Taub–NUT space

Phenomena

Spacetime
Kepler problem Gravitational lensing Gravitational redshift Gravitational time dilation Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole
Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge

Equations
Formalisms

Equations
Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein
Formalisms
ADM BSSN Post-Newtonian
Advanced theory
Kaluza–Klein theory Quantum gravity

The Taub–NUT metric (/tɔːb nʌt/,^[1] /- ˌɛn.juːˈtiː/) is an 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 homogeneous but anisotropic cosmological models formulated in the framework of general relativity.^{[citation needed]}

The underlying Taub 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 (or equivalently line element)

g=-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

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 $U=0,t=m+(m^{2}+l^{2})^{1/2}$ , 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 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,^[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]

A simplified 1+1-dimensional version of the Taub–NUT spacetime is the Misner spacetime.

References

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

Notes

Newman, E.; Tamburino, L.; Unti, T. (1963), "Empty-space generalization of the Schwarzschild metric", Journal of Mathematical Physics, 4 (7): 915–923, Bibcode:1963JMP.....4..915N, doi:10.1063/1.1704018, ISSN 0022-2488, MR 0152345
Taub, A. H. (1951), "Empty space-times admitting a three parameter group of motions", Annals of Mathematics, Second Series, 53 (3): 472–490, doi:10.2307/1969567, ISSN 0003-486X, JSTOR 1969567, MR 0041565

Relativity

Special
relativity

Background	Principle of relativity (Galilean relativity Galilean transformation) Special relativity Doubly special relativity
Fundamental concepts	Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration Relativistic mass
Formulation	Lorentz transformation Textbooks
Phenomena	Time dilation Mass–energy equivalence (E=mc²) Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Ladder paradox Twin paradox Terrell rotation
Spacetime	Light cone World line Minkowski diagram Biquaternions Minkowski space

General
relativity

Background	Introduction Mathematical formulation
Fundamental concepts	Equivalence principle Riemannian geometry Penrose diagram Geodesics Mach's principle
Formulation	ADM formalism BSSN formalism Einstein field equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation
Phenomena	Black hole Event horizon Singularity Two-body problem Gravitational waves: astronomy detectors (LIGO and collaboration Virgo LISA Pathfinder GEO) Hulse–Taylor binary Other tests: precession of Mercury lensing (together with Einstein cross and Einstein rings) redshift Shapiro delay frame-dragging / geodetic effect (Lense–Thirring precession) pulsar timing arrays
Advanced theories	Brans–Dicke theory Kaluza–Klein Quantum gravity
Solutions	Cosmological: Friedmann–Lemaître–Robertson–Walker (Friedmann equations) Lemaître–Tolman Kasner BKL singularity Gödel Milne Spherical: Schwarzschild (interior Tolman–Oppenheimer–Volkoff equation) Reissner–Nordström Axisymmetric: Kerr (Kerr–Newman) Weyl−Lewis−Papapetrou Taub–NUT van Stockum dust discs Others: pp-wave Ozsváth–Schücking Alcubierre In computational physics: Numerical relativity

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