A five-parameter class of solutions to the vacuum Einstein equations

We present a new five-parameter class of Ricci-flat solutions in four dimensions with Euclidean signature. The solution is asymptotically locally flat (ALF), and contains a finite asymptotic NUT charge. When this charge is sent to infinity, the solution becomes asymptotically locally Euclidean (ALE), and one in fact obtains the Ricci-flat Plebanski-Demianski solution. The solution we have found can thus be regarded as an ALF generalisation of the latter solution. We also show that it can be interpreted as a system consisting of two touching Kerr-NUTs: the south pole of one Kerr-NUT touches the north pole of the other. The total NUT charge of such a system is then identified with the asymptotic NUT charge. Setting the asymptotic NUT charge to zero gives a four-parameter asymptotically flat (AF) solution, and contained within this subclass is the completely regular two-parameter AF instanton previously discovered by the present authors. Various other limits are also discussed, including that of the triple-collinearly-centered Gibbons-Hawking solution, and an ALF generalisation of the C-metric.

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