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arXiv:hep-th/0312285v2 7 Jan 2004
hep-th/0312285
WATPHYS-TH03/09
Nuttier (A)dS Black Holes in Higher Dimensions
Robert Mann and Cristian Stelea
Department of Physics, Waterloo University, 200 University Avenue West, Waterloo,
Ontario, Canada, N2L 3G1
Abstract
We construct new solutions of the vacuum Einstein field equations with cosmolog-
ical constant. These solutions describe spacetimes with non-trivial topology that are
asymptotically dS, AdS or flat. For a negative cosmological constant these solutions
are NUT charged generalizations of the topological black hole solutions in higher di-
mensions. We also point out the existence of such NUT charged spacetimes in odd
dimensions and we explicitly construct such spaces in 5 and 7 dimensions. The exis-
tence of such spacetimes with non-trivial topology is closely related to the existence
of the cosmological constant. Finally, we discuss the global structure of such solutions
and possible applications in string theory.
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1 Introduction
Ever since the seminal papers of Bekenstein and Hawking it has been known that the en-
tropy of a black-hole is proportional to the area of the horizon. This relationship can be
generalized to a wider class of spacetimes, namely those whose Euclidean sections cannot be
everywhere foliated by surfaces of constant (Euclidean) time.
if the Euclidean spacetime has non-trivial topology: the inability to foliate the spacetime
leads to a breakdown of the concept of unitary Hamiltonian evolution, and mixed states with
entropy will arise [1, 2]. Spacetimes that carry a NUT charge are in this broader class.
Intuitively the NUT charge corresponds to a magnetic type of mass. The first solution
in four dimensions describing such an object was presented in ref. [3, 4]. Although the
Taub-NUT solution is not asymptotically flat (AF), it can be regarded as asymptotically
locally flat (ALF). The difference appears in the topology of the boundary at infinity. If
we consider as example of an AF space the Euclidean version of the Schwarzschild solution
then the boundary at infinity is simply the product S2× S1. By contrast, in the presence
of a NUT charge, the spacetime has as a boundary at infinity a twisted S1bundle over S2.
Only locally we can untwist the bundle structure to obtain the form of an AF spacetime.
The bundles at infinity are labelled by the first Chern number, which is in fact proportional
to the NUT charge [1]. The presence of a NUT charge induces a so-called Misner singularity
in the metric, analogous to a ‘Dirac string’ in electromagnetism [5]. This singularity is only
a coordinate singularity and can be removed by choosing appropriate coordinate patches.
However, expunging this singularity comes at a price: in general we must make coordinate
identifications in the spacetime that yield closed timelike curves in certain regions.
There are known extensions of the Taub-NUT solutions to the case when a cosmological
constant is present. In this case the asymptotic structure is only locally de Sitter (for a
positive cosmological constant) or anti-de Sitter (for a negative cosmological constant) and
we speak about Taub-NUT-(a)dS solutions. In general, the Killing vector that corresponds
to the coordinate that parameterizes the fibre S1can have a zero-dimensional fixed point set
(we speak about a ‘NUT’ solution in this case) or a two-dimensional fixed point set (referred
to as a ‘bolt’ solution).
Generalizations to higher dimensions follow closely the four-dimensional case [6, 7, 8, 9,
10, 11]. In constructing these metrics the idea is to regard the Taub-NUT space-time as a
U(1) fibration over a 2k-dimensional base space endowed with an Einstein-K¨ ahler metric gB.
Then the (2k + 2)-dimensional Taub-NUT spacetime has the metric:
These situations can occur
F−1(r)dr2+ (r2+ N2)gB− F(r)(dt + A)2
(1)
where t is the coordinate on the fibre S1and A has a curvature F = dA, which is proportional
to some covariantly constant 2-form. Here N is the NUT charge and F(r) is a function of r.
The solution will describe a ‘NUT’ if the fixed point set of e0= dt+A (i.e. the points where
F(r) = 0) is less than 2k-dimensional and a ‘bolt’ if the fixed point set is 2k-dimensional.
We can consider in the even-dimensional cases circle fibrations over base spaces that can be
factorized in the form B = M1×···×Mkwhere Miare two dimensional spaces of constant
curvature. In this case we can have a NUT charge Nifor every such two-dimensional factor
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and in the above ansatz we replace (r2+N2)gBwith the sum?
Note that the factor in front gBof is never zero unless we go to the Euclidean section.
Hence when we shall consider the possible singularities of the above metrics we shall focus
mainly on their Euclidean sections, recognizing that the Lorentzian versions are singularity-
free (apart from quasi-regular singularities [12]), i.e. scalar curvature singularities have the
possibility of manifesting themselves only in the Euclidean sections.
In this paper we generalize the above construction to odd-dimensional space-times.
We use a base space that is odd-dimensional and construct an S1bundle over an even-
dimensional K¨ ahler space M that is a factor of the odd dimensional base space. Specifically
we assume that the base space can be factorized in the form B = M × Y and employ the
following ansatz for the metric of the odd-dimensional Taub-NUT spaces:
i(r2+N2
i)gMi. In particular,
we can use the sphere S2, the torus T2or the hyperboloid H2as factor spaces.
F−1(r)dr2+ (r2+ N2)gM+ r2gY− F(r)(dt + A)2
Here gM is the metric on the even-dimensional space M while gY is the metric on the
remaining factor space Y . We explicitly construct NUT-charged spaces in 3, 4, 5, 6 and 7
dimensions. These solutions represent new generalizations of the spacetimes studied in refs.
[6, 9, 10, 11].
Our conventions are: (−,+,...,+) for the (Lorentzian) signature of the metric; in d
dimensions our metrics will be solutions of the vacuum Einstein field equations with cosmo-
logical constant λ = ±(d−1)(d−2)
equivalent form Rij±d−1
(2)
2l2
, which can be expressed in the form Gij+λgij= 0 or in the
l2gij= 0.
2 Taub-NUT-dS metrics in 3 dimensions
The only consistent way to construct a three-dimensional spacetime as a U(1) fibration
over a two-dimensional base space with constant curvature is to use as the base space an
hyperboloid H2. In this case we obtain the following NUT-charged metric which is a solution
of the vacuum Einstein field equations with negative cosmological constant λ = −1
l2
16n2(dt + 2ncoshrdθ)2+l2
l2:
ds2= −
4(dr2+ sinh2rdθ2) (3)
where n is the NUT charge. The signature of this metric is (−,+,−).
Lorentzian signature (−,+,+) then we must restrict the values of the cosmological constant
to be positive (λ =
solutions are given by:
If we require a
1
l2) and so we must analytically continue l → il in (3). Two other
ds2= −
l2
16n2(dt + 2nsinhrdθ)2+l2
4(dr2+ cosh2rdθ2)(4)
ds2= −
l2
16n2(dt + 2nerdθ)2+l2
4(dr2+ e2rdθ2)(5)
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If we analytically continue the coordinates t → iχ, θ → iθ, r → it we obtain a time-
dependent metric of the form:
ds2=
l2
16n2(dχ + 2ncostdθ)2+l2
4(−dt2+ sin2tdθ2) (6)
Another possibility is to analytically continue θ → it and n → −in in (3) to obtain (with
t → θ and λ < 0):
ds2=
l2
16n2(dθ + 2ncoshrdt)2+l2
4(dr2− sinh2rdt2) (7)
ds2=
l2
16n2(dθ + 2nsinhrdt)2+l2
4(dr2− cosh2rdt2) (8)
ds2=
l2
16n2(dθ + 2nerdt)2+l2
4(dr2− e2rdt2) (9)
which are all solutions of the Einstein field equations with negative cosmological constant.
The above metrics correspond to the Lorentzian versions of some the so-called Thurston
geometries [13]. These geometries are so named because of Thurston’s conjecture that a
3-manifold with a given topology can be decomposed into a connected sum of simple 3-
manifolds, each of which admits one of eight geometries: H3, S3, E3, S2×S1, H2×S1, Sol,
Nil and SL(2,R). In our case we can apply a T-duality along the Hopf S1direction and
untwist the circle fibration to the product space H2×S1[14]. Hence the spacetimes that we
have obtained are T-dual to some of the eight Thurston geometries.
