞) satisfying the cubic
equation with respect to r2
− A2r6+ 4r4− 2(aA − 2)((aA − 2)(m + q) + 2q)r2+ (aA − 2)2q2= 0,
where the constant A is defined by
2a(q2− (2m + q)r2
r6
(44)
A = −
∞)
∞− a2(q2− 2(m + q)r2
∞).(45)
10
Page 11
⁄
S'
m
q
mq
= −
mqa
=+2
2
m
qr
−
ra
q
=
+
2
−
∞
∞
()
()
22
2
2
m
qa q
a
= −
+
2
22
2
2
h r ( )
+= 0
(,)
−a a
22
(,)
−∞
r
∞
r
22
(,)
rar
∞∞
−
222
2
FIG. 3: In the region Σ, since h(r+) > 0, the solution has no CTC everywhere outside the horizon,
but in the region Σ′, since h(r+) ≤ 0, it has CTCs outside the horizon. The curve which enter only
the region Σ′and has endpoints at the points (0,0) and (−a2,a2) denotes h(r+) = 0.
In fact, for r2
+< r2< r2
∞, this equation has a single positive root within the region of
parameters Σ, the boundaries ∂Σ−= {(q,m) | m + q = 0,− r2
{(q,m) | m−q−2a2= 0, −a2< q ≤ r2
the region of parameters (10)-(14) and ∂Σ±.
∞≤ q < −a2} and ∂Σ+=
∞−2a2}. Thus, there is always an ergo region within
IV.VARIOUS LIMITS
1.
r∞→ ∞
In the limit of r∞ → ∞ with the other parameters fixed, where the size of an extra
dimension becomes infinite, the function k(r) takes the limit of k(r) → 1. Then the metric
coincides with that of the asymptotically flat solutions obtained by Cvetic. et.al. [22]. In
the specified case of q → 0, the solution reduces to the five-dimensional Myers-Perry black
hole solution [23] with two equal angular momenta.
11
Page 12
2.
q → 0
In the limit of q → 0, the solution coincides with the one obtained by Gibbons et.al. [24,
25]. This solutions has only a angular momentum in the direction of an extra dimension.
As mentioned previously, Wang regenerated it [27] via the squashing transformation for the
five-dimensional Myers-Perry black hole solutions.
3.
m → −q
Taking the limit of m → −q with introducing new coordinates (˜t, ˜ r) and the parameters
(˜Q,R∞,˜J) defined as
r2=4(R2
∞˜ r + R∞˜Q)
˜ r + R∞
R2
∞
R2
∞= 4R2
, (46)
t =
∞−˜Q
∞,
˜t,(47)
r2
(48)
q = −4˜Q,
a = −2˜J
(49)
˜Q,
(50)
we obtain the following metric
ds2= −H−2?
d˜t +
˜J
R2
∞
Hkσ3
?2
+ Hds2
T−NUT,(51)
where ds2
T−NUTis the metric on the Euclidean self-dual Taub-NUT space written in the
Gibbons-Hawking coordinate and is given by
ds2
T−NUT= Hk(d˜ r2+ ˜ r2dΩ2
S2) + R2
∞H−1
kσ2
3
(52)
H and Hkare harmonic functions on the three-dimensional Euclid space and are expressed
in the coordinate system as follows
H = 1 +
˜Q
R∞˜ r,
(53)
Hk= 1 +R∞
˜ r
.(54)
This coincides with the metric of the supersymmetric black hole solutions with a compactified
extra dimension on the Euclidean self-dual Taub-NUT space in Ref [3]. This spacetime has
12
Page 13
an ergo region. Moreover, in the case of r∞→ ∞, the metric can be written as follows
q
R2
ds2= −
?
1 +
?−2?
dt −
aq
2R2σ3
?2
+
?
1 +
q
R2
??dR2+ R2dΩ2
S3?, (55)
where R2= r2+ q and dΩ2
S3 is the metric on a unit three-sphere. This is the metric of the
BMPV black hole solutions written in terms of the Gibbons-Hawking coordinate.
4.
a → 0
The case of a → 0 corresponds to the metric of the static non-BPS Kaluza-Klein black
hole solution with a squashed horizon obtained by two of authors [26].
M, Q, J,
M, Q,
M, J,
M=Q, J, r1
r1
r1
r1
Gaiotto-Strominger-Yin
Gibbons-Wiltshire
Dobiasch-Maison
Ishihara-Matsuno
M, Q, J
M, Q
Cvetic-Lu-Pope
Cvetic-Youm
M=Q, J
Breckenridge-Myers-Peet-Vafa
M, J
Myers-Perry *
(5-dim. Reissner-Nordstrom)
Tangherlini
Squashing Transformation
Asymptotic Flat Black Holes
Kaluza-Klein Black Holes
@
FIG. 4: Various limits: This figure shows the relation between black hole solutions with asymptot-
ically flatness and Kaluza-Klein black holes. * We consider, here, the Myers-Perry solution in the
special case of two equal angular momenta.
5.
m → q + 2a2
In the case of m → q+2a2, two horizons degenerate, although this is not a BPS solution.
