# Taub–NUT space

The Taub–NUT metric (/tɔːb nʌt/[1] or /tɔːb ɛnjuːˈt/) is an exact solution to Einstein's equations, a cosmological model formulated in the framework of general relativity.

The Taub–NUT space was found by Abraham Haskel Taub (1951), and extended to a larger manifold by Ezra T. Newman, Louis A. Tamburino, and Theodore W. J. Unti (1963), whose initials form the "NUT" of "Taub–NUT".

Taub's solution is an empty space solution of Einstein's equations with topology R×S3 and metric

${\displaystyle ds^{2}=-dt^{2}/U(t)+4l^{2}U(t)(d\psi +\cos \theta d\phi )^{2}+(t^{2}+l^{2})(d\theta ^{2}+(\sin \theta )^{2}d\phi ^{2})}$

where

${\displaystyle U(t)={\frac {2mt+l^{2}-t^{2}}{t^{2}+l^{2}}}}$

and m and l are positive constants.

Taub's metric has coordinate singularities at ${\displaystyle U=0,t=m+(m^{2}+l^{2})^{1/2}}$, and Newman, Tamburino and Unti showed how to extend the metric across these surfaces.

When Roy Kerr developed the Kerr metric for spinning black holes in 1963, he ended up with a 4 parameter solution, one of which was the mass and another the angular momentum of the central body. One of the two other parameters was the NUT-parameter, which he threw out of his solution because he found it to be nonphysical since it caused the metric to be not asymptotically flat,[2][3] while other sources interpret it either as a gravomagnetic monopole parameter of the central mass,[4] or a twisting property of the surrounding spacetime.[5]

## References

1. ^ McGraw-Hill Science & Technology Dictionary: "Taub NUT space"
2. ^ Roy Kerr: Spinning Black Holes (Lecture at the University of Canterbury, 25. May 2016). Timecode: 21m36s
3. ^ Roy Kerr: Kerr Conference (Lecture at the New Zealand Residence in Berlin, 4. July 2013). Timecode: 19m56s
4. ^ Mohammad Nouri-Zonoz, Donald Lynden-Bell: Gravomagnetic Lensing by NUT Space arXiv:gr-qc/9812094
5. ^ A. Al-Badawi, Mustafa Halilsoy: On the physical meaning of the NUT parameter, from ResearchGate