New Infinite Series of Einstein Metrics on Sphere Bundles from AdS Black Holes

Abstract

A new infinite series of Einstein metrics is constructed explicitly on S^2 x S^3, and the non-trivial S^3-bundle over S^2, containing infinite numbers of inhomogeneous ones. They appear as a certain limit of a nearly extreme 5-dimensional AdS Kerr black hole. In the special case, the metrics reduce to the homogeneous Einstein metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein metric on the non-trivial S^{d-2}-bundle over S^2 from a d-dimensional AdS Kerr black hole. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on CP^2\sharp\bar{CP^2}. Comment: 15 pages, remarks and minor corrections added

Full text

Page 1

arXiv:hep-th/0402199v2 1 Apr 2004
OCU-PHYS 207
hep-th/0402199
New Infinite Series of Einstein Metrics
on Sphere Bundles from AdS Black Holes
Yoshitake Hashimoto†, Makoto Sakaguchi⋆and Yukinori Yasui∗
†Department of Mathematics, Osaka City University
⋆Osaka City University Advanced Mathematical Institute (OCAMI)
∗Department of Physics, Osaka City University
Sumiyoshi, Osaka 558-8585, JAPAN
Abstract
A new infinite series of Einstein metrics is constructed explicitly on S2×S3, and
the non-trivial S3-bundle over S2, containing infinite numbers of inhomogeneous
ones. They appear as a certain limit of a nearly extreme 5-dimensional AdS Kerr
black hole. In the special case, the metrics reduce to the homogeneous Einstein
metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein
metric on the non-trivial Sd−2-bundle over S2from a d-dimensional AdS Kerr black
hole. Our construction is a higher dimensional version of the method of Page, which
gave an inhomogeneous Einstein metric on CP2♯CP2.
†hashimot@sci.osaka-cu.ac.jp
⋆msakaguc@sci.osaka-cu.ac.jp
∗yasui@sci.osaka-cu.ac.jp
1

Page 2

1 Introduction
Anti-de Sitter (AdS) spaces have attracted renewed interests after the AdS/CFT corre-
spondence conjecture [1], which relates the properties of the supergravity on AdS and
those of the strongly coupled gauge theory on the AdS boundary. For example, the
Hawking-Page phase transition [2] between AdS and AdS Schwarzschild black holes was
interpreted [3] as the phase transition between confining and deconfining phases of the
dual gauge theory. Motivated by this, the study of AdS black holes has been extended in
various directions. Among them, AdS Kerr black holes with two angular momenta in 5-
dimensions, as well as ones with one angular momentum in d-dimensions were constructed
in [4].
On the other hand, an inhomogeneous Einstein metric on CP2♯CP2was constructed
by Page [5]. This metric is of cohomogeneity one with principal orbits S3. It was obtained
as a certain limit of the 4-dimensional de Sitter black hole together with the Wick rotation.
It should be emphasized that this metric is the first example of inhomogeneous Einstein
metrics. Furthermore, B¨ ohm proved the existence of an infinite series of Einstein metrics
of cohomogeneity one with positive scalar curvature on SN(5 ≤ N ≤ 9) and SN1+1×SN2
(5 ≤ N1+ N2+ 1 ≤ 9, N1> 1, N2> 1) [6].
Combining these two observations, we explicitly construct new Einstein metrics with
positive scalar curvature on sphere bundles, applying the method developed in [5] to the
5-dimensional AdS Kerr black hole with two angular momenta and the d-dimensional
AdS Kerr black hole with one angular momentum constructed in [4]. In summary, we will
construct
• an infinite series of Einstein metrics on S2×S3and S2? ×S3(the non-trivial S3-bundle
the parity of k1+ k2; It is trivial if k1+ k2is even, and is non-trivial otherwise.
When k1?= k2, the metrics are inhomogeneous (see Theorem 1). When k1= k2,
they are homogeneous Einstein metrics on S2× S3(see Theorem 2).
• an inhomogeneous Einstein metric on S2? ×Sd−2, the non-trivial Sd−2-bundle over S2
It should be noticed that the metrics on S2? ×S3in Theorem 1 obviously are not
apparently different from the ones which are proved to exist in [6]. The metrics in Theorem
2 coincide with the homogeneous Einstein metrics on M1,1
studied by Wang and Ziller [7]. The metric in Theorem 3 is of cohomogeneity one with
principal orbits S3× Sd−4if d ≥ 5. In the case of d = 4, it reproduces the Page metric
on S2? ×S2= CP2♯CP2with principal orbits S3.
2
over S2) parameterized by a pair of integers (k1,k2). The bundle type depends on
(see Theorem 3).
included in the case of B¨ ohm’s existence theorem and that the metrics on S2× S3are
k,1, circle bundles over CP1×CP1

