In this chapter, we describe what is widely known as the Taub–NUT solution. This was first discovered by Taub (1951), but expressed in a coordinate system which only covers the time-dependent part of what is now considered as the complete space-time. It was initially constructed on the assumption of the existence of a four-dimensional group of isometries so that it could be interpreted as a possible vacuum homogeneous cosmological model.

This solution was subsequently rediscovered by Newman, Tamburino and Unti (1963) as a simple generalisation of the Schwarzschild space-time. And, although they presented it with an emphasis on the exterior stationary region, they expressed it in terms of coordinates which cover both stationary and time-dependent regions. In addition to a Schwarzschild-like parameter m which is interpreted as the mass of the source, it contained two additional parameters – a continuous parameter l which is now known as the NUT parameter, and the discrete 2-space curvature parameter which is denoted here by ∈. It is only the case in which ∈ = +1, which includes the Schwarzschild solution, that was obtained by Taub. The cases with other values of ∈ are generalisations of the other A-metrics.

We will follow the usual convention of referring to the case in which ∈ = +1 as the Taub–NUT solution. However, there are two very different interpretations of this particular case. Both of these have unsatisfactory aspects in terms of their global physical properties. In one interpretation, the space time contains a semi-infinite line singularity, part of which is surrounded by a region that contains closed time like curves.