3 Taub-NUT-AdS/dS-like metrics in 4 dimensions
In four dimensions we can use as base spaces any Einstein metric. For simplicity we shall
consider the following cases: the sphere S2, the torus T2and the hyperboloid H2. These
metrics are solutions of the vacuum Einstein field equations with a cosmological constant
λ = −3
[16]. We shall refer them as topological Taub-NUT-AdS/dS spacetimes since in general the
base manifold will be a compact space that is not simply connected.
l2 in 4 dimensions and their rotating versions have been presented and discussed in
• U(1) fibration over S2.
The metric is given by:
ds2= −F(r)(dt − 2ncosθdφ)2+ F−1(r)dr2+ (r2+ n2)dΩ2
where dΩ2= dθ2+ sin2θdφ2is the metric on the sphere S2and
(10)
F(r) =r4+ (l2+ 6n2)r2− 2mrl2− n2(l2− 3n2)
l2(n2+ r2)
(11)
Notice that when the NUT charge n = 0 we recover the Schwarzschild-AdS/dS solution
with
F(r,n = 0) = 1 −2m
r
−r2
l2
(12)
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• U(1) fibration over T2.
In this case we use as base space the torus T2. The toroidal Taub-NUT-AdS/dS
solution is given by:
ds2= −F(r)(dt − 2nθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ dφ2)
Notice that in this case A = 2nθdφ for which the curvature 2-form is proportional with
the volume element of the torus. The function F(r) is given by:
(13)
F(r) =r4+ 6n2r2− 2ml2r − 3n4
l2(n2+ r2)
(14)
When n = 0 we recover the toroidal Schwarzschild-AdS/dS solution, with:
F(r,n = 0) =r2
l2−2m
r
(15)
• U(1) fibration over H2.
The solution is given by:
ds2= −F(r)(dt − 2ncoshθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ sinh2θdφ2)
where:
F(r) =r4+ (6n2− l2)r2− 2ml2r − n2(3n2− l2)
(16)
l2(n2+ r2)
(17)
Notice that for n = 0 we recover the hyperbolic Schwarzschild-AdS/dS solution, for
which:
F(r,n = 0) = −1 −2m
r
−r2
l2
(18)
Here m is a mass parameter, and the spaces are realized as non-trivial fibrations over
a compact two-dimensional space. As shown in [15] one can have NUT and bolt solutions,
with the exception of the one that corresponds to a circle fibration over H2, in which case
there are no NUT solutions.
4Taub-Nut-dS/AdS spacetimes in 5 dimensions
In even-dimensions the usual Taub-NUT construction corresponds to a U(1)-fibration over
an even-dimensional Einstein space used as the base space. Since obviously this cannot be
done in odd-dimensions, we must modify our metric ansatz in such a way that we can realize
the U(1)-fibration as a fibration over an even dimensional subspace of the odd dimensional
base space. In five dimensions our base space is three dimensional and we shall construct the
NUT space as a partial fibration over a two-dimensional space of constant curvature. The
spacetimes that we obtain are not trivial in the sense that we cannot set the NUT charge
and/or the cosmological constant (now λ =
6
l2) to vanish.
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Consider first a fibration over S2. The ansatz that we shall use in the construction of
these spaces is the following:
ds2= −F(r)(dt − 2ncosθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ sin2θdφ2) + r2dy2
The above metric is a solution of the Einstein field equations with cosmological constant λ
provided
F(r) =4ml2− r4− 2n2r2
l2(r2+ n2)
and
n2=l2
(19)
(20)
4
(21)
Let us consider next the Euclidean section of the above solution (obtained by making
the analytical continuations t → iχ and n → in):
ds2= FE(r)(dχ − 2ncosθdφ)2+ F−1
where
FE(r) =r4− 2n2r2+ 4ml2
l2(r2− n2)
and the constraint λn2= −3
cally continue l → il for consistency with the initial constraint on λ and n2.
In order to get rid of the usual Misner type singularity in the metric we have to assume
that the coordinate χ is periodic with period β. Notice that for r = n the fixed point of the
Killing vector
solution. However, for r = rb, where rb> n is the largest root of FE(r), the fixed point set is
three-dimensional and we shall refer to such solutions as bolt solutions. Note that for either
situation the the period of χ must be β = 8πn to ensure the absence of the Dirac-Misner
string singularity.
In order to have a regular NUT solution we have to ensure the following additional
conditions:
E(r)dr2+ (r2− n2)(dθ2+ sin2θdφ2) + r2dz2
(22)
(23)
2holds. Since we analytically continue n we must also analyti-
∂
∂χis one dimensional and we shall refer to such a solution as being a NUT
• FE(r = n) = 0 in order to ensure that the fixed point of the Killing vector
one-dimensional.
∂
∂χis
• βF′
singularities at r = n (in other words, the periodicity of χ must be an integer multiple
of the periodicity required for regularity in the (χ,r) section; we identify k points on
the circle described by χ).
E(r = n) = 4πk (where k is an integer) in order to avoid the presence of conical
It is easy to see that the above conditions lead to k = 1 and mn =
precisely for this value of the parameter m that the above solution becomes the Euclidean
AdS spacetime in five-dimensions.
Let us now turn to the regularity conditions that we have to impose in order to obtain the
bolt solutions. In order to have a regular bolt at r = rbwe have to satisfy similar conditions
as before, with rb> n:
n4
4l2 =
l2
64. It is
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• FE(r = rb) = 0
• βF′
The above conditions lead to rb=kn
E(r = rb) = 4πk where k is an integer.
2and
m = mb= −k2l2(k2− 8)
1024
(24)
To ensure that rb> n we have to take k ≥ 3; as a consequence the curvature singularity at
r = n is avoided. We obtain the following family of bolt solutions, indexed by the integer k:
ds2= FE(r,k)(dχ − 2ncosθdφ)2+ F−1
where
E(r,k)dr2+ (r2− n2)(dθ2+ sin2θdφ2) + r2dz2(25)
FE(r,k) =256r4− 128l2r2− k2l4(k2− 8)
256l2(r2− n2)
2. One can check directly that the bolt solution is not simply the AdS space in
disguise by computing the curvature tensor of the bolt metric and comparing it with that of
the Euclidean AdS space.
We can obtain NUT spaces with non-trivial topology if we make partial base fibrations
over a two-dimensional torus T2or over the hyperboloid H2. We obtain
(26)
and n =
l
ds2= −F(r)(dt − 2nθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ dφ2) + r2dy2
for the torus, where
F(r) =4ml2+ r4+ 2n2r2
(27)
l2(r2+ n2)
(28)
where now the constraint equation takes the form λn = 0 where λ = −6
consistent Taub-NUT spaces with toroidal topology if and only if the cosmological constant
vanishes. The Euclidean version of this solution, obtained by analytic continuation of the
coordinate t → it and of the parameter n → in has a curvature singularity at r = n. Note
that if we consider n = 0 in the above constraint we obtain the AdS/dS black hole solution
in five dimensions with toroidal topology.
If the cosmological constant vanishes then we can have n ?= 0 and we obtain the following
form of the metric
l2; we can have
ds2= −F(r)(dt − 2nθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ dφ2) + r2dy2
(29)
where
F(r) =
4m
r2+ n2
(30)
The asymptotic structure of the above metric is given by
ds2=4m
r2(dt − 2nθdφ)2+r2
4mdr2+ r2(dθ2+ dφ2+ dy2) (31)
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If y is an angular coordinate then the angular part of the metric parameterizes a three torus.
The Euclidian section of the solution described by (29) is not asymptotically flat and has a
curvature singularity localized at r = 0. However, let us notice that for r ≤ n the signature
of the space becomes completely unphysical. Hence, for the Euclidian section, we should
restrict the values of the radial coordinate such that r ≥ n.