Page 3

This paper is organized as follows. In section 2, we apply the method [5] to the 5-
dimensional AdS Kerr black hole with two angular momenta, and obtain the first class
of the Einstein metrics. We will derive the second class of the Einstein metrics from the
d-dimensional AdS Kerr black hole [4] with one angular momentum in section 3.
2 5-dimensional Einstein Metrics
The 5-dimensional Euclidean de Sitter (dS) Kerr metric may be extracted from the
Lorentzian AdS Kerr metric [4] by the substitution t ?→ −iτ,
l ?→ −il (AdS ?→ dS),
∆r
ρ2
Ξα
Ξβ
+∆θcos2θ
ρ2
Ξβ
−1 − r2l2
r2ρ2
Ξα
+ρ2
∆θdθ2,
where
a ?→ −iα,b ?→ −iβ with
g5 =
?
dτ −αsin2θ
dφ −β cos2θ
βdτ +r2− β2
dψ
?2
+∆θsin2θ
ρ2
?
αdτ +r2− α2
Ξα
dφ
?2
?
αβdτ +β(r2− α2)sin2θ
dψ
?2
?
dφ +α(r2− β2)cos2θ
Ξβ
dψ
?2
∆rdr2+ρ2
(2.1)
ρ2
= r2− α2cos2θ − β2sin2θ,
=
r2(r2− α2)(r2− β2)(1 − r2l2) − 2M,
= 1 − α2l2cos2θ − β2l2sin2θ
∆r
1
∆θ
(2.2)
with parameters Ξα= 1−α2l2and Ξβ= 1 −β2l2. The radii of the horizons are given by
the roots of ∆r= 0. If there is a double root r0, the black hole becomes an extreme black
hole, in which the number of the free parameters reduces. The parameters r0, M, Ξαand
Ξβare written in terms of l and the dimensionless parameters
ν1= α/r0,ν2= β/r0,(2.3)
as‡
r0 =
?1 − ν2
(1 − ν2
1ν2
1− ν2
1)2(1 − ν2
2(2 − ν2
1(1 − ν2
2 − ν2
2(1 − ν2
2 − ν2
2
2 − ν2
2
?1/2
l−1,
M0 =
2)2(1 − ν2
1− ν2
1ν2
1− ν2
1ν2
1− ν2
1ν2
2)
2)2
l−2,
Ξ0
α
= 1 −ν2
= 1 −ν2
2)
2
2)
,
Ξ0
β
2
.(2.4)
‡The suffix 0 is added for extreme parameters.
3

