Gravity and Strings

270 The Taub–NUT solution
4. This metric does not have curvature singularities and is perfectly regular at r = 0.
However, it has the so-called “wire singularities” at θ = 0 and θ = π where the metric
fails to be invertible. These coordinate singularities cannot be cured simultaneously.
Misner [697] found a way to make the metric regular everywhere by introducing
two coordinate patches.
(a) One patch covers the region θ ≥ π/2 around the north pole. In this region we
change the time coordinate from t to t (+) defined by
so
t = t (+) − 2Nϕ, (9.7)
ds 2 (+) = f (r) dt (+) − 2N(1 − cos θ)dϕ 2 − f −1 (r)dr 2 − r 2 + N 2 d 2 (2) .
(9.8)
(b) The second patch covers the region θ ≤ π/2 around the south pole. In this region
we change the time coordinate from t to t (−) defined by
so
t = t (−) + 2Nϕ, (9.9)
ds 2 (−) = f (r) dt (−) + 2N(1 + cos θ)dϕ 2 − f −1 (r)dr 2 − r 2 + N 2 d 2 (2) .
(9.10)
In the overlap region t (+) = t (−) + 4Nϕ and, since ϕ is compact with period 2π, then
both of t (±) have to be compact with period 8π N.
5. The metric admits three Killing vectors whose Lie brackets are those of the so (3)
Lie algebra. When the period of the time coordinates is precisely 8π N this local
symmetry can be integrated to give a global SO (3) symmetry and the metric is indeed
spherically symmetric [587]. Furthermore, the Taub–NUT spacetime now has a very
different topology: the hypersurfaces of constant r are 3-spheres S 3 constructed as a
Hopf fibration of S 2 , the fiber being the time S 1 .Thus, Taub–NUT has the topology
of R 4 .
6. This way of eliminating the wire singularities is identical to the way in which we
eliminated the string singularity in the vector field of the Dirac monopole because
the mathematical problem is identical. The Dirac quantization condition translates
into a relation between the periodicity of the time coordinate and the NUT charge.
This relation is more than just a coincidence: in Chapter 11 we will generate by
compactification of the Euclidean time of the Euclidean version of the Taub–NUT
solution a magnetically charged black hole. For this reason, the Euclidean Taub–NUT
solution, which we will study later, is also known as the Kaluza–Klein monopole.