The Taub–NUT metric (,

McGraw-Hill McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes refere ...

''Science & Technology Dictionary'': "Taub NUT space" ) is an exact solution to

Einstein's equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...

. It may be considered a first attempt in finding the metric of a spinning black hole. It is sometimes also used in homogeneous but anisotropic cosmological models formulated in the framework of

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

. The underlying Taub space was found by , and extended to a larger manifold by , 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 (or equivalently

line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...

) :

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)

where :

U(t)=\frac

and ''m'' and ''l'' are positive constants. Taub's metric has coordinate singularities at

U=0, t=m+(m^2+l^2)^

, and Newman, Tamburino and Unti showed how to extend the metric across these surfaces. When

Roy Kerr Roy Patrick Kerr (; born 16 May 1934) is a New Zealand mathematician who discovered the Kerr geometry, an exact solution to the Einstein field equation of general relativity. His solution models the gravitational field outside an uncharged ...

developed the

Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...

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,Roy Kerr:
Spinning Black Holes
' (Lecture at the University of Canterbury, 25. May 2016). Timecode
21m36s
/ref>Roy Kerr:
Kerr Conference
' (Lecture at the New Zealand Residence in Berlin, 4. July 2013). Timecode
19m56s
/ref> while other sources interpret it either as a gravomagnetic monopole parameter of the central mass,Mohammad Nouri-Zonoz, Donald Lynden-Bell:
Gravomagnetic Lensing by NUT Space
' arXiv:gr-qc/9812094 or a twisting property of the surrounding spacetime.A. Al-Badawi, Mustafa Halilsoy:
On the physical meaning of the NUT parameter
', from ResearchGate A simplified 1+1-dimensional version of the Taub–NUT spacetime is the Misner spacetime.

References

Notes

* * Exact solutions in general relativity {{relativity-stub