Taub–NUT Spacetime

Lorentzian Taub–NUT is a two-parameter type-D vacuum family whose NUT parameter behaves locally like a gravitomagnetic monopole. It provides an unusually sharp lesson in the difference between local regularity and global causality: patching away the axial Misner strings requires a periodic time identification, but that identification makes the stationary time orbits closed and timelike in ordinary stationary regions.

Required background. Frames, Coframes, and Anholonomy supplies principal frames and patch transformations.

Helpful background. The Einstein Equation supplies Ricci flatness; The Riemann Curvature Tensor supports invariant regularity checks; Exact Solutions of Einstein’s Equations explains how to specify exact spacetime solutions; Causality and Global Structure supplies causal and extension language; Killing Fields and Isometry Groups supplies the stationary and axial actions.

Local metric and type-D structure

Let $m=GM$ and let $n$ be the NUT parameter, both with dimensions of length. A common Lorentzian form is

$$ds^2=-F(r)\left(dt+2n\cos\theta\,d\phi\right)^2 +F(r)^{-1}dr^2+(r^2+n^2)d\Omega_2^2,$$ $$F(r)=\frac{r^2-2mr-n^2}{r^2+n^2}.$$

Here $0<\theta<\pi$, $\phi\sim\phi+2\pi$, and $t$ is initially noncompact. Direct calculation gives $R_{\mu\nu}=0$. An adapted null tetrad has only $\Psi_2$ nonzero, so the Weyl tensor is type D.

The round metric $(r^2+n^2)d\Omega_2^2$ belongs to the two-dimensional base, or orbit space, over which the time direction is twisted. It is not the generic induced metric on a fixed-$r$ spatial two-surface. Indeed, imposing $dt=dr=0$ in one local gauge gives

$$ds^2\big|_{t,r} =(r^2+n^2)d\theta^2 +\left[(r^2+n^2)\sin^2\theta-4n^2F(r)\cos^2\theta\right]d\phi^2,$$

which is neither generally round nor everywhere positive definite. After the regular north and south patches are glued with periodic time, a constant-$r$ hypersurface is instead the total space of a circle fibration over the round $S^2$ base; with the standard period its topology is $S^3$. Misner states both the time-circle and fixed-$r$ $S^3$ topology explicitly in Misner 1963, abstract, printed p. 924.

The roots

$$r_\pm=m\pm\sqrt{m^2+n^2}$$

are simple for $n\ne0$ and can be crossed by Eddington-like coordinates. Curvature scalars remain finite at those roots. The original vacuum extension and algebraically special character are given in Newman, Tamburino, and Unti 1963, pp. 915–923.

Misner strings and patching

The one-form $dt+2n\cos\theta\,d\phi$ resembles a monopole connection and is ill behaved on an axis if represented in one angular gauge. Near the north axis define

$$t_N=t+2n\phi,$$

so the one-form becomes $dt_N+2n(\cos\theta-1)d\phi$, regular at $\theta=0$. Near the south axis set

$$t_S=t-2n\phi,$$

giving $dt_S+2n(\cos\theta+1)d\phi$, regular at $\theta=\pi$. On the overlap,

$$t_N=t_S+4n\phi.$$

Since $\phi$ changes by $2\pi$ around the axis, consistent patching identifies time with period $8\pi|n|$. The apparent string is then absent from both local connection forms, and constant-$r$ hypersurfaces acquire the nontrivial circle fibration just described. This periodic bundle is a global choice, not a coordinate transformation of the spacetime with noncompact time.

The causal tradeoff

Where $F(r)>0$, the stationary Killing field $\xi=\partial_t$ has norm

$$g(\xi,\xi)=-F(r)<0.$$

If $t$ is periodically identified, every orbit of $\xi$ at fixed $(r,\theta,\phi)$ is a closed timelike curve in that region. Removing both Misner strings by the standard global patching therefore introduces closed timelike curves everywhere the stationary field is timelike.

If instead $t$ remains noncompact, the closed orbits of $\partial_t$ are avoided, but the north and south gauges cannot define the periodic circle bundle above: one retains an axial Misner-string representation or adopts an additional covering or transparency prescription. Closed timelike curves can still occur where the axial norm becomes negative. For the gauge family $dt-2n(\cos\theta+C)d\phi$, this neighborhood is characterized explicitly in Clément, Gal’tsov, and Guenouche 2015, § IV, printed p. 6, Eq. (4.20), Open PDF. Thus nonperiodic time does not by itself restore global causality.

The asymptotic metric retains the cross term $-4n\cos\theta\,dt\,d\phi$ rather than approaching Minkowski space in a single global chart. It is often called asymptotically locally flat under appropriate identifications, but the precise asymptotic bundle and time choice must be stated. The NUT parameter is consequently not an ordinary asymptotically flat angular momentum.

Limits and extensions

Setting $n\to0$ at fixed $r>2m$ gives Schwarzschild, and the patch transition becomes trivial. Setting $m=0$ with $n\ne0$ does not give flat spacetime; the NUT curvature remains. The denominator $r^2+n^2$ has no real zero for $n\ne0$, so the simple Schwarzschild scalar blowup at $r=0$ is removed, but causal and extension pathologies remain.

There is no identification-independent object called the maximal Taub–NUT extension. With periodic time, the standard analytic construction can be extended through only one of the two horizons and remain geodesically incomplete, or through both at the price of a non-Hausdorff maximal completion. With nonperiodic time, Clément, Gal’tsov, and Guenouche instead treat the Misner string as transparent to geodesics and argue for a Hausdorff continuation through both horizons that is geodesically complete Clément, Gal’tsov, and Guenouche 2015, § I, printed p. 2, and § V, printed pp. 7–8, Open PDF. That conclusion uses a different global prescription; it does not remove the axial string from the metric representation or eliminate the closed-timelike-curve region.

Calling the family “regular” merely because scalar invariants stay finite would therefore conceal the chosen topology, time identification, axis prescription, horizon extension, and causal behavior. Those data must be stated alongside the local metric.

Exercise

Derive the required time period from the north–south patch transformation and show directly why it creates closed timelike curves where $F>0$.

Solution

On the overlap $t_N=t_S+4n\phi$. Increasing $\phi$ by $2\pi$ changes the right side by $8\pi n$, so the period has magnitude $8\pi|n|$. At fixed $r,\theta,\phi$, the time circle has tangent $\partial_t$ and squared norm $-F$. Whenever $F>0$, this tangent is timelike, making the identified orbit a closed timelike curve.

References

• Clément, Gérard, Dmitri Gal’tsov, and Mourad Guenouche. “Rehabilitating Space-Times with NUTs.” Physics Letters B 750 (2015): 591–594. DOI: 10.1016/j.physletb.2015.09.074. Open PDF.
• Misner, Charles W. “The Flatter Regions of Newman, Unti, and Tamburino’s Generalized Schwarzschild Space.” Journal of Mathematical Physics 4, no. 7 (1963): 924–937. DOI: 10.1063/1.1704019.
• Newman, Ezra T., Louis A. Tamburino, and Theodore W. J. Unti. “Empty-Space Generalization of the Schwarzschild Metric.” Journal of Mathematical Physics 4, no. 7 (1963): 915–923. DOI: 10.1063/1.1704018.