Page 4

In terms of these parameters, we have
∆r
= −(r − r0)2˜∆(r),
˜∆(r) =
r2(r + r0)2(l2r2− ν2
1
1ν2
2). (2.5)
Let us consider a nearly extreme black hole; it has two horizons located at r = r1≡
r0− ε and r = r2≡ r0+ ε, where the parameter ε represents a small deviation from the
extreme black hole. In the region between two horizons, Ω(ε) = {r|r1 ≤ r ≤ r2}, we
introduce a new radial coordinate χ (0 ≤ χ ≤ π) by
r = r0− εcosχ. (2.6)
The ∆r= −(r − r1)(r − r2)˜∆(r) restricted to Ω(ε) is then given by
∆r= ε2˜∆(r0)sin2χ + O(ε3), (2.7)
where
˜∆(r0) =4(1 + ν2
1ν4
2 − ν2
2+ ν4
1ν2
1− ν2
2− 3ν2
2
1ν2
2)
. (2.8)
We consider the limit ε → 0 of the metric (2.1) in the region Ω(ε). In this limit, the term
proportional to ∆rvanishes and so leads to a singularity of the metric. To avoid this, we
define a rescaled time coordinate
η =
ε˜∆(r0)
1)(1 − ν2
r2
0(1 − ν2
2)τ + O(ε2). (2.9)
It is also convenient to define the azimuthal angles
φ1= φ +
αΞα
r2
1− α2τ,φ2= ψ +
βΞβ
r2
1− β2τ. (2.10)
The new coordinates (η,χ,φ1,φ2,θ) are then well-behaved local coordinates in the limit
ε → 0:
∆r
ρ2
Ξα
Ξβ
∆θsin2θ
ρ2
Ξα
→r4
ρ2
0
Ξ0
α
∆θcos2θ
ρ2
Ξβ
→r4
ρ2
0
Ξ0
β
(a)
?
dτ −αsin2θ
?
0(1 − ν2
dφ −β cos2θ
dψ
?2
→
ρ2
0
˜∆(r0)sin2χdη2
(2.11)
(b) αdτ +r2− α2
dφ
?2
1)2
∆0
θsin2θ
?dφ1
dψ
−
4ν1(1 − ν2
(1 − ν2
2)
1)˜∆(r0)sin2χ
2dη
?2
(2.12)
(c)
?
βdτ +r2− β2
?2
0(1 − ν2
2)2
∆0
θcos2θ
?dφ2
−
4ν2(1 − ν2
(1 − ν2
1)
2)˜∆(r0)sin2χ
2dη
?2
(2.13)
4

Page 5

(d)
1 − r2l2
r2ρ2
→r4
?
αβdτ +β(r2− α2)sin2θ
Ξα
dφ +α(r2− β2)cos2θ
Ξβ
dψ
?2
0(1 − ν2
ρ2
1)(1 − ν2
1− ν2
2)
0(2 − ν2
2)
?ν2(1 − ν2
−4ν1ν2ρ2
˜∆(r0)r2
1)
Ξ0
α
sin2θdφ1+ν1(1 − ν2
2)
Ξ0
β
cos2θdφ2
0
0
sin2χ
2dη
?2
(2.14)
(e)
ρ2
∆rdr2→
ρ2
∆θdθ2→ρ2
ρ2
0
˜∆(r0)dχ2
(2.15)
(f)
0
∆0
θ
dθ2, (2.16)
where
ρ2
0
= r2
0(1 − ν2
1cos2θ − ν2
1 − ν2
2 − ν2
2sin2θ), (2.17)
∆0
θ
= 1 −
1ν2
1− ν2
2
2
(ν2
1cos2θ + ν2
2sin2θ). (2.18)
Finally, we obtain a metric with two parameters ν1and ν2
g = h2(θ)dθ2+
2
?
i,j=1
aij(θ)ωi⊗ ωj+ b2(θ)gS2. (2.19)
Here, gS2 represents the standard metric on S2,
gS2 = dχ2+ sin2χdη2.(2.20)
The 1-forms ωi(i = 1,2) are defined by
ωi= dψi+ kicosχdη,(2.21)
where
k1 =
4ν1(1 − ν2
(1 − ν2
4ν2(1 − ν2
(1 − ν2
2)Ξ0
α
1)˜∆(r0)
1)Ξ0
2)˜∆(r0)
=ν1(1 − ν2
1 + ν4
=ν2(1 − ν2
1 + ν4
2)(2 − ν2
1ν2
1)(2 − ν2
1ν2
2− ν2
2− 3ν2
1− ν2
1ν4
1ν2
1ν2
1ν2
1ν2
2)
2+ ν2
1ν4
2
,
k2 =
β
2)
2+ ν2
2− 3ν2
2
,(2.22)
and ψiare introduced as
φi=1
2(ψi+ kiη).(2.23)
The metric components are found to be§
h2
=
1 − ν2
1cos2θ − ν2
∆0
2sin2θ
θ
,(2.24)
§The metric is symmetric with respect to the exchange (ψ1,ν1) ↔ (ψ2,ν2). Taking account of this
symmetry, we often discuss only the one side.
5