Gravity and Strings

9.1 The Taub–NUT solution 269
More details on the Kerr solutions can be found in most standard textbooks on GR and
in the monograph [741]. Our subject now is the Taub–NUT solution.
If, asymptotically,
gtϕ ∼ 2N cos θ, (9.4)
the solution describes an object with NUT charge N. Wewill discuss soon the meaning of
this new charge, for which there is no Newtonian analog. The simplest vacuum solution
with this kind of charge is the Taub–NUT solution [724, 879]
ds 2 = f (r)(dt + 2N cos θ dϕ) 2 − f −1 (r)dr 2 − r 2 + N 2 d 2 (2) ,
f (r) = (r − r+)(r − r−)
r 2 + N 2
,
r± = M ± r0, r 2 0 = M2 + N 2 ,
(9.5)
which is a generalization of the Schwarzschild solution with NUT charge, and reduces to it
when N = 0.
Let us list some immediate properties of this spacetime.
1. The solution is non-trivial in the M → 0 limit, in which it may be interpreted as the
gravitational field of a pure “spike” of spin [167, 329].
2. The mass of the solution can be found by standard methods and it is M.Inparticular,
we know that we can determine the mass by studying the weak-field expansion and
making contact with the Newtonian limit. The Newtonian gravitational potential is
given in this approximation by φ ∼ (gtt − 1)/2 =−M/r. The Taub–NUT solution
has other non-vanishing components of the metric. The diagonal components are still
related to the gravitostatic Newtonian potential φ, but the off-diagonal ones gti are
related to a gravitomagnetic potential A according to Eq. (3.141). In the coordinates
that we are using, we see that the Taub–NUT gravitational field has, as non-vanishing
component of the gravitomagnetic potential,
Aϕ = gtϕ = 2N cos θ. (9.6)
This is essentially the electromagnetic field of a magnetic monopole of charge proportional
to N. Thus, the NUT charge N can be considered as a sort of “magnetic
mass” [297] and so the Taub–NUT solution can be interpreted as a gravitational dyon
[328].
3. This metric is not asymptotically flat but defines its own class of asymptotic behavior
(asymptotically Taub–NUT spacetimes) labeled by N, which is associated with the
non-vanishing at infinity of the off-diagonal gtϕ component of the metric and, as we
are going to see, with the periodicity of the time coordinate. The reason for this periodicity
is the desire to avoid certain singularities and to have a spherically symmetric
solution. Thus, let us first study the singularities.