In the case of a fibration over the hyperboloid H2we obtain:
ds2= −F(r)(dt − 2ncoshθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ sinh2θdφ2) + r2dy2
where now λ = −6
F(r) =r4+ 2n2r2− 4ml2
(32)
l2,
l2(r2+ n2)
(33)
and the constraint n2=l2
The Euclidean section of these spaces is described by the metric
4holds.
ds2= FE(r)(dt − 2ncoshθdφ)2+ F−1
where n2=l2
E(r)dr2+ (r2− n2)(dθ2+ sinh2θdφ2) + r2dy2
(34)
4and
FE(r) = −r4− 2n2r2+ 4ml2
l2(r2− n2)
(35)
and it is a Euclidean solution of the vacuum Einstein field equations with positive cosmolog-
ical constant. The coordinates θ and φ parameterize a hyperboloid, which after performing
appropriate identifications becomes a surface of any genus higher than 1. In general the
metric has a curvature singularity located at r = n =
which mn = −l2
disguise.
In order to discuss the possible singularities in the metric first let us notice the absence of
Misner strings, the fibration over the hyperbolic space being trivial in this case. Moreover,
if we impose the condition that there are no conical singularities at r = rp, where rpis the
biggest root of FE(r), then we must set the periodicity β of the coordinate χ to be
. If we take rp= n we obtain β = 8πn and m = −l2
is the dS space in disguise.
In order to determine the bolt solution one has to satisfy the following conditions:
l
2with the exception of the case in
64when the space is actually the five-dimensional Euclidean dS space in
4π
|F′
E(r=rp)|
64, which means that the NUT solution
• FE(r = rb) = 0
•
we identify k points on the circle described by χ.
4π
E(rb)|=8πn
|F′
k
where k is an integer and the period of χ is now given by β =8πn
k; again
The above conditions lead to rb=kn
2and
m = mb=k2l2(k2− 8)
1024
(36)
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We must take k ≥ 3 to ensure that rb> n, which again avoids the curvature singularity at
r = n. We obtain the following family of bolt solutions, indexed by the integer k:
ds2= FE(r)(dχ − 2ncoshθdφ)2+ F−1
where
E(r)dr2+ (r2− n2)(dθ2+ sinh2θdφ2) + r2dz2
(37)
FE(r) =−256r4+ 512n2r2+ k2l4(k2− 8)
256l2(r2− n2)
and n =l
2.
5 Taub-NUT-AdS metrics in 6 dimensions
In this section we shall describe Taub-NUT-like solutions for the vacuum Einstein field
equations with cosmological constant. In 6-dimensions the base space that we can use is
4-dimensional and we shall use all the possible combinations of products of S2, T2and H2.
5.1 ‘Full’ fibrations over the product base space
• U(1) fibration over S2× S2
The metric is given by:
ds2= −F(r)(dt − 2n1cosθ1dφ1− 2n2cosθ2dφ2)2+ F−1(r)dr2
+(r2+ n2
1)(dθ2
1+ sin2θ1dφ2
1) + (r2+ n2
2)(dθ2
2+ sin2θ2dφ2
2)
(38)
where:
F(r) =3r6+ (l2+ 5n2
2+ 10n2
1)r4+ 3(n2
3(r2+ n2
2(l2+ 5n2
1)(r2+ n2
2l2+ 10n2
1)(r2+ n2
1n2
2+ n2
1l2+ 5n4
1)r2
2)l2
+6ml2r − 3n2
3(r2+ n2
1n2
1)
2)l2
(39)
Here the above metric is a solution of vacuum Einstein field equations with cosmological
constant (λ = −10
(n2
l2) if and only if:
1− n2
2)λ = 0(40)
Consequently we see that differing values for n1and n2are possible only if the cosmo-
logical constant vanishes.
In order to analyze the possible singularities of these spacetimes we shall consider the
corresponding Euclidean sections, obtained by analytic continuation of the coordinate t → iχ
and of the parameters nj→ injwhere j = 1,2.
If the cosmological constant is zero then one can have two distinct values of the parameters
n1and n2. Let us assume first that n1= n2= n. Then the NUT solutions correspond to
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the fixed-point set of
mn=4n3
3
while the periodicity of the coordinate χ is 12πn. However, even if we avoid the
conical singularity at r = n we still have a curvature singularity located at r = n, as one can
check by computing some of the curvature invariants.
For the bolt solution we shall still impose the periodicity of the coordinate χ to be12πn
while the fixed-point set is four-dimensional and located at r = rb =
this implies that only the values k = 1,2 are relevant. The mass parameter is given by
m = mb=
6k3
.
In the case in which the values of the two NUT charges n1and n2are different, one can
assume without loss of generality that n1> n2. In this case in the Euclidean section the
radius r cannot be smaller than n1or the signature of the spacetime will change. Generi-
cally, there is a curvature singularity located at r = n1; however for a certain value of the
mass parameter m this curvature singularity is removed. The NUT solution in this case
corresponds to a two-dimensional fixed-point set located at r = n1. Removal of the Dirac
string singularity forces the periodicity of the coordinate χ to be 8πn1. The value of the
mass parameter is m = mb=
3
, and only for this value of the parameter m the metric
is well-behaved at r = n1. This is a similar situation to that in the five-dimensional case;
the curvature singularity at r = n disappears if and only if m is a specific function of the
NUT charge. For any other values of m (as in the bolt solutions) the curvature singularity
is still present.
The bolt solution corresponds to a four-dimensional fixed-point set located at r = rb=
2n1
k, for which the periodicity of the coordinate χ is given by8πn1
parameter is m = mb=
12
. Since rb> n1, we must choose k = 1, thereby avoiding
the curvature singularity.
If the cosmological constant is non-zero then the above constraint equation will impose
the condition n1= n2= n. The Euclidean section of the metric has a curvature singularity
located at r = n. The NUT solution corresponds to a zero-dimensional fixed-point set of the
vector
4n3(l2−6n2)
∂
∂χat r = n and we obtain the value of the mass parameter to be
k,
3n
k; since rb > n,
n3(k4+18k2−27)
n3
1+3n1n2
2
k
and the value of the mass
n1(12n2
2−4n2
1)
∂
∂χlocated at r = n. The periodicity of the coordinate χ is given by 12πn while the
value of the mass parameter is m = mb=
to a four-dimensional fixed-point set at
3l2
[10, 11]. The bolt solution corresponds
r = rb=
1
30n(l2±
√l4− 180n2l2+ 900n4)
while the mass parameter is given by:
mb=−3r6
b+ (15n2− l2)r4
b+ 3n2(2l2− 15n2)r2
6l2rb
b+ 3n4(l2− 5n2)
In order to avoid the singularity at r = n we shall impose the condition rb > n which,
together with the condition that rbhas only real values leads to [10]:
n ≤
?
3 − 2√2
30
?1
2
l
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Let us notice that we obtained two bolt solutions, corresponding to the different signs in
the expression of rb.
• U(1) fibration over S2× T2
The metric is given by:
ds2= −F(r)(dt − 2n1cosθ1dφ1− 2n2θ2dφ2)2+ F−1(r)dr2
+(r2+ n2
1)(dθ2
1+ sin2θ1dφ2
1) + (r2+ n2
2)(dθ2
2+ dφ2
2) (41)
where:
F(r) =3r6+ (l2+ 5n2
2+ 10n2
1)r4+ 3(n2
3(r2+ n2
2(l2+ 5n2
1)(r2+ n2
2l2+ 10n2
1)(r2+ n2
1n2
2+ n2
1l2+ 5n4
1)r2
2)l2
+6ml2r − 3n2
3(r2+ n2
1n2
1)
2)l2
(42)
Here the above metric is a solution of vacuum Einstein field equations with cosmological
constant if and only if:
(n2
1− n2
2)λ = 2 (43)
Notice that if either the NUT charges are equal or if cosmological constant vanishes,
the above U(1) fibration over S2× T2is not a solution of the vacuum Einstein field
equations.
In order to analyze the singularities of the above metrics let us consider their Euclidean
sections, obtained by analytic continuation of the coordinate t → iχ and of the parameters
nj → inj with j = 1,2. If λ = −10/l2then the above constraint equation becomes n2
n2
located at r = n1. The periodicity of the coordinate χ can be shown to be 8πn1(ensuring
the avoidance of string singularities) while the value of the mass parameter is:
1=
2+
5
l2. Then the NUT solutions corresponds to a two-dimensional fixed-point set of
∂
∂χ
m = mb= −n1
15l2(3l4− 40n2
1l2+ 120n4
1)
For this value of the mass parameter the metric is well-behaved at r = n1as one can see by
calculating some of the curvature invariants.
