Gravity and Strings

9
The Taub–NUT solution
The asymptotically flat, static, spherically symmetric Schwarzschild and RN BH solutions
that we have studied in the two previous chapters were the only solutions of the Einstein
and Einstein–Maxwell equations with those properties. To find more solutions, we have
to relax these conditions or couple to gravity more general types of matter, as we will do
later on. If we stay with the Einstein(–Maxwell) theory, one possibility is to look for static,
axially symmetric solutions and another possibility is to relax the condition of staticity and
only ask that the solution be stationary, which implies that we have to relax the condition
of spherical symmetry as well and look for stationary, axisymmetric spacetimes. In the first
case one finds solutions like those in Weyl’s family [949, 950] which can be interpreted
as describing the gravitational fields of axisymmetric sources with arbitrary multipole momenta
1 or Melvin’s solution [692] (which has cylindrical symmetry and was constructed
earlier by Bonnor [165] via a Harrison transformation [499] of the vacuum), among many
others. In the second case, we find the Kerr–Newman BHs [617, 723] with angular momentum
and electric or magnetic charge and also the Taub–Newman–Unti–Tambourino
(Taub–NUT) solution [724, 879], which may but need not include charges. The Taub–NUT
metric does not describe a BH because it is not asymptotically flat. In fact, the only stationary
axially symmetric BHs of the Einstein–Maxwell theory belong to the Kerr–Newman
family of solutions (see e.g. [532, 533]).
The Taub–NUT solution has a number of features that are particularly interesting for us,
which we are going to discuss in this chapter. In particular, it carries a new type of charge
(NUT charge), which is of topological nature and can be viewed as “gravitational magnetic
charge,” so the solution is a sort of gravitational dyon and its Euclidean continuation (for
certain values of the mass and NUT charge) is the solution known in other contexts as
a Kaluza–Klein monopole. This is a very important solution with interesting properties
such as the self-duality of its curvature and its relation to the Belavin–Polyakov–Schwarz–
Tyupkin (BPST) SU(2) instanton and the ’t Hooft–Polyakov monopole. In Chapter 11 we
will study how it arises in KK theory. Here we will describe it as a self-dual gravitational
instanton and we will take the opportunity to mention other gravitational instantons.
1 Forareview see [793].
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