View source for Taub–NUT space

{{Short description|Exact solution to Einstein's equations}}
{{Technical|date=January 2025}}

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The '''Taub–NUT metric''' ({{IPAc-en|t|ɔː|b|_|n|ʌ|t}},<ref>[[McGraw-Hill]] ''Science & Technology Dictionary'': "Taub NUT space"</ref> {{IPAc-en|-|_|ˌ|ɛ|n|.|j|uː|ˈ|t|iː}}) is an [[exact solutions in general relativity|exact solution]] to [[Einstein's equations]]. It may be considered a first attempt in finding the metric of a spinning black hole. It is sometimes also used in [[Homogeneity (physics)|homogeneous]] but [[anisotropic]] [[cosmological model]]s formulated in the framework of [[general relativity]].{{Citation needed|date=June 2018}}

The underlying Taub space was found by {{harvs|txt|first=Abraham Haskel|last=Taub|authorlink=Abraham Haskel Taub|year=1951}}, and extended to a larger manifold by {{harvs|txt | last2=Tamburino | first2=Louis A. | last1=Newman |author1-link=Ezra T. Newman |first1=Ezra T. | last3=Unti | first3=Theodore W. J. | title=Empty-space generalization of the Schwarzschild metric | doi=10.1063/1.1704018 |mr=0152345 | year=1963 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=4 | pages=915–923}}, whose initials form the "NUT" of "Taub–NUT".
==Description==
Taub's solution is an empty space solution of Einstein's equations with topology '''R'''×'''S'''<sup>3</sup> and metric (or equivalently [[line element]])
:<math>g =-dt^2/U(t) + 4l^2U(t)(d\psi+ \cos\theta d\phi)^2+(t^2+l^2)(d\theta^2+(\sin\theta)^2d\phi^2)</math>
where 
:<math>U(t)=\frac{2mt+l^2-t^2}{t^2+l^2}</math>
and ''m'' and ''l'' are positive constants.

Taub's metric has coordinate singularities at <math>U=0, t=m+(m^2+l^2)^{1/2}</math>, and Newman, Tamburino and Unti showed how to extend the metric across these surfaces.
==Related work==
===Kerr metric===
When [[Roy Kerr]] developed the [[Kerr metric]] for spinning black holes in 1963, he ended up with a four-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,<ref name="kerrtube1">Roy Kerr: ''[https://www.youtube.com/watch?v=LeLkmS3PZ5g&t=21m36s Spinning Black Holes]'' (Lecture at the University of Canterbury, 25. May 2016). Timecode: [https://www.youtube.com/watch?v=LeLkmS3PZ5g&t=21m36s 21m36s]</ref><ref name="kerrtube2">Roy Kerr: ''[https://www.youtube.com/watch?v=jji2pgfq7oE&t=19m56s Kerr Conference]'' (Lecture at the New Zealand Residence in Berlin, 4. July 2013). Timecode: [https://www.youtube.com/watch?v=jji2pgfq7oE&t=19m56s 19m56s]</ref> while other sources interpret it either as a gravomagnetic monopole parameter of the central mass,<ref name="arxiv1">Mohammad Nouri-Zonoz, Donald Lynden-Bell: ''[https://arxiv.org/abs/gr-qc/9812094 Gravomagnetic Lensing by NUT Space]'' [[arXiv:gr-qc/9812094]]</ref> or a twisting property of the surrounding spacetime.<ref name="researchgate1">A. Al-Badawi, Mustafa Halilsoy: ''[https://www.researchgate.net/publication/225442437_On_the_physical_meaning_of_the_NUT_parameter On the physical meaning of the NUT parameter]'', from [[ResearchGate]]</ref>
===Misner spacetime===
A simplified 1+1-dimensional version of the Taub–NUT spacetime is the [[Misner space|Misner spacetime]].

==References==
{{reflist}}

==Notes==
*{{Citation | last2=Tamburino | first2=L. | last1=Newman | first1=E. | last3=Unti | first3=T. | title=Empty-space generalization of the Schwarzschild metric | doi=10.1063/1.1704018 |mr=0152345 | year=1963 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=4 | issue=7 | pages=915–923|bibcode = 1963JMP.....4..915N }}
*{{Citation | last1=Taub | first1=A. H. | title=Empty space-times admitting a three parameter group of motions |mr=0041565 |jstor=1969567 | year=1951 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=53 | issue=3 | pages=472–490 | doi=10.2307/1969567}}
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