For the bolt solutions the fixed-point set is four-dimensional and it is located at:
r = rb=
1
20n(l2±
?
l4− 80n2
1l2+ 400n4
1)
while the value of the mass parameter is given by:
m = mb= −15r6
b+ (10l2− 75n2
1)r4
b+ (3l4− 60n2
1l2+ 225n4
30l2rb
1)r2
b+ 3n2
1(l4− 10n2
1l2+ 25n4
1)
(44)
Again, we have two kinds of bolt solutions given by the two roots r = rb.
11
Page 12
• U(1) fibration over S2× H2
The metric is given by:
ds2= −F(r)(dt − 2n1cosθ1dφ1− 2n2coshθ2dφ2)2+ F−1(r)dr2
+(r2+ n2
1)(dθ2
1+ sin2θ1dφ2
1) + (r2+ n2
2)(dθ2
2+ sinh2θ2dφ2
2) (45)
where:
F(r) =3r6+ (l2+ 5n2
2+ 10n2
1)r4+ 3(n2
3(r2+ n2
2(l2+ 5n2
1)(r2+ n2
2l2+ 10n2
1)(r2+ n2
1n2
2+ n2
1l2+ 5n4
1)r2
2)l2
+6ml2r − 3n2
3(r2+ n2
1n2
1)
2)l2
(46)
Here the above metric is a solution of vacuum Einstein field equations with cosmological
constant if and only if:
(n2
1− n2
2)λ = 4(47)
Again, if the cosmological constant is zero or if the NUT charges are equal, the above
U(1) fibration over S2× H2is not solution of vacuum Einstein field equations.
As before we analytically continue t → iχ and nj→ inj, where j = 1,2. The constraint
equation becomes in this case n2
dimensional fixed-point set of the vector
parameter is given by:
m = mb= −2n1
and the periodicity of the coordinate χ is 8πn1. Notice that for this value of the mass
parameter the metric will be well-behaved in the vicinity of r = n1.
The bolt solution will correspond to a four-dimensional fixed-point set located at:
1= n2
2+2l2
5. The NUT solution will correspond to a two-
∂
∂χ, located at r = n1. The value of the mass
5l2(l4− 10n2
1l2+ 20n4
1)
r = rb=
1
20n(l2±
?
l4− 80n2
1l2+ 400n4
1)
while the value of the mass parameter is given by:
m = mb= −15r6
b+ (10l2− 75n2
1)r4
b+ (3l4− 60n2
1l2+ 225n4
30l2rb
1)r2
b+ 3n2
1(l4− 10n2
1l2+ 25n4
1)
(48)
Notice that we obtain two bolt solutions, that corresponds to the different signs in the
expression of r = rb.
• U(1) fibration over T2× T2
The metric is given by:
ds2= −F(r)(dt − 2n1θ1dφ1− 2n2θ2dφ2)2+ F−1(r)dr2
+(r2+ n2
1)(dθ2
1+ dφ2
1) + (r2+ n2
2)(dθ2
2+ dφ2
2) (49)
12
Page 13
where:
F(r) =3r6+ 5(n2
2+ 2n2
1)r4+ 15n2
3(r2+ n2
1(n2
1)(r2+ n2
1+ 2n2
2)r2+ 6ml2r − 15n4
2)l2
1n2
2
(50)
Here the above metric is a solution of vacuum Einstein field equations with cosmological
constant if and only if:
(n2
1− n2
2)λ = 0(51)
As in the case of the circle fibration over S2×S2, if the cosmological constant is zero,
we can have two different parameters n1and n2. If the cosmological constant is not
zero then the above U(1) fibration over T2× T2is a solution of the vacuum Einstein
field equations if and only if n2
1= n2
2.
If we consider the Euclidean section in this case, notice that there are no Misner strings
and that FE(r) becomes zero only if r = 0. The solution is singular at r = n, where n is the
greatest of the NUT charges n1and n2.
Let us consider now the more interesting case in which the cosmological constant is non-
zero. Then the constraint equation imposes n1= n2= n. There is a curvature singularity at
r = n. However it can be readily checked that if the mass parameter is given by m = mb=
−8n5
bolt solutions have a four-dimensional fixed-point set located at rb> n and we find that in
this case the mass parameter is given by:
l2 then the metric is well-behaved at r = n, and it corresponds to a NUT solution. The
mb= −r6
b− 5n2r4
b+ 15n4r2
2l2rb
b+ 5n6
Regularity at the bolt requires the period of χ to be:
β =
4πl2rb
5(r2
b− n2)
• U(1) fibration over T2× H2
The metric is given by:
ds2= −F(r)(dt − 2n1θ1dφ1− 2n2coshθ2dφ2)2+ F−1(r)dr2
+(r2+ n2
1)(dθ2
1+ dφ2
1) + (r2+ n2
2)(dθ2
2+ sinh2θ2dφ2
2) (52)
where:
F(r) =3r6+ 5(n2
2+ 2n2
1)r4+ 15n2
3(r2+ n2
1(n2
1)(r2+ n2
1+ 2n2
2)r2+ 6ml2r − 15n4
2)l2
1n2
2
(53)
Here the above metric is a solution of vacuum Einstein field equations with cosmological
constant if and only if:
(n2
1− n2
2)λ = 2 (54)
and so for vanishing cosmological constant or equal NUT charges this U(1) fibration
over T2× H2is not a solution of the vacuum Einstein field equations.
13
Page 14
To analyze the singularities of the above spacetimes we shall consider their Euclidean
sections, which are obtained by the following analytical continuations t → iχ and nj→ inj,
with j = 1,2. The constraint equation will become in this case n2
there is a curvature singularity localized at r = n1. However, if the value of the mass
parameter is given by:
mp= −4n3
the the metric becomes well-behaved at r = n1. This corresponds to the NUT solution for
which the fixed-point set of ∂χis two-dimensional and located at r = n1. Solutions with
bolts correspond to four-dimensional fixed-point sets located at rb> n1, for which the mass
parameter is given by:
1= n2
2+l2
5and generically
1(6n2
1− l2)
3l2
mb=−3r6
b+ (15n2− l2)r4
b+ 3n2(2l2− 15n2)r2
6l2rb
b+ 3n4(l2− 5n2)
(55)
and the periodicity of χ is:
β =
4πl2rb
5(r2
b− n2)
(56)
• U(1) fibration over H2× H2
The metric is given by:
ds2= −F(r)(dt − 2n1coshθ1dφ1− 2n2coshθ2dφ2)2+ F−1(r)dr2
+(r2+ n2
1)(dθ2
1+ sinh2θ1dφ2
1) + (r2+ n2
2)(dθ2
2+ sinh2θ2dφ2
2) (57)
where:
F(r) =3r6+ (5n2
2+ 10n2
1− l2)r4+ 3(−n2
3(r2+ n2
2(l2− 5n2
1)(r2+ n2
2l2+ 10n2
1)(r2+ n2
1n2
2− n2
1l2+ 5n4
1)r2
2)l2
+6ml2r + 3n2
3(r2+ n2
1n2
1)
2)l2
(58)
Here the above metric is a solution of vacuum Einstein field equations with cosmological
constant (λ = −10
(n2
l2) if and only if:
1− n2
2)λ = 0(59)
Hence either n2
1= n2
2or the cosmological constant vanishes.
