# Nuttier (A)dS Black Holes in Higher Dimensions

**ABSTRACT** We construct new solutions of the vacuum Einstein field equations with cosmological 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 dimensions. 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 existence 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. Comment: latex, 30 pages, added references

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**ABSTRACT:**We construct new charged solutions of the Einstein Maxwell field equations with cosmological constant. These solutions describe the nut-charged generalisation of the higher-dimensional Reissner Nordström spacetimes. For a negative cosmological constant these solutions are the charged generalizations of the topological nut-charged black hole solutions in higher dimensions. Finally, we discuss the global structure of such solutions and possible applications.Physics Letters B 01/2006; 632(4):537-542. · 6.02 Impact Factor - SourceAvailable from: Chong Oh Lee[Show abstract] [Hide abstract]

**ABSTRACT:**We consider a finite action for a higher dimensional Taub–NUT/Bolt–(A)dS space via the so-called counter term subtraction method. In the limit of high temperature, we show that the Cardy–Verlinde formula holds for the Taub–Bolt–AdS metric and for the specific dimensional Taub–NUT–(A)dS metric, except for the Taub–Bolt–dS metric.Physics Letters B 12/2008; 670(2):146–149. · 6.02 Impact Factor - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**I show that there are no SU(2)-invariant (time-dependent) tensorial perturbations of Lorentzian Taub NUT space. It follows that the spacetime is unstable at the linear level against generic perturbations. The difficulty of formulating a physically reasonable thermodynamics for Taub NUT space is discussed.Classical and Quantum Gravity 06/2006; 23(11):3951-3962. · 3.10 Impact Factor

Page 1

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.

Page 2

1Introduction

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

2

Page 3

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.

2Taub-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)

3

Page 4

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)

4

Page 5

• 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.

5

Page 6

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

6

Page 7

• 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)

7

Page 8

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)

8

Page 9

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.

5Taub-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

9

Page 10

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

10

Page 11

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].

5.2 Fibrations 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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- Available from Cristian Stelea · May 16, 2014
- Available from arxiv.org