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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.
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• 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)
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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.
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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
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