Consider now the Euclidean sections of the above spacetimes obtained by analytical
continuation of the coordinate t → iχ and of the parameters nj→ inj, with j = 1,2. If the
cosmological constant is zero one can have different values for the NUT charge parameters
n1 and n2. Let us assume that n1 > n2. Then we obtain a NUT solution for which the
fixed-point set of the isometry
the mass parameter is given by mb= −n1(3n2
∂
∂χis two-dimensional and localized at r = n1. In this case
2+n2
1)
3
and the periodicity of the coordinate χ is
14
Page 15
given by 8πn1. One can easily check that for this value of the mass parameter the metric is
well-behaved at r = n1. The corresponding bolt solution has a four-dimensional fixed-point
set located at r = rb=
coordinate χ is given by8πn1
rb> n1hence we must have k = 1. The mass parameter is then given by mb=
If the cosmological constant is non-zero then the above constraint equation will impose
n1= n2= n. There is a NUT solution which corresponds to a zero-dimensional fixed-point
set of the
is 12πn and the mass parameter has the value m = mb= −4n3(l2+6n2)
singularity at r = n as can be checked by calculating some of the curvature invariants at
r = n.
The bolt solution corresponds to a four-dimensional fixed-point set, which is located at:
2n1
k, where k is an integer number such that the periodicity of the
k. In order to avoid the curvature singularity at r = n1we impose
n1(4n2
1−15n2
12
2)
.
∂
∂χisometry, the NUT being located at r = n. The periodicity of the coordinate χ
3l2
. There is a curvature
r = rb=
1
30n(−l2+
√l4+ 180n2l2+ 900n4) (60)
while the mass parameter is given by:
mb= −3r6
b− (15n2+ l2)r4
b+ 3n2(2l2+ 15n2)r2
6l2rb
b+ 3n4(l2+ 5n2)
(61)
Finally, we can consider circle fibrations over CP2. In this case the metric is given by:
ds2= −F(r)(dt + A)2+ F−1(r)dr2+ (r2+ n2)dΣ2
The explicit form of A and dΣ2in this case was given in [10]:
(62)
dΣ2=
du2
?1 +u2
6
?2+
u2
4?1 +u2
6
?2(dψ + cosθdφ)2+
u2n
2?1 +u2
u2
4?1 +u2
6
?(dθ2+ sin2θdφ2) (63)
and
A =
6
?2(dψ + cosθdφ) (64)
while the expression for F(r) is the same as in the S2× S2case. A singularity analysis of
this metric was given in [10].
5.2Fibrations over ‘partial’ base factors
Another class of solutions is given for base spaces that are products of 2-dimensional Einstein
manifolds and the U(1) fibration is taken over only one of the components of the product.
• (U(1) fibration over S2)×S2
The metric is written in the form:
ds2= −F(r)(dt − 2ncosθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ sin2θ1dφ2
1) + αr2(dθ2
2+ sin2θ2dφ2
2)(65)
15
Page 16
where to satisfy the field equations we must have
α =
2
2 − λn2
(66)
F(r) =3r5+ (l2+ 10n2)r3+ 3n2(l2+ 5n2)r − 6ml2
3rl2(r2+ n2)
(67)
The metric (65) is a solution of the vacuum Einstein field equations for any values
of n or λ = −10
ensure that α > 0, which translates in our case λn2< 2. We have given above the
form of F(r) using a negative cosmological constant and in this case this condition
is superfluous; however we can also use a positive cosmological constant (we have to
analytically continue l → il in F(r)) as long as the above condition is satisfied. The
Euclidian section is obtained by taking the analytical continuation t → iχ and n → in.
The NUT solution corresponds to a two-dimensional fixed-point set of the vector
located at r = n. The periodicity of the χ coordinate is in this case equal to 8πn and
the value of the mass parameter is fixed to mb=
curvature singularity present at r = n disappears and we obtain the AdS spacetime.
The bolt spacetime has a four-dimensional fixed-point set of
l2. However, in order retain a Lorentzian metric signature we must
∂
∂χ
n3(l2−4n2)
3l2
. Furthermore, if n =l
2then
∂
∂χlocated at r = rbwith:
rb=l2±√k2l4− 80n2l2+ 400n4
20n
(68)
while the value of the mass parameter is:
mb= −3r5
b+ (l2− 10n2)r3
b− 3n2(l2− 5n2)rb
6l2
(69)
where the periodicity of the coordinate χ is8πn
k, where k is an integer.
• (U(1) fibration over S2)×T2
In this case we have
ds2= −F(r)(dt − 2ncosθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ sin2θ1dφ2
1) + αr2(dθ2
2+ dφ2
2) (70)
where
F(r) =−3r5+ (l2− 10n2)r3+ 3n2(l2− 5n2)r + 6ml2
3rl2(r2+ n2)
(71)
and the vacuum Einstein field equations with cosmological constant are satisfied if and
only if:
α(2 − λn2) = 0
Since α cannot be zero we must restrict the values of n and λ such that λn2= 2,
forcing a positive cosmological constant. The Euclidian section is obtained by taking
the analytical continuation t → iχ and n → in and l → il with l =√5n. Notice that
(72)
16
Page 17
in this case the NUT solution corresponds to a two-dimensional fixed-point set of the
vector field
equal to 8πn and the value of the mass parameter is fixed to mb=n3
of the mass parameter the solution is regular at the nut location. The bolt spacetime
has a four-dimensional fixed-point set of
k3n3(20−3k2)
960
where the periodicity of the coordinate χ is
where k is an integer. To ensure that rb> n we have to take k > 3 and in this way
the curvature singularity at r = n is avoided as well.
∂
∂χlocated at r = n. The periodicity of the χ coordinate is in this case
15. For this value
∂
∂χlocated at rb=
kn
2and the value of the
mass parameter is mb=
8πn
k,
• (U(1) fibration over S2)×H2
The metric is written in the form:
ds2= −F(r)(dt − 2ncosθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ sin2θ1dφ2
1) + αr2(dθ2
2+ sinh2θ2dφ2
2) (73)
where
α =
2
λn2− 2
(74)
F(r) =−3r5+ (l2− 10n2)r3+ 3n2(l2− 5n2)r + 6ml2
3rl2(r2+ n2)
(75)
are necessary for the vacuum Einstein field equations to hold. The condition α > 0
implies that λn2> 2, again yielding a positive cosmological constant. The Euclidian
section is obtained by taking the analytical continuation t → iχ and n → in while in
order to keep α > 0 we must continue l → il . The NUT solution corresponds to a two-
dimensional fixed-point set of the vector
χ coordinate is in this case equal to 8πn and the value of the mass parameter is fixed
to mb=
3l2
. Furthermore, if n =l
disappears and we obtain the six-dimensional AdS spacetime. The bolt spacetime has
a four-dimensional fixed-point set of
∂
∂χlocated at r = n. The periodicity of the
n3(l2−4n2)
2then curvature singularity present at r = n
∂
∂χlocated at r = rbwith:
rb=kl2±√k2l4− 80n2l2+ 400n4
20n
(76)
while the value of the mass parameter is:
mb= −3r5
b+ (l2− 10n2)r3
b− 3n2(l2− 5n2)rb
6l2
(77)
where the periodicity of the coordinate χ is8πn
k, where k is an integer.
• (U(1) fibration over T2)×S2
The metric is written in the form:
ds2= −F(r)(dt − 2nθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ dφ2
1) + αr2(dθ2
2+ sin2θ2dφ2
2) (78)
17
Page 18
where now
α = −
2
λn2
(79)
F(r) =3r5+ 10n2r3+ 15n4r − 6ml2
3rl2(r2+ n2)
(80)
While the above metric is a solution of the vacuum Einstein field equations for any
values of n or λ, the condition α > 0 implies that λ < 0.
The Euclidian section of this solution is obtained by analytical continuations t → iχ,
n → in and l → il (in order to preserve α > 0). The circle fibration is trivial: in this
case we obtain two-dimensional fixed point sets of ∂χ(or ‘nuts’) for mn=
bolt solutions have a four-dimensional fixed-point set located at rb> n and we find
that in this case the mass parameter is given by:
4n5
3l2. The
mb=3r5
b− 10n2r3
b+ 15n4rb
6l2
The Euclidian regularity at the bolt requires the period of χ to be:
β =
4πl2rb
5(r2
b− n2)
• (U(1) fibration over T2)×T2
The metric is written in the form:
ds2= −F(r)(dt − 2nθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ dφ2
1) + r2(dθ2
2+ dφ2
2) (81)
where
F(r) =3r5+ 10n2r3+ 15n4r − 6ml2
3rl2(r2+ n2)
(82)
and the vacuum Einstein field equations force the condition λn2= 0 where λ = ±10
Hence a U(1) fibration over T2exists if and only if λ = 0.
l2.