In this case, the metric can be written in the form
ds2= −Γ
r4
?
dt +ω
2Γσ3
?2
+r4
∆k2dr2+r2
4
?
k(σ2
1+ σ2
2) +∆
Γσ2
3
?
,(56)
13
Page 14
where the functions ∆, Γ, ω and k(r) in the metric are given by
∆ = (r2− (q + 2a2))2,
Γ = r4− 2(q + 2a2)r2+ q2,
ω = a((3q + 4a2)r2− q2),
k(r) =(r2
(r2
(57)
(58)
(59)
∞− (q + 2a2))2
∞− r2)2
.(60)
This spacetime has an ergo region. It is noted that from Eq.(37), the surface gravity of the
black hole vanishes.
6.
ρ0→ 0 with ρ±finite
Here we consider the limit of ρ0→ 0, where the function k(r) → 0. We introduce the
new parameters ρ±defined as
ρ±=
r2
±
r2
∞− r2
±
ρ0.(61)
In terms of r±, the constant ρ0defined in Eq.(16) is written in the form
ρ2
0=(r2
∞− r2
+)(r2
4r2
∞
∞− r2
−)
.(62)
Then, the metric can rewritten and as
ds2= −¯V dT2+ Udρ2+ ρ(ρ + ρ0)dΩ2
where T = 2ρ0t/r∞, and the metric functions¯V ,U,W, H in Eqs.(18)-(21) are rewritten in
S2 + Wσ2
3+ 2HdTσ3,(63)
the form
¯V =
?
1 −ρ+
ρ
??
1 −ρ−
?
1 −ρ−
ρ
?
−m + q
2r2
∞
?a
ρ0
?2?
1 +ρ0
ρ
?2
,(64)
U =
?
1 +ρ0
ρ
?
1 −ρ+
∞ρ3− a2(ρ + ρ0)2(q2(ρ + ρ0) − 2(m + q)r2
4r4
H = −(ρ + ρ0)(r2
4r3
From Eq.(61), the following equations holds
ρ
??
ρ
?,(65)
W =r6
∞ρ)
∞ρ2(ρ + ρ0)
,(66)
∞(2m + q)ρ − q2(ρ + ρ0)
∞ρ2
?a
ρ0
?
.(67)
r2
∞− r2
+=r2
∞
ρ+ρ0,
r2
∞− r2
−=r2
∞
ρ−ρ0.(68)
14
Page 15
In order that ρ±are finite in the limit of ρ0→ 0, it is necessary that r±→ r∞. Therefore,
from Eq.(30), a pair of parameters (q,m) must take either limit of
(q,m) → (−r2
∞,r2
∞) (69)
or
(q,m) → (r2
∞− 2a2,r2
∞).(70)
Furthermore, from Eqs.(30) and (68), the parameters (q,m) must behave as
(q,m) ≃ (−r2
∞+ β1ρ0+ β−
2ρ2
0, r2
∞− β1ρ0+ β+
2ρ2
0),(71)
or
(q,m) ≃ (r2
∞− 2a2− β1ρ0− β−
2ρ2
0, r2
∞− β1ρ0+ β+
2ρ2
0),(72)
respectively, where the constants β±
2satisfy β−
2+ β+
2= β2and the constants (β1,β2) are
given by
(β1,β2) =
?
2(ρ++ ρ−),2(ρ+− ρ−)2
4ρ+ρ−− a2
∞− β1ρ0+ β+
dT2+
?
?4ρ+ρ−− a2
2ρ
?
2ρ2
dρ2
??
dTσ3,
.(73)
(i) In the case of (q,m) ≃ (−r2
∞+ β1ρ0+ β−
2ρ2
0, r2
0), the metric becomes
ds2= −4(ρ − ρ+)(ρ − ρ−) − a2
4ρ2
1 −ρ+
ρ
1 −ρ−
ρ
?
+ρ2dΩ2
S2 +4ρ+ρ−− a2
4
σ2
3+ a(74)
where the coordinates ψ and T are transformed as
ψ → ψ −
a(ρ+− ρ−)
√ρ+ρ−(4ρ+ρ−− a2)T,
4ρ+ρ−
4ρ+ρ−− a2T.
(75)
T →
?
(76)
A coefficient of σ2
3, i.e., the size of the S1fiber, takes the constant value.
(ii) In the case of (q,m) ≃ (r2
∞−2a2−β1ρ0−β−
2ρ2
0, r2
∞−β1ρ0+β+
2ρ2
0), the metric reduces
to
ds2= −16ρ2
+ρ2
−(ρ − ρ+)(ρ − ρ−) − a2(a2− 6ρ+ρ−)2
16ρ2
+ρ2
S2 +(4ρ+ρ−− a2)(2ρ+ρ−+ a2)2
16ρ2
−ρ2
dT2
+
dρ2
??
?
1 −ρ+
+aa2− 6ρ+ρ−
2ρ+ρ−ρ
ρ
1 −ρ−
ρ
? + ρ2dΩ2
?
+ρ2
−
σ2
3
(4ρ+ρ−− a2)(2ρ+ρ−+ a2)2
16ρ2
+ρ2
−
dTσ3
(77)
15
Download full-text