9.1 Examples of gravitational instantons 193
To see that the metric is complete set ρ
2
= r
2
%
1 − (a/r)
4
&
. Expanding the
metric near r = a and fixing (θ,φ) gives
g ∼
1
4
*
dρ
2
+ ρ
2
dψ
2
+
.
In the standard spherical polar coordinates ψ has a period 4π on SU(2). In
our case the period of ψ is 2π to achieve regularity. Therefore the surfaces
of constant r are real projective planes defined by identifying the antipodal
points on the sphere, RP
3
= S
3
/Z
2
. At large values of r the metric looks like
R
4
/Z
2
rather than Euclidean space. The Eguchi–Hanson metric is an example
of the asymptotically locally euclidean (ALE) manifold.
The Eguchi–Hanson example motivates the following definition:
Definition 9.1.1 A complete regular four-dimensional Riemannian manifold
(M, g) which solves the Einstein equations (possibly with cosmological con-
stant) is called ALE if it approaches R
4
/ at infinity, where is a discrete
subgroup of SO(4).
If contains only the identity then asymptotically g is Euclidean and Ricci-flat.
The positive action theorem [183] implies that any such g is a flat metric on R
4
.
The Eguchi–Hanson metric corresponds to generated by a single reflection.
The following example also has ASD Riemann tensor, but its asymptotic
behaviour is rather different from that of the Eguchi–Hanson metric.
r
Example. Taub-NUT gravitational instanton. Consider the metric [77]
g =
1
4
r + m
r − m
dr
2
+ m
2
r − m
r + m
σ
2
3
+
1
4
(r
2
− m
2
)
*
σ
2
1
+ σ
2
2
+
. (9.1.4)
At r = m the three-sphere of constant r collapses to a point – an example of a
NUT singularity (see below). The change of variables r = m + ρ
2
/(2m) gives
near r = m
g ∼ dρ
2
+
ρ
2
4
*
σ
2
1
+ σ
2
2
+ σ
2
3
+
,
where we set r + m ∼ 2m in all numerators. Thus the metric extends
smoothly over this point, and therefore is defined on the whole of R
4
. This
metric is not ALE as the coefficient of σ
2
3
approaches a constant as r →∞
while the coefficient of (σ
2
1
+ σ
2
2
) grows like r
2
. Thus the circle generated
by σ
3
separates from the three-sphere at large distances. The infinity has
a topology of the S
1
bundle over S
2
. Examining the transition functions
shows that this is in fact the one-monopole bundle considered in Section
6.2.1. This kind of behaviour is referred to as asymptotic local flatness. It
implies flatness in the three-dimensional sense. The fourth (imaginary time)
dimension is periodic.