Gravity and Strings

272 The Taub–NUT solution
This solution is known as the (Sorkin–Gross–Perry) Kaluza–Klein (KK) monopole [483,
860]. The 1-form A satisfies the Dirac-monopole equation (8.149), which we know has to
be solved in two different patches.
9.2.1 Self-dual gravitational instantons
If we use the above form of the solution as an Ansatz in the vacuum Einstein equations,
we find that we have a solution for every function H that is harmonic in three-dimensional
space:
−dσ 2 = H −1 (dτ + A) 2 + Hdx 2
3 ,
A = Aidx i , ɛijk∂i A j =±∂k H,
∂i∂i H = 0.
(9.13)
In fact, we know that the Laplace equation is the integrability condition of the Diracmonopole
equation, ensuring that it can be (locally) solved. Now it is possible to have
solutions with several KK monopoles in equilibrium by taking a harmonic function H with
several point-like singularities (Gibbons–Hawking multicenter metrics [437]):
H = ɛ +
k 2|NI |
. (9.14)
|x3 −x3 I |
I =1
If we choose ɛ = 1, we have the multi-Taub–NUT metric. If all the NUT charges NI are
equal to N, then all the wire singularities associated with each pole can be removed simultaneously
by taking the period of τ equal to 8π N. Asymptotically the topology is that of a
lens space:anS 3 in which k points have been identified, and so they are not asymptotically
flat in general.
If we choose ɛ = 0, the wire singularities can be eliminated by the same procedure, but
the NI scan all be made equal by a rescaling of the coordinates. The topology is the same as
in the ɛ = 1 case, but the metrics are asymptotically locally Euclidean (ALE), i.e. they are
asymptotic to the quotient of Euclidean space by a discrete subgroup of SO(4). The k = 1
solution is just flat space. The k = 2 solution is equivalent [787] to the Eguchi–Hanson
solution [348], which is usually written in the form
−dσ 2
= 1 − a4
ρ4 2 ρ
4 (dτ + cos θdϕ)2
+ 1 − a4
ρ4 −1
dρ 2 + ρ2
4 d2 (2) . (9.15)
This solution has an apparent singularity at ρ = a that can be removed by identifying
τ ∼ τ + 2π.With this identification, all the ρ>a constant hypersurfaces are RP 3 (S 3 with
antipodal points identified).
All these solutions are gravitational instantons, the gravitational analog of the SU(2)
BPST Yang–Mills (YM) instantons discovered in [102], i.e. non-singular solutions of the
Euclidean Einstein equations with finite action, i.e. local minima of the Euclidean Einstein