The Euclidian section of this solution is obtained by analytical continuations t → iχ,
n → in. Since FE(r) =
the radius r. Generically there is a curvature singularity located at r = n at which
point the fixed-point set of ∂χis two-dimensional. The ‘bolt’ solutions correspond to
four-dimensional fixed-point sets located at rb> n (which also avoids the curvature
singularity). In both cases there are no restriction of the periodicity of χ.
2m
(r2−n2)rwe notice that FE(r) cannot be zero for any value of
• (U(1) fibration over T2)×H2
The metric is written in the form:
ds2= −F(r)(dt − 2nθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ dφ2
1) + αr2(dθ2
2+ sinh2θ2dφ2
2) (83)
18
Page 19
where in this case
α =
2
λn2
(84)
F(r) =−3r5− 10n2r3− 15n4r + 6ml2
3rl2(r2+ n2)
(85)
The above metric is a solution of vacuum Einstein field equations for any values of n
or λ. Again the condition α > 0 implies λ > 0.
The Euclidian section of this solution is obtained by analytical continuations t → iχ,
n → in and l → il (in order to preserve α > 0). The circle fibration is trivial in this
case we obtain two-dimensional fixed point sets of ∂χ(or ‘nuts’) for mn= −4n5
bolt solutions have a four-dimensional fixed-point set located at rb> n and we find
that in this case the mass parameter is given by:
3l2. The
mb= −3r5
b− 10n2r3
b+ 15n4rb
6l2
(86)
The Euclidian regularity at the bolt requires the period of χ to be:
β =
4πl2rb
5(r2
b− n2)
(87)
• (U(1) fibration over H2)×S2
The metric is written in the form:
ds2= −F(r)(dt − 2ncoshθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ sinh2θ1dφ2
1) + αr2(dθ2
2+ sin2θ2dφ2
2) (88)
where
α = −
2
2 + λn2
(89)
F(r) =3r5+ (−l2+ 10n2)r3+ 3n2(−l2+ 5n2)r − 6ml2
3rl2(r2+ n2)
The above metric is a solution of vacuum Einstein field equations for any values of n or
λ. Now the condition α > 0 is equivalent to λn2< −2 and so we must have a negative
cosmological constant.
(90)
The Euclidian sections of this solution correspond to the analytical continuations t →
iχ and n → in. However, in order to keep the sign of α positive we have to analytically
continue l → il and impose the constraint relation 5n2> l2. The NUT solution has a
two-dimensional fixed-point set of ∂χ(a ‘nut’) located at r = n and the value of the
mass parameter is mn=
3l2
, the periodicity of χ being in this case 8πn. For this
value of the mass parameter the spacetime geometry is smooth at r = n. The bolt
solutions have a four-dimensional fixed-point set of ∂χlocated at rb> n where:
rb=kl2±√k2l4− 80n2l2+ 400n4
n3(4n2−l2)
20n
(91)
19
Page 20
the mass parameter being given by:
mb=3r5
b+ (l2− 10n2)r3
b− 3n2(l2− 5n2)rb
6l2
(92)
while the periodicity of χ is given by
obtained two bolt solutions each corresponding to the different signs in rb’s expression.
8πn
k, k being an integer. Notice that we have
• (U(1) fibration over H2)×T2
The metric is written in the form:
ds2= −F(r)(dt − 2ncoshθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ sinh2θ1dφ2
1) + r2(dθ2
2+ dφ2
2) (93)
where:
F(r) =3r5+ (−l2+ 10n2)r3+ 3n2(−l2+ 5n2)r − 6ml2
3rl2(r2+ n2)
The above metric is a solution of vacuum Einstein field equations with a cosmological
constant if and only if
2 + λn2= 0
(94)
(95)
Hence we have to restrict the values of n or λ such that λn2= −2 (that is λ < 0).
The Euclidian section of this solution is obtained by analytical continuation t → iχ,
n → in and l → il such that 5n2= l2. The NUT solution has a two-dimensional
fixed-point set of ∂χlocated at r = n, the mass parameter is given by mn= −n3/15
and the periodicity of χ is 8πn. Notice that for this value of the mass parameter the
geometry is smooth at the nut’s location. The bolt solution has a four-dimensional
fixed-point set at rb=kn
2, the mass parameter is given by:
mb=k3n3(3k2− 20)
960
(96)
the periodicity of χ being 8πn/k. Since rb > n we must restrict the values of the
integer k such that k ≥ 3.
• (U(1) fibration over H2)×H2
The metric is written in the form:
ds2= −F(r)(dt − 2ncoshθ1dφ1)2+ F−1(r)dr2
+(r2+ n2)(dθ2
1+ sinh2θ1dφ2
1) + αr2(dθ2
2+ sinh2θ2dφ2
2) (97)
where the vacuum Einstein field equations imply
α =
2
λn2+ 2
(98)
F(r) =3r5+ (−l2+ 10n2)r3+ 3n2(−l2+ 5n2)r − 6ml2
3rl2(r2+ n2)
(99)
20
Page 21
for any values of n or λ. In this case the restriction α > 0 leads to λn2> −2. We can
have a negative cosmological constant as long as this relation is satisfied; alternatively
we can have a positive cosmological constant (by analytically continuing l → il in the
above expression for F(r)).
The Euclidian section of this solution is obtained by analytical continuation t → iχ and
n → in. We can choose to analytically continue the parameter l to obtain a solution with
positive cosmological constant, however in this case we must impose the restriction α > 0
which amounts to 5n2< l2. Let us consider first the solution with negative cosmological
constant (that is, we do not analytically continue l in the initial solution). If we try to make
r = n into a fixed point of ∂χ, we find that FE(r) becomes negative for r close enough to (and
bigger than) n. This means that FE(r) vanishes at some value r > n and instead of a nut we
find a bolt. It is a situation analogous with the four-dimensional case, where it was found in
[15] that there are no hyperbolic nuts. The bolt solution corresponds to a four-dimensional
fixed-point set of ∂χlocated at rb> n. The value of the mass parameter is given by:
mb=3r5
b− (l2+ 10n2)r3
b+ 3n2(l2+ 5n2)rb
6l2
(100)
where
rb=−l2+√l4+ 80n2l2+ 400n4
20n
(101)
while the periodicity of χ is 8πn.
Let us consider now the Euclidian section obtained by further analytical continuation
l → il. The NUT solution corresponds to a two-dimensional fixed-point set of ∂χlocated at
r = n and the value of the mass parameter is given by mn=
mass parameter the spacetime geometry at the nut location is smooth. The bolt solution
corresponds to a four-dimensional fixed-point set of ∂χlocated at:
rb=l2+√l4− 80n2l2+ 400n4
n3(l2−4n3)
3l2
. For this value of the
20n
(102)
the mass parameter being given by:
mb= −3r5
b+ (l2− 10n2)r3
b− 3n2(l2− 5n2)rb
6l2
(103)
and the periodicity of χ is 8πn. If we require rb> n then:
n ≤
?
1
10−
√3
20
?1
2
l.(104)
21
Page 22
6 Taub-Nut-dS/AdS spacetimes in 7 dimensions
In seven dimensions the base space is five dimensional. We shall factorize it in the form
B = M × Y , where M is an even-dimensional Einstein space endowed with an Einstein-
K¨ ahler metric. We have now two possibilities: the factor M can be two-dimensional or
four-dimensional. If M is two-dimensional we have three cases, which we can take to be a
sphere, a torus or a hyperboloid. In the four dimensional case we can use products M1×M2
of two-dimensional spaces or we can use M = CP2. Let us consider next all these cases.
