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The Wikipedia article about the Taub-NUT spacetime says that it was a first attempt in finding the Kerr solution. Since the Kerr spacetime is a stationary solution, meaning that it admits an asymptotically timelike Killing field (that is, near the future and past null infinities $\mathscr I^\pm$), I wonder if that is true also for the Taub-NUT spacetime.

In particular, what is its stationary limit (the so-called ergosphere)? Is it finite or infinite? Any ideas?

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A common definition of a stationary spacetime is a spacetime that has a timelike Killing vector (see e.g. Wald, General Relativity, p. 119, or Stationary and Static). In this meaning both the Kerr spacetime and the Taub-NUT spacetime are stationary.

Regarding the second question: consider the Taub-NUT metric in the following form (Ortin, Gravity and Strings, p. 269) $$ds^2= f(r)(dt+2N\cos{\theta}d\varphi)^2-\frac{dr^2}{f(r)}-(r^2+N^2)(d\theta^2+\sin^2{\theta}d\varphi^2),$$ $$f(r)=\frac{(r-r_{+})(r-r_{-})}{r^2+N^2},$$ $$r_{\pm}=M\pm\sqrt{M^2+N^2}.$$

Timelike Killing vector $\xi^i=(1,0,0,0)$, and $\xi_i\xi^i=f(r)$. The ergosphere is located where the Killing vector becomes null, it happens on spheres $r=r_{\pm}$.

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