If M is two dimensional we can take it to be a sphere, a torus or a hyperboloid. Then the
base factor can be factorized in the form B = M ×Y with Y a three dimensional riemannian
space, which we shall take for simplicity to be an Einstein space. For each choice of the
factor M we can take Y to be a sphere S3, a torus T3or a hyperboloid H3. The ansatz for
the Taub-NUT metric that we shall determine is given by:
ds2= −F(r)(dt + A)2+ F−1(r)dr2+ (r2+ n2)gM+ αr2gY
where gMis the metric on the two-dimensional space M and gY is a riemannian metric on
the three dimensional space Y . We shall consider the following cases:
(105)
• M = S2then Y can be S3, T3or H3. The metric is given by:
ds2= −F(r)(dt2+ 2ncosθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ sin2θdφ2)
+αr2gY
(106)
where:
F(r) =4r6+ (l2+ 12n2)r4+ 2n2(l2+ 6n2)r2+ 4ml2+ n4(l2+ 6n2)
4l2r2(r2+ n2)
(107)
The cosmological constant is given by λ = −15
and λ are constrained via the relation as follows:
l2 and the values of the parameters α, n
α(5 − 2λn2) = 10s (108)
where s = 1,0,−1, for S3, T3and H3respectively. We must have α > 0 , which in turn
imposes a joint constraint on λn2, which can be satisfied in various ways depending
on the value of s. Solutions for a positive value of λ can be obtained by analytically
continuing l → il in the expression (107) for the function F(r).
The Euclidian sections of the above solutions is obtained by the analytical continuations
t → iχ and n → in. We are free to make the continuation l → il as long as the
condition α > 0 is always satisfied. Since the singularity analysis of the metric does
not depend on this positivity condition we shall consider the two cases: in the first
case we analytically continue n in the above expression of F(r) (which corresponds to
λ < 0) and in the second case we perform the analytical continuation of n and l (which
corresponds to a positive cosmological).
22
Page 23
If the cosmological constant is negative λ = −15
to a three-dimensional fixed-point set of ∂χlocated at r = n, for which the value of
the mass parameter is mn=
fixed-point set of ∂χlocated at r = rbwhere:
rb=l2±√l4− 96n2l2+ 576n4
l2 then the NUT solution corresponds
n6
2l2. The bolt solutions correspond to a five-dimensional
24n
(109)
the value of the mass parameter is
mb= −4r6
b+ (l2− 12n2)r4
b+ 2n2(6n2− l2)r2
4l2
b+ n4(l2− 6n2)
(110)
the periodicity of χ being 8πn. In order to have rb> n we have the condition:
n ≤
?
1
12−
√3
24
?1
2
l. (111)
Consider now the case in which we make the further analytical continuation l → il
(we obtain a solution with positive cosmological constant). If we try to make r = n
into a fixed point of ∂χwe find that FE(r) becomes positive for r close enough to (and
bigger than) n. This means that FE(r) vanishes at some value r > n and instead of a
nut we find a bolt. Hence in this case there are no nut solutions. The bolt solutions
correspond to a five-dimensional fixed-point set of ∂χlocated at r = rb> n. Their
mass parameter is given by:
mb=4r6
b− (l2+ 12n2)r4
b+ 2n2(l2− 6n2)r2
4l2
b− n4(l2− 6n2)
(112)
the periodicity of χ being
4πl2rb
l2−6(r2
b−n2).
• M = T2then Y can be S3, T3or H3. Then the metric is given by:
ds2= −F(r)(dt2+ 2nθdφ)2+ F−1(r)dr2+ (r2+ n2)(dθ2+ dφ2) + αr2gY
where (with λ = −15
F(r) =2r6+ 6n2r4+ 6n4r2+ 4ml2+ 3n6
(113)
l2) we have
2l2r2(r2+ n2)
(114)
along with the constraint
αλn2= −5s(115)
where s = 1,0,−1, for S3, T3and H3respectively as before. Note that if Y = H3we
must analytically continue l → il in the expression for F(r) so as to have a positive
cosmological constant.
23
Page 24
If s = 1 then initially we have a negative cosmological constant and when we go to
the Euclidian section (by means of the analytical continuations t → iχ, n → in)we
must continue as well l → il to keep α > 0. Then the NUT solution corresponds to
a three-dimensional fixed-point set of ∂χlocated at r = n with the mass parameter
mn= −n6
located at r = rb> n with the mass parameter
4l2. The bolt solutions correspond to a five-dimensional fixed-point set of ∂χ
mb=2r6
b− 6n2r4
b+ 6n4r2
4l2
b− 3n6
(116)
while the periodicity of χ is given by
2πl2rb
3(r2
b−n2).
If s = 0 then the cosmological constant must vanish. Notice that in this case FE(r)
does not vanish anywhere and that the solution is singular at r = 0.
If s = −1 we obtain the Euclidian section by means of analytical continuations l → iχ,
n → in and l → il (preserving the condition α > 0). We obtain a NUT solution with
a three-dimensional fixed-point set of ∂χlocated at r = n with the mass parameter
mn =
singularity located at r = n. We also have bolt solutions with a five-dimensional
fixed-point set of ∂χlocated at r = rb> n with the mass parameter
n6
4l2. Notice that for this value of the mass parameter there is no curvature
mb= −2r6
b− 6n2r4
b+ 6n4r2
4l2
b− 3n6
(117)
while the periodicity of χ is given by
2πl2rb
3(r2
b−n2).
• M = H2then Y can be S3, T3or H3. Then the metric is given by:
ds2= −F(r)(dt2+2ncoshθdφ)2+F−1(r)dr2+(r2+n2)(dθ2+sinh2θdφ2)+αr2gY (118)
where as before λ = −15
F(r) =4r6+ (−l2+ 12n2)r4+ 2n2(−l2+ 6n2)r2+ 4ml2+ n4(−l2+ 6n2)
4l2r2(r2+ n2)
l2,
(119)
and the values of the parameters α, n and λ are constrained via as follows:
α?5 + 2λn2?= −10s (120)
where s = 1,0,−1, for S3, T3and H3respectively as before; should a positive value of
λ be required we must analytically continue l → il in the above expression (119) for
F(r).
If s = 1 we obtain the Euclidian section by making the analytical continuations t → iχ,
n → in and l → il (again preserving the positivity of α). The NUT solution corresponds
to a three-dimensional fixed-point set of ∂χ located at r = n with the mass parameter
mn= −n6
2l2. The periodicity of χ is 8πn and there is no curvature singularity at r = n. The
24
Page 25
bolt solutions correspond to a five-dimensional fixed-point set of ∂χlocated at r = rbwith
the mass parameter
mb=4r6
b+ (l2− 12n2)r4
b− 2n2(l2− 6n2)r2
4l2
b+ n4(l2− 6n2)
(121)
and
rb=l ±√l4− 96n2l2+ 576n4
24n
(122)
while the periodicity of χ is 8πn. If we require rb> n we obtain
n <
?
√6n and to obtain the Euclidian section we have
1
12−
√3
24
?1
2
l (123)
If s = 0 we have the constraint l =
make the analytical continuations t → iχ, n → in. If we try to make r = n into a fixed-point
of ∂χwe find that FE(r) becomes negative for r close enough to (and bigger than n), which
means that FE(r) vanishes at some value r > n and instead of having a nut we find a bolt.
The bolt solutions correspond to a five-dimensional fixed-point set of ∂χlocated at r = rb
while the mass parameter is:
mb= −2r6
b− 9n2r4
b+ 12n4r2
12n2
b− 6n6
(124)
and the periodicity of χ is
If s = −1 then we can have two cases, corresponding to a positive or a negative cosmolog-
ical constant. If we analytically continue t → iχ and n → in we obtain an Euclidian section
with negative cosmological constant; however α < 0 and so the solution is unphysical. If we
analytically continue t → iχ, n → in and l → il we obtain a Euclidian section with positive
cosmological constant. In order to ensure the positivity of α we must require that 6n2> l2.
The NUT solution corresponds to a three dimensional fixed-point set of ∂χlocated at r = n,
the mass parameter is mn= −n6
a five-dimensional fixed-point set of ∂χlocated at r = rb, where:
rb=kl ±√k2l4− 96n2l2+ 576n4
4πn2rb
r2
b−2n2.
2l2 and the periodicity of χ is 8πn. The bolt solutions have
24n
the mass parameter is given by:
mb=4r6
b+ (l2− 12n2)r4
b− 2n2(l2− 6n2)r2
4l2
b+ n4(l2− 6n2)
and the periodicity of χ is8πn
The second possibility in the ansatz (105)is to take the space M four dimensional and
the space Y one-dimensional. In this case we can further factorize M into products of two-
dimensional spaces with constant curvature M = M1× M2, or we can consider M = CP2.
k, k being an integer.
25
Page 26
Let us begin with the second case, namely M = CP2. The metric is
ds2= −F(r)(dt + A)2+ F−1(r)dr2+ (r2+ n2)dΣ2+ r2dy2
where A and dΣ2are given in (63) and (64) and λ =15
(125)
l2. We obtain
F(r) =2ml2− r6− 3n2r4− 3n4r2
l2(r2+ n2)2
(126)
where the above metric is a solution of the vacuum Einstein field equations with positive
cosmological constant if and only if 2λn2= 5. The Euclidian section is obtained by analyti-
cally continuations t → iχ and n → in (with l2= 6n2). The NUT solution corresponds to
a one-dimensional fixed-point set of ∂χlocated at r = n. The value of the mass parameter
is mn =
24
and the periodicity of χ is 12πn. The bolt solution has a five-dimensional
fixed-point set of ∂χlocated at rb=kn
−n4
3, the value of the mass parameter is given by:
mb= −r6
24n2
b− 3n3r4
b+ 3n4r2
b
while the periodicity of χ is12πn
The other possibility is to take M = M1× M2. We could also consider the possibility
of having two different NUT charges n1and n2that correspond to the circle fibrations over
M1and respectively M2. However, it turns out that in order to have consistent solutions
using the above ansatz we are forced by the field equations to consider only the cases where
M1= M2and moreover (with one exception - as we shall see below) we must have n1= n2.
We have then only three possibilities:
k. To ensure that rb> n we require that k ≥ 4.
• M = S2× S2and the metric is given by:
ds2= −F(r)(dt + A)2+ F−1(r)dr2+ (r2+ n2)(dΩ2
where:
1+ dΩ2
2) + r2dy2
(127)
A = 2n(cosθ1dφ1+ cosθ2dφ2)
dΩ2
(128)
(129)
i= dθ2
i+ sin2θidφ2
i
while
F(r) =4ml2− r6− 3n2r4− 3n4r2
l2(r2+ n2)2
(130)
The above metric is a solution of the vacuum Einstein field equations with a positive
cosmological constant λ =15
by the analytic continuations t → ıχ and n → in (with l2= 6n2). We obtain a solution
with negative cosmological constant. The NUT corresponds to an one-dimensional
fixed-point set of ∂χ located at r = n, the mass parameter is mn = −n4
periodicity of χ is 12πn. The bolt solution has a five-dimensional fixed-point set of ∂χ
located at rb=kn
l2 if and only if 2λn2= 5. The Euclidian section is obtained
24and the
3, the mass parameter being:
mb= −r6
b− 3n2r4
b+ 3n4r2
24n2
b
where the periodicity of χ is12πn
k. To ensure that rb> n we require that k ≥ 4.
26
Page 27
• M = T2× T2and the metric is given by:
ds2= −F(r)(dt + A)2+ F−1(r)dr2+ (r2+ n2)(dΩ2
where:
1+ dΩ2
2) + r2dy2
(131)
A = 2n1θ1dφ1+ 2n2θ2dφ2
dΩ2
(132)
(133)
i= dθ2
i+ dφ2
i
while
F(r) =
4m
1)(r2+ n2
(r2+ n2
2)
(134)
The above metric is a solution of the vacuum Einstein field equations for any values of
the parameters n1and n2. We obtain the Euclidian section by analytic continuation
t → iχ and nj→ inj, j = 0,1. Notice that FE(r) does not vanish for any value of r
and that there is a curvature singularity located at r = n, where n is the maximum
value of n1and n2.
• M = H2× H2and the metric is given by:
ds2= −F(r)(dt + A)2+ F−1(r)dr2+ (r2+ n2)(dΣ2
where:
1+ dΣ2
2) + r2dy2
(135)
A = 2ncoshθ1dφ1+ 2ncoshθ2dφ2
dΣ2
(136)
(137)
i= dθ2
i+ sinh2θidφ2
i
while
F(r) =4ml2+ r6+ 3n2r4+ 3n4r2
l2(r2+ n2)2
(138)
The above metric will be a solution of the vacuum Einstein field equations with a
negative cosmological constant λ = −15
section is obtained by analytic continuation t → iχ and n → in, with l2= 6n2.
The NUT solution corresponds to an one-dimensional fixed-point set of ∂χlocated at
r = n, with the mass parameter mn=n4
bolt solution has a five-dimensional fixed-point set of ∂χlocated at rb=kn
parameter:
mb=r6
l2 if and only if 2λn2= −5. The Euclidian
24and the periodicity of χ being 12πn. The
3, the mass
b− 3n2r4
b+ 3n4r2
24n2
b
and the periodicity of χ is
that k ≥ 4.
12πn
k, k being an integer. To ensure that rb> n we require
27
Page 28
7 Conclusions
We have considered here higher dimensional solutions of the vacuum Einstein field equations
with (or without) cosmological constant that are locally asymptotically de Sitter, anti-de
Sitter or flat. These solutions are constructed as circle fibrations over even dimensional spaces
that can be in general products of Einstein-K¨ ahler spaces. The novelty of our solutions is
that, by associating a NUT charge n with every such circle fibration over a factor of the base
space we have obtained higher dimensional generalizations of the Taub-NUT spaces that can
have quite generally multiple NUT charges. We have also generalized the Taub-NUT ansatz
to the case of odd-dimensional spacetimes and we have explicitly constructed the form of
the Taub-NUT solutions in five and seven dimensions.
We found that depending on the specific form of the base factors on which we con-
struct the circle fibration we can have cases in which the cosmological constant can have
either sign. Another interesting characteristic of our solutions is the unexpected relation
between the NUT charges and the cosmological constant. We found that quite generally,
in dimensions higher than four, it is impossible to make either all the NUT charges or the
cosmological constant vanish independently. In other words our solutions do not have non-
trivial asymptotically flat NUT-charged limits that are obtained by simply taking the limit
λ → 0.
Even if, for space reasons, we have given the solutions up to seven dimensions, our solu-
tions can be generalized in an obvious way for dimensions higher than seven. For instance,
in eight dimensions, the full base space is six-dimensional and in general we can factorize
it in product manifolds constructed out of combinations of two-dimensional spheres, tori or
hyperboloids. For each such two dimensional factor one can associate a NUT charge in the
final circle fibration. We can also consider the base factor as a product of a four-dimensional
manifold (like CP2or more generally an Einstein manifold of constant curvature) with a
two dimensional one (which could be a sphere, a torus or a hyperboloid). For every factor
that is an Einstein-K¨ ahler manifold we can associate a NUT charge parameter.
In our work we have given the Lorentzian form of the solutions however, in order to
understand the singularity structure of these spaces we have concentrated mainly on the
their Euclidian sections. In most of the cases the Euclidean section is simply obtained using
the analytic continuations t → it and nj → inj. However, given the special relationship
between the NUT charges and the cosmological constant, in some cases we also have to
analytically continue l → il in order to obtain a consistent solution with Euclidian signature.
When continuing back the solutions to Lorentzian signature the roots of the function F(r)
will give the location of the chronology horizons since across these horizons F(r) will change
the sign and the coordinate r changes from spacelike to timelike and vice-versa.
Leaving the study of the thermodynamic properties of these solutions for future work,
it is worth mentioning that our solutions can be used as test-grounds for the AdS/CFT
correspondence. For our solutions the boundary is generically a circle fibration over base
spaces that can have exotic topologies and this could shed some light on the study of the
conformal field theories (CFT) on backgrounds with such exotic topologies. In particular
one could be able to understand the thermodynamic phase structure of such conformal field
theories by working out the corresponding phase structure for the our supergravity solutions
28
Page 29
in the bulk.
This work was supported by the Natural Sciences and Engineering Council